TENS VEC ENCE

& V RTUALPH LOSOP

f rI

lin

Intensive Science and Virtual Philosophy cuts to the heart of the ph osophyof Gilles Deleuze and of todays science wars

o L ndexpl 11

Manuel DeLanda began his career in experimental film. became acomputer artist and programmer, and is now Adjunct Professor ofPhilosophy at Columbia University. He is author of the best-sellingbooks, War in the Age of Intelligent Machines and A Thousand Yearsof Non-Linear History.

Int n tve SCI nc and VIrtual PhilosOpll'V I wnnen for lanti-Deleuzi n f phitcso h r for an i-phil () hIt III h n 1 r . f Y V J hm

J th st r of t

PHILOSOPHY I CULTURAL & MEDIA STUD IES I SC ENCE STUD ES

continuum manuel delanda

rPV'".Q(7'-- - - ------ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ......__IltANS\, I:HSAI.SNEW I>JH LCTI ONS IN PIII LOSOPIIY

ERIE EDITOR

Keith Ansell Pearson, University of Wan i .k

CO NSULTANT EDIT O R

Eric Alliez , Richard Beardsworth, Howard Caygill, Gary Gcnosko ,

Elisabeth Grosz, Michael Hardt, Diane Morgan, John Mullarkey, Paul

Patton, Stanley Shostak, Isabelle Steng ers, James Williams, David

Wood.

Transver sals explores the mo st exciting collisions within contempo rary

thought as philo soph y encounte rs nature, materiality, tim e , technology,

science , culture , politics, art and everyday life. Th e ser ies aims to

pr esent work which is both theoreti cally innovativ e and challenging,

while retaining a commitme nt to rigour and clarity, and to the power

and precision of thought.

INTEN SIVE S(~ II: NeEAN l) V11{~r UAL

PHILOSOPHY

MANUEL DELANDA

Intensive Science &.. Virtual Philosophy

Felix Guau ori: an Aberrant Introduction

Political Physics: Deleuze, Derrida and the Body Politic

FORTHCOMING

Manuel DeLand a

Gary Genosko

John Protevi

Philosophy in the ABe if Science &.. Capital Gregory Dale Adam son

II

ContinuumThe Tow er Building, 11 York Road, London SEI 7 IX370 Lexington Avenue, New York , NY 10017- 650 3

www.continuumbooks.com

First published in 2002

© Manuel DeLanda 2002

All rights reserved . No part of this publication may be reproducedor transmitted in any form or by any means, electronic or mechanical,

including photocopying, recordin g or any informati on storage or retri evalsystem, without permi ssion in writin g from the publishers.

British Lib rary Cata loguing-in-Pu blicat ion DataA catalogue record for this book is available

from the British Library.

ISBN 0- 8264 - 5622- 7 (hardback)0-8264 - 5623-5 (paperback)

Typeset by CentraServe, Saffron Walden , EssexPrint ed and bound in Great Britain byMPG Books Ltd, Bodmin, Cornwall

C tit .nt

lntroduct ion: IJl, ll'uze's W orld

The Mathemat ics of th e Virtual: Manifolds,

Vec to r Fields and Transformation Groups

2 T he Actualizat ion of th ' Virtual in pace

3 The Actualization of th e Virtual in Time

4 Virtuality and th e Laws of Physics

Appendix: Deleuze's Words

Notes

Index

9

45

82

117

157

181

241

'1'" Jltlil"! ;t "" rl ,<I :1 l'illl "LI .

\1'11<) ra uoh t s" Illl l ('!l abo u t ti ll' worl. ]Co

lntroduction: Dclcurc 's World

Th ere arc always dan gers in writing it book with a specific a ud il' lKt' in

mind . The most obvious one is the danger o f missing the targt'l

audit-nee co m plete ly I eithe r because the subj ect matter fails to gr.lb ihattention or because the sty le of presentation docs not nu -vt its

standards or e xpectations. Th en there is th e associated danger of Iw,ing

readers wh o , had not that particular target been chose n, would IM H '

formed the real audience or the book . A book may end up this \\ .I~'

without an~' read ership at all. In the world or W estern philosophy . 1,,,'e xample, history and geog raphy have co nspired to divide th is world

into tw o almost mutually exclusive camps, the Anglo~Amcrican .111<1the Continental camps, each with its own style , research priorities and

long traditions to defend , A phi losophical book which refus es [ 0 I.,k,·

sides, attem pting, for example , to present the work of a philosopher

of on e cam p in the terms and sty le of the other, may end up heing a

ho ok witho ut an audience: too Anglo~American for the Continentals.

and too Cont inental for the Anglo~Amcricans .

Such a danger is evident in a hook like this, which attempts to

present the work or the philosopher Gilles Dclcu zc to an aud ience or

analytical philosophers of science , and of scientis ts interested ill

philosoph ical questions. W hen confronte d wit h Dcleuze' s original te xts

this audience is bo und to be puzzled , and may even be repelled by t IH.',

superficial sim ilarity or these texts with books belonging to what has

cume to he known as th e ' post -mode rn' traditio n . Although as I arglH.~

in these pages Dcieuzc has absolutely nothing in com mo n with that

tradition, his expe rimenta l sty le is bound to create that impression .

Another source of difficulty is the philosophical resources whi ch Dclcuzc

brings to his project. Despite the fact that authors like Spinoza and

Leibniz, Nietzsche and Bergson. have mu ch to offer to phil osophy

tod ay. they arc not ge nera lly perceived by scientists o r analyt ical

philosophers of science as a legitimate resource. For thi s reason wh at I

"11,, IH" h lI"t ,,111 •• , 1111"1",1.11'''11 tIt 11,1,'1/, 10\\ 1'\11 .1

r, '<llI/ llrl/,/I"" 01 III' 1'1111"'''1'1". 11'"1 .1~ ,,,111,1, ,hll",,,t tllI"I"","re source S ,11111 lines 01 .lrgullwllt. 1111' 1'''.111 lIt 1111 , •• "" IIIHU,," is

1I0t jus t to make his ideas sc,'m Il'gitim.III' 10 III} illll'lIdl'd audience.

bu t also to sho v that his co n .lusions do not depen d 011 his particul : r

cho ice of resources , or th e particul ar lines of arg ume llt he uses , but

th at they arc robust to changes in theoretical assum ptio ns and st rategi's .

Clearly , if the same conclus io ns can be reached from enti rely different

points of departure and following ent irely d ifferent paths , th e valid ity

of those conclus ions is thereby stre ngthe ne d.

I must qualify thi s state ment , however, because what I attem pt

here is far from a com pre hens ive recon struction of all of Delcu zc 's

philosophical ideas. Instead, I focus on a particular ye t fundamental

aspect of his work: his ontoloBY' A philosopher ' s o nto logy is th e set

o f entit ies he or she assumes to exist in reality, th e typ es of entit ies

he or she is comm itted to assert actually exist. Although in th e history

of philosophy th ere ar e a great vari ety of ontological com mit me nts,

we can very roughly classify th ese into three main groups. For some

phil osophers reality has no ex istence ind ependently from th e human

mind that perceives it, so th eir ontology consists mostl y of mental

entities , wh ether th ese are thought as transcendent obj ect s or , on th e

cont rary , as linguisti c representati on s o r soci al conventions. Other

philosophers grant to the objects of everyday expe rience a mind­

ind ependent existe nce, but remain unconvinced that th eoretical ent it ­

ies, whether unobservabl e relations such as ph ysical causes, or

unobservable ent ities such as el ectrons, possess suc h an ontological

autonomy . Finall y, th ere are phil osophers who grant reality full

auton omy from the human mind, disr egarding th e differen ce between

th e obse rvable and th e un ob servable , and th e anthropocentri sm this

distinction implies. These phil osophers are said to have a realist ontol ­

0BY . Delcuzc is suc h a reali st ph ilosopher, a fact that by itself sho uld

distinguish him from most post -m od ern phil osophies wh ich remain

basicall y non-reali st.

Reali st philosophers, on th e other hand, need not agree about th e

co nte nts of thi s mind-independent reality . In particular, Deleuze rejects

several of th e ent it ies taken for gra nted in ordinary for ms of realism .

To tak e th e m ost obv ious exam ple, in some realist approaches th e

2

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,h"llI " 'lilt', 01 ,111\ othl'l Ir.JIl (( '11.1"111 ,11111 v , 0 III III plllio "1'1.\

"",..111111' 1'1 " is m'I"It 'd t" explain "h.lt I .H ohl " t IIII'll 100.,,,tll

IIId "h.11 1>!"l'SI'n '.,s lhis idcntit th rouoh tinu- . Hricllv, th is sll"wthi,,', ," lsI' is d maun cal proll: J '\. , OIl1l' of th ese pr"cl'ssI'S arc ma n-ria l . lId

" ,wr Idie, so me ar e not , but eV1'1I the latter remain immane nl 10 tlH'

wor ld of matter and enl' rgy. Thus, J) ' le uze's proc 'ss onto log, hn '"k

\ ith the essent ialism th at charac te r izes naive I' .alism and, sim ul

t.1I1 -ously, re moves on' of the main o bjec tions wh i h non -real ist s IIJ.l k,

against th e postulat ion of an auto no mous reality. T he ex te nt to whic h

he indc d deprives non -r ealist s fro m thi s casy way out dcp mds, o n lh ,

othe r hand, on the det ails o f his account of how th e mt it ies th .lt

populat e realit y are produced without the need for any thing trans e nd­

cnt, For thi s r eason I will not be co nce rned in thi s recon struction with

th e textual so urce of Delcuze 's ideas, nor with his sty le o f argumenta­

tion o r his usc of language . In sho r t , I will not be co nce rned with

De lcuze.'s words only with Dcleuzc 's world.

T he basic plan of th e book is as foll ows. Chapte r I introdu cs the

forma l ideas needed to think about the abstract (o r rather virtual)

struc ture of dynamical processes. I draw on th e same mathemat ical

resources as Deleu ze (different ial ge ometry , gro up th eory) but , unlike

him , I do not assume th e reader is already familiar with th ese field s ,

Deleuzes grasp of th e technical details involv ed is, I hope to show,

co m ple tely adequate (by anal yti cal philosophy standards) , but his

discu ssion of technical details is so co m pressed, and assumes so much

on th e part of th e read er , that it is bound to be misinterpreted .

Chapter 1 is written as an alt ernative to his own presentation of th '

subject, guidi ng the reader step by ste p th ou gh th e different math ­

emat ical ideas involved (man ifo lds , transformati on groups , vec to r

field s) and giving exam ples of th e application of these abstract ideas to

th e task of modelling concre te physical processes. Despite my efforts

at unpacking as much as possibl e th e contents of Deleuzes highl y

co m pressed description s, however, th e subject matter remains techni ­

cal and so me readers may st ill find it hard to foll ow. I recommend

that suc h readers skip this first chapte r and, if need be, co me back to

3

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.1 p p lk.HiclIl' III Ic ·s, .,h..u-ac t m.1I1t '1 II I tilt IfllIll\\ III lll.lph·1

C hdptl'rs and J deal \\ ith thlo prodlll l ion III Ih.. c11 11'T"1I1 t'n li tit'S

that populate Dclcu zes world. Tlu- h'l."iic them e i, lit" t , \\ ithin .l n·d lis l

pe rspective, one does not ge t rid of essences until one repl aces them

with so me thing else, This is a burden which affect s only till' realist

philosopher given that a non -r eali st can sim ply decl are esse nces mental

entit ies or reduce them to socia l conventions. One wa)" to think about

esse nt ialism is as a theor-y of the genesis of form, that is, as a theory

of morphogenesis, in which physical ent ities are viewed as more or less

faithful realizatio ns of idea l forms. The de tails of the process of

realization arc typica lly ne ver given. lisscnccs arc thou ght to act as

models, ete rnally maintaining their identity, while part icular ent it ies are

co nceived as mere copies of these models , resem bling them with a

higher or lowe r degree of perfect ion . Dcleuze replaces the False gen C'sis

implied by these pre-existing forms which remain th e same for all t ime ,

with a theory of morphogen esis based on the notion of the d!fferent. He

co nce ives differen ce not negathoely, as lack of resemblance. but

positiv ely or productively, as that which drives a dynamical process .

The best examples are intensi..e d!ffirences, the differen ces in tempera­

ture. pressure , speed, chemical concentrat ion , which are key to the

scientific explanation of the genesis of the fonn of inorganic crysta ls,

o r of the forms of organic plants and animals. Chapter 2 is concerned

with the spatial aspects of this inte nsive genesis while Chap ter 3 deals

with its temporal aspec ts .

After reconstructing Delcu zes onto logy I move on in Chapte r 4 to

give a brief acco unt of his episiemoloqy, For an)' real ist philosopher

these two areas mu st be , in fact. int imately related , This may he most

clearly seen in the case of naive realism, where truth is conce ived as a

relation of correspondence between , on on e hand, a ser ies o f facts about

the classes of entit ies populating reality and , on th e o the r , a ser ies of

sente nces exp rl.~ssing those facts. If one assumes that a class of entit ies

is defined by the esse nce wh ich its members share in com mon , it

becomes relatively simple to conclude that these classes are basically

given, and that the)' exhaust all there is to know about the world . The

ontological assumption that the world is basically closed , that entirely

novel classes of e ntit ies cannot emerge spontaneously, may now he

4

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mu ..t n 'l"tl ",Hh on I ' 0 1 Iltl·"" .'''U111ptiolt .uul 't 'pl.H"I' tl)l'lI1 \\Itll

di llt-n'n l om-s.

\Vhilt· in III.., fir..t thn'e ..hapu-rs I J.IIt'mpt 10 ~' li ll1i l1,lh' lIu· c rrom-ou

.I, ..umprion o f.1 d ost,c1 \\ o rld . in Chapter 4 I t r)' to repl 'ln · 1I0t Cl llh

tlu- idea of .1 sim ple co rrespo ndence hut . h C)'OIHI that, Il) J,ohlillt' ,"

\'(':Y iJ"ll cj.truth . In ot her words, I wi ll .1 rgUt· tha t even if otu - ,Kn'pt

th,lt there are true se-n tences ex prl.'ssing rea l facts it can stil l Ill"

mai ntai ned that most of these factual sentences arc lririal , TIlt" rolt, 01till' th inker is no t so m uch to utter truths or establish facts. hut to

di stinguish among thc large population of true fact s those that ,In·

import ant and relevant from those th at arc not. Importance anJ relevance,

not truth , arc the key co nce pts in Dcleuzc 's l' pistc molog)". the task of

realism be ing to gro und these co nce pts preventing them from hdug

red uced to subject ive e valuations or soci al co nve ntions . This point can

be made cleare r if we co ntrast Dcleuzcs position not with the

lingu ist ic version of co rre spo nde nce theory but with the mathcmaucal

one. In this case a relation of correspo nde nce is postulated 10 e xist

between the sta tes of a physical objec t and the solut ions to mathematical

models capturing the essence of that ohje ct. By contrast , Dcl cuve

st ress es thc ro le of cor rectly posed problems, rather than the ir true

solutions, a problem being well posed if it captures an objective

distribut ion of the important and the un im portant , or mo re mathemat­

ically , of the sinq ular and the ordinaly .

Chapte r 4 explores th is problematic epistemolo8Y and compares it with

the more fami liar axiomatic or theorematic versions which pred ominate

in the physical scie nces. To anticipate the main conclusion of the

chapte r , while in an axiomatic episte mo logy one st re sses the roll' of

qeneral laws, in a probl ematic on e laws as such disappear but without

sacrificing the obj ectivity of physical knowledge , an objectivity now

captured by distribution s of the singular and the ordinal). If such a

co nclusion can indeed be made plausible , it fo llows that despite the

fact that l reconstruct Dcl cuze to cater to an audience of scienti sts and

analyti cal philosoph ers of scie nce , nothing is yielded to the orthodox

5

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p!lf ,de.ll ~d('nn ,J .lIH I ,u IoI IJ l il',11 plai lo..oph)' "II11'l'g" 11'.1I 1'ilol"lll( ·;I· from

this enc-ounte r with Ik' k ul.t" till' rOnllCr rt'l 'lini ng its o l> j(·l·th·il)' hut

losing the laws it holds so ch-ar, the lat ter maint aining its rigo ur and

clarity but losing its ex clusive focus on fact s and so lutions . And more

importantly, the world itself eme rges transformed : the vcry idea that

there can be a set of true sente nces whi ch give us the facts once and

for all, an idea presupposing a d osed and finished wor ld , gives way to

an open world full of divergent processes yielding novel and unex­

pected entities , the kind of world that wo uld not sit st ill lon g eno ugh

for us to take a snapshot of it and present it as the final truth .

To conclude this introduction I must say a few words conce rning

that other audience which my reconstruction may seem to overlook:

Deleuzian philosophers , as well as thinkers and artists of different kinds

who are interested in the philosophy o f Dcleuze . First of all, there is

much more to Dcl cuze 's books than just an ontology of processes and

an epistemology of prob lems . He mad e contr ibutions to such diverse

subjec ts as the nature of cinema, painting and literatu re, and he held

very specific views on th e nat ure and genes is of subjectivity and

language. For better or for worse , these are the subjec ts that have

captured the attention of most readers of Del euze , so it will come as

a surprise that I will have nothing to say about them . Ne vertheless , if

I manage to reconst ruct Dcleuze 's world these other subjec ts should

be illuminated as well , at least indircctl y: once we understand

Dcl euze ' s world we will be in a better position to und erstand what

co uld cine ma , language or subjectivity be in that world .

On th e other hand, if th is reconstructi on is to be faithful to

Dclcu ze 's world it is clear that I mu st rel y on an adequate intcrprc ta ­

tion of his wo rds . Therc is a cer tain violen ce whi ch Deleu zes texts

mu st endure in order to be reconstructed for an aud ience they were

no t int ended for , so whenever I break with his own way of presenting

an idea I ex plain in detail the degr ee of rupture and the reason for it

in a footnote. A different kind of violence is invol ved in wren ching his

ideas from his collaboration with Feli x Guattari. In this reconstructi on

I use Dc!euze ' s ontology and episte mology as expose d in his ea rly

text s, and use onl y those parts of his collaborative work which ca ll he

directl y traced to those ear ly texts. For thi s reason I always ascribe the

6

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n t' lI \\ 11t II 'Juolln ' h OIll rlu-u 101llt 1I")d 1 11 1.111 ) . tlWlt' I th t \ lfl k i ll t

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hxillg hi'i 1t'l'l11illolog)· will 'i( '( '1II 10 'iOllh' .lkin to pin ning .10\\ II .1 Ii\'(,buth'rlly, As an .1IItidllt" I otl~'r .111 .ippc ndix whe-re I n ,I.,tt' till' h 'l"IlI "

u'I,d in Ill)' n -ron struction to all the diffe- rent h ·rmiuo!o git·s Ill' U'it'" III

his ow n t('x ts and ill his co llahorativc work, se tt ing his words li ",'('

linn' ,lgain afte r they have served their pu rpose of givi ng LIS his wurb l.

The hope is that this wo rld will ret ain all its ope nness and din'l'gl'lIn ' ,

so that the inten se cx prcss ivity and eve n madn ess so often at t ribut.-d

to De lcuzc's wo rds rna)' be see n as int egral properties of the wo rld

itse lf.

7

CIIAI' I LR I

7h !!'Iathemat ic C!I th e Virtu al:

Manifolds, Vector Fields and Traniformation Groups

or all the oncepts which populate the wo rk of Gilles Deleu ze there

is on ' that stands out for its longevity: th e conce pt of mult iplicity . T his

oncept makes its appearance in his early books and remains one of

central im po rta nce, with almost unchanged meanin g and function, until

his linal work. I Its formal definition is highly techni cal , including

·Ieme nts fro m seve ral different branches o f mathematics: differential

ge ome try, gro up theory and dynamical syste ms theory. In this chapte r

I will discuss the techni cal backgr ound need ed to define this important

concept bu t some preliminary informal remarks will prove help ful in

sett ing the stage for the formal discussion . In the first place , one may

ask what ro le the conce pt of a multiplicit y is suppose d to play and the

answer would be a re place me nt for the mu ch older phil osophical

concept of an essence. The es ence of a thing is that wh ich ex plains its

identity, that is, those fundame ntal traits without which an object

wo uld not be w hat it is. If such an esse nce is share d by man y objects ,

then possession of a co mmo n esse nce would also explain the fact that

these objec ts resemble each other and, indeed, that they form a distinct

natura l kind of things .

Let 's take one of the most trad itional illustrations of an essence .

When one asks what makes omeone a member of the human species

the answer may be , for example, be ing a 'rational animal'. The exact

definitio n of the human essence is not what is at issue here (if

rat ionality and animal ity are not co nsidered to be essential hum an

pr operties some other set will do). T he imp ortant point is that there

be some set of defining characte rist ics, and that this set explain bo th

the identity of the human species and the fact that particul ar memb ers

of the species resemble eac h ot her. In a Dcl euzian ontology, on the

othe r hand , a species (or any other natural kind ) is not defined by its

9

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consti tuted c lltit it's, tlw rescmhlanc« of the-ir uu-tubc-r: n.p lai,wd hyhaving undergone co mmon pron ·sst.·s of natu ral sch-ctlon . .1IId the

endur ing identity of the speck-s itself guar.m lt·t.·d hy the fad th.u it has

become reproductively isolated from other species. In short , whil e an

essentialist account of speci es is basically sta tic, a morphojjenctic

account is inherently dynam ic. And while an essentialist account may

rely on factors that transcend the realm of matter and energy (et ernal

archetypes, for instance), a morphogen eti c account ge ts rid of all

transcendent factors using exclusively form -gen erating resources wh ich

arc immanent to the material world .

Animal and plant species are not, of course, th e onl y natural kind s

traditionally defined by essences. Many other natural kinds, the

chemical clements or the set of elementary particles, for e xample , arc

also typically so defined . In eac h of these cases we would need to

rep lace timeless cate gorics by historical proccsses . Yet , even if success­

ful thi s replacement would take us only half-way towards our goal.

The reason is that e ven if the details of a given process account for the

resemblance am on g its products, the similarities which make us classify

them as members of the same kind, there may be similarities c!f process

wh ich still demand an explanation . And when accounting for these

common features we may be tempted to reintroduce esse nce s through

the back door. These would not be essences of object s or kind s of

obj ects, hut essences of pro cesses, yet essences nevertheless . It is in

order to break this vicious circle that mult ipliciti es are introduced .

And it is because of the ten acity of thi s circle that the concept of

multiplicity mu st be so care fully constructed , justif)'ing each ste p in

the construction by the wa)' it avoids the pitfalls of esse ntialism . To

anti cipate the concl usion I will reach after a lon g and techn ical

definitional journey: multiplicities speci fy the structu re c!f spaces c!fpossibilities, spaces whi ch, in turn , explain the regularities exhibited by

morphogeneti c processes. I will begin by defining an appropriate

notion of 'space ' , a notion whi ch must not he purely geometr ical hut

also capable of bein g linked to qu estions of process.

The term ' mult iplicity ' is close ly rel ated to th at of ' ma nifold' , a

term which designates a geome tr ical space with cert ain characte rist ic

1 0

pillpltlil 10 'f.1 P \\Iltl 1 I"tlll ,IIUlllI Ifllllllllial (.11lt1 \ liltII ltllltl 1111 (Ollllplt .mothlll' \ lllcl, 11111.11111I)11\\11111( tllltll

to 'I\ t ., hlll 'f .Hl t tunl 01 Il III tl"h.,1 1111'11I hilt III II 1111' lit ' 01

1"(lIl1t.tllt .,1 I_nul o.lnn ... lltl" IIH~ 'HI)U1UlII 01 IIIIII,lt'm IS .HI ,lIHllll1

pr .H tin ' IIlht·rih·tI fr o m t ilt' Cn'l·k!l, tht' es tcn ..i\t· u r- of ltll \t. .1Ilt!

tr. 'jl.c to rh·... in thl' formulalioll of .1 , .,rid) of plaf'ik .ll prohl t'l11!l from

tht. ... ixh't"llth n' lI l u ry on mad e it TWC(·ssar y 10 d('velop ru-w pl'ol,h'lIl

snh·illg n· ...ourccs. \Vith thi s ill mind, Hew," l k sc.,rh·s and Ph·H.· lIl'

l-cr'mat inven te d lIll' now familiar method of t'mht,tlding curves into .1

rwn .dinu-nsional sran~ on which arbitrar y axes co uld lu- fixed . Ont (.

so embedded . the fixcd axes allowed the assigIlIlH.' nt of .1 pJ.ir of

num bers, or coordinates, to every point of the curve, so that the

gl'OIlll't riC rel ation s between points could now he expres sed as n '!at ion

be tween numbers, a task for which the newly developed algehra \\ .1"

perfl.ctl y suite d. This translation sche me, in short , allowed the comhi

natorial resources of algebra to be brought to hear on the so lution of

gt·ome trical problems.The term ' ma nifold' does not l>dong to the analyt ical geo ll1 l'l ry of

Descartes and Fermat, but to the d1Jerent ial aeometry of Friedrich Gauss

and Bernhard Riemann , but the basic idea was the same : tapping into

a new reservoir of probl em-solving resources, the reservoir in thi s case

bei ng the differential and int egral calculus. In its original application

the calculus was used to solve problems involving relations between

the changes of two Of more quantities. In particular, if the se rel ations

were expressed as a rare <if change of one quan tity relative to ano ther,

the calculus allowed findin g the instantaneous value for that rat e . For

example , if the changing quantities were spatial position and tim e , one

co uld find instantaneou s values for the rate of change of one relat ive

to the other , that is, for velocity. Using this idea as a resource in

geome try involved the realization that a geom etrical object , a curved

line or surface , for instan ce, co uld also be characterized by the rate at

which some of its properties changed , for example , the rate at wh ich

its curvcrure change d between different points. Using the tools of the

calculus mathematicians could now find ' instantaneo us' values for this

rate of change , that is, the value of the curvature at a given

infinit esimall y small point.In the ear ly nin eteenth century, when Gau ss began to tap into these

II

dll it It 1111••1 I t IIUI'I ', I tll l\ld 1\ \ 11 1111111 11 1111111 III l.u t \\.1 tudn-d1l"lIlg Ih(' old l ' .l rh·'I.lIl 1I1dhod : 1111' "'Il llI.H(· \\,1 t IIIhnld,d 111 .1 Ih n 't'

dilll t'II,,,iOIl,tl sp.tn' l'o m p lt"lt' \\ itll it" 1) \\ II fl'(t'd "(' 1 01 .lX t ·" : then , thillg

tllO~1' axvx, coo rd inates wou ld he .t~sjg ,wd to (,\(T) ' point of the

~lI r1 al'C"; finall)', . the gl'ometric links between points deh>rmining the

form of the surface would be expressed as algebraic n -latlons between

the numbers. Rut Gauss realized that the calculus , focusing as it docs

o~ infinites~mal points on the surface itself (that is, op(~rating entirely

With local mformation), allowed the stud), of the sur face without anI

rejerence to a alobal embedding space. Basically , Gauss developed a method

to implant the coordinate axes on the surface itself (that is, a method

of ' coordinatizing ' the surface) and, on ce points had been so translated

into numbers, to use dilTerential (not algeb raic ) equat ions to character­

ize thei r relations. As the mathematician and historian Morris Kline

obse rves , by. g<·lting rid of the global embedding space and dea ling

With the surface through its own local properties 'Gauss advanced the

to ta lly new concept that a suiface is a space in itself ,2

The idea of st lld)'ing a surfa ce as a space in itself was further

developed by Riemann . Ga uss had tack led the two-d imensional case

so one would have ex pecte d his disci ple to treat the next case, three­

dimen sional curved surfaces . Instead, Riem an n wen t on to successfully

attack a m uch more gene ra l problem: that of N -d ime nsional surfaces

or spaces . It is these N-d ime ns iona l curved structures , defined excl u­

sively thro ugh the ir int rin sic features, that were originally referred to

by the term ' ma nifo ld ' , Riemann' s was a vcry bold move, one tha t

took him into a realm of abst ract spaces with a varia ble number of

dimen sion s, spaces which could be studied withou t the need to embed

them intu a high er-dimension al (N't- L) space . As Morris Kline puts it :

' The geometry of space o lTered by Riemann was no t just an extension

of Gauss 's d ifTerentia l geometry . It reconsidered the whol e approach

to thc study of space . " And we co uld add that this new way of posina

spatial problems would, a few decades later in the hands of Einstein and

othe rs , co mpletely alter the way physicists approached the qu estion ofspace (or more exactly, of spacetime) .

A Dclcu zian multiplicity takes as its first defining feature these tw o

traits of a manifold : its variable number of dimen sion s and, more

importantly . the absence of a supplementary (higher) dimension impos-

1 2

1I1~ .1Il t 11111 It 100ltllll.lll/oI llIlIl , III.! III lit I , 1m ,,,,11\,",11. ,I./IuulIUIlO , I I I II lilt \\l llt · • ' tl hl l'lI l l " 11111 1 lit II l it l'll .llt •• 1 11111111

u.umu ell 11' 1' 1II .1Il \ .lIltl t ilt 0111 , hili , .ttl ll I an 01 ' .11111.1110 11 lu1011 ' II I'

III ti ll ' IIMIl\ .t lIl h, \\ hh h la.l IIU Ill·"d \\ h,ll"lll '\(" til ullit) ill on lt'l

10 IOflll .1 ",,, It'm ,' " 1' " , 'lH t ' ''l, 0 11 the- othvr h.ultl, do p ll ..'t· .....t dd illlllg

u nitv (r.g. J tlw uuitv 01 r,l lio ll.t lil) ' and ,mil1lollily de fining till' lunu.m

t'S'it'; ln') ,lilt!. lIlon ': )\'('[ ' arc takr-n 10 vxi ..l in ,I t ranscx-nd r-nt "1'.1' ~\\ hi," ".. rvc-s i.1S J container for them or in which lhe) an' (· m b ,· c1d ,·t l.

A mult iplicity, on the other hand , 'h owever man y dinu -nsion ... il 111.1)

Ii.l n ' , . , . ne\'er has a supplementary dimension 10 that which tran

"pin's upon it. This alon e mak es it natural and immanent. ' S It 111.1)' he'o bj ('Cll'" that these an' pllrl'i )'J ormal dilTl'n'n ces between concq >t s• .and

that as such , they do not ne cessaril y point to a deeper ontological

din~'[t.·I lC'e . If we~ arc to repl ace esse nces as the explanat ion of till'idcntitv of materi al obj ects and natural kind s we need to specif)' tlu ­

wav in wh ich multiplicities relate to the ph ysical processes which

gL' l;e rate those mat erial obj ects and kinds,

Achieving this goal implies establishing a more int imate relation

between the geometric properties of manifolds and the propcrt h-s

which define morphogenetic processes. The resou rces in this case co me

from th e theory of dynamical systems wh ere the di mensions of a

manifold ar c used to represent pr op erties of a particul ar phys ical

process or system, while the ma nifold itsel f becomes [he space ifpOSSible

states which the ph ysical system can have. " In other words, in this

theo ry manifolds are co nnec te d to material reality by thei r USl : as

models of physica l processes. When one attem pts to mod el the dynam ­

ical be haviou r of a particular physica l object (say , the dynamica l

behaviour of a pendul um or a bicycle , to st ick to relat ively sim ple

cases) the first step is to determine the number of relevant wap in

which such an ob ject can change (these are known as an object 's deqtees

iffreedom), and then to relate those changes to one another using the

difTcrential calculus . A pendulum , for instance , can change on ly in its

position and momentum, so it has two degrees of freedom. (A

pendulum can, of course, be melted at high tem peratures , or be

explode d bv dvnamite . These ar e , indeed, other ways in which thi s

object can ~ha~ge, the)' sim ply are not relevant ways from the point

of view of dynamics. } A bicycle , if we consider all its moving parts

13

( .HI

dlllll II IOIMI In.Ulll nld ( \ " II, d In!JUlllllll I \ llh " 1' '' ' 1 I I" It 11111111 III I

III III. h. 1I.l\ nuu 01 till H"I' t 1111 It I ,11 11 1 li lt I II I. I.IU.I n pll 1111

oll tu.il t 'lll ' o f (,tlc " lit ,I pll\ h .11 '1111I , .1 1.11 It mlllll 'l h I 111 th ,

ht'h.l\ uurr of tllt' 1111\"lt "I s, tvrn it (·It .·, ,Sillgul.lriti t's 11M)' inlhlt'Jl(\" bt,h,l\'iour h) ,u ting ,\ S aftra(t ~T\ .Ii .,r 1111

Ir,\jc'llorit':-. . \ Vllal this IIW.lIlS is th.n ,I I.ug\· numlx-r of dlllen'nl

tr,Ijt't'lorit's, :-.t,uti ng thei r cvo lutio u at \'t'r)' d ill't' rent placx-s ill ti lt'uianifoh l, m ol)' end lip in t'xactl)' th e same final sta te (t he at tractor) . .IS

long a.s all of them hl'g in somewhere within the 'sphere of inlhn-no­

of the auractor (the basin oj aurdcri on). Given that , in this SI' T1St',

din....rent trajectories rna)' be attracted to the same final state . singu lar .

itivs arc said to reprl'scnt the inherent or intrinsic long-term tendencies

of a svstcm, the sta tes which the system will spontaneously tt'nd to

adopt "'in the long run as lon g as it is no t co nstrained hy other fon~c..;s,

Some singularities arc topological points , so the final sta te they d ('(IIlt.'

.1S J dest iny for the traj ectories is a stead)' state . Beside these, Poin care

also found that ce rta in closed loop s acte d as attractors and called them

"lim it cvcles' , The fina l state which traject ories att rac ted to a limit

cyc le {or periodic at tractor) arc bound to adopt is an osci llato ry state .

But whether we are dealing with steady-state, periodic or o ther

attractors what ma tters is that they are recurrent topoJo8ical features,

which means that different sets of equations, re presenting quite

different physical systems , ma y possess a similar d istribution of aurae­

tors and hen ce, sim ilar lon g-term behaviour.Let me give a sim ple example of how singularit ies (as part of wh at

defines a multiplicity) lead to an entirely different wa )" of viewing the

genesis of physical forms. There are a large number of difTerent

physical st ructures which form spontaneously as their co mponents try

to meet certain energetic requirements . T hese com ponents rna)' be

constrained , for example , to seek a point of minimal free energy, like

a soap bubble , wh ich acquires its spherical form by minim izing surface

tension, or a co mmon salt crysta l, which adopts the for m of a cube by

minimizing honding energy. W e can imagine the state space of the

process whi ch leads to th ese forms as st ruc ture d by a single point

attractor (re presenti ng a point of minimal ene rgy). O ne way of

describing the situatio n would be to say that a capo/anical fo rm (a

singu lar poi nt in a manifo ld) guides a process which results in many

.. I mhh .lIl el Ih.Ihl '1\1 P,lIt ~

(11,111111111,11 • 1I.1I1t ,,1111 I. Il ,lIIl t 1i.1I1l '1 .11 \\1111 I1\\01'('11.11,) h,l u-n cI,·gr.t 01 I"'t·dum (C·.1l1a 01

I h,Ulgt· in b01h po,i1ioll .1Ite l 1Il0uH'1I1Um) ,

Ncxt , one maps 1',1('11 d'·gn't· of frn,dolll into one 01 lht, dirrn -usiun s

of a mani fold , A pendulum ' s span' of possihilit it's will m-cd a twu .

dimen sion al plane , but the bicycle: ' vill involve a ten -dimen sion al s p,lCt' ,

~fter thi~ mapping ope rat ion , the state of the object at any gi vl' ll

Instant of time becomes a single point in the manif old , whi ch is now

called a state space. In addition, we can capture in this model an

object's changes ef state if we allow the rep resent at ive point to move in

this abstract space, one tick of the clock at a tim e , describing a curve

or traj ectory . A physicist can then study the changing behaviour of an

object by st udying the behaviour of these representativ e trajecto r ies . It

is important to noti ce that eve n thou gh my exam ple Invo lves tw o

objects , what their state space captures is not their sta tic properties

but the way these properties change, that is, i t captures a process. As

with any mod el, th ere is a trade-off here: we exchange the co mplexity

of the object's changes of state for the complexity of the modelling

space . In other words, an object's instantaneous sta te, no matter how

co mplex, becomes a single po int, a great simplification, but the space

in which the object's state is embedde d becomes more complex (e .g .

the three-dimen sional space of the bicycle becomes a te n-di mensiona lsta te space) .

Besides th e great simplification achieved by modelling co mplex

dyn amical processes as trajectories in a space of possible states, there

is the added advantage that math ematicians can bring new resources to

bear to the study and solution of the physical problems involved . In

particul ar , topoioqtcal resources rna)" be used to analyse ce rtai n features

of these spaces, features wh ich det ermine recurrent or typical behaviour

common to man)" different models, and b), ex te nsion, co mmon to

man)' physical processes. The main pion eer of this approach was

ano the r great ninet eenth-century mathemati cian , Henri Poin care . Poin­

care began his study not with a differential equation modelling a real

physical syste m , but with a vcry sim ple equation, so sim ple it had no

physical applicat ion, but whi ch nevertheless allowe d him to ex plore

the recurrent traits of an)' model with two degrees ef freedom. He

discovered and class ified certain special topological features of two-

'4 I ~

dlllt"t 'l1l ph) 11.11101111. lIullllhlll ph~' II ' !'i ntl (11)1('. l.hll llllt \\1111t1 ifl ~ ' n ' 1I1 .'I( (lm rtru propt"rti~''' ' 'I Ill' i.. \\ h.n 1 )..1 " \1/ 1' nu-ans \\ [u-u Ill'sa)'s th.u singular-i lil's are like ' im plicit for ms th.n df\" Illpologil.ll r.ulu-rth an gl·OlTIl· t ric ' ,'* Thi s rna )' hi' cont ras ted to the ('sSt" lIt i.llis t dpproadlin whi ch th e ex planation for the spherical 10fl11 of soap bubbles, forinstan ce , would be framed in terms of the esse nce of sphe r icity. tha tis, o f ge ome trica lly charac te rized essences acting as ideal Forms.

I will di scuss in a moment th e mean ing and re levance of th etopological nature of singularit ies . What matters at thi s point is thatsingularities, by determining long-term tendenci es, st ructure the possi­bilities which make up state space , and by ex tension, str uc ture th epo ssibilities op en to th e phy sical process modell ed by a state space. Inaddition, singu lari t ies tend to be recurrent that is th ey tend to, "characte r ize processes independently of their particular physical mcch -anism s. In th e exa mple above, the mechanism whi ch lead s to th eproducti on of a soap bubble is quite dilTerent from th e one leading toa salt crys tal, ye t both arc minimizing proce sses. This mechanism~independence is wha t makes singular it ies (or rather th e multiplicitiesth ey define) perfect cand idates to replace essences ."? As I said befnre,however, we mu st be care ful at thi s stage not to make sing ularit ics th eequivalent of thc esse nce of a process, To avoid this erro r I will discusssome additional formal properties of multiplicities distingui shing themfrom essences and then, as above , I will discus s th e way in whi ch th esepurely co nceptual differences connect with qu estions of physicalprocess.

The formal difference in qu estion has to do with the way essencesand multiplicities arc specified as ent it ies, Whil e essences are tradition­ally regarded as po ssessing a clear and distin ct nature (a clarity anddisti nct iveness also charac te rizing th e ideas whi ch appear in the mindof a phil osopher wh o grasps one of th ese esse nces) , multiplicities arc ,by design. obscure and disrinct: th e singularit ies which defin e a m ult i­plicity co me in sets . and th ese sets are not given all at on ce but arest ructure d in such a \\'ay th at th ey pro8ressiYcIy specify th e nature if amult iplicity as th ey unfold foll owing recurrent seque nces. II What thismean s may be illustrat ed first by a metaphor and then given a precisetechni cal definition . Th e metaphor is that of a fertilized egg pri or toits unfolding into a fully developed orga nism with di fferentiated tissues

16

.u II I 0 1 1.1lI ( \ l"Otl ~1l0\\11 ,1 ",I".t'l/ II \' ) \\1111. 1111 11111 .111 tlIItt'Pltl .lllon 0 1 I mill \11 'I li t I II III Illd ol .m .HI "pI H' fdl "hl"I, t '.It!\ gl\ I 'll III tlw I' I I ( t'f('/~mll(tJ, • ~ It \ \ ct ,,\1 11 1 III li t t 11,1\ III I d t 1.- .11

,lIu l di..tim I 1l.lhln') llIo 'll billlngl '" IlId ,,) h.I\t.' g l\ (' 11 Ill' plt' lOllIlI"illt,II lt I .h t t"ph"d IIH' idt.'., th.u dIlItTI'llli.tlt·d ..'rll~ um-c l' l1 ll' rg l' P~I~ "I·..i\l.h' .IS ti ll' ('g.g <I t"\\·lop,... TIH' l"gg is 1I0t, 01 nH lrSl'•.111 untllllt'n 'lIli,'h';1 mass : it P()~St' SSI · S au obscure )l'l t1istillt:l ..rruc-turc dt"lilu,d h)zones of biod ll' l11 il"al comTlllr.llioll and hy po lari t h-s l's lolhlistH't1 h)' till '.IS\, I11 111 t.. t r il..ll positi on of till' yolk (o r nucleus), But e-ven though itde:t's POSSt'SS th e necessary biochem ical materials an ti ge netic i ll f~) r~llJ.tion, th ese materials and info rmat ion do not co nta in a clea r ,mel (hst llU'thiucprint of th e final organism . 11 .

Although th e egg me taphor docs provide a vivid illu st rat ion 01 d.wdis ti nct ion I am trying to draw here, it is nevertheless just a wwlul,,m alogy. Fortunate ly, th ere are technical wa)'s of dcfinin.~ th e hit',]. 01f,oHressil'e J1JCrent ialion wh ich do not rely on metapho rs. I [u- technicalresources in th is case co me from ano ther crucial ninctl'el1lh ~lTl1ll1r)'inn ova tio n. th e theory of groups. a field of mathemat ics which. tikt'th e d ifferen tial ge ome try I discussed before, e ventua lly becam e anintegral part of th e basic mathematical technolo~)' ()f.t~\'cn t ic ~ h-c('n t u~yphysics. The term 'gro up' refers to a set of entltl~s . (WI~~ specialpr op erties) and a rul e of combinat ion for th~se cntl~lcs . .1he most,important of th e pro pe rties is th e one nam ed closure , whi ch meanstha t wh en we usc th c rul e to combine an)' two entit ies in th e set, th eres ult is an e nt ity also belonging to th e set. For exam ple . the set ofposi tive integers displays closure if we usc addi tio n as a comhin~ti.onr ule : add ing together any two positiv e int egers yield s anothe r posltl\'Cinteger. that is , another clement in th e original set. H . .

Although sets of numbers (o r many othe r m ath ematical objects)rnav he used as illustrations of g roups , for th e purpose of ddlnmgpr~grcssi\'e differentiation we need to co nside r groups ~\' h~se m embersare not objec ts but tranifo rmations (and th e co mbinat ion rul e , aco nsecut ive applicat ion of th ose transformations) , For e xam ple, th e, setco nsisting of rot ati ons h)' nin et y degrees (that is. a set conta in ingrotations hy 0, 90 , 180. 270 degrees) forms a group , since any .twoco nsecut ive rotations produce a rotation also in the gro up. provided360 degr ees is taken as zero . The importance of gro ups of transforma-

17

1'011 tI•.•1 till III 1'1 II I" 101 , I I II '1 OIlI lt 11 11 11 '111 ' II till II

/II I(.lr1</III\ : II \\, ' p," 101 Ill!''' Olll 01 till I , Ollp 10111 11'11 O il I IIh, . .in

observer who did Ilol w it m -ss till' Ir.II I, lo rJll,ll io ll \\Iurld not ", ,1" 1,' ( 0

noti ce that allY change had .1·tually O("('lI'T,'d (th.u is, tilt' iS lI.11

appca ranc of the cube wo uld rem ain invarian t rc l.uivr- to this

observe r). O n the othe r hand , the cube would not remain invariant

und er ro tatio ns by, say, 45 degrees, but a sph ire wou ld. Inde d, a

sphere rem ains visually un changed under rotations by an)' amount of

degrees. Mathematically this is ex pressed by aying that the sphere has

more ~mmet1J' than th e cube relative to the rot ation transformation,

That is, degree of symme try is measured by the number of transforma­

tions in a gro up th at leave a property invariant, and relation s bet ween

figures may be established if the gro up of one is included in (o r is asubgro up of) the gro up of the other.

ClaSSifying geometrical objects by their degrees of symme try repres­

ents a sharp departure from the trad itional classificati on of ge ome trical

figures by th eir essences . While in th e latter approach we look for a

set of properties co mmon to all cubes, or to all spheres, gro ups do

not classify these figures on the basis of their sta tic prop erties but in

terms of how th ese figures are affect ed (o r not affect ed ) by acti ve

transformations, that is, figur es are classified by their response to events

that occur to them. I f Another way of putting this is that even though in

this new approach we are still claSSifying entit ies by a prop erty (their

degree of sym me try), thi s property is never an intr insic prop erty of

the enti ty being classified but always a property relati ve to a specific

transformation (o r group of transformations). Additionally, the sym­

metry approach allows dynamic relations to ente r into the classification

in a different way. When two or more entities ar e related as the cube

and the sphe re above, that is, when the group of transformations of

on e is a subgro up of the other, it becomes possible to envision a process

which converts one ~ the entit ies int o the ocher by losing or gaining

symmetry. For example , a sphere can 'b ecome a cube' by loosing

invariance to some transformations, or to use the technical term, by

und ergoing a ~mmet1J'-breakin8 transiti on. While in the realm of pure

geometry this transmutation may see m some what abstract , and irrelev­

ant to what goes on in th e worlds of physics or biology , there arc

18

111.111\ dlll·.II.III"" II I \1111 111It \ I.. 1~1I 11 ' 1.1111111111 III til 11'"1'

1 1'11 1 I t Ie 1II Ifl1,IIII',

I II ph Il .d III 01 I "III 111111 .111 1111 Ih, OIl 'h hi o ~. 11 11l 1l1l 1/\ 11 10.

,,,. III , 1111 " .unpl,-, III till 101111 III rho,,' 1'.," '1/ ,111 I'h.1 I (, .111 111 1111

11" ,'\, '11 1 "hi. h I.lk. · pl.Il" .11 '1Ilil .11 \,1111\' ot OIlW p,II .IIIIt'I. I

II, IlIpl'loIIUI" ', lill'" .nuph-) w it lUll' ,I pll\ ital "It'1Il IIo III tJ/1l 1.11.

\., .mo t lu-r , like the, riti ca] POlllls Ill' tl'mlwr,llllrt' al "llIlh \ .1"'1

, !l .11I , , 'S [rom ice to liquid, or rro m liquid to str-am , '1Ill' "rokl II

'.\ nutu-tr aspect here can be clcar l S 'I' lI il' WI: cOIllI),\/"l' tilt' g.lS .lIId

""Iid states o r a material , and if' Ior simpli 'it , W I ' .ISSUn1l' p!'r rl'etl )

Illdlill'lll gases and perfect rystal arrange ments. In these idl',ll COli

,lit ions, the gas would d i 'play invariant prop ert i , . under all I rans la

t 1< iu s, rotations and reflections, while thc solid wo uld be invarian t [Il

" lIly a subs et of these transformations. For xamplc, while tlu- 'il

cou ld h disp laced by any amo unt and r main basi ally the same (that

is, an observer wo uld b unable to tell whether a clisplacc m -nt

lie urrcd at all) the so lid wo uld rem ain visually unchanged onl und er

displacem ents which moved it one unit crystal at a time (or multip les

of that unit). In other words, the gas has more symmctry than till'

so lid , and can become the so lid by undergoing a symmetry-breaking

phase tra nsitio n . I S The metaph ori cal example I gave above, that of a

rel'tilized egg which differentiates into a fully formed organism, can

now be made quite literal: the progressive di fferentiat ion of the

spherical egg is achieved th rough a complex cascade of symmctry­

br eaking phase transiti ons. 16

Let me now incorporate the idea of progressive di fferen tiation into

the co nce pt of multiplicit y by showing how it can be translated int o

state-space terms. I said before that for th e purpose of defini ng an

entity to repl ace esse nces the aspect of state space that matter ed was

its singulari ties . O ne singular ity (o r set of singular it ies) may undergo a

sym met ry-breaking transition and be co nve rted into another one.

T hese transitions are called bifurcations and may be studied by add ing

to a particular state space one or more 'control knobs' (technically,

control param et ers) which det ermine the stre ngth of externa l shoc ks

or perturbations to which the system being mod elled may be subject.

These control param et ers tend to display critical values, threshold ' of

IIIkll:'4l1, .11 \\Iull. ,I p ,lI III 111 ,11 hllllll .llIoll LIll' pl." I ' I lll".LIll I till III luI

S}llIrndry of lilt' ")""h'I II . J\ " Li ll o "p.ll t' "t1 lh llll n l It) 0 111' 1)1 Ii lit

nt tr-ac-tor, 1(11' cxn m p lc, 111.\) ' hifi.l n -.l tt' in to ,lIIlltht'r w ith 1\\C1 ~HH h

at tractors, or a point an -ac to r may hil'urcat l' into ,\ IH 'riodi c o tu -, losing

some of its ori ginal symme try . 17 Much as at t rac to r-s cor m - in recu rrent

forms, so bifurcations may define recurrent sequen((~.'i o f' such forms.

There is a sequence , for instan ce , that begin s with a point an -acto r

which, at a critical value of a control param et er, becom es unstabl e and

bifurcates into a periodic attractor. Thi s cycl ic singular ity, in turn, can

become unstabl e at another crit ica l value and und ergo a serlue ncc of

instabilities (several period-doubling bifurcations) which transform itinto a chaotic ott-actor.

Thi s symmetry-breaking cascade of bifurcations can , in turn, be

related to actual recurring sequences in physical processes . There is,

for example , a realization of the above cascade occurring in a well ­

studied series of distinct hydrodynamic flow patterns (steady-sta te ,

cycl ic and turbulent flow) . Each of these recurrent flow patterns

appears one after the other at well -defined critical thresholds of

temperature or speed . The seguencc of phase transit ions may be

initiated by heating a water container from below. At low temperatures

the flow of heat from top to bottom , referred to as thermal conduction,

is simple and steady , displaying onl y a hland, featureless overall

pattern , having the degree of symmetry of a gas. At a cr it ical point of

temperature , however, thi s ste ady flow sudde nly disappears and

another on e takes its place, thermal con vection, in which coherent roll s

of wat er form, rotating either clockwise or anti -clo ckwi se . Th e water

conta iner now has struct ure and, for the same reason , has lost some

symme try. As the temperature continues to intensify another threshold

is reached, the flow loses its orderly periodic form and a new pattern

takes over: turbulence. The cascade that yields the sequence conducti on ­

convection-turbulence is, indeed, more complicated and may be

studied in detail through the usc of a special machine called the

Coue tte-Taylor apparatus , which speeds up (rather than heats up) the

liquid mat erial. At least seven different flow patterns are rev ealed bythi s machin e, each appe aring at a specific critical point in speed, and

thanks to the simple cylindrical shape of the apparatus , each phase

20

11 .111 111011 111.\\ I II dill I I " I I Lilt d t il '" 111 1l~ 1 II \ I lllii t 11 ' III II.. I I ti ll I'fll 11 .111 lelllll .1111l1I 111 11 11 I , llIltll l I

'\ 'Ii 1.111 h.' 1'1 '11 It 0 11I 1111 1' ox.un p h-, .1 1 .1 t,ltl., III halUll ,llilln, 111.'\ lit1.lith lu lh n ·,IIi/t·d ill .1 )'11\ !'o i l ,II "!'itt' ll l, I hi, n ·.lli/.'1 Ion , ho\\ t" r-r, ht·.I1 !'1

Ilt l rt·,,· ;nhl.ll1l t' to till' 1I~lol tlh ' IlI :\lk,, 1 ( '.l ~(,ldl' . III p,lrt h Il I,H' , un like ti ll"

1,IUI '1" " hkh is mt.·dllll1/HII -lntlqll·nJl·rH, tlal' php ic,t' n '.,lil'"tio lt ill'"ht '

"I'l'd lic nwch,mi.'ims. To hl'gin with tln-n- ,11"1 ' causa l inu-r.nuons .uul

tlu-ir t' ITl'ets . To re t u r n to our e xam ple, till: How o f 1H'"t into ti ll't'tlllt"inl'r causes ,1 graded dl'l1sity ditl l'n'IKe to form , gin' l1 th'lt wau -r

t'xlMnds wlu-n heated (that is , becom es less den se) . This tklls it)

gradient , in turn , interacts with other forces like the viscosity or ti ll'wate r, their balance of power det ermining whether a s)'SIt'Tn swi ld le"

fro m one flow patt ern to the next. For exa mple , the de nsity gradit·nt

will tcnd to amplify small differen ces in movem ent (fluctuat ions) \\ luch

cou ld add so me detail to the bland ste ady-sta te How, but whk-h art'

damp ed by the viscosity of the fluid. As the flow of heat is intvnsifi...l,

how e ver , the syste m reaches a cri t ical po int at wh ich th e dl'n sity

gradiellt is strong enough to overcome viscosity, leading to till'am plification of fluctuations and allowing th e formati on of co here-nt

ro lls . Thus , a very specific seque nce of e vents underlies the transition

to convection. On the other hand, as the biologist Brian Goodwin has

pointed out , portions o f this hydrodynamic seque nce may be obse rved

in a complete ly different process, the co mplex morphogenet ic

Sl'llue llce whi ch turns a fertilized egg into a fully developed organism.

Afte r describing another instan ce of a sequence of flow paltl'rns in

hydrod ynami cs Goodwin says:

Th e point of the description is not to sugges t that morphogen et ic

patterns originate from the hydroJynamie properties of living

orga nisrns . .. What I want to emphasize is simply that many

patt ern-generating processes share with developing organisms the

characterist ic that spatial detail unfolds progressivel y simply as a

result of th e laws of the process. In th e hydrodynami c exa mple we

see how an initi ally smoo th fluid flow past a barri er goes through a

symme try- brea king event to give a spat ially periodic pattern , fol ­

lowed by the elaboration of local nonlinear detail whi ch devel ops

2 I

hilt ti ll') do ~o .,1 po int ' . ti ll tilt' , 'e1 1 t · ~••md umh-r g lllllllll' l ill '

whic h IH'\('r havv llll' u n ilelr ln it ) clf .1 n.uu rol light . ()n (·.It h

Ill t .l , ion , ohscurit it·l'i .Hlel 1 111l1' S of ,Jude,,\ co rn" pofl( l 10 tlu-u

c1 i,linl"l ion . IMultiplidt il' ''il .H·t' c1islingub.lwd from on e .urorln-r , but

nu t at .111 in the same manne-r as form s and tlu- terms in "hkh thc' '''

.m - incarnated . Thcv are ohj l'ctin'h' mad e and unmade according tl l

till' co ndit ions that : lct e rl1l inl' thl'i; fluent synthes is. This is IWI". 11'1',tht;)' co mbine the greate st power o f hdng difTcrenrialt·d with an

inahility to be dttfcrcnciated .JO

Altho ugh I will not stick to thi s subtle typographical distin ction ,

I ),ol,'ule distingui shes the progressive unfolding of a multiplicit j'

Ihrough broken symmetries (diffc renrioltion), from the progn'''isht'

"i lwcification of the continuous space formed by mu ltip licities as it gin'''i

rise to our world of discontinuous spatial structures (differl.·nciatio n) .

Unlik e a transcendent heaven which exists as a separate dimension from

n 'aHty, Dcleuze asks us to imagi ne a conti nuum of mul tiplicities whic-hdlj]crenciaees itself into our familiar three-dime nsional space as wel l as

its spatially struct ured conte nts ,Let me explain in what sense a co ntinuous space may he said to

beco me progressively defined giving rise to discontinuous spaces . First

of all, a space is not just a set of points , but a set together with a way

of binding these poin ts together into neiohbouThoods th rough well ­

defined relati ons of proximity or continuity . In our famili ar Euclidea n

gcomet ry these rel ations are specified by fixed lengths or distance"

which det ermine how close pOinL'i arc to eac h othe r. The concept of

' length ' (as we ll as rel ated ones, like 'a rea' or 'volume' ) is what is

called a metric conce pt , so the spaces of Eucl ide an geome try are known

as meeric spaces.1. There ex ist other spaces, however. whe re fixed

distances canno t define proximities since dis tances do no t remain fixed.

A topological space, for exa mple, may be stre tched without the

neighbourhoods which defin e it changing in nature . To cope with such

exotic spaces. mathematicians have devised ways of defi ning the

property of 'be ing nearby' in a way tha t does no t presup pose any

olll 01 llit ' PI ' rlOelh It) Illlhn ollH d ,· \ (·IClI"I ll'1l1 10110 \ \ .1 111 11 1.11

(llI.l lit.l l in · COurw: illit i.,lly Mllooth p'II1"" .IX'·... . dIC·IlI' (·h ,·... tlH'res ult of Sl>.ll ial bifurcatton from .1 uniform st.ll t·, bifun:.llt· to

spa tially periodic patt ern s such as sl'gnwl1 ts [iu J.1l Insec-t hodYI.

wi th in which fine r de ta il develops . . . through .1 progn'~si\'c

expression of non linearities and successive bifurcations . . . The ro ll.'

of ge ne products in such an unfolding is to stabilize a parttcular

morphogenetic pathway by facili tating a sequence of pattern transi­tions, res ulti ng in a particular morphology, l q

From a Deleuzian point of view, it is this uni versality (or mechanism­

independen ce) of mul tip licities whi ch is high ly significant. Unlike

essences which are always abstract and general entit ies, multiplicities

arc concreee universals. That is. concre te sets of attractors (realized as

tenden cies io physical processe s) linked together by bifurcations

(r ealized as abrup t transitions in the tendencie s of physical processes) .

Unlike the generality of essences, and the resemblance with which this

gene ra lity endows instan tiations of an essence , the universality of a

multiplicity is typically diverqem: the different rea lizations of a multi ­

plicity bear no resemblance \..-hatsocvcr to it and there is in prin ciple

no end to the set of po tential divergent forms it may ado pt. Thi s lack

of resemblance is amplified by the fact that multiplicities give form to

processes, not to the final product, so that the end res ults of processes

realizing the same multipl icity may be highly dissimilar from each

othe r , like the spherica l soap bubble anel the cubic salt crys tal which

not only do not resemble one anothe r. but hear no similarity to thetopological point guiding th eir production.

The co nce pt of progressive differentia tion whi ch I have just defined

was mea nt , as I said , to dist inguish the obscure yet distinct natu re of

mult ip licities from the clear and disti nct identity of essences, as we ll

as from the clari ty afforded by the light of reaso n to essences grasp ed

hy the mind . A final di stinct ion must now be made : un like essences,

wh ich as abstract genera l en tit ies coe xist side by side sharply distin ­

gu ished from one another, concrete universa ls must be thought as

meshed tonether into a continuum. This further blurs the identity of

multipliciti es, creating zones of ind iscernihilitv where they blend into

each othe r , fonning a continuous immanent space very different from

22

•• I I II\1ltl 0 1 t Il 111011 .lId l' " pit Ol I t

\ t1 It11'11\ 1111 , I I). I. 1111 \\ l i lt •

li ll lilt I Ollll 'pl , hili 0111\ 110111111 111 1 t IIllt t pi I.kl ' 1111111111' 1I11••1 ,Ill t '

nl "'~!". lion ,'\ vr nih' I II.II",H. h ', it t'''' It . lilt ' til tu« tmn 111'1 \\ ITII II/n t ll IlIlt '

nonmctric J/,I1US is fuu d.uucutal in .1 I)"I"llli.Hl olllol,,!:) . rJ ,\1o n'o 't' r .

and this is the cr ucial point, there arc \\TIJ ·dl' f iIWd u-clmir-al \\.I\'S of

linking metric and no nmct ric span:s in such .1 wa v t hat the fo~m"-'r

become the product of the progr essive dmcr('ntiJti (J~ of the latte r . To

ex plain how such a svm mctrv- brcaking cascade would wo rk in this, ,case , I will need to tak e a bri ef det our throu gh the history ofnineteen th-century geometry.

Althou gh in that ce ntury most physicists and mathematicians though t

the struct ure of physical space was captured by Euclidean geome try.

man y othe r geome tr ies , with very different properties, had co me int o

existe nce. Some of them (such as the non -Euclidean geome try de­

ve loped b)· Lohatch evskv] shared with the geometry of Euclid the

propert), of being metric . There wer-e , however , o the r geome trics

where metri c co nce pts we re not in fact fund am ental. The differential

geometry of Gauss and Riemann wh ich gave us the co nce pt of a

manifold is one exam ple, bu t there were se veral o the rs (pro jec tive

geometry, affine geometry , topology). Moreo ver, and despite the fact

that Euclidea n geometry reigned supre me, some mathem aticians

realized that its basic conce pts could in fact be derived from the

non metric co ncepts whi ch formed the foundation of the newcomers.

In par ticul ar, ano ther influential ninet eenth-century math em ati cian

Felix Klein , realized that all the geome tr ies known J to him could be

categorized by the ir invariants under gro ups of t ransformations, and

that the different gro ups were em bedded one int o the o the r .23 In

mod ern te rmino logy this is equivalent to saying that the different

ge ome tries we re re late d to each ot her by rel at ion s of bro kensym me try.

In Euclidean geome try , fo r example , lengths, angles and shapes

remain unalt ered by a group conta ining rotation s, translat ion s and

reflecti on s. This is called the gro up of riBid traniformations. T hese

metric prop ert ies, howe ver, do not remain invariant under the groups

of transforma tions characterizing othe r geomctrit·s . T here is onc

geometry , called affine Beomet')', which adds to the gro up characte riz ing

Eucl idea n geometry new transfonnation s, called linear traniformations,

under which pro perties like the parall elism or the straightness of lines

It 111.111 1 urv .ur.mt , IHIt 11111 till II It Il·d. 111111 till II I 1"11/11 '" IIlnl/lCff.,

wlm h ,.dd In II l id .mel )1111'.11 11 .111111111I .111011 tlllI I 01 1'1 0lt ·tlloll ,

tCll"Il "pon ding III , 111I1111,111111011 ,I 111 1'11 ot hlru , .11Il1 , (·t I IOIl, I I",

n plh.l)t·nl o f intt 'rn'pti ng liuN' ligh t I .l ) ·"i 011 ,I "it I (T I1. (Mort· In hui

("111 )', thi s gl'onw lr) .ldcls t ra n- Io rm.u illllS t-,lllt·d ·pro jl·t l i\'il it's' . ) lln-sc

rr.msforrnat ions do 1101 nt·n ·ss.lr ily h-a ve l .uc-l id r-an or atfi m- propt'rti t·

undl.1ng('d. d S CJ Il h.., ('asH)' pictured if we inl.lgilH' .1 film pro jed or

(which typically inc reases the mJgnitudl' of length s) and .l projr-rtion

" ..-n-cn at an angle to it (which distorts parallel lines) .

If we picture these three gt.'ometr ies as forming the leve ls of a

hierarch)' (projcct i\"c-affine-Euclide an) it is easy to sec that tlu­

tra nsfo rmat ion group of each level includes the transformations of tln­

leve l below it and adds new ones . In o the r words, each level POSSt'SS('S

more sym metry than the leve l bel ow it . This suggests that, as we

1110 \"C down the hierarch )' , a sym me try- breaking cascade sho uld pro ·

dun' pr ogressively more differentiated geometric spaces, and, vice

versa, that as we move up we sho uld lose differen tiat ion . For exam ple,

as we asce nd fro m Eucl idean geometry more and more figur es become

eq uiva lent to one another, fo rm ing a Jesser number if distinct classes,

Thus , while in Euclidean geome t ry tw o triangles are equivalent on ly if

their sides have the same length, in affine geometry all tri anglcs an '

the same (regardless of lengths). In other words, as we move up the

class of equivalent triangles becomes less differentiated . Or to take a

dillerent exam ple , whil e in Euclidean geometry tw o co nic sec tions

(the family of curves co ntaining circl es, ellipses, parabolas and hyper­

bolas} are equivalent if th ey are both of the same type (both circles or

both parabolas) and have the same size, in affine geometry they only

need to be of the same type (rega rd less of size) to be equivalent , whil e

in pr oject ive geometry all co nic sections , without further qu alificati on.

are the same .H In sho rt, as we move up the hierarchy figures whi ch

used to be fully differentiated fro m one another become progressively

less distinct event ually blending into a single one , and vice versa, as

we move down, what used to be one and the same shape progressivcl)'

d iffere ntiates into a vari ety of shapes .

This hierarchy can be expanded to include other geo me tries, such

as differential geometry and topo log}'. The latter , for exam ple , may be

ro ughly said to co nce rn the prop erties of geometric figures wh ich

2~

I t 111.1111111 \ .111.\111 !llIdll IUlldlll '. 1111111111 • II I .1 .1 01111111 ' II II Il lI1I1.l

tn ut x, llloll i . tr,m"iIOI"I1l.ltloll \\lu, II do lIul t 1I' .,le IW\\ pOlllt " 0 1 IU "'l'

t'xis l ing ones . (fvl(ln' "'(,I<'t l)'. lcll)ll log )' ill\I,I\( ' tl.III ,lllrnlolliIJn ... , I.IIIt,(1' homeomo rp hisms ', which conrcrr n(drb) POints lilt .. nt'utb) ['Oint! .mel

wh ich can be reversed or be continuously 1II1dOlW,) Llnck-r these

transformati ons many figun~s which arc complete ly distil1(,t in Euclid­can geome try (a tri angle, a square and a circ le. for exa mp le) become

one and the same figure . since they can be deformed into one another.

In thi s sense . topology may be said to he the least d1fe rentiateJ

geome try. the one with the least number of distin ct equivalence

classes, the one in which many discontinuous forms have blended into

one cont inuous one ." Metaphorically, th e hierarchy ' topologica l­

dilTerential-projecti ve- affine-Euciidean ' may be see n as re presenting

an abstract scenario for the birth of real space . As if the metric space

wh ich we inhabit and that physicists st udy and measure ",as born from

a nonmetric, topological continuum as the latter differentiated and

acquired struct ure following a ser ies of symme try· breaking transitions,

Thi s morphoceneuc view of the relation between the difTerent geo­

metries is a metaphor in the sense that to math emat icians these

relations are purely logical , useful because theorems whi ch are valid at

on e level are automatically valid at the levels below it. ' 6 But thi s

cascade of broken symme tr ies rna)' be also given an ontoloqtcol dimen­

sion. O ne way in which thi s scenario for the birth of metric space can

be mad e less metaphorical and more directl y onto logical, is through a

co mparison between metric and nonmctric ge ome t rical properties , on

one hand, and extensive and intensive phy sical properties, on the other.

Extensive properti es incl ude not only such metric propert ies as length ,

area and volume, but also quantities such as amount of ene rgy or

entropy , They are defined as properties which are intrinsically divisible:

if we divide a volume of matter into two equal halves we end up with

tw o volumes, each half the exte nt of the original one . Intensive

pr operties, on the other hand, arc properties such as temperatu re or

pressure, whi ch canno t be so divid ed , If we tak e a volume of water at

90 degrees of te mperature, for instan ce, and break it up into tw o

equal parts, we do not end up with two volumes at 45 degrees each ,

bu t with two volumes at the original temperature .?"

Dcleu ze argues , however , that an inten sive propert}' is not so mu ch

26

flil l Ihlt I 111111\1 1111. 1.II1 IIl U \\ llh lll.UII'., ~ .1I . hl J "hill" ""I .I. lntl ...d"ml/( III klllJ. 1111 111111" '1 .1111I 1 III .• ' 1\1 11 \ IIIII III I III 11llll id \\.11_ 1,

III, " .lIll pll, t ,m mdl 'l ,d III ' d l \ II II' d ' 11\ 111 ,11111 t lilt' tlll1l .IIIII " [rr un

Illuk l lll,.It ll l U·.l llfl' ,I lI 'IIlP, 'I .lllllt· dlll ll l'll t l 11('1\\ t ' . '1I tilt' t llP .uHIho llum portion' of th e w an-r , )'t 'l. \\1.11" pnor to tlu' Iw,ltlllg till

, h' m is .11 cquililu-ium , 011('(' th t' h '111I H'r,lIU l'l' di lll'n'm I' i.. I fl ·.lh ,d

til(' '1)"sk m will [n - aw.1)' from equilibri um , th.u is , wv con di\ iell ' it ll

Ic- mpt.' rat lln· hut in so doing we dl.mgt' the sph'm Cl ll.l!it.lIh c1) .

Indt'('cI, .IS we just saw, if tilt' h 'mpe ratu rc- c1ifl~ ' n'I )(' I ' is m.ul t' inll'n"i.·

(' lIough the SYStl' l1\ will und ergo .1 ph ase transition, losing S)' I1II1h' II")'

.,,11 1 changillg its d ynamics, dl'vL·loping the periodi c p.lltL'rn of Iluidmotion which I referred to above as 'convec tion". T hus. in a \'t' r )" n -al

w nsc, phase transit ions do divid e the temperature scale hut in so doing

Iht,), mark sudde n changes in the spatial symml't ry of a material.

Using these ne w concepts we can define the sense in which tht'

me tri c space we inhabit emerges fro m a no nme tric continuu m through

.1 cascade of broken symme tries. Th e idea wo uld he to view this

g" m'sis not as an abstract mathemat ical pr ocess but as a co nc- re te

phpical process in which an undi fferen tiated intensive space (t hat is, a

s paCl~ defined by co nti nuous inten sive properties) progressivcl )' differ­cn tiatcs, eventually giving rise to extensive structures (discontinuous

structures with definite metric properties) , W e can take as an illus­tration of thi s po int some recent developments in quantum field

theories. Although th e conce pt of spo ntaneo us sym me try breaking,

.1I1d its connec tio n with phase transitions, devel oped in rath er humble

branches of physics, like th e fields of hydrodynam ics and conde nsed

matter physics. it was e ventually incorporated into the main strcam .!"

Today, thi s conce pt is helping unify the four basic forces or physics

(gravitational, elect romagnetic, strong and weak nuclear forces) as

physicists realize that , at extremely high temperatures (the ex tre me

conditions probably pre vailing at the birth of the universe) , these

forces lose their individuality and blend int o one, highly symme tr ic ,

force . The hypothesis is that as the universe ex pande d and cooled , a

se ries of phase transitions broke the original sym metry and allowed the

four forces to differentiate from one ano thcrv '" If we conside r that , in

relativity theory, gravity is \,shat gives space its metric properties

(mo re exac tly , a gravitational field co nstitutes the metric structure of

27

,I 111111 d llll1'l1 11111," 11I,lIl1ll1ld>. 111'( II \\1 ,11 1.1 III 1111 11. ,11 ' I I II II, Itl' lIl1' rgl' . ,IS .1 (lisli llll fi l/'l\' ,11,' P '\1111 II/I" ," pllllll III II 1111, II 1\'

prop 'rty (te m p .raturc ) , the idl',l 01 .1 11 intvusrv « p,lI " ' 1\ 111' 1IIIIh 10

exte nsive ones through progrcssivl' din~'rcnl i .1t i on h" l'olll l' · 1110r, ' than

a suggestive metaphor. I I

Let me pause for a moment to sum marize th e argument so Iar . I

began by es tablishing some purely formal d ifferen ces bet ween th e

co ncepts of 'essence' and of ' m ult iplici ty' : whil e th e former co ncept

implies a unifi ed and timeless identity, th e latter lacks unity and implies

an id entity which is not giv en all at on ce but is defin ed progressively ;

and while essences bear to th eir instantiations th e same relation whi ch

a model has to its copies, that is, a relation of greater o r lesser

resemblance, multiplicities imply divergent reali zations which bear no

sim ilar ity to th em. These formal differences, I said, are insuffici ent to

character ize th e distinction between essences and multipliciti es as

immaterial ent ities whose job is to account for th e genesis of form:

replacing etern al archetyp es involves supplying an alternative expla­

nation of morphogenesis in th e world. Unlike essences whi ch assume

that matter is a passive receptacle for exte rnal forms, multiplicities ar e

immanent to material processes, defining th eir spontane o us capacity to

generate pattern without exte rn al intervention. I used cer tain features

of mathematical models (state space s) to defin e th e nature of multipli ­

cities: a multiplicity is defined by distributions of singular ities , defining

tenden cies in a process; and by a se r ies of crit ical transitions which can

take several such di stributions em be dded within one another and

unfold th em. Finally, I said that a population of suc h conc re te

universals forms a real dimen sion of th e world , a nonmetric co ntinuo us

space whi ch progressively specifies itself giving rise to ou r familiar

metric space as well as th e discontinuous spat ial st ruc tures that inhabit

it.

No doubt, despite m y effo rts th ese remarks remain highl y meta­

phorical. First of all , I have defin ed multipliciti es in terms of attractors

and bifurcations but these ar e features of mathematical m odels . Give n

th at I want th e term 'm ult iplicity' to refer to a conc re te universal ( to

replace abst ract ge ne ral essences) th e qu esti on m ay ar ise as to th e

legitimacy o f taking features of a model and reifying th em into th e

definin g traits of a real entity . Second , th e relation between a

28

'"11111111 11 111 " I 1IIII IIIpill Ill. III I II.. ,h 1111 111111 " I 111 1 .11 I .1.1 I'"

I I I I 1" " .11 1 11 111 . 1 1. 11 11111 ' 11 1. '11111 11II 11111 • • I ' " I ,

111 .,11. '11' ,1111 " '''II III" 111111 ,11" 1111 1,IId , III "111111 1111 III I dlt 11111 I.

1,,1. li-m. 111111111 .11111 ' Ih.. 1I1 ,'llphlllll ." '"11 1' 11 1 \\ i1 1 \ll\ lIh ,' 11,,111111

,I Ihllrllu ,h <l lllo lo 111 ,,1 \II.lh i 01 I,ll' p,lI" "Ih,'l II ', '/, ,'/ (I.,/, c,'/

IIl1 c"i<1 /)/1 1..111 III S(' ll.Ir.lll'd 11'0111 its \.1 1"1.11.1 .. 111,111,, '111.1111".11111111<'111 , hilI

111 addiliol1, J d, 'l.lih,d discu ssion 0 1 ho\\ 1111'S" tllp" logil.11 111\.11 i.urt

111.1 • Ill' WO\'l'1I to gl'lhl'r to C(I/I.\/[ II I ,I continullus, '('\ Ill'll'ro " '111'0 11 ,

SIl.1 Cl' , In t lu- fo llowi ng chr pll..r I w ill show in technical (k l,lil hem 11 11

l tl l1st ruet io n can bl' carried o ut and ho w th e I' sultinv cu nt inuum . 11.1

replace th e top o r least m ct ri ' level in th e hierar ch y of gl'ollll"ll'i, ' . I

\\ ill also di s uss how th int rm ,<Iiate I rve ]s may he rl' p l.H I'd b\

inl -nsiv pro cesses of ind ividuat ion whi ch yield as the ir fina l prtllhll l

tlu- fully differ ntiat ed m tric st ruc tures th at popUlal l' ~h l' ,bo tlOIl\

lvvc l, At the e nd of chapte r tw o the metaph or of a ge nes Is 0 1 nu -tr«

spa c th rough a cascade of broken sym m ' t r ies sho uld have been most!

e lim inated , and a literal acco unt taken its place.

Meanwhile, in what rem ain s of thi s cha pter I would lik 10 mak.. ,I

1I10re detailed analysis of th e nature o f multipliciti es . T he fir st s('\ 01

issues to be d iscussed will invol ve th e technical details of Del C U/." ' .

ol1to logical interpretation o f th e co ntents of sta te space. His approach

is very un orthodox as will be sho wn by a co m pariso n with th e stau ­

space o ntologies proposed by anal yti cal philosophers, T he n I wi ll 1110\'1'

on to a sec ond se t of issu es co ncern ing th e modal srcrus o f multipliciti cs.

Modal logic is th e bran ch o f ph ilosophy w hich deals w ith th e rdat i or~ s

between th e possible and the actua l , Here th e qu esti on to be answered IS

if state space is a space of possibl e states what is th e status of attra ' to rs

and bifurcations in relation to th ese po ssibilities? Can multiplicities 1)('interp re ted in terms of th e traditional modal catego ries, th e possibk.

and the necessary , o r do w e need to postulate an origina l form 0 1

physical m odality to characte r ize th em ? Fina lly, a third se t of issues.

that needs to be dealt with is related to th e speculat ive dimension o f

Deleuze's proj ect. Replacin g essences with soci al co nve nt ions or

subject ive beliefs is a relativel y safe mo ve , but putting in their place a

new set o f object ive ent it ies inevitabl y invol ves philosophical sp' u­

lation . What gu ides thi s speculat ion? One wa y of looking at thi s

ques tion is to see Deleuze as engaged in a co ns truc t ive project guided

, . ( . 11.1111 1" 1' "/'''\ '''''' /' 1/1''. II.. I I . ( " " 1111111 Ill e I. u ll 111111 11 ,,1

\\ h.11 10 do hili \\ h,ll 10 .1\ oltl .10111 ' (lilt lit I. "" 11.11111 I 0 1 (1111 c', ,to avoid the- I'-"f> 01 ('sSI'llli.llislII, hili Ihl rl' .11( ' o IIIC' I ,11I.1 till' (' Ill'l'dto be d iscussed.

Let me begin with Del zuz ·'s ollto logie. 1 .m.lly.sis or state 'pa 'e .

Many philosophers arc today look ing at these abstract spaces as ob]e ts

of study and reflecti on . A recent shift in the analyt ical philosophy of

scie nce, for example, moving away from logic (and s t theory) and

towards an analysis of the actual math ematics used by scient ists in the ir

eve ryday pra cti ce , has brought the importance of sta te spaces to the

foreground. 32 Yet non e of the philosophers invol ved in this new

movement has attempte d such an origina l analysis of state spac e as

Del euze has. In particular, analytical philosophers see m unaware of (or

at least uncon cerned with) Poin care 's topological studies and of th e

onto logical differen ce that may be posited bet ween the recurrent

features of state space and the traj ect ori es these features det ermine.

Given that this onto log ical differen ce is key to the idea of a Deleu zian

multiplicity, I will need to ex plain how state spaces are co nstructed.

First of all , it is important to distingui sh th e different ope rato rs

invo lved in this co nstructio n. As I aid above , given a relat ion between

the changes in two (o r more) degrees of freedo m ex pressed as a rate

of change, one ope rato r, differentiation , gives us the instantaneous

value for such a rate , such as an instantaneous velocity (also kno wn as

a velocity vector). The o the r operator, integration , performs the opposite

but compleme ntary task: fro m the instantaneou s values it recon stru ctsa full traject ory or series of states .

These two operators are used in a par ticul ar order to generate the

stru ture of sta te space. Th e modelling process begins with a choice of

manifold to use as a sta te space . The n from ex perime ntal observations

of a system's changes in time, that is, fro m actual series of states as

obse rve d in th e laboratory, we create some traject ori es to begin

populating this manifold . These trajec tories, in turn , serve as the raw

mat erial for th e next step: we rep eatedl y apply th e di ffere ntiation

operato r to the trajectories , eac h application generating o ne veloci ty

vecto r and in thi s way we gene rate a velocity vectorfield. Fina lly, using

the integration operato r, we genera te from the vector field further

trajecto ries which can function as predi ctions about future observa tio ns

3°

II III I 111 ' 11 '1" 11111. I (111,,1lit Ih. 1'111 1111 ,. t II I'll' I I '

I • I • I II I II. 11 11 • I ), I. 11/1 111.1 ' "'"1I II 1' 1,1 ( 1'''111.111 II l'

1.///" /1111,,,11 .1, "//(" 1'// " ," // /1" trill 01 ,." till ' 1'1'( ,II III li lt 1 h .1 (

I I I J /1, " 1/1" I' 1.1, 11 11 ti ll o lh( Il'0rll.1I1 II • \ 11111 . Oil o ll( 10111( , "//1 «

\\'h ill' .1 p,lll i 111 .11" II"II( 11 01 or mu - '1.11curv I' ) IlIodl'l .1 UU ' 11111 (II

,111 11.11 st.lll'S 01.1 \sll' lIl ill Ihl ph~' si .JI \\ odd , Ihl \Idor 1I1·\l1 c ,111Il1I1

•

Ih.. inherent n-ndcn cics or man ' such tr: j.·cloril's, and hellll' 01 111.11'

.1 tual svstcm s, to ln-have in C(·rt.l in \\'a~'s , As mcnt ion ..d .11 1lI\(' , th.. I

tendencies are represented hv singularit i .s in th .. vector field, ,lIld I

1kleuz' not .s, lit-spite the fact tha t the precise narure of cac h sin iul.u

point is w ·1I d ·fin d on ly in th e phase portrai t (b the [orm till

ira] c tories take in its vicinity) rhe e.t islence anJ dimihullon 0\ till ('

singularit ies is already complet ly given in the vector (or din'( l ion)

field. In one mathematician " words:

Th e geometrical interpretation of the theory of different ial · (11l·11iUl~

\car ly places in evidence two abso lutely distinct rea lities :. then' I

the field of directions and the tcpoloqica! accidents wh ich 11101 )

suddenly crop up in it , as for example the ex iste nce of . .. singlll.lr

points to which no direction has been attached; and then' an', tho

integral curves with the form they take on in the vic~nity 01 Ihl

singular ities of th e field of directi on s . , . T h ~X l st ' I~ e and

distributio n of singularities are notions rela tive to the field 01 vc ' tors

defined by the differential eq uation . T he for m of the int gra l curves

is relative to the solution of thi s equatio n. The two problem ' arc

assuredly complementary, since the nature of the singu larities of.th...

field is defined by the form of th e curves in their vicinity. But It IS

no less true that the field of vec to rs on one hand and the int 'gral

curves on th e other are two essentially distinct mathematical realiti es. 14

T here are seve ra l other features of singular it ies, or more spccif ally,

of attractors, which are cr ucia l in an on to logica l analysis of state 'pa "

and which furthe r differentiate its two 'distinct mathematical rea liti 's' .

As is we ll known , the trajectories in thi s space always approach an

attractor asymptotically, that is, they approach it indif/nitely close bur

nerer reach it. 35 This means that unlike trajecto ries, which re present the

actual sta tes of objects in the wo rld, attrac to rs are never actualized,

3 I

--

II". 1111 p.1I1l1 01 .1 II Jt I 1111 \ I \t I II ,It III' IIll .u t r .u nu II f II It I 111

l h i , M' II 'iI ' th,lt ' 1Il I u l. lI ll it ' , n 'ptt ' c u t flllh Iht ' ICII1~ It rill h ,lltkllt II " of

a ,"}Stl' III, neve-r its J.d ll.ll 'il.\h' " , I )t" IHIt' t hl' i, I." k 01 .lllll.lli l\,

at tractors are ucvcrth clcss n -al am] h.\,,' dl'fillilt' dlt'lts on .\('tll~ienti ~i~s . In part icular, the)' co nfer on trajc-ctoru-s .1 cc-rtnin dt 'grn' of

sta~lh t)', called a~)'mprolic stability. it, Sma ll shol'ks rna)' d i sl ()dg(~ a

trajectory from its att ractor but as lon g as the shock is not too large

to push it out of the basin of attracti on , the traject ory will naturally

return to the stable state defined by the at tractor (a stca dv sta te in the

case of point attractors, a stable cycle in the case of periodic attract orx,

an~ so o~) , Another important feature invo lves not the stability of the

trajecto ries but that of the distribution of attractors itself (its structural

stability). Much as the stability of trajectories is measured by their

resistance to small shocks , so the sta bility of a particular distribution

of attractors is checked by submitt ing the vector field to perturbation s,

an effec t achieved by adding a small vector field to the main on e and

checking wh ether the resulting distribution of attractors is tapaJoo;caJIy

equivalent to the original one, J7 Typ ically, distributions of attrac to rs arc

str uctura lly stable and thi s, in part , is what accounts for their

recurren ce among different physical systems. On the othe r hand, if the

perturbation is large enough a distribution of attractors may cease to

be struct urally stable and change or bifurcate into a different one, Such

a bifurcation event is defined as a continuous deformation of one

vector field into another topologically inequivalent one through astruct ural instability, 38

Using the technical terms just introduced I can give now a final

definition of a multiplicity, A multiplicity is a nested set if' vector fields

related to each other by symmetry-breakino bifurcations. [oaether with the

distribut ions of allraclors which define each if' its embedded levels, This

definiti on separates out the part o f the model wh ich carries informati on

abo ut the act ua l world (t rajectories as ser ies of possible states) from

that part wh ich is, in principle, never actualized, This definition

presupposes only the two co nce pts of 'd ifferential relation ' and

'singularity ' , J will return in the next chapter to a discussion of what

~urther philosophical traniformati on these two conce pts need to und ergo

In order to be truly detached from their mathematical realization. At

this point, granting that the definition I just gave could specify a

3 2

l oltt II II , 11111v , \\ t 111 ,1\ .1 ~ \ .. h ,lt 111I1111l) ' II .,1 1.11" li t h .111 t 11111 \

.... "lIld h,I'" ( 10 1)1,11...1 I .lIef 01 p.ltlllli 01 ll\dllllhll.lIllll 1111 .... lidII I IMth 'l'lI IIll'llIhl )lllolh'll t! e"I·lop"lt·III .1 tll\(1 11111 r('"I" IIl /ll1l Ill .,

llI1i" 'I', .,1 lIIultlpli, il I mi,lt ,.\tlll1g 'illll l' II ll~l' t th,ll ,llt "I' IM'It -lll

an - re-al, \\ hill' till' Illllitiplid') iht,lI i, 11111. So I h'lt-tl/l' '1)I'"l.. nlll 01

' t!' ,lli/ ,l ' io ll ' but of dtllllJludlWn, .lIltl inlrotlu,,''i J no"'1 oll lologil ,ll

,""h-gor)' to refer to the status of lnu lti plit-itil 'S tlll'm""I,"" : II"u"hr ,

l'hi-, te- rm does not rc-h-r, of co ursc-, to the virtual n ,.llity .....hich digl t.,1

..Imulatio ns han ' made so fnuil iar , hut to a real l" ir'"tJlit)' fo rm ing .1

\ it.ll ('o mp(Hlent of th e ol*'cti\"l' world, As he writes:

TIH~ virtual is not opposed to the real hut to the actual. The d rtlllJl

isJ ulJ.y real in soJar as it rs rirw cJI , , . Indeed , the virtual mu st [u­

defined as st ric t ly a part of the real obj ect - as thou gh till' ohjt'l t

had one part of itself in the virtual int o \v·hich it plunged as thou gh

into an object ive dimen sion , , , The realit y of the virtual co nsists

of the differential elem ents and relat ions along with the singul.u

points wh ich co rrespo nd to them, The reality of the virtual is

struct ure, W e mu st avoid giving the elements and rel ations that

form a struct ure an actuality whi ch they do not have, and withdraw­

ing from them a reality whi ch they have. ?"

\Vhat is the modal status of the virtual? If state space traj ectories han'

the sta tus of possibilities (possible ser ies of states) what modality do

virtual multiplicities represent? This is not an casy qu estion to answer

givcn that the ontological status of even the familiar modal categoril's

is a thorny issue. So before dealing with virtuality let me discu ss the

ques t ion of possibility. Traditionally, ontological discussion of possi­

bilit ies has been very controvers ial du e to their elusive nature , and in

part icular , to the difficulty of giving a clear crite rion for individuatino

them, that is, for telling when we have one instead of another

poss ibility. As a famous cr itic of mod al logic, the philosopher Willard

Van Orman Quine , jok es:

Ta ke , for instance, the possible fat man in the doorvvay; and again ,

the possible bald man in the doorway . Are they the same possible

man, or tw o pos sible men ? How do we decide? How many possible

33

"WII lht 1( ' ••r III Ih.lt dIHlr\\.I\ / II till It 1I1t'11 ' l" I 11.1, 111111 lUll

th..1II lat OIIt'S ? 110 \\ " Mil) of du-m .Ir, ' ,. l lk,· ? (h \\ ould tlwil Ih'lIlgalike llIake th em o ne? ,\ n ' not two pos.•dhh, things .,Jill"( I ~ Ihb th l'

same as saying that it is impossihl e fc)r tw o thin g,'i to h,· .,Iikl·? Or,

finally, is th e concept of ide nt ity simply inapplicable to unactualiz,«]

possibles? But what sense can be found in talk ing o f ent it ies whichcanno t be meaningfully said to he iden tical wi th themsel ves anddistinct from one anothcr-P'"

Most approach es to modal logic concentrate on langua ge , or more

specifically, on an anal ysis of se nte nces whi ch express what could hare

been, sentences such as 'If j.F.K. had not been assassinated th en th e

Vietnam War would have ende d sooner.' Given that human beings

seem capable of routinely using and making sense of these countcr fac­

tual sentences, the modal logician's task is to explain this ordinary

capability." However, th e fact that linguisticall y specified possible worlds(like th e po ssibl e world wh ere j .F.K. survived) are so devoid of

st r ucture , and allow so mu ch ambiguity as to what distinguishes one

po ssible world from another, is what has prompted cr it icisms such as

Quine's . But as some philosophers have suggested, the problem here

would seem to he ,..'ith linguistic representations and their lack of

resources to st ructure possibl e worlds, and not with possibilities as

such . The philosopher of science Ronald Giere, for instance , thinks th e

extra const raints which st ruct ure state space can overcome the limi­tations of other modal approaches:

As Quine delights in pointing out, it is ofte n difficult to individuate

possibilities . . . [But] many models in whi ch th e syst em laws arc

expressed as differential equations provide an unambiguous cr ite r ion

to individuate the possible histories of the model. They ar c the

trajectories in sta te space co r res ponding to all possibl e initial

condit ions. Threatened ambiguities in the set of possibl e initial

condit ions can be elim inated by explicitly restrict ing th e set in th edefinition of th e th eoret ical mod el. 42

G,iere argues that state spaces may be viewed as a way of specifying

possibl e worlds for a g iven physical syste m , or at least , possibl e

34

11I1lJlll b u 11 ,1,111111'1111 IIllhl phil pUllIlIIIl"1 I lit III I 11I11

pu Ibll [u lui H .11 111'11 '11\ I III t ,Ilt 101., .. It '" 01 p,nl I I ht

illdh idtl.llil\ 0 1 tht' d llll'lt'lIl ptl"l"l lhll ' IUIOllt' \\1111111 Lilt· P·ltl I

ddilH'd I I) ' /111\'\, t·xpn ·....,·d h ti ll' dilh-n 'llti ;d l ·clll.lll lln tll ,11 lUlu tio u.dl)

rc-l.ue tilt' sni t(,!H 'S dl'gn 'l ' s of frlTdom, .Is \\,· 11 .1"1 b)' HlltllJI IIIIIJIlWf!\,

the specific ..ta tv , or po int in till' manif old . \\ Iwn ' .I "ph' III Iwgill!l it

evolut ion. Given a specific iuu ial co ndition .md a dcn-rm ini tic 1.1\\

(such as those or classical physics] OIl(: and only tIIW tr.lj ITt or) i

individuated , a fact that may he lIsc.'d to Ch.l l1(, l1gc.~ Quilw 's sn ·p t il.,1

stance. The phase portrait of any particular stale space will l»- typic,llIy

Fil led with man)' such individual traject ori es, one for each po ssihll'

in it ial condit ion. O ne may reduce thi s number bJ adding other 1.1\\s

wh ich forbid ce rta in co mbinat ions of values for th e degrees of freedom ,

that is, which make some initial cond it ions not available for J gin'lI

svstc rn , but st ill, one enos up with many possibl e histories .o4l

The problem for the phil osopher becomes what omo loq tcal SloW' tn

assign to th ese well -defin ed possibilities. One onto logical stance , which

Giere calls 'actualism" , deni es any reality to th e pos sibl e trajcctorics ,

however well individuated they ma y he. A mathematical model , in thi s

view , is simply a tool to help us in the control of particular phy sical

syste ms (that is, th e manipulation in th e laboratory of th e beh aviour:,f

real syste ms) as well as in th e predicti on of their future beha\'l(:ur. h:r

this limited purpose of predi ction and control all we need to Judge IS

the empirical adequacy of th e model : we generate one trajectory for J

given initial condit ion , then try to reproduce that particular combi ­

nation of valu es for the degrees of freedom in the laboratory, and

observe wh ether th e seque nce of actual states matches that pr ed icted b)'

th e traject ory. Give n th e one trajectory we associ ate with th e actual

seque nce in an expe riment , th e rest of th e population of traject ories is

merely a useful fiction , that is, ontologically unimportan~ . ·4 As Giere

argues , however, thi s ontological stan ce misses th e lact th~t th e

population of trajectories as a whole displays certain reBularities In tI~t.~

possible histories of a system, global regularities whi ch play a rol e m

shaping any on e particul ar actual history. :" To him , understandin~ a

syste m is not knowing how it actually behaves in thi s or that specific

situat ion , but knowing how it lVould behave in conditions which may in

fact not occur . And to kn ow that we need to use th e global information

t"lllboclh d 111 lilt ' popUJ..lioll oj pC " 'lhlt III IOlw • 1II101l1l.111011 whuh I

Im.l if we COlln'ntr."t ' on tilt ' 0Ilt' t l"' ln to , ) \\hhh I tclIllp,Ht,c1 withreal Sl'(IUC IICt'S o f sta tes. 4t>

As sho uld hl' d ear from ti ll" discussion in th is d I.1JlIt"r .. Dt"It'lI/c was

not an "actualist ' . He held a realist position to ward s the mod al

st ructure of sta te space but would have disagrcl'd wilh Git'n .' in his

int erpretati on of what co nst itutes that mod al st ructure . In part icular ,

in a Dcleuzian ontology one mu st em phasize that lhe regul arities

displayed . by the different possib le traj ectories are a consequence Vf the

sIngu lantles that shape the vector field . The well -defined nature of the

poss ible histories is not to be approached by a mere mention of laws

ex pressed as dlffcrcntlal equations, but by an understanding of how

such equations in fact individuate trajectories , Each pos sible sequence

of states, each possible history, is ge nerated by following at each point

of the trajectory the directions specified by the "ect or field , and any

regularities or propensities exhibited by the trajectories sho uld indeed

be ascribed to the topological accide nts or singularities of the field of

directions , As Deleu ze puts it, ' the singular it ies preside over the

gencs is' o f the trajectories. "? In o the r words, Giere is right in thinking

that state space offers more resources than language to individuate

possibilities (thus sides tepping Quine's crit icisms) but wrong in his

assessment of how the process f!f indi viduation takes place . To leave the

vector field out of our ontological analysis (that is, to mak e it int o an

auxiliary const ruct ion or yet another useful fiction) hides the real

source of th e regularities or propen sities in the population of possiblehistorics. r"

This point tends to be ob scured in traditional philosophi cal analyses

by the use of examples involving the sim plest typ e of cquation , a linear

equation . Despite the fact that of all the types of equations availab le to

physicists the linear typ e is the least typical, it happen s to be the I)'pe

that becam e dominant in classical physics. Th e vect or fields of these

differential equatio ns are extre mely simple, "the only possible attractor

of a linear dyn am ical syste m is a fixed point. Furthermore, this fixed

point is unique - a linear dyn am ical s}"stc m cannot have mo re than one

basin of attracti on . '49 In o ther cases (in co nse rvative s}"stl' IllS which are

qua si-isolated from their surro und ings) there may be no alt rac to rs at

all , only traj ectorics.Thux, in a linear conse rvative syste m (such as the

Ia,UllloIlU n t all.\llIl II' d .1 .111 I .11 11 1'1. iI\ (,I I I. ) 1111 \n 1111 ht'lti 1 ' 0

b.u •.1\ '\11uc tUII·t! tl..11 It 111 ,1\ . 101 lUll t l" ,H tll ,ll pur pn,~ ' s, ht ' ignCln'et,. I

. l .1 'CHlit t ' of loll,tr:unt s in the IIldi\ iduation 01 Ir.lit'dorics , On t u-

o ther h.uul, tilt, mort' typic<11 ~'Cfualio lls (nonlinear equations) have J

mort' t,l.,horah' distr-ibu tio n of singuIJri til's. the sta te space bein g

normal ly par-titioned in a ce llular fashion b)' many altractors and their

basin«, and these multiple at t ractors may be of different types. In these

mo rt' co m mo n cases, the vector field has too mu ch structure to be

igno H'(1.'>0 •

This argument, however .. establishes only that there arc In state

.span : othc-r constraints for the individuation of possib le histo ries, bl~t

not that they should be given a sl'parate modal status , W e could , It

wo uld see m , take singulari t ies to belong to the realm of the possibl e

and save ourse lves the trouble of introducing novel forms of phy sical

mod ality, such as virtuality. One way of doing thi s would be to take a

basin of attract ion to be merely a subse t of points of state space. Given

that sta te space is a space of possible states , any subset of it will also

he just a co llec t ion of possib ilities, 51 Yet, as I mention ed before,

despite the fact that the nature of singularities is well defined only in

the phase portrait of a system, th eir existence and distr;bution is a~read)'

give n in the vector field .swh erc they define overall flow tendencies for

the vectors, It may see m plausible to think of point attractors, for

exa mple , as just one marc point of state space , but this sing ular point

is not an available pos sibility for the system since it is never occupied

by a traj ectory, only approached by it asymptotically. Trajectories wi ll

tend to approach it ever closer but never reach it, and even when on e

speaks of the end state of a trajectory, in reality the curve is fluctuating

aro und its at t racto r , not occupying it. Strict ly speaking, as I said above,

attrac to rs arc never actualized,Thus, it see ms, a more co mple te analysis of sta te space does see m

to de mand a form of physical modality that goes beyond mere

poss ibility , Rut could not that o the r tradi tion al modal catcgor)',

necessity, do the job? After all, in classical physics' models a gennal

law rel ates all the successive points of a traj ect ory in a necessar y or

det erministi c way, and wh ich specific trajectory is gene rate d is ncccs ­

sarily det ermined given a particular initial sta te .S2 This is, i.ndecd, .t~u.e ,

but the relative importance of gene ral laws and particular 111111.11

37

39

I I I ' ,,, llI'd \ \\ ' knm that the cells will, - Ithl' IltH .l \ III I , ,t\s SOOIl as , I . , .t to strict dctermlll ism .

I ' I ' 111111 I th"n,lor ' su ) I~C •appear: t liS P WIlIlIll\ , ' f tl ' cells [clock - or ant i-

I I I rot lion 0 11.: •In co ntrast, t il" ir e: /l Oll II' II blc a ni chance, in th e

10ckw iseJ is ullpredict.lhlc and uncontro a e. y , '1 d at theI ' h t ay have pre\ al e

form of till' particul: I' pertur )at~ol ln ]t ~ 1m

hether a given ce ll is(' I ) . r imcnt W I necrc e w

mom ent 0 t 11' (X I ' _ t remarkable cooperationI I' l l d W e thus arnve a a

right - or c t lane ' . . . .d ore formally, severalI I de term nllsm . , State m

b .twccn c lance anc I Chance alon e will'I I for th same parameter va ue.

solutions arc pOSSI ) e . 51

decide which of these solutions is realized .

f nt for a different inte rpre ta tion of th e mod~1This line 0 argume . _ f t Ocleuze ' s own , alth ou gh It

f t te space IS III act, nostructure 0 sa ' 'I I ' Oeleuze own argum nt s

. I fl ' ontologlca ana YSls. .follo ws direct y rom llS . f h _ ible and the necessary are ol

h d ategones 0 t e POSSI eagainst th e ort 0 ox c 54 d linked dir ectlv with the

hil I' I nature an are j ,

a more gene ral p I oso~ nca d d ' b . di cussed in the remaind 'r 01f ' I saId nee e to e IS

third set 0 Issues . , uid e Oc!euze ' s speculation aboutthi s chapter: the co nstramts ~hat dg h constraint to avoid at all

I I lady mentlOn e one sue 'virt uality. lave a re I " I' . . ete rn al esse nces . Meeting

I" ' . t al mu tIp icttrc s ascosts conceptua Izmg VII' U.. h f h t modal logiC has to say

, . . ectmg mu c 0 w a ,this const ramt reqUIres reJ , th t the postulation 01

" ., d it The reason IS aabout POSSibIlity an necessl y. I ld as Quine and oth I'

ld . " longside th e actua wor ,Possible wor s eXlstmg a . I' plies a commitment to

f ' ked almost a ways imcritics have 0 ten Icmar ' . I' 55 A d it should b emphasized,

h f m of essentla Ism. n ,one or anot cr or d I hil ophe rs but also to th os

. " . I' t only to mo a p I osthi s cn tlCISm app res no . " h . t nee of alt ernate parall Iphysicists who ser iously believe m t e exts e

univ er ses. II I . es both philosophers and. ki b t th ese para e urnvers I

When thin 'mg ah"" . e of f ully fo rmed individuals populatin g th

ph ysicists assume t e eXlstenc.. di t I raises a number of qucs-ible 'orlds ThIS irnm e ia e y I

different POSSI e \\. . I' htl alte red in othe r work s?h 'ndividual eXist, s Ig y, I

tio ns: Can t e same I . ny worlds afte r scvc rah . tai thi s identIty across rna ,

Can he or s e main am I d? C Id we identify him or her aft erslight alteratio ns have accumu ate. ou

I <Illdll 10III 11..111 'I "'10 I \, 10101 111 1111.11111' t )11 0111 "'Ild , tI" 11111

III .111\ 1' .11 tHIII .1I 111111., 1 1.111' I 111 .ld\ dlllllli l I" d II". 111.111\ IIl1tl .II

I OllditiollS (all tho s« tli.u .111' IIHlud,·d \\ltlllll.1 I),\ IIHII I.II h.1 III) will l«:

eCJ uivall'lIt .I S far as the l'lld st.lle of ti ll" tr ,ljl'l tol ) IS ('ollll'rIIed. 'l he

states a traj 'c tory adopts on its way to till' I'lld stall', what l'ngilll' C'rs

call its transient sta tes and which co nst itute the bulk of the trajc tory,

may be of interest some times, but lcarl y will not be as important as

the stabl e end state , since the syste m will spend most of its tim

fluctuating around that state . On the other hand , the role of the

gen eral law will also be diminished because the behaviour of the

traj ectory at its end state , a steady-state or a cyclic beha viour, for

example , will be determined not by its pr evious states (defined by the

general law), but by the typ e of the attractor itself.

Thi s argument, again, establishes the need to consider additional

factors in the individuation of possible histories but not the need for

additional modalities , After all, is not the end state of a traj ectory

necessary? In this case too, the complexity of the distribution of

singularit ies makes a great differen ce in our interpretation of the modal

structure of state space . A state spac e with a single attractor , and a

sing le basin encompassing the entire spac e, has a unique end state for

the evolution of th e syst em. Concentrating on this atypical case ,

therefore, can mislead us into thinking that det erminism implies a

single necessary outcome, On the other hand, a space with multiple

attractors breaks the link between necessity and determinism, giving a system

a 'choice ' between different de stinies, and making the particular end

state a syste m occupies a combination of determinism and chance . For

instance, which attractor a system happen s to be in at anyone tim e is

det ermined, in part, by its contingent history: a traj ectory may be

dislodged from an attractor by an accident, a strong-e nough exte rn al

shock pushing it out of one basin and into the sphe re of influen ce of

another attractor. Furthermore, which specific distribution of attractors

a yste m has available at anyone point in its history, may be changed

by a bifurcation . When a bifurcation lead s to two alt ernative distribu­

tions, only one of which can be realized, a det erministic syste m faces

further 'c ho ices' . Which alt ernative obtains, as nonlinear scientists lIya

Prigogine and Gregoire Nicolis have been arguing for decades, will be

decided by chance fluctuation s in the enviro nme nt . Speaking of the

\ ' Ill I I I III ,

\\ riu -:

III "'ll ,, \1 1111 "II ,I I'h' II "' It11111 , till IIII h ili

..1111'1'1°,1,."", 11,,1,1, I I I"'t 1.1, 11111 . tllll' 111" ,101 p .Utltll.II,

•tn ' lIIt1 0dlll n l to c!"l ll it til' Idllllll ) 01 till I 1I1111\lIlu.ll .u rd to

g U.U-,lII l t '(· it s pn'M'I"' ''lliOI1 .HTO" \\l,dtl" J IWl t ' .11" ! I,I", iL III )' t wo

dilli.'n·nl tec'hnical wa),s or.u ili,·\' jng th is d ltO( I. ( ) II W it' 1I.lIu l, OIH' C•in

cla im that t ran sworld identity is insured h} tilt.' pmis..oss ion or d particular

essence, that is, the propert)' of bein g thi s part icula r indi vidual. O n the

o the r hand . on e can deny that there arc, in fact, such transw orld

individuals. and speak sim ply of counterparts, that is, o ther possib le

individuals which closely resemble their real co unte rpart, but arc not

identical to it (in particular, they do not share the esse nce of being

precisely th is ind ivid ual) . T hese counterparts , ho wever I would share a

general esse nce . (Such as bei ng ' rat ional animals' , in the case of humanheings. '6)

The alternative o ffered by Del euze is to amid taking as gi ven full)'

formed individuals, o r what am ounts to the same thing, to always

account for the eenesis of individuals via a specific individu ation process ,

such as the developmental process which turns an embryo into an

organism. Thi s emphasis on the objective producti on of thc spatio­

temporal structure and boundaries o f individuals stands in stark

contrast with the comple te lack of process medi at ing between the

poss ible and th e real in orthodox modal th inking. T he category of the

possible assumes a set of predefined forms which retain their identity

despite their non-exi sten ce, and which alread y resemble the forms

they will adopt once they become realized. In other words, un like the

indi vidu ati on process linking virtua l multiplicit ies and actua l structures,

realizing a possib ility docs no t add anything to the pre -exi sting formbut mere reality. As Deleuze writes:

What difference can there be between the existent and the no n­

existent if the non-existent is alr cady possible, already included in

the co nce pt and having all the characterist ics that the co nce pt

co nfers up on it as a possibility? . . . Th e possible and the virt ual arc

. .. distingui shed by the fact that on e refers to the form of identity

in the concept , wh ereas the other designates a pure multiplicity .. .

which radically excludes the iden tical as a prior condit ion . . . To

the exte nt that the possible is open to ' realizat ion ' it is understood

as an image of the real, whil e the real is supposed to resemble the

40

po 'ilhlt I h.u I \\ II 11 I dllh , tllt III U ll til I 1.11111 \\ 11.lt , I 1t "IU I

.Itld ttl tlu (1IIll' pt "til II 11 II dill 1 douhh' lilt' "ith lilt ' .. .,\ l'1 l1 .lli, .lt io ll III (' .lk \\ It II II·M'mhl.ull ,. ,1 '\ .1 p r on 'ss no lr-ss th .111 it

doc-s with itk nt il) .u .l prim iph- . In this sense, act ua lizat ion o r

dif1~'n'ndatioll is .1lwa)'s a gt'l1uinc creation . Actual terms never

rcsernhle the singulari tit's they incarn ate . . . For a potential o r

virtual object to he actua lized is to crea te divergent lines whi ch

co rres po nd to - without resem bling - a virtual m ultip licity.n

Besides the avo idan ce of esse nt ialist thinking, Deleuze ' s speculation

abo ut virtuality is guid ed hy the closely related const raint of avoiding

lyp oloBical thinking, that sty le of thought in which ind ividuation is

achieved throu gh the creati on oj classifications and offormal criteria for

membership in those classifJcations. Although some classificat ion s are

csse ntialist, that is, use transcendent essences as the crite rio n for

membership in a class, this is not always thc case. For exam ple , unlike

Platonic esse nces whi ch are transcendent entit ies, Aristotle ' s 'natural

states ' those sta tes towards which an individual tends, and which

would' be achieved if there ",'ere not interfering forces , are not

t ranscendent bu t immanent to those individuals. But while Aristo telian

philosophy is indeed no n-essent ialist it is st ill completely typological ,

that is, co nce rned with defining the cr ite r ia which group individuals

into species , and species into gene ra. 58

For the purpose of discussing the constraints guid ing Deleu zc ' s

co nstructive project, on e historical exam ple of typological thinking is

particularl y useful. This is the classificatory pra cti ces which were

co m mo n in Euro pe in the seventeenth and eighteenth ce nturies , such

as those that led to the botanical taxonomies of Linnaeus. Simplifying

some what , we may say that these classificat ions took as a point of

departure perceived resemblances am on g fully formed individuals, fol­

lowed by precise co mparisons aimed at an exhaustive listing 01 what

differed and what stayed the same amo ng those individual s. This

amounte d to a translation of their visible features int o a lingui st ic

re presentation , a tab ulation of differences and identities which allowed

the assignment of individuals to an exact place in an orde red table .

Judgmen ts of analoBl between the classes included in the table we re

used to gene rate higher-order classes , and rel at ions of opposition were

4 1

t 1.111 11 hnll lt 1\\ 1'. II 11111. 11.1". III \11 ,1.1 dilltollllllli 0 1 111111" , '1.1bor

.llt' hit'r.ln !Iii ' 01 t )" ' :0; , I ill' n " ultll1g IHCl lllgh .1I 1.1 . ClIIOlllil 's wvrc

SlIj> j>IISt'll lo n 'con,lrucl ,1 nat ur.rl ordor- \\hi ,'h \\ .l"' j lUd l1t1d (On t m lw lH,

rcganll ,'ss o f th e I:let that historical ac'c-k lcn ts m.rv h.Wt' b ro ken that

continuity . In othe r words, given the fixity of tilt.' ':io)ogkal types, time

itself did not play a constructive role in the gene ration of types, as itwould later on in Darwin 's theory of the evolution of species, ~q

Dclcu ze takes the four elements which inform these classificat ory

practices, resemblance, identity, ano lo8)' and opposition (o r co ntradict io;)

as the four categories to be avoid ed in thinking about the virtual.

Dclcuze, of course, would not den y that there arc objects in the world

whi ch resemble one another, or that there arc entit ies wh ich manage

to maintain their identity through tim e , It is ju st that resemblances and

identities must be treated as mere results of deeper physical processes,

and not as fundamental cat egories on which to base an ontology ,60

Similarly, Dcleuze would not deny the validity of making judgments of

analogy o r of establishing relations of oppos it ion, but he demands that

we give an account of that whi ch allows making such judgments or

establishing those relations. And thi s account is not to be a story about

us , about categories inherent in our minds or conventions inherent in

our socie ties , but a story about the world, that is, about the objectiv e

individuation processes which yield analogous groupings and opposedproperties. Let me illustrate this important po int.

I said before that a plant or animal species may he viewed as defined

not hy an essence hut by the process which produced it. I characterize

the process of speciation in more detail in the next chapter where I also

discuss in what sen se a species may be said to be on individual, differing

from organisms only in spatia-temporal scale, The individuation of

species consists basically of 1\\'0 separate operations: a sort ing operation

performed by natural select ion, and a conso lida tion operation per­

formed by reproductive isolation, that is, by the clos ing of the gene

pool of a spe cies to exte rnal ge ne tic influences, If selection pressures

happen to be uniform in space and constant in tim e, we will tend to

find more resemblance among the members of a population than if

those select ion forces are weak or changing . Similarl y, the degree to

whi ch a species possesses a clear-cut identity will depend on the degree

to wh ich a part icular reproductiv e community is effecti vely isolated.

M ,1I1\ 1'1.1111 I II I U , 101 I .l1 l1l'll. II LUll till II I II' ,It 11\ to In,hlld"l1I1111llgl.out 11\('11 11\ (tlln (,Ill' tl. .lIl " tclIdh 1ll.11t1 1.11 \\ltll olllt'l

p l,lll t ' 1H'('j, ·:"i) .111.1 IWII( t ' pU'I 'U" " ,I It- I h-.u ,Ill 1('lwth Ukll l lt )' th.m

1',.rI,'( II)' I"t'protlm th ,') i,ol.lh' d .iniru.•ls. III ~Il Cl r.t , 1I,It" d.~gn: , ' 01

Tt'\I'mhl.mn' atu l id" ntit)' lh'pt'ml, on ('(lntingt'n t Ill,t orll':,1 dd .lll , Clltht' pron'ss o f ind ivid ua t ion, .u HI is tlt,'n -fon ' not 10 be taken lor

gr,mh' d . For the same n ·.lSOI\ , rt.'st.·mhlann· and idt.·TlIity sho uld ~10 1 ,1)(used .1S fundamental co nce pts in an ontology, hut only as (h ' r1\'.ltl\('

not ions,In addition to sho wing , case b)' case, how similar it}' and idt.·ntil)' .1ft ' ,

("(Hl t ingcnt on the details of an individuation process, th e rejection 01

static categories and esse nces mu st he e xte nde d to all natu ral kmds , not

just bio logical ones . W e mu st sho w, also case b)' case , how terms

wh ich purport to refer to natural categories in fact refer to hisIOr;ct11lj

cOnSlilUled individuals. In a way terms like 'human' arc th e easiest to

de.essentialize given that Darwin long ago gave us the mean s to think

abo ut species as historical entit ies , But what of terms like 'gold' wl\t.'n~

the esse ntialist account seems more plausible? After all, an samples 01

go ld must have ce rtain atomic properties (such as having ~ specific

ato mic number) whi ch , it can be plau sibly argued , const u utc lIwesse nce of gold. Part of the answer is that all atoms, not onl y go ld

atoms, need to be individuated in processes occurring within stars

(nucleosynthesis) , and that we can use these processes to specify what

go ld is instead of, say, giving its atomic number."! But amon'

compelling reason to re ject essentialism here wo uld be to deny that a

given sample 'of gold large enough to be held in one's hand can be

conside re d a mere sum of its atoms, hence reducible to Its atorrnc

properties. . .,In particular, much as between individual ce lls and the individual

organisms which the)' compose there are several intermediate st ruc­

tures bridging the two scales (t issues , organs, organ ~ystems) ~o

between individual atoms of go ld and an individual bulk pIece of solid

material there are intermedi ately scaled structures that bridge the

micro and macro scales : individual atoms form crystals; individual

crystals form small grains; individual small grains form larger g rains ,

and so on. Both crystals and gra ins of different sizes are individuated

following specific cau sal processes, and the properties of an individual

H

hulk ',11111'1 ( 'l1h" I' lrOIll 1111 ' ("II .r] IIJIl r t llllll Itd\\( I 11 till" (

iuu-rnu-di.m - ,,(rllt(lI n · . 'Llu-r r- .ln~ '1If111 proper lit ' II! '0ld, 'llth.H

h,ning a spt.Tilic melting point , I()r vx.unpk-, \\ hid . hy dC 'hnition do

not bel ong to individual go ld ato ms sino- single atoms do not me lt .

Alth ough individual gold crysta ls rna) be said to melt . in rca litv it

takes a population of crysta ls with a minimum critica l size (a so .cal led

'rnicroclu ster ") for the melting point of the bulk sam ple to emerge .

Moreover, the prop erties of a bulk samp le do not emerge all at once

at a given cr it ical scale but appear on e at a time at different scales.v!

In conclusion, avoiding essentialist and typologica l thinking in all

realms of rea lity arc basic requirements in the construction of a

Dc lcuaian ontology. But besides these negative constraints there must be

some positive resources which we can use in thi s construction . I will

develop these resources in the following chapter from a more detailed

analysis of the intensive processes of individuation which actualize

virtual multiplicitics. The virtual, in a sense . leaves behind traces of

itself in the int ensive processes it animates, and the phil osopher' s task

may be seen as that of a detective wh o foll ows these tracks or co nnects

these clu es and in the proccss, creates a reservoir of co nce ptual

resources to be used in com plet ing the project whi ch this chapte r has

only started . This project need s to include , besides defining multiplici­

ties as [ did above, a description of how a population of multiplicities

can form a virtual continuum. that is, it needs to include a theory of

virtual space. Similarly, if the term 'virtual multiplicity' is not to be

just a new label for old timeless essences , th is project m ust include a

theory of virtual time, and specify the relations which this non-actual

temporality has with actual history . Finally, the relationship between

virtuality and the laws if'phy sics need s to be discu ssed, ideally in such a

way that ge ne ral laws are replaced by univ ersal multiplicities whil e

preserving the objec t ive conte nt of physical knowledge . Getting rid of

laws, as well as of esse nces and reificd categories, can then justif)' the

introduction of the virtual as a novel dimen sion of realit y. In othe r

words. while introducing virtuality may see m like an intlati cnarv

ontological move, apparently burd~ning a reali st phil osophy with a

co m plete new set of entit ies , wh en see n as a replacement for laws and

essences it actually becomes deflationary, leading to an ultimatelyleaner ontology . '

C I I,\ I''[' [,I{ 2

The Actualization if the Virtual in Space

T he picture of a relatively undifferentiated and co nt inuo us topological

space undergoing di scontinuous transitions and progressively acquiring

detail until it conde nse s into the measurable and di visible metric space

which we inhabit , is a powerfu l metaphor for th e cosmic genesis of

spatial structure. I attempted before to remove some of its metaphorical

co nte nt by comparing the rel ation between topological and metric

spaces to that between Int ensive and extensive properties: the latt er ar c

divisible in a simple way, like lengths or volumes are, whil e th e former,

exem plified by properties like temperature or pressure , arc continuo us

and relatively indivisible . T he cascade of sym metry-breaking events

which progressively differentiates a topological space was, in tum ,

com pared to pha se transition s occurr ing at critical values of intensit y. I

gave an e xample from co nte m po rary physics where such a sce nario is

becoming literally true but the fact is th at. as a description of the

genes is of space, thi s picture remains just that, a picture.

It is time now to givc a less metaphorical account of how the

intensive can engender the extensive , or more exactly, how processes

of individuation characterized by intensive properties can yield as their

final product individuals with specific spatial structures . In the first part

of this chapte r I will discuss two different aspects of the int en sive ,

eac h illustrated 'with a specific individuation process. First I willdescribe th e process whi ch individuates biological species and from thi s

description I will extract tw o of the main co nce pts which characterize

inten sive thinking: populations and rates if'chanae. I will also show how

these co nce pts can be used to repl ace the two main feat ures of

esse ntialist thinking: fixed cypes and ideal norms. Then I will move on to

our second task, a discu ssion of how the e xte ns ive or met ric features

of individuals emerge from processes whi ch are, at least in an

approximate sense, nonmetric or topological, using as illustrat ion the

process which yields as its final product indi vidual organi sms, A more

45

tld .u lt,c1 d" l.lJli 11111 0 1 t' II1 I"\1I11 11I I \\111111\01\1 II.. til I tlt'p.UIlIlI

from o u r gt'o!Ut' lric Illd.lpho r gl\'"!l th.t! II 1' 1011111 I .111' ddllH"c1 ntlC

on9' h)' e:Ucm lC leJ but also b) qualma. In olhl 'l \\ ol d, ..111 olg,lII i'm is

defined both hy its spatial ar('hihTture, as \\"( ·11 , IS b\' tilt' di fferent

materials {hone, muscle) which gin ' thai archih '('u;n' its spl'cific

mechanical qu alities. The intensive will then Iw revea led to ht.~ behind

both the e xte nsive and the qualitative.

Let 's begin with the process of individuation of species. First o f all,

in what sense can we speak of 'individuation' here ? For centuries

biological species were one of the main exam ples of a natural kind.

Whether on e thought of natural kind s as defined by a transecndent

essence , as Plato did, or by an immanent (natural state' as did

Ari stotle, an imal and plant species provided the exe mplar of what an

abstract aeneral entity was suppos ed to be . I Charles Darwin of course

broke with this tradition by showing that species , far from bein~

eternal archetypes, are born at a particular historical time and die

through extinction in an equally historical way, but th e idea that species

ar~ individuals, not kinds, has only recentl y (and still controversially)

gain ed ground . Much of the cre dit for the new view on speci es go es

to the biologist Michael Ghi selin who has been arguing for decad es

that a species , formed th rough the double process of natural selection

and reproductive isolation, does not represent a hiBher om oloqical

calegory than the individual organisms that compose it. 2 Unlike the

relation between a natural kind and its members, which is one of

~xe,~pli fi cation or instantiation, the relation of individual species to

individual organisms is one of whole and parts, much as the relation

between an organism and the individual cells that compose it. More­

over , unlike the relation between a particular instance and a gcneral

type, the relation of parts to whol e is causal: the wh ole emerges from

the ,causal interaction s between the component parts. J A new species,

for Instance , may be said to be born when a portion of an old species

becomes unable to mate with the rest . This reproductive isolation is a

causal relation between the memb ers of two sub-populations, and

morc~"er, it is a relation which must be maintained through time .

Anything that breaches the ge ne tic, mechanical o r geographical barriers

mamtaining this isolation will co mpromise the enduring gc netic identity

of a species.

l 'l l fl r h ,t1ul t " ,HI 1I1.III\chlll lllltl "11\\1111 1" l l t ,lIldll lfJllI III ,

tilt' mo ; 0 11\ ion Wh' 1 1I' 1 1l ~ dl lli I t"" t ' III I.Ilc · ' IMlI" II)', .• "'WI. il. '"

h." ., muc h I.,rgl·r r' " '11" 0 11 lh,1Il all orgol l1i'l1I sinn' it is t)'pically

t'011lprist.d of SI'\l' I",1 rt'protitH rive l'OmmulIitil-s inh.lhilillg gt.·ographi­

("III )" st'paralt.·d t't.'osplt'llls. Temporally, a species also operates at

much largt.·r scales, its a\"cragl' life span heing much greater than the

lih-cvc lcs of organisms. But the fact that species arc construc te d

t hro~lgh a historical process suggests that they are, in fact , just ano ther

individual entity, one which operates at larger spatia- temporal scales

than organisms , but an individual ent ity nevertheless. One philosoph­

ical consequencc of this new conce ption of species must be emphasized:

while an onto logy based on relation s between general types and

particular instances is hierarchical, each level representing a ~li fTercnt

ontological category (o rganism , species, genera), an approach III terms

of interacting parts and cmergcnt wholes leads to a fiat ontoloBY, one

made exc lusively of unique, singular individuals, diffcring in spatio­

temporal scale but not in ontological status! On the othe r hand, the

new appro ach demands that we always specify a process through whi ch

a whole emergcs, a process which in a Del euzian ontology is character­

ized as intensive . The process of speciation may he said to he intensive ,

first of all, because its description involves the hasic ideas of population

and heteroBeneity, tw o fundamental co ncepts which characterize a mode

of biological explanation known as population thinking. What makes thi s

form of thinking different fro m esse ntialist and typological thought is

expressed in a famous quote by one of the creators of the mod ern

synthesis of evolution and genetics , Ernst Mayr:

[For the typologist there] are a limited number o f fixed , un change­

able ' ide as' underlying the observed variability lin nature], with the

eidos (idea) bein g the only thing that is fixed and real , whil e the

observed variability has no more reality than the shadows of an

object on a cave wall . . . [In contrast}, the populationist stresses

the uniquen ess of everyth ing in the organic world .. . All organisms

and organic phenomena arc composed of unique features and can be

described co llect ively only in sta t ist ical terms. Individuals, o r any

kind of organic entities , form populations of which we can deter­

mine the arithmetic mean and the statistics of variation. Averages

4 7

.111 ' 1111 H,I, t III I I I• ua .1111 IltOIl. nllh fhl 111(11\1.111 .t1 III \\huh till

PClPU l.l tl Cl II ~ .m- t OlllpO""t ·«! h. l\ I' r.·,,111 \ I I I I 'I I . . U Ulllll.ll( tll'h thlCHI'i 0 1

t )(' popu anun th inker .md Iht" t\ I .Fo I . ' I . " po ogl\l .m- pnT"t·l,. 11..· 0ppo... iu-.'II r , I ll I),",o ogl st the I)'PC (ddos) is n-al J. ut! tht" varia tion .111I usron W lilt· for th e I "

l. ', { popu auonrst, the typt.' (tlw an ' rage) is an

a sst racuon and only" th . . .c vanation IS real . No tw o W J ) 'S of lookingat nature co uld be more different. "

When one views species as natural kinds wh ose memb ers sharecom mon s t f Id ' I ' ath o l e o I entica properties, th e inevitable variat ion between

Po~n7e7 "" of a

fcdlass canno t be hut an accident of history. Fro m the

o VI C'\' 0 ctc rm ining thd f fi c co mmon set of pr op erties whichFe nes a I ~ed ar,chctype, thi s vari ation is indeed quite unim~ortantor popu ation thmkers on the o th hI ' , 'that is C' f b " . er anc , vanauon , genetic variation

I Jar rom cmg ummp rta ' h f Iadaptive d 'ffi 1 0 nt IS t e ue of evo lution: witho ut. bl I cr.cnc~s >etween organisms natural select ion ,...ould bemcapa e of yieldin 'II

g an)' Improve ments in the population let Ia ow novel for r t ' a oneh , ' : s 0 emerge. Put differen tly, for populat ion th inkerseteroa~nell)' IS t estate we should expect to exist spontaneousl under

:~yst :1T~umstahncesb' while bomopeneny is a high I)' unli kely stat: whi che ro ug t a out only under ve 'fi I .abno rm II . 'C' ' ry speCl c se ect ron pressuresa } unuorrn 111 space an l ti ~ 6 M 'thinks of th " , rr ttme , oreover, whil e the typologist

e genesIs of form 111 terms of the e xpression of sin Ie tfor the populationist the forms of ' I a ypes ,1 ' orgamsms a ways evolve withico Iecttviues (re produc tive co m m unities cor I ) I I m

I. ,.< exam p e as se 1" I

ac vantageou s traits with differc t ., . ec i vc ypopulation . n ongms propagate through the

essPOtpullat ion thh inking el iminates one of the two und esirable aspects ofen ra Ism, t c existe nce f ' ,id ' f 0 pre -exlstmg arche types defin ing the

I entity 0 specie Th h, es, e ot er aspect the role which h hI , id I ' sue arc ety'pesp a) as I ea norms which their instantiat ions a '

:~ss perfect degree, is elim ina ted by ano the r k:;:::~:;::;~:r:7~:;~e norm if reaction . To illust rate this co nce pt let 's " .

diff ducti Imagll1e tv..-oerent repro ucuve com m unit ies b I · th. h bi , li e ongmg to e same spec ies but

III a ltmg ( iflcrcn t ecosystems T h [ react ithat there j h . c norm 0 react io n refers to the factere IS eno ug flexibility' i th . 'b dil ' h " n e co nnect ion between ge nes and

o I y tra its t at differen ces in th .e environment can yield diffe rent

t 1I.1I .1t 11'11 Iii till 1I1l' 1\.. 0 '011111111111111 I I \, II ,!lIlU ,11 lilt" .111 till till

.mu- 'IH't It' . l-o t t' .lIlI pll. ell l" ndlll' 0 11 tilt I .lll· 01 .1\.111.lhihl) 0 1 .1

IMrtitul.lr n '''Ollllt' (..unh ght . 11I1' r-: .lIu ph' , or .1 lMrtinlbr nutr-ie-nt] tilt.'

rates of grow th of the org,lni..ms in tht' two n )lnl1lllnitit's may in:

<Ii1lt'n'nt , with o ne co nsis ting of sll1.l lll'r o rganisms than the o the r. In

this case, rln-n- would be no point in saying that one co mm unit)'

f('prcsl'nts the no rma l, ideal, fixed phen otype, o r that it approxima tes

it to a gn'ate r degree of perfect ion . Since the phen otypes are flexible

within ce rtain lim its, all realiza tions of the genoty pe arc normal within

those limits." The concept of norm of reaction repl aces the idea of

decrees of peifection wi th that of relations between rates of chanae (in our

ex am ple , rates o f nutrient availability coupled to rat es of growth) .

Dcleuze credits Darwinism with thi s double blow to esse ntialism .

challenging stat ic classification s and the mod e of th inking th ey im ply

with a dynamic form of thought which is at once populational ond

d1!e renl ial . As he 'wri tes:

First . .. the form s do not preex ist the populati on, they are more

like sta tist ical re sults . The more a pop ulation assumes divergen t

fo rm s, the more its multiplicity divides int o mult ipliciti es of a

di fferent nature . . . the more efficie ntly it distri butes it sel f in the

mi lieu , o r divides up the milieu , ' . Second. sim ult aneously and

under th e same condit ions . . . degrees are no lon ger measured in

terms of increasing perfecti on . . . but in terms o f differential

relation s and coefficie nt s such as selection pressure, catalyt ic act ion,

speed of propagation, rate of growth, evolution , mutation . ..

Darwinism ' s two fundam ental co ntr ibutions move in th e direction

of a scie nce of multiplicities: the substitution of populat ions J or types,

and the substi tu tion of rates OT differential relat ionsJOT dCBrees.8

I said before that between organisms and the cells that are thei r

working parts there arc int ermediatel y scaled indiv idual st ru ct ures,

such as ti ssues or organs, Simi larly, be tween these organisms and the

species th ey compose there are halfway individuals called demes:

concre te reproductive com munit ies inhabiti ng a given eco system ." The

intensive prop erties of these demes, such as how den sely thei r

com ponent organisms are packed in their habitat , arc characterized hy

4 9

1\ , II" IIld. 11 111, "I \.,. :' "' I ,I I I., " . I

11 ' ti lt. II d,'lIl\l'd III

I" I' I' II ,I. • I II ' •111111" , 1111 " ,,1," ,,1"1,1 111 111 " II' II " I I

11\1" 1111111'1\1 ' 111.\1111". ,tll"II·, tll.,1 11 1....1 I I

.11, tll,tl .11 1' tI\I"l l 1111,11 pilltllli I. '" Ill ' ,'Il l I " I

11.11111,11 n'pl.ll '\'IIIl 'lIt ~1\"11 Ih,11 • lit 1'\ . 1., •

u -rms or .'ss.'nn's,Thl'S!' would hc , in .1 nutslll'II, till Ih1"1'1 , nntolo ,jc.11 dillll'lI sion s

which onstit ute th« I i-lcuzian world: thl' vi rt ual , the intcnsi vv and

Ihl' actual. r to phrase this in te rm ' of thl' m 'taphor that 01' .ncd thi s

-haptcr (and n >glecting for a mo ment the t ' mporal dim -n 'ion) the

indivi duals populat ing the actual wo rld would be like the discontinuous

spatial or metric structures which condense out of a nonrn t ric , vir tu. I

continuum. These metric individuals would ex ist at different spatial

scales, since populat ions at one scale ma y form larger emergent

ind ividuals at another scale, but altogether (fro m the smalle r indi vidu al

particles to the largest cosmic indi viduals) they would co nsti tute tho

familiar, measurable and divisible space of th e actual world. At this

point, however , we mu st make our first departure from th e geometr ic

met apho r : actual indi viduals differ fr,om each other not only in th eir

extensity (spatial st ructure and scale) but also in th eir qualities. A

species, for ex ample, possesses both an exte nsive aspect defining its

distribution in space (its division into seve ral reproductive communities

inhabiting distinct ecosyste ms) as we ll as a qualitative aspect defined by

populati on-Ierel qualit ies, distinct from th ose of individual organisms,

such as playing a particular role in a food chain or having a particular

reproductive strategy ,13 This means that inten sive individuation pro­

cesses must be described in such a way th at the or igin of both

ex te nsit ies and qualities is accowlte d for .To illustrate thi s important point I would like to move to a different

level of scale, do wn from species to organisms, and discuss two

examples of inten sive pr ocesses in embryogenesis, one related to th e

pr oduction of ex tensities, th e othe r to th e pr oducti on of qualiti es. .or

more speci fically, I would like to discuss two different embryologIcal

processes, one behind the spatial structu ratio n of organisms throu gh

cellular migration , folding and invagination , and the othe r behind the

qualitatil'e dlj]'erentiation of neu tral ce lls into fully specialized muscle,

bon e, blood , nerve and othe r ce ll tvpes. !" Met aph orically, an egg may

be compared to a topological space which und ergoes a progressive

Ill' , \ 1..' ) I ,ll. "I • h,lIlI. '" till • I ' I III I.. th, I lit " I ""I Ih 0 1

tIl,' d"IlIl ' , w hu h I I" h., .II 1111 ' U I hi'll 110111 till ' Il n Ih I,ll. ' or

individual Ol"g,lIliSlllS I just nu-m iom-d. I h., r.II., 01 IIII\\th or .111

individual ,dcmc depend s on thl' hir th, death ,lI1d llligr,lI ion rat's

prevalen t In the co mmunity, as wel l as on the rate or • vailabi lit \' or

res~urces (some times referred to as the carrying apa ity ot its

environment.) A demc so defined is, ind cd , a dynam ical system, and

as such may exhibit endogeno usly ge nerate d sta ble state (attractors)

a~ well as abrupt transiti ons between stable sta tes (bifurcations), In

s llT~ple m~dels, for instance , the system consist ing of a deme coupled

t~ Its enviro nme nt ex hibits an unstabl e steady state (o ne with popula­

non at zero numbers, meaning extinc tion) as well as a sta ble steady

state where population numbers match the carry ing capacity .!" More

complex attractors, such as stable cycles , app ear the moment we add

nonli~earities to the model. This may be done , for exa mple, by making

the birth-rate term more realistic to reflect the fact that there are

always nonlinear delays between the moment of birth and the moment

of sexual maturity, When the growth dynamics of a dem e are gove rned

by a periodi c attractor, the numbers characte rizing its population will

tend not to a fixed stable value but will oscillate between values , II

This simple example is meant only as an illustration of the sense in

which a dynamical process occurring in populations and defined by

coupled rates of change may be said to be inten sive , How is such an

inte~sive process rela ted to th e vir tual multipliciti es I discussed in the

pr evIOus chapte r? As I said, multiplicities consist of a structure defined

by differential rel ation s and by the singularities whi ch characte rize its

unfolding levels , These two elem ents of the virtual find their co unter­

part in th i~ten,s~ve , Th e coupled rat es of birth, death , migration and

resource availability corres pond with out resemblan ce to the differential

rcl~tions that charac ter ize a multiplicit y. The co llec tive ly stable sta tes

aV~'lable to po~ula~ions (steady-s tate or periodic, in my exam ple)

c.OI resp~nd , agam WIthout any sim ilar ity, to a distribution of singulari­

~Ies , ~h,s correspo~dence, in turn , is explained by th e fact that a given

mte~sl,ve ~rocess of individuation embodies a multiplicity , and the lack

of slmtl~rtty b tween the virtual and the int ensive is explained in terms

of the dl~ergent characte r of this embo dime nt, that is, by the fact that

seve ra l different processes may embody the same multiplicity, 12 Finally,

5"0

IMII 'II .1lul'plollll ,III\1 111111'1111... 11011 III lillI/ill! ' ,h. 11111,,, p,U.'n 'pn '''c-Il11 'c1 h) .1 111 11) IUIIII.·d o lg.lI ll'1 l1 l . 11 \11 In \ \ h,11 l ' II'~ I ' 1,111 "ggs

and org.lllisms hI' ,'i.l id to Ionu ,"p.ln·,..? A'i I s" id ill t ilt' pn' \'iOliSI h.ipn-r- ,

dll' d istinction betwee- n me-t r- ic ,lII d uotu m-t rh- sp.ln-s hoi! , d o w n to till'~V.l)' in whi ch neigh bourh oods (o r the linkagl's bet ween thl' points thai

form a space) are defincd~ eithe r throu gh exact Il·ngth s o r through

non -exact topological relations o f proximity. In thi s sense , the fc rtil izt.d

egg . defined mostly hy chemical gradients and polar it ies, as well as the

early emhryo defined hy neighbo urhoods with fuzzy bord ers and ill­

defined qualities, may ind eed be viewed as a topol~gical space which

acquires a rigid I)' metric anatomical SlTucture as tissues, organs and organ

s),stems become progressively better defined and relatively fixed inform .

, Let' s begin with the creation of distinct spatial st ructures, starti ng

with the aggrega tion of individual cells int o difTerent neighbourh oods

or collect ives via a variet ), of adhesion processes, These neighbourhoods

do not have a well -defined metric structure . Within an)' one neigh ­

bourhood, the exact location of a cell is immaterial as long as there are

sufficiently man y cells with a shared history located nearby. Simil arly.

the exact number of neighh ours is not impo rtant and , at any rate. it is

always subject to statistical fluctuations, What is important arc the

local, adhesive interacti ons between ce lls (or between cells and their

ext ra-cellular matrix during migration ) . int eract ions which are t),pically

both nonlinear (sma ll changes may lead to large consequences) and

statist ical. " As the biologist Gerald Edelman has shown . the se local

int eracti ons yie ld tw o stable sta tes for co llectives: ce lls may be tightly

linked to each other by adhesion molecul es int o sheets (called epithelia)

or be loosely associated via minimal int eract ion s int o migratory groups

(referred to as mesenchym e) . Th ese two stab le states arc related to

each other by a transformation wh ich closely resembl es a phase

transition . and which leads to two different types o f cellular motion :migrati on and folding . '.

While ce llular migrations move entire co llectives into new places,

where they can inte ract with differen t co llecti,·es. ce llular folding and

m\"agmatJOn cre ate a large \"ar iety of three -dimensional struct ures

which constitute the external and int ernal spatial boundaries of an

organism , Just where a co llective migrates and what extensive struc-

,2

IIl1t ,,01 hllltitl \\alll" !tlllJltd I d.ltlfllill 01 1111'11 II Illhlll\1

1l , 1.111t11l ~ : not o llh th l' I ,lit ' 01 \ fill•. I 11111 tit ',.ul,IIIlJll 01 Ih.ditkl'l'nt .1(111"',;&011 ' 1Il Il II ' l ll l l " (., 111.1111 1 11'1 111.111\1 1IIIIIIht ' I III ' lit II

11101,'1 uh-s, \\lakh in (urn IIl l ·el i ,l lI · tllC' pll.l. u au 111011 ht'l\\('I'n th l'

t \\ () st.lhll' sta te-s}, hut .l lso th., birth .mtl d",llll 1'.1(( " of n ,lI" \\ ithin .1

('olln't in' , I? Tlu-rc is 11 0 de-tailed gl'Tll' tic cnuu-ol of till" l'x ., d numbe-r

of n·1I d ivisions, or of the exact number uf n· 1I deat hs. but ra tlu-r .1

nonlinea r feedback rela tion between birth and death rat es and till'

processes of migration and fo lding : these pr ocesses an: affected hy till"

rate at which new cel ls are born and die and, vice versa, the rates .m­

strongl)" place-dep enden t and hence affected hy migrato ry and foldi ngmo tions . III

T he intensive (populational and differential) aspl~cts of th is proCl'ss

rna)' be said to be nonmetric in the following sense. Dcl euzc ofte n

spea ks of the cnexccr yec riaorous sty le of tho ught whi ch may I,,·

neccssar )' when ever we need to think abo ut non metric ent it ics .!" A

good example would be the way Edelman approa ches his ce ll colk-c­

tives, where the exact number of members or their exact position is

immate r ial. Thi s attitude to wards quant itath'e exactitude is not a sign

that biologists. unli ke physicists, are less careful nr disciplined . It

indic ates, on the contrary ~ the presen ce of a more sophisticat ed

topo logical style of thought. To quote another biologist whose work

will be discussed in the following chapte r , Arth ur Wi nfree:

T he science s of life have never been admired for quantitati ve

exactitude . . _ But it cannot be said that living things are at heart

sloppy , fuzz)', ine xact , and unscientific. How docs an oceanic salmon

find its way home to spawn on the ver)" rivulet it left in O regon

three years earlier? How is a meter-long sequence of billions of

nucl eotide base-pa irs reversibly coiled without entang lement int o a

nucleu s no more than a few thousand base -pairs in d iameter? _ , .

Such miracles bespeak of reproducible precision . But that precision

is not the kind we know how to write equations about, no t th e

kind we can measure to eight deci mal places, It is a more flexibl e

exactitude whi ch evades quant ifying. like the e xact itude of a ce ll's

plasma membrane dividing the un iverse into an inside and an outside

with not even a viru s-sized hole lost some where in all that

'1111\ I ,lUll d I IMII I Illpe do 'It .•1 t .11 Ililldl , 11I1 111/(((11I "t I/ Uelll ' II, , " I 4"

Jntlt/\ of ..lr.rp fon c , ,mel 11I111' , 'q

Thus, then- is J wcll-rh-Iincd St·nSt· in whk -h the SIJ.lli.1 1 rvl.uio us

characterizing an egg or the still devel oping pJrts of an em bryo arc,

indeed, anexact yet rigorous, As migration and fo lding hegin lo yieldfinished anatomical structures, however, these non mel ri c relations

becom e progressively replaced by a less flexib le set of metric ones .

The finished product is a spatial st ructure adapted to specific fun cti ons.

Like a building or a bridge , for example , an animal mu st he able to

act under gravit)· as a load-bearing structure. On the other hand, the

spatial architecture of an organi sm is not the only factor that deter­

min es its capacity to bear load s, the qualities of the mat erials making

up that archit ecture also matter: the qualities of muscle that allow it

to bear loads in tension , for instance, or the qualities of bone that

allow it to bear them in com pression. The intensive processes that

create these materials are another example of a process of progressive

differentiation, one which starts with a population of relatively undif­

ferentiated cell s and yields a st ruc ture chara cterized by qualitatively

distinct cell types.

When ce lls begin their em bryological development they arc pluripo­

tent, that is, they are capable of becoming any of the dilTerent typ es of

ce lls whi ch characterize the adult individual. This number vari es from

two in bacteria, to twenty or thirty for je llyfish, to about 254 for

human beings. 21 Co ntact between different cells (or between dilTerent

cellular collect ives) leads to the important phenomenon of induction.

This term refers to a complex process in which co llectives exchange

che m ical sign als which lead to the enhance ment or suppression of

cel lular dilTerentiati on . However, as the biol ogist Stuart KaulTman has

shown, these inductive signals act as non-specific sti mula (o r perturba­

tions) which switch a cell am ong a variety of int ernally available stable

states. The basic idea in KaulTman' s model is that the regulatory gen es

within a cell form a complex network in which ge nes, interacting via

their products, can turn one another on or ofT. Kauffman has found

that there are certain recurrent patterns of ge ne activit)' within these

networks, patterns which exhibit the kind of homeostatic stability

.1 tlll.ll,d \\111. .1II r .u l or III1 Itl I.d 111111 tu 1..11,\1 tllll . IIl lllt ,I ,

, '.H II .l tl l.1l III' 11I.1\ 1)( (1 til 111, I (d 'I) It 1'1 I I III .. I t I \II I ( III t 1·11 I " l"Kauflm.m' 1111111. ·1 dlh'l1Il't III I'l l .Ih I IIllt IIllh lilt IHII II Ill'I 01

dilh'rl'nl n ·1I I)rH' ill ,I gl\l'11 01 ' ,111 1 m , bUI ilion' lIl1porl.lIlt!) Irum

our poinl of \ h-w, II", numla-r of n ·1I t) p" \\ hi, II .1 particular (( ·11 , '.HI

t.!i rec'lj' «1!D~·r(flrl(J '/.· into. (;i n' n .1 n 'lI \\ uh <I Slh·dfic hbl or)', ,lilt! .1

certain inductive sig'l<1 1 which can ('It,lUg" its ratt', llw outcome of tlH' ir

int eracti on will depend Oil how man)' other att r-actors exist n/.·urh)' in

the state span~ of the cell (o r more exacl ly, in the stan - sp,ln ' of til"netw ork of ge nes within the cel l) , In other words, far from din'cd }

determining the qualities of a differentiated ce ll, inducti ve signals ad

as trioo ers causing cells to swi tch from one attractor to another Iwarh)'

one, guiding a process of qualitative differentiation which follow s

attracto rs as so many stepping-stones . This property of st imulus­

independence must be added to the mechanism-ind ependence I discussed

before as part of what defines ' the 'signature ' or the virtual, or put

differ ently, as part of wh at defines the traces which the virtual leav..s

in the intensive . But relative autonomy from specific stimula can he

achie ved only if the internal dynamics of a cell (o r co llec t ivities of

cells) arc rich eno ugh in endogeno usly genera ted stable states, This

co ndition is by no means guarantee d and depends on certain inten sive

properties of a network, those defining its connectivity: the number of

ge nes directl y or indirectl y influen ced by each single gen e or the

number of steps needed for the influen ce of one gen e to be propagat ed

to other genes . At critical values of connectivity a phase transition

occurs leading to the crystallization of large circuits of genes, each

displaying multiple attractors. : "Edelm an's and Kauffman's models illustrate the sense in which the

int en sive may be said to be behind the genes is of both the exte nsive

and the qualitative. Yet, neither one is a literal rendering of a simple

cascade of broken symmetries . While the ce llular neighbourhoods in

Edelman's model do illustrate how non -rigidly metric spaces may he

transformed into fixed spatial struct ures, the connection with topology

is indirect. This is even more true in Kauffman's model where the

connection with nonm etric questions is completel y indirect, mediated

by the topological invariants (such as co nnect ivity) of abstrac t spaces

0 1 I'll.. 111I11I 1c" d l 1111111 J till" .I\M!.,"I. qll .•IIIII ' , .. 1111It 1.111 , hlllll t ' .un

pit'!'> ..!luuld I ll ' '1 '1'11 Il lli .1\ till I ', t1 ) IIi WOl rau u I hUI .. fl"l'Jlll "'.'I p.lI h IIf

t he simple sym l1l\' t r}'-brt'.lk ing C.l"' ''ldt · . It is through "Ill II pin.' 11 1\",11

rc placc nu-n ts that lite ral co ntent m .l)' 1)(· unparu-d to, .lIId nu-t.rphoru-alcontent re moved from, our guiding imagl' fo r th e ac-t ualiza tion of till'

virtual in space . There is one more aspe('t of ,'mhryog"lll'sis fromwhich we can der-ive further resources to co ntinue this process of

progressive literalization. It involves looking at a dC\'eloping cmbrvo as

a process c1 assemblj' of organisms, a process which must yield individualswith the capacity to em/reo As an illustration of this point I will co ntrast

two different assembly processes , the process behind the creation o f

industrial products , as it takes place in an assembly-line factory , for

example , and th e process taking place wi thin and amo ng living ce lls

whi ch results in th e asse mbly of t issues and orga ns .

The parts of an object put together in an assembly line are typi cally

fully Eucl idean, hav ing rigid metric properties such as sizes, shapes and

positi on s, a fact that limits the kind of procedures that may be fo llowed

for their assembly. These procedures must include a rig idly channe lled

transport system (using conv eyor belts or pipes to transport rawmaterials, and wires to transport energy and informatio n) as we ll as

sequences of rigid motions to correctly position the parts relative to

one another. By con trast, the component parts used in biologica l

assembly arc defined less by rigid metric properties than by thei r

topoloqtcal connect ivity : the specific shape of a cell's membrane is less

important than its con tinuity and closure, and the specific length of a

muscle less important than its attachment points. This allows co mpo ­

nent parts to be not inert but adaptiv e, so that muscle lengths can changeto fit longer bones, and skin can grow and fold adaptivcly to cove r

bo th . It also permits the tra nsport processes not to be rigidly

channe lled, using simple diffusion through a fluid medium to hring th e

different parts together. Components ma y float around and randomly

co llide, using a lock-and-key mechanism to find matching patterns

with out the need for exact positioning.

All of thi s has consequences for th e capacit}' to evo lve through

mutation and selection which each of these tw o assembly processes

may have . If putting together organi sms followed an assembly-line

pattern, random mutations would have to occ ur simultaneo usly in

1I1,lll11l1I' p.lIt . ll. .l1l11ll .lI l1l p l lI t l . 11I11 III IIl d, t III "II,ld .1 \I .lhll

('1I1It) (111 whu h n.u ur.il ..t ,ll·llltlll ttllllt! (11"1 Itt l l« r u r ut t r-tn c- 0 1

"lIt h .t I,trgt' number or .. irnu lt.uu -uu 1111111111111 I I (II l tl ll l "I' , .l highl}'

iml'roll,lhl,' (' \\'1 11 . III hio logic;\1,ls"<t 'lIlllh, 011 1111 otht 'l" 1I.IIH I, muta tions

do not h.tn· to h(~ so (:oo rdin.H,·tt .11It I till gn',ldJ e nhanc-es tlu­

possi bilities for evolutionary cx pcrinu-nt.uiou. As till' scien t ist Ericl Irvx ler w rites:

Because ce lls and organisms make widespread usc of diffusive

transport for energy, information and molecular parts, the evolut ion

of new processing entities (enzymes , glands) is facilitated , A genetic

change that introduces an enzyme with a new function can have

immedi at e favorable effects becau se diffu sion automatically links th e

enzyme to all other enzymes , energy sources and signal moleculesin the sam e m embrane compartment of th e cell (and often beyond) .

No new channels need to be built . .. [and ] no special space need

be set aside for the enzyme, because device placement isn't

geometric, Changes in the number of parts . . . become easy. There

are no strong geometric or transport constraints; this often allowsthe numbe r of mo lecular parts in a ce ll to be a variable, statistical

quantit)" With many copies of a part, a mutation that changes the

instruct ions for some co pies is less likely to be fatal . . . At the

level of m ultice llu lar organisms, th e striking adaptability of t issues

and organs ensures that basic requirements for viability, such as

continuity of skin and vascularization of tissues, continue to be met

despite changes in size and structure . If skin and vascular systemswere inert parts , they would require com pensating adjustments forsuch changes .25

Thi_s example illustrates another indirect way in which the metric

may be said to emerge from th e nonm etric . Unlike . a developing

embryo , a finished organism has more specialized tubes and channels

and some of its com ponents lose adaptabilit y and rigidify. T his

"metr-iza rion' is, of course, never co mplete , even when an organism

reaches maturity . But what is very significant is that, at least in thecase of multi -cellu lar animals, if organisms were not individuated in an

intensive environment which is not rigidly metric, their capacity to

n

nol\(' \\0111.1 ht ' 11·. l tI) d 1111111 I Iud Ih.lIlk IUlth lodlliu 1\ 111 .111 pllit .

lock .unl kt') lII.thlllng .1 ('lIIbl) . IlIpldogll .11 .II H I .ld.lpll\l · IMlt • 0 11

one h.md, .1S we-ll as stimulus indt 'p"nd"IH e, 011 tht ' utiII'I , cvolutu m

has an opl'n space in which to carry out its hlind ~.;t'.Irdl for 11('\\ I ~)rllls,

Put differently, hiological e volution can lx- din'rgl'nl Jud It'dd to J

prol iferation of novel ties thanks to the fact that the elements it uses to

try out new combinat ions are neit her rigid l)' connected (to specific

stim ula, to specific channels) nor intolerant to hctl' rugen d ty and

variation .

Let me summa rize what thi s discussion of embryogenesis has taught

us about the actualization of the virtual in space . Inten sive processes

possess nonmctric properties in subt le and complex ways: som etimes

they in volve the spatial co ntinuity and indi visibility of pr operties like

temperature, pressure or den sit y; other tim es the anexact yet rigorous

way in which ce llular spatia l neighb ourhoods are defined ; so metimes

what is invol ved is nothing specifically spatia l, but rather that wh ich

remains topologically invariant in a spatial process: and other t im es

spec ifically spatial capacit ies arc conce rned, such as the capability of

adaptive components to fold , stre tc h or bend . Simila rly, the final

product of an intensive process is no t ju st metric geome trically

spea king: extensive properties include some geome tric ones (like

length or volume) but also seve ral o the rs that have nothing geome tric

about them, like entropy o r amount of ene rgy; the n there are

properties which are metric, such as channelled tran sport or rigidity of

parts, but which expand the co nce pt from structure to function; lastly,

a finished product is characterized by qualities, whi ch also result from

inten sit ies but whi ch are metrically indi visible like int en sities. T hus ,

the re lation between the metric and the non metric in a process of

indi vid uation is not as simple and stra ightforward as the metaphor of a

'topo logical egg' progressively differentiating int o a ' Euclide an organ­

ism ' would suggest . But what this comparison has lost in simplicity it

has, I be lieve, gained in literal adequacy.

Ha" ing clarified the relations bet ween the int ensive and the non­

metr ic , in th e ne xt part of thi s chapte r I woul d like to probe more

deepl y into the nature of inten sities. Altho ugh as I said in Chapter I ,

th e term ' inte nsive property' belongs to thermod ynamics, it may be

exte nde d to co ver other areas. Indeed, my usc o f the word ' intensive'

III 1111 tI, ( 1 11'11011 0 1 Ihl 11 111 1\ 1ll1l1l lll11 II I I' II 111.1 or .111 1 III ' .1

.1111'.1(1\ .111 " h' llI lt- d II .11(' . \tl 11'1 III t I , ~ III till 1(11011 \\111 ht 10

1)('( il; till' (fl ll lle'( 11011 ht't\\(" ,u lilt l .lUli ml til 11111 111111 ,md it '(' \(' 1.,1

c xu-u srons , Afh'r thi , l OIH t'ptu,ll ( I.'nla, ,111011' l tl l1lp l t'tt'd I ,\ ill 111o,

on to di" 'u',, 011(' o f I h,lt-u/t ·' .. 11Il....1 imporLutt Iht" t' s n~ganling till"

inu-nstve. The h,)~i(' idl',) is Ih.ll 0110' .1 prol ('SS or iudividuauon is

cornph-u-d, till' iun-nsivc factors whivh dl·fim·d this procl'ss disJppt',lr

or lx-come hidden underneath the e xtensive and qualitative prOpt'rlit's

of the final produ ct. Or as Dc lcuzc puts it, ' we know intc usitv on l), .IS

already de veloped withi n extensity. and as covered over by qualities' , 110

Thi s theme of the disguising of pr ocess under product is key to

Dc lcuzc's phil osophy since his philosophical method is, at least in p.ut ,

designed to ove rcome the objective illusion fostered hy thi s

concea lme nt .Let 's begi n this discu ssion with the textbook definit ion of til<"

distinction between the inten sive and the extensive : "The rmodyuarnic

properties can be divided into two ge ne ral classes, namely intensive

and exte nsive properti es. If a quantity of matt er in a given sta te is

div ided into two equa l parts, each part will have the same value of

intensive properti es as the original, and half the value of the exte nsive

prope rt ies . Pressure , temperature, and den sity are examples of intens­

ive properti es. Mass and total volume are exa mples of ex te nsive

properties. "?" Although this definit ion does point to a basic differen ce

between inte nsit ies and ex te nsities, its emphasis on divi sibility allows

it to equally appl y to qualities, such as colour or texture . But as we

just saw, a crucial part of Dclcu zc' s argument hinges precisely on the

distin ct ion between th e inten sive, on one hand , and the ex te nsive and

qualitat ive, on the othe r. Co lours are, ind eed, not di visible in ex te n­

sion: _a certain patch of material of a given co lour does not yield , whe n

hroken into equal halves, two sma ller patches with half the value of its

co lour (half the hue and half the brightness). T his lack of di visibil ity

has misled some philosophe rs into failing to distinguish qualities, or

even subjectively ex perien ced intensities, such as pleasure , from

ob jective inten sive properties ,2~ Thus, we nee d a characte ristic other

than indiv isibili ty in ex te nsion to distingui sh objective inten sities fro m

qu aliti es.T here is, indeed , another way in which physicists sta te the distinc-

11011 1u- 1\\1 " II dll mu-u 1\" , li lt I till I ' tc-u 1\( \\1111. 1\\ 11 ' It 'll'il\t'

prolH'l'til" ,Hid lip ill ,I , illl pl(· \\ ,,)' (I\\O,U( ' ,1 "tid lip til , I III 0 1' " 1 t11l11o\1I)

lol rg t' r arc-a), inu -nsive propt 'rti t" do no t oldcl up hilt r.ulu-r dlt""Hc. This

avcraging ope ratio n is an objective operation . in till' !'it'n,',' th.u pladng

into co ntact tw o bodies with ditlcrcm t.'mp,·ratu n ·s will lr igger a

spontaneous diffu sion process which will equalize the tw o tl'mpt'raturcs

at some intermedi ate valu e. "? Thi s capaci t)' to spo ntaneo usly reach an

average value explains wh y temperatures or pressures canno t he

divided in exte nsion . A particular value of temperatu re or pressure ,

hein g an average, will remain the same when the bod )' possessing th ese

properties is broken into tw o or more part s. But beyond that, it points

to a dynamical aspect of inten sive properties not shared by qu alities:

differences in thermodynamic int en sities ar c capable of drh'ing a

process of equilibration in a populati on of molecul es, a process in

which these differences will tend to average themselves out. The

int ensiv e would then be distingui shed from the qualitative by the fact

that d!fferences in intensity, though not in quality, can drive flu xes of

matter o r ene rgy .

Intensive differen ces may be sharp or gradual (in which case they

arc referred to as 'g radients') bu t in either case they are nothing like

the exte rn al differences which distinguish one fully formed individual

from another. In static typologies one confronts the diversity of obj ects

in the world by a careful tabu lation of that which stays the same and

that which differs among them, Th e exte rn al difTerences between

diverse objects ar c viewed simply as a lack of similar ity so the conce pt

of differen ce plays a purely negative rol e . Int ensive or int ernal differ­

ence s, such as a temperature or pressure grad ient within one and the

same body I arc, on the co ntrary , positive o r productive, forming the

basis of simple processes of individuati on . The soap bubbles and salt

crystals I mention ed in the last chapte r , for instance . arc equilibrium

structures whi ch e me rge from a process dri ven hy intensive gradic nLo;;,

o r more exact ly, from th e spontaneous tendency of the molecula r

co mpo ne nts of bubbles or crys tals to minimize a pot ential (o r minimize

an int en sive differen ce) . Given thi s morphogen eti c rol l' , it is not

surp rising that Del eu ze makes int ensive dilll 'rl'n n 's J kt} clement in

his onto logy , As he writes:

60

llll/"' '''IH 1\ li t " dun '0 II 'f It II hili dill . I t I" I I tlMI h\

whn h tlJ(' '1\' II I 11\111 11,11 , I I II I 1101 pill 111111ll"1I01l II1Il 1111nllflllWlloll , Ill 1' ''11 III Ihl pili nOIll. 111111 I \ 1'1 \ Ihm t \\ hit h h.lp

pen' ,lIul ('\, '1' IIl1n I \\lIh" '1l'IH,I I ltllrt 'l.ltt'd wit h ordt'r~ 01dil1 ~ 'r t'l1 cl' s : di lfl ' ll 'lll t " III 1t·\( ,I, 1t'llIp,'r,llUn', pn's~un', n-nvion ,

pou-nt ial, dint'n'Il('t' of ill1t 'Il,il ) , III

The first modification which mu st he made to the standard definit ion

of intensive pro perty is, the n. that the inten sities definin g a parti cu lar

physical syste m may indeed he 'divided ' but the differen ces that result

change the syste m in kind (fro m an equilibri um syste m, where

differences are cancelled, to a non-equilibrium one) . Moreover if th l"~t'

d iffere nces are made int en se enough a crit ical thres ho ld rna)' be reached

and the physical syste m in quest ion will undergo a phase transit ion . its

ex te nsive properties suffering ~ radical change in nature . Thus, rather

than indivi sibility, the key conce pt in the definit ion of the int ensive is

productive d1Jerence, as well as the rel ated conce pts of endogeno us stable

sta te (such as a thermod ynamic equilibrium state) and of cri t ical

transitions between states . How does this rel ate to th e tw o conct'Jlts

whi ch I said defined the inten sive in biology, populations and rates?

The answe r is relatively straightforward : intensive gradients arc rncas­

urcd by rat es of change , and th e fluxes of matter and energy these

differen ces drive are eithe r the migratory mov ements of a molecular

population , or mov em ents of energy through such a populat ion , H In

this sense , the thermodyna mic defin it ion is di rectly related to the one

I used in biology, bu t I also made several depart ures fro m it.

W hen I descri bed population thi nkin g in evolutionary bio logy a key

issue was the role of genetic differen ces. W hile in essentialist or

typological thinking uniformity is the natural state and diffe rence what

need s special explanation , for pop ulation thinkers it is differen ce that

is unprobl ematic. Thi s use of the conce pt of differen ce alread y

constitutes an exten sion of the orig inal noti on of intensive gradient,

hut it is nevertheless related: a hiological population where ge netic

dlffercn ces have been el iminate d is as unproductive as a thermodynam ic

syste m where differen ces in temperature or pressure have been

cance lled throu gh equilibration, " Yet, the biological examples I gave

6 .

., lt ll \ I' il l\oh, ' .1 1II 0l t' l .ldh .ll lkp.lItlll l II l1l n till' 011 '11 1.1 1 d d ll ll ll oll II I

till' inu-nslve . In p.u-tic-ul.u- , unhkr- till ' 111011'1 ul.u- pClplll.lt ioll ' ludil,d in

thermodynamics, the meml u-rs of biologic.ll popul.uion s [rave .1 I.lrgl'r

repertoire of ways 10 interact with eac-h other. I ike a thermodvuamk­

system, a biological population ma)' exhibit auractors (and tims hl'

defined in part bJ the tenden cies with which these singularities endow

it) but in addition its members will typi cally display com plex capacities

for interact ion which have no counterpart in the physics of heat.

An individual may be characterized by a fixed number of definite

propert ies (e xtensive and qualitative) and }'et possess an indefinite

number of capacities to c1Jeet and be c1Jeeted by other individuals . Thedegree of openness of this set of possible interactions will vary from

individual to individual. In the realm of chemistry. for instance,

different chemica l elements have different capacities to fonn novel

combinations with other elements. the capacities of carbon, for

instance , vastly outperforming those of the inert gases. In biolog)', as

we jus t saw , the flexib le capabilities of adaptive parts or the capability

to transport and match co mponents without rigid channels or position ­

ing procedu res, lead to even more ope n combinatorial spaces . This

ope nness is also re lated to the virtual as can be glim psed from the fact

that it dem and s fro m us th e use of modal terms (such as 'un limited

possibilities ). Deleuze , in fact , always gives a two-fo ld definition of

the vir tual (and the inten sive), using both singularities (unactua lized

tendencies) and what he calls c1Jeccs (unactualized capacities to affectand be affecte d) ,"

Unlike sing ulari ties , which arc relativ ely well studied thanks to the

developmen t of the topological approach to state space , the for ma l

study of affects is relati vely underdeveloped . Several scientists who had

previously focused on the study of singularities, howe ver, have recen tly

switched to the study of a different type of formal system which allows

the exp loration or cons tructive capacities. Stuart Kauffman and Walter

Fontana, among others, view the capacity to form novel assemblages

when objects are put into functional relations with one another as a

problem which is complement ary to that of state space , a problem whichmay also lead to the discovery of universal features analogous to those

reveal ed by classifications of attractors. Alth ough the formal s)'Ste ms

they have designed to stud)' affects ( Kauffman's random grammars,

6 2

1 0 Ill .IIl ," "l lolltl\l1\lt dW II1 I " \) III I. \ \1 II lind. I 1111 III I!J.IIl lilttllnl tlld to Iiltlth 11 1 III .lIH 1' • tI'l\ hoi, . .,Ir' ,llh \llhltd \ .•IIl ,llt',II I I ,ht ~ 1I1 tO cl lI" tlC 'llI ~ 0 1 111111 t ll lll "l lllh 11., 110 11, 1111 ludru I tl u ' ell " 0\1 '1")

of .• I't'\\ (("lurrm' t1 \\t'nrh~ {'Imam (IIt h .l ,,"tll,.l t ., I) lit 1001''1 ) ,, !lil h

11M \' turn out to hI' univ t'r~,,1. '4

\Vhilt' th e re lation [u-twcc-n inn-n..itit' s .Hld ..ingul,Irit it·s does 110t

invo lve anv dt'p,lrtufl' from th e thermodynamic dcfimuon of ' iuk U'

in" , Jddil;g capacit ies implies ('xtl'nding that definition . Let uu- fi ....tgin' a more detailed characterization of capacities and then show ho\\

the original definition may be naturallv e xte nded to include them . Anindividual organism will 'typically exhibit a variety of capabilit ies to

form assemblaaes with other individuals, organic or inorganic . A good

example is the assemblage which a walking animal form s with a pil·n·

of solid gro und (which supplies it with a surface to walk) and with a

graVitat ional field (which endows it with a given weigh t). Although the

capac ity to form an assemblage depends in part on the emcrg('nt

properties of the interacting individuals (an imal, ground, field ) it is

nevertheless not reducib le to them. We may have exhaustive know ­ledge about an individual' s properties and yet, not having observed it

in interaction with other individuals, know noth ing about its

capacit ies. 35

The term 'capaci ty' is elosely related to the ter-m 'a fforda nce'

introduced by James Gibso n within the context of a theory of

ecological interact ionsv'" Gibson distinguishes betw een the intrinsic

properties of things and their a tTordances . A piece of ground docs have

its own intrinsic properties determining, for example, how horizontal

or slanted, how flat, concave or convex , and how rigid it is . But to be

capable of affording support to a walking anima l is not just ano the r

intrinsic property, it is a capacity which may not be exercised if thereare no animals around. Given that capacities are relational in this sense,

what an individual affords another may depend on facto rs like their

relative spat ial scales: the surface of a pond or lake may not afford alarge animal a walking medium, but it docs to a small insect whic h can

walk on it because it is not heavy enough to break th rough the surface

tension of the water. Affordances are also symmetric, that is, the)'involve both capacities to affect and be affected . For exam ple , a ho le

in the ground affords a fleeing animal a place to hide , but such animal

l o uld .d tI d.· II "" II hllll , Ihu 111. 1 1111 ' " ' I h,lI' ' II I' Ih. ' 1tlulld

itsl II, ~illlll .l ri •.111 .1Il II 11oI1 '",1\ III I bl'l ,llI I .1 I'" .1,1111' 11111' d II d.1Il "' I

but it its..lf aftlmls nut r it ion 1;1Ih. pn ·d.lllll , ·

W e may expand th ' rnl'anillg of the krill •intr-nsi , I" 10 ill' lud l' th '

properties of assemblages, or mol" exact ly, of the prou'ss,'s wh ich

give rise to th em. An assembly process may be said to ln- charac t .r izcd

by intensive properties when it articulates het eropcncous cI .rn cnts as

such. i" In the assemblage formed by a walking animal, a pic e of

ground and a gravitational field, three heterogeneous individuals ar e

joined together as such without the need for any homogenization ,

More generally, the interactions which organisms have with the organic

and inorgan,ic components of an ecosyst em are typically of th e int ensive

kind (in the enlarged sense), an ecosystem itself being a complex

assemblage of a large number of heterogeneous components: diverse

reproductive communities of animals, plants and micro-organisms, a

geogra phical site characterized by diverse topographical and geologica l

features, and the ever diverse and changing weather patterns. Similarly,

the meaning of 'extensive ' may be enlarged to refer to the properties

of processes, such as the assembly-line process I mentioned before,

where hotnoqeneous components ar e linked together. The enlarged

meaning of 'intensive' is related to the standard definition in the

crucial role played by d!fJerence. Much as a thermodynamic intensive

process is characterized by the productive role which differences play

in the driving of fluxes, so in the enlarged sense a process is intensive

if it re lates diffirence to d!fJerence. 39 Moreover, as the exam ple of

assembly processes based on adaptive components showed, th e flexible

links which these components afford one another allow not only the

meshing of differences, but also endow the process with th e capacity

of divergent evolution, that is, the capacity to further d!fJerentiate

diffirences.

Armed with this more adequate definition of inten sive pr oc ss we

can move on to the second set of issues I said need d to be discu ssed:

the concealment of the intensive under the extensive, as \ ell as the

concealment of the concrete universals (singularities and afTect s) which

animate intensive processes . To anticipate th con .lusion I will reach

in a moment, in the case of singularities th existe nce of th e virtual is

manifested in those situations where int ensive diftt'r ' 11(,(" are not

c in. vl! ,I IIll d ll l . III tilt ' I ' II I II h,l l II

I I Illbl .1 '. lilt lit ddt. I I III' U hili .111 " , 11111 ' Ih, III

tllI lllI .h hOlllll 'I'III/ .IIIIIII , Ih.II I 1111 .1\ til " I II • I III I'll Ihlllll<'

1,IIIill' 11II .111 I ' 1'1.111 .1111111 ill \1'1 111 III \ 11111,111 t ti ll \1 ' 1 ,,1\• ..tl ll\\ III '

di l'fl'r('JI('"s ill inll 'nsi t \ III I . 1.1111. , 111.1 III ..11I1I1ll.llin ' d illi 'n 'lI' I'

Ih rou gb uni fllrmiz,llilll; , I' n~' l"l ivI · l IlId.,s Ih., \ irt u.il ,1I11 1 Illakl's till'd isappearance of process 1I11lkr product SI'(' III k-ss pr ohl crnat ic .

Althoug h thi s co nce alme nt is partly tlu- result of human int crvvnt ion ,

o f laboratory practi ces whi ch focu s on the final ' 'l uilib r ium stale or

which 'yst ' matically homogeni ze materials, for exa m pi . , it is also. n

obj .ct ive phenomen on . Any ar ea of th e world whi ch is in thermo­

dynamic equi lib riurn , for instance, is an ar ea wh ere intensi c dill er­

cnccs have cancelled th emselv es out, and hence an area whi ch co nn', I.th e virtual without the need for human intervention. These ar eas of

the world, in short , would constitute an objecti ve illusion.

Deleuzc argues, for exam ple , that des pite th e fact that classical

th crmodynamics yielde d valuable insights into the importance of th e

inten sive, thi s branch of physics did not provide th e foundation need ed

for a th eory of individuation given its exclusive focu s on th e final

equilibr ium state of a s)"te m . The problem with concentrat ing on the

final state is that only during th e difference-driven process can th e

equilib r ium state be see n as a virtual attractor, a state which is not

actualized yet but which is neverthel ess real since it is actively

attract ing th e successive states of th e syste m towards itself. But while

it is true that classical thermodynamics tends in this sense to under­

est imate the virtual and th e intensive, 'this tendency would lead

nowhere if intensity, for it s own part, did not present a corresponding

tendency within th e extensity in which it develops and under th e

quality which co vers it. Intensity is difference, but this differen ce tends

to deny or to can cel itself out in extensity and underneath quality' .-w

In other words , while certain scie nt ific practices tend to systematicall y

down-grade th c intensive and conceal the virtual, th ese practices only

amplify an illusion which is obj ective and which is, therefore, much

harder to overcome.One way of allowing th e virtual to manifest itself is to design

expe riments or to study phenomena in circumstances wh ere intensive

differen ces are not allowed to cancel th emselv es, This is what is done

III lIu" 1.'h'l \t ' l Inll 0 1 Iht " ti t Ilt t 0 1 h.-.u , ti ll held 0 1 IIJr IfI'mcl/ llI hhn UIII Iht ·nllOd)"ll.llJlICS, \,ht'n' .111 Inlt 'n ,,' lIem of m.u n -r .1Ilt!

t' lIl'rgy cont lnuouslv lr.l\"t'rsl'S lilt' svsn-ru undt 'l" stud v .Kling a.. J.- - ,constrain t maintaining intensive- dil1~'n'nn's JliH·...• I said ill Iht' pn-vi -

ous cha pter that the varictv of atl ractors which J s)'s ll 'rn n1.1)' h.wl'

depend s on wheth er its dynam ics arc linear or nonlinear. Whi k linear­

systems possess the simplest dis tribution of sing ularities, a singll~ glohal

optimum structuring the whole of state space, nonlinear ones typicallyhave multiple att racto rs (o r put more techni cally. nonlinear equations

allow for multiple so lutio ns) . To the mathemat ical distin ct ion bet ween

the linear and the nonli near , therefore , we mus t now add a thermo­

dynam ic one , that betw een s)'ste ms near and Jar from equ ilibri um . As

Prigogine and Nicolis pu t it "witho ut the maintenance of an appropriate

distance from equilibrium, nonlineari ty canno t by itself give rise to

mul tipl e solutio ns. At equilibrium detailed balance introduces a further

co ndi tio n that restricts and even un iquely fixes ' the solutio n." ? In other

words, to exhibit their full co mplexity nonlinear systems need to be

driven away fro m equilibr ium, or what amounts to the same th ing,

appropr iately large differen ces in inte nsity need to be maintained by

e xte rna l co nstra ints and not allowed to ge t cance lled or be made too

small. In this sense, as these authors say, 'no nequilibrium reveals the

potentialities hidden in the nonlinea rities, potenti aliti es that remaindormant at or near equilib rium' .H

Thi s is important in the presen t co ntext because it explains the

physical source of the objecti ve illusion Deleu ze ta lks about. Take for

example a linear syste m wi th a sing le attractor . As I just said, wh ile

the system is on its way to thi s attracto r the unactualized end state is

indeed there alr eady, actively att racting the process toward s itsel f. At

this point its virt uality is relatively easy to grasp. But once the process

is over it becomes easy to over loo k the virtual nature of the end state,

even thou gh a system will never actually reach the attracto r, only

fluctuate in its vicinity. A nonlinear syste m with multiple attractors,

on the other hand, continues to display its virtuality even on ce the

syste m has settled into one of its alternative stab le states, because the

othe r alt ernatives are there all the tim e, coexisting with the one that

happen s to be actua lized . All one has to do to reveal their virtual

66

I'l l I li t I I 10 IIU ., I II II 1 111111 111 IIl1t l I II Itt. II III 111 pll h II IItl1

ol on 1"' ''111 I II ,l lt l,H !lli ll , ll1d IlIll l .lIl l ltllll ( 1 11 11 \\t tllUld. "I ttHlI I ,

n -h-r t o lilt.. ,. It , · l l loIl i \ I ' , I" hl. " I, l tt ' ,I J",nl/"I" ..." IIn1 \ 1I111.•lIlit ·"l llll I1M\(' ,.In·.ul) ,1fglll·t1 for 1hv w,.·d to h·pl.hI ' lilt pO"lhlc \\ ith ,l mUI('

.l t k 'lU.lI t· form of physical rnod.llit ),. )

A systeTll with multiple auractors, in shor t , has J gn'ah'r t"' IMdt)·

to l'xpn'ss or reveal t111~ virtual. But this I,'xpn'ssin' cap.wit)' will

depend , in turn, on the thermodynamic "zone of inu-nsity in which

the syS1l'm orl'rJtl~s: at low inten sit ies (ncar equi librium) a nonl inea r

system will in effect he linearized, that is, its pot ential complexbehaviour will no t be revealed . This procedu re has, in fact, lx-comc

ro utine in physics whe neve r trou blesom e nonlinear effects ru-ed to he

eliminate d: one simply studies the syste m in question at very low

inte nsity values for the trouble-making variable.-4-4 How ever I by fo!lo\\, ­

ing pro ced ures like this and systematically neg lecting the high inu- nsit yvalues at wh ich nonlinear effects arc fully expresse d , physicists promotl'

an illusion which is originally objective but wh ich now becom es

subjectively amplified . O n the othe r hand , study ing syste ms wh ich are

bo th nonlinea r and nonequ ilibrium , systems where the objective

illusion is at it wea kest , opens up windows into the virtual.

One of the tasks of a philosopher attempting to create a theory of

virt uality is to locate those areas of the world where the virt ual is st ill

ex presse d, and use the unactualized tendencies and capacit ies one

discov ers there as sources of insight into the nature of virtual

multiplicities. More exac tly, Deleu ze recommends following a very

specific philosophical me thod in which, as he says , it is

necessary to return to the interior ef scientific states ef~ajrs or bodies in

the process of belne consti tuted, in orde r to penetrate int o co nsiste ncy ,

tha t is to say, into the sphe re of the virt ual, a sphere that is only

act ualized in them. It would be necessary to 90 bock up the porh rhor

science descends, and at the very end of which logic sets its camp.:"

In other words, unli ke the linear and equili briu m approach to science

\.... hieh conce ntrates on the final product, or at best on the process of

actualizatio n but always in the direct ion of the final product , philosophy

It I' I 11111.1 Ilk, 10 ,Ic I. II III 1111111111 I, .. 1111 nlll ' 10 till 1111 I. 1'1.111

" III(h 0P,"11 d 1111 ,I.,'pl'" I 1°1'0111 'II ,.I 1',11 I hll II dill, I' IItl,II. ",.1

di"II !.' il '01111111111,.1 II 1ll" Ollll" 111 0 11<" I'd, mil',' Il'ldl 1111'1'11J '

[ollowi n J .1 (.be.H I" 01 sv rruru t rv hi ".lkin I ,'v, '111 ., ,I: . u -nsivv struct ures wo uld co nstituu tl u- co unh 'rl' rt or thl' holl Oll 1

lc 1'1, while inte nsive pr o ·( ' SS 's wo uld he Ihl' co unterpart 01 till'

int rrn ccliatc leve ls, -ach one representing a g 'o llll' l ry which is no t

fully mctri but whi h can, in fa t , be me:tricized .n Th · top Il'v"I , .111

ideally co ntinuo us and relatively undifferentiated spa " would III till'co unte rpart of the virtual . I us ' terms like ' to p' and ' bottom' hl'r, '

informally, with no sugges tion that these spac s a tu ally form ,I

hierar hical structur . A better image here would b a nested set III

spaces, with the cascade acting to unfold spaces whi h arc -mbcddcd

into on e another. Another important qualifi cation is that each on orthe spaces that comprises thi s' nested set is classified not by its

exte nsit ies or its qualities, but by its affects, that is, by its invariants

under a transformation (or group of transformations). In othe r words,

what matters about each space is its way of being affect ed (o r not

affect ed ) by specific ope rations, themselves characte rized by their

capacity to affect (to translate, rotate, project, bend, fold, st rc t h) .

Without this caveat, we could run th e danger of circularity, since th o

exte nsive properties of the bottom level would be used to define tlu­

other levels as well.Thi s metaphor supplies us with a target for a theory of th e virtual:

we need to conce ive a cont inuum which yields , through progressive

differentiation , all the discontinuous individuals that populate the a tual

world. Unlike the metaphor, however, this virtual continuum canno t

be conceived as a single , homogen eous topological space , but rather as

a heterogeneous space made out of a population of multiplicities, each

of which is a topological space on its own . The virtual continuum

would be, as it were, a space if spaces, with each of its compone nt

spaces having the capacity of progressive differ entiation. Beside this

multiplication of spaces , we need a way of me shing th ese together into

a het erogen eous whole. Delcuzc, in fact , refers to th e virtual contin­

uum as a plane if consistency, using the term 'consiste ncy ' in a unique

I.ollid 1111'" III till 01'1'"11. dlll,lllIlI '111111 '11111111 III 11111111 .

I II ti ll' 1111 ,'11 1\1 1" 0" " " ,dllc i. 1'," dIlC' tl1I'11I , lid 1111111 till It 10 th,

virt u: I.Let me give a con, rl'll' cxa m pl« or w h.n il " wil d III' ,II' to re- turn

to the interior of a bod y in the pr ocess or being co nstituted . Biological

categories, particularly those above '1' cies, tend to be cr iatcd by

observing similarities (or techni cally, homologies) among the anatom ­

ical parts of fully formed organisms, To the exte nt that the pr ocess

which generates these organisms is ignored these static classifications

conceal the virtual. But the development of a nonlinear, non equilib­

rium approach to embryology has reveal ed a different, more dynamic

way of creating classifications. A good example is provided by a new

approach to the study of the tetrapod limb, a structure which can take

many divergent forms, ranging from the bird wing, to the singl e digit

limb in the horse, to the human hand and its opposed thumb, It is

very hard to define this structure in terms of the common properties

of all the adult forms, that is, by concentrating on homologies at the

level of the final product. But focusing instead on the embryological

processes that produce this structure allows the creation of a more

satisfactory classification. As on e author puts it, this new classificatory

approach 'sees limb homology as emerging from a common process

(asymmetric branching and segmenting), rather than as a pr ecisely

rep eated archetypal pattern '. 46

Returning to the int erior of the tetrapod limb as it is being

constituted would mean to reveal how on e and the sam e 'virtual limb'

is unfolded through different int ensive sequences, some blocking the

occurrence of particular bifurcations (those leading to the branching

out of digits, for example), some enabling a full ser ies to occur,

resulting in very different final products. This step in the method,

however, can only con stitute a beginning. The reason is that it st ill

relie s on the notion of similar ity or homology, even if this now

characterizes processes as opposed to products. A second step needs to

be add ed to explain the source of these process homologies. Or to put

this differently, once we have rev ealed th e intensi ve process behind a

product we still need to continue our ascent towards the virtual

structures that can onl y be glimpse d in that process but which explain

68

II I' ," 1.11 11It III 101< I 11'1'111 ' III I 11,11111 ,I dl I II loll III till 1.1111.1

• Illhl , '.I,ll I..,tt.. I ••11I.1 I ,h. I I I I I' II

01 .1 h. I 10 "111 011 1,,"111111111"

llu III t 1.1 k I • till II , I" •• I tI, ' "11' I" ItII It I" I. 1I11 1l\lJpl

[roru 11I.l tl lI' lIl .l l il (.11111111111••1 II I 111'" , 1II I1ILlIII ) .1IId 'll lid ot .111\

trao- or .Kl lI,l li t) 11t.11 till , '0111 pI 11l.1\ 11111.... 11 di -spite tl w i! .111'1',11"

h i ' h ly a bstracl n.u urv . III Il,Ir II ul.n , 1I01H' or th ese ('olln' p l 1'.111

I" '5IIPPOSC tndividuation, T hey lH'l'd to he u-ansformccl t o IWl"OIl H' fllih

pre-individual nonons so that they 'an for m th logical and pit sical h .ISI

for th e ge nesis of indi vidu als. When physicists o r mathemati ians spI·.lk

of 'di ffere nt ial relati on s ' , for example , they have in mind a part icul ar

math ematical object whi ch embodies th ose relation s : a J unclion. Such

an objec t ma y be viewed as a device whi ch maps one domain or

numbers (o r othe r ent ities) into another , or to use a more technolo

gical metaphor, as a device whi ch rec eives some inputs and maps them

into an output .?" As suc h, functions defin e mathematical individual i n

processes . For example , when a function is used to model a physical

syste m , its inputs (or ind ep endent variables) become th e dimensions of

state space , while its output (dependent variable) individuates a parti u­

lar state in that space. (A series of such states forms a trajectory.)

Although Deleuze do es defin e virtual ent it ies via differential relation s

(that is, as relations between changes or differences) it is clear that Ill'

cannot co nceive of th ese relations as possessing th e form of a functi on ,

since thi s would presuppose individuality. In othe r words , th e differ­

ent ial relations defining multiplicities canno t involve th e asymmetry

between dep endent and independent variables (or input and output).

If anything, th ese relations must be like ' form less functions' , wh re

inputs and outputs are not yet distingui shed, wh ere th e relation is not

a rate of change of on e quantity relative to an other, but th e rate at

whi ch two quantities change relative to each other. As Deleuze puts

, it , virtual relations must involve a purely reciprocal determination

between th eir elements, a reciprocal synthesis between pure changes

or differen ces which should not presuppose any prior individuation .so

A philosophical transformation is also need ed to lift th e virtual conte n t

from th e mathematical concept of singularity . Mu ch as virtual differ­

ent ial relations must be distinguished from individuatine Juncti ons ,

virtual singularit ies should be distinguished from indi viduated sta tes .

Attract ors, for example , ma y be defin ed as special subse ts of state

I' ll v , 111.1 III 1'.11 111 11 1.11 , III ,I I II (. 1..1\ Ill' 1I0t/1I1l' 10 .I" III. 10 111.,1

IO IlSiS ll' IH), Ih.11 is, " ill. 11.<, .lbs<, III' · 0 1,, "'111.11111011. H.t/III, 'Oil i

l:nc)" is defined as the syntlicsts 1 h CI l'r0H '1ll?/I1C\ (,/\ \/II h .'

T here arc tw o se ts of issues that mu st Ill' d iscu ssi«] b,.ro n · \\1' c. n

move beyond thi s metaph o r . Both arc issues r ·Iat ing to till" CII I it ies th . t

populate the virtual. First o f all, Chapter l ' description of multiplicit ­

ies left unresolv ed th e qu estion of th eir nature as co nc re te unive rsal

ent it ies . In other words, I used ce rtain features of math imatical models

(the vector fields of state spaces) as a source for th e noti ons that defin e

a multiplicity but 1 did not discuss how th e properties of an actual

entity, a mathematical model, can be made into th e properties of a

virtual on e. This is a task which will involve a specific philosoph ical

transiormation of the mathematical concepts involved, a m eans of

detaching th ese concepts from th eir mathematical actualization, so to

speak. In addition to this, th e first part of this discussion need s to add

to the last chapter's characterization a description of what makes

multiplicities capable of being meshed together. 1 will argue that by

extending eac h singularity into an iriflnite series, and defining th ese

series without the use of m etric or quantitative concepts, multipliciti es

can become capable of forming a heterogeneous continuum .

The second set of issues involves going beyond singularities and into

a discussion of affect s, I said before that there ar e two special cases of

intensive processes that cry out for explanation in terms of virtuality

(or at any rate, in terms of some kind of physical modality. ) The first

case was exem plified by physical systems with multiple attractors,

syste ms which for ce on us th e problem of accounting for th e mod e of

existence of th e available ye t una ctualized tendencies. The sec ond case

was flexible assembly processes whi ch lead to an open se t of potential

combinat ions. When a process leads to a clo sed set of assemblages,

thi s set may be given by exhaust ive enumeration (that is, it ma y be

defined extensionally) elim inat ing th e need to bring in a modal

explanat ion . But if th e set is divergent (as in th e case of biological

evo lut ion) th en no exhaustive enumerati on will do since there will

always be novel assemblages not included in th e list . The qu esti on now

is, if multiplicities and their singular ities co rrespond to multiple stable

state s, what corresponds to these unactualized capacities in th e vir tual

co ntinuum? Is there another virtual entity embody ing th e capacity to

7° 7 I

'l p .III', tll.11 I ••1 ltmn (Iu't, (01 1111111 t'1 of ~ 1. 1 1t ) But II \ III t III1'IJI

.1S s l.H I'" \\ OI lld IIl1ph rh,u tl U'\ .l ln' ,llh Pel ,,"," ,) dduillt 1I I(I I\ ulll.1I1" .

lien ee n el l·tl / e....s ide:' that tht.... pn .. iIHli, idu.l1 .lSIW I I (II III l u l.l n th'" c~nonly he grasped be fore the) .lC<lu in° a wd l·dl'fillt'd iell·nt it ), in a state

space full of t raject ori es, that is, when they arc onlv ' ".lgu l'h" defined, , ,by their existence and distributi on in a vec tor field . Llnlikc trajectories , a

vect or field is not composed of Individuated sta tes, but of instantaneous

values for rates of change . Individually, these instantaneous rates (o r

infinites ima ls) have , in fact , no reality, but collec t ively they do exhibit

topological inva riants (sing ulari t ies) , and it is these invariants that

should be given ontological significance . O ntologically, however , an

invari an t of a vector field is just a topolog ical accident , a point in th e

field which happens to be stationary (more tech nically, a point at which

the zero vector is attached). Dcl euze proposes that these topologica l

accidents should be given the ontological status of an event, but given

their universality or recurren t nature, these events sho uld be seen as

ideal, not actual. A similar point applies to the bifurcations which

unfo ld th e em bedded levels of a multiplicity: each one of these

sym metry-breaking transitions sho uld be see n as an ideal event , and

not, of course , as an actual pha se transiti on . As Delcuze writes:

W hat is an idea l event ? lt is a singulari ty - or rather a set of

singularit ies or of singular points charact erizing a mathematical

cu rve , a physical sta te of affairs, a psycho logical and moral person.

Singul arit ies arc turning points and poi nts of inflect ion; bo tt lenecks ,

knots, foyers, and centers; points of fusion, condensation and

boiling; points of tears and joy, sickne ss and health, hope and

anxi cty, ' sen sitiv e points' , , , [Yet , a singularity] is cssentially pre ­

individual, non -persona l, and a-con ceptual. It is quite indifferent to

the individual and the co llec t ive, the person al and the imperson al,

the particular and the gen eral - and to their opposition s. Singularityis neut ral ,Sl

To com plete the characte rizat ion of multipliciti es as entit ies we now

need to discuss the capacities for int eraction which these complex

events may be expected to exhibit. Eacb of the singularit ies defin ing a

mult iplicit y must be thought as pos sessing the capaCity to be extended or

7 2

I"."IIIIHI",J th tin '''lUlU " I J I» I. III. 'I" r III rlu \u ll",1pl Ul l' ''' ., .1 \0111 1111 111.111111 IIllullllll. II I I IJr h i 11\1 Jl

nwt .lpl.ori l .11 c1" 'llpllllll lit till p'Utl .11 11 1 dU ll ~I \ I' i" " ',hIlH,,1d l' lill il io ll, TIlt" I1 H't .1ph lll I Ihl Ot c ut rr lilt 01 ,I ph.l ( t r a nvi fio u in .111

a Clll.11 n1.lh.'rial SUI h .1' " .1"''' . \\'hl n It '.IIn i ' H IOI"d clown 10 .1 lTith .,1poi nt (about 100°C at !'oe.1 It·H·I) it will spo ntaneously dlang" natun­

and condense into a liquid, b ut as we co ntin ue to dl·ITt'.lSe tlU'

tl·mpcratllrc . the singular e vent wh ich occurred at the crit ical poinl

will be foll owed by a series '!fordm ory el't~·nts (each additional low l'ring

of tcmp('rature will have only a linear cooling effect on the !i<luid

water) , a ser ies which exte nds up to the neighbourhood of another

singulari ty (O°C, w here the nex t critica l e vent, free zing, occ ur s) . Asim ilar idea would apply to the virtual: the singularit ies dl'iining .1

multiplicity would become the origin of se ries of ordinary idea l evc-n tx

extending up to the vicinity of other singularit ies bel on ging to o thvr

multipliciti es , Unlike the metaphor, however, these series of ide-al

e vents would not form a sequence in time but rather a se ries of

coexist ing clements, (I will expand on this in the next chapte r wh en Idiscuss the form of tem po rality of th e virtual. " )

To get rid of the metaphorical content and to show in what sense

the ser ies cxtc nding from singularit ies arc nonmetric (thus capable of

forming a virtual co ntinuum) I will need to introduce one more

techn ical term, that of an irifJnite ordinal series. Unlike an infinite ser ies

of cardinal numbers (o ne , tw o , three . . .) an ordinal seri es (firs t ,

second, third . .. ) does not presuppose th e ex istence of fully indi­

viduated numerica l quant ities , To be defined an ordinal series demands

only ce rtain asymmetrical relations betw een abst ract clements, rcl a­

tions like that of beina in between two other clem ents, In other words,

it is only the order in a sequence that matters, and not the nature

(numer ical o r otherwise) of the eleme nts so orde red . Bertrand Russell ,

wh ose thought in th ese matters has influenced Deleu ze, argues that

mu ch as non metric geomctries eventually provided the foundation for

the older metric ones , so ordina l ser ies became the foundation for our

vcr)' noti on of numeri cal quantity.54 There is, in fact, a direct

relation ship between metric spaces and cardinal numbers, on the one

hand, and nonmetric spaces and ordinal numbers, on the other. Two

metric ent it ies , two lengths, for example . can be d ivided in a simple

73

\\ l\ 1/11 0 1>.1" 1111/111 I " d 111111 1111 II " \ II,. /II , ,, I.. \ .11I"(Illllll.lred silll I \\ I 1,IIl .. sl.. I>h h U/I,\I/lh"IIOU " 11.. III"'" 1/, .,1 ..1<'1111 / ,

of the two Il'nglhs, Ordi/l.l l SI'I iI's, on till' orlu -r hand , hI h.1\ ' ilion

like topologica l snaccs , whe-re WI' can rigorousl) I'stahlish th: I .1 poinl

is nearby another , but not by cxa .t1 y how mu ch (givcn that their

separa tio n may b stretched or co mpr .ssccl) .

Russell introduced the term distance (or int ensit y) to define relati on s

of proximity betw een th e elements of an ord inal se r ies . 5'; As a relation,

an ord inal di stance canno t be divided , and its lack of div isibi lity int o

identical units implies that two ordinal distances can never be exactly

compared altho ugh we can rigorously establish th at one is greater or Ie s

than another. The d!lJerence between two distan ces, in o the r words,

cannot be cancelled through numerical identity, so th e resul ts of th ese

co m parisons are always anex act ye t rigo ro us . In sho rt, o rdinal distances

ar e a nonmetric or non-quantitative co nce pt. Dcleu ze adopts these

ideas from Russell but break s with him at a crucial point : he do es not

co nce ive of th e priority whi ch th e ordinal has over th e card inal as

bein g purely logical or conceptual, but as bein g ontological. In othe r

words, Dcleu ze establishes a genetic relationship between se rial o rde r

and its defining nonmetric distances, on one hand, and numerical

quantities , on th e othe r. An ord inal se ries which is den se (that is,

where between any two clements there is always another one) would

form a one-dimensional continuum out of whi ch cardina l numbers would

emerge through a sym metry- bre aking discontinuity.56

L t' s return to th e problem of assembling virtual multipliciti es into

a plan e of consiste ncy. As I said , each on e of th e singular idea l events

defining a multiplicity need s to be imagin ed as being extende d into a

series of ordinary events which are st ill virt ual o r ideal but that , unl ike

sing ularities, alre ady possess a minimal actualization .57 Each of th e

series which emanates from a singulari ty sho uld be im agined as being

den se and defined exclusive ly by ord inal distan ces, thus constituting a

one- d imensional co ntinuum. A heterogen eou s co nt inuum co uld th en

be woven from the many se rial co ntinua spring ing from each member

of th e population of mu ltipli citi es. To ensure that multipl icit ies are

meshed together by th eir di fferen ces, Deleu ze arg ues th at the relations

amo ng th ese se ries mu st be both convergent and diverqent, In othe r

wo rds, the series m ust be mad to co me togeth er and communi cate but

74

d ., I , . , "'''/' '/1111"'/'/"" II II" 1.0 ,I.,. 1.1111" 11 , ,,

" " 1\\ 1'\ 1111 /1.1 .11\ 11 " Ill ' dll /101 1" \ 111'1'... /I " I 1111 11'\ '011'

III \\ 1 111 10 .l\ olel ( lell ll ll l\, 1I11t11111 ,.11 ,.11.,.\ .1111 1 \' '' 111 11 111 111 ",).11 11 1

/11,1) Ill' us,·d 10 ' 1 111'1.1"' ,.1 1'1 fllldol l \ 1 " /1 ' 'IU,'/I" , tl1l' lllod ,tI

(" t(' Jo ril'S <pClSsibilil y) Ill' \\ bill'S 10 n -pl.u« . ' j

At Ihis poin t .1Il import.lnt qua lificarion should I" made . Multipli. it

iI'S , houk] nol b,' co nce-ived • S possl'ssi ng tlH' '. pae it to activ , I '

interact with one • nether th rough these s .r ies, De-le uze thinks about

them as en dow d \ ith only a me re apacity to be affected , since t1 ll'y

ar c, in his words, 'impass ive en tit ies - impassiv results , ,(,() TIll

ne utrality or steri lity of mu ltip licities may be ex plained in th e fo llowing

way . Although their divergent universalit y ma kes them indepe nde nt of

any particular mechanism (the same mult ip licity may b actua liz ,d b .

several causa l mecha nisms) they do depend on the empiricalJact that orne

causal mechanism or another actua lly exists," T his is merely to say that

th ey arc not transcendent but immanen t ent it ies. But beyond thi s ,

unlike ete rna l and fixed ar chetypes whi ch have no hist orical o rigin,

Deleu ze views multipliciti es as incorporeal 1Jects c!f corporeal causes, that

is, as historical resul ts of act ual causes possessing no causal powers of

their own. O n th e othe r hand , as he writes, ' to th e exte nt that they

diffe r in nature fro m th ese causes, th ey ente r, w ith one ano ther, into

relations of quasi-ca usality . T ogether th ey ente r into a relation wi th a

quasi-cause whic h is itself incorporeal and assures th em a very special

ind ependen ce . .. ' 6 2

I said before that th e co nst ruction of a vir tual cont inuum invol ves

co nside ring not only th e role of singularit ies but also of affects . Unlike

actua l capacities, which are always capaci t ies to affec t and be affected ,

vir tual affec ts ar e sharply divided into a pure capaci ty to be affec te d

(d isplayed by impassible multipliciti es) and a pure capacity to 4fect. T his

capacity, as I hinted above, is ex hib ite d by ano ther incorporeal enti ty

which Delcu ze refers to as a 'quasi-ca use' . At thi s point , int ro duci ng

more entities may strike us as ar t ificial, o r at least as inflati onary,

enc umbe ring an already unfamiliar onto logy with further un famili ar

features. But thi s int ro d uctio n is far from being artificial . A key

co ncept in th e definition of a mult ip licity is that of 'i nva r iant', but

invari an ces are always relati ve to so me tra nsforma tion (o r group of

transformations) . In othe r words, whe never we spea k of th e invar iant

P lllP" lllt ' ol.lII "11111 ) \\t o .110 Iltnllo .II ' llillt .1I1 1l1'l l lor , 01 1'OUpo f 0 p.' r. ltllr!'\, ,.11),11.11 ' II I Pt' I IO l"ll1l llg lot.,lIOIl . 11 .111 I.Itious, IHOII'll io n'i.

fc'ldings .1Ild .1 \'.l d,·')' o f o ther t''.ln ,fo rm.lt ioll ''i on It..lt cutitv. So the

o nto log ica l co nten t o f the virtual mu st also IH' l'Uri, hl 'd wi til at le.lsl

one ope ra to r. Th e qu asi-cause is. indeed, thi s 0p"'rato r and it is dcfjncd

not by its giving rise to multipl iciti es but by its capaci ty to affect them .

'T he qua si-cause does not create , it ope ra tes", as Dclcu zc says.v'

T his new entity must be as care fully co nstructe d as multiplicit ies

were : C\ 'CI1' ste p in the const ruct ion mu st meet the co nst ra int o f

avoiding esse ntialist and typological categories , and all the conce pts

involved in its definition must be shown to be pre- indi vidual. Roughly,

the task which the quasi-causal ope rato r mu st acco mplish is to create

am ong the infinite seri es springing from each singularity ' resonances or

echoes', that is, the most ethe real or least corpo real of rclati ons.t" The

techni cal aspects of this task may be specified using conce pts from

abstract co mmunication theor-y, In co mmunication theory , the actual

occurrence of an event is said to provide information in proportion to

the probabilities of the event 's occurrence: a rare event is said to provid e

more information on being actualized than a commo n one.es These

events , each with its own probability of occ urrence , may be arranged

in a se ries. \Vh en two separate series of events are placed in

communication, in such a way that a change in probabilities in one

series affects the probability distribution of the other, we have an

iriformation channel. A telegraph, with its coupled series of events

(electrical events defining letters in Morse code at both sending and

receiving ends of the transmission line), is an example of an informa ­

tion channe l. But in the abst ract version of communication theory

nothing whatsoever is said about the physical realization of a channel ,

such as the length of the transmission line, or the type of code used .

Simil arly, no mention is mad e of information flowing through a

channel : an emission of a 'quantum ' of information is associated with

an)' change in probabilities in one series relative to the othe r series.

(Technicallv, the tw o series are 'connecte d' onlv through a conditional" "

probability matrix. )""

T his definition of an infonnation channe l appea ls to Delcuzc pre ­

cisely because of its highly abstract nat ure , presupposing no thing abo ut

det ails of physical impleme ntation,"? But ma the ma tica l models using

dlllt-lIllll,llld.lllllll II HIII'lh 1'11111 tid I I \'o l 1\ \ 1111 \ III \llllu

)( ' " lI11pl ) llllil ' 1IC1(IlIIl ~ \,hhl. Il 111 11 pl l 1lIll l\ lcllI.ll, \ ollll IlII"I 'pl

01 .HI .Ih , tr.l( t 111 101111.1111111 (l. ,lIUli I ( I I ,11111' ' Ollll1l 11 lli( .Ilioll" .1111011

~ l' r i ('~ of idt'.ll ('\(' lIt s 1I1t 1"i1 IH huthl I 1I .1Il 1IIIIIwd to hlTOI1l(' tr ill)

pro -individua l. I w ill 11ll'l1li0 I1 h" I(' (111)' the 1110"t im por-tan t rrq uin­

nu -nt , ah l10ugh Dl·leu /.l' d i..(u~.,,·~ S(·\t·r,l l more: the idea l ('\( 'n1'oo

for ming a virt ual se-ries must not bl' conn'in'd as having nUn/t'n ca /

probabilit il's of occ urrence associated with them ; th l')' mu st b,·

.uranged in se ries using only ordinal distances, and he distingui sht'd

from onc ano ther exclusively by the di fference between the singular

and the ordinary , the ra re and the common, without furt her specific-a

tion . In other words, the co upled changes in distribution s wh kh

co nstitute an informati on transfcr sho uld not be co nceived as chang"s

in conditional probabilities, but simpI)' chanaes in the distribution of th e

sina ular and the ordinary within a series .b K

I will return in the next chapte r to a more complete characte rization

o f the rela tions between th ese three elements of the virtual (multipli­

cities, qua si-causal operator, plane of consiste ncy) . But to conclude the

present chapter I would like to address a possible obj ecti on to thi s

scheme, What motivates the postulation of a qua si-causal ope rato r?

After all, we feel confide nt postulating th e existe nce of multipliciti es

to the exte nt that we can study in the laboratory certain phen omen a

(such as the se ries of flow patterns conduction- convect ion- turbulencc)

which embody such a progressively determinable entity , Moreover, we

can also check empirically that a portion of the same symme try­

breaking cascade is exhibite d by other processes (embryological pro­

cesses , for ex ample) which depend on such different causal mechanisms

that they almost demand we postul ate a mechanism -ind ep endent entity

as part of th eir explanation . Hut what evide nce do we have that there

arc int en sive processes which can spontaneously pciform iriformation

transmission operauonsi I will argue in a moment that the answer to thi s

qu estion is that th ere arc in fact such processes, and that they provide

the justificati on for thinking that such ope rations may indeed be

performed virtually. But before doing that let me add that thi s reliance

on 'evide nce ' fro m int en sive processes (more exac tly, a rel iance on

traces left b), the virtual in the inten sive) would co nstitute one of the

main characteristics differentiating a theory of thc virtual from a theor-y

77

01 r-tet u ••1 ,111.1 11I1II11It.ihll 1"'1 III I , lI l1 llk 1111 d 1'''011 11'" I' 01 I 'I 'm I '

in human thought po,tlll.lll'd Ii ) lho ,~ ' ,, 1.0 hl,lIn, III ' ''I ii " lI l1li,'s.

there wou ld lu- <.In rml'",C'hlJl '!J th l' ' /fIU.,I, 'I Ill' lOI I~ 'l· Pb. of " in ' lol l

multiplicit y, quasi-causal opt 'rat ur and pl.lIH· of ul lI:o; ish"lC)' would he .

in thi s sen se, concrete empirtco-ideal notions, not abst ract catl'gor ies. f,9

Is there any e vidence motivating the postul ation of a qu asi-cau sal

operator? There is, in fact, a relati vely new field o f nonlinear scie nce

dedicated to the study of 'emerge nt co m putation' , that is, to the stud)'

of physical processes in 'which th e int eracti on s am on g co mpo nl'nts can

exhibit the capaci ty for non-trivial informati on processing ."? Th e

mean ing of the term 'com putat ion ' in the co nte xt of natural phenom­

ena is relat ively easy to grasp if we th ink about DNA and the ce llular

machinery for its transla tion, since thi s invo lves the rel atively unprob­

lematic idea that bio logica l mechanism s have been evolved for the

purpose o f sto ring , transferring and processing information . But I want

to focus my discussion on a more gen era l se t of physical phenomena

that do not involve any specialized hardware and ye t can be said to

transmi t informati on . W e need to keep in mind that informatio n

transfer need not involve any com pute r- like mec hani sm, bu t only the

establishm ent (by whatever means) of a correlat ion between the

pr obabilit ies of occur rence of two seri es of events. As the phil osopher

Ken neth Sayre puts it, we can conceive 'as an instance of information

transmission any process in which the prob abili ty of one or more

members of an ensemble of event s or sta tes is changed as the result of

a change in pro babili ty of an eve nt or state outside the ensem ble . Thus

conceived , information transmi ssion occurs with every physica lprocess , ' 71

The simplest non -biological instance of spontaneous correlation

betw een the probabilities of events is the behavi our of materials near

phase transitions. In th is case the tw o se ries of e vents forming the

information channel are, in a way , co llapsed into on e, since the

co rrelations arc established between th e probabilities of occurre nce of

spatially separated events in on e and the same syste m .72 More exactl y,

mat eri al syste ms can be characte rized thermodynamically by certain

variab les wh ose values are not fixed (even at equilibrium) but rath er

fluctuate (w ith definite probabilities) around a given sta te. It is these

/IUCtlltlllt11U1".tIIUIIII1UII th, t .111 '''," hhl.'IIII,I .llllln 11101\ 1,('

(""...1- 11 "1"1. ,'I 1'1 1111 11"1111111 ,11 .. t1uo 1",111111 HI It. 11.111\ 1·(ltliIHOI.,thlc' ,

or put dill.... n ·ntl), thn .11' 1110 II 11It( lit II I,.tl .! It intortu.tuon

t r.m sm iscion occur". Bil l .1 ., \ I. III "I'IHO.I' II,· .1 ph.IS'· t r .m su io u .

t1 Il' S ~' lluctuati ons I)(.'gill to di"pl.l ) ,orn·l.llion" till' corre latio n lenglh

{the d istance across wh ich ,'\','n l" iulhu-ucv ".Iell o ther's pr ob ahilit icv)

incfl'asing the closer the s)'Slt:m gl'ts 10 the crit ical point. In the ricinU)

oj the bifurcation the capacity to transmit iriformation is maximized. Thi:o;

pheno me no n does not de pe nd on the ph ysical mechanism s und erl ying

the phase tra nsition: the same idea applies to a metall ic mat eri al

switching fro m the magn et ized to the unmagnetized state , o r to .1

material switching from th e gas to th e liquid state . In o ther words , 111l'

phenomen on of strong co r rela tio ns between fluctuation events in till'neighbourhood of a ph ase transit ion displa ys divergent uni versalit y." !

To scientists working in the field of emerge nt co mputa t ion thi s

univ ersality is highl y significan t. Some even think that thi s univ ersal

capacity for information transmission is accompanied by com plement­

ary capacities to store and process information associated with other

characterist ics of phenomena ncar phase transit ions.?" This has led to

the hypothesis that the specialized hardware which living organisms usc

to process information may have required that evo lut ionary forces kept

early organisms poised at the ed8e of a phase transition , or 'what amounts

to the same thing , away from any stab le altractor. Chr istopher

Langto n, a pioneer in th is field of research, puts it this way:

Living systems are perhaps best characterized as systems that

dynamically avoid att ractors . . . O nce such syste ms emerged near

a cr it ical t ransition, evolution see ms to have discov ered the natural ­

information processing capacity inherent in these near-critical

dynamics, and to have taken advantage of it to further the abi lit y of

such s}"stems to ma intain themselv es on essentially open -ended

transients . . . There is ample evidence in living ce lls to suppo rt an

intimate connection between phase transitions and life . 1\13n)' of the

processes and structures found in living ce lls are being maintain ed

at o r near phase transit ion s. Examples include the lipid membrane,

whi ch is kept in the vicinity of a so l-gel transition; th e cytoske leto n,

79

III \ 111.11 II" . 11 01 0 1 11111 I o IIII.III. " ' lit 101 11 Ih . 1'"1111 I.. 1\ " n

~IO \\ l h .lIld dl ~~ollli io ll : .l lld II" 1I.111II.11 101I .11 II I d. n .ll lII l ll l lI l (III'

ping and unzipping) of' tlu - ('lIl11 p ll·1I11·nl.lr · sl r.lIl1 l. IIf I) ,

Kauffman 's networks of' regulatory ge nes \ hich, as I discussed

above, may form th e basis of processes of difl cr int iat ion in populations

of ce lls, arc also poised systems of this typ . T hat is, in this as ', to o,

the maximum information transferring capacity is achie ved when the

network is poised at the brink of a threshold, a threshold beyond

which this capacity melts away . It is much too early in th e development

of this research programme to assess the full significance of these

claims . Some of th e early formal results (using cellular automata) have,

in fact, be en challenged .?" But the basic claim that th e vicinity of phas e

tran sitions is a specia l place wh en it comes to th e emerge nce of

spontaneous information transmi ssion (as opposed to processing or

storage) is st ill valid. And it is the existe nce of this emergent capacity

in systems which come vel)' close to but do not actualize the phase transition,

which justifies us in pos tu lating such an entity as a quasi -causal

op erator.

In conclusion I would like to add that, as un familiar and ap parently

complicated as De leuze 's scheme for the prod uct ion of a virtual

continuum may seem, he must at least be given cre dit for working out

in detail (however speculat ively) th e req uirements for th e eliminat ion

of an immutable world of transcendent ar chetypes. Giv en that essences

are typica lly po st ulat ed to explain th e existence of individua ls or of

natura l kinds, eliminating them involves giving an alt ernat ive explanation,

not ju st reducing th ese individuals and kinds to social conventions.

First , we must give a detailed description ef the int ensive processes efindi viduation which generate aetua lJorms. econd, we must sho w in detail

in what sen se the resources involved in individuation pro cesses ar e

immanent to th e world of matter and ene rgy , that is, we mu st not

sim ply deny transccndentality in gcn eral but describe concrete mechanisms

ef immanence to explain how th e virtual is produced out of th e actual. The

two halves of this chapter ar e merely a ske tc h of how these two task s

are to be performed . The third and final requirement will invol ve

discussing th e temporal dimension of Deleu zc 's ontology. This will

com plete the elimination of essences we have bcgun here, ensuring

80

Ih.11 111111111'111 III'" p''''' ''' tlWII 0\\11 111,10111 11\ ,11101 1""\I 'lIll1lg tI" '11I

horn IH'ill~ l'OlIhl,, ·d \\111i (·... 1"11.11 .1I·.Ill'I\I"·" 1111' I' .1 ''''III'I' ·IIll'1I1.11'\

task III Ih~)sl' pl'l'lt)l'IIlI·d ill Ihis Ih.ll'lI'r : dn·..I0I'"lg .1 lIll'ol'y of t iuu:

with .ll'lu.\1 .lIld virtual IMl'l.s , thl' t wo dissinul.u: h.11 v", Iillkl'd Ihrough

a properly intensive form of temporalit y. It is to this other task that I

now turn .

8 1

CIIAI'IIH ~

The Actualization ~f the Virtual in Time

T here is a conflict at the heart of physics, a confli t between two

for ms of scie ntific temporality. O n one hand, the re is the conception

of t ime that develop ed in the most prest igiou s branches of phy ics ,

classical mechanics and later the special and general theories of

relat ivity. On the o the r, the concept of time born in humble areas of

applied physics, such as engineering and physical che mistry, a conce pt

wh ich eventually becam e the time of classical thermod ynami cs. T he

main difference between these two forms of time , beside their di ffer ent

degrees of int ellectual prestige , is that whil e in classical and re lat ivist ic

physics th ere is no arro w of time, th e time of th e scie nce of heat

co ntains a fundamental as)'mmetry betw een past and future . T his

asymmetry is exe m plified by the fact that thermod ynami c syste ms have

a preferential direction always tending to approach thermal equilibrium

as their final sta te . As lon g as these two co nce ptions of time simply

coexiste d side by side , as they did for most of the nin eteenth ce ntury ,

their contradic tory relations did not cause any major foundational

co nflicts in the scientific co m m unity . But wh en the physicist Ludwig

Boltzm ann attem pte d to unite classical physics and thermodynami cs

into one un ified th eory (stat istical mechanics) , the co ntradiction

between reversibility at the microscopic level, at the level of the interac­

tion s between the molecul es that make up a gas, for ex ample, and

irreversibility at the macroscopic level, at th e level of co llective qu antities

like temperature or entro py, co uld no longer be avo ide d .'

Th e term ' revers ibility of time ' has nothing to do with the idea of

time flowing backwards, that is, with a flow of time go ing from the

future towards the past. Rather it refers to the fact that if we took a

certain process, see n as a ser ies of eve nts, and reversed their sequential

o rde r, th e relevant properties of the process would not change. 2 A

sim ple ex am ple fro m classical physics would be the moti on of an

object in a frict ionless medium , such as a ball thrown up wards in a

82

, 111111111 1011 0 lof It II .Ill II II of Illll 'I II I 11111 III II til II 111111 II

l'" 111011 :\ 111011 011 1"<l U' 1 of 1111 1" IlII ouiol 1,,0 I <lh ti lt ,1111'

if pi 0 lll ,, ·d 111 II" l' r" , ()II II" 011", 1..11,,1, 1110 I p r ll" 't' III

1I11'I'IIHlll)II .II11ics, :-;1 1( h .1 dillusiou 0' lit .i t «OI" hl' 111111 , .11', ' not 1'1" "I

ibk in this svnsc. Diffusion, for «x.uuph-, tl'lId. to homo ', 'niz,' sm.rll

difference s or lIu .tuat ions, that is, u-nds to damp them. But if w, '

revers,' the sequcnce of event ' we gl' t th ' oppo 'it e effect . a clampin I

pron' " turn ing into a proces ' of amplificat ion of fluctuations, I Math

c m atically, these ideas abo ut processes are expresse d in terms of t lu­

invariance if the laws governing a process: while the laws of .lassica]

and relat ivisti physics remain invar iant un der a time-reversal t rans

formation, the laws of thermodynamics do not."

I will arg ue in the foll owing chapte r that most of the object ive

co ntent of class ical physics can be recov ered in an onto logy without

laws. But in the traditional ontology of physics, laws ar e clea r ly th ..

single most important enti ty . Thus, given their ontological ce nt ra lity

and their invariance under time-reversal, it is not surprising th at for

mo st physicists the resolution of the conflict has tak en the form of

keeping the sym me try of the laws whil e explaining irrev rsibility

away. " On the other hand , the emerge nce of new co nce pts in the

nonlinear branches of classical physics, as well as the ex te nsion of

thermod ynam ics to situations far fro m equilibrium, has added new

mod els and new phen omena displ aying irreversible temporal behav­

iour, forcing a re-evaluation of the co nflict's resolution . lIya Prigogin e ,

a leading pract it ion er in both th ese fields, has been one of the most

vocal cri tics of the atte m pts to elim inate irreversibility. As he argu e ,

if reversing the seque nce of eve nts which makes up a process has no

effect what oever on the nature of time , then tim e becomes a mere

co ntaine r for eve nts happening in it:

Conseque nt ly, as Henri Bergson and othe rs em phasized , everything

is given in classical physics: change is nothing but a denial of

becoming and time is onl y a parameter unaffected by the trans­

formation that it describes, The image of a stable world , a world

that escapes the process of becoming , has remained until no w th e

very ideal of theoreti cal physics . . . Today we know that ewto­

nian dynami cs describes only part of our physical expe rience ...

[hut It·l.tti,it .lI lt l l jl l.t ll t UIII pIa, 1t "lllI llC'llhcl till 1.1. 1 111 1\\ 111111.111

php .il"s: ., static- uuiv erst ', ., univ ('rs,' of bt·"'.'1 \\ Itlll HII I-c j l'tI"",t/."

T he Dclcu zian ontolugy I have dc scr lhed in tlw sl' IMgl's is, on thecontra ry, one charac te r izing a uni verse o f bl-'comins without bdnH. Ormore exact ly a universe where individual beings do e xist hut only asthe outcome of bccomings, that is, of irreve rsible pr ocesses ofindividuation . This is, of course, not a coinc ide nce , since Dclcuze wasgreatl y influen ced by those philosophers (such as Henri Bergson ) whowere the harshest critics of the reversible and uncreative temporalit yof classical scien ce. Ne vert he less, th e theory of tim e created byDeleuze, a theory which I will attempt to reconstruct in thi s chapter ,goes beyond the conflict between reversibility and ir reversibility . Theproblem of time in a Deleuzian ontology needs to he approached inexactly the same terms as that of space: we need to conce ive of anon metric tim e, a temporal continuum whi ch through a sym mc try­breaking process yields the familiar, divi sible and measurable time ofe veryday exper ience. In particular, we cannot take for granted theexiste nce of a linear flow of time alread y divid ed into identical instantsbearing such clos e resemblance to one another that the flow Illay beregarded as essentially hom ogen eous.

In the first part of this chapte r I will introduce the ideas need ed tothink about extensive and int ensive time. The term 'exten sive' may beapplied to a flow of time already di vided into instants of a s h'enextension or durati on, instants whi ch may be counte d using any devicecapable of performing regul ar sequences of oscillat ions. These cyclicsequences may be maintained mechan ically, as in old clock-w orks, orthrou gh the natural osci llatio n of ato ms , as in newer versions, but inei ther case sequences ifcycles of different extens ion arc used to measurestretches of time of different sca les : seconds, minutes, hours, days.Thi s idea , on the other hand , may be extrapo lated from the measuringprocess to the very process which gives birth to tim e. I will discuss atheory by the nonlinear physicist Arthur Iberall according to which themeasurable flow of tim e of our everyday experi ence is in fact a productof a metrirat ion or a quantization of tim e int o instants. Between thefastes t vibra tions of subato mic particles and the e xtre mely long life­eyell 's o f stars and other cos mic bodi es, Iberall imagin es a nested set

01 II l l l b t lt lll pul .1111 1 ' I 1111 I I I III ,I 1"" '1 I 111111 , .11 , · .. IJl Il\ Ul lll '111m' \\Itll its nu-tru u u, 1111 1 1111 1.11 1, 1.1 IU UI I, .' .... uuu-s Ih.lt urn,­i.. n jl( Iik,' th.tt 01 ll., .. Il.11 ph\ It • Ih.lt I ~. t111ol1l"llt 'd h th ,' pron ·....t ·

.lIld tr.U1sIIJrl1lation ... ml.UrllII' \\1111111 I I.

Afu-r rl'vi('wing II H'I"'lI's tlwo r} .1111 1 showing how it rvlau -s 10Ik lt-UZl" s, I will move on to discuss some of till' Inten sive l·h.lraclt ·ristics of rime, those relating to the individuation of the stable osci llator..which co lh-ctivclv create a metric temporali ty . I will describe th« wo rk, ,o f the nonlinear biologist Arthur Winfree who pioneered a method tost ud)' the birt h and death of oscillati ons, o r more exactly, a method tolocate the sensitive point in an oscillation at which an ex te rnal shoc k ofthe right int en sity and duration can completely annihilate it. 11 <, h.lSalso inn'stigat cd the opposite phen om enon, how a stimulus of the rightint en sity and timing can givc birth to se lf-sustained osci llat ions. WIMtWinfree ' s work sho ws is that the sequences of osci llat ions at d ifferentscales making up metric tim e cannot be view ed as co mposed ofidentical instants. Rath er, each seque nce will e xhibit a distribution ofsinsular and ordinary instants bearing witness to their intensive origin.W infree 's conce pts of critical l imina , durat ion and intensity will play acrucial rol e in defining the int en sive or nonmetric aspects of time ."

Let' s begin then with th e qu estion of exte nsive tim e. A nested setof cycles or different temporal scales would see m to offer the rightform of temporalit y for the flat ontology of individuals I proposedbefore . In thi s ontology, individual organisms are component parts ofspecies, mu ch as indiv idual ce lls are parts of the organisms themselves,so that ce lls, organisms and species form a ne sted set of individuals atdifferent spatial scales . But clearl y, each of these individuals alsoope rates at a different temporal scale so that somc thing like a nest edset of cycles would he need ed to complete the picture . On the otherhand, to think of species, organisms or ce lls as possessing a sing lccharacte rist ic spatial scale is too simplified . As I said, between the ce lland the organ ism there are a variety of spatial st ruct ures (tissues ,organs, syste ms of organs) bridging the two scales. A species, in turn,is typically co mposed of severa l reproduct ive communities (de mes)inhabiting different ecosystems, each community constit ut ing an indi ­vidual ope rating at a inte rmedia te spat ial scale between that oforganism and spe cies.

11111111 1' "1/11 11'1'1.. 10 10 1111'"1 d 11111 I 1'" ,III

dl\pl.l) 1/1 1 .1 I', l/ WIII oj /111I "II 1.111 IIldl 1, 111 I 0 1 ' 111 1 III , I"" . .un p l. p O SM ' SS iut vrna] 10, b \\ hi, h " 1.,1111 h 1111 " I tI" II 1"lIlpOI •.IS 'ales (th .ir sln' l> awake cycle), but 111.1" ,llso h. " , 1II01llhh ,md vi-ar l ., ..cycles and e ven longer Diles , like lilt' 1"IIgih o r t im, 1ll',·d,'d 10 achieve

sex ual maturity (r product ive ycl .s) . They also pll"sess lllallY short ' r

cycles displa yed in d iffere nt types of rhythmic be haviour: br eath ing ,

masti cati on, locomotion . Thi s means that act ual time , rath er than bein g

a simple nesting of cycles , may include overlaps between th multi ­

plicity of temporal scales asso ciated with each le vel of ind ividuality. In

th e present conte x t , however, it wi ll be more expe dient to assume a

simple embe dding of time scales. For thi s purpose we can assign (by

convention) a particu larl y prominent time scale to each indi vidu al

level, such as th e cycle wh ich measures the maintenance if th eir identity:

th e length of time after which all (o r most) of th e individual ce lls in

an organism have been rep laced by new on es without affecting the

organism ' s identity, or the length of time after whi ch all th e indi vidual

organism s that form a species have died and new ones have taken th eir

place, th ereb y preserving th e co nt inuity of th e species' own identity.

This simplified nested set of cycles wi ll constitute my working model

of ex te ns ive or actual time . The qu estion now is whether thi s metri c

temporality can be accounted for in th e sam e way as metric space , that

is, as the product of a sym metry-bre aking event .

Nonlinear dynamics, in fact , allows a natural approach to th e

quantization or metrization of time in terms of spontaneous broken

sym metry. In particular, there is a well -studied bifurcati on , the Hop'!

bifurcation, which converts a ste ady state attract or into a periodic one ."

To see in what sense this bifurcation implies a broken time sym metry

we can use a spatial analogy, I said before that th e phase transiti on

from a gas to a crys talline state offere d an ex ample of a loss of

invariance under spat ial displacement. While th e pattern of distribution

in space for th e gas remains basically th e same under all di splacements

(if we imagin e th e gas store d in an infinite containe r) a regular

arrangement of crys tals loses so me of thi s invariance and remains

visually un changed only for a specific number of displa cements (those

matching th e length of individual crys tals, or multiples of th at length) .

Sim ilarly , the time distribut ion of a proces ' caught in a steady state

86

It, ,,1 111 d, pi I III III I 1111 II I , hlll It . I

11,,1" II,hll ' ,IIIOII 0 111 0111 III lilt11'1. " I ti ll 1>tIIl HI ( 0 1

dur.iuon) 01 till' ," I. III I h 111111 dl II r1'1111011 11I1l h.in ,,·d, ,.II

olh ' I S \\ ill en',ll, ' ,I I 'I"' II ' \ " I ,', Ii Ih,ll IS 0 11I (If I}h(/~l' \\ ilh Ih,

ori rina ] one , As I' r i O'llll ,llId i, "I.s plll it, ,1 process 'i n tilt' n - 'i nl!

of uniform stcac lv stall' . . , ignores t imc . But once in till' pl'r iod i,

re 'ime, it sudd 'n ly "dis overs" time in th phase of the periodic

motion ... We refer to th is as the breakinB if temporal symmetry. ' 10

Unl ike linear osci llato rs (those most pr valent in c1assi al and

rel ati visti c physics) , a non linear oscillator born from a Hopf bifurcati on

d ispl ays a characteristic period (and amplitude). By co ntrast, the peri od s

and amplitudes of linea r osci llators (typically modell ed as sinusoi da l

osci llat ions) are not intrinsic but dep end on co ntinge nt det ails abo ut

thei r initial co nditions. 'I Arthur Iberall uses thi s ide a of an intrinsi c tim

scale not de pe nde nt on extrinsic co nst raints as a basis for his th eory of

th e quantizat ion of time . As he puts it , suc h a th eory should be bas 'd

on

. . . th e m ath ematics of sequences if pulses urifoldinB in time as

dist inpuisbed from sustained sinusoidal oscillations. The basic idea is that

each pul se of acti on , in a nonlinear syste m embe dde d in a real

univ erse, emerges as a new cre at ion out of its past . It is th

sustained linear instability in th e local env iro nment [which cause d

th e Hopf bifurcati on in the first place) th at ensures th e rep etiti ve

quality of th e acti on. On the other hand , in th e ideali zed lossless

[i.c . co nse rv at ive ) linear isochron ou s syste m, with its characte ristic

susta ined sinusoidal oscillation, causality for th e acti on would be

yoked irrevocabl y to th e endless past and to an un ending fut ure . 12

Iberall argu es that , given that nonlinear osci llato rs have a characte r ­

istic tim e scale, ranging from th e very sho rt cales of atomic oscillators,

to th e intermediate scales of biological osc illato rs, to th e very long

Iifecycles of stars and othe r cos m ic bodi es, we may view th em as

forming a nested set of levels. This em bedde d set would ensure 'the

unfolding of time , pul se by pulse ... Time is not a uni versal unity for

all levels of organizatio n . Yet le vel s are nest ed within one another and ,

with in limi ts , are re ferable to each ot he r .' 13 In other words, ra ther

I N T E N S I V E SCIENCE AND VIRTUAL PHILOSOPHY

than assuming th at t ime ex ists as an alre ady quanti zed flow (divide d

into uniform, iden tical instants ) we should accountJor th is metric structure

using the embedde d set of differently scaled osci llat iuns . In a sense ,

each oscillation would 'synthesize a pul se of me tric time , many nest ed

sequences of th ese pul ses yielding the famili ar form of time which we

hum ans can measure using a variety of chrono me te rs. This conccpt of

time is remarkably clo se to that of Deleuzcs for whom each of th ese

pulses of acti on would constitute a synthes is of ' present tirn e ' (the

'lived present ' of atomic, biological and cosmic osci llato rs) , a synthes is

that wo uld work by contracting an immediate past and futu re into a

living present. He refers to thi s metric or ex tensive time by th e name

of 'Chronos ", and writes:

In acco rdance to Chronos, onl y the present exis ts in time. Past ,

prescnt and future are not three dimen sion s of tim e; onl y the

presen t fills tim e, whereas past and future are two dimensions

rel ative to the present in time . In other words, whatever is future

or past in relation to a certain present (a certa in ex te nsion or

durati on ) belongs to a more vast present whi ch has a gre ate r

extension or duration . There is alway s a mo re vast present which

abso rbs th e past and the future. Thus, th e rel ativity of past and

future with resp ect to the present entails a relativity of the presents

themselves in relation to each oth er . .. Chronos is an encasement , a

coW ng up tifrelati ve presents . . . 14

Let mc ex p lain in wh at sense each cycle would constitute onl y a

present, and not a past or a future . Given an osci llato r at a particul ar

scale (a biological clock , for instan ce) , what is immediate past and

future for such an enti ty would still be part of the ' lived' present of

..In osci llato r ope rating at longer tim e scales , at the level of geo log ical

or ste llar dynamics, for exa mple . Co nve rsely, the minimum living

present for a biological oscillator already includ es man y past and future

eve nts for osci llators operating at atomic and sub-ato mic scales. Metric,

ex tensive time would the n be fundame ntally cycl ical and 'composed

only of interlocking pr esents' .I 'i I mu st emphasize at th is point that,

desp ite the refe rence to a 'Hvcd pr esen t ' , this acco unt of tim e has

nothing to do with psychological tim e. It is t ru e that Dclcuz c

TH E A CT UA LIZATI ON O F T H E V I R TU A L IN T IME

some times presen ts his theory of the syn thesis of th e presen t by

contraction of immedi ate past and future , as a psychological theory,

bu t thi s is simply a matter of convenience of presentation and not

fundamental to his account. 16

The idea that it is not subject ive ex perience but the objective t ime

scale of osc illato rs th at matt ers may be further illustrated with a well ­

known example fro m rel ativity theory 1 an example which has some­

times led to confusion du e to a mistaken psychological interpretation.

The example concerns two twin brothers one of which stays on earth

whil e the twin travels in a spaceship at a speed clo se to that of light.

The rel ati vistic conclusion that th e twin on the spaceship would age

much less than the one who staye d on earth has some times been

challenged on the gro unds tha t th e differen ce between the two

situations is a matter of subjective conve ntion : while the twin in th e

spaceship may be said to be moving forwards rel ati ve to th e one on

the ea rth , it is also possible to say th at, taking the spaces hip as our

frame of reference, it is the ear th that is mo ving backwards relative to

the ship , so that the sit uation is strict ly symme tric. Given this

symme try, the shrinkage of time wo uld be an illusion, similar to th e

appare nt shrinkage in size which observ ers expe r ience as they ge tfurther away from each othe r. 17 This conclusion is, of co urse , false. As

the philosopher Han s Reichenbach argued lon g ago, the sit uation for

the two twin s is not symmetric. To see this, however, we must go

beyond the psychol ogical t ime of the observ er to the time scale of the

osci llators tif which the observer is composed , not only the biological

oscillato rs defining metabolic cycles at the cellular scale , but also the

ato mic oscillators of wh ich the cells th emselves are made. It is th ese

osci llators that ar e objectively ~cled in the case of the rapidly moving

twin, slowing down and hence retarding the aging process, but not in

the case of his eart hbound counterpart. IS

A better way of explaining in what sense we may speak of the 'lived

present ' of a particul ar osci llato r is through the relati ons between

objective time scales , on one hand , and the resulting capacit ies to

alTect and be affecte d, on the othe r. I said in Chapte r 2 that what one

individual may afford another may depen d on their relative spat ial

scales: the surface of a lake affords a wa lking medium to a small insect

hut not to a large mammal. A similar poin t applies to time scales. Each

I NTE NSIVE SCI EN CE AN D VI RTUAL PHILOSOPHY

level of temporal scale defines wh at oscillators at that level 'perceive .'

as relevant chance: certain cycle s are simply too slow for them to appear

as changing or moving relati ve to a faster level , and vice versa , certain

oscillations arc much too fast for them to even count as existing for

oscillators ope rating at longer time scales . Subject ive human time, our

psycho logically lived present with its expe rie nced duration , would

become in this interpretat ion a particular case of these objective

rel at ions of mu tual relevance between the affordances of osci llators,

Indeed , we may generalize this po int to include physical phenomena

which cannot be character ized as periodic . W hat matters for this

arg umc nt is th e existence of characteristic time scales, whether on e thinks

of these in terms of the intrinsic period of cyclic attractors or, more

gene rally, in terms of the relaxation time associated with any kind of

attractor.An example of what is meant by 'relaxat ion tim e' is the time taken

by a radio transmitter to settle into a stable pe riodic sta te after being

turned on , what engineers refer to as ' t ransient behaviour ' . These

transients occur in man y phenomena and in each case they disp lay a

charac te rist ic t ime scalc .!" In state-space terminology this can be

ex plained as follows . As I said before , all trajectories within a particular

basin of attraction will be deterministica lly drawn to the att rac to r .

O nce there they may be temporar ily dislodged from the attractor by

an ex ternal shock but as long as th e shock is not int en se enough to

expe l them from the basin , they will return to the attractor-. In thi s

case , the tim e taken for the trajectory to return to its att ractor is its

re laxa tion time . How this relates to the qu estion of affordances may

be illustrated with an example adapted from Arthur Iberall . There arc

some solid materials, refer red to generically as 'glasses' , wh ich un like

their crystalline counte rparts, do no t have a well-defined phase transi­

tion from the liquid state. In a sense, glasses are 'arres ted liquids' , that

is, they retain thc amorpho us spatia l arrangement of molecul es that a

liquid displays but flow much more sJo wly. Roughly, the distinction

bet ween the glass and liquid states can be mad e in terms of relaxation

times: these arc relatively long for glasses and rel ati vely short for

liquids.lhcrall argUl~s that whet her a particular hody dppcaTs solid or liquiJ to

d Hil'cn observer w ill depend on the rati o lu-twccn re laxat ion and

THE A CTUALIZATION OF T HE VIRTUAL I N T I M E

obse rv ational tim e scales , in the sense that for sufficie ntly long

obse rva tional tim es th e glass will appear t o the observer as a flowing

liquid. "? Thc inclusion of the observ er in thi s description may give the

wrong impression that some thing psycho logical is being discussed , but

this impression disso lves once we realize that ' observation' is sim ply

on e particular instance of ' inte raction ' , In other words, what counts

here is th e ratio ef relaxation tim e to int eraction time, a ratio that can be

defined witho ut inclu ding a human observer in the picture. In particu­

lar , we can let the liquid and glass interact with each other and spe ak

of how solid th e glass 'a ppears' to the liqui d , and vice versa. The glass,

given its long relaxation time scale relative to the scale of interaction

with the liquid, will be have as a solid , affording the liqu id, for instance,

an obstacl e to its flow, or affording it a channel in which to flow . The

flowing liquid , in turn, wi ll afford eros ion to the glass. In short , what

capacit ies the glass has to affect and be affect ed by the liqu id will

depend on their rela tive time scales , the characteristic durations of

their relaxation to equilihr ium .

T he objectiv e relativity of afTordance s with respect to te mp oral

scales mak es them the ideal candidate to define the ' lived present ' of a

particular indi vidual, that is, what this individual 'perceives' within its

o wn time scale as th e rel evant capaci t ies of the other individuals

interacting with it. It is in th is sen se that Deleu zc allirms , quite

literally , that even ino rganic things 'have a lived ex perien ce ' . 2 1 To

summarize th e ma in conclusion of this sec tion: materia l and ene rge tic

processes give time its metric and measurable form by their possession

of a characte rist ic time scale , specified either through relaxation times,

or as I will do in the rest of thi s section, through the intrinsic per iod

of nonlinear oscillations. To phrase thi s conclusion in Deleu zc' s words,

at anyone of these embedded time scales the present is 'cyclical ,

measures the movement of bodies and depends on the matter that

lim its it and fills it out ' .22

Having ske tched ho w exte nsive time should he conce ived in a

Dcleuzian ontology I would like to move on to discuss the ideas

needed to think abo ut the int ensive aspects o f temporality , In th is book

qu estions of intensity have been mo st ly related to the problem of the

ge nes is of individuals. In the case of the non linear osci llato rs wh ich

<Iuanti i",e tim e Arthur Winfn 'c 's uxpc rinu-ntal and theoretical work

I N T E N S I V E SCIEN C E AND V IFIT UAL PH I L OS OPHY

gives us, as I said, the means to explore the intensive propertiesinvolved in the birth and death of osc illat ions. Winfree ' s best -known

work deals with populations of uiological osc illato rs (the internal clocks

of fruit flies or mosquitoes , for instance) which he isolates from their

surroundings to perform controlled ex periments on their reaction toshocks of different timing, duration and intensity. Winfree ' s main

result is, basically, that a stnquiat, critical st imulus applied at a smqu lar,

sensit ive moment has a destructive effect on the sleep- awake cycle of

organisms , giving a popul ation of mosquitoes, for example, pennanentinsornnia .:" The stim ulus itself need s to be of the right duration and

intensity in order to act as an annihilating shock , but it neverthel ess

acts not as a direct cause of the death of an osci llation but merely as a

trigger. What effect the shock will have will dep end on the internal

in tens ive structure of the osci llator itself.For exam ple, if the osci llation is go verned by a period ic attractor

which contains within it a stable steady-state attractor (what Winfreecalls a ' black hole ' ) then the crit ical st imulus will co mpletely annihilate

the osc illat ion .H On th e other hand, th e result of the st im ulus may be

not steady-state, atemporal behaviour but arrhythmic, ambiguous

temp oral behaviour, if the periodic attractor is associated with a set ofsta tes (called a 'phaseless set') bounded uy a phase singu!arity ." In

addition to these results related to the extinction of oscillations,

Winfree has studied the complementary problem of wh at gives rise to

these osci llations in the first plac e. Basically, he has found tbat bychanging the expe r-imental conditions he can transform an annihilating

stimulus into a conj urina stimulus, that is, a critical shock that can create

osci llations, the phase singularity in this case becoming an organizingcent re for temp oral str uc turcs .?" Winfree' s results display many of the

traits that we have found characterize intensive processes, in particular,

mechani sm-independent tendencies. The tenden cy to be annihilated by acritical shock, for example , is not limit ed to the temporal behaviour of

animals with nervous systems but is also exhibited by thc behaviour of

much simpler oscillators , ranging from )'east cells to inorganic chemical

rca crions .?"

O ther aspects of W infree ' s work on osci llators illustrate a difi'erent

fC.' .ltUfl· of the Intensive: the ability o f nonlinear osci llators tu synchromre

or entrain one anothcrs tem poral be haviour, I said in Chapter 2 that

T HE A CT UALIZAT I ON OF TH E V IFIT UAL I N TI M E

the definition of 'intensive ' may be expanded to include capacit ies, and

in particular, the capacity of an individual to form assemblages withindividuals very different from itself, Unlike the quantitative or

qualitative properties of an individual, which as emergent properties

refer to an individual' s inside (that is, to the interactions among the

lower scale individuals which co mpose it) , an intensive property in theex panded sense refers to 'an adequate outside with which to assemble in

heterogeneity' , as Deleu ze puts it. 211 The capacity of nonlinear osci lla­tors to entrain one another's temporal behaviour is a particularly

striking example of this other aspect of the intensive, allowingbiological osci llators, for instance , to synchronize their sleep-awake

cycles with cycles outside themselves, sucb as the day-night cycle of the

planet. Entrainment is another phenom enon which Winfree has studied

in det ail, partly because of the need to prevent it from happ ening

whil e studying the effects of annihilating st imula. O nly if mosquito or

fruit fly populations are isolated from the effects of the Earth 's rotati on

will their int ernal clocks di splay their intrinsic duration or period . This

period varies for different animals , from twenty-three hours for

mo squitoes to twenty-five for humans, explaining the name 'circadian'

given to these clo cks, a term meaning 'nearly a day's length' ,When not in isolat ion, circadian docks becom e entrained with

the planet's own rotational period of tw enty-four hours, a synchro ­

nizing capacity with obvious adaptive value since it allows a flexiblecoord ination of internal rhythms and seasonally cbanging day lengths.

Thanks to entrainment, biological oscillators can mesh, or form

a het erogen eous assemblage , with the daily and seasonal rh ythms

of their ex ternal environment. Entrainment displays the typicalcharacteristics of an intensive process, st im ulus- independence and

mechanism- independence . Synchronization o f temporal behaviour is

t rigge red rather than caused uy relati vely weak co upling signals

which may be optical, chemical or mechanical. The exact nature of

the signals serving as stimuli is not as important as their intensity:

these signals must be maintained at a critical threshold of strengthelse the synchronization will abruptly stop.?" A similar indifference is

displayed towards the mechanisms implementing osci llating behaviour:

e ntrainmen t occurs in populations of purdy physical oscillators. suchas the vihrating <.'omporwnts of lase r light. in inorganic chemical

INTENS IVE SCIENCE AND V IRTUA L PH ILOS OPHY

reacti ons, and in a large varie ty o f hiological osci llators , including themen strual cyc les of humans. ?"

The theory of metric time in terms of a nested set of cycles which I

sketched above involves a kind of temporality whi ch is inherent ly

sequential , each individual life being a linear sequence of osci llat ions.

T he first part of W infr-ee ' s work sho ws th at th ese linear seque nces arc

not , in fact , homogen eous series of identical mom ents or instants.

There are, in eac h series. a distribution of singula r and ordinary moments

and this distribution implies that there exist relati ons of critica l tim ing

be tween the sensitive points of osci llato rs and exte rnal shoc ks . The

second part of his work displa ys a different aspect of int en sive time.

an aspect which tak es us beyond sequential and int o parallel temporal

structu res. The phenomen on of ent rainme nt allows many indep endent

sequences of oscillatio ns to act in unison, to become in e ffect a sing le

para llel process. The most dramatic and well -studied example of thi s

phenom enon is perhaps the slime mould Dicty ostetium. The lifecycle of

this crea ture involves a phase where the organism s act as individual

amoebae , the behaviour of each constituting an independent sequential

proCl~ss . At a cr it ical low point of availability of nutrients, however,

we witness the spontaneous aggregation of an entire population of

these amoeb ae into a single field of parallel osc illato rs , eventually

leadin g to their fusing together into a single organi sm with differenti­

an-d parts. As on e scientist has remarked, witnessing this phenomenon

'one may reall y be watching a replay of the basic kind s of events

responsible for the appe arance of the first mult icellular organism s. ' JI

In the next sect ion of thi s chapte r I would like to extend these ideas

about cri tica l duration and timing as well as parallelism to more

co mplex processes of individuation than those exe mplified b), the slime

mo uld . But let me first summar ize what I have said about the birth of

metric or ex te nsive tim e. I gave before an exam ple of how eac h of the

embedde d cycles making up thi s form of temporality may be said to

I", bo rn th rough a symmetry-brea king event (a Hopf bifurcati on ). Thi s

was , howe ver, a purely forma l example leaving out the details of

prol'l'ss wh ich constit ute the subs tance of the intensive. Addi ng to th is

forma l model W infree ' s experime ntal results mitigatc but do no t

l'ompll'tt' ly so lve tilt' problem , \ Vc can compare this simplified model

of tilt' birth of me tric tlrnc to the me taphor I lISt,1! in the J.lst chapter

THE ACTUALIZATION OF T H E V IR TUAL IN TIME

to illustrate the birth of metric space. T he neat picture of a symme try­

breaking cascade transforming a topological space into a metric one

had to be co mprehensively reworked to make it physically plau sible :

the non metric aspects of int ensive processes t urned out to be subtle

and co mplex, as did the metric aspects of the ex te nsive products;

mo reover the least met ric level of the embedded set had to be,replaced with a virtual continu um whose description required yet

another set of co mplex concepts.A sim ilar complcxification is now in order to put some Ilesh on the

rath er skeletal formal model of a Hopf bifurcation . I will return to my

two e xamples of individu at ion processes (the genes is of organisms and

species) not only to add detail to W infree 's ideas abo ut cr it ical t iming

and parall elism , but more importantly, to show how int ensive tempo­

rality may be crucia l to the eme rge nce of novelty in biological evo lution .

The process of embryogenes is, for instance. involv es the parallel

development of many simulta neous seque nce s of events , the relations

between these sequences det ermined in part by the relative duration

of these processes with respect to one another, and by the relative

timing of the on set or cessation of onc process relative to another. At

this scale, as I will argue in a moment, the eme rgence of brand new

design s may come about through relative accelerations in these parallel

processes. A different source of novelty may be illustrated by moving

up in scale to a discussion of ecos ystems, which as individuation

enviro nme nts may be said to play rel ative to spe cies the role which an

egg or a womb play for individual organism s. In thi s other case too ,

relative accelerations in the tempo of evolution may lead to radi cal

innovations. Unlike the temporality of the embryo, however, where

the term ' inte nsive' has its original mean ing, ecosyste ms will involve

the e xpanded meaning, that is, the source of accelerati on and inn ova ­

tion in thi s case is the assemblage of het erogen eous species in the

process known as symbiosis.Let me begin with the temporal aspects of the genes is of organisms .

In the last chapter I emphasized the role of rates of change and co uplings

be tween sepa rate rates as key to understanding embryologica l devel­

opml·nt. Altho ugh a rate of change docs not need to involve time (we

may he interested in the rate of change or pressurt' relat ive to oceanic

depth or at mos pheric ht'ight . for example}, tinu: docs en te r into the

I NTEN SIV E SC IENCE A ND V IRTUAL P H ILOSO P HY

formulation of many important rates. These rates of change display th e

same int erplay between characte r istic time scale and alTordances which

I mentioned before in connec tion to rel axation times (the latter ar e , in

fact , nothing but rates of appro ach to equilibr ium) . A process may

change too slo wly or too fast in relation to another pr ocess, the

relationship between th eir temporal scales det ermining in part their

respective capacities to affect one another. Even when two processes

ope rate at similar scales, the result of their interaction may dep end on

their coupled rat es of change. For example , the graphic patterns which

man y organisms display in their skins (e .g. zebra stripes or leopard

spots) may be explained as the result of the var iable conce ntration of

che mical substances, a conce ntration which depends on the rat es at

which substances react with each other relative to the rates at which

the products of such reaction diffu se through an embryo's sur faces.

Different patterns may be achieved by contro lling th ese relative rates,

a task performed by gen es and gen e products (enzymes).

As the physicist Howard Pattee has conv inc ingly argued , in the

developing organism we find an int erplay between rate-dependent

phen om ena (like che mical reaction and diffu sion effects) and rate­

independent phen om ena . While the formation of sel f-o rganize d patterns

of chemical concentration do es dep end on th e relati ve rates of diffu sion

and reacti on , the information contained in genes does not depend on

the rate at whi ch it is decoded . On the other hand , thi s rat e­

indep endent information, once translat ed into enzymes , act s by control­

line rates. t? Enzymes are catalysts , and the latter are defined precisely

as chemical elements capable of accelerating or decelerating a chemical

r 'actio n. The fact that embryo logical development is all about rat es of

change which are coupled or un coupled through th e action of ge nes

and gene pr oducts, sugges ts that th e processes underlying embryologi­

cal developmen t may be view ed as a kind of 'compute r program ' . But

this met aphor sho uld be used care fully because th ere are different

kind s of compute r programs pr esupposing diffe rent forms if time, some

lIsing seq ue ntial or serial tim e , others departing sharp ly from these

linear forms of temporal ity. As Stuart Kauffman puts it :

It is a major initial point to realize that, in whatever sense the

gen omic regulatory system constitutes something like a develop-

T HE A C TU ALI Z A T ION OF THE V I R TU AL IN T I M E

ment al program , it is almost ce rtainly not like a serial-processing

algorithm . In a ge nomic syste m, each ge ne responds to th e various

'products of tho se ge nes whose pr oducts regulate its acti vity. All the

different genes in the network may respond at the same time to the

output of th ose genes which regul ate them . In other words , the

ge nes act in parallel. The network, in so far as it is like a compute r

program at all , is like a parallel-processinp network. In such net works,

it is necessary to conside r th e simulta neous activity of all th e genes at

each moment as well as the temporal proeression if th eir activity

patterns. Such progression s constitute the integrated behaviors of th e

parallel -processing genomic regulatory syste m . 3.3

Thinking about th e temporality involv ed in individuation processes

as embodying the parallel op eration of many different sequential

processes throws new light on the question of the emergence of

novelty. If embryological processes followed a st rict ly sequential order,

that is, if a unique linear seguence of eve nts defined the production of

an organi sm , then any nov el structures would be const rained to be

add ed at the end if the sequence (in a process called 'terminal addition' ) .

On th e contrary , if embryo nic development occurs in parallel, if

bundles of relatively inde pe nde nt pr ocesses occ ur simultaneously, th en

nell' desiens may arise ]rom disenBaBinB bundles, or more pr ecisely, from

alt ering the duration of one process relative to another, or th e relative

timing of the star t or end of a process. Thi s evo lutionary design

strategy is known as heterochrony, of whi ch the most exte nsively studied

case is the pro cess called 'n eoten y'. 34

In neot eny the rate of sex ual maturation is disengaged from the rate

of development of the rest of th e body, ind eed, accelerated rel ative to

somatic development, resulting in an adult form which is a kind of

'g ro w n-up lar va ' . 35 Neoteny illustrates that nov elty need not be the

effect of terminal additi on of new features , but on th e contrary, that it

can be the resul t of a loss of certa in old features. Humans, for example ,

may be regard ed as ju venalized chimpanzees, that is, primates from

wh ich a developmental stage (adulthoo d) has been eliminated . More

ge ne ra lly, the loss of a feature mad e possible by the uncoupling of

rates o f change may pr ovide an esca pe rout ' from morphologies that

have becom e too rigid and specia lized allowing organisms to ex plore

IN T E N S I V E SC IENCE AND V IRTUAL PHIL OS OP HY

new developmenta l pathways.J" T o Del eu ze thi s aspcct of indiv iduation

pro ces, es (an aspect which must be added to populati on thinking to

co mplete the Darwinian revol ution) is highly significant because it

el iminates the idea that e volutionary processes possess an inherent

dr ive towards an increase in co mplexity, an idea which reintroduces

teleology into Darwinism . As he writes, "re lat ive progress . , , can

occ ur by formal and quantitative simplification rather th an b)' compli­

cation, by a loss of co mpo nents and syn theses rath er than by acquisit ion

. . ' It is through populations that one is formed, assumes forms, and

through loss that one progresses and picks up speed .' 17

The flexibi lity with whi ch parallel processes endo w e mbryo logical

dev elopme nt may be said to come to an end once the final organism

acquires a more or less fixed anatomy, That is, at this poi nt the

inten sive becomes hidden under th e exte nsive and qualitative . Yet ,

anato mical feat ures arc never fully fixed even in adulthood . Many parts

of the body retain their capacity to sel f-re pair , and in some animals

eve n thc capacity for complete regeneration . Additionally, even if

relative to the flexibili ty of an embryo the anatomical propert ies of a

finished organism are ind eed rigid , its behavioural properties may no t

he, parti cul arly if such an organism is endowed with flexibl e skills

beside its hard -wi red reflex es and behaviou ral routines . At any rate,

even the mos t anato mically and behaviou rally rigid individual , eve n the

1110st extensive of finished product s, is im med iately caught up in larger­

scale indiv iduat ion processes where it becomes part of othe r int en sities,

such as the inten sive properties characte rizing ec osystems.

O ne of the most importan t facto rs conside red in studies of ecos)'s­

terns is changes in the population density of each of the interact ing

species. Populat ion den sity, like temperatu re or pressure , is an inten s­

ivc property th at canno t be divided in e xte nsion. Hut like other

intensit ies it ma y be divid ed by phase transiti ons. In particular, there

are critical th resholds at which the sta te of a populat ion changes in

kind, such as min imal values of den sity (so me times called 'nucleation

thresholds") below whi ch a populat ion goes extinct. '· Similarly, mu ch

as a populat ion of molecul es will spo nta neously te nd to relax, afte r a

certain characte r-istic timc , to an eq uilibrium valu e for its temperature,

so populat ion d"'nsity will exhibit a characte rist ic re laxat ion tim e afte r

he ing suhjected to an envi ronme ntal shock, sudl as a part icularly harsh

THE ACTUALIZATION O F T H E V IR TUAL I N TIME

win ter. The ecologis t Stuart Pimm argues tha t this rate of return to

equilibrium characterizes a population ' s resilience to shocks: sho rt rates

of return to equilibrium signal a robust population, that is, one capable

of recovering rapidly afte r a shock, wh ile lon g rel axation times betray

poor resilien ce and hence , vu lnerability to ex tinction . Given that

extinct ion mean s the death of a species as an indiv idual , and that the

ex tinct ion of one species may mean the rapid birth of others to occupy

the vacant niche , these int en sive properties may be said to partly

characte rize processes of individuat ion at thi s scale .

Ecosys te ms involve processes opera ting at several simultaneous time

sca les. O ne factor affect ing population den sity is int ernal to a species,

that is, det ermined by the birth and death rates of a population . This

fact or disp lays a relatively short time scale of re turn to equilibr ium.

When the densit ies of several populations are co upled in parallel, as

,...·hen a population of plant s, hervibo res and carnivores is coupled into

a food chain . relaxation times become longer: when the density of a

predator popul ation affects tha t of its prey, and thi s, in tum, the

dens ity of the plants it consumes , re-equ ilibrat ion afte r a shock may be

delayed until the cascading effects stop. This lon ger t ime scale of

recovery is determined by the degree of connectivity whi ch one species

has to other species, that is, by the length of thefood chain to whi ch the

species belongs. Finally, there are even lon ger-term pr ocesses det er­

mined by non-biological factors such as the rate of availability of

mineral nutrient s in an ecosystem during reco vcry fro m a catas trophe,

such as the effects of the onset or cessation of an Ice Age . 39 Given th e

importance of resilien ce as protection against extinction , and given the

key ro le which the degree of connect ivity plays at intermed iate time

sca les, an ecosyste m may also be conside red a parallel-processing

net work in which changing relationships offi cness (between predators

and prey, or hosts and parasites) propagate at different rates th roughout

the network influencing both the eme rge nce of new , and the disap ­pearance of old, individual specics .t"

Relations betw een population den sities, however, give us a ni)' a

rough ide a o f the co mplex temporal structu re of an ecosyste m , Co n­

sidered as a network in which the fl esh (o r hiomass) of plant s and

anima ls circulate, an CCOs)'stc m will display a varict )' of temporal

rhythms d larackrizing eac h of its alime ntary co uplings, these rh yth ms,

I N T E NS I V E S C IEN CE AN D VIR TU AL PHI L O S OP H Y

in turn, associated with the spectrum of osci llatory behaviour at

d ilTerent scales ex hibited by every organism . But considered as an

individuation environment there is a particular rhythm which must be

sing led out : the evolutionary rates of each of the co upled species.

Evolutionary rates used to be thought as basically uniform, characte r ­

ized by a linear and gradual accumulation of gene t ically code d beneficial

traits . This rat e of accumulation would vary from species to species,

du e to their different generation times, but within each species it was

supposed to be basically un iform. Today we know that thi s picture is

incomplet e given that for a variet)' of reasons there occur accelerations

and decelerations in these evolutionary rates. (The very large time

scales invo lved in evo lution means, ho wever, that even an accelerated

rate will st ill characterize a very long process, one between 5000 and

50,000 years, for e xam ple!')

As in the case of em bryological developmen t where loss of a

part icu lar process or com ponent may lead to the emergence of novel

features, in an ecosystem losses may also lead to accel eratio ns in

evo lutionary rates and rapid spread of novel designs. An extinct ion

event , for exam ple, may eliminate a set of spec ies and vacate their

niches, leadin g in t urn to an ex plosion of new design s by ot her spe cies

(an ada ptive radiation) to occupy the vacant positions in the food

chaln .:" A di ffe rent exam ple of events leading to accelerated evolut ion

and rap id emergence of new capacit ies is 9'mbiosis. Altho ugh tradit ion­

ally the term 'sy mbiotic relat ion ship ' refers to a partic ular kind of

alimentary co upling (one in which both partners ben efit from the

assoc iatio n) the difficulty in defining and estahlishing mutually beneficial

re lat ions has led to a new view of its nature and function. Today

symbios is is defined as an assemhlage of heterogeneous species which

persistsJor lona periods, relativ e to the generation times of the int eracti ng

organisms, and wh ich typically lead s to the emeraence if norel metabolic

cdpelbili t ies in at least one of the partners. r" The em phasis on long

dura tion is due to the need for coel'olulion between the partners. both

of wh ich need to have e xerted selection pressures on eac h othe r biasing

till' long-term accumulat ion of the ir genes and bodily traits . (Given

that some membe rs of an ecosyste m rna)' have arrived through recent

invasions or co lonizat ions. not all interact ing co uples in a food chain

nt·t·e! to han ' ('(u.·voln·d.)

THE A C T UALI Z AT I ON OF T H E V IRTUAL IN T I M E

Symbiosis as a source of evolutionary innovation oc cu rs at many

level s of scale . At the ce llular leve l, for exam ple, two of the key

capacit ies at the basis o f food chains may have eme rge d through an

assembly of heterogeneities. Phot osynthesis , the ability to 'bite ' int o

solar rad iation to produce che mical ene rgy sto red in sugars , and

respiration, the abi lity to tap int o a reservoir of oxygen as fuel to burn

these sugars, are both thought to have emerge d through cellular level

symbioses with micro-organisms.'" At larger scales, examples include

the auton omous com munities of micr o.organi sms which line the guts

of hcrviborcs allowing th em to digest ce llulose , the bact eria that allow

legumes to fix nitrogen , and the fungi wh ich permit man y plant roots

to ge t access to phosph orous . In all these cases, novel capabilit ies to

e xplo it otherwise unavailable resources have co me about no t through

a slow and gradual accumulation of favourable mutations but through

an accelerated process: mes hing the capabilit ies of tw o or more

het erogeneous populations of organ isms followed by th e subseq uent

coe vo lution of the partners.H

W hen discussing inten sive processes Dclcuze usually di vides the

subject into singu larit ies and affects , but some times he uses an

alte rnativ e and equivalent formulati on in terms of spee ds and affects :

speeds if becoming and capacities to become.": The many parallel processes

which define a developing em bryo, fo r example , are defined by their

rel ativ e speeds , and by the accelerati ons and decelerat ion s th ese may

und ergo resulting in the product ion of no vel forms . In Delcuzian

terms, such an indiv iduatio n environm ent wou ld be characterized in

part by relation s of 's peed and slowness, rest and movemen t , tardiness

and rapidity ' .4 7 As I said , changes in these relative spee ds may be used

as an evolutionary strategy (he te rochro ny) allowing an organism an

escape route from an over-specia lized design . Eco s)'stems also display

relations of relative speed between para llel processes but in this case

the emerge nce of novelty depends more on the capacity to Join in with

a heterogen eous partner in a com mon cocvolutionarv line of flight.

o r as Delcuzc puts it . on ' a co mpos ition of speeds and affects involv­

ing entire ly different individuals, a sym biosis ' .'HI To phrase this in

Prigogin e ' s term s of being and h(.~coming : whereas emlJryogenesis is

a process through wh ich a yet unform ed ind ivid ual becomes what it is,

acqu iring a well -defined inside (the intr insic propt·rti t·s defining its

I N T E N SI V E S C IEN CE A N D VIRTUAL PHILOSOPHY

being), symbiosis represents a process through which a fully formed

bein g may cease to be what it is to become somethinq else, in association

with something heterogen eous on th e outside .

This description of more com ple x forms of inten sive temporality

was intended as a comple me nt to th e simpler formulation in terms of

th e ind ividuation of oscillations. Questions of cr itical timing and

duration, as w ell as of parallelism , are st ill prominent but have acquired

a subtle r form . Similarly, the problem of th e metrization or quantiza­

tion of time, which also had a sim ple formulation in terms of a nest ed

set of seque nces of oscillation s, need s now to lose so me of that

simplicity. In particular, for th e sake of ease of presentation I have

arti ficially se parate d issues related to time and spa ce, but in reality we

arc always confronted with complex spatio- ternporal ph enomena. Even

the sim ple oscillators st udied by Winfree ar e nonlinear spatia-te m poral

osc illators where th e spat ial and temporal aspects interact. For thi s

reason, the qu estion of th e emerge nce of metric o r exte nsive prop erties

sho uld be treat ed as a sing le process in which a continuous virtual

spacetime progressively differentiates itself into actual discontinuous

spatio- tc mporal st r uc tures operating at different scales. In other words,

the emerge nce of a metric space t ime involves the ent ire flat ontology

of ind ividuals, each nested level of scale co nt ribut ing to th e metrization

of space and time simultaneou sly.

I would like to conclude thi s chapte r with a more detailed discussion

of this virtual space time . In Chapter 2 I described th e elements whi ch ,

accord ing to Deleuze, constitute th e content of a nonmetric contin­

uum : changing populations of virtual multipliciti es (co nce ived as

complex ideal eve nts) and a quasi-causal op erator whi ch asse m bles thi s

heterogeneous population into a plane of consiste ncy. This particular

breakdow n of th e co nte nts o f th e vir tual is, of co urse, speculative, and

as su h, it may very well turn out to be wrong. There is, as I said , an

.m piricism of th e virtual, even if it does not (and should not) resemble

the e mpirical study of th e act ual. But whil e th e specific solutio n which

Dclc uzc prop ose may turn out to be inadequate , he sho uld ge t cre di t

for having aclcquat ' Iy posed th e problem. In o rder to ge t rid of essent ialist

and t Ipo lug ical thinking it is not eno ugh to den oun e th transcendent

and aflirm the imm: ncnt , R ' pia ing Plato ' s transc mdc nt ss inc ' S with

risto l1l" · im ma ne nt natural sta tes, for c ample. gets us out of

THE ACTUALIZ ATION OF THE VIRTUAL IN TIME

esse n tialism but not of typ ological thought. One mu st also give

mechanisms c1 immanence (however speculat ive) to explain th e ex iste nce,

relat ive auto no my and ge ne t ic power of th e virtual."? Let me first

sum ma rize wh at I said before abo ut the quasi-causal ope rator, the

mann er in whi ch it meshes multiplicities by th eir differen ces, since thi s

co nstitutes th e first immanen ce mechanism. I will th en describe the

second task whi ch Deleuze ascribes to thi s virtual ent ity : to Benerate the

multipliciti es by ex tracting th em from actual inten sive processes.

T ogether, th ese two tasks ensure that th e resulting virtual space t ime

does not have th e form of a transcendent space filled with tim elessesse nces .

I described th e first task of th e quasi-causal ope rato r as that of giving

vir tual multipliciti es a minimum of actualization by prolonging th eir

sing ularities into se ries of o rd inary ideal events , and establishing

relations of co nve rgence and divergen ce between th ese se r ies . I said

that to specify how th ese immaterial linkaBes between se ries arc

established Dcleu ze borrows from th e mo st abst ract vers ion of co m ­

munication theory th e concept of transmission of information in a

channel (a sign / signal syste m , in his terms,). An information channe l

(signal) exists when ever two heterogen eous se ries of events ar e

co upled by chang ing probability distributions. No referen ce need s to

be mad e to eithe r a causal mechanism or to anything actually flowing

in th e channe l. Quanta of information (signs) ma y be said to pass from

one ser ies to another wh en ever a change in the probability distribution

in on e seri es is correlat d to a change in th e other on e . Such a linkage

of se ries of events through signs occurs spontane ously in some intensive

syste ms , suc h as syste ms poised at the edBe of a phase transition. Even

whe n suc h poi sed syste ms are inorganic, that is, even in the absen ce of

specialized biological hardware , th ey can cohe re n tly transmit informa­

tion as long as th ey manage to remain in the vicinity of th e crit ical

event without actually crossing th e threshold.

The em bryo logical and ecolog ical indi viduation processes I have just

discussed , at least when modelled as parallel -p rocessing networks,

dis play th is emergent abi lity in th e neighbourhood of a critica l point c1connectiVity. Stuart Kau ffman argues, for example, th at the many food

.hains that form an ecosyste m mu st not exceed a ce rta in riti al length

(typica lly o f four sped": a plant , a hcrvibore , a pred ator , and a

I NTENSIVE SCIENCE A N D VIRTUAL PHILOSO P HY

predator of th e predator) for th e parallel network to display complex

behaviour. 50 This sensitive valu e ma y be achieved via the coevo lut ion

of th e members of a food chain . Similarly, th e parallel network form ed

by gen es and gen e products whi ch const itutes th e informational

backbone of a developing embryo also need s to keep its degree of

co nnect ivity near a crit ical value . Kauffman explicit ly compares thi s

crit ical value (not too low but not too high ) to th e singular zone of

inten sity exist ing at th e phase transiti on between a gas and a so lid (that

is between states with too little and to o mu ch orde r , respecti vely) and

argues that embry os and ecosys te ms may need to be poised at the edge

in orde r to maximize th eir emergent co mputational capacities.F'

Unlike actual poi sed syste ms , however, where information trans­

m ission takes the form of co rrelations between the numeri cal probabi li­

ties of occurance of two ser ies of eve nts , virtual ser ies must exclusively

involve changing distributions of th e singular and the ordinary, given

th at vir tual se ries and th e space th ey form cannot presuppose any

me tric o r quantitative notion without begging th e qu esti on. In particu­

lar , vir tual se r ies mu st be conce ived as den se ordinal ser ies whi ch, as I

argued, arc logi cally and ge ne t ically prior to alr eady quantized numer­

ical se ries and can be regarded as on e-dimensional nonmetric cont inua .

In addit ion , th e requirement of not presupposing any notion to whi ch

th e virtual is suppose d to give rise implies that th e statist ical distribu­

tion s invo lved in an information channe l canno t be conceived as fixed

(or 'sede ntary' ) like th e famous Gaussian or bell -shaped distributions

characterizing th e stat istical properties in many actual population s.

Unlike these familiar equilibrium distribution s whi ch refer to alread y

individuate d populations occupying a metric space , Deleu ze designs th e

quasi -ca usa l operator to produce mobile and ever-changing (' no mad ')

dis tribut ions in th e virtual series, establish ing both conve rge nt and

div 'rgent relation s between them. 52

In sho rt, th e first task of th e quasi-cau sal op erator is what Deleuze

calls a condensation if singularit ies, a process invo lving th e continuo us

creatio n of co mm unicat ions between th e se ries em anat ing from every

singularity, linking th em together through non-ph ysical resonances,

while simu ltaneously rami fying or di ffere nt iat ing th e se r ies, ensuring

they are link ed together only by th eir differen ces .5 I T he mesh of on ' ­

dimens ional co nt inua that results would co nst itute th e spatia l aspl' t o f

THE A C T U A LI Z A T I O N O F THE VIRTUAL IN TIME

th e vir tual. To thi s, a temporal dim en sion , whi ch Deleu ze call 'Aion",

should now be added . As he wr ites, th e specification of th e virtual

implies, on th e one hand, a space of nomad distribution in which

singulari ties ar c d istributed (T opos) ; on th e other hand , it impl ies a

tim e ifdecomposition whereby th is space is subdivided into sub-spaces. Each

one of th ese sub-s paces is successively defin ed by th e adjuncti on of

new points ensuring th e progressive and co m plete determination of

th e domain under conside rat ion (Aion) . There is always a space

which co nde nses and precipitates singularit ies , just as there is always

a time whi ch progressively co mpletes th e event through fragments

of future and past eve nts.v'

Deleu ze borrows th e term 'adjunc t ion ' from th e mathematician

Evariste Galois, th e cre ato r of gro up th eory. I will return in the next

chapte r to th e work of thi s pion eer, but at thi s point it is enough to

say that th e ope rat ion Gal ois defined as 'adjunc tion of fields ' is an

abstract ope rat ion ve ry clos ely related to the idea of th e progressive

differentiation of a space through a cascade of sym me try -bre aking

transiti on s. In othe r words, th e successive det ermination of sub-s paces

to whi ch Deleu ze refers is sim ply th e progressive unfold ing of multi ­

pliciti es through a se ries of symmetry-bre aking events. T he form of

temporality involved in thi s unfolding, however, sho uld be conceived

in a very different way from tha t in whi ch actual bifurcati on events

occur . The latter invol ve a temporal sequence of events and stable states ,

th e seque nce of phase transiti on s whi ch yields th e se ries of stable flow

patterns co nduct ion-eonvection-turbulence, for exam ple . Moreover,

as eac h bifurcati on occurs, only one of th e several alt ernatives available

to th e syste m is actualized. For example , in th e transiti on to th e

co nvection regime , eithe r clo ck or anti -clockwise rotating convec t ion

ce lls may emerge , but not both . Indeed, at eve ry bifurcati on th ere ar e

alt ernatives th at ar c phy sically unstable (unlike the two options for

co nvection ce lls both of whi ch are stable) whi ch means that even if

they are actua lized th ey will not la t very long and wi ll be destroyed

by any de tab ilizing fluctuation. ss In a virtua l un folding , on th othe r

hand , the symm etry-brcaking events not only Jullj' coexist with one

ano ther (as opposed to follow eac h othe r), but in add it ion, eac h brok en

INTENSIVE SCIENCE AND VIRTUAL PH ILOSOPHY

symmetry produces all the alternatives simulta neously, regardless of

whethe r th ey are physically sta ble or not.

T his virt ual form of t ime, involving the idea of absolute simulta neity

(or abso lute coexistence) would see m to vio late th e law s of re lativi ty.

In relati vist ic physics two events cease to be sim ultaneous th e moment

they become se parated in space, th e dislocati on in tim e becoming all

the more evident th e larger the se pa rating distan ce .56 There are two

reasons, however, why thi s sho uld not be an object ion to De leuzc's

conception of vir tual time . The first and m ost obvio us reason is th at in

vir tua l space there are no metric distances, only ordinal distances which

join rather than separate events . Mu ch as the noti on s of spat ial ' length'

or 'a re a' lose th eir meaning wh en we move away from Eucl idean

geometry to othe r ways of specifying th e relati ons of proximity

defi ning a space , so sho uld the notions of 's t re tch' or ' lapse' of time

separating non-simultaneous events be meaningless in the co ntext of a

no nmctric temporalit y. But there is a second and more import ant

reason why relativisti c co nstraints on absolute sim ultane ity, suc h as th e

constraint on th e maximum speed at wh ich causal signals may travel ,

sho uld not apply to th e vir tua l. T he temporality of th e virtual sho uld

not be co mpare d to that of th e processes governed by th e laws of

relat ivity, but to the temporality if the laws themselves. Unlike ex pe r imen­

ta l laws (like Boyle ' s law of ideal gases) whi ch simp ly record laborato ry

reg u larities , fundamental law s (such as Newto n's or Einstein's) arc not

m re mathematical re -descript ions of ex pe r ience .57 Although physicists

do not usually speculate abou t the onto log ical sta tus of fund amental

laws, to philosophe rs th ese laws ar e supposed to be eternal, and to be

valid simultaneously through out th e un iverse . In other w ords, in phil o­

sophical discussions fundamenta l laws enjoy th e same form of timeless­

ness as immutabl e essences. And it is th is form of time th at th e virtual

is supposed to re place.

Nevertheless the ques tio n re ma ins, what form of temporality would

allow the absol ute coexiste nce of virtual events? Or what amo unts to

th e same thing, how should we co nceive of a non m etric fo rm of time?

It clearly can no t be any presen t tim e , however long , since the very

concept of' a present assumes that of' a stretch or lapse of' ti me of a

particu lar chara tc rist i sca le. But it cannot be a t imeless d imen sion

either if we an' to avoid the tr ap pings of essen tia lism. The so lutio n

THE A CTUALIZATION OF T HE VIRTUA L I N TIME

which Deleu ze prop oses to esca pe th ese alterna tives is inge nious .

Unlike a transcendent heaven inhabite d by pure beings without becoming

(unchanging esse nces or law s with a permanent identity) th e vir t ual

needs to be po pulate d excl usively by pure becomings without being .

Un like act ual becomings wh ich have at most an intensive form of

temporality (bund les of seque ntia l processes occur ring in parall el) a

pure becoming mu st be characte rized by a parallelism without any trace

if sequentiality, or even directionality , Deleuze finds inspi rat ion for thi s

co nception of time in phase transit ions, or more exactly, in th e cr itical

events defi ning unactualized transi t ions. W hen see n as a pure becoming ,

the cr itic al point of temperature o f o-c, for exa m ple, m ar ks neither a

melting nor a freezing of' wate r, both of which are actual becomings

(becoming liquid or so lid) occurring as the cr itica l threshold is crossed

in a definit e direction. A pure becoming, on th e other hand , would

involve both dir ecti ons at once, a m elting- fr eezing event which never

actually occurs, but is ' alwaysforthcoming and already past. ' 58

The events invol ved in th e constructio n of vir t ual space, th e

progressive un folding of virtual multipliciti es as well as the stretc hing

of th eir singularities into series of o rdinary points , need to be th ought

as pure becom ings in thi s sense. In th is co nst ruction, as Deleu ze says,

•Time itself urifolds . . . instead if things urifolding within it . . . [T imeIceases to be cardinal and becomes ordinal, a pure order of time ."!"

Unlike actual time, which is made exclusive ly out of presents (what is

past and future relat ive to one t ime scale is still th e living present of a

cycle of greater duration) , a pure becoming wo uld imply a temporality

which always sidesteps the present , since to ex ist in the present is to be ,

no longer to become. This temporality must be conceive d as an ordinal

conti nu um urifolding into past and f uture, a t ime wh ere nothing ever

occurs but where every t hing is endless ly becoming in both unlimited

d irections at once, always 'already happ en ed ' (in th e past direction )

and always 'a bout to happen ' (in th e future di rect ion). And unlike

actua l t ime which is asymmetric relat ive to th e direction of relati ve pasts

and fut ures, a pu re becoming would imply a temporality which is

perfectly symmetric in thi s respect , the d irectio n of the arrow of time

em rgi ng as a brok en symmet ry on ly as the virtual is act ua lized. "?

I said in Chapter 2 that multiplicities, being in o rpo rca l ffc ts of

mate ria l causes, are impassi ble or ausa llv steri le enti ties. T h· t ime of

INTENSIVE S C IEN CE AND V I RT U A L PHIL O S OPH Y

a pure becoming, always already passed and ete rnally ye t to come ,

forms the temporal dimension of thi s impassibility or ste rility of

mult lplicitl es."! But I also said that the quasi-causal ope rato r, far from

be ing impassible , is defined on the contrary by a pure capacity to

alTect, act ing in parallel with physical causality in the production of th e

virtual. In particular, th e quasi-cause mu st be capable of weaving

m ult iplicit ies into a het erogeneous co ntinuum and to do so co nstantly

so as to endow th e latter with a certain auto no my fro m their co rporeal

causes.b1 \Vhat temporal aspect would co rrespo nd to the exe rcise of

this capacity? Here again , we cannot presuppose any metric co nce pts ,

that is, we cannot assume that thi s performance occurs in an)' present

st retch of time , however short . This othe r time must ind eed be

conceived as instanta neous, As Del euze writes :

Corpo real causes act and suffer throu gh a cos mic mixture and a

uni versal present which produces the incorporeal event , But the

qu asi-cause operates by doubling thi s physical causality - it em bodies

the event in the most limited possible present which is the most

precise and the most instantaneous, the pure instant grasped at the

point it d ivides itself into future and past ."!

In wh at sen se would a temporality charac te rized by a instant which

unfolds itself into past and fut ure he nonmetric? Actual time , as I said ,

ma)' he sec n as the product of a metrization or qu ant izat ion performed

hy a nested set of presents with characte rist ic t ime scales. Whether

one views the latt er in terms o f relaxation times or in terms of the

intrinsic peri od of nonlinear osci llat ions , th e processes occur ring in

actual time always have a time scale of limited Juration and )'et are

potentia lly irifinite, in the sense that a particular seque nce of cycles

l11a)' go on pu lsing for eve r . Virtual timc , on the o the r hand, would

he.' nonmct r ic in the sense that it is unlimited in the past and future

directions in wh ich it unfolds, bu t alwaysfinile like the insta nt without

thickness tha t pe rforms the un folding ." T he time of the virtual would

he.' consti tuted cn tirelv bv wh at , from the point of view of metr ic, ,tu nc , canno t he hut !Oingular ities : a maximum and a minimum, events

of unhmllcd Juration (the unfo ldi ng of mu ltiplicit ies) and events of cero

THE ACTUALIZATI ON O F THE V IRTUAL IN T I M E

duration (t he operat ion of the qu asi-cause ) . The quasi-causal operato r

would have to

bring about the corresponde nce of the minimum time wh ich can

occur in the instant with the maximum time whi ch can be thought

in accordance with Aion . To limi t the act ualizat ion of the event in a

present without mixture, to make the instant all the more intense ,

taut , and instantaneou s since it ex presses an unlimited future and an

unlimited past. bOO

No doubt, this description o f the temporal aspect of virtualit y lacks

the precision of its spa tial co unterpa rt. The latter has the advantage of

ove r a century of mathematical work on the nature of nonmctric

spaces and their broken sym me try relations to metric ones , whereas

similar formal treatments of time do not really exist. Moreov er, even

if we disregard time and focus only on space, Deleuzc ' s description of

the virtual co ntinuum goe s beyond the resources available from those

formal theories and may therefore see m mu ch too speculative and

com plicate d. Why, on e may ask, go through so mu ch trouble to

speci fy the immanen ce mechanisms through which a virtual continuum

is const ructe d when it is simpler and more natural to assume that the

entit ies revealed by nonlinear mathematics (att racto rs , bifurcations) ar e

of the same t)'pe as our more familiar Platoni c entities? A leading

figure in the theory of dynamical syste ms , th e mathematician Ralph

Abraham , for example , phrases his evaluation of the merits of the field

this way:

The ben efits of using dynami cal co nce pts at the present stage of

devel opment of sclf-organi7..atio n theory fall in two classes: perman­

ent ones - the acquisition of conce pts to be em bedded in morpho­

dynam ics , guiding its development ; and temporary ones - the

prac tice of new patterns of thought. In the first ca tegory I would

place the att rac to rs , the stable bifurcations, and their global bifurca ­

tion diagrams, as esse ntia l features of morphod ynami cs. These rna)'

he.' fl'ga rdcd as guidel ines, exclusio n rul es and topolog ical rest rict ion s

on the full complex ity of morphodynamic se.''1uences , .. I Sl'C' [the

INTENSIVE SCIENC E AND VIRTUAL PHILOSOPHY

importance of dynamicism] for self- organizing syste m th eory as

temporary and preparatory for a more co mp lete morphod ynamics

of th e future . And yet, dynamicism eve n now promises a permanent

legacy of restrictions, a taxonomy of lega l, universal restraints on

morphogen etic processes - a Platonic ideali srn .t"

Deleu ze would agree with much of what is ex pressed in thi s

passage, particularly th e characte r izat ion of th e rol e of virtual entit ies

as to po logical restricti on s or co nstraints, that is, as quasi-causal rela­

tions whi ch com plement causal ones in th e determination of a given

sel f-organizing or inten sive process. On th e other hand , to view th e

set of top ologi cal restrictions discovered so far as forming some kind

o f fixed , ete rn al taxonomy, would seem to him to defeat th e very

point of po stulating such const rain ts in the first place . No doubt, it is

much simpler to assume the existe nce of Platonic en tit ies than to

define a co mplex ope rat ion through whi ch th ese entities ar e meshed

into a co ntinuum th ereb y acquiring a ce rtain auton omy from actual

events. T he preferen ce for simplicity here , however, has less to do

with the elim ination of redundant features (the legitimate use of

simplicity arguments, as in O ccam's razor) and more to do with

f amiliarity . Arguments based on th e latter , as physicist s conce rned with

the co nce ptual foundations of their sub ject ar e aware, make an

ilkgi timate use of simplicity."? In th e present conte xt , it see m s to me,

to espo use a Platonic ideali sm on th e basis that it is a more familiar

thesis would be misguided . Given that no philosopher (o r scie ntist) has

ever before specified mechanisms of immanen ce, our lack of familiarity

with th e latter sho uld be seen merely as a co nt inge nt fact about

int e llect ual history not as a basis to reject a new theory.

I emphasize thi s point about sim plicity because however complex

th e description of th e virtual may see m so far, it is onl y half th e story.

In parti ular , we ma y grant that th e above description is a reasonable

specificat ion of how a nonmetric space t ime continuum ma y be built

NiI-cn a populat ion of virtual multipliciti es and st ill demand to know

where these multiplicities come fro m. Clearly , they canno t be simply

.iss umcd to exist on thei r own since this would make th em into ent it ies

hard ly d ist ingui shabl e fro m immutabl essenc . Ther is , in fact,

anot lu-r task which th e quas i-ca usa l ope rato r mu st perform, anothe r

THE ACTUALIZAT ION OF THE V IRTUAL IN T IME

immanen ce mechani sm which accounts for th e very ex iste nce of

multiplicities. As Deleu ze ays, th e quasi-cause 'extracts sinBularitiesf rom

the present, and from indi viduals and persons which occupy thi s

present ' . 68 This extraction operation, recovering a full multiplicity from

a partial spat ia- te m poral actualization, defin es th e second immanen ce

mechanism. Del eu ze so metimes usc a geometric characte rizatio n of

thi s operation, describing it as th e ext ract ion of a section or slice.

O rdinar ily, thi s mathematical ope rat ion sim ply reduces th e dimension­

ality of th e object to whi ch it applies . A slice of a th ree -dimensional

vo lume , for exam ple , is a two-dimensional surface, whil e th e vo lume

itsel f ma y be viewed as a slice or sec t ion of a four-dimensional

hypervolume. The anal ysis of attractors in state space, particul arl y

strange or chao tic attr actors , makes exte nsive usc of thi s op eration (a

' Po incare sec t ion') to ex t ract information from a complex topological

shape and displa y it in a way wh ich is easier to study."? Deleuze,

however, has a more elabo ra te ope rat ion in mind, one that docs not

have a co unte rpart in mathemati cs.

T o see wh at thi s or iginal slicing ope rat ion am ounts to let 's re t urn

to th e example of th e sequence of flow patterns co nduc t ion-convec­

tion-turbulen ce. Let ' s imagin e a co nc re te physical syste m in a state of

convec t ion , that is, actualizing one of th e available flow patterns (a

periodic attractor) . In thi s case , th e virtua l com ponent (the attractor)

exists merely as an effec t of actual causes , such as re lations between

temperature and density differen ces or compe tition between gravita­

tional and viscous forces, causal relation s whi ch account for th e

emerge nce and maintenance of co nvection ce lls . Dc lcuze 's hypothesis

is that suc h an actual syste m ma y be 's ampled' or 's liced through ' to

obtain its full qu asi-causal co mpone nt, th e entire set of attractors

defining each flow pattern and th e bifurcati on s whi ch mediate between

patterns. In other words, a Deleuzian sec tion would not consist in a

mere reduction o f th e original dimensionality, but in an elimination of

every detail o f th e actual event except its topoloqi cal im'arian ts: th e

distribution of its singularit ies, as well as th e full dimensionality o f its

sta te space .

Let me spe ll out th e details of th is important idea. I aid in Chapt r

1 that Del uze bor ro ws fro m Riemann th co n ept of an - (limen­

siona l mani fold whi h doc ' not n ccd to be mb cdded in a 'pace of

I NTE NS IV E S C IENC E AND VIRTUAL PH I L O S O P H Y

+1 dimensions to be studied , but that constit utes a space on its own ,

each one of its dimen sions defining a rel evant degree of freed om of,

or a relevant way of changing for, a given dynamical system . Each

multiplicit y extrac ted or sampled from actual inten sive processes would

possess a definite dimen sionality (a specific value for the N variable)

since the process it governs is capable of changing in only a finite

number of relevant ways. This finite number of dimensions would

co nstitute a key characte rist ic defining the virt ual multiplicit y as a

conc re te universal entity, and this finite number would vary for

diffe re nt multipliciti es extracte d from different processes. In other

words , the population of multiplicities would be dimensional ly hetero­

geneous. Given that the plane of consiste ncy mu st assemble multiplici­

ties together by their differences, thi s 'plane' cannot be conce ived as a

two-dime nsional surface but as a space of variable dimensionality,

capable of bringing a dimen sionally diverse virtual population into

oex iste nce . As Del euze writes:

It is only in app earance that a plan e of this kind ' reduces' the

numbe r of dimen sions; for it gathers in all th e dimen sion s to the

ex te nt that fiat multiplicities - which nonetheless have an increasina or

decreasinp number if dimensions - are inscribed upon it ... Far from

redu cing the multiplicities' number of dimen sion s to two, the plan e

I?I consistency cuts across them all, inters ects them in orde r to bring

into coexiste nce any number of multiplicities, with any number of

di me nsions . Th e plane of consiste ncy is the intersecti on of all

concre te forms . .. The only questi on is: Does a given becoming

reach that point? Can a given multiplicity flatten and co nse rve all its

dime nsio ns in th is way, like a pr essed flower whi ch remains just asalive dry?70

Dclcu zc so me times phrases his description as if the qua si-causal

operator was th e agent performing th e extraction or sec tion ope ration,

some other times ascribing this agency to the plane of consi ten cy

itse lf, The di ffercn e bet ween the two formulations is, I believe,

unimportant . What is impo rtant, on the othe r hand , are the det ail of

till' operat ion and thei r justification . In particul ar , th e fact that eac h

mu lt iplicity dd llH's a space of its o wn, that is, the absence cj' a space if

THE ACTUALIZATION O F THE VIRTUAL IN TIME

N+ I dimensions wh ere they would be embedde d, is key to the task of

conce iving a virt ual space which does not unify multipliciti es, that is, a

space composed by th e coe xisting multipliciti es them selves in their

het erogeneity. Similarly , the qua si-causal operator is oft en referred to

as a ' line' but not because it would be a on e-dimen sional entity .

Rather, th e qua si-cause would ope rate at N- ] dimensions, unlike a

transcendent source of unity which mu st op erate from a suppleme ntary

(e .g. N+ 1) dimension . In Deleu ze 's wo rds:

Unity always operates in an empty dimen sion supplementa ry to that

of th e syste m co nside re d (overco ding) . . . [But a) multiplicity never

allows itself to be ovc rcodc d , never has availabl e a supplementary

dimen sion over and abov e its number of lines [or dim en sions) . . .

All multipl iciti es are flat, in the sense that th ey fill or occupy all of

their dimen sions: we will therefore speak of a plane of co nsiste ncy

of multiplicities, eve n though the dimen sions of this ' plane' increase

with the number of co nnec tions that ar e mad e on it. Multiplicities

ar e defined by the outside : by the abstract line , the line of flight

. . . according to whi ch th ey change in nature and connect with

other multiplicit ies . . . The line of flight marks: the reality of a

finite number of dimen sion s that the multiplicity effectively fills; the

impossibility of a supplem entary dimension, unless the multiplicity

is transformed by the line of flight; the possibility and necessity of

flatt ening all of the multiplicities on a single plan e of consiste ncy or

ex te rior ity, regardless of their number of dimen sions.7\

Let me summar ize what I have said about th e two immanen ce

mechanisms. Th operator's first task, to assembl e multiplicities together

by cre ating converge nt and divergent relations among th e ordinal seri es

emanat ing from th em, may be conside re d a pre-actualization . It would

endow multiplicities with a minimum of actuality and, in thi s sense, it

would represent the first broken sym me try in th e cascade that culmi­

nat es in fully formed actual beings. The second task of the quasi-causal

ope rator, to ex t rac t vir tual events from inte nsive pr ocesses may, in

turn , be see n as a veritable counte r actua liza tion since it wo uld follow a

direct ion oppos ite to that which goes from the vir tual to till' inte nsive,

and from there to the ex te nsive and qualita tive. 71 'o untc r-actualizat ion

I N T E N SIV E SCI E NC E AN D VI R TU A. L PH IL OS OPHY

would, in fact, compleme nt pre -actualization : while the former e xtracts

flat (o r folded) m ultiplicit ies from act ually occurring events , the latter

would take these and 'unflatten ' them, that is, it would allow them to

progressively unfold and differentiate without fully actualizing them.

Each of these two operations would possess a temporal dimension : the

quasi-causal operator would sample or section all actual events, at alldilTerent time scales, instantaneously; then, each flat multiplicity would

be immediately unfolded in two unlimited directions at once, past and

future, distr ibuting the singularit ies which define each of the unfo lding

levels on both sides of the instant at once, 'in the manner of a pod

which releases its spo res' . 7 l

The operation of pre -actualization would give multiplicities not only

a ce rtain autonomy from the intensiv e processes acting as their realcauses , it wo uld also endow these impassive and stcrile effects with

whatever mo rphogenet ic power they enjoy .7 4 In other words , pre ­

actualization would not on ly explain how an unactualized singularity

bel onging to a physical system with multiple a!tracto rs wou ld subsist

as a potential alternative state , it would also explain how the singularity

that is actualized gets its power to attract in the first place . To the

ex te nt that linking multiplicities together and endowing them with

productivity foreshadows the intensive processes which follow down

the symm etry- breaking cascade , the quasi-causal operator is referredto as a 'dark precursorT" The operation of counter-actualization, on

the other hand, would operate in the oppos ite direction, up the cascadefrom the inte nsive towards the virtual. I said in Chapter 2 that some

areas of the world, those defined by processes which are nonlin ear and

which operate far from equ ilibrium, do not conceal the virtualunderneath extensities and qualities but rather reveal it, or allow it to

express itself.?" These areas woul d represe nt a spontaneous movement

toward s the virtual wh ich is st ill physica l and co rporeal but whi ch may

I" , given a boost making it reach the level of a pure virtuality. T o th e

e xtent that co unter-actualizatio n accelerates an escape from actuality

which is already present in some intensive processes, the qua si-causa]

0pt'rator is referred to as a 'line of flight '. 77

In conclusion, I wo uld like to repeat that whatever the merits of

Dl'lcuzc 's particular proposals for the implementation of the quasi­

t',l U~.l l Opt'rator, we should at least credit him with having e lucidated

THE A C T UA L IZ A. TI ON OF T HE VI R TUA L I N T IME

the overall constraints that any implem entation wou ld have to meet. If

we arc to get rid of essent ialist and typo logical thought we need some

process through which virtual mu ltipliciti es are derived from the actual

world and some process through which the results of this derivation

mal' be gh'en eno ugh co here nce and autono my . Deleu ze himself gave

seve ral different model s for each one of these tasks, a fact that shows

that he did not think he had achieved a final solut ion to the problem,

on ly its correct formulation . On the other hand , he clea rly thought

that the problem itself was worth posing, regardless of its particularsolutions . That this is indeed the case may be glimpsed from the fact

that Dclcuzc ' s description of his co nstruc tivist method in philosophy

close ly mat ches the two tasks whi ch the ope rato r is supposed to

accomplish : creat ing virtual events (m ultiplicit ies) by extrac t ing them

from actual processes and laying them out in a plane of consistency .711

This methodology, moreover, is what in his view would distinguishphilosophy from science . As he writes :

It co uld be said that science and philosophy take oppos ed paths,

because philosophical concepts have events for consistency whereasscientific functions have states of affairs or mixtures for ;eferencc :

through con cepts, philosophy cont inually extracts a consistent event from

the states if eif!airs . .. whereas through functions, science continually

actualizes the event in a state of affairs, thing, or body that can bereferred ro. ?"

It matters little whether we describe thi s method as involving two

separate operations (to extract ideal events and to give them co nsist­

ency) o r as a single one (to extract a consistent e vent). The importa ntpoint is that Dcleuze con ceives of pre-actualization and counter­

actualization, howe ver implemented, a') defining an object il'e movement

wh ich a phil osopher mu st learn to grasp . As he puts it, we phil oso ­

phers must invent devices to allow us to becom e 'the quasi-cause

of wh at is produced within us, th e O pe rato r"."? Spelling out the

details of Dcleuvcs methodology will involve co nnecting the resultsof his onto log ical analysis with ques tions of' epistemology . In cp istc ­

mological terms to extract an ideal event from an actually oecuningone is, hasicall v, to dcfilU' what is problemauc about it , to grasp what

INTENSIVE SC IE NCE AND VIRTUAl. PH Il.O SOPHV

abo ut the eve nt objectively stands in need oj explanation . This involves

discerning in the act ual event what is relevant and irrelevant for its

e xplanation , what is important and what is no t. T hat is, it involves

correctly grasping the objective distribution ojthe singular and the ordinary

defi ning a well-posed probl em. T o give consiste ncy to these well ­

posed probl ems, in turn , mean s to endow them with a ce rtain aut on­

omy fro m th eir particular solutions , to show that probl ems do not

disappear be hind th eir solutions , just like virtual multiplicit ies do not

disappear behind act ualized indiv idu als. T he ep iste mo logical side of a

Dc lcuz ian ontology is co nstitute d by such a philosophy of problems

and this will form the subject matter of the followi ng chapter.

C H A PTE R 4

Virtuality and the Laws if Physics

In a flat ontology o f indi vidu als, like the one I have tri ed to de velop

here , there is no room for rcificd to talit ies . In particular , th ere is no

room for ent ities like ' society ' or 'culture ' in ge nera l. Institutional

organi7..ations, urban centres or nation states are in th is ontology not

abstract totalities but concre te social individuals, :\"ith th e same o~~olo­gica l sta tus as indiv idu al human bein gs but ope rating at larger spatio­

temporal scales . Like organisms or species these larger socia l

indiv idu als are products of co ncrete historical processes, having a date

of birth and , at least potentially, a date of death or ex tinction . And

like organ isms and species, th e rel ations betw een individuals at each

spatia -te mpo ral scale is on e of parts to wh ole , with each individual

eme rging from th e causal int eractions am on g the members of popula­

tions of smaller scale individuals. Althou gh the det ails of each individu­

ation process need to be described in det ail, we can roughly say that

from th e interactions among individual decision -makers, institutions

eme rge ; from int eractions among institutions, citi es t~merge; and from

urban interacti on s, nation states eme rge _1 The population serving as

substra tum for the eme rg ence of a larger whole may be very hetero­

ge neo us or, on the contrary , highly homogen eous. But even in those

cases where the degree of homogen eity at different scales is high

eno ugh to suggest the existe nce of a single 'culture' or 'society' , the

temptation to postulate such to talit ies mu st be resisted, and the degree

of homogen eity whi ch motivated such postul ati on must be given a

co ncrete histori cal explanation .

Thus far I have used the term 'science ' as if its use was unprobl em ­

atic, but given the requirements of a flat o nto log)' it is clear that this

te rm shou ld no t be used since it refers to an abstract totalitv and

moreover, to a to ta lity defined by an essence . Instead , we mus; : trh-e, ,to ide ntitY the specific processes which have gin>n rise to inJiI"idual

SCIentific fi elJs, which like an) other indivi dual , mus t he conceived as

I N T E N S I V E SC IE NCE AND V IRTUAL PHILOSOPHY

com posed of populations of entit ies at a smaller scale . In the case of

the field of classical mechanics, for ex ample. these co mponents are ,

ro ughly: populations of mathematical models and techniques for the

indi viduation of predi ctions and e xplanations ; populations of phenom­

ena produced in laboratories and population s of machin es and instru­

me nts wh ich individuate and measure those phen om ena; populati on s

of ex pe rime nta l skills , theoretical conce pts and institutional pract ices.

Like an organic species , the degree to which an individual scie ntific

field has a well -defined identity will dep end on co ntinge nt historical

facts such as its degree of int ernal hom ogen eity and its degree of

isolation from othe r fields. Simil arl y, the degree to whi ch several fields

resemb le each othe r should be given a historical explanation , such as

one field serving as exemplar for the const ruction of ano the r . or the

ex port of inst ruments and techniques from one field to another, or th e

shari ng of institutional compone nts among different fields. This way

the qu estion of whether th ere is such a thing as 'scien ce ' in ge neral

becomes an empirical question, one which, I believe. sho uld receive a

nega tive answer . Many conte mporary ana lysts do ind eed seem to thin k

that, as a matter of empirical fact, science d isplays a deep and

characte rist ic disunity .2

In the first par t of thi s chapter I wo uld like to develop the ideas

needed to think abo ut individual scienti fic fields, using classical mech ­

anics as a concre te example, but also to review some of th e traditional

philosophical obstacles which have historically prevented a correct

assessme nt of the di sunity, heterogen eit y and divergent development

of 'sc ience". At thi s point it should come as no surprise that in my

view the main obstacle has be en th e entre nchme nt of essentialist and

ty po logical thought in philosophical st udies of scientific practice . Many

philoso phe rs in the past have tak en the essence of classical mechanics

to be its exceptionless laws. Thi s is particularly true when fundam ental

laws, such as Ne wton's law s, ar e view ed as gene ral truths from wh ich

"\'l'rything else[ollows mechanically , tha t is, by simple logical deduction.

\VI1l'n species arc view ed no t as indi vidual entit ies but as gene ral

c.ltegorit."s, the productive or ge net ic processes whi ch yield these

indi vidu als te nd to he ignored . Similarly. the view of law s as general

t ru ths has tended , historically, to eli minate from philosop hical discu s-

V IRTUALITY A ND THE LAWS OF PHYSICS

sian the productive or Benetic connections invol ved in the physical

processes go verned by those law s.

More specifically , the esse ntialist view of laws has co ncealed the

producti ve po we r of causal connections, that is, the fact that events

act ing as causes actually produ ce their effects. Co ntrary to a popular

miscon ception , ph ilosophical approaches to scie ntific practice have

thrived, from the seve nteenth century on , in a world devoid of causes

and ruled e xclusive ly by laws sta ting constant regu larities. J Part of what

made possible the re placement of causes by laws was a view of

causa lity as an inherently linear relation, such that , give n a particular

cause, the same effect was bound to be produce d. Clear ly, if causality

always exhibited thi s simple form , if effects always followed mechani ­

cally and necessarily from their causes, postulating a separate produc­

tive power of causes distin ct from the exce ptionless laws governing

their ope ration would be redundant. But more co mplex forms of

causality do exist , nonlinear and sta tistical causality, for instance, and

these arc involved in all the inten sive production processes which I

have described in previous chapte rs. Hen ce a crucial task for a

Dcleuzian episte mologist invo lves rescuing these ge netic links between

events from the limbo wh ere general laws have cast them.

Besides concealing productive relations behind static categories , the

traditional philosophical approach to laws may be crit icized for subor­

dinat ing mathematica l models to Iinaui stic statements . Much of what I

have argued in this book depends on treating mathematical models in

thei r specificity, that is, as disp laying a certain beha viour which is crucial

for thei r successful application to scientific tasks . The most obv ious

example is the tendency of solutions to an equation to approach an

attrac to r , a tenden cy which is not displayed by linguistic translati ons

o f the conte nt of the equation but which dep end on the speci fic

math ematical form of both the equation and the ope rato rs that act on

it. Thus, a second task for a Deleu zian episte mologist is to rescue

models and their dynamic behaviour from static linguistic renderings

of law s. These tw o related errors , elimination 1" causes and subordination

to lanau one (and deductive logic) arc the basic characte rist ic of essen­

tialist approaches to classica l physics , and their criticism will form the

suhj('ct matt er of the first sec tion of this chapt e r , Let ow begin with

I NT E NSI V E S C I EN CE AN D VIRTU A L P HI LOS O P HY

tlu- dismi ssal of productive causes in favour of constant regularities. As

th l' ph ilosopher of scien ce [an Hacking puts it:

Ilume notoriously taught that cause is only constant conjunct ion .

To say that A caused B is not to say that A, from some power or

characte r within itself, broupht about B. lt is only to say that th ings

uf type A are req ularlv followed by things of type B ... Hume is in

1:1 t not responsibl e for th e widespread philosophical acceptance of

a co nstant -co njunction attitude towards causat ion. Isaac Newton did

it, unintentionally. The greatest triumph of th e human spirit in

l lu rnc ' s day was held to be th e Newtonian theory of gravitation

. .. Immediately before Newton , all progressive scientists thought

that th e world must be understood in terms of mechanical pushes

anti pulls. But gravity did not see m ' mechanical' , for it was action

at a distance .. . For em pirically minded peopl e th e post -N ewtonian

alt itude was, th en, this: we should not see k for causes in nature,

hut only regularities . .. The natural scie ntist tries to find universal

state me nts - th eories and law s - which cover all ph enomena as special

cases. To say that we have found th e explanat ion of an event is only

to say that th e event can be deducedfrom a aeneral repul aruy,"

Ii a king argues that this elimi nat ion of producti ve causes in favour

of state ments o f regularities (and deducti ve relations between those

si.ucmc nts) is characte rist ic not of physics in general, but only of

phi loso phies of physics whi ch concentrate exclusively on th e th eoretical

t ompo nent of a field at th e expe nse of its expe rimental compone nt .

T he day to day practi ce of expe r imental physicists, consist ing as it does

in specific causal int erventi ons in reality , is much too rich and complex

10 he rcdu cd to logical relations between state ments. The expe riment­

alist is d irectl y invo lved in productive relation s, whether th ese involve

till creat ion o f an apparatus to individuate ph enomena or th e use of

inst rum -nts to produce individual measurements of properties of those

ph -no rncna . It is only th eory-obsessed phil osophies, whether held by

physicisls o r professional philosophers, that can afford to forget about

1..ur sa] co nnec tions and co n e ntrate ex lusively on logical relation s. The

ult imatc ex press ion of thi s esse nt ialist stance is a model of s icntific

c-xplnnnt ion de velop ed in the twentieth c >ntury whi ch takes th e

VIRTUAL ITY A N D TH E LAW S OF PH Y SI C S

Humean reduction of cau ses to linguistic statements of regularities to

an extreme.

In this episte m ological th eory, known as th e deduai ve-nomolopical

approach , scientific explanat ions ar e treated as logical arguments consist­

ing of several propositions, on e of which must be an exceptionl ess

law . The term 'proposition ' refers to th e meaning of declarative

sentences , that is, to what two sentences in different languages,

expre ssing th e sam e state of affairs, have in common. In this model,

to explain a particular laboratory phenomenon is to deduce it from a

set of propositions: from a linguistically stated law (su ch as 'two bodies

are gravitationally attract ed to each other in direct proportion to th e

product of th eir mass es, and in inverse proportion to th e square of

their distance ') and a set of propositions describing initial (and other)

co nditions, we derive further propositions which may be treated as

predictions to be tested for th eir truth or falsity in a laboratory. [I' th e

behaviour of th e ph enomenon conforms to th ese predictions we can

cla im to have explained it , not, of course, by having given causal

mechanisms for its production, but in th e way one explains things in a

ty pological approach: subsutninp it as a particular case under a aen eral

catea0T)'. Although hardly any w orking physicist would accept that his

or her co m plex explanatory practi ces are captured by this simplistic

theory , th e deductive-nomological approach has dominated much of

twentieth-century philosophy of scien ce and continues to have many

defenders in thi s field. 5

When on e accepts thi s model of explanation th e structure of th e

th eoretical component of a scientific field takes th e form of an

axiomatic: from a few true statements of general regularities (the

ax io ms) we deduce a large number of consequences (theorems) which

are th en compared to th e results of obs ervations in th e laboratory to

check for their truth or falsit y . Given that deduction is a purely

mechanical way of transmittina truth or falsity, it follows that whatever

truth on e ma y find in a th eorem mu st hav e already been contained in th e

ax ioms. It is in thi s sense that axioms are like esse nces. To counter

thi s essentialist concept ion , a new gen eration of philosophers has

d eveloped an alt ernative charac te rizat ion of what a th eory is, reintro­

du ing productiv causa l rel ations as an int egral part of explanat ions,

as well as rejecting th e Iingui sti haracterizati on or explanator

I N T E N S I VE S C IENC E AND VIRTUAL PHILOSOPHY

practices . In th e view of these phil osophers, ex planations , rath er than

being simply logical arguments, involve a complex use of mathem atical

mo dels of different types: mod els of gene ra l relation s, models of

partic ular experimental situations, as well as sta tistical models of th e

raw data gathe re d in laboratori es. O ne of the de fenders of this new

view, Ronald Giere , puts it this way:

Even just a brief examination of classical mechanics as pr esented in

modern textbooks provides a basis for some substantial conclusions

about the overall structure of this scientific th eory as it is actually

understood by th e bulk of th e scientific community . What one finds

in standard textbooks may be described as a cluster (or cluster of

clu sters) of models , or , perhaps better, as a population if models

consisting if related famili es if models. The various families ar e

constructed by combining Newton's laws of motion, particularly

the second law , with various for ce functions - linear functions,

inverse square functi ons, and so on . The models thus defined are

then multiplied by adding other force functions to th e definition .

T hese define still further families of models. And so on."

Giere emphasizes the point that, despite the fact that some members

of this population of mod els (Newto n's laws of moti on ) serve to

generate the various branchin g famili es , the relation bet ween a funda­

mental model and those deri ved from it is not like that between axioms

and theore ms. Far from being a mechanical process of deduction , the

com plex modelling practices which have historically gene rated th ese

families invol ve man y judicious approximations and idealizati ons,

gUided by pri or achievements serv ing as exe mplars ." I will return to

this ques tion in a moment but for now I would like to add that th e

basic idea of th inking of a physical theory as a population of models

fits we ll with th e onto logical stance I am defending . Such a population

is easi ly conce ived as the product of a historical accumulation, subject

to all the co ntinge ncies of such histori cal pr ocesses, and hence with no

pr ·t n e that it represen ts a co mplete or final set of mod els. At any

rate, the completeness or clos ur of the set becomes an empirical

matt 1', not .orncthing to b assumed at the outs t as in axiomatic

trva t rncnts . ~ rtai n popul ations (like those of the sub -fi lei of classica l

V IR TUAL IT Y AND T H E L A W S OF PHYS I CS

dynamics) may see m to have achieved closure at a ce rtain point in

history only to be reop en ed later giving rise to a new round of

accumulation, as when compute r- dr iven developments in nonlinear

dynami cs reop en ed what was widely conside re d a closed field . As I1ya

Prigogin e puts it : ' Unfortunate ly, many co llege and uni versity text­

bo oks pr esent classical dynamics as a closed subject .. . [but] in fact ,

it is a subjec t in rapid evo lution. In the past twenty years , [physicists]

have introduced important new insights , and further developments can

be ex pected in the near future. ' 8

The philosopher of scie nce Nancy Cartwright has pr op osed a set of

distinctions that may be used to describe the non-axiomatic stru cture

of this population of models. Som ewhat parado xicall y, she argues that

the fundamental laws of physics, those laws which in axiomatic

treatmen ts ar e assume d to be the highest truths, are indeed false. Th e

laws of physics lie , as she puts it . What she means is that a fundamental

law achieves its generality at the expense if its accuracy. A fundamental

law , such as Newton 's law of gravity, is strict ly speaking true onl y in

the most artifi cial of circumstances, wh en all other forces (like

electromagne tic forces) ar e absent, for instance , or when there is no

fr iction or other nonlinearities. In other words, the law is true but

on ly if a very large 'a ll othe r things being equal' clause is attached to

it .? W e can compe nsate for the sho rtcomings of fundamental laws by

adding to the basic equation other equations representing the action of

other forces or th e complex causal interacti ons between forces. But

then we lose the gene rality that made the orig inal law so appealing to

essentialists. Th e mod el becomes more true, describing with increased

accuracy the structure of a given expe rime ntal phen omenon , but for

the same reason it becomes less gen eral. In short , for Cartwright th e

objec tive conte nt of physics do es not lie in a few fundamental laws,

but in a large number of causal models tailored to speci fic situations.

(Giere does not speak of 'causal models' but of 'hypo these s ' linking

th e abstract models and the world , but th e overall thrust of his

argument is very close to that of Cartw right .10)

The esse ntialist may obj ect that , given that the speci alized causal

mod els are d rive d fro m the fundamental laws , th y mu st inherit

what ver degree of truth they have fro m thos law s. But Ca rtwright

(l ike Giere) rcpl ic that this oversim plifies the description of the

INTENSIVE SCIEN CE AND V IRTUA L PHI LOSOPHY

modelling practices of real physicists. The causal models are not

logicall y deduced from th e general laws, but construct ed from them

using a complex set of approximation techniques which cannot be

reduced to deductive logic. As Cartwright says, the content of th e

causal models 'we derive is not contained in th e fundamental laws that

exp lain them. ' II In short, the population of models whi ch con stitutes

the theoretical component of classical mechanics may be roughly

divided into two sub -populations: a large number of causal models

closely adapted to particular expe r imental situations, and a few

fundame ntal models corresponding to basic laws from which branching

families of other abstract models ar e derived . This breakdown of the

ontents of the population leaves out a different class of models,

statistical models if the data, which is also very important. Positivist

philosophers used to think that the predictions deduced from axioms

and auxiliary premises (those describing initial conditions) were con ­

fro nted directly with observations in a laboratory, that is, with raw

data. But for at least two hundred years physicists have used statist ical

mo dels to organize the raw data, and, in particular, to attempt to

capt ure the distribution ifmeasurement errors in th e data. 12 Beside ignoring

th is important kind of model , th e po sitivi st emphasis on 'the observer'

is misleading because it reduces to a subjective phenomenon what is in

fact a complex practi ce of data gathering, involving not passive

observations but active causal interventions.

Leaving aside the expe rimental side for a moment, what ar e we to

think of the few fundamental laws ? Is it correct to say that they lie, or

is it more accurate to say that they are not the kind of mathematical

objects that can be true or false? Cartwright suggests that the function

of these laws is to un!fj and oraanize th e rest of the population. 13 This

is, I beli eve, a ste p in the right direction but we cannot simply take

this unifying capability for granted; we must at least try to account for

it. Histori cally, the unification of the different branches of classical

mechanics was achieved by a ser ies of physicists and mathematicians,

start ing with the work of Leonard Euler in th e mid -eighteenth century

• nd culminating a hundred years later with that of William Hamilton.

It may be said that, togeth I' with other important figures (Maupc rtuis,

I ,lgr.1I1gl'), th esc scie ntists transformed classical me han ics from a

VIRT UA L I TY AND TH E LAW S O F PH Y S ICS

science of forces to one of sinq ularities. In the words of the historian

Morris Kline:

Hamilton 's principl e yields the paths of falling bodies, th e paths of

projectil es, the elliptical paths of bodies mo ving under th e law of

gravitation, the laws of reflection and refracti on of light, and the

more elementary phenomena of electricity and magnetism. How­

ever , the chief achievem ent of the principl e lies in showing that the

phenomena of all these branches of physics satisfy a minimum

principle. Since it relates these phenomena by a common mathemat­

ical law, it permits conclusions reach ed in one branch to be

reinterpreted for another. Hamilton's principle is the final form of

the least -action principle introduced by Maupertuis, and because it

embraces so many actions of nature it is the mo st powerful single

principle in all of mathematical physics. 14

Th e history of minimum principles, the idea that, for example , light

moves along the path that minimizes travelling distance, is ind eed a long

one having roots in Greek antiquity and medi eval philosophy. IS In th e

seve nteenth century , Pierre de Fermat cre ated th e first application of

this idea in th e conte xt of ear ly modern physics, the Principle of Least

T ime govern ing the behaviour of light in geometrical optics. For much

of its history the principl e carried strong theol ogical overtones as it

was assoc iate d with the beli ef that it reflect ed the econo my of thought

of a Cre ator. Maupertuis eve n went as far as to state that his Least

Action principl e was the first scientific proof of the existe nce of God.

Event ually the theological connection was lost , as scientists realized

that what mattered was not th e ideological interpretation but the

math ematical technology that was create d around these ideas: the

calculus if variations. Thi s was the first technology ever to deal directly

with singular it ies and it rivals in importance, as far as its effects on

ninet eenth- and tw entieth -century physics, the other mathematical

fields I have discussed in this book (differential geometry, group

thcorvj.!"

One way of looking at the calculus of variations is as a novel way of

I'0sinfl mechanical problems. Instead of looking at a problem in physics as

I N T E N SIVE S C I E N C E AND VIRTUA L PHI LOSOPHY

a problem of the causal effccts of forces, one looks at it as a pr obl em

of finding, am on g the many possible pr ocesses that ma y change a

physica l systc m from one sta te to ano ther, the actual process. More

exactly, the techniques develop ed by Euler and Lagrange allow th e

construction of a set of possibiliti es (fo r exa mple, a set of possible

paths which a light ray might follow) and supply th e resources needed

to sort these possibilities into two gro ups, on e of ordinal)' and one of

sinpulat cases . As it happen s, the results of expe riments show that the

singular cases (a minimum or a maximum) are the ones that are in fact

act ualized ."? Although the singularities uncovered by the calculus of

variations are not, st ric t ly speaking, attractors, its cre ators did see m to

thi nk that they played a similar role . Attractors are described as

defining the long-term tendencies of a syste m , that is, the state the

syste m will adopt if we wait long eno ugh to allow it to settle down.

Thi s emphasis on th e final sta te sugges ts that one way to look at the

difference be tween attractors and causes is through the old distinction

mad e by Aristotl e between final and qpcient causes. Euler him self,

when introducing his var iatio nal techn ology, used this Aristotelian

distinction:

Since the fabr ic of the universe is most perfect , and is the wo rk of

a mo st wise Cre ator, nothing whatsoever takes place in the universe

in whic h some relat ion of maximum and minimum does not appear.

W herefore there is absolutely no doubt th at eve ry effect in the

univ erse can be explained as satisfacto rily fro m final causes, by the

aid of the meth od of maxim a and minim a, as it can from the

effective causes the mselves .. . Therefore, two method s of studying

effects in Nature lie ope n to us, one by means of effective causes,

which is co mmo nly called the direct method, the other by means of

fina l causes . . . O ne ought to make a speci al effort to see that both

wuys if approach to the solution if the problem be laid ope n ; for thus

not only is one solutio n greatly stre ngthe ned by th e othe r, bu t ,

more than that , fro m the agreement between th e tw o so lutions we

sec ure the very highest satisfaction. 18

In a Dc lcuz ian ontology final causes wo uld have to be replaced by

quasi -causes in order to avoid ascribing teleological or goal-seeking

VIRTUAL IT Y AND THE LAWS OF PHYSICS

behaviour to physical systems . But the impo rtant poin t for my

argument is that it was precisely the ability to pose a pr obl em not in

te rms of specific efficient causes (forces) but in a way which by-p assed

causal det ails, that allowed the variational version of classical mechani cs

to play a unifying and organizational role in th e population of models.

The singulari t ies whi ch th e calculus of variatio ns uncovered rep res­

en te d, in my terminology, a mechanism -indep endent reality. On th e

other hand, as Euler himself acknowledged , thi s method was comple­

mentary not exclusive to the causal one . O ne may know that a given

classical mechanical pr ocess will tend to minimize some quantity, but

the full ex planation of the process will also invol ve a correct descrip­

tion of the causal mechani sm s that achieve such minimizati on . Thi s

other task, how ever, must be performed by other mod els, less ge neral

and more specifically tailored to th e details of an ex perime ntal

situatio n .

To summar ize the arg umc nt of thi s sec tion , far from being mere

mathematical express ions of linguistic truths, laws must be viewed as

models fro m which th e m ath em atical form cannot be eliminated . Th e

un ificatio n brought about by the calculus of variat ions, for ex ample,

cannot be understood otherwise since its techniques do not appl y to

ling uistica lly stated laws. These irre ducibly mathematical models form

a gro wing and heterogen eous populati on , some members of which

carry causal informatio n about pr oductive relati ons between events,

ot hers embody quasi-causal relations between singular itics . In other

words, the populat ion of mod els making up the th eoreti cal compone nt

of classical mechanics contains a large number of speci fic causal models

which are th e vehicles for truth (the part of the population that

inte1aces with the actual world), and fewer models whi ch do not refer

to the act ual worl d (hence are neither true nor false) but wh ich

nevertheless do inteJjace with the virtual world by virtue of being well ­

posed problems. For Deleu ze a probl em is defined precisely by a

distribut ion of the singular and thc ordinary, th e important and the

un importan t , the relevant and the irrelevant. A we ll-posed probl em

ge ts these distribu tion s right, and a solut ion always has th c truth it

des erves acco rding to how well specified the co rrespo nding pr obl em

is .!" In these t rm s N iwton' s a hi vem nt wo uld co n ist not in having

discovered gene ral tru ths about the uni v ersc , but in having correct ly

INTENS IVE S C IEN CE AND VIRTUAL P HIL O SOPH Y

posed an objective problem defined by the simplest distribution of

singular ities (unique minima or maxima). This interpretation preserves

the obj ectivity of Newton ' s law s but it deflates his achievement

somewhat, in th e sense that , if th e insights of nonlinear dynamics

abo ut multiple attractors ar c correct , th e single minimum problem is

not th e most general on e .

T his conclusion assumes, however, that the traditional axiomatic

appro ach to physics can be replaced by a problematic approach, that is,

that problems can replace fundamental law statements. But this

replacement needs more justification given that it go es again st th e grain

of th e traditional ontology of physics . Hamilton 's Least Action prin­

ciple , for exam ple, is still interpreted by most physicists as an axiom

ex press ing a general truth from which many particular truths in physics

follow mechanically. As Morris Kline puts it:

To th e scient ists of 1850, Hamilton's principle was th e realization

of a dream ... From th e time of Galileo scientists had been st r iving

to deduce as many phenomena of nature as possibl e from a few

fundam ental physical principles ... Descartes had already expressed

the hope that all th e law s of science would be derivable from a

sing le basic law of th e universe. i?

And, I sho uld add , thi s hope for a single law state ment from which

every thing else follows has displayed a consid erable resilien ce and

long vity, st ill animating th e dream for a final theory among some

onte mpo ra ry physicists. Therefore the task for th e next section of this

chapter will be to describe in more detail the extra-propositional and

sub-representative nature of th ese distributions of th e important and th e

unimportant which arc supposed to replace law statements as well as

esse nces . In Dcleuzc 's words:

It will be said that th e essence is by nature th e most 'important'

thing . This, however, is precisely what is at issue: whether notions

of' importance and non -importance ar c not precisely notions whi ch

co nce rn events or accidents, and arc mu ch more 'important' within

accid ents than th e crud, oppos it ion between esse n e and acc ide nt

itself. Th e probl em of th ought is ti d not to esse n cs but to th

VIR TUALITY AND THE LAW S O F PHY S ICS

evaluat ion of what is important and what is not, to th e distribution

of th e singular and regular, distinctive and ordinary points, whi ch

takes place entirely within th e un essential or within th e description

of a multiplicity, in relation to th e ideal events that const itute the

conditions of a probl em. 21

I will focus first on a particular kind of problem, explanatory problems,

to show th e rol e which th e cau sal and th e quasi-cau sal play in th e

explanat ion of physical phenomena. As Ian Hacking has argued, th e

same positivist biases which promote th e beli ef that causality is not an

obj ective relation also promote th e downplaying of explanation as an

epistem ological activity, that is, promote th e po sitivist thesis that

'explanations may help organize phenomena, but do not provide any

deep er answer to Why questions ... ' 22 To th e non-positivist philo­

sophe rs who arc reviving th e study of causality, on th e contrary,

questions as to why a phenomenon occurs are crucial since they require as

answers more than a mere description of regularities. Answering a

Why question typ icall y demands supplying a causal explanat ion , per­

hap s in th e form of a causal model of a mechanism. In addition, I will

argue that th ese qu estions sometimes require supplying a quasi-causal

factor to explain whatever regularity there is in th e beha viour of th e

mechanism s, that is, to capture the m echanism -indep endent aspect of

th e ph enomen on .P Despite th e fact that qu estions and answers ar e,

indeed , linguistic enti ties , Why qu estions involv e as part of th e

co ndit ions that make th em answerable, or well -posed, a non-linguisti c

or ex t ra-propositional aspect whi ch is properly problematic: a distri­

bution of the relevant and th e irrelevant. Let me begin this new

sec tion with a quote from th e philosopher Alan Garfinkel who has

developed an original approach to these matters:

When Willie Sutton was in prison, a pri est who was trying to

reform him asked him why he robbed banks . 'Well , ' Sutton repli ed,

' that's whe re th e money is.' There has been a failure to connect

here , a failure of fit. Sutton and th e pri est are passing each other by

. . . Clea rly th ere arc different values and purposes shaping th e

qu stion and answer. Th 'y take different thing to be problemat ic or

stand in n cd of ex planat ion. For th pri e t , wh at sta nds in need of'

IN TEN S I VE SC IENCE AND V I R T U AL P H ILOSOP HY

explanation is the decision to rob at all. He do es not really care

what . But for Sutton, that is the whole question . What is problem­

at ic is the choice of what to rob. 24

Garfinkel suggests that requests for explanations may be modeled as

questions having the form ' W hy did event X (as opposed to Y or Z)

occ ur?' with the clause in parenthesis constituting what he calls a

contrast space. The misunderstanding between the thief and the pri est in

his example is du e to the fact that each is using th e same question but

with different contrast spaces . While for the thief the question is 'W hy

rob banks?' (as opposed to gas stations or retail stores) for the priest

the question is 'Why rob banks ?' (as opposed to making an honest

living). The thief's answer is ind eed a true answ er, but as far as th e

pri est is concerned, it is an irrelevant answer, a fact that suggests that

the rel evancy and valid ity of an explanation is relative to a particular

contrast space . These spaces capture both what is presupposed in a

question (Gi ven that one must rob , why banks?), and hence considered

to be not in need of explanation, as well as the rel evant explanatory

alternatives . Garfinkel argues that characte rizing contrast spaces invol ves

going beyond the resources of language , even in cases (like the thief

and pri est exam ple) wh ere the situation is mostly linguistic. As he puts

it:

T hese contrast spaces are still not well-understood obj ects. Th eir

structure is not readily identifiable with any of th e traditional objects

of logic, for example . They have some similarities with ' possible

worlds', for instance, but th ey are not simply spaces of possible

wo rlds . Th ey are more like equivalence classes of possible worlds

(unde r the relation 'differs inessentially from') with almost all

possible worlds excluded altogether from th e space. (Contrast spaces

are typically quite small.) .. . Basically, these spaces are similar to

what physicists call stat e spaces. A state space is a geometric

representation of th e possibilities of a system; a parametrization of

its states , a display of its repertoire .25

I have already discu ed why linguisti cally sp Hied possible worlds

f: iI to break with esse ntialism, and how bringing in math matical

V IR TUA L ITY AND THE LAWS OF PH YS I CS

entmes (such as state spaces and th eir attractors) can eliminate the

need to characte rize rel evant alte rnatives (equivalence classes) through

relations like I differs inessentially from ' . In a typical nonlinear state

space , subdivided by multiple attractors and their basins of attraction ,

the structure of th e space of possibilities depends not on som e

extrinsically defined relation (specifying what is an inessential change)

but on the distribution if sinpularities itself. Th e traj ectories in state

space , defining possible sequences of states, are spontaneously broken

into equivalence classes by the basins of attraction : if the starti ng point

or initial condition of two different traj ectories falls within a given

basin both traj ectories are bound to end up in the same state, and are

equivalent in that respect. Garfinkel, in fact, acknowledges the rol e

which attractors may play in structuring the contrast spaces of physical

and biological explanations. As he says, 'What is necessary for a true

explanation is an account of how the underlying space is partitioned

into basins of irrelevant differences, separated by ridg e lines of cr iticalpoints. '26

How does a distribution of singularities obj ectively define th e

correctness or truth of a problem? Th e answer is that, as Del euze says,

'there are problem s which are false through indet ermination, others

th rough overdetermination'. 27 In other words, a problem may be false

or badly posed if the alternatives which str ucture a contrast space are

roo sharply difined , since in that case th e validity of the explanation

becomes too dependent on the occurrence of precisely those events

(overde te rmination). On the contrary, the problem may fail to be true

if it is so vapuely difined that it is impossible to tell whether an actually

occ ur ing eve nt belongs to on e or another of the relevant alternatives

(inde te rminat ion) . Let me give an example of a problem which is not

w II posed du e to its conditions being overdetermined. Garfinkel

illustrates this case with a well-known ecological phenomenon, the

rhythmic or periodic changes in the overall numbers of coupl ed

populations of prey and predators (rabbits and foxes, in his example) .

As the population of rabbits increases th e foxes' numbers also increase

du to the ex tra available food. But at some point, there are too man y

fox 's 0 that th population of rabbits is reduc d. This, in turn, brings

down the number of fa xes, which allows th rabbit population to

recover and sta rt the cycl again. Thi s cycl ic beha viour of the ouplcd

INTENS IVE SCI ENCE AND VIRTUAL PHIL OSOPH Y

populations is what is ecologically problematic about th e situation, that is,

what demands an explanation . 28

W e may pose the problem in two alternative ways, on e at the level

of intera ctions between individual rabbits and foxes, which gives an

ove rde te rmined contrast spac e with too many alternatives, and another

at the level of the overall density of th e populations yielding a well­

posed problem. To put this in linguistic terms, if we posed the

pr obl em 'Why was this rabbit eaten?', one answer may be framed at

the population level (because of th e large number of foxes) and another

at the organism level (because it passed through the capture space of a

speci fic fox at a specific time) . In other words, on e problem is 'Why

was thi s rabbit eaten (as opposed to not eaten)?' while the other is

I Why was this rabbit eaten (by this particular fox as opposed by this

or that other fox)?'. The second contrast space includes much that is

irrelevant to the question since , given a high enough density of foxes, if

this rabbit had not been eate n by this fox it would have been eaten by

ano ther. In other words, there is a ce rtain degree of redundant causality

ope rat ing at the micro-level, so that framing th e question at that level

is bound to yield the wrong distribution of the important and the

unimpo rtant. 29 The second way of framing th e question is, as Garfinkel

says, explanatorily unstable:

The gene ra l crite rion in th e cases we are dealing with is that an

objec t of explanation should be chosen whi ch is stable und er small

perturbations of its conditions. In the whole microspace of the foxes

and rabbits syste m there is a point corresponding to the death of

that rabbit at th e hands of that fox, at that place and time, and so

forth. Now imagine a kind of mesh laid over the space, which

det ermines what is to count as relevantly the sam e as that event.

[This is, in effect , the contrast space of the explanation.] If the mesh

is very fine, the resulting causal relations will be relatively unstable .

P .rturbing the initial conditions slight ly [say, making the rabbit pass

not so near that fox] will result in a situation which is different,

in '(Iuival -n t . [The rabbit not being eaten by that fox.] If however,

w · choose a mesh large enough (and cleverl y nou gh ) we can

·apturt· a sta ble rel ation , lik the on b twc m high fo population s

VIRTUALI TY AND THE LAWS O F PHYS ICS

and high likelihoods of rabbit deaths. [Where changing the path of

the rabbit st ill results in its being eate n but by another fox. 30]

Using the notion of explanatory stability , Garfinkel develops an

application of contrast spac es to differentiate the validity of expla­

nations operating at different scales of reality. In the context of a flat

ontology of individuals this differentiation is crucial since we would

like to have objective criteria to tell when an explanation is valid at

the level of individual organisms , for example , and when we need an

explanation at the spatio-t cmporal scale of an individual species. In th e

example just mentioned, a population-level intensive property (density)

can furnish a more stable explanation of the cyclic behaviour of the

pr ey-predator system than an organism-level on e. Similarly for expla­

nations of social phenomena, some will be adequate at th e scale of

individual subj ects, others will serve to answer Why questions at the

scale of individual institutions, and yet others will capture the relevant

causal effects of individual citi es or nation states.

In short, causal problems should be fram ed at th e correct level

given that each emergent level has its own causal capacities, th ese

capacit ies being what differentiates these individuals from each other.

But what about quasi-cau sal factors, how do they affect the success or

failure of explanations? To return to our example , if the properties of

the cyclic dynamics of the prey-predator syste m , the duration of the

cycle , for example , are not stable, that is, if exte rn al shocks can easily

change this duration, th en there is no need for quasi -causal factors.

But , on the other hand, if such shocks only temporarily change the

dura tion and th e cycle spontaneously returns to its original period,

then there will be an aspect of the dynamics not explained by the

causal model, a mechanism-independent aspect which still demands

explanation. Population biologists have in fact observed such stable or

ro bust cycles both in the field and in the laboratory, a fact that has

influen ced the introduction of attractors as part of their explanatory

mod cls.! '

I sho uld cmphasiz that, despite my choi ce of example , there i

nothing speci fically biological about this argument. Th e ex act same

ideas apply to syst im s of causally int ern ting populations of inorg. nic

I NTE N S I V E SC IEN CE AND V IRT UAL P H I L O S O P H Y

entities. I have mentioned seve ral tim es th e regim es of flow of

convection and turbulence . When explaining such phenomena one has

to frame the problem at the correct level so as not to introduce

irrelevant differences. Given a convec t ion cell and its cohere nt cyclic

behav iour, for example , there are a large number of micro-causal

descript ions (of indi vidual mol ecules colliding with on e another) which

are irrelevant to its explanation. In other words, there is a larg e causal

red undancy at the micro-level , with many collision histories being

.ompatible with the same macro-level effect: a coherent cyclic flow

pattern. Here th e proper level of explanation will involve macro-causal

factors: temperature and density gradients, competition between grav­

itational and viscous forces, and so on. Moreov er, th e existence of

ritical thresholds recurring at regular values for th e gradients (struc­

tural instab ilities) and th e robustn ess of the recurring flow patterns to

shocks (asymptotic stability) will call for additional qua si-causal factors:

bifurca tio ns and periodic attracto rs. (O r, in the case of turbulence ,

chaotic attractors.)

Let me pause for a moment to bring th e different lines of the

argument together, and th en link the conclusions to those reached in

previous chapte rs . I argued first that the axiomatic approach to classical

mechan ics, exem plified here by th e deducti ve-nomological model of

ixplanation , views laws as the main car rie rs of objective truth , a truth

which is then mechanically transmitted to th eorems via deduction.

Exp laining a given phenomenon is modelled as a logical argument ,

subsuming the truth of a theorem describing the phenomenon under

the trut h of a law. An alte rnative approach, a probl ematic approach,

I' jects the idea that fundamental laws express gene ral truths and views

th ern instead as posing cor rec t problem s. Problems are defined by

th ir presuppos itions (what is not being explained) as we ll as by their

contrast spaces (defining what the relevant options for explanation

ar i) . In the particular case of ex planations in classical physics, where

the laws are expressed by differential equations, th e presuppositions

arc th physical quantities chose n as relevant degr ees of freedom

(w hich makc up th different dimensions of a state space) while the

co nt rast spa is defined by a distribution of singularit ies in sta te space,

that is, by a part i ular partition of possibiliti es into disti nct basins of

attraction. As the xam pl of hydrod ynamic reg imes of 1I0w shows,

VIR T UA LI T Y A ND T HE L A WS OF PHYSICS

how ever, a contrast space may have a more complex structure: a

cascade of symmetry-breaking bifur cation s may link several such spaces

in such a way th at a problem may aradually specify itself as the different

cont rast spaces it co nta ins reveal themselves, one bifurcation at a time .

These conclusions are directl y connected with the onto logical ideas

I explo red before, but to see this connec t ion we must expand the

conce ption of probl em s beyond those involving scientific explanations.

In Deleuze 's approach the relation between well -po sed ex planato ry

problems and their true or false solutions is th e episte mological

co unte rpart of th e onto logical relation between the virtual and the

actual. Expl anatory problems would be the co unte rpart of vir tual

multiplicities since, as he says, 'the vir tual possesses the reality of a

task to be performed or a probl em to be solved' . 32 Individual solutions,

on th e other hand , would be the counte rpart of actual individu al

beings: 'An organism is nothing if not th e solution to a problem, as

are each of its differenciated organs, such as the eye which solves a

light probl em .P ? Let me illustrate th is idea with a simple example I

used before : soap bubbles and salt crystals, viewe d as the emerge nt

result of int eractions between their constituent molecules. Here th e

problem for the population of molecules is to fi nd (or compute its way

co) a minimal point of ene rgy, a probl em solved differently by the

molecules in soap films (which collec tively solve a minimization

problem state d in surface-te nsion terms) and by th e molecules in

crystalline structures (w hich co llectively solve a bonding ene rgy prob­

lem). It is as if an ontologica l problem, wh ose conditions are defined

by a unique singular ity, 'explicated' itself as it gave rise to a variety of

geometr ic solutions (spherical bubbles, cubic crys tals). 34

This intimate relation between episte mo logy and ontology, bet ween

problems posed by humans and self-posed virtu al problem s, is charac­

teristic of De1euze. A true problem, such as th e one which Newton

posed in re latively obscure geometric terms and which Euler , Lagrange

and Hamilto n progressively clarified , would be isomorphic with a real

virtual problem . Similarly, the practices of ex pe rime ntal physicists,

whi h includ e amo ng other thin gs the skilful use of machin es and

instruments to individu ate phenomena in the laboratory, wo uld b

isomorphic with the intensiv proc sses of individuation which solv

or .xpli at' a virtual problem in rea lity. This co nce ptio n of th task of

INTENSIVE SCIENCE AND VIRTUAL PH ILOSOPHY

theoretical and experimental physicists runs counter to the traditional

realist picture which views it as that of producing a corpus of linguistic

propositions expressing true facts which mirror reality. In this old and

tired view, the relation between th e plan e of reality and that of physics

wo uld be one of similarity . Yet, as Deleuze says, there is 'no analytic

resemblance, correspondence or conformity between the two plan es.

But their indep endence do es not preclude isomorphism . . . ' 3S Indeed,

as I said in the conclusion of the previous chapter, there is a further

isomorphism which must be included here: the philosopher must

become isomorphic with the quasi -causal operator, extracting problems

from law-expressing propositions and meshing the problems together

to endo w them with that minimum of autonomy which ensures their

irreducibility to their solutions.

In the second part of this chapter I would like to discuss the details

of these isomorphisms, one involving the experimental, the other the

theoreti cal component of classical physics . This will imply dealing with

both sides of the relation, that is, not only the laboratory and modelling

pra cti ces of physicists, but also the behaviour of the material phenomena

and machinery which inhabit laboratories as well as the behaviour of the

mathe matical models with which the theorist makes contact with the

virtual. I will begin with a discussion of how the capacity of material

and ene rge tic systems to self-organize and self-assemble, a capacity

which reveals a properly problematic aspect of matter and energy , is

co ncea led when physici sts or philosophers focu s on linear causality at

the xpe nse of more complex forms. Yet, I will also argue that even if

a material system under study has been fully linearized and domest ­

icated, the causal relations between experimentalist , machines, material

phenomena and causal models are still nonlinear and problematic. Indeed,

the physics laboratory may be viewed as a site where heterogeneous

assemblages form, assemblages which are isomorphic with real intens­

ive individuation processes.

I will th in move on to questions of quasi-causality and compare

Dclcu z ·'s episte mological approach to state space, an approach that

emphasizes the singularit ies that define the conditions of a theoretical

pr obl em, to thos of analytical philosophers who stre ss the solut ions

to the problem, that is, who sec not the singularit ies but the

trajectori es in state spa c as the conv 'yors o f theoreti cal knowl edge .

VIRTUALITY AND T H E L A W S O F PHYSICS

While trajectories bear a relationship of geometric similarity to

quantities measured in the laboratory, the singular it ies defining a

problem in physics are isomorphic with those defining the conditions

of a virtual multiplicity. Here too, I will argue that it is the behaviour

of linear equations that conceals the problematic aspect of mathematical

models. In short, whether we are dealing with causes or quasi -causes,

with experime nta l or theoretical physics, the crucial task is to avoid

the subordination if problems to solutions brought about by th e search for

simple linear behaviour. Let me begin with a quote from the philo­

sopher of science Mario Bunge on the conception of matter brought

about by excessive concentration on linear causes:

Before atoms, fields and radioactivity became pieces of common

knowledge, even scientists could be found that shared the belief that

' brute matter' is a homogeneous, unorganized and quiescent strif! entirely

lackinn spontaneity - th e matter, in short, dreamt by immaterialist

philosophers. From th e fact that every experiment is an encroach­

ment on matter, they jumped to the Aristotelian conclusion that

matter is nothing but the barren receptacle ifforms - a beli ef still held

in esteem by those quantum theorists who hold that it is the

experimente r who produces all atomic-scale phenomena. I"

And, I could add, st ill held in esteem by those cr itics of scien ce

who think that all phenomena are socially constructed. This conception

of matter as basically inert is directly linked to the defining character­

istics of classical causality, the most important of which is the simple

additivity of the effects of different causes. This apparently innocent

assumption is indeed full of consequences, some of which are fatal for

the philosophical project which I have sketched in these pages. In

particular, a flat ontology of individuals assumes that, at every spatio­

temporal scale, there are properties of a whole which cannot be

ex plained as a mere sum of the properties of its component parts, but

which emerne from their causal interactions. Without stable em ergent

pr operties, and the novel causal capacit ies these, in tum, give rise to,

the co ne pt of a larger scale individual collapses .

T he id a of additive causes becam dominant in physics for th

appare nt Simplicity \ ith which it endo w a syste m lind r study."? In

IN TEN SIV E SCIENCE AND VIRT UA L PHI LOSOP H Y

from the Aristotelian concept of efficient cause: externality . In this

view , causes are taken to be exte rn al agents operating on relatively

passive targets, hence being solel y responsible for whatever effects are

produced . The previous four traits of linear causality presuppose

exte rnality to the extent that they break down precisel y when the

body being acted upon ceases to be a mere patient . A failur e of uniqueness

occ urs whenever one cause can produce several effects depending on

the tendencies of the body it acts upon, and similarly for the case in

which th e same effect can be triggered by a variety of cause s. The

elimination of necessity in favour of enhanced probability and the

different probabilities of achieving an effect which a causal process may

transmit also depend on the probabilities to be affected carried by the

target of th e cause . And, of course, the failure of uni -di rectionality

and proportional ity are directly linked to the fact that the bodies acted

upon by causes are not passive but can rea ct back and exe rcise theirown causal powers. t "

Th e flat ontology of individuals I have defended in these pages

dep ends crucially, as I said, on th e elimination of linear causes , or,

at least , on cutting them down to size by showing them to be

speci al limiting cases. In this ontology individuals alwa ys exist as part

of populations in which the most meaningful and rel evant causal

r elations are of the statistical or probabilistic kind. None of the e

indiv iduals is ever a passive receptacle for extern al causal influences

sin e their int ernal causal structure always plays a part in determining

th final effect . Th e lack of uniqueness and uni-directionality is further

stre ngthened by the existence of quasi-causal relations. If the internal

dynamic of an individual is such that several alternative stable states

arc availabl e to it, it is hardly surprising that the same effect (a switch

bctwe n two attractors, for example) may be brought about by a

vari ty of causes , and conversely, on e and the same exte rn al cause

may trigger different effects depending on how close an individual is

to a bifurcation , or to th e border of a basin of attraction.

In short , whil e linear causality makes th e response of a material

syste m to an external cause basically unproblematic (given the cause ,

the re is nothing lse in th e effect that demands explanation), nonlinear

ami statisti al cau ality re-problemoti ze material syste m, showing th em

capable of s ·If-organization and self-assembly, with man y thin gs left

V IR T UALI T Y AND TH E L A W S O F PH YS I CS

un explained in the effect afte r the mere citation of an exte rn al cause .

In addition , linear and nonlinear causality impl y two different models

for the relationship between matter and form. Additivity and ex te rn al­

ity presuppose, as I said , a matter obedient to laws and constitu ting an

inert receptacle for forms imposed from the outside. Matter under

nonlinear and non-equilibrium conditions is, on the other hand,

intensive and problematic, capable of spontaneously giving rise to form

drawing on its inherent tendencies (de fined by singularitie s) as well as

its complex capacities to affect and be affected. As Deleuze says, the

first model :

assumes a fixed form and a matter deemed homogeneous. It is the

idea of the law that assures the model' s cohe re nce, since laws are

what submits matter to this or tha t form, and conversely, rea lize in

matter a given property deduced from the form . . . [But that]

model leaves many things , activ e and affective, by th e wayside. On

the on e hand , to the formed or formable matter we must add an

entire ene rgetic materiality in mo vement, carrying sing ularities . . .

that are alread y like implicit forms that are topological, rather than

geome trical, and that combine with processes of deformation : for

example, th e variab le undulations and torsions of th fibers guiding

the operations of splitting wood. On th e other hand , to th e essential

prop erties of matter deri ving from the formal essence we mu st add

variable int ensive eifJect5, now resulting from the operation , now on

the cont rary , making it possible : for example , wood that is more or

less porous, more or less elasti c and resistant. At any rate, it is a

question of surrendering to the wood, then following where it leads

by connec ting operations to a materiality instead of imposing a form

upon a matter . ..44

Although Del euze is referring here to artisans (carpenters in this

exa mple , but also blacksmiths) sim ilar conclusions appl y to exper i­

mental physicists . As Ian Hacking has forcefu lly argu ed, experime ntal

physics, far from being a mere app endage of theoretical phy ics

(supplying tests to confirm or disconfirm prediction s from formal

mod ' Is), has in fact a lif,· of it own . For example, the exp rim entalist

must indi viduate in a stable and r peat abl way laboratory phenomena .

INTENSIVE SCIENC E AND V IRTUAL PH ILOSOPHY

Rath er than being a mere by-product of theoretical knowledge of laws,

the indi viduation of phenomena involves, as Hacking says, 'a keen

ability to get nature to behave in new ways' .4-5 In the traditional

interpretation, thes e material and ene rgetic phenomena were supposed

to be unintelligible outside a th eoretical framework , but Hacking

shows that , on the contrary, laboratory phenomena (such as polariza­

tion of light, the photoelectric effect , Brownian motion) typically

survive the birth and death of new theories, or what amounts to the

same thing, the switching from on e to another incommensurable

theoreti cal paradigm. Many times the individuation of a phenomenon

not only precedes the development of a theory that will explain it, but

it remains in this problematic state, crying out for an explanation, for

man y decades. r"

Beside individuating phenomena that mayor may not occur nat­

urally , experimental physici sts must develop techniques and procedures

to isolate, identify and manipulate entities which have been individu­

ated by obj ective processes occurring outside the laboratory. In this

as too, it is a question of connecting op erations to a materiality

instead of deducing the form of the entities in question from a

theoret ical law . As Hacking argues, physicists individuate entities like

elect rons by int ervening causally in the world, int eracting with real

electro ns so as to determine their mass (as was done by Thompson in

1897), or th eir charge (as performed by Millikan around 1908 ), as

w II as othe r of their properties. t? The individuation of electrons (as

we ll as other formerl y theoretical entities) is even more complet e

when exper ime ntalists move beyond their properties to study their

apaci ties. W e learn from electrons, we acquire expertise about them,

by making them part of heterogeneous assemblages where they affect

• nd are affecte d by other ent it ies , and it is this causal know-how more

than anything related to general laws, which gives us confidence that

these individuals actually exist . As Hacking writes:

T he re are an enormous number of ways in which to make instru­

ments that rely on th e causal properties of electrons in order to

produ c d ired effects of unsurpassed precision . . . W e do not

mak instrum nts and th n infer the reality of the lectron , as

when we test a hypothesi , and th 'n b Ii -ve it hccaus ' it pass d th e

V I R TU A LI TY AND THE LAWS O F PHY SI C S

test. That gets the time-order wrong. By now we design apparatus

relying on a modest number of home truths about electrons, in

order to produce some other phenomenon that we wish to investi­

gat e . .. W e spend a lot of time building prototypes that don 't

work. W e get rid of innumerable bugs . . . The instrument must

be able to isolat e, physically, the properties of entitie s that we wish

to use, and damp down all the other effects that might get in our

way. We are completely convinced if the reality if electrons when we

reoularly set out to build - and iften eno up ]: succeed in buildino - new

kinds ifdevice that use various well -understood causal properties ifelectrons

to inteifere in other more hypothetical parts ifnature.4 8

It is in the context of these complex laboratory practices that the

causal models I mentioned before (th e part of the population of models

that interfaces with the actual world) are deployed. As the sociologist

of science Andrew Pickering has argued, experime ntalists , machines,

causal models and electrons (or other material entit ies) form, in the

conte xt of a particular experime ntal project, a heterogeneous assem ­

blage . Each of the se distinct components retains its heterogeneity but

they are meshed to on e another in a complex process in which causal

mod es are fine tuned to better adapt to the results of an experime nt ,

machines and procedures redesigned to change the way they affect and

are affected by phenomena, and skills sharpened to cope with 'unfore­

seen difficulties. In this assemblage each of the component parts plays

a role interactively stabiliz inq the whole. As Pickering writes, 'Scientific

knowledge should be understood as sustained by, and as part of,

inte ractive stabilizations situated in a multiple and heterogen eous space

of machines, instruments, con ceptual structures, disciplined practices,

soc ial actors and their relations, and so forth. ' 49

Following Del euze we may think about these complex assemblages

as the epistemological counterpart of the intensive in ontology. Much

as virtual multiplicities (view ed as self-posed ontological problems)

depend on intensive assemblages like ecosystems to progressively give

rise to ontological solutions, so expe rime ntal problems must first be

embodied in an int ensive assemblage prior to their being solved. In

I 'a rn ing by doing, or by interacting with and adjusting to material s ,

machin . and mod els, xperime ntalists proorcssivcly discern \ hat is

IN T E N S I V E SC I E NC E A ND V IRT U A L P HI LOSOP HY

relevant and what is not in a given experiment. In other words, the

distribution of the important and the unimportant defining an expe r i­

mental problem (what degrees of freedom matter, what disturbances

do not mak e a difference) are not grasped at a glance the way one is

supposed to grasp as esse nce (or a clear and di stinct idea), hut slow ly

hrought to light as the assemhlage stabilizes itself th rough the mu tual

accom modation of its heterogeneous components. In this assemb lage

the sing ularities and affects of the ex perimentalist 's body are meshed

with those of machines, mo dels and material processes in order for

learning to occ ur and for embodied exp ertise to accumulate. so On the

other hand , besides th is expertise (w hich may he applied in the design

and performance of other experiments and which, therefore, remain sintensive) there are also extensive or formal products of laboratory

practices: individual pieces of data, individual facts, individual solu ­

tio ns , whi ch take th eir pla ce in th e corpus of accumulated knowledge.

As Deleuze writes, 'Learning is the appropriate name for the subject ive

acts carried out when one is confronted with the objectivity of a

probl em . . . whereas knowledge de signates on ly the gen erality of

concepts or the calm possession of a rule enabling solutions. ' 5 1

To summarize , there are two different ways of subordinating

problems to solutions in the causal realm . One involves the eli mination

of the nonlinear causal capacities of the material systems under studyeithe r by homogenizing them or hy focusing on low-int ensity cquilib­

rium situations. In either case, one studies a matter so obedient to

laws that the productive aspect of causal connections may be disreg­arded and he reduced to a constant regu larity. What makes a material

system problemati c, what continuously demands new explanations, is

precisely the opcn -endedness of the assemhlages it may form, or the

multiple stable states in which it may exist and the abrupt transitions

it may undergo . But if we assume that there is always a unique stable

state, or that a cause always produces one and the same effect , wemay forget about the problem and focus on the sol ut ion: the constant

n'gularity itself as descr-ibed hy a law . O n th e other hand , one

subordinates problems to solutions when the complex causal interven ­

tions in reality which the ex perimentalist must perform, as we ll as themutual adjustments between machines, skills and 'a large number ofinh'r!ocking low level generalizations', S2 arc relegated to a secondary

VI RTU ALITY A ND THE L A W S OF P H Y S I C S

place and the formal cogn it ive products of thi s assemhlage are tak en as

the only worthy objects of phil osophical reflecti on . O nce detached

from their intensive individuation co ntext, where the experimental

learning of relevances and irrelevances takes place, these individual

items of knowledge become significant only hy reference to a theoret­

ical framework of laws and abstract concepts.

Let me turn now to the subordination of problems to solutions in

the realm of the quasi-causal. As I said before, the par t of the

population of models which interfaces with the virtual is not the one

composed of detailed models of causal mechanisms but the one

including the mu ch simpler on es expressing fundamental laws. Unlike

the case of co mplex causal models, the relation of problems to

solutions in the case of basic law s (and models directly derived from

them) may he approached using the results of Deleuze' s ontological

analysis of state space . State-space ideas do not apply to causal models

for two reasons. On e is their sheer co mplexity: the mathematical

techniques need ed to ana lyse sta te space are typically valid only for

models with a few degrees of freedom, defining a state space with a

low dimensionality, and are not at present sufficiently developed toapply to more com plex cases. This lim itat ion may be lifted on e day as

these techniques improve but there is a more important reason why

they w ill sti ll he of limited valu e to the experimenta list : state spaces do

not capture any iriformati on about causal processes.

Let me exp lain. In som e interpretations of state space the series of

poss ible states whi ch populate it (that is, the trajectories or sol ution

curves) are erroneousl y endowed with causal significance, with each

successi ve state viewed as the cause of the following one (or in some

interpretations, the initial state is taken as the cause while the final

state is the effect) . This is, indeed. a mathematical expression of thepos it ivist redu ction of the productiv e or genetic aspect of causes to a

process if" uniform succession (another version of Humc ' s regular conjunc­

tion). But as critics of positivism have pointed out. only actual events

can perform the genetic role of causes. As Mario Bunge argues . 'states

cannot have a productive virtue of their own. The state of a material

system is a syste m of qualities , not an eve nt or a string o f events .Evcry state is the outcome of a set of determiners . . . Consequently

then' can he no action of one state upon another state of a given

I NT E NS IVE SC I E NC E AND VI RT U AL PH I L O S O P H Y

systemj in particular, there can be no causal links among states, nor among

any other system of qualities . -ss

On the o the r hand, whil e the analysis of the state space of a model

may not provide us with causal information, it can be made to )ield

insight about quasi-causal relations . This epistemo logical result , how­

ever, depends on a particular ontological interpretation of the contentsof state space. Deleuze , as I said , does not view the differential

relations defining a model as expressing a law go verning the generation

of the serie s of sta tes that make up a trajectory, but as defining a

vec to r field whi ch captu res the overall tendencies o f th e syste m as a

distribution of singularities. 'Beneath the general operation of laws' as

he says 'there always remains thc play of singularities.' 'i'' These

singular ities define the conditions of the problem, ind ependently of its

so lutions, while each solution curve is the product of a specific

individuation process guided at every point by th e tendenci es in the

vector field :

Already Leibniz had shown that the calculus . . . expre ssed problems

which co uld not hitherto be solved or, indeed, even posed ... One

thinks in particular of the role of the regular and the singular points

which enter into the complete determination of the specie s of a

curve . No doubt the specification of the singular points (for

example, dips, nodes, focal points, centres) is undertaken by meansof the form of integral curves, which refers back to the solutions of

the differential equations. There is nevertheless a complete deter­

rnination with respect to the existence and distribution of these

points which depends upon a completely dilTerent instance, namely,

the field of vectors defined by th e equation itself . . . Moreover, if

the specificat ion of the points already sho ws the necessa ry imman­

ence of the problem in the soluti on, its involvem ent in the solution

which covers it , along with the existence and distrihution of points,

test ifies to the transcendence of the problem and its directive rolein relation to the organization of the solutions themselves.ss

To bring out the originality of Deleuze ' s ana lysis it will help to

co ntrast it with the analyses perform ed by analytical phil osophers wh o

focus exclusively on the epistemo logical role pla)'{'d hy trajectories . In

V IR T UALITY AND T H E L A W S OF PH Y S I C S

one approach, for example , the role of the trajectori es is to be used as

predictions about the specific sequence of values "vhich the relevant

properties of the syst em being mod elled will follow. The first ste p in

the procedure , according to this approach, involves making measure ­ments of the properties of a real system in a laboratory and plotting

the resulting numerical values as a curve. If the laboratory system isprepared in such a way that it starts its evolution in the same initial

conditions as the model, thcn this curve and the co rresponding state ­

space trajectory should be oeome<ricolly similar. A perfect match

be tween the two, with the state-space trajectory exact ly tracking the

plotted values, co uld then be inte rpre ted as meaning that the model is

true to the modelled system. Given that, du e to empirical limitations,

we cannot prepare a laboratory system to start at precisely the sameinitial conditions as an abstract mod el , the relation between plotted

values and predicted trajectories will not be a perfect match, so that

their relation will be one of approximate truth . Neverthel ess, it is the

geometrical similarity, or approximate similarity, between the twocurves that matters for epistemological purposes.56

An alternative view would disregard this extrinsic resemblance

between metric objec ts, and emphasize instead the common possession oftopological invariants. As one physicist puts it,

For present purposes, a system may be viewed both as a field of

phy sical phenomena in which a class of elements exhibits its

functions or behaviors in space and time , and as an abstract

description which presumably may be isomorphic with the physical

field . . . Two systems will be viewed as fun ctionally isomorphic

over a dynamic range if they have the same sing ularit ies of motion, inthe stability sense, over that range . 'i7

This would be the co rrect stance to adopt in a Deleuzian anal ysis. The

episte mo logical valu e of state space would be to reveal a topoloqicol

isomorphism between singularities in the model and singularities in the

physica l system bein g mod elled . T his isomorphism , in tum, would be

ex plained by sho wing that the model and the physical system are co­

actualizations of the same virtual multipli city (or o f part of the same

mult iplicity t given that the isomorphism is valid only within a rangl') .

I N T E N SI V E SCIE NCE AN D V I R T UA L P HI LO S O P H Y

Dcleuze's approach does not exclude the possibility that there can be

sim ilarities between traj ect ories and plotted values, but this resemb­

lance must itself be explained as a result of the common topological

properties of the systems producing the curves . The repl y that

possession of common properties is what makes a mod el and a real

ystem similar is, as th e phil osopher Nel son Goodman argued long

ago, redundant. As he put it, 'to say that two things are similar in

having a specified property in common is to say nothing more than

that they have that property in common'. 58

There is another way of stating th e differen ce between th ese tw o

philosophical approaches to the episte mology of state space . In the

analytical appro ach , the main episte mological relati on is that between

laws (ex pres sed by differential equations) and the traj ectories obtained

as solutions to tho se equations. This relation is one of Beneral to

particular. In other words, if we ignore the rol e which the vector field

plays in the individuation of traj ect ori es, it see ms natural to view laws

as stat ing a Beneral rule govern ing the volution of series of states , and

to see eac h t rajectory as th e result of appl ying that rul e for a particular

initial condition . In th e Deleuzian approa ch , on the co ntrary, th e

particu lar sta te at which a trajectory starts becomes irrelevant, given

that ma ny different start ing points within the same basin of at traction

end up in th e same place , the at tracto r. In othe r wo rds, it is th e

distribu tio n of sing ularities itself that det ermines what changes in initial

conditions are relevant (relative to the end state) and wh ich are

irrelevant , O n the other hand, the gene rality of the law (of which a

giv'n tra jec tory and plot of real values are particular instances) is

r 'p laced by the uni versality of virtual multipliciti es of wh ich both model

.1IIt! real syste m are divergent actualizations. As Deleuze wr ites,

'S ingularity is beyond particular proposition s no less than uni versality

is h 'yond ge ne ra l prop ositions. '59

The subse rv ience of problems to so lutions in the analysis of state

SIl.\CC is but one example of an error with a rather long history, a

' long perversion ' wh ich Deleuze traces back at least to Aristotl e ."?

O riginally, the subo rdinat ion deri ved fro m the habit of thought of

think ing abo ut problem s as if th ey were proposition s, that i , from

missin J the non -lingui st ic and ex tra-proposi t ional nat ure of th ir

VI R T UALI T Y A ND TH E L A WS O F PHYSICS

co nditio ns (contrast space). But in more recent times, in the historical

peri od when classical mechani cs developed , the surrende r to solutions

took a more specific, more mathematical form . To Del eu ze , math­

ematical problems are subordinated to their solutions when ever the

we ll-posedness of a problem is approached in terms of its solvability

(the possibility of finding a solution) . In the final sec tion of th is chapter

I would like to discuss two episode s in the history o f mathematics

where this traditional subordination was inverted , with solvability

becoming a consequence of the well -posedness of a problem . As I will

discuss in a moment, thi s inversion has for Deleuze revolutionary

consequences whose impact has not been gen erally appreciated . One

episode invol ves the history of algebraic equations, and the reversal of

th e subordination had, as one of its con sequences, the birth of group

theory. The other episode is more familiar, relating to the history of

differential equations , having as a result th e birth of the th eory of

dynamical syste ms, whi ch is th e source of the modern approach to

sta te space.

Let me begin by describing in very rough form the techni cal issues

invo lved in questions of so lvability in the case of algebraic equatio ns .

T here are two kind s of solutions to equations, particular and Beneral. A

partic ular solution is given by numerical values whi ch, whe n used to

re place an equation's unknowns, make the equation come out true .

(Fo r ex ample, an algebraic equation like x2 + 3x - 4 = °has as its

nu me rical so lutio n x = I .) A gene ral or exact so lution , on the othe r

hand, does not yield any specific value or set of values but rather the

Blobal pattern if all particular solutions. Thi s gen eral pattern is typi cally

give n by another equation or formula. The above example , which

may be written as x2 + ax - b = 0, has th e gen eral solution

x = V d h + b - ~ . When mathematicians speak of the solvability of

an equation th ey usuall y mean its exact solvability, and the subordi­

nation of problems to solutions ste ms from th e demand that a well ­

posed problem have an exact solution , not just numeri cal ones. By th e

sixteenth -ce ntury mathematicians knew that exact solvability was an

achi vable goa l, at least with equations wh ere the unknown variable

was raised up to th fourth pow er (that is, thos including x2 , x I and

x"). But th n a crisis ensued. Equations raised to th fifth power

INTENSIVE S CIENCE AND VIRTUAL PH ILOSOPHY

sed to yield to th e previously successful method . Was this lack of

:t solvability indicative that there was something wrong with the

olern as it was posed by the fifth degree equation?

'he answer cam e two centuries later when it was noticed that there

a pattern to the solutions of th e first four cases, a pattern which

ht hold the key to understanding the recalcitrance of the fifth,

wn as the quintic. First Joseph -Luis Lagrange and Neils Abel, and

I Evariste Galois, found a way to approach the study of this pattern

g resources that today we recognize belong to group theory . In a

.hell we can say that Galois ' showed that equations that can be

cd by a formula must have groups of a particular typ e, and that

quintic had the wrong sort of group' .6 1 I cannot go here into the

mical details of Galois's work but what he achieved was to invert

subordination of problems to solutions: rather than general solv­

ity defining the correctness of a problem, the form if the problem

me the explanation ifBeneral solvability. In other words, whil e before

exact solvability of the first four ca e was tak en for granted (as a

Jerty which problems mu st have) it now became something that

ld be explained by a uni versal feature of th e problem which these

, cases posed. This is what Del euze means when he says that 'it is

th e solution which lends its generality to the problem, but the

rlem which lends its universality to the solution ' , 6 2 a universality

ured in this case by a group of transformations. But how exactly

> a group of transformations capture th e universal conditions that

ne a problem as a problem, that is, independently of its solutions?

'0 answe r this question let me first take a different example, th e

of transformation groups to study the invariants of physical laws.

) of the mo st typical transformations in this case are displacem ents

pace or time. Given a law -governed physical process that can be

'oducc d in a laboratory, if we simply move it in space (for instance ,

-cproducing it in another, far away laboratory) we can expect the

liar aspects of its behaviour to remain invariant. Similarly, if we

)Iy hange th tim e at which we begin an experime nt , we can

ct this tim e displacem ent to be irrelevant as far as the regularity

he pro ess is on c rned . It is onl y the difference in time b tween

first and final states o f the process that matt rs, not the ab olute

at which the fir t tat e 0 ur s. Thus, via transformation appli ed

V IRT UA L I T Y AN D T H E L A W S OF P H Y S I C S

to the equat ions express ing laws, we can discover those types of

change to which th e law is indifferent, that is, the types of changes

whi ch do not matter as far as the law-like process is concerned . The

sense in which the group of an equation captures th e conditions of a

probl em is th en that it reveals distributions of the rel evant and the

irrelevant, th e irrelevance of using absolute time or absolute position

as inputs to a law for instance. It may be asserted without exaggeration

that understanding this connection had profound implications in the

history of physics playing a crucial rol e , for example, in the develop­

ment of the general theory of relativity. 63

Similarly, Galois's analysis of algebraic equations relied on the use

of certain transformations (substitutions or permutations of th e solutions)

which, as a group, showed what changes were relevant to the validity

of the equation (or more exactly, to th e validity of the relations

between solutions), More specifically, wh en a given permutation of

one solution by another leaves the equation valid, the two solutions

become, in a sense, indi stinpuishable as far as this validity is concerned.

The equation is indifferent to the switch. As Morris Kline writes, 'The

gro up of an equation is a key to its solvability becau se th e group

expresses the degree of indistinguishability of the [solutions]. It tells us

what we do not know about the [solutions]. '64 Or as Deleuze would

put it , the group reveals not what we know about the solutions, but

the object ivity if what we do not knoll' about th em, that is, the obj ectivity

of the problem itself. 6 5 Moreover Galoi 's method involves the equi­

valent of a symmetry-breaking cascade in that the solutions to the

equation become increasingly 'more accurately defined as the original

group gives rise to sub-gr oups which proBressively limit th e substitu tions

leaving the relations invariant. In other words, through a cascade which

unfold s th e original group, the problem itself becomes progressively

better specified and, as a by-product if this se!f-specification, individual

solutions emerge . As Dcleuze writes:

We cannot suppose that , from a technical point of view, differential

calculus is the onl y mathematical expression of problem s as such

... More r cent ly othe r procedures have fulfilled this rol e better.

Recall the cir I in which the th ory of problem s was caught : a

problem is .olvabl only to the ex te nt that is is 'tru ' but we always

I N T E N S I V E SCIENCE A ND V IRTUAL PHI LOSOP HY

tend to define the truth of a problem by its solvability . .. The

mathematician Abel [lat er followed by Ga lois) was perhaps the first

to break thi s circle : he elaborated a whole method according to

which so lvability must follow from the form of a problem. Instead

o f see king to find out by trial and error whether a given equation isso lvable in general we must determ ine the conditions of the problem

which progressively spe cify the field s of solvability in such a way

that the stateme nt contains the seed of the solution . This is a radical

reversal of the problem- solution relation, a more considerablere volut ion than the Copernican .w

T he reversal of the problem - solution relation also had revolutionary

co nseque nces in the case of differential equat ions . Although very

different from their algebraic coun terpart, equations in the calculus

alsn have particu lar and gen eral solut ions, both produced by th e

inh.'gration opcrator. As it happens, mo st differential equations cannot

he so lved by integration in a general or e xact way . Today we get

around this limitation by using computers to generate a population of

m,lny numerical so lutions , a popu lation which may be used to discoverlh,' general pattern . In the eighteent h century, wh en the physics whi ch

Ne wton and others had created was first given differentia l form, this

way out of the difficulty was not, of course , available . One conse ­

qucnce was the neglect of models who se constituent equations could

1I0t be solved exactly, given that without a way of knowing the ove rall

pattern of particular so lutions, physicists could not learn very muchfrom a model. Thus, in a vcry real sense, the solvability of a problem

was what made it worthy of study . As the mathematician Ian Stewartwrites :

Th e math emati cians of the eighteenth century ran headlong into a

problem whi ch has plagu ed theoretical me chanics to this day: to set

up the equations is one thing, to so lve them quite another . . . Thet'ightcl'nth ce ntury's main achievements were in setting up equations

to model physical phen omena . It had much less success in solving

them ... A process of se lf-selection set in, whereby cquations thatco uld not he solved were automatically of less interest than thosethat could."?

VIR TUALI TY AND THE LAWS OF PHYSICS

One can hardly blame these ma themati cian s and physicists for falling

prey to this process of self-s election, since they were operating within

the limits imposed by the mathematical technology of their t ime. On

the othe r hand , the long. tern l effccts of subordinat ing th e cho ice of

problems to their solvability did influen ce their (and their successors")

wor-ld view, biasing it towards a clockwork picture of reality. Thereason for this was that the equations that could be exact ly so lved

happened to be the lin ear equations. The mathematical difference

between linear and nonlin ear equations is ex plained in terms of thesuperposit ion prin cipl e. which states that g iven tw o d ifferent so lutions of

a linear equation, their sum is also a valid so lution. In oth er word s.

once we have discov ered a fcw solutions to an cquation many more

can he obtained for free via the superposition principle. In an eracharacte rized by the gene ral scarcity of exact solutions, such a principle

mu st have see me d like a gift from th e optim izing rationality of God .

Conversely , failure to obey thi s prin cipl e promoted the negle ct of

nonlin ear cq uationst" In the term s I have been using in this chapter

we may say that superposition, that is, a property of the behaviour ofsolutions, biased the process if accumulation that created the population

of models making up the theoretica l component of classical mechanics.

The requireme nt of exact solvability prom oted the accumulation of

linear model s at the expense of nonlinear ones , and even the fewnonlinear models allowe d to become part of the population were used

only in a linearized form . ( Linearization is achieved by using non linear

models only for very low int ensities of the recalcitrant variables. ) AsStewa rt puts it :

Classical mathematics co nce ntrated on linear equations for a sound

pragmatic reason: it co uld not solve anything else . . . So doci le arelinear equatio ns, that classical mathematicians were willing to

co mpromise their physics to get them . So the classical theory dea ls

with shallow waves, low-amplit ude vibrations, small temperaturegradients [that is, linearizes nonlinearities). So ingrained became the

linear habit that by the 1940s and 1950 s many scientists and

enginee rs knew litt le el se . . . Linearity is a trap. The behaviour of

linear equa tions . . . is far from typical. But if you decide that only

linear equations are worth thinking about, se lf-ce nso rship sets in.

I N T E N S I V E S CIENC E AN D V IRT UA L PHILOS OPHY

Your textbooks fill with triumphs of linear analysis, its failures

buried so de ep that the graves go unmarked and the exis te nce of

the graves goes unremarked . As the eighteenth century beli eved in

a clockwork world , so did the mid -twentieth in a linear one ."?

T he co unte rpart to Abel' s and Galois' s reversal of the problem-.

solution relation is represented by the work of Henri Poin care on the

qualitat ive (or topological) study of differential equatio ns . His was a

novel approach crea te d, like the group-theo re tic approach to algebraic

equa tions, to break through the barrier of a recalcit rant pr obl em: the

three body problem, the problem of modelling the mutual int eractions of

three solar system bodies (such as the sun , the earth and the moon).

Altho ugh other mathematici ans had already approached the st udy of

so lutions by analysing their behaviour in the neighbourhood of singular

points, Poin care approached the wider questi on of the way in which

the existence and distribution of singularities organized th e space of all

so lutions. In other words , like Galois, Poincare by-passed exact

so lvability as a way to get global information and instead used a novel

method to investigate the space difining the problem itself, that is, he used

the distributions of singular points as a way to gain qualitativ e

information about the tendenci es in the behaviour of all solutions. "?

Poin care' s phase-portrait approach to state space has, of course,

been the basis of mu ch of what I have said in this bo ok abo ut the

on to logy of the virtual and the problematic. But Galois' s approach has

also been crucial since it provided the idea of a progressive spe cificat ion

of virtual multiplicities through symme try -breaking cascades . In short ,

a theory of virtuality as has been pursued in these pages depends

fundamentall y on the results of th e reversal of th e problem- solution

relat ion , and conversely , subordinating problems to solutions may be

see n as a practi ce th at effective ly hid es the virtual , or th at promotes

tln- illusion that th e actual world is all that must be explained. Thus

construed , this subordination joins th e axiomatic treatment of classical

physics as a barrier to a more satisfactory probl emati c approach." In

.uldi t ion , there are th e obstacl es posed by the linearity of causes in

ex pe rime nta l physics, and the linearity of models in theoretical physics ,

both of which arc intimately related since the former' s addit ivity is

t'Cj ui\',lll'n t to the latt er' s supe rpos it ion. Additivity and supe rposi tion

VI R T UA LI TY AND THE L A W S O F PHY S I C S

characterize an unprobl emat ic world , or at best, a world whi ch is only

temporaril y probl ematic or in need of ex planation, but whi ch will

eventually yield to a supe r-law or a th eo ry o f everything whi ch will

leave nothing unexplained . On th e othe r hand, nonlinear model s and

the ir multiple at t ractors, as well as nonlinear causes and their co mplex

capacit ies to affect and be affected , define a world capable of surprising

us through th e emergence of unexpect ed novelty, a world where there

will always be some thing else to explain and whi ch will therefo re

re main forever problematic . As Mario Bunge writes :

If the joint acti on of severa l causes is always an exte rnal juxtaposi­

tion, a supe rpos it ion , and in no case a synthesis having traits of its

own, and if the hypothetical pati ents on which the causal agents act

arc passive thin gs incapabl e of sponta neity or se lf-activity ~

incapable, in short, of adding some thing o f their own to the causal

bond - then it follows that, in a sense, e}Jects preexist in th eir causes.

According to thi s extre me but consistent doctrine on the nature of

causation, only old things come out of cbanpe; processes can give rise

to objects new in number or new in some quantitative respe cts, not

however new in kind ; or again , no new qualiti es can emerge . A

,v-orld running on a str ictly causal pattern [i.e. a linear pattern] is

such as yog is, Thomists and eighteenth-ce ntury Ne wto nians ima ­

gined it, namely, a uni verse without a history . . .71

Unlike this linear world, the ontology I have devel oped in this book

is fully histori cal. Each of the individuals which populates thi s other

world is a product of a definite historical pro cess of individuation and ,

to the extent that an individual ' s identity is defin ed by its eme rge nt

propert ies and that these properties depend on th e continuing causal

inte ractions among an individual' s parts, each individual is itself a

historical causal process. The realm of the quasi-causa] is also fully

histori cal but, as I ex plained in the previous chapte r , it possesses its

own o riginal form of temporality and thus bear s no resemblance to

causal history. In ot her wo rds, in a Deleu zian onto logy there ex ist tw o

histo ries, one actual and one virtual, having complex int eraction s with

on e ano the r. O n one hand there is a historical series of act ual eve nts

gt'lwticall)' involved in the producti on of other events , and on the

I NTENSI V E SC IE NC E AN D V IRTU AL PHILOSOP H Y

other, an equally historical series of ideal events defining an objective

realm of virtual pr obl ems of which each actualized individual is but a

speci fic solution. To conclude with Del euze 's own words ,

It is correct to represent a double series of events whi ch develop in

two planes, echoing without resembling each other: real events on

the level of the engende re d solutions, and ideal events embedded in

th e conditions of the problem, like the acts - or, rather, the dreams

- of the gods who double our history. 7 ~

Appendix: Deleuze's Words

Gilles Del euze changes his terminology in every one of his books.

Very few of his conce pts retain their names or linguistic identity. The

poin t of thi s terminological exuberance is not merely to give the

impression of differ ence through the use of syno nyms , but rather to

develo p a set of different theories on the same subject, th eories whi ch

are slightly displaced relative to one another but retain eno ugh over­

laps that they can be meshed together as a het erogen eous assemblage.

Thus, the different nam es wh ich a given conce pt ge ts ar e not exact

synonyms but ncar synonyms, or sometimes, non-synonym ous terms

defin ing closely related conce pts . In th is book I delib erately homo­

genized th e tenninology for the sake of clarity but giving a list of

ncar synonyms will now prove useful to the read er as he or she

mov es back from my sim plified presentation of Deleuze' s ontology to

his original ones . In fact, beyond providing a mere list I will try to

map the connections between the different terminologies and discuss

the different ways in whi ch the onto logy is conce ptualized and artic­

ulated in each of the books. As I map th ese tenninological connec­

tions I will use the following abbreviations of Deleu ze ' s books,

followed when necessary by a page number (chapte r numbers refer

to the pr esent book):

Anti-Oedipus

A Thousand Plateaus

Difference and Repet ition

l.oOic if Sense

What is Philosophy ?

AOATP

D&R

LOS

WIP

The main sou rces used in my reconst ru cti on were D&R, where

the theory of multipli it ies and the virtual co nt inuum they form is

most clearly articul ated , and L S which pr es .nts the most detailed

A P P EN D I X : D E L E UZE 'S WORD S

description of the qu asi-causal op erator. I will begin this app endix with

a list of the compone nts of Deleuzc 's ontology (D&R, 277-8) . [ will

then expand the description of each of the seven compone nts of this

'o nto logical list ' , not onl y to relate them to the terminology used in

Illy pr esentation, but also to add details wh ich I left out for the sake

of simplicity but which are now necessary in order to rel ate the items

in thc ontological list to those in other books. Finally, [ will take three

hooks, ATP, AO and W [P, and map each component of the list to

their co unte rp arts there .

T HE ONTOLOGICAL LIST

( I) the depth or spatium in which inten sities ar e organizcd;

(2) the disparate series th ese form , and the fields of individuation that

th 'y outline (individuation factors) ;

(3) the ' dark pr ecursor ' which causes them to communicate ;

(4) thc linkages, internal resonances and forced movements which

result ;

(5) the constitution of passive selves and larval subj ects in the system ,

and the formation of pure spatio-ternporal dynamisms;

(6) the qua lities and exte nsions . .. whi ch form th e double differen ­

ciation of the syste m and cover over the preceding factors;

(7) the cent res of envelopment which neverthe less testify to th e

persisten ce of these factors in th e developed world of qualities and

ex te nsit ies .

l , Inten si ve Spatium

This term refers to th e virtual continuum formed by multipliciti es. In

this hook I used the term 'plane of consistency' to refer to it , a term

used throu ghout ATP. Other near synonyms include ' plane of imman­

l'll 'c' (W [»), ' body without organs' (AO , ATP) , 'rnachini c phylum '

(AT I'), and ' ide al or met aph ysical surface' (LO S). A possible sourc of

co nfusion h .rc is the term ' inte nsive ' which in my presentation was

lIsed ill relati on to indi viduation processes, not the virtual co ntinuum .

I)l, leu:l.l' uses the term in thr 'e se nses :

AP PEN D I X : DE LEUZE 'S W O RDS

a) Its original , thermodynamic sense in which it refers to inten sive

properties, like pr essure, temperature or density. Differen ces in

these quantities have a morphogen etic effect (they drive fluxes

of matter or ene rgy, for example) and when not allowed to get

cance lled (as in non-equilibrium physics) display the full potential

of matter-en ergy for sel f-o rganization.

b) A second derived sense in which it refers to the assembly of

different compone nts as such , that is, the cre at ion of het ero­

geneous assemblages in which the compone nts ' differences are

not canc ell ed through homogenization.

c) A third derived sen se in whi ch it refers to the properties of

ordinal series , Th ese ser ies ar e con sti tuted by the differences

between their terms, that is, by asymmetrical relations such as

'in between' . When we conside r more than one term between

two other s, this ser ial relation is called a 'distance ', although this

term must be qualifi ed (Deleuze speaks of 'non-de composable

distances' ) to distingui sh it from its non -te chni cal meaning where

it refers to a metric concept (such as ' length ' ) . Finally, there are

th e uncance llable differences, or constitu tive inequalities , which

ordinal ser ies present when compared to one another (on ly

judgments of greater or lesser are possible, not of exact equal­

ity) . lt is mainly in this third sense that the term is used in the

expression 'intensive spatium ' as the following quote shows:

Differe nce , distance and ineq uality are th e positiv e charact eristics of

depth as intensive spatium (D&R, 238),

2. Multiplicities and Divergent Series

Altho ugh the term 'm ult iplicity' is not used in the list above, it is clear

that it belongs in this entry since the 'disparate series' mentioned are

no thing but the effect of expanding in a serial form the singularities

d 'fining each unfolding level of a mu ltiplicity. The term has some near

s Ilonym s: 'partial objects ' (AO ); ' philosophical co nce pts' (W [P);

' idea l eve nts' (LOS). Some times Dcleuze refer s to multipliciti es

indirectly via their compo nents , suc h as ' nomadic singularit ies' and

APPE NDI X : DELEUZE 'S W O R D S

'noe matic attr ibutes ' (LOS) , or 'vague essences' and 'becomings'

(AT P) .

Th e term 'disparate ' means ' d ifference of difference' (D&R, 241 ) .

To speak of 'disparate ser ies ' is another way of expressing the idea

that the ordinal ser ies which form the nonmet ric continuum mu st be

rela te d to one another via '!ffirmative divergence, so that not only are the

se ries mad e up of differences, their divergent relations further differen­

tiate these d1Jerences:

Di fference mu st become the element , the ultimate unitv; it must,therefore refer to other differences which nev er identify it but

rath er differentiate it. Each term of the series , being already a

differen ce, must be put into variable relations with other terms,

th ereb y co nstit ut ing other se ries devoid of cente r and convergence.

Divergen ce and decentcring must be affi rme d in the se ries itself.

(D&R, 56)

3. Dark Precursor

T his term refers to what in my reconstruction I called the 'quasi-causal

operator '. Its near synonyms include: 'quasi-cause", 'al eatory or para­

doxical point' and ' nonsense ' (LOS); 'line of flight ' and ' abstract

machine ' (ATP); 'desiring machines' (AO) ; 'conceptual person ae'

(W IP); 'o bject = x ' (D&R, LOS) .

4. Resonances and Forced Movements

This entry includes the effects which the quasi-causal operator has on

the multipliciti es and their series . In my reconstruction I used an

info rma tion-theore tic model for these effects (in terms of emissions of

signs or informati on qu anta) but Del euze also uses an alt ernative physical

model in terms of resonances (D&R, LOS , WIP) . The te rms ' resonance'

and ' forced movem ent.' should not be taken as mere physical met aphors.

Rath er , we sho uld think about resonance as positi veftedback, a gene ric

pr on'ss which implies one or o the r form of mutually stimulating couplin8s

k.g. .autocatal ysis} inducing resonance s amo ng het erogeneous elem ents,

as well .lS the ampljfica t ion iforiginal djfferences (forced mov em ents) .

AP PEND IX : DELE UZE 'S WORDS

The crucial idea is that th e qua si-causal op erator m ust couple the

ordinal series ema nating from multiplicities so as to weave these into a

nonmctric co ntinuum. Resonances are the means to effect couplings,

while the resulting forced movem en t produces the continuum (LO S,

239-40). As I have just said, th e couplings between ser ies must ensure

the ir affirmative divergen ce , keeping the continuum op en and in

co nstant variation . But also , as a separate operation (what I called ' pre­

act ualizatio n ' in Chapter 3) , it must induce some convergences in the

se ries, since it is in these cent res of co nvergence that th e process of

actua lizat ion begins :

To be actualized . . . mean s to extend ove r a ser ies of ordinary

points; to be selecte d according to a rul e of converge nc e ; to be

incarnated in a bod y; to become th e state of a body; and to be

ren ewed locally for the sake of limited new actualizations and

ex tensions . (LO S, I 10)

5. Passive Selves and Spatio-Temporal Dynamisms

T his entry contains the two components of what in m y reconstruction

I referred to as 'inten sive individuation processes' . The first meaning

of the term 's patia-tem poral dynamism ' is straightforward, referring

to the phenomena of sel f-organizat ion which occur in many non­

equilibrium systems. Self-organizing dynamics ar e typically governed

by the singularities (at t racto rs and bifurcations) which chara cteri ze

differential relations (that is, coupled rates of change or relations of

re lat ive rapidity and slowness.) In thi s se nse, th e term relates to the

first sense of the word 'intensive', as in a non-equilibrium material

where inten sive differen ces have not been cancel led . But the term also

refers to 'a ffects', or th e second sense of 'inten sive ' , that is, to the

capaci ties and dynamisms whi ch produce heterogeneous assem blages.

That the tw o senses are intimately connected is clear from the

''''lowing:

It is no longer a qu estion of imposing a form up on a matt er but of

d ahorating an increasingly rich and co nsistent materi al , the better

to tap incr easingly int en se forces . What makes a matt' rial incrcas-

A PPEND IX: D ELE UZE 'S WORDS

ingly rich is the same as what holds heterogen eiti es togcther without

the ir ceasing to be heterogeneous . (ATP, 329)

Unlike spa tio- te m po ra l dynamisms, the terms passive sel f' and

'larval subject' recei ved very little elaboration in my recon struction ,

mostly because I wanted to keep the description of Deleu zes ontology

as free from anthropocentrism as possible . The first term is related to

the 'passive synthes is' whi ch forms the core of Dclcu zc ' s theory of

time , the synthesis of 'living presents' which metricize or give measure

to time . In his theory, this synthesis is directly related to the gen esis

of subjectivity (it is a co nte mplative subject who contracts instants into

a present ) but, as I ex plained in Chapter 3, these 'co ntemplations'

occur everywhere, in the form of proto-perception s and proto-feelings

wh ich even microscopi c individual entities may be said to have . Hence,

we not only contract instants to synthesize our psychological sense of

present , we are made out of micro-contractions and their presents:

W c arc made of contrac ted water , earth , light , and air - not onl y

prior to the recognition or representation of these, but prior to

their being sensed . Every organism, in its recept ive and perceptual

el em ents, but also in its visce ra, is a sum of contractions, of

retentions and expec ta t ions . (D&R, 73)

The term 'larval subject' is clo sely related to these ideas, referring

to the 'vo luptuous consumption' of the intensities which drive spa tio­

tempo ral dynamisms . The best ex am ple here is the developing embryo

as it experiences the intensive fold ings , migrations, and oth er transfor ­

mations which will eventually turn it into a fully formed organism .

Indeed, unlike my recon struction where the term 'individual' refers to

the final produ ct (o rganisms, species, etc.) in Deleu zc 's work it refers

to the lar val subjec ts themselves. It ofte n has the mean ing of a

Lc ibnizian 'mon ad', and it is said to be born during pre -actualization ,

that is, from the ce nt res of co nvergence whi ch occur in the virtual

series :

A world already envelops an infinite syste m of singularities selected

through co nve rgence. Within this world, howe ver, individuals are

APPEND IX : DELEUZE 'S WORDS

co nstituted which select and envelop a finite numb er of the singular­

ities of the system . . . An individual is therefore always in a world

as a circle of co nvergence, and a world ma y be formed and thought

on ly in the vicinity of the individuals whi ch occupy or fill it. (LO S

109-10)

To avoid co nfusion, I will usc the term 'intensive individual ' to

refe r to these monad s, and 'individual' without qualification to refer

to the ex tended and qualified actual entities whi ch form my flat

ontology of indi viduals .

6. Extensities and Qualities

These are the two characte rist ics which define the realm of the actual,

the fully co nstituted world of ex te nde d and qualified individuals. In

ATP these two characteristics are referred to as 'substances ' and

'forms' respecti vely . To sec the connec tion one needs to think , on the

one hand, of a substance without any oth er characte ristic than its

manner of occupying space (its extension), and, on the other hand, of

the form s or structures wh ich endo w this substance with specific

qua lit ies (such as its mechanical or optical properties) . Given that no

act ua l substance is ever purely ex tensional, these two characteristics

arc 'not really distinct. They are the abstract components of every

articu lat ion ." (AT P, 50 2)

7. Centres of Envelopment

This co nce pt was not discussed in my reconstruction. I introduce it

here not only bec ause it appears as the last item in the listing of

ontological components under discussion , but also because its defin ition

relates to aspects of the theory of the actual whi ch bear on questions

of te rmino logy. The different spheres of the actual (roughly, the

physico-chemical, organic and cultural spheres) need to be conce ived

witho ut presupposing a teleological devel opment or ' any kind of

rid iculous cos mic evolutionism' (AT P, 49) . There are , on the other

hand, very real distinction s bet ween these spheres . In particular, unlike

the physico -che m ica l sphere wh ere the 'co de' that underlies form s o r

A P P E N D I X : DELEUZE 'S W O R DS

q ua lities is distributed throughout the three-dimen sionali ty of a struc­

ture , in the organic sphere this code becomes detached as a separate

one -dime nsional structure: the linear seque nce of nucleic acids consti­

tuting the gene tic code, The ge net ic code, in Deleuze 's view, repres­

ents an interiorization if the intensive individuatinq factors whi ch in

physico-chemical st rata remain ex te rn al to indi vidu als. Thi s int eri ori ­

zation, which characte r izes the increase in complexit y of living syste ms ,

is what is referred to by th e term 'centres of envelopme nt' :

Th e functi on of these centres may be defined in se veral ways

we claim that complex systems increasingly tend to interiorize their

co nstitutive differen ces: the centres of envelopment carry out this

interiorization of the indi viduating factors. (D&R, 256)

Summary

Let me now summarize what I have just said about the co nte nts of the

ontological list. Items 1, 2 , and 3 co nstitute the eleme nts of the virt ual:

the co nti nuum , the multipliciti es and the quasi-causal ope rato r. Items

4 and 5 may be made to cor res pond, with a bit of tw eaking , to th e

intensive . The reason why some tweaking is necessary is that it invol ves

s 'parating the divergent and the conve rgent relati ons between the

xer ic , the former belonging to the virtual and the latter (as a kind of

pre-actualization) to the int ensive. Centres of conve rgence would

orrcspond to wh at some scie ntists call ' mo rphogenetic fields ' , o r what

Dc leuzc calls ' fields of indi vidu ation '. Although Deleu ze includes as

part o f Item 2 'fie lds of indi viduation ' , and the resonances of Item 4

also produce di vergen ces, it will prove useful to keep the tw o Items

.lpart and define the inten sive both by the fields of individuation and

th . spa tio -tc rnpo ral dynami sms that perform th e actualization of these

field s. Pinally, Items 6 and 7 form the conte nts of the actual. Precisely

because the vir tual, the int en sive and th e actual are aspects of one and

til(' same process, or th e different mom ents o f a cascade of progressive

d iffer mtiation , some Items (4 and 7) represent areas of ove rlap

(so l!wthing of th virtual, onv rg nc , within the inten sive ; so me thing

of the inte nsive, cnv .lo pmc nt ntrcs, in th actual). Let me now show

hm\ tln- virt ual, the int en sive , and th actual arc trc: ted in other books.

APPEND IX : DELEUZE 'S WORDS

A THOU SAND PLAT EAU

In ATP the different spheres which make up the actual world (physico­

chemical, orga nic , cultural and so on) are called 's trata' . The term

's tratification' is near syno nymo us with 'actualization'. The different

extensit ies and qualities which characte rize the actual world are

referred to as 's ubstances' and ' forms' , and also as ' te r ritorialit ies' and

' codes ' . Thus, Del eu ze writes that strata ' proceed simultaneously by

code and by territoriality ' (AT P, 40) . The inten sive processes which

give rise to strata, and which become hidden und er strata, ar e th erefore

called 'territorializat ion ' and 'coding' . Given that some parts of the

wo rld may be pu shed away from their equilibr ium state , ther eby

revealing the hidd en inten sive factors, the terms 'de ter ritor ialization'

and 'decoding ' ar e used to refer to th ese departures from the rigidity

of strata, or rather, to the inten sive movem ents which animate strata

fro m within. In D&R, Deleuze had alre ady introduced the notion of

'dc-differe nciation' (D&R, 249) but it is onl y later that thi s notion

aC'luires its full importance and that it is divid ed along the two

components of actualizati on.

Indeed , as I argued in Chapter 3, th e quasi-cau sal ope rato r may be

said to accelerate th ese departures from actuality in an ope ration called

'counter-actualization ' . In ATP , Deleu ze speaks of ' re lative det errit­

orializations' to refer to moveme nts away fro m the actual toward s th e

intensive , and of 'abso lute det erritorializati on ' to refer to counte r ­

actualization , the acce leration of these movemen ts allowing them to

reach all the way int o the virtual. The three compo ne nts of the virtual

(the continuum , th e multiplicities that co mpose it and th e quasi -causal

ope rator which effects the composit ion) have exact counte rparts in

T P as the following ex tract illustrates:

There was a first gro up of notions: the Body without Organs or

dcstrat ified Plane o f Co nsiste ncy ; the Matter of the Plane, that whi ch

oc .urs in the bod y or plane (sing ular, non segm ented multipliciti es

composed of int en sive cont inuums , emissions of particle-signs, con­

junctio ns of flow ); and the Abstract Machine , or Abstract Machines, in

Sl) far. s they co nst ruct that bod y or draw the plan e or 'diagram' what

oc urs (line. of flight , or • hsolut det errit orialization) . ( T P, 72)

APP E NDIX : O E LEUZ E 'S W OR D S

Multiplicities ar e said to ' occ ur ' in the plane of consiste ncy because,

as I argued , they are ideal events or becomtnqs. The term ' no nseg mente d '

should be read as near synonymous with ' nonmetr ic", and ' inte nsive

continuum ' as 'ordinal co ntinuum' . The ' emi ssion s of particle-signs '

arc the resonances that couple the multiplicities, and the 'conjunctions

of flows' correspond to mutual amplifi cations or forc ed movements,

The quasi-causal operator, here called the 'abstract machine', is

characterized in terms of 'lines of flight' which refer to th e process of

counte r -actualization, and is said to 'draw th e plane ', that is, to extract

ideal eve nts from what actually occu rs and to mesh these multipliciti es

into a heterogeneous cont inuum . As Deleu ze writes 'the plane of

consiste ncy does not preexist the movements of deterritorialization

that unravel it, the lines of flight that draw it or cause it to rise to the

surface , the becomings that compose it ' (AT P, 270). Finally, the

' centre s of envelopme nt ' are not given a special nam e but they are

referred to indirectl y when it is asserted that ' the abstract Machine

exits simultaneously devel oped on the destratified plane it draws, and

enveloped in each st ratum whose unity of composit ion it defines . .. '

(AT !' , 70 ; my emphasis) .

This is, roughly, the mapping from one set of terms to another .

But in ATP we witn ess an elaboration of the original ontological

components and thi s introduces new terms and ideas . In particular,

in ATP the actual world is not defined simply in terms of extensit ies

.1nd qualities, but of vcry spe cific articulations of th e exte ns ive and

the qualitati ve . As I discu ssed in my reconst ruction, the actual consists

exclusively of individual entit ies , eac h individual at a given level of

scale emerging from the interactions of populations of smaller scale

individ uals. Deleu ze refers to these two scales of every stratum as the

' mo lecular ' and the ' molar ' . Stratificati on co nsists in producing popu ­

lations of ' mo lecules ' and organizing them into ' molar' , or large

scale, aggregates . (Clearly, 'molecul es' may be cell s or even organ­

isms, when the molar scale is that of the organism or the species,

rcspcctivelv.} Thus, every stratum needs a double articulation, a

doub] o play of substances and form s, of exte nsit ies and qu aliti es, one

at the level of mol ecu lar populations and another at the level of molar

.lgg reg.lt es :

AP P EN DI X : DE L EU Z E 'S WO RDS

The first artic ulatio n chooses or deducts, from unstabl e particle­

flows, meta stable mole cular or quasi-m olecular units (substances)

upon which it imp oses a statistical order of connect ions and

successions (forms) . The second articulation establishes functi onal ,

compact , stable structures (/orms) , and constructs the molar com­

pounds in which structures arc simultaneously actuali zed (substances) .

(AT P 40 -1 )

This process is called a 'double articulation ' . Although the term

'double differen ciation ' alr eady occurs in the ontological list , it refers

only to the pair substance and form , not to thi s more elaborate

inte rpl ay of territorialiti es and codes . A similar elaboration is evident

in Del euze 's treatment of th e inten sive . As I argued in Chapter 2,

eve n the most rigidl y metric (or ' most stratified ') indi vidual still has

unactualized capacit ies to affect and be affected, and ma y not be

limi te d to a sing le stable equilibrium but have a vari ety of unactualized

stable states available to it. Th ese two aspect s of the int en sive , 'affects'

.1I1d 's ingularit ies' , become further developed int o 'parastrata ' and

'cpist rata' in ATP . On one hand, affects endow individuals with the

capacity to establish novel connections with alien mili eus, as with the

evolution of the capacity to tap into a reservoir of ox ygen, or other

non -alimentary ene rgy sources . Organisms may also have the capacity

10 act ively shape thei r environme nt , as spide r webs or beaver dams

illustrate . These capacities are what Delcuze calls 'parastrata", the

(·.lpad ty to connect with an 'anne xe d or associated milieu' (ATP, 5 1) .

O n the other hand, a fully formed individual may be capable of a

\ ,u'kty of stable states whi ch may be actualized by crossing crit ical

points and give rise to ' variat ions that ar e tol erated below a certain

threshold of iden tity ' (AT P, 50) . Th ese ' intermed iate states or milieus'

.m- what Deleuze calls 'epistrata". As he writes, even ' a single chemical

substance (sulfur or carbon , for example) has a number of more or

h'ss dcu -rrit orializcd sta tes' (AT P, 53) . The relations of the different

t('l'ms for int en sive factors can then be summa r ized like this :

Fo rms re late to codes and proccsst.~s of coding and decoding in the

IMf.l-"t rata; substances , l»..·ing formed matt ers, relate to tcrr'itorialities

APPENDIX : DELEUZE 'S W O R D S

and mov em ents of territorialization and deterritorialization on the

epistrata. (AT P, 53 )

Finally, there is a term which refers to the actualizatio n (or

effectuation) of th e quasi-causal ope rato r itself. I did no t discuss this in

d .tail, but I did give an ex ample in Chapte r 2 of the neighbo urhoo d of

a pha e transiti on (or 'e dge of chaos'). Deleu ze 's own example is not

crit ical points in a line of values, but crit ical sUIfaces in objects with

volume (LOS , 103) . (In both cases the quasi-cause opera tes at an N-Idim ension, as discussed in Chapte r 3) . In ATP, the organic membrane

as a critical surface is kept as an instance of the quasi-cause as it exists

effect uated in the actual , organizing the division of epist rata and

paras trata (AT P, 49-50). But now a spec ial term is coined for this

actualized qua si-causal ope rator : ' machinic assemblage' . As he writes:

'The most important probl em of all : given a machini c assemblage ,

what is its relation of effectuation with the abstract machin e? How

do cs it effectuate it, with what adequation ?' (AT P, 71 ) .

Much as th e quasi-cause o r abstract machine endows the vir tual

continuum with consiste ncy , the machinic assemblage endows actual

ent ities with consiste ncy , 'What we term machini c is precisely this

synthesis of heterogen eiti es as such' (ATP, 330) . The machini c assem­

blage performs the different ope rations invo lved in stratification, suc h

as articulating a stratum with whatever serves as its subs tratum (e.g.

the pr e-biotic soup for organic strata), as well as doubly ar ticulating

the different extensit ies and quali ties, substances and forms, which

defin e a give n stratum (AT P, 71 ) . But also , as an actualized quasi­

cause , the machini c assemblage is the agent behind co unter­

actualizatio n:

T he assemblage has two poles or vectors: on e vec tor is oriented

toward s the strata, upon which it distributes territorialities, relative

dct c rr itori alizati ons and rct erritori alizations: the othe r is oriented

toward s the plane of consiste ncy or destratificati on , upon wh ich it

co njugatcs pr ocesses of det crrit orialization , carry ing them towards

till' abso lute of the eart h. (AT P, 145)

APPENDIX : DELEUZE 'S WORDS

A TI -O EDIPU S

In th is book the mapping of the items of the onto logical list is less

straightforward . In particul ar , the virtual and the intensive are gro uped

togethe r in a pro cess whi ch is referred to as ' mo lecular' (in the sense

just mention ed ), while the actual is referred to as ' the molar '. Unlike

ATP, wh ere all kinds of strata ar e co nside re d, in AO only the

actualization of human soc ieties is dealt with , so the molar see ms to

beco me synonymous with ' large social aggregat es', such as stable

persons, govern me ntal or economic insti tution s, agricult ural or indus­

trial machin es. But it sho uld be kept in mind that thi s narrowing of

the meaning of ' the molar ' is a matter of focus and not a change in the

underl ying theory.

W ith some care , in fact , th e different elem ents of the onto logical

list can be paired with their counterparts in AO. The virt ual and the

intensive processcs of actualizati on are referred to as 'des ir ing produc­

tio n' and defined as consist ing of three separate ' passive syntheses'

(AO , 26). These are referred to as ' the connective', ' the disjuncti ve '

and ' the conjunctive' syntheses. (This three-p art classificati on first

appears in LO S, 174. ) Th e disjunctive synthesis involves th e crea tion

of divergent relations among se ries , and it is said to occ ur on the bod y

without organs (AO, 13). It therefore re fers to th e virtual continuum,

'a pure fluid in a free sta te, flowin g without inte rruption, streaming

over the surface of a full body' (AO , 8) . The conjunctive synthes is, in

turn, invo lves the cre ation of conve rgent relat ions am ong series, an

operation whi ch as I said above , forms ' individuation field s ' which

already prefigure the intensive (pre- actualization). Thi s synthes is cap­

tures one of the aspec ts of th e inten sive , the emergence of a larval or

passive subjec t, 'a strange subject with no fixed identity, wandering

abo ut ove r the body without organs .. . being born of the [int en sive]

states that it consumes , . " (A0, 16) . Finally, th e connec t ive synthesis

captures ano the r aspect of th e inten sive, the machini c assemblage. It

connects or couples together het erogcn eous 'partial objects or organs'

thro ugh the emiss ion of 'e ne rgy flows' (AO, 323) . Here the term

' partial' is not used in its xt in ive sense but in the sense of matter

filling spa • to a give n d 'g r of int ensity. 'The ye, the mouth , the

.lJlII S degrees of matter" ( 0, 309) .

APPENDIX : DELEUZE ' S WORDS

This interpretation of the three syntheses gives us one of the

elements of th e virtual (the plane of co nsistency or bod y without

organs), and tw o of the intensive (larva l subjects, assembl ages), but

leaves several things out. In particular, the other two elements of thevirtual, mult iplicities and the quasi-causal operator, don 't see m to be

included. Multipliciti es appear in AO as ' partial objects' when these

'a tt ach themselves to the body without organs as so man)' point s of

disjunction between which an entire network of new syntheses is nowwov en marking the surface off into coordinates , like a grid ' (AO, 12) .

This co rresponds to the idea that multiplicities exist in the sphere of

the intensive embodied in self-organizing processes, but may beextracted from these as 'Rat multiplicities' or 'pure events' and

deployed as such on the plane of consistency. The quasi-causal operatoris, in turn, referred to as a 'desiring machine':

Insofar as it brings together - without unif}'ing or uniting them ­

the body without organs and the partial objects, the desiring

machine is inseparable both from the distribution of partial objects

on the body without organs, and of the leveling [i.e. flatt ening]

e ffect exerte d on the partial organs by the body without organs ,

wh ich results in appropriation. (AO, 327)

The desiring machine is said to have 'chains' as its apparatus oftransmission (AO, 327). The term 'chain' is used instead of 'series'. It

has the meaning of a 'Markov chain' (AO, 39), a series of events in

which the probability of occurren ce of an y event depends only on the

previous one in the series . In other word s, a 'chain' is a partially

aleatory series. This co rresponds to one of the effects of the quasi­

cause, bri efly discu ssed in Chapte rs 2 and 3, of injecting chance in the

distrihutions o f virtual singularities to create 'nomadic' distributions,

,1S opposed to the 'sede ntary' probability distributions which character­

ize population s in the actual world. This is also expres sed by saying

that the quasi-cause must affirm all of chance with every throw of the

dk-c ( LO S, 59 -60) . The term 'chain ' is also used as in the expression

'signil),ing chain' hut without any reference to a fixed code , linguisticor otherwise . Rather these heteroge(wotls chains arc mad e of 'Hying

br-ic-ks , . . containing within [them] not only an inscription with signs

APPENDIX : DELEUZE'S WORDS

from different alphabets , but also various figures, plus one or seve ral

straws, and perhaps a co rpse' (AO , 40).

There is onc more detail to be discussed which provides an

important bridge to the nex t book to be deciphered (W IP). Much as

multipliciti es are woven into a virtual continuum through their diver­gences, but also form individuation fields when their series converge,

' the points of disjunction on the body without organs form circles that

co nverge on the desiring machines; then the subject , , . passes through

all the degrees of the circle, and passes from one circle to another'(AO , 20). The term ' passing' is used here as synonymous with

'becoming', and the 'degrees of the circle' are 'intensive quantities in

their pure state ' (AO , 18). The idea here is that this larval subj ect

w ith out identit y can move about the plane , from one individuation

field to another , becoming now this and now that intensive individual

depending on the intensities it consumes . This is the key idea behindthe process which in AO, ATp and WIp is referred to as 'becoming­

animal' (as well as 'becoming-w oman", 'becoming-mo lecule' , etc .).

The co nce pt app ears first in D&R , 254:

W e sho uld not say tha t individuals of a given species arc distinguished

by their participation in other species: as if, for exampl e , there was

ass or lion, wolf or sheep, in every human being, There is indeed

all that and metempsychosis retains all its symbolic truth. However,

the ass and the wolf can be considered species only in relation to the

fields of individuation . . . lit is true that sorncone's soul) never

change d bodies, but its bod y could be re -env elopcd or re -irnplicated

in order to enter, if need be, other fields of individuation . . .

In other words, becoming-animal is an operation which cannot be

perfo rmed within the actual , by a transformation from a fully consti­

tuted individual of one species to another of a difTercnt speci es. But if

we move towards the virtual, towards those circles of convergence orfields of individuation where there are still communications between

not-yet -actualized species, one can become 'rc -cnvcloped' in another

field . T his theme is elaborated in AO, 86 and in ATP , 238 and

becomes a key component of Dcl euzc's theory of artistic practice as

dis{'uss('d in WIP.

APPEND IX : DELEUZE 'S WORDS

WH AT IS PH ILOSO P H Y?

Muc h as AO narrows the focus of the onto logy and deals only with

the act ualizat ion of social structures, WIP deals ex clusively with the

rela tions bet ween the virtual , the int ensive and th e actual, on one

hand, and the different forms which thou8h t assumes in certain societies

(p hiloso phical, artistic and scientific forms of thought). The virtual

appears here as ' the plane of immanen ce' explore d by philosophical

thought; the int ens ive as ' the plan e of compos it ion' as it app ears in

artis tic tho ught ; and th e actual as ' the plan e of referen ce ' as it is

investigated by scientific thought. Let me discuss each one of these

'planes' star ting with the actual world,

One way of thinking about the plan e of referen ce is as a flat

ontology of indi vidual s. Th e subject matter of scie nce would be , in

th is interpretat ion , the world of fully consti tuted individuals and the

metric and measurable spacetime they form. In other words, actual

indivi duals would form the referen ce of scientific statements, and all

ref rents wo uld form a 'plane' precisely in the sen se that, ontologically

at least, they do no t have a hierarchical str ucture but remain a ' flat'

set , \'arying only in spatio-temporal scale . In Chapters 1 and 2, where

I discussed the philosophical conce pt of 'multiplicity', I emphasized that

the scientific ideas involved (differe nt ial relations, singularities) had to

he detached fro m th eir original context wh ere they are related to

mathematicalJ unctions. Th e justification I gave for this transformation

was that functions, as they are ordinarily used , presuppose indi vidu a­

tion . Indeed, in some of their uses (as in their use to create state or

phase spaces) they define procedures for the individuation of states

within these spaces . T hese states of affairs constitute a re ferent, and

I he use of functions the re fore foll ows the line wh ich goes fro m the

virtua l to its act ualization , retaining only the final product.

T his is part of what Deleuze mean s when he asserts that the objec t

of science is 'functions which are presented as propositions in discursive

sysu-ms' (W IP, 117). I will return below to th e question of wh ether

one can cha racterize scie nce in th is way. As I said in Chapte r 4, 1 do

no t think there is such a th ing as 'science ' in genera l, so I reject many

of the det ails of the characte rizat ion given in WIP . Nevertheless, the

IMrt of it that I do kee p is the assertio n that most s icntilic fields tend

APPEND IX : D ELEUZE 'S W OR D S

to study the world in the direction of act ualization , sometimes

co nce ntrating on the final pr oduct and disregarding the pr ocess (e .g .

equilibrium th ermod ynamics) , sometimes studying the process but

always in the directi on of the final product.

Art, on the other hand , may be said to study, or engage with, the

inte nsive itself. The term ' inte nsive' is used in a varie ty of senses

only some of whi ch are relevant to th is characte rizatio n. One of the

co mpo nents of th e inten sive given in th e onto logical list was the

lar val subject who consumes int en sities as such, and is born and

reb orn of th ese voluptuo us consum ptions . In thi s case , the inten sive

state co mes first or it is prior to the individual that lives it (AO , 20) .

In other words, obj ecti ve inten sities do not constitute psychological

sensations bu t the very 'be ing of th e sensible' (D&R, 140) , a being

which is itself imperceptible psychologically given that intensities

become hidd en underneath qualities and exte nsities (D&R, 230), In

W IP this being of the sensible is divided into two co mpo nents ,

'perce pts' and 'affects';

By mean s of the material [e .g . paint , canvas, brush], the aim of art

is to wrest the percept from perceptions of objects and th e states of

a perceiving subject , to wrest the affect fro m affecti on s [e. g .

feelin gs] as th e transition fro m one state to another: to ex tract a

bloc of sensations , a pure being of sensat ions . (W IP, 167)

Simplifying somewhat , we may say that 'pe rcepts' are related to the

passive selves involved in the synthesis of living presen ts at all scales of

reality, in th e organi c and inorgani c world. Even though these presents

are constituted by 'conte m plations' or 'contracti ons of past and future

instants', they do not refer to a psychological reality . As Deleu ze

writes :

Th e plant conte mplates by co nt racting the ele me nts from whi ch it

originates - light , carbon, and the salts - and it fills itself with

olors and odo rs that in each case qualify its var iety, its co mpos ition:

it is sensatio n in itsel f. It is as if flowers sme ll themsel ves by

sme lling what composes them .. . before being perceived or eve n

sme lled hy an agent with a ner vous system and a br ain . (WIP, 2 12)

APPENDIX : DE LEUZE'S WORDS

On the other hand , affect s re fer to sta te transm ons whi ch mu st be

und erstood as ' becomings', in the sense of a becoming-animal or

becoming-plant discussed above . Th e artist must reach th at inte nsive

state where one can leave one individuation field to enter ano ther,

where one can reach 'a zone of indet ermination , of indiscerni bilitv , as

if things, beasts , and persons . . . endlessly reach that point that

immed iately preced es their natural differentiation ' (W IP, 173) . Finally,

having reached the very being of the sensible , the artist mu st place

these percep ts and alTects in their own plane , a plane of compos it ion,

a bloc or compo und of sensations whose 'only law of creation is that

the co mpo und mu st stand on its own' (W IP, 164) .

T hus, in a very lit eral sense , art is conce rned with making perceptible

the usuall y hidd en realm of the intensive . Similarly, philosophy mu st

ma ke the virtual intelligible. Philosophy must go beyond the centres of

convergence wh ere th e lar val subjects of percepts and affect s und ergo

intensive becomings, to reach the virtual in its full divergen ce and

difference, its continuous or ' inse parable variations' (W IP, 126).

Philo sophy cannot perform this task via a set of propositions whi ch

rifl er to the virtual, but rather, it must const ruct a thought wh ich is

isomorphic with th e virtual. The re fore, any philosophy mu st be con­

structed out of the three compo ne nts of th e vir tual: multipliciti es,

qua si-causal ope rato r, and the continuum . In WIP these three compo­

n .nts are referred to as 'conce pts', 'conce ptual persona e ' , and ' plane

of immanence', respecti vely.

Th term 'concept ' do es not refer to a semantic entity , that is, to

cone pts in th e ordinary sense, a sense in which there would also be

s i mtific conce pts (e .g. entropy) . Rather, it is defined as an enti ty

whi h wo uld be isomorphic with virtual multipliciti es.

[A concept is] a multiplicity, an absolute su rface or volume [e .g . a

ma nifold I ... mad e up of a ce rtain number of inse parable inten sive

vari: tions according to an orde r of neighborhood, and traversed by

a po int in a sta te of survey . (W IP, 32)

To say that a conce pt 'orders its co mpone nts by zo nes of neighbor­

hood ' (W IP, 20) is to say that the relations it invo lves ar nonmetric

or ordinal. This re fers to the third sense o f 'i nte nsive' as defined

APPEND IX : DELEUZE 'S WORDS

above , and to the definition of to po logical spaces in Chap ter I , and is

also ex pressed by saying that a concept 's co mponen ts are 'i nte nsive

ordinates' (WIP, 20) . Concepts, therefore, are not to be thought of

sema ntica lly, bu t literally as sta te or phase spaces, that is, as spaces of

possibili ties st ructured by singularities and defined by their dime nsions

or intensive ordinates. As Deleu ze writes, 'Every conce pt therefore

has a phase space, altho ugh not in th e same way as in science' (W IP,

25) . For example , the Cartes ian conce pt of ' the Cogito' wo uld be a

space with three dimen sion s (doubting, thi nkin g and being) each

divide d by singularities into phases (e .g. perceptual , scie ntific, obses­

sional doubting , as different phases of doubt, as oppose d to different

species of the genus doubt) .

The idea of a 'point in a state of survey' refers to an op eration of

the quasi-cause whi ch I did not describe in my recon structi on. Much

as multipliciti es mu st be meshed together into a continuum whil e

preserving th eir dilTeren ces ('exo-consiste ncy'), so the heterogen eou s

components of a multiplicity must th emselves be meshed by a 'po int

of abso lute survey' (W IP, 2 1) which continuously traverses them at

infinite speed ensuring th eir 'endo-consistency' . Exo-consiste ncy is

explained in WIP in terms of resonances between divergent series:

Co nce pts which have only [endo-]con sisten cy or inten sive ordinates

outside of any coordinates, freely ente r into relation ships of non ­

discursive reson ance . .. Con cep ts ar e cente rs of vibrations, each in

itself and everyone in relat ion to all the othe rs . This is wh y they

all resonate rather than cohere or corres pond to each othe r . ..

They do form a wall, but it is a dry -stone wall, and everyth ing

holds together only along diverging lines. (W IP, 23)

The qua si-causal ope rator behind these effects of endo- and exo­

consistency is referred to as a ' con ceptual persona ' . Thus, Deleuze

writes: 'The conce ptual persona is need ed to cre ate concepts on th e

plane , just as the plan e need s to be laid out. But these two ope ratio ns

do not merge in the persona , which itsel f app ears as a distinct ope rato r'

(W IP, 76). Co nce ptual person ae are endo wed with all the characte r­

ist ics of the quasi-cau 011 operato r. Mu ch as th latter mu st inject as

mu h hancc into the distribution ' of th singular and till' ordinary in

APPEND IX : DELEUZE 'S WORDS

virt ual series, ' the persona establishes a corresponde nce between each

throw of the dice and the inten sive features of a conce pt ... ' (W IP,

75 ) . And mu ch as the ope rato r is said to ex tract ideal eve nts fro m

what act ually occ urs (that is, to perform co unte r-actualizat ions or

' counter-effectuations'), in philosophy 'i t is precisely th e conce ptual

persona who co unte r-effectuates th e event' (W [P, 76).

But why the term ' pe rso na'? A clue to the meaning of thi s

expression may be glimpsed fro m some rem arks in LO S. As [ have

just said , in the circles of conve'8ence defined by pr e-actualized multi ­

plicities an int ensive indi vidual develop s (larval subject) , an indi vidual

which ex presses the world which conve'8ent ser ies form. Similar ly, in

the divergent series a ' virtual person ' develop s, a person who ex presses

what is common to man y different worlds (LO S, 115 ). A more

eI tailed ex planation , however, emerges from a discussion in D&R .

Much as a larval subject is born from percepts and affects which do

no t refe r to psychological phen om ena, but ar e the very being of th e

s msiblc, so personae are intimat ely connecte d with what constitutes

the very being of the int elligible (D&R, 141 ). Differen ce in inten sity

is the being of the sensible (sentiendum') and simultaneously that

which cannot be sensed (by fully actualized indi viduals) since it is

normally covere d by ex te nsities and qualiti es (D&R, 144). imilarly,

the being of the intell igible (cogitandum') is what can only be tho ught

and at the same time that which marks the impossibilit y of though t

(again, imposs ibility from the point of view of a fully actualized

think ' 1') , Hen ce the need to invent a conce ptual person a to capture

these cogitanda or ' thought-events', a persona who ' lives inte nse ly

within the thinker and forces him to think ' (W IP, 70 ).

Finally, there is the third compone nt : the virtual co ntinuum itsel f

or the 'plane of immane nce ' of a philosophy. Thi s refers to the

presu pposit ions of a philosophy, the main one of whi ch is an assumed

' im: ge of thought' (W IP, 37), in other words, a pre-con ceptual

intuitio n of w hat it is to think: 'E very philosophy dep ends upon an

int uitio n that its co nce pts constantly develop through slight differen ces

flf intensity .. . ' (W IP, 40). O ne way of und erstanding what thi s

means is to think of the relation b tween co nce pts and the plane of

immanc n l' as that between so lut ions and problem s. As I discussed in

'hapu-r 4, problems ar not reelucibl ' to th ir so lutio ns bu t rather arc

APPENDIX : DE LEUZE 'S WORDS

defined by their conditions : a give n distribution of the singular and the

ordinary , the impo rta nt and the unimportant. As such, pr obl em s are

inherently 'obscure yet distinct' and only acquire clarity in the pr ocess

which progressively specifics each of their so lutions . The intuition

referred to above wo uld refer to the grasping of a pr oblem as such, as

d istin ct and obscure (as oppo ed to grasping an esse nce, or a clear and

distinct idea) , an intuition which can only reveal itse lf progressively as

co nce pts are create d as cases o f so lutio n :

If th e conce pt is a solution, the conditions of the philosophi cal

problem are found on th e plane of immanence presupposed by th e

conce pts . . . and the unknowns of th e problem are found in the

conce ptual persona e that it calls up . . . Each of these three instances

is found in the others, but they are not of the same kind, and they

coexist and subsist without one disapp earing into the other . . .

[T]he three acti vities making up [the philosophical method] continu­

ously pass fro m one to the othe r, support one another, some times

pr eced e and sometimes follow each other, one creating co nce pts as

a case of solutio n , ano the r laying out a plane and a mo vement on

the plane as the conditions of a prob lem , and the othe r inventing a

persona as the unknown of the pr obl em . (W [P, 8 1)

In my reconstruct ion of Deleuze ' s onto logy I used as a guiding

constraint the avoid ance of the categories of typological thou ght:

resemblance, identity, analogy and contradiction. But [ co uld have as

we ll said that what guides this construct ion is the avo idance of the

image of thou ght implied by these categor ies: ' a natu ral capacity for

th ought endo we d with a capaci ty for truth or an affinity with the

true ... ' (D&R, 131) . This image which, Dcl euze argues, haunts the

history of philosophy, has the result of turning the plan e of immanen ce

int o a plane of transcenden ce, Or what amounts to the same thing, to

trap philosophy within the plan e of referen ce, linking it to linguistic

propositions whi ch are either true of or false of their referents. This

manoeu ver, of course, closes the road to th e virtual or the problematic.

[f, on the co ntrary, the image of th ought leads to a plan e of

im mane nce, th n phil osophy 'docs not consist in knowing and it is not

inspired by truth. Rath er it is categories like Int I' sting, Rem ark able ,

APPENDIX : D E LEUZE 'S WORDS

or Important that deter-min e success o r failure ' (W IP, 82 ). The image

of tho ught th at has thi s problematic e ffect is on e in whi ch thought is

horn from the violent shock of an encounter with pure intensiv e

difl ercn ces (being of th e sensible), a shock whi ch a philosopher may

th en be capable of comm unicating to his or her other faculties, leading

all th e way to pure virtual dilTeren ces (being of th e int elligible) (D&R,

140).T his is not th e place to argue for or against thi s view of philosophy.

Whether o r not all phil osophical systems may ind eed be analysable in

terms of the three components o f the virtual remains an open question.O n the o the r hand, I must take issue with the imagc of science which

WI!' develops, particularl y because my disagreement with it bears not

just on narrowly scientific quest ions but on deep ontologica l matters .

Speci fically, my main divergen ce from Deleuzc 's ontology occurs atth e level of th e flat ontology of individuals. I m entioned above that I

broke with Dclcuze 's terminology by using the term 'individual' for

extended and qualified actual beings, while he reserves it for intensivebei ngs (larval subjec ts) . But the break is more than just terminological.

Altho ugh a flat ontology meshes well with many of Delcuze' s ideas

(his th eory of actual time as a nested set of cycli c presents of different

durations, for example) , it is unclear to what extent he subscribed to

suc h a view. In part icular , in a flat ontology as I have developed here

th ere is no room for totaliti es, such as 'society' or 'science' in general.But Dclcuze does not seem to mind such entities . For example , while

I would never speak of a virtual multiplicity co rresponding to all of

society (i.c, a 'social Idea' or 'social multiplicity ' ) he does so without

hesitat ion (D&R , 186).

In the case of 'science ' as defined in WIP, that is, in term s of

functions working as discursive propositions, the problem is that the

image invoked is one too clos e to that created by Anglo.American

philosophers of science of the first hal f of the twentieth century. All

the examples o f ' func t ives" (the components of functions) given in

\VIP co me from classical mechanics . No mention is made, for instance,

of the operators of quantum physics , which use functions themselvesas inputs ,1I1d outputs . And, of course, the question of what chemical

or biological functions arc is left most I)' unspecified . This amount s toclt'flning scie nce as if its ' e ssen ce' was classical mechanics . Furthermore,

APPEN DIX ; DELEUZE'S WORDS

mu ch as o ld-scho ol anal yti cal phil osophers disregarded th e actual

mathematical models used by ph ysicists and focused exclusively on se t

theory, so Dcl euze view s set theory as the too l which constitutes the

plan e of reference of scien ce (WIP, 121). My analysis in Chapter 4- o f

classical me chani cs (as an indi vidual field) broke with all this . It

preserved the idea that clas sical physics (as many other scie nt ific fields)

is mostl y co ncerned with the plane of reference (actual beings , metricspaces) but it uses a very different co nception of how referen ce (o r

the fix ing of reference) is achieved, placing more em phasis on caus al

interventions than on representati ons. Similarly for my treatment of

mathematical models, which are not reduced to linguistic entiti es(func t ions as propositions) hut tackled in th eir specificity.

On th e other hand , my ana lysis of classical physics meshes we ll with

Dclcuze's views on scien ce as developed elsewhere . The requirement

of avo id ing th e categories of typological thought to prevent th e plane

fro m becoming a plane of transcenden ce may also be expressed by

saying that we must avoid the ' classical im age of thought , and th e

st riating of mental spa ce it elTects' (AT P, 379) . Th e term ' striate d

space ' refers to a metri c space , while non metric spaces , 'vectorial,

proj ective, or topological' (AT!', 361 ) are referred to as ' smooth ' .

The transformation of thought itself into a metric space is not,however, an internal affair of philosophy , but on the contrary, it's

directly linked to th e relations between individual phi losophers (e .g .

Hegel) and indi vidual State or Royal institution s. It is these intitutions

whi ch first st riate or metricize real space (e .g . agricultural lands, urban

areas), and later perform the same operation on mental spaces . The

opposite transform ation, to create a nonmetric space for thought is

pe rform ed by philosophers (e .g. Spinoza) wh o operate outs ide of th eState .

A simi lar distinction is made between scientific fields or even,among the different practices (theoretical as opposed to expe rimental)

within one field, We have. on one hand, 'Royal science ' (the science

of th e great Ro yal Societies o r Academies at th e se rv ice of th e Stat e),

and. on the other, the 'minor sciences' operating in less prestigioussurroundings. Roughly, the distinction is between scientific practices

which arc axiomatic or theoremat ic , as opposed to problematic; that

0lll'ra l l~ within metr ic and exactly measurabl e spaces, as opposed to

APPENDIX : D E L E U Z E ' S W O RD S

d >aling with anexact yet rigorous nonmetric on es; that focus on the

simple behaviour of matter, as in ideal solids or gases, as opposed to

confronting the complex behaviour of liquid s (e .g. turbulence); and

that st ress constant and homogeneous laws, as opposed to becomings

and heterogeneiti es (AT P, 361) . My account of classical physics, which

is clea rly at odds with the Royal and legalisti c image which that field

has of itself, ma y be seen as an account from the point if view if min or

science. But for the same reason, it mak es the distinction whi ch WIP

establishes between science and philo soph y pass right through the

middle of science itself. Thi s, it seems to me, is the 'more Deleuzian '

approach to the subj ect.

Notes

TH E MATHEMATI CS OF TH E VIRTUAL:MANI FOLDS, VECTOR FIELDS AND

TRANSFORMATION GROUPS

1. The term 'multiplicity' makes its first appearance, as far as 1 can tell , in

1966 in Dcleuze 's book on Bergson, Gilles Deleuze, Berpsotiism (Zo ne Books,

New York, 1988), p. 39 . Its final appearance occurs in Deleuze' s last book

in collaboration with Felix Guattari, Gilles Deleuze and Felix Guattar i, What

Is Philosophy? (Co lumbia University Press, New York, 1994), p. 15.

2. Morris Kline, Mathematical Thouqh: fro m Ancient to Modern Times, Vol. 3

(Oxford University Press, New York, 1972), p. 882. (My emphasis)

Making surfaces into spaces, by eliminating the supplementary dim ension ,allowed the differentiation and study of different metric geometries . As

Morri s Kline wri tes:

Thu s if the surface of the sphere is studied as a space in itself, it has

its own geo metry , and even if the familiar latitud e and longitude are

used as the coo rdinates of points, the geo metr y of that surface is not

Euclidian ... However the geometry of the spherical surface is Euclidian

if it is regarded as a sur face in three-dim ensional space. (p. 888)

For the details on Gauss coordinatization pro cedure, which is what

guarantees th is absence of a supplementary dim ension or embedding space,

see Lawrence Sklar, Space, Time, and Space-Time (University of CaliforniaPress, Berkeley, 1977 ), pp. 27-42.

3. Kline, Mathematical Tboupbr, p . 890 .

4 . Gilles Deleuze, D!lJerence and Repetition (Columbia University Press, New

York , 1994), p . 182. O n page 183, for example, he says: ' In all cases the

mult iplicity is intrinsically defined, with out externa l reference or recourse

to a uniform space in which it would be submerged .' See also Gilles Deleuze

and Felix Guattari, A Thousand Plateaus (University of Minnesota Press,

Minneapolis, 1987), pp . 8-9,

Unity always operates in an empty dimension supplem ntary to that ofthe system considered (ovcrcoding) . .. [But aJ multiplicity never allows

r

NOT ES

itself to be overcoded, never has available a supp leme ntary dim ension

over and above its number of lines, that is, over and above th e

multiplicit y of numbers attached to those lines.

.5 . Deleuze and Guattari, II Thousand Plateaus, p . 266 . Th e rem ark quoted is

made about th e 'plane of consiste ncy ' not about multipliciti es. But th e

form er is nothing but the space formed by the multipliciti es th emselves , as I

will exp lain in detail in the next chapte r .

6. Wh en Dcl euze defines his multipliciti es he always see ms to be referring to

manifolds whose dimensions are used to represent degrees of freedom (or

indep endent variabl es) of some dynamic, and not to manifolds as mere

geo me tric objec ts . Thus, in his first introducti on of th e term he says,

Riem ann defined as 'm ultiplicities ' th ose things that could be det ermined

by th eir dimen sion s or their independent variables. He distinguished

between discrete multiplicities and continuous multipliciti es. Th e former

contain the principle of th eir own metrics . . . Th e latter found a m etrical

principle in some thing else , eve n if onl y in ph enom ena unfolding in th em

or in th e forces acting in th em. (Bcrasonism, p. 39)

And else where he says, using the word ' Ide a' to refer to concre te univ ersals

or multiplicities as repla cem ents for esse nces ,

An Idea is an n-dimen sional , continuous, defined multiplicit y. Colour ­

or rath er, the Idea of colour - is a three dim ensional multiplicit y. By

dim en sion s, we mean th e variables or coordinates up on whi ch a phenom­

eno n depends; by continuity , we mean th e set of relations between

changes in th ese variables . . . by definition, we mean the elements

reci procally determined by th ese relations, eleme nts which cannot change

unless the multiplicit y changes its order and its metric. ( D!iJerencc and

Repetit ion , p. 182)

7 . I take th is rather Sim plified description fro m Ian Stewart. Does God Play Dice?

The Mat hematics ifChaos (Basil Blackwell, Oxford, 1989), Chapter 6 .

H. Loo king for relation ship s between th e differe nt solution curves [i.e .

tra jectories ] of the same differential equation, Poin car e began with a local

analysis and examined th e beha vior of these curves in th e neighb orhood

o r a singular point . . . He sho we d that there were four possible different

types or singular points and classified them by the behavior of the nearby

so lutio n cu rves: nccuds (no des), through which an infinite number of

sol utio n curves pass; eols (sadd le points), th rough which only tw o so lutio n

curves pass ... .foyers (fo i) , which th e so lution curves approach in the

NOTES

manner of a logarithmic spiral; and centres (ce nte rs), aro und whi ch th e

so lutio n curves are closed, envelo ping one another. Having used direct

algebraic co mputat ion to sho w that th ese four types necessaril y exist, he

studied th eir distribution. He found that in the gene ral case only three

types pr evailed - nod es, saddle points and foci - with cen te rs arising in

only exceptio nal circ umstances. (June Barrow-Green , Poincare and the

Three Body Problem [American Mathematical Societ y, 1997J, p. 32)

Roughly, we can say that Poincar e discover ed not only the existence of ce rta in

recurrent ' to po logical forms' which are bound to app ear in a large class of

differ ent physical models, but also that some of th ese forms are 'more

gene r ic' than othe rs, that is, th at if we study the distr ibution of singularities

in many different models some of them (cente rs) are less likely to occu r

than oth ers. See also discussion of th e term 'gene ric' , a technical term

whose meaning is still evolving , in Ralph Abraham and Chr isto phe r Shaw ,

Dynamics: The Geometry if Behavior, Vol. Three (Aerial Press, Santa Cruz,

198.5) , 1'1' . 19-34.9. Deleuze and Guattari , A Thousand Plateaus, p. 40 8.

10. 'To rever se Platoni sm ' , as Del eu ze says, we need ' firs t and for em ost to

rem ove esse nces and to substi tute events in th eir place, as jet s of singu lari­

ties ' (Gilles Deleuze , Loqic c1 Sense [Columbia Uni versity Press, New York,

1990], p. 53) .11. Speaking of the image of the light of reason (or of rationalit y as a faculty

capable of graspin g the essential truth of thin gs) Deleuze says,

Th e very conce ption of a natural light is inseparable from a ce r tain value

supposedly attached to the Idea - namely, 'clarity and distinctness' ...

Th e restitution of the Idea in the doctrine of th e faculties requires th e

explosion of the clear and distin ct , and th e discovery of a Dion ysian value

according to whi ch th e Idea is necessarily obscure in so fa r as it is distinct, all

the more obscure th e more it is distinct . ' (Em phasis in th e original;

Gilles Deleu ze , D!iJerence and Repetition , p. 146)

Th e term 'Id ea ' here refers to multipliciti es , and th e fact that Deleuz e uses

that Platoni c term shows he mean s to repla ce essences with multipliciti es,

Ideas are by no means essences . In so far as problem s are th e object of

Ideas, probl em s belong on th e side of events, affecti ons, or accidents,

rather than o f theorematic essences . . . Co nsequently the domain of

Ideas is that of th e inessential. (I" 187)

12. Self-ass mhl y during [the ear ly stagcs of) em bryo nic development is not

mediated by direct gene int erv ention . Wh en all the tran scriptions have

NOTES

been prevented [thro ugh the use of an inhibitor] the regular cleavage

patt erns are re tained. However, the polarity of molecular organizatio n of

both the egg's cytoplasm and its nucleus ... are essential for normal

development. Hence the main features of [earl y] embryogenesis - ce ll

differentiation, indu ction, det ermination of pattern form ation - all ste m

from the ooge netica lly originated, spatial distribution of preformed

informatio nal macrom olecules. Th e initial conditio n of embryogenesis is

ooge nesis. The epigenetic.~ of embryo nic development is built on the

topo logical self-organization and orienta tion of macromolecules of the

total egg. (Vladimir Glisin , ' Molecular BioloBJ in EmbryoloBJ. The Sea Urchin

Embryo", in Se!f-0rsanizins Systems. The Emerpence eif Order, ed . Eugene

Yates [Plenum , Ne w York 1987], p . 163)

The term 'oogenesis' refer s to the pro cess which creates th e egg in the first

place .

13. Joe Rosen, Symmetl) ' in Science (Springe r- Verlag, New York, 1995), Chapter

2.Besides closure, a collec tion of enti ties togeth er with a rul e of comb i­

nation needs to display associativity, and possession of identity and inver se

elements. The set of positive integers (including zero, and using addition as

a comb ination rule) displays associativity because the result of adding two

numbers first, and then addi ng a th ird one is the same as that of adding the

first to what results from adding the last two. It also conta ins an ' identity

cle ment' , that is, an eleme nt whi ch added to any other leaves the latt er

unchanged (in this case the identity elem ent is the number zero) . But it fails

to be a gro up because it lacks inverse elements , those which when compose d

with certain othe rs yield the identity element. For instance , the number

'-3' when composed with the number '+3' does yield zero (w hich is the

identity eleme nt) but '-3' is not part of the set of positive integer s. Thus,

for the integers to for m a group we must also include negati ve numbers in

the set.

14. T his dyna mic aspect of sym me try- based classifications is obscured in standard

presentations of the subject by the fact that the emphasis is not placed on

the t ransfor mation as an event , but on its input and output. That is, the

t ransformatio n is a pro cess but all that matter s math ematically is the init ial

and final states of the object transformed. See Ian Ste wart and Martin

Go lubits ky, Fea1ul Symmetry (Blackwe ll, Ox ford , 1992), PI" 32-3.

15. JIM, p. 97.Besid s assuming ideal solids and gases, th is illustra tion of broken

s)"lnn1l'try assumes that the gas containe r and the crysta l lat ti care infinit

ill all direct ions. The use of an 'obs rve r' to define invar iancc is just a

NOTES

convenience. The subjective po int of view can , in fact , be avoided. See Joe

Rosen , S)'mmetl)' in Science, PI" 173- 4 .16. Stewart and Golubitsky, Fea1ul Symmetl)', Chapter 7 .

17. Ralph Abraham and Christopher Shaw , ' Dynamics: A Visual Int roduction ' ,

in Se!f-Orsanizing Systems, ed. Yates, p. 576.

18. Stewart and Golubitsky, Fea1ul Symmeuv, Chapte r 5. See also , Gregoire

Nicolis and lIya Prigogine , Exp/orins Complexity (W. H . Freeman, Ne w York

1989), pp . 12-1 5.

19. Brian C. Goo dwin, 'The Evolution of Ge neric Forms', in Orsanizational

Constraints on the Dynamics eif Evolution, ed. J. Maynard Smith and G. Vida

(Mancheste r University Press, Manchester 1990), PI" 11 3- 14 .20. Dele uze, Difference and Repetit ion, p. 187.

Altho ugh Deleuze does not ex plicitly use the term 'symme try- brea king

cascade ', he docs refer to an 'e mbedding of groups' (p. 180) precisely in

the contex t of explaining how a multiplicity may be pr ogr essively deter ­

mined . Unfortunately, his brief discussion of gro ups uses a very obscure

aspect of Galo is's meth od , the originato r of group theory, called the

'ad junction of fields'. The two formulations are, nevertheless, equivalent,

fields of numbers and groups being two related ninet eenth-cen tury abst ract

objects. An algebraic problem , specified progressively as its field is com­

pleted by successive adjunctions, is the eq uivalent of an abstract smooth

space being specified by a progr essive series of broken symmetries, yielding

increasingly mor e differe ntiated, more striated spaces . Deleuze 's discussion

of Galois is correct technically, but it is not as clea r and intuitive as the

equivalent formulation in terms of 'embedding of !,TfOUps' . Hence in this

reconstruction I will stick with the clearer alte rnati ve. But wheth er one uses

fields or groups, it is clear that some form of prosressil'e differentia tion is a key

component of the concept of a Deleuzian multiplicity.

2 I . What distingui shes a pace as opposed to a mere set of points is some

concept that binds the points togeth er . Th us in Euclidea n space the

distance between points tells how close points are to each othe r . . . As

Frechet [a pioneer in the development of topol ogy] pointed out, the

bind ing pr oper ty need not be the Euclidea n distance function . In

particular he generalized the noti on of distance by int roducing the class

of metric spaces. In a metric space, which can be a tw o-dim ensional

Euclidean space, one speaks of the neighborhood of a point and means all

those points whose distance fro m the point is less than some quantit y

. . . However , it is also possible to suppose that the neighb orh oods,

certain subse ts of a gh'en set of poin ts, are speci fied in some way, even

without the introduction ~f a metric. uch spaccs are said to have a

NOTE S

neighborhood topology. (Mor ris Kline , Math emat ical Thouqht ; p. 1160;

my em phasis)

1 will use the term ' me tric space' and 'no nmetric space' throughout th is

book in the sense in which th ey are defined in this quote but 1 will take

some liberties. I will spe ak of top ological spaces , for example, as th e ' least

metric ' and of Euclid ean as th e 'most metric' , even thou gh it would be

more techni cally co rrec t to differentiate fla tures if spaces that do or do not

depend on an)' strictly metric property.

22. Dclcuze usuall y speaks (follow ing Bergson ) o f tw o different t)'p es of multi­

pliciti es, metric and nonmetric, whi ch he calls ' striated' and 's mo oth'. For

the purposes of ensuring th e co r re ct int erpretation of Delcu ze 's position

her e it would have been ver y useful if he had ever discussed Felix Klein 's

work, thereb y clarifying the relations between the metric and the nonmetric

as one of group inclusion . Unfortunately, as far as I can tell, Dcl euz e never

discusses Klein . On the oth er hand, Deleuze is perfectly aware of th e

ex iste nce of several nonmetric geo me t ries and uses a sinnle term (' smooth

space ') to refer to all of th em:

It is the difference between a smooth (vectorial , projecti ve, or topolonical )

space and a striated (metriC) space: in th e first case 's pace is occupied

without co unting' and in the second case 's pace is counte d in orde r to be

occupied'. (De lcuze and Guattari , A Thousand Plateaus, P: 361 ; my

emphasis)

T he definitions given in the extract are his own, but are linked to th e

more orthodox definitions. A metric space is counted in order to be

occupied in th e sense in which sede ntary cultures divide the land int o

measured (or counte d) plots in orde r to inhabit it:

Good sense is . .. agricultural , inseparable from the agrarian problem,

th e establishme nt of enclosure s, and the dealings of middle classes th e

part s of whi ch are supposed to balan ce and to regulate on e another. Th e

ste am engine and livestock , but also properties and classes, are th e living

sources of good sense , not onl y as facts that spring up at a particular

peri od, but as ete rn al archetypes. (Deleuze, Loqic if Sense, p. 76 )

To the sede ntary way of metricizing space, of dealing with it as esse ntially

exte nsive, Dcleuz opposes an int ensiv e way of oc upying space the way a

liquid do cs, that is, occupying it without Jividing it or co unting it. Thi s

alternative h· calls a ' no madic d istribution ' . Th e distin cti on bctwc n scdcnt­

.lTV .md nomad ic distribution s is first mad , in DilJ;'rence and Repetit ion ,, .

N O T E S

pp . 36-7, in relation to questions of typolog ical thinking, but is taken

further in an actual co mpariso n of nomad and sede ntary cultures

. . . eve n though th e nomadic traj ect ory may foll ow trails or customary

ro utes , it do es not fulfill th e function of the sede ntary ro ad, wh ich is to

parcel out a closed spaee to people, assigning each person a share and

regulating the com munication between shares . Th e nomadic traj ect ory

does th e opposit e : it distributes people (or anima ls) in an open space .. .

sede ntary space is striate d [i.e. metricized], by walls , enclosures and

roads between enclosures, whil e nomadic space is smooth [i.c. non­

metric], marked on ly by 'traits ' that ar e effaced and displa ced with th e

traj ect ory. (De lcuzc and Guattari , II Thousand Plateaus, p. 380; emphasis

in the original)

23. Morris Kline, Math emat ical Tboupht, p. 917.

24. David A. Brannan, Matthew F. Esplen, Jerem y J. Gra y, Geometry (Cambridge

University Pre ss, Cambridge, 1999 ), p. 364.

25. This way of describing the subject oversimplifies things some what. First of

all , th e actual relati ons between the different geome tr ies are more complex

than the Simplified hierarchy 'topological-differential-projective-affme­

Euclidean geome tries' may sugges t. For th e detail s of Klein 's orig inal

classificati on see ibid., P: 919.

My friend the math ematician Andreas Dress (pe rso nal com munication)

summarizes Klein 's programme (called th e Erlange r Program) like this,

Th e Erlanger Program by Felix Klein is based on the fact that depending

on whi ch (bijective) transformations you need to deal with (isome trics

keeping distances invariant , similarities scaling all distan ces by th e same

fact or and, hen ce, keeping rati os of distances invariant, affine maps

keeping rati os of distances of points on parallel lines invariant, proj ectiv­

ities keeping cro ss-ratios of distan ces invariant , differential transforma­

tions respecting infinitesimal straightness, hom eomorphisms respecting

nothing but infinit esimal closeness) , it always makes sense to ask ( 1)

which features of configurations within th e space of int erest do remain

invariant , and (2) wh ether a basic famil y of such features can be found so

that every other such feature can be expressed as a function of those basic

ones.

26 . Morris Klin " Math ematical Thouqlu , p. 9 21 . Th er e are imp ortant exce ptions

to this state me nt. Some mathematicians, like Riemann himself, but also

\ illi: m ' IilTord, did see an ollto logica l connec tion between the metric and

N O T E S

nonme t:ric prop erties of spaces. As one historian of twentieth-century physics

writes,

[RiemannI asserted that space in itse lf was nothing more than a three ­

dim ensional mani fold devoid of all form: it acquire d a definite form only

through the mat erial co nte nt filling it and det ermining its metric relations

. . . Riemann' s anti cipation of such a dep end en ce of the metric on

physical data later provided a justifi cation for avoiding the noti on of

absolute space wh ose metric is ind epend ent of physical forces . For

example , more than sixty years later, Einstein took Riemann 's em pirical

conce ptio n of geome try using it as an important justification for his

gene ral theory of relati vity.

(Tia n Yu Cao, Conceptua l Development if Twenti eth -Century Field Theories

[Camb rid ge University Press, Cambridge, 1997], P: 373)

27 . Gordo n Van W ylen, Thermodynami cs (j ohn Wil ey & ons, New York , 1963) ,

P: 16.

28 . Wh at is the significance of these indivisible distances that are ceaseless ly

transformed and cannot be divid ed or transformed without their eleme nts

changing in nature each time? Is it not th e int ensive characte r of this type

of multiplicity' s elem ents and the relations betw een them ? Exact ly like a

spee d or a temperature, which is not co mpose d of oth er speeds or

te mperatures , but rath er is envelo ped in or envelops othe rs , each of

which marks a change in nature. The metrical principle of these

multiplicities is not to be found in a homogen eous milieu but resides

elsewhere , in forces at work within them , in physical phen om ena

inhabiting them . . . (De leuze and Guattari , A Thousand Plateaus,

pp . 3 1-3)

T he term 'd istance' is used as if it was a nonmetr ic property, though in its

usual meaning it certainly den otes something metric. Deleuze takes this

specia l inte nsive mean ing of 'distance' from Bertrand Russell as I will discuss

in de tail later in the next chapte r . O n distances as int ensive magnitudes, or

as 'i ndivisible asymme tr ical relations ' see Deleuze, Difference and Repet it ion,

p. 237 . Deleuze does not explicitly give phase transitions as examples of

'c hanges in kind ' . But one of the very few illustrations he does give is indeed

a symmet ry-brea king transition , 'For exa mple , one can divide movem ent

into the gallop, tro t, and walk , but in such a way that what is divided

changes in natu re at each moment of the di vision ... ' (Dc lcuzc and

C uauari, /1 Thausatul Plat eaus, p. 483).

NOTES

O n phase transitions in animal movem ent as broken symme tries see,

Ste wart and Golubitsky, FeOIjul Symmetry, Chapte r 8.

29 . Cao, Conceptual Development ,?! Twentieth-Centu rJ Field Theories, p. 283.

30. Th e essen tial idea of grand unified theories . .. [is] the general form of

hierarchical symme try br eaking: an und erl ying large gauge symme try of

all int era ction s is brok en down in a success ion of ste ps, giving a hierar chy

of brok en symme tr ies . (ibid., p. 328)

31. It is beyond the sco pe of this chapter to analyse Einste in's use of differential

mani fold s in technical detail. But I sho uld at least mention the way in which

his usage differs from that of Deleu ze. In Einste in's theory a gravitational

field const it utes the metr ic struc ture of a four-dimensional manifold

(spacetime), and to thi s exte nt, the metric properties of space (rathe r,

space time) are ind eed connected to the physical processes wh ich occ ur

within it. However, as the philosoph er of science Lawre nce Sklar reminds

us, despit e the fact that Einste in's field equation does rel ate the metric of a

manifold to the distribution of mass and energy, the relation between the

two is not genet ic: the metric is not caused by the mass-energy distribution ,

it is only associated with it in a lawlik e way. ee Sklar, Space. Time, and

Space-Time, pp . 50- I .

32. Th e mo ve away fro m metamath em atics (set theory) and back to the actual

mathem atics used by scientists was initiated by the philosopher Patrick

Suppes . Yet the credit for the introducti on of state space into mod ern

analytica l philosoph y, as we ll as the cr ed it for em phasizing physical mod ality

in the analysis of that space , goes to ano the r philosoph er, Bas Van Fraasen .

See Bas Van Fraasen, l.aws and Symmetry (C lare ndo n Press, O xford , 1989) ,

Chapte r 9.

33. Ralph Abrah am and Chris to phe r Shaw, Dynamics: The Geometry cd' Beha vior,

Vol. 1 (Aerial Press, Santa Cruz, 1985 ), pp. 20- 1. My description is merely

a paraphrase of the foll OWing description:

Th e modeling pro cess begins with th e cho ice of a particular state space

in which to represent the system. Prolonged observations lead to man y

tr ajectories within the state space. At any poin t on any of these curves, a

veloci ty "ector may be deri ved [using the differentiation operato r ]. It is

useful in descr ibing an inherent tenden cy of the system to move with a

habitu al velocity, at part icular po ints in the state space. Th e prescription

of a veloc ity vect or at each point in the state space is called a velocity

vector .fielJ. T he sta te space , filled with trajectories, is called the phase

p"r/mit of till' d -narn ical system. The velocity vecto r field has been

NOTES

derived from the phase portrait by d!fTerentiation . . . Th e phrase dyruunical

Sj'stem will specifically denote thi s vector field . (Emphasis in the original)

l4. Albert Lautman, quoted in Gilles Deleuze, Loqic if Sense (Columbia Univer­

sity Press, ew York , 1990) p. 345. (My emphasis)

Lautman 's Le Probleme du Temps (fro m which thi s ext ract is taken) and

'Essai sur Ie otion de tructure et d ' Existence en Math ematiques ', are

Dclcuzc 's main sources on the ontological analysis of stat e space. Deleu ze

paraphrases Lautrnan 's description in other books, but given the ce ntrality

of these ideas in his work 1 prefer to qu ote Lautrnan ' s own words.

15. Abraham and Shaw, Dynamics: The Geomeuy ifBeha vior, pp. 35-6.

36. Nicolls and Prigogine, Explorina Complexitv; pp. 65-71 .n . Abraham and Shaw, Dynamics: The Geometry ifBehavior, pp . 37-41.38. Abraham and Shaw, Dynamics: A Visual Introduction, p. 562.

~9 . Deleuze, D!lJerence and Repetition, pp. 208-9. (Emphasis in the original. )

Deleuz e borrows the ontological distinction of the actual and the virtual

from Bergson . See Deleuze, Berpsonism, pp . 96-7.

40. Willard Van Orman Quine , quoted in Nicholas Rescher, 'The Ontology of

the Possible', in The Possible and the Actual, ed . Michael J. Loux (Cornell

University Press, Ithaca , 1979), p . 177.

41 . For a brief account of the recent history of modal logic, see Michael J.Loux , ' Introduction: Modality and Metaphysics', in Loux, The Possible and th e

Actual, pp . 15-28 .4 2. Ronald N. Giere, 'Constructi ve Realism ' , in lmap es if Science. Essays an

Real ism and Empiri cism with a Reply by Bas C. Van Fraasen, cds . Paul M.

Churchland and Clifford A. Hooker (University of Chicago Press, 1985),

p. 84 .4 1. Bas Van Fraasen, Laws and Symmetry', p . 223. Van Fraasen discusses the tw o

standard typ es of laws, laws of succe ssion (which gov ern the evolution of

trajectories, and are exe mplified by Newton' s laws) and laws of coexiste nce

(which restri ct position in state space, and are illustrated by Boyle 's law for

ideal gases) .

44 . Exactly mat ching initial conditions in the laboratory and the model is not

possible, so we normally deal with bundles ?f traj ectories in state space. Th e

statistical distribution of a small population of initial states in the model is

mad e to mat ch that of the errors which the exper imente r may have made in

pr 'paring the real syste m in a parti .ular initial condition. In what follow s

thi s point will not make mu ch differen ce so I stick to the simpler case of a

single trajectory.

4S . t ;i(,rt, rgues that the regularities exhibited by the possible histories reveal

'om, thin g about the w USCl I reqularitles in the real ph)'Sical s 'sl .m:

NO TE S

For the modal realist, the causal stru cture of the model, and thu s, to

some degree of approximation , of the real syste m, is identical with the

modal structure. For any real syste m , the functional relation ship among

the actu al values of [the degrees of freed om] are causal not because they

hold among the actua l values in all such real systems but because they

hold for all possible values of thi s particular system . (Consrrucrrre Realism ,

p. 84; emphasis in the original)

See also Ronald N . Giere , Explaintnq Science. A Coanit i l'e Approach (Univer­

sity of Chicago Press, 1988), Chapte r 4. Gier e is, in this case , wrong. State

space, as I will argue in Chapter 4 , provides no causal information about the

modelled processes.

46 . One's attitude towards modalities has a profound effect on one's whole

theory of science . Actualists . . . must hold that the aim of scienc e is to

describe the actual history of the world . For [modal realistsI . . . the aim

is to describe the structure of physical possibilit y (or propensity) and

necessity . Th e actual history is just that one possibilit y that happ ened to

be realiz ed .. . (Giere , Constructi ve Realism, p. 84)

47. Deleuze, Loaic if Sense, p . 54.48. Considering that Deleuze 's analysis hinges on the differen ce between the

differ entiation and int egration operators of the calculus, it will be necessary

to remove on e traditional obj ection to the very idea of giving an ontological

dime nsion to these operato rs. Thi s objection is that the output of the

differentiation op erator (instantaneo us rates of change or infinit esimals)

cannot be thought of as anything but mathematical fictions . ot to do so has

led in the past to man y ste ri le speculat ions and controversy . However,

alth ough a vector field is ind eed com posed of man y of these instantaneous

rates of change, what matters to us here arc not the 'instants ' themselves,

taken on e at a tim e, but the topoloqical in variants which those instants displa y

collect ively , that is, the singularities of the field .

49 . Ste phe n G. Eubank and J. Doyne Farmer, ' Intr oduction to Dynamical

ystems ' , in Introducti on to Nonlinear Physics, ed . Lui Lam (Springer-Verlag,

New York, 1997), p. 76.50. Abraham and Shaw , Dynamics: The Geometry if Behavior, pp. 7- 11.51. Attractors ar e indeed defined as a 'limit se t ' with an open inset (its basin).

But the word 'limit' in the definiti on mak es all the difference in the world ,

since it refers pr ecisely to the tendenci es of traj ectories to approach the

att rac tor in the limit. See ibid. , p. 44.

S2. 'Intuitively, according 10 Russell, a syste m is det erminist ic exactly if its

N O TES

pr evious stat es determine its later states in the exact sense in which the

argum ents of a function determine its values. (Van Fraasen, Laws and

Symmetl)' , p. 251)

See Van Fraasen 's discussion of the relation between the modal category

of physical necessity and deterministic laws in Chapters 3 and 4 of Laws and

Symmetry :

53. Nicolis and Prigogine, Explorinq Comple xity, p. 14. (Emphasis in the original. )

54. For example , the way Deleuze approaches the question of necessity is by

splitt ing the causal link : on one hand , processes of individuation are defined

as sequences of causes (every effect will be the cause of yet anoth er effect)

while singularities become pure incorporeal ifJeas of tho se series of causes; on

the oth er hand, these pure effects are viewed as having a quasi-causal capacity

to affect causal processes. By splitting causality this way, Dcleuze manages

to separate the det erminism which links causes to causes, from strict

necessity . See Lopic t?f Sense, p. 169 .

Deleuze uses the word 'de te rminism' as synonymous with ' necess ity',

and uses the word 'des tiny' instead for the modified link between causes . I

keep the word 'de te rminism' to avoid introdu cing neologisms, but empha­

size the break with strict necessity. Anoth er way of expres sing Delcuzc ' s

conceptualization of this modality is from D!lJerence and Repetition , p. 83,

Destin y never consists in step- by-step deterministic relations between

presents which succeed one another . . . Rather, it impli es between

successive presents non-localizable connections , actions at a distance, systems

of replay, resonances and echoe s . . . which transcend spatial locations

and temporal successions.' (My emphasis)

Th e idea of 'non-localizable connec tions' is the key conce pt her e and can

be und er stood by refer ence to convec tion cells. Whil e the causal intera ctions

between the cell 's components are localizable collisions (billiard- ball style

causality) , the source of cohere nce in the flow pattern (the periodic attractor)

is, indeed, nowher e specifically in space or tim e. Th e attractor establishes

connec tions (e lse there would be no coherence in the flow) but not

localizable ones.

')5. Willard Van Orman Quine , ' Reference and Modality', in From a Loqical Point

'!f' Viell' (Harper & Row , New York, 1965 ) , p. 155. Even though most

modal analyses deal with purely linguistic phenomena, such as counte rfactual

sente nces , the mom ent one approaches such sente nces as referring to the

real world (tec hnically, the mom ent we quantify over possible entities) we

arqu ir« an onto logical commitme nt to the existence of ess mces. In othe r

NO T ES

words, we commit ourselves to affirm that objects possess some of their

pr operties necessarily while others only contingently .

56. The first option (ensuring transworld identity through particular essences or

hacceiti es) is exemplified by Alvin Plantin ga, 'Transworld Identity or

Worldbound Individu als?', in Loux, The Possible and the Actual, pp. 154-7 .

The seco nd option (co unte rparts linked through general essences) is

illustrated by David Lewis, ' Counte rpart Th eory and Quantified Modal

Logic', in The Possible and the Actua l, pp. 117- 21.57. Delcuze , D!lJerence and Repetit ion , pp . 211- 12. See also Deleuzc , Berqsonism,

p. 97 . Deleuze does not, in fact , refer to the virtual as a physical modality,

but the fact that he explicitly contrasts virtua lity and possibilit y (following

Bergson ' s lead ) does indicate that he is thinking in modal terms.

58. I take this description of Arist otelian philos oph y from Elliot Sober, The

Nature t?f Selection (MIT Press, Cambridge , 1987), pp. 156-6 I .

59. Deleuzc, Difference and Repetiti on, p. 29. To avoid falling prey to the dangers

of representationalism (or as I call it typological thinking) Deleuze follow s

Michel Foucault 's analysis of classical representation, wh ich according to the

latter forms an episte mo logical space with four dim en. ions or 'degrees of

freedom ' : identity, resemblance, analogy and opposition, P: 262 .For a discussion of this aspect of Foucault 's thought from the point of

view of an analytical philosoph er see Gar y Gutting , Michel Foucault 's Archae­

oloBY rif Scientific Reason (Cambridge Univer sity Press, 1993), Chapter 4 .

In what follows I Simply take the idea that there are recurrent features in

these classificatory practices (rese mblance, identity, etc .) but not that these

form a global entity called an 'e pisteme' . I do not believe such global entities

or totalities exist as will becom e clear in the followi ng chapte rs .

60 . 'The first formula posits resemblance as the condition of differ ence . It

ther efore und oubtedl y demands the possibility of an identical concept for

the tw o thin gs that differ on condit ion that they are alike . . . According

to the other formula, by contrast, resemblance, identity, analogy and

opposition can no longer be conside red anyth ing but effects of a primary

difference or a primary system of differences . (Dc lcuzc , D!fJerence and

Repetiti on, p. 117)

Dclcuze, in fact , does not speak of 'c onstraints guiding a construct ive

project ' . He rath er affirms his desire for creating a ph ilosophy '!f' difference,

and then denoun ces the categories of typological or represent ational thinking

as obstacles to reaching that goa l. Th e differences he has in mind are not the

e ucrnal diffe rences between thinq« that are part and parccl of classificatory

pract ices, bUI productive differcnces perhaps best illustra ted by inccmil'e

NOTES

d!fJerences, dilTerences in temper ature, pr essure , etc. within one and the

same system, which are mark ed by thres holds of intensity determi ning phase

tra nsitions. See p. 222 .61 . Ronald F. Fox , Eners)' and the Evolution if Life (W . H. Free man, New York,

1988), p. 8.

T he mechanisms by which the chemical clem ents come into existe nce is

stella r nucleosynthesis. The processes involved are an example of how ener8Y

./1011' pr odu ces complex states of matter from simpler constituen ts. A

combination of gravitational energy and nucl ear energy converts vast

quant ities of hydr ogen gas, the simplest ele me nt, into the nuclei of other

more complex cleme nts . Nucleosynthesis involves nuclear reaction cycles

and happ ens in stages that corre late stro ngly with changes in ste llar

structure . (Emphasis in the original)

62 . Philosopher s tend to imagine that a piece of bulk material is simply a

collect ion of individual crystals arranged so perfectl y that , for all practi cal

pur poses, th e properties of the bulk sample are simply a sum of the

properties of these crystals. In oth er words, they imagine we can divide the

hulk sample in extension and , given the packing arrange ment of the crysta ls,

we will alwa ys end up with a similar if smaller sample . But in realit y, we

do not have perfectly regular crystal lattices (the irregularities playing a

crucial ro le in the stability of the stru cture) and we canno t divide a bulk

sample beyond a given size without losing some eme rge nt pr operties:

Like the biologist , the metallurgist is conce rned with aggregates and

assemblies in which repeated or ex te nded irrepularities in the arranged

atoms becom e the basis of major structural features on a larger scale ,

eventually bridging the gap betw een the atom and things perceptibl e to

human senses . (Cy ril Stanley Smith , 'Structure, Substructure, and Super ­

structure ', in A Search for Structu re [MIT Press, Cambridge, 1982), p. 54 ;

my em phasis)

See also, in the same volume , Smith, ' Grain Shapes and other Metallur­

gical Applicat ions of To pology'. O n the eme rgence of bulk prop erti es at

di fferent critical scales, see Michael A. Dun can and Denn is H. Rou vray,

tIIicroclu.ltw (Scientific Ameri can , Dece mber, 1989), p. 113.

2 THE ACTUALIZATION OF THE VIRTUAL IN SPACE

I . Michael T . Ghisclin, Metaphysics and the Oriqin l?I Species (State University of

New York Press, Albany, 1997), p. 78.

NOTES

2. A good history of this de bate, explaining the ro le which Michael Ghiselin

played in it , can be found in David L. Hull , Science as a Process (University of

Chicago Press, Chicago , 1988) , Chapter 4.

3. Ghiselin , Metap hysics and the Oriq in if Species, pp . 37-41.4. It is unclear to what extent Deleuze subscribes to th is idea of a flat onto logy

of singu lar individuals. Some parts of his theor y (for example, his theory of

tim e involving a nested set of larger and larger temporal scales) seem to

dem and such an onto logy. Yet , elsewhere , he does seem to talk of tot alities.

Thus, while I view the rea lm of the socia l as a flat onto logy (made of

individual decision -makers, individual instituti onal organizations, individual

cities, individual nation states) and thu s would never speak of 'society as a

whole' or 'culture as a whole ' , Deleuze does talk of 'society as a whole '

and spec ifically, of a virtual multiplicit y of soc iety . See, for example, Gilles

Dele uze, D!lI erence and Repetit ion (Co lumbia University Press, Ne w York,

1994), p . 186. There are also terminological problem s that need to be not ed

give n that Dcleuze uses the term ' individual' in a very idiosyncratic way. In

parti cular, he does not use 'actu al entity' and 'individual' as synonyms as I

do. For Deleuze the term ' individual' refers to an entity in the process '!f

actu alizati on , that is, before it acquires its final qualiti es and extensities . For

example, a fully develop ed hum an being would be an actu al entity , but the

embryo as it is being unfolded and develo ped wo uld be an individual. On e

would be an ex tensive being, the othe r an intensive one. (See , for example,

pages 247 and 250.) I will use the word ' individual' in the sense in which it

is used by Ghisclin to link it to anti-essenti alist thought, but this should not

cause mu ch distortion to Dcleuze.

O n the other hand, I do break with Deleuze 's use of the term 'species '

which does not seem to impl y that species are also individuals, and hence ,

the produ ct of an indi viduati on pro cess disti nct from the one that gives rise

to organic individuals during embryoge nesis . He does no t see m to keep the

tw o levels of scale separate (as I think they should be) and speaks of 'species'

and 'parts ' as the organic expression of qualities and exte nsities respectively

(p 25 1). Yet , he does acknowledge in passing the role of rep roducti ve

isolation in th e individuation of species. He writes,

A kineti cs of population adjoins, without resembling, the kinet ics of the

egg; a geog rap hical process of isolation may be no less formative of

species than intern al genetic variations, and sometimes precedes the

latt er . (p. 2 I7)

5. Ernst Mayr, quoted in Elliot Sober, The Nature of Selection (MIT Press,

Cambridge, 1987), p. 156.

NOTES

6. Ibid. , p. 159. Sober makes some corrections to Mayr 's way of explaining the

reve rsal of Aristotelian esse ntialism . He believes it is incorrect to compare

averages and essences, as Mayr do es in the extract , since averages may be

taken to be real properties at the populationa l level. So the reversal is

characte rized in terms of the rol e of variation : while for Aristot elians

hom ogeneity is the natura l state and variation is what needs special

explanation , for population thinker s it is variation which is nat ural , while

homogeneity, when it exists, is what needs to be explained .

7. lbid. , p. 160.

8. Gilles Deleuze and Felix Guattari, A Thousand Plateaus (University of

Minnesota Press, Minneapolis, 1987), p. 48. (My emphasis)

9. Niles Eldredge, Macro-E"olutionary DynamiCS (McGraw- Hill, New York ,

1989 ), pp. 155- 7 .

10.1 . D. Murray, Mathematical BioloBY (Springer-Verlag, Berlin 1989), pp . 1-4.

II. Ibid., pp. 8- 1l.

12. In both organism and cellular populations, for example, we are concerned

with rates of birth (rates of cell division ), rates of death , as we ll as migration

rates. These rates of change , in turn, define in both cases a dynamical

pro cess which disp lays threshold effects as we ll as asymptotic stabl e states.

Divergent uni versa lity also implies that these organic phenomena may share

dynamical feat ures with inorganic ones. Some processes, like the formation

of concentration patterns due to an interaction between the rate at which a

chemical react ion proceeds and the rat e at which the pr odu cts of that

reac tio n diffuse, occur in both embryological processes and non -biological

chemical processes (like the famous Belou sov-Zhabotinsky reaction), a fact

which suggests that a virtual multiplicity can be divergently actualized in

both organic and inorganic mo lecu lar populations. Indeed , the mathematical

techniques and analytical methods which are used to model intera ction s

between animal and plant populations (such as predator-prey systems) are

direct ly appli cable to reaction kinetics, that is, to the dynam ical models of

interacting populations of mol ecules, organic or inorganic. Sec ibid., p . 63 .

13. For a discussion of population -level qualiti es see Sober, Nature ef Selection,

p. 167 .

14. How does aetuali zation occur in things thems elves? Beneath the

actu al qualities and exte nsities [of things them selves] ther e are spatio -

tempor al dynami sms. Th ese arc the actualizing, differ enciating agencies .

Th ey must be surveyed in every domain , eve n though they are ordinarily

hidd en by the const ituted qualiti es and exte nsities. Embryology shows

that the division of the egg is secondary in relati on to more significant

Illorphogeneti c rno vcm mts: the augm ent ation of free surfaces, stre tching

N O TE S

of cellular layers, invagination by foldin g, regional displacement of

groups . A whole kinematics of the egg appears which implies a dynamic.

(Deleuze , D!lJerence and Repetition, p. 214)

IS. Gerald M. Edelman, Topobioloqy, An Introduction CO Molecular Emb'J'oloBY (Basic

Books, New York, 1988) , pp. 22- 4 .

16. As a result of epithe lial-mesenchymal transformation , two kinds of motion

can arise that differ to some degree in scale . The first invo lves the

obvious cel l migration that can take place after conversion to mesen ­

chyme, as well as its cessation following cond ensation of mesenchyme

into rounded epithe lial masses. Th e second . . . is the folding, invagina ­

tion or evagination of whole tissue shee ts to form various st ruc tures ,

including tubes . In both cases, new ce llular environments are created ,

leading to the possibility that different inductive Signals will be released.

(Ibid., p. 70)

17. lbid., p. 94 .

18. lbid.; pp. 80 -1.

19. The phras e 'an exact yet rigorous ' is used on several occasions by Dele uze to

refer to a style of thought, but also to a characte ristic of topological

manifolds themselves. O ne occasion is the discussio n of Bertrand Russell's

concept of 'ordinal distan ces ' which I will discuss later in the main text.

See, Dele uze and Guattari, A Thousand Plateaus, p. 483. Another use of the

phrase occurs while discussing Husserl's notion of ' vague and material

essences ' , topologieal essences which are assimilated to singularities (events)

and affects (p . 407) .

20. Arthur T . W infree, When Time Breaks Down. The Three-Dimensional DynamicS

ef Electrochemical JVa ves and Cardiac Arrhythmias (Princeton University Press,

Prin ceton, 1987), p. 253 . (My emphasis)

21. Stuart Kauffman , The Orioins ef Order. Se!f0roanizat ion and Selection in

Evolution (Oxford University Press, New York , 1993), p . 461 .

22 . lbid., p. 44 2.

23. Th e expec ted network connec tivity features exhibit stro ng self-organiza ­

tion properties analogous to phase transitions in physics, as the number

of regu latory connec tions , M, among N gen es increases. If M is small

rel ative to N, the scrambled geno mic system consists of many small

genetic circuits , each unconn ected to the remainder. As the number of

regulatory connect ions, M, increases past the number of gen ' S, N, large

connec ted circuits form . The crystallization of large circuit s as M increases

is analogo us 10 a phase transition. ( tuart Kauffman, 'Sc lf-O rganizatlon.

NOTES

Selective Adaptation and its Limit s', in Evolution at a Crossroads, eds .

David. J. Depew and Bruce H. Weber [MIT Press, Cambridge , 1996),

pp . 180)

24. In Deleuze's philosop hy th e connec t ion between multiplicities, on one hand,

and quali ties and extensities, on the other, is more intimately defined , with

differential relations corresponding to qualities and singularities to

extensities.

[A) multiplicity such as that of co lour is constituted by the virt ual

coe xiste nce of relati ons between genet ic or differential eleme nts of

a part icular order . Th ese relations are actualized in qualitatively dis­

tinct colours, while their distinctive poin ts are incarna ted in distinct

extensit ies , which correspond to tho se qualit ies ... W e have see n

that eve ry pro cess of actualizatio n was in this sense a double differencia­

tio n, qualitative and exte nsive. (Deleuze, D!iJerence and Repetit ion ,

p. 245)

25. K. Eric Drexler, ' Biological and Nanomechanical Syste ms : Contrasts in

Evolut ionary Capacity', in ArtifiCial L!fe , cd. Christo pher G. Langton (Addi­

son- Wes ley, Redwood City, 1989) , p . 510.

26. Dele uze, D!iJerence and Repetit ion, P: 223.

Intensity cre ates the extensities and the qualiti es in which it is ex plicated;

these ex te nsities and qualities are differ enciat ed . .. Crea tion is always

the production of lines and figures of difler enciation. It is neverth eless

true that intensity is explicated only in being cance led in this differ en ­

ciated syste m that it cre ates. (p. 255)

27. Van Wy len, Thermodynamics, p. 16.28. Bert rand Russell , Principles if Mathemati cs (W. W. Nort on , New York) ,

p. 104 (for remarks on pleasur e) and p. 171 (for remarks on colour) .

Dc lcuze wo uld not co unt pleasure as an intensive quantity part of mental

irulivuluatinq processes . He see ms to view pleasur e as an effect of the cance lling

of intensive differe nces:

Bioph ysical life imp lies a field of individuati on in wh ich differ ences in

intensity are distributed her e and there in the fo rm of excitations. Th e

quanti tat ive and qua litative process of the resolution of such diffe rences

is what we call pleasure . (Deleuze, D!iJerence and Repet ition , p. 96)

29. Marti n II . Krieger , DoinS PhySiCS. How PhySicists Take /-101.1 C!l th e World (Indiana

l.ln ivcrsity Press, Bloom ington and Indianapolis, 1992), p. 130.

10. n,·J..uz<', D!I]crmce lind Repa it ion , p. 222. (My emphasis)

NOTES

In this extrac t, 'd ivers ity' refers to the wo rld of actu al phenomena and

their exte rna lly defined differe nces (that is, to difference as subordinated to

rese mb lance) while intensive differe nces define the in-itself (nuo mena) of

the world, the positive and prod uct ive differences which create or generate

phenom ena.

31. lIya Prigogine and Isabelle Stengers , Order out l' Chaos. Man 's Nell' Dialoque

with Na tu re (Bantam Books, Ne w York, 1984) , p. 135.32. Deleuze explains the relation between intensive differences and gene tic

differences b), saying that 'complex systems increasingly tend to interiorize

thei r constituent di ffere nces ' , that is, thei r individuating factors (D!iJerence

and Repetit ion , p . 256). See also Deleuze 's discussion of Darwini an differ­

ences on pp . 248-9.33. Wh en discussing the virtual and the intensive, Deleuze usually divides the

subject into two areas, although the terminology varies . Someti mes he speaks

of 'singu larities and affects' , other times of 'speeds and affects', yet in other

places he speaks of 'events and att r ibutes' . All these formulat ions are , I

believe , equivalent . See furthe r discussion and references in Chapte r 3,

footnote 46.

34. O n this new class of formal spaces which complements state space, see

W alter Fontana, ' Functional Self-Organization in Complex , Syste ms' , in

1990 Lea ures in Complex Systems, eds, L)'nn Nadel and Danie l Ste in (Addison­

W esley, Redwood City, 1991); and, in the same volume , Stuart Kauffman,

' Random Grammars: A New Class of Mod els for Functional Integration and

T ransfor mation in the Biological , Neura l and Social Sciences' .

35. W e know nothing about a bod)' until we know what it can do, what its

affects are , how they can or cannot ente r into composition with other

affec ts, with the affects of ano the r body, either to des troy that bod y or

to be destroyed by it , eithe r to exc hange actions and passions with it or

to join with it in composing a mor e pow erful body. (Deleuze and

Guatta ri, A Th ousand Plateaus, P: 257)

36. James J. Gibson, The Ecoloqical Approach to Visual Percept ion (Houghto n Mifllin

Company, Boston, 1979), pp. 15-1 6.37 . lbid., p. 132.38. orne of the rec ur rent assemb ly patt erns that have been discover ed (and

which may tu rn out to be universal) are of the type that articulates

hete roge neous elements. Stuart Kauffman has coined the term ' meshwo rk'

to refer to th is type of assem hlage.. ee Stua rt Kauffman, Random Gramma rs,

p.428.I haw mad' -xtcnsivc usc of Kauffm an 's meshwork s, and of the ir

op posite, hierarchies, as recur ren t assernhly patt erns for the analysis of

NOT ES

human history in Manu el DeLanda, A Thousand Years if Nonlinear History

(Zone Books, New York, 1997). A similar distin ction (or a spec ial case , that

of centralized and decentralized decision -making systems) as well as a relat ed

set of recurrent assembl y patterns (clockworks, motors and networks) is

d iscussed and appli ed to history in Manuel Dcl.anda, War in the Age ifIntelligent Machines (Zon e Books, New York , 1991).

~9 . It is no longer .a question of imp osing a form upon a matter but of

elaborating an increasingly rich and consistent mat erial , the better to tap

increasing ly intense f orces. What makes a mat erial increasingl y rich is the

same as what holds heterogeneiti es topether with out their ceasing to be

heterogeneous. (Del euze and Guattarl , A Thousand Plateaus, p . 329; my

emphasis)

40. Delcuze , D!fference and Repetition, P: 22 3.

There is an illusion tied to intensive quantities. This illusion, however, is

not int ensit y itself, but rather the movem ent by which difference in

intensity is canceled. Nor is it onl y apparently cancele d . It is really

canceled , but outside itself, in extensity and underneath quality. (p . 240;

my emphasis)

41 . It is now an easy matter to extend our discussion to nonequiltbrium states

. . . They can be transient . . . But they can also be permanent if we

establish and maintain appropriate conditions, which we refer to as

constraints. Thus , a temperature difference appli ed between two sections

of a slab .. . will result in nonequilibrium situations in which the syste m

is never allow ed to identify itself with its enviro nment. We should not

conclude from these examples that non equilibrium is an artificially

imposed condition ... we see non equilibrium states in mu ch of our

natural environme nt - for example , the state of the biosphere which is

subjected to an energy nux that arises from the balance of radiation

between th e sun and the earth . (Emphasis in the original; Gregoire

Nico lis and lIya Prigogine, Exploring CompleXity [W. H. Freeman, New

York 1989], p . 56)

42. lbul ., p. 59.4L lbid. , p. 60.

44. David Acheso n, From Calculus to Chaos. An Introduction to Dynamics (O xford

University Press, Ox ford , 1997) , pp. 54-6.

45. Delcuze and Guatta ri, Whar ;s Philosophy ?, p. 140. (My emphasis)

4fl. Richard Hin 'hlilTe , T oward a H omol ogy 01' Pro cess: Evolut ionary lmpli ca-

N O T ES

tions of Experime ntal Studi es on the Generation of Skeletal Pattern in Avian

Limb Development ', in Organi zational Constraitus on the Dynamics ifEvolution ,

cds. J. Maynard Smith and G. Vida (Mancheste r University Press, Man­

cheste r 1990), p . 123. (Emphasis in the original)

The biologi st Brian Goodwin, who has taken the br oken symmetry

approach to classification to its extreme, argu es that these insights about

specific organs may be generalized to explain the dynamical origin of all the

morphological features behind our static classifications:

There are several consequences of this view of morpbogenesis. First , it is

evident that morphology is gen erated in a hierarchical manner, from

simple to complex , as bifurcations result in spatially ordered asymmetri es

and periodicities, and nonlinearities give rise to fine local detail. Since

th ere is a limited set of simple broken symme tries and patterns that are

possible (e.g. , radial, bilateral, periodic) , and since developing organism s

must start off laying down these elem ents of spatial orde r , it follows that

these basic forms will be most common among all species. On the oth er

hand, the finer details of pattern will be mo st variabl e between species ,

since the pattern-generating process results in a combinatorial richness of

terminal detail , and specific gen e products in different species stabilize

traj ectories leading to one or another of these . . . Th e fact that Virtually

all the basic organismic bod y plans were discovered and established

during an early evolutionary period, the Cambrian, is oft en remarked

with surp rise , but it is just what one would expect on the basis of the

above argument. (Brian C. Goodwin, 'The Evoluti on of Gen eri c Form s' ,

in Organizational Constraints on the Dynamics clj' Evolution, eds. Maynard

Smith and Vida , pp . 114-[5)

See also Brian Goodwin, How the Leopard Changed its Spots (Simon & Schuster,

New York 1996), Chapter 5.

47 . When I introduced dilTerential geometry in Chapte r 1 I said that one of

Gauss 's achievements was to get rid of an embedding space by coordinatizing

the manifold itself. This allow ed him to define the equivalent of metri c

lengths (and oth er properti es) in this differ enti al space. This coordinatization

is an example of what I mean when I say that a nonmetric space is

metricized . DeIeuze also refers to this operation in his discussion of the

relation betw een metric (str iated) and smooth spaces in Deleuze and

Guatta r i, A Thousand Plateaus, p. 486.

48 . Co nsiste ncy necessarily occ urs between heterogeneit ies, not because it is

the hirth of a diff erentiation. hut because heterogeneit ies that were

limnerl )' conte nt to coex ist or succcc l one another becom e hound up

NOTES

wit h one ano the r through the 'conso lidatio n' of th eir coexiste nce or

succession . . . W hat we term machinic is pr ecisely this synthesis of

het erogen eities as such . (ibul.; p. 330)

T erms like "self-consistent aggregate' and ' mac hinic assemblage' are used

synonymously in this book.

49. Although there are a few math em ati cal functions which produce seve ra l

outputs, the majority of them have a sinole outpu t. That is, some functions

map inputs and outputs (ar!:.'1Iments and values) in a one- to-one fashion,

others in a man y-to -on e fashion , and a few in a one- to-many form. See

Russe ll, Principles rif Mathematics , pp . 265-6. Deleu ze 's reciprocal det ermina­

tion , I believe , would imply a many -co-many mapp ino, and a mapping such as

this would be useless as a functi on . On th e othe r hand , this ' useless '

mappi ng wou ld capture th e desir ed idea for a multiplicit y, an organization

of the ' many' as such, without the need for th e 'one ' .

50. Dcleuze , D!lJerence and Repetiti on, pp . 172 - 4 .

In ' W hat is Philo sophy' , th e distinction between virtual multiplicities

(there referred to as 'conce pts') , and functions is mad e the basis of Dclcu zc' s

critique of science's inability to gra~p the virtual. Unfortunately, the analysis

there is obscu re d by his introduction of unfamiliar terms like ' functive'. Sec

Dcleuze and Gu attari , What is Philosophy ?, pp . 117-1 8.

I think its is clearer to see his rejecti on of functions as mod els for the

virtual in terms of th e pr e-indi vidual nature of the virtual coupled to the

fact that functi on s may be taken to represent ind ividu ation processes. Thi s

way it becomes clear why function s without this individuation aspect

(wi thout a distinction between dep endent and inde pe nde nt variables) can

indeed be mad e part of th e virtual , Reference to ' form less func tions' as a

defining ele ment of concre te uni versals (o r as th ese are some times referred

to, 'a bst ract machin es ' ) can be found in Deleuze and Guattari, A Thousand

Plateaus, p. 141.

51. De leuze, Loqic rif Sense, P: 52. Let me elabo rate thi s point (eve nts as pr e­

indi vidu al ent it ies) by first conside r ing the typ e of individ uality of actual

crcnts. Co mpare d to th e individuality of an organi sm or a species (to mention

only the two entit ies for which I have given individuation pr ocesses) an

actua l eve nt has a more fleeting and changing individuation . Del iuze argues

that eve nts have th e ind ividu ality of a haecceity , exe mplified by the ' thisness'

or un ique Sing ulari ty of a mom en t. As he says

T h re is a mode o f ind ividu ation very differe nt fro m that of a person,

subject, thing , or substance. W e reserve the name haccceity for it. A

season, a Winter, a summer, an hour , a date have a perfect individua lity

1'lCking nothing, even ulOugh th is individua lity is di ffere nt from that of a

NOTES

th ing or a subject. T hey are haecceities in the sense that th ey consist

entire ly of relation s of movement and rest between molecules or

particles , capacities to affect and be affecte d. (Deleuze and Guattari , A

Thousand Plateaus, p. 26 1)

Some of th e het erogeneou s assemblages I mentioned before, such as th e

assemblage of a wa lking anima l, a piece of ground and a grav itational field,

have this individuality. Thi s is particularl y clear if we do not picture an

abstract case but thi nk instea d abo ut a concre te eve nt: this animal walking

on this hot and humid summe r day. (' This should be read without a pause :

th e animal-stalks-a t-five-o' cloc k' , p . 263). Thi s event consists of affects, not

only the affordances of anima l, gro und and fie ld , but also the capacities of

the othe r individuals involved, including degrees o f heat and humidity. (' A

degree of heat is a perfect ly individuated warmth distin ct from th e substance

or subj ect that receives it . A degree of heat can ente r int o com position with

a degree of whiten ess, or with another degr ee of heat , to form a third

unique ind ividuality . . .' , p. 253) T he event also consists of relations of

rapidity and slowness : th e gro und affords the animal a so lid sur face only

because relative to the spee d or temporal scale of change of the animal, the

ground changes too slow ly. At geological t ime scales th is piece of so lid

ground would indeed be mu ch more Iluld .

T o apply th is to ideal eve nts . Th e singularit ies wh ich populate the virtual

are also haecceities , but the tw o definin g features (spee ds and affects) are

distributed differe ntly: a singulari ty is nothing bu t an accidental feature in a

field of speeds (o r velocit y vectors) its indi viduality consisting enti rely of its

inva riance , that is, its capac ity of not being affect ed by ce rt ain transforma­

tio ns which affect th e rest of th e field .

52. The term •conde nsat ion of singu larities' to refer to the expansion of

singulari t ies into series, and th e establishme nt of converge nt and divergent

relation s betw een series, is used for exa m ple in Deleuze, D!1Jerence and

Repetit ion, r- 190 .

53. Multipliciti es (o r Ideas) are referred to as 'complexes of coe xiste nce' in

ibid ., p. 186.In othe r wo rds , unlike th e singu larit ies which define an int en sive process

which may be actualized only one at a time (e ither because th e bifurcation s

need to be crossed sequentially, or because only one am ong alternati ve

attracto rs may be occupied) virtual singularit ies all coe xist within their own

special temporalit y. Wi thin th e int ensive ' the Ideas, relations, variati on s in

these rela tio ns [embed ded levels] and dist inct ive point s [singularitiesl arc in

a sense separated: instea d of coexisting they enter states of simultanei ty or

succession' (p. 252).

N OT E S

54. Th e impo rtance of orde r , from a purely mathematical standpo int, has

been immeasurably increased by man y modern devel opments. Dedekind,

Canto r , and Pean o have shown how to base all Arithmeti c and Analysis

upon series of a ce rtain kind . . . Irration als are defined . . . entire ly by

the help of orde r . . . Proj ectiv e Geom etry [has] shown how to give

points, lines and plan es an orde r indep end ent of metrical considerat ions

and of quantity ; whil e descriptive Geom etry proves that a very large part

of Geom etry demands only th e possibility of serial arrangem ent. (Russell ,

Principles ifMathematics, p. 199)

55. lbid.; pp . 157-9 . Actually, Russell uses the more gene ral term 'magnitude '

to refer to th ese indi visible int ensiti es , and 'distance' as a speci al case of a

magnitude. A terminologi cal confusion should be avoid ed here. Russell uses

the term 'magnitude' to oppose that of 'quantity' (one involves onl y serial

orde r, the othe r cardinal number) . But wh en Deleuz e co mme nts on Russell' s

wo rk (as wcll as Mein ong 's ), he uses ' mag nitude ' as synonym with 'quanti ty'

and opposes both to 'distance' . See Deleuz e and Guattari , A Thousand

Plateaus, p. 483.

This terminological conflict should not be a probl em here since I will not

be using th e term 'magnitude ', and I will always use the term 'ordinal

distance' instead of just 'dis tance ' to distingui sh th e latter from ' me tric

d istances' or length s. Although Russell introduces distances as inte nsive, that

is, as indivisible in exte nsion, he then devises a sche me which allow s him to

speak of distanc es as divisible (by reducing them, via a conve ntion, to

extensive 's tre tc hes') and thu s abandon s any hope of linking th e int ensiv e

and the ex te nsive morphogenctically. (Russe ll, Principles if ,tlathematics,

pp . 180 - 2)

')6 . Ordinal constr uction do es not imply a supposed same unit but only .

an ir re ducible noti on of distan ce - th e distances implicate d in th e depth

of an intensive spatium (orde re d distan ces). Identical unity is not pr esup­

pose d by ordinatio n ; on the co ntrary, this belongs to cardinal number

. . . 'We sho uld not , therefore , believe th at cardinal number results

unaIyticalIy from ordina l, or from th e final terms of finite ordinal series

. . . In fact, ordinal number be com es cardina l only by ex te nsion , to the

exte nt that the distances [are] develop ed and equalized in an ex tensity

established by natu ral number. W e sho uld therefore say th at , from th e

outset, the concept of number is synthetic. (Deleuze , D!fference and

Repetition, p. 233; m), emphasis)

Russell , on the other hand , establishes between magnitudes and numbers

on I)" .1 Iogic'lI re lation, that bet ween the ge ne ral and the part icular : qu antities

NOTES

are magnitudes which are particularized by spatio -te mpo ral positi on (Russe ll,

Principles ifMathematics, p. 167) .O ne way of bringing up the differen ce between Deleuzc ' s and Russell' s

appro aches to series and numbers, is by contrasting th eir analyses of the

th eory of ir rational numbers of Dedekind . Arguing that there were gaps in

th e co mpact series of ratio nal number s, Ded ekind introduced th e noti on of

a 'cut', a way of segmenting a dense cont inuum int o two , mutually

excl uding, parts. His idea was to define th e conce pt of number in terms of

such cuts performed on purely ordinal continua. Some of these discontinui­

ties yield rational numbers, but others, he po stulated , mu st yield irrationals.

Russell , for whom the den sity of the rati onal s see ms to be eno ugh, objects

to this merely postulated existe nce of irrational cuts, and equates irrational s

with one of th e classes of rati onals crea te d by th e cut . Thi s, in effect ,

ex plains one exte nsive conce pt (num ber) in terms of anothe r, equally

exte nsive one (class or set). Deleuz e, on th e cont rary, sees in th e conce pt

of a cut a way to ex press the genesis of numeri cal quantity out o f int ensive

non -numeri cal co ntinua: ' In thi s sense , it is the cut which constitutes th e

next genus of number, the ideal cause of continuity or the pu re eleme nt of

quantitati vity' (De leuze , D!fference and Repetition, p . 172).

57. Dcleuze , LoOic cj"Sense, p. 109.

58 . Divergen ce and disjun ction are , on th e cont rary , affirm ed as such . But

what does it mean to make divergen ce and disjun ct ion th e objects of

affirmation ? As a gene ral rul e two th ings are simultaneou sly affirm ed only

to the exten t that their differen ce is deni ed . . . W e speak, on the

contrary , of an ope rat ion according to which tw o thin gs . . . are affirm ed

through th eir difference . . . to affirm their distance as that wh ich relates

one to the other insofar as th ey are different . . . Th c idea of a positive

distance as distance (and not as an annulled or overco me distan ce) appea rs

to us esse ntia l . . . Th e idea of positi ve distance belongs to top ology and

th e sur face . (Ibid., p. 172 )

59. Co nverge nt and divergent relation s defin e th e modal stucus of vir tual relation s.

Following Leibniz, Deleuze calls th ese virtual relati on s compossibility and

incompossibilitv:

T wo eve nts are com poss ihle wh en the ser ies which are organ ized aro und

their singularit ies extend in all direction s [that is, co ll\'erge ]; th e)' are

incornpossihle wh en th e ser ies diverge in the vicinity of constitutive

singularities . Converge nce and divergence are ent irely original relat ion s

which cover th e rich domain of alogica l com patibilit ies and incompatib il­

irics. (Ihid., p. 172 )

N O TE S

The modal status of the virtua l may be more easily grasped by contrast ing

it with othe r modal rel ations, such as the relations which modal logicians

postulate to exist between possible worlds. Th e modern theor y of possible

worlds is also based on the ideas of Leibniz, but disregards these alogical

capacities or affect. Briefly, the key relation between possible worlds is that

of accessibili ty: one world is accessible from anoth er possible one , if eve ry

situation possible in one is also possible in the othe r. Given this relation,

possible worlds may be grouped togeth er into families or equ ivalence classes.

Wh enever situations in one class are imposs ible in anoth er one, that is,

when ther e exist logical or physical contradict ions between them , worlds

belonging to one arc inaccessible from those belongin g to the oth er (Michae l

J. Loux , ' Intro duction: Modality and Metaphysics ', in The Possible and the

Actual , pp. 20- 8).

Dclcuze would accept these ideas but argue that contradic tions between

possible worlds are a deriva tive phenomenon . In other words, that distribu­

tions of possibl e worlds, and their fully individuated contents, depend on

deeper relations of compossibility and incompossibility between pre-indi vid­

ual multiplicities: where the series emanating from multiplicities converge ,

a family of accessible possible worlds would be defined; wher e they diverge,

an inaccessible family of worlds would begin . See Gilles Deleuze , The Fold.

Lcibntz and the Baroque (University of Minn esota Press, Minn eapolis, 1997),

p. 60 . ee also Deleuze , D!fJerence and Repetiti on , p. 48, wher e he adds 'the

noti on of incompossibility in no way reduces to that of contradiction and

docs not even impl y real opposition: it impli es only divergence ... '

60 . Dcleuzc, Loa ic rif Sense, p. 5.

6 I . peaking of the particular case of catastro phe theory, wher e the limitation

to pot enti al-driv en syste ms with four degr ees of freedom makes a full

classification of attractor s and bifurcat ions possible, Alexander W oodcock

and Monte Davies write

In any syste m governed by a pot ential and in which the system 's behavior

is det ermined by no more than four differ ent factors, only seven

qualitatively differ ent types of discontinuity [bifurcation] are possible . In

other words, while ther e is an infinit e number of ways for such a system

to change continuously (staying at or near equilibrium) , there are onl y

seven structura lly stable ways for it to change discontinuously (passing

through non -equilibrium rates), Other ways are conce ivable, but un­

stable; th 'y are unlikel y to happ en mor e than once . . . The qualitati ve

type of any stable discontinuity does not depend on the spec ific nature of

the pote ntial involved , merely on its existence. It do 's not depend on the

specific conditions reg ulating behavior , merely on their number . It does

N O T E S

not de pend on the specific quantitative cause-and-effect relati onship

between the conditions and the resulti ng behavior merely on the empirica l

Jaa that such a relationship exists. (Alexande r W oodcock and Monte Davies,

Catastrophe Theory (E. P. Dutton , Ne w York , 1978), p . 42; my emphasis)

Th ere are tw o imp ortant ideas expressed here. Th e first is related to the

question of uni versality: as long as differ ent equation ' or differ ent physical

syste ms share the same topological invariants (the same number of singulari­

ties, the same number of dim ensions) the detail ed nature of the equations

or of the syste m (the speci fic type of intensive differ ence driving th e process,

or the speci fic quantities which define the process) does not make mu ch

difference in the spec ificatio n of their long -term tend encies. Th e second idea

relates to the question of immanence: the long-term (asympto tic) tend encies

of a process may he independ ent of specific causes , but they do depend for

their very existe nce on ther e being some causal process or another.

62. Deleuze, Loa ic rifSense, p. 169. (My emphasis) Deleuze adopts this approach

from the Stoics who wer e the first to split the causal link : on one hand,

processes of individuation are defined as sequences of causes (every effect

will be the cause of yet another effect) whil e singularities becom e pure

incorpor eal effects of those series of causes ; on the other hand, these pure

effects are viewed as having a quasi-causal capacity to endow causal processes

with cohere nt form. By splitti ng causality this way, Delcuze manages to

separate the determinism (or destin y) which links causes to causes, from

strict necessity.

63 . lbid., p. 147.64. Th e image of echoes and resonan ces as that which links multipliciti es recurs

throughout Deleuze 's work. See Chapte r 3, footn ot e 53 for an explanatio n

and examples.

65 . Kenn eth M. Sayre, Cybernetics and the Philosophy rif Mind (Rout ledge and

Kegan Paul, Lond on, 1976), p . 23.

66 . lbid., pp. 26- 30.67. Th er e is a close relation between communication theory and thermodyn­

amics. Much as in the latter the equilibrium state (for an isolated system) is

defined as the one characte rized by maximum disorder (maximum entro py),

the state achieved once differ ences in int ensity have been cance lled, so in

the former equilibrium corresponds to a situation where the differ ences

within series have been cance lled , where all the events have becom e

equiprobable. In such state no information may flow in the channel (ibid . ,

pp. 38- 43).Deleuze uses th is connectio n between the intensive and the informational

to define the relations between the series of ideal vents. As I have said, he

NOTES

re fers to an information channel as a 's ignal' , and to th e information quanta

as 's igns' ,

Such systems, constit ute d by placing dispara te elements or het erogeneou s

series in cc:nmunicat ion, arc in a sense qui te common. They are signal­

sign syste ms. The signal is a structure in which differen ces in po tentia l

arc distributed, assuring the comm unication of disparate co mpone nts: th e

sign is what flashes across the boundary of two levels, between tw o

comm unicating series. Indeed, it see ms th at all phen omen a respond to

th ese conditio ns inasmuch as they find th eir ground in a co nstitutive

dissymmetry, di fferen ce, inequality . All physical syste ms are signals, all

qualities are signs. (De leuze, Logic c1 Sense, p . 26 1)

Sec also De leuze, Difference and Repetition, pp . 20 and 222 .

68. If we examine th e singular ities co rres po nding to th e tw o im portant basic

se ries we see that th ey arc distin gu ished, in both cases, by their

distribution . From one to the othe r, certa in singular points disapp ear or

are divide d, or und ergo a change of nature and funct ion. Th e moment

th e two series resonate or co mmunicate we pass fro m one distri bution to

another. (Dele uze , Logic c1 Sense, p. 53)

69 . It is in difference that . . . phen om en a Hash th eir meaning like signs. Th e

inte nse world of differen ces . . . is pr ecisely th e object of a supe rio r

em piricism . This empiricis m teaches us a strange ' reason' , th at of

the mult iple, chaos, and differen ce . (Deleuze, Difference and Repetition,

p. 57)

T here is in addition a temporal dim ension of the virt ual, which I will

discuss in the next chapte r, whi ch also defines thi s othe r empiricism.

An Idea, in this sense , is neither one nor mu ltipl e , but a mu lt iplicit y

constituted of d ifferentia l ele ments, differe nt ial re lations betw een th ose

clements , and singularit ies co rrespo nding to those relation s .. . All three

are projected in an ideal te mpo ral dimen sion which is th at of pr ogressive

determination . There is, therefore , an empiricism c1 the Idca . . . (p . 278;

my emphasis)

On the concepts of mu ltiplicity and quasi-causal operator (and rela ted

ideas, like 'perplication' , 'complication', etc.) as ernpirico-idcal notions, see

r- 84 .70 . Stephanie For rest, 'E mergent o rn putationr .'elf-organizing, o llcctive and

Coop erat ive Phen om en a in Nat ura l and Art ificial om puting ctwo rks", in

NOTES

Emerpent Computation, ed . Ste phanie Forrest (MIT Press, Cambridge, 1991),

p. 2.71. .lyre, Cybemetics and the Philosophy c1 Mind, p. 30.72 . W hen th e two series of events are co llapsed into one we get what is called

a 'Markov process'. See ibid., p. 29.

73. David L. Goodstei n, States C!I Matter (Dover, New York, 1985), pp . 468 -86.

See also Nicolis and Prigogine, Exploring Complexity , pp . 168- 85.74 . T hese other charac te r istics arc a 'c r it ical slowing down ' (rel axation tim es

become lon ger as th e singularity is approached) and 'sensitivity to size ' (the

dynamics o f a syste m can take into account de tails abo ut boundary con­

di tion s). However , the link between th ese phenomena and information

pro cessing and storage has been esta blished only with in the narrow field of

'cellular auto mata' mod els of comp uta tion. See Christophe r G. Langto n,

'Computa tio n at th e Edge of Chaos ', in Etnerpent Compu tation , ed. Forres t,

pp . 32-3.

75 . Chris to phe r G . Langton , 'Li fe at the Edge of Chaos', in Artificial Life II, eds.

Christo phe r G . Langton, Charles Tay lor, Doyne Farmer and Steen Rasmus­

sen (Addison-Wesley, Redwood City, 1992), pp . 85-6.

76 . Melanie Mitchell , James P. Crutchfield and Pet er T. Hraber , ' Dyna mics,

Co mputat ion , and the "Edge o f Chaos": A Reexam ination ' , in Complexity:

Afetaphors. Models. and Reality, cds . George A. Co wa n, David Pines and David

Meltzer (Addiso n-Wesley, Redwood City , 1994), p . 5 10.

Th e resu lts present ed here do not disprov e the hypothesis that co mputa­

tional capability can be co rrel ated with phase tr ansit ion s in [cellular

automata ] rule space. Ind eed , this general phenom ena has alrea dy been

not ed for other dynami cal syste ms . .. More generally , the co mputa tio nal

capacity of evolving syste ms may very well require dynamical properties

characte ristic of phase tr ansit ions if th ey are to incr ease their complexity .

3 TH E ACTUALIZATIO N O F THE VIRTUAL IN TIM E

1. O n the history of these conflicting conce ptio ns of tim e and a philosophical

discussion of the different ways in which th e conflict has been approached in

both physics and philosoph y of scie nce, see Lawren ce Sklar , Physics and

Chance. Philosophical Issues in the Foundations C!f Statistical Mechanics (Cambridge

University Press, Camhri dge, 1995), Chapte r 10. And Robert B. Lindsay

and Henr )' Margen au , Foundations tif Physics (Ox Bow Press, W oodbridge ,

1981), Chapte r 5.

2. Joe Rosen , Symmcrry in Scicnce (Springer. Ver lag , Ne w York, 1995), p. 141.

In add ition to r('vc rsing the o rde r of th e temporal seque nce, a time

NOTES

'reflection' transformat ion changes the sign of any variable (such as velocit y)

that depends on th e time variable . This introduces some subtle ideas that

matter in a se rio us analysis of th e symmetry properties of laws. ce

discussio n of this point in Sklar, Physics and Chance, pp . 246-8.

3. Gregoire Nicolis and lIya Prigogine, Explorinq Complexity (W. H. Freeman,

ew York 1989) , p . 52.

4 . Euge ne P. Wigner, ' Invariance in Physical Theory', in Symmetries and

Rifleetions, cds. W alter Moore and Michael Scriven (Ox Bow Press, W ood ­

bri dge, 1979), p . 4.

As the physicist Euge ne W igner rem ark s, if physical regularities had not

displayed this minimal am ount of invariance , we would probably never have

discovered th em at all simply because they would not app ear to us as

regularities. lnvariance und er transformations can also reveal subtle assump­

tions behind a law . For instan ce , to say that a law is invariant und er spatial

or temporal displacem ent implies that, as far as the regulariti es describ ed by

th e law are conce rned, space and tim e are homoqeneous. Similarly, to say that a

law is invariant und er rot ation in space is to say that th e absolute or ientatio n

of the states of the pro cess mak es no differen ce in th e pro cess' s behaviour,

but it also means that we assume space to have uniform prop erties in all

directions (te chnically, we assum e it to be isotrop ic) .

5. T here arc several strategies for ex plaining ir reversibility away. Some

physicists, for example, think th e inh erent directionality of th e arrow of

time, so evident in macr oscop ic pro cesses, is merely a subjective effect (an

effect of our ignorance of all th e micr o details) . T o othe rs th e direction alit y

of time is not reducible to psychology but it is nevertheless den ied the status

of a t rue law , being merely a continge nt sta tistica l result. As the physicist

John Wheeler puts it, th e real molecular int eraction s arc 'time-symmetric

with on ly th e sta tistics of large numbers giving it the app earan ce of

asymmetry' (j ohn A. W heeler, 'T ime T oda y ' , in Physical Oriains if Time

A~mmet'J' , eds , Jon athan J . Halliw ell, Juan Perez-Mercader and W ojciech H.

Zurek [Cambridge Uni versity Press, Cambridge , 1996), p . 1) .

In gene ra l, th e authority of th e old reversible tim e has been pr eserv ed

and th e tim e of classical thermodynamics has disapp eared from the structu re

of the edifice of physics. As Wheeler puts it ,

The expansion of th e empire of tim e has elevated the conce pt, human

born as it is , to platform up on platform upon platform of authority.

R 'gularit ies of sun and seaso n raised the first foundation . On top of it

cw to nian dynam ics erecte d a second and tight er platform ; special rela­

tivity a third , terraced further in and up; and gene ral re lativ ity stands at

ti ll' sum mit, the final level of authority. ot cxc pt out of the mou th of

NOTES

Einste in 's 1915 and still standard theory of spacetime can one hear th e

ge nerally agreed acco unt of all tha t 'time' now means and measures. (p. 6)

6 . lIya Prigogine , From Beina to Becomino (W. H. Freeman . ew York, 19S0),p. 19.

7, Arthur S. lberall , Towards a General Science if Viable Systems (McGraw- Hi li,

ew York 1972) .

Iberall' s onto logy is based on individuals which he calls ' atomisrns' (a

category of which atoms would be only one instance) . He co nce ives of these

in general as auto no mo us, nonlinear osci llators. Thanks to their nonlinearity

th ese atomi sms are show n capable of int eracti ve orde ring (via ent rainme nt,

for example) and capable of forming a continuum at a larger scale . Th ese

cont inua, in turn , are sho wn to und ergo symme try- breaking bifurcations

which fragm ent them (or quantize th em ) to yie ld super-a to misms, that is,

ind ividuals at a larger spatio- te rnporal scale. Ibcrall sho ws in detail how this

alternation of ato m ism and continuum can be used recursively to account

for man y features of physics, che mistry, biology and eve n sOciology. He also

shows, on th e other hand, how mu ch this picture breaks with those of

classical and, more importantl y, quantum physics, given tha t the latter docs

not give a morphogen etic acco un t of quantization .

S. Winfr ee does not use the terms ' intensive' or 'nonme tric', Yet, in the previous

chapte r I qu ot ed W infree's ideas abo ut top ological thi nking when appli ed to

biology and his ideas are ind eed very close to those of Deleuze . Using my

terminology, we can sa)' th at an anexaet yet riaorous approach characterizes

W infree 's research on the birth and death of oscillations, a process which

also exhibits divergent un iversality or m echani sm -indepen den ce . In his

words,

As a resul t of these co llec tive efTorts, th e reality of phaseless sets, phase

singularities , tim e crystals, and so on became firmly established. Th eir

physiological ' me aning' is less clear . . . But that deficien cy is in a way

the most int eresting aspect of these findin gs: because their pr edi ction was

in no way dependent on the mechanistic underpinn inqs of circadian physiology,

the same principl es mig ht find appli cati ons in other areas of physiology

and bio chemi stry . Th ese prin cip les m-e not 'mathematical', in the familiar

sense of 'mathematics ' as 'moving symbols around on paper' or ' mov ing

numbers aro und in co mpute rs' . Th ey are, rath er, [top ologica l] concepts

about conti nuity that could be used in diverse contexts with s!ifJicient riBor

to precisely infer biological or che mical eve nts that had not been

observed. (Ar thur T. Win free , II'hen Time Breaks Down. The Three­

Dimensional Dynamics if Electrochemical lI'al 'cs and Cardiac Arrhyt hmias

[Princet on Unin 'rsit)' Press, Princet on , 19 71, pp. 64 - 5; m), emphasis)

N O T E S

9. Ian tewart and Martin Golubitsk y, Feaiful Symmetry (Blackwe ll, O xford,

1992), pp . 66 - 7.

10. Nicolis and Prigogine, Explorinq Complexity , p. 21.

11. Ibid . ,p.103.

12 . Iberall, Towards a General Science ?f Viable Systems, p . I S3.

J 3. lbid. , p. 161.

14 . Gi lles Dcl euze, Loa ic rifSense (Columbia Uni versity Press, Ne w York, 1990),

p. 162. (My emphasis)

1S. lbid., p. 62 .

16 . Gilles Deleuze , D!iJerence and Repeti ti on (Co lumbia Univ ersit y Press, ew

York, 1994), PI'. 70 -1. In these pages Deleuze , following Hume, does

indeed pr esent this contraction whi ch synthesizes pr esent time as a faculty

of the mind : a cont ractile power rif contemplation or imaqination which retains a

past and ant icipates a future. But a few pages later (p . 73 ) he says that ' we

are made of contracte d water , earth , light and air - not merely prior to th e

recognition or representation of th ese, but pri or to th eir being sensed ' .

Clear!y, this remark mak es no sense within a purely psychol ogical int erpre­

tat ion , but it does if we think of thi s contraction as involving a metabolic

cycle with a characte rist ic tim e scale. He goes on to ascribe to habi ts (or to

the co ntraction of rep etitive , habitual behaviour) a similar power of synthesis

(I" 74) but again , this appli es not only to the habi ts of human beings but to

any re pe titive, cyclic beha viour at all scales.

A soul sho uld be attributed to the heart, to th e muscles, nerves and ce lls,

but a conte mplative soul wh ose entire functi on is to cont ract a habit.

T his is no mystical or barbarous hypothesis. On the co ntrary, habit here

manifests its full gene rality: it conce rns not only the senso ry -mo to r habits

that we have (psycho logically), but also, before th ese, the primary habi ts

that we are; the th ousands of passive syntheses of wh ich we are orqanica lly

composed. (My emphasis)

17. T he philosopher who argued again st th e relativisti c conclusions regarding the

co ntrac tion of tim e in th e twins' case is, of course, Henri Bergson. Bergson

was wrong in assuming that th e case for the two twins is sym me tric , or as

he put it, a purc 'effect of perspectiv e' similar to that of tw o observ ers

looking at eac h othe r at a distance and seei ng each other shrunk in space.

• l'l' li lT .xamplc his repl y to criticisms by Andre Metz in Henri Bergson ,

'Pici itiou s T imes and Real Time ", in Berqson and th e Evolution ?f Physics,

cd. P. . Y. Gunte r (University of Tennessee Press, Knox ville , 1969),

pp. 169 7 1.

This volume co ntains man y of the pice s written abo ut till' de bate

incl uding the exch.mge lx-tw ccn III rgson and Einstei n himsel f. If one focuses

NOTES

on Bergson 's excessive ly psychological int erpret ation of relati vity th en one

mu st grant that he lost thi s de bate. O n th e other hand, if instead one sees

him as arg uing for the need of an acco unt of me tric time (which must

emerge fro m a nonmet ric , virt ual tim e) then the outcome of th e debate is

less clea r. Th is is Deleuze 's own interpret at ion. He sees Bergson as

cr it icizing Einste in for not having und erstood th e difference between the

actual and th e virt ual, th e difference between metric and nonm etric

multipliciti es.

Bergson thu s brought to light tw o very different kind s of multiplicit y, one

qualitative and fusional , continuo us, th e othe r numeri cal and hom ogen ­

eo us, discret e ... Th e confro ntation between Bergson and Einstein on

th e topi c of Relativity is incomprehensible if one fails to place it in the

context of th e basic theory of Riemannian multipliciti es , as modified by

Bergson. (Gilles Dcl euz e and Felix Guattari, A Thousand Plateaus [Univer­

sity of Minnesota Press, Minn eapolis , 19871, p . 484)

ee also Gilles Dcleuze , Berpsonism (Zone Books, New York , 1988), Chapter

4.

18 . Hans Reichenbach, The Philosophy rif Space and Time (Dover, Ne w York,

1958), p. 194 . An exam ination of the relations between the theori es of tim e

in nonlinear and relati vist ic physics is beyond the sco pe of this book, but

neve rt heless tw o concl usions follow rather directl y. One is th at there is no

inco mpat ibility between th e two and ind eed th e nonlinear theory may

compleme nt that of relativity by giving a morphogen eti c account (via

conce pts like the Hopf bifurcation ) of th e eme rge nce of the osci llato rs

(clocks, elec tromagnetic vibrations) used in the expos ition of relativity. On

the othe r hand , once we realize th at th e metric of tim e is eme rge nt , that is,

that oscillato rs operating at differ ent scales literally quantize time, th e

shri nkage o f time at veloci ties near th e speed of light becom es less counte r

intuiti ve: an eme rge nt metric, as opposed to an intrinsic one, is easier to

visualize as subject to int en sive transformations that do not pr eserv e certain

of its properties invariant.

19. In his care ful exam inatio n of foundational qu esti on s the philosopher Law­

rence Sklar shows that besides th e need to deriv e the tim e-asymmetric

macro scopi c beha viour of a thermodynamic syste m from the tim e-symmetric

microscopi c laws, th er e are two additional fundamental question s in th e

[ound ation s of sta tistical mechanics: to show that th e final equilibrium state

of a syste m is indeed an attraetor for its initi al and all its other int ermedi ate

states, and that th t ime scales of approach to equilibrium in math em at ical

mo de ls r .Ilcct the tim e scales obse rved in the lahoratory. klar argu l's th at

tlu s tw o <llIcslions arc 0f'~n problems in equilibrium tlu-rm od 'namics:

NOTES

physicists have not yet rigorously demonstrated that equilibrium states

att rac t, nor explained why the relaxati on tim e exhibits a characte ristic scale

(Sklar, Physics and Chance, pp . 156- 8, 189 and 216) .

klar, how ever, neglects to mention that both of these open probl em s

have indeed been given a mor e precise formulation , if not solved, in far­

from-e quilibrium thermodynamics. In this field on e gets th e asympto tic

approach to a particular state as an integral part of one's mod el, whil e in

conservative system s without attractors the asymptotic stability of the final

equilibrium state needs a speci al explanation. A similar point appli es to

relaxat ion tim es. Unlike the conservative system case , in non -cons ervative

syste ms we have an explanation , in terms of the 'area ' covered by the basin

of att raction , which is an integral part of the mod el. Sklar does discuss

Prigogine 's work to some exte nt , but not the specific points raised here

(pp. 269- 76) ..?O. Ibcrall discu sses this issue in more tec hnical terms (including terms like bulk

viscosity and bulk modulus needed to define the relaxation tim e of internal

mod es) which are beyond the scope of this book to explain. Yet I believe

his basic point is captured by my simplified example . ee his discussion in

Towards a General Science l!f Viable Systems, pp. 122-6.

2 I. . . . the inte ract ions between bodi es condition a sensibility , a proto­

perceptibility and a proto-affecti vity . . . What is called 'pe rce ption ' is

no longer a state of affairs but a state of the body as indu ced by another

bod y, and affection is the passage of thi s state to another state as increase

or decr ease of potential -power through the acti on of othe r bod ies .. .

Even when they ar e nonliving , or rath er inorgani c, thinas have a lived

e.tperience because they are perceptions and affections. (Gilles Deleuze and

Felix Guatt ari, What is Philosophy ? [Columbia University Press, New York,

19941, p. 154; my emphasis)

Elsewhere he is even more explicit about this . W e saw before that

the actualization of the world relies on int ensive processes of self­

organization (such as co nvection cells or the mi!,JT"ation and folding of

embryo nic ce lls). He refers to these phenomena as 's patio- te rnporal dyna ­

misms' and says

N O T E S

22. Deleuze, Loq ic ifSeme, p. 62.

23. W infree , When Time Breaks D OII'n , p. 22.

24 . Leon Glass and Michael C. Mackey, From Clocks to Chaos. The Rhythms if Life

(Prince to n University Press, Princet on , 1988), p. 94 . (This text contains a

discussion of Winfree 's work , and references to black holes.)

25. Winfree , When Time Breaks DOlin , p. 99 .

26. ue., Chapte rs 7 and 8.

27. Thro ugho ut we will discover again and again , in a sur prising diversity of

contex ts the same paradox ical enti ty : a moti onless, tim eless organizing

cente r called a phase singular ity . Th is is a place where an otherwise

perva sive rhythm fades int o ambiguity - like the outh Pole , where the

24 hourly tim e zon es converge and the Sun merely circles along the

horizon . (Ibid., p. 5)

Our [topological] inferenc es seldom involved speculation about adaptive

values , molecular mechani sms, or neural pathways. But they led us to

ever sharpe r focus on expe rime ntal condit ions in which somethinq stranae

was 8 uaranteed to happen: return of metamorphosing flies to the timeless

condition of the newl y fertilized egg, perpetual insomnia in mosquitoes,

abrupt suspension of pacemaking in othe rw ise perfectl y healthy and

capable heart mu scle, vortex ce nte rs of arrhythmia in elec trically rhythmic

tissue, chem ically tim eless rotors seque ncing reactions aro und their

perimeters, and che mical clocks made of shifting patt erns of color

topologically locked into three dim ensional organizing cente rs. (Ibid.,

p. 254; my emphasis)

28. Deleuz e and Guattari , A Thousand Plateaus, p. 24. (My emphasis)

29. O ne of the more ro bust and striking predictions of thc theory of mutual

synchronization was that it should fail abruptly below a critical coupling

stre ngth . John Aldridge and E. Kendall Pye tri ed this expe rime nt with

yeast and found exactly that : when the cells get more than about tw enty

diam eters apart , the amplitude of their collec tive rhythm falls abruptly.

(Arthur T. Winfree , Bioloqical Clocks [Scientific Ameri can Library, New

York, 19871, p . 128)

Actu alizati on takes place in three series: space , tim e and also conscious­

ness. Every spatio -te mpo ral dynamism is accompanied by the eme rge nce

of an e1em ' ntary consciousness which itself tra ces dir ecti ons, doubl es

movem ents and migrations, and it is born on the threshold of the

conde nsed singular itks of th bod y or objec t whose conscio u 'ness it is.

(Dclc uzc, D!Dcrence and Repetiti on , p. 220)

30. Populations of cric kets entrain each other to chirp coherently. Populati ons

of fireflies come to cohere nce in flashing . Yeast cel ls display cohere nce in

glyco lytic oscillati on. Populations of insects show cohere nce in their

cycles of c1osion (e me rge nce from the pupal to the adult form) . . .

Popul ations of wom en living together may show phase entrainme nt of

their ovulatio n cvcles. Populations of secreto ry ce lls, such as till' p ituitar "

N O T E S

pancreas , and other organs, release their hormones in cohe re nt pul ses.

(Alan Garfinkel , 'The Slime Mold Dict yostelium as a Mod el of Self­

Organization in Social Syste ms', in Self- Organizing Systems. The Emergence

i!I Order, ed . F. Eugen e Yates [Plenum Press, New York, 19871, P: 200)

\ 1. M. Cohen, qu oted in ibul. , P: 183.

P. Il ow ard H. Pattee, ' Instabilities and Information in Biological Self­

O rganization' , in Sell-Organizing Systems, p . 334.

I \. Stuart Kauffman, The Origins eif Order. Self- Organizat ion and Selection in

Evolution (Oxford Univ ersity Press, New York, 199 3), p.442. (My

emphasis)

\.1. Rud olf A. Raff , The Shape eif L!Ie. Genes, Development and the Evolu tion eifAnimal Form (University of Chicago Press, Chicago, 1996), p . 260. Unlike

terminal addition, which implies that ear ly stage s of the development of an

embryo resem ble (or recapitulate) early stages of species (or higher tax a)

d .velopment , the type of heterochrony involved in parall el networks destroys

any similari ty between th e tw o.

IS . lbid.; p. 255 .

Ill. IIJ id . , P: 337.

Dissociation app ears paradoxical as a cre ator of developmental novelty

because nothing new is add ed. In the case of some het erochronic

dissoc iat ions, such as neoteny in th e axolotl, a novel developmental path way

and life history have result ed from th e loss eif a featu re of th e ances tral

system . (My emphasis)

n. Dc lcuze and Guattari , A Thousand Plateaus, p. 48 .

IS. W. H. Zure k and W . C. Schieve, ' Nucleatio n Paradigm : Survival Thresholds

in Population Dynamics' , in Self-Organization and Dissipative Stru ctures: Applica­

li ollS in the Physical and Social Sciences, eds . William C. Schieve and Peter M.

Allen (University of Texa s Pr ess, Austin, 198 2) , pp . 20 3-22 .

It). . tu art L. Pirnm, The Balance eif Natu re. Ecological Issues in the Conservation eifSpecies and Communit ies. (University of Chicago Press, Chicago, 1991 ),

Chapte rs 2 and 3.

·10. Kauffma n, The Origins ?f Order, p. 256 .

·\ 1. Rud olf A. RafT and Thomas C. Kauffman ; Embryos, Genes, and Evolution

(Indiana Univ ersit y Press, Bloomington , 1991 ) , p. 40 .

42 . T he fastest evolutionary rat es fall in th e last, and perhaps most int eresting

category, tachytely .. . Tachytel y resembles th e punctuation of Eldredge

and Gould in that both rely on exce pt ional high rat es of evolutio n .

However whil e Eldrl· dge and Gould focused on a speci ation mod el .. .

N OT E S

[Simpson] suggest ed that the primary concomitant of tach ytely is a shift

in a population from on e major adaptive zone to another . . . Thus

tachytely is possible during early radiation s of new groups expanding into

vacant adaptive zon es. During th e rapid radiation all lineages are relatively

poorl y adapted and not mutually compe tit ive. Th e result .. . is the

production of divers e lines that qui ckly becom e ex tinct as other lines

conso lida te their position s in the adaptive zone at the expe nse of th eir

less-efficient cousins. (lbid., p. 44)

4 L Angela E. Dougl as, Symbiotic Interactions (O xford Uni versity Press, New

York , 1994) , pp . 7-9. This author em phasizes the eme rge nce of novel

met abolic capabilities related directl y to th e flow of biomass in food chains .

As she says, ' Nutritional int era ctions are fundamental to most symbioses,

beca use the metabolic capabilit ies most com monly acquired through sym bi­

osis relate to nutrition ' (p. 56 , and see Chapter 7 for an evaluat ion of the

eco logical impact of symbiosis).

44. We rner Schwe m rnlcr, 'Symb iogenesis in Insects as a Mod el for Cell

Differe ntiation, Morphogenesis, and Speciation', in Symbiosis as a Source eifEvolutionary Innovation , cds . Lynn Mar guli s and Ren e Fester (MIT Pre ss,

amb ridge , 1991 ) , p . 195.

4') . Deleuze places great emphasis on symbiosis as a means of becoming.

Coevolutio n , as in th e aparallel evolution o f th e wasp and th e orchid it

pollin ates, is a well -known example . See Deleuze and Guattari, A Thousand

Platea us, p. 10. But more generally, the very definition of a het erogen eous

assemblage as a 'rhizome ' has its origin in symbiosis. Though his introductory

exa mple of rhizome is bulbs and tubers, that is, plants without an arborescent

root syste m , he immediately acknowledges that ' plants with roots or radicles

can be rhizomorphic in other respects alt ogether' (p. 6) . Thi s other respect

may be illustrated by the formation of th e so-called rhiz osphere, th e und er­

ground food web composed of th e plant ro ot s of different species together

with the diverse micro-organisms that form symbiotic couplings with th em

and interface th em to th e flow of und erground nutrients .

46 . Throughout this book I have used his first formulation, singularities and

affec ts, hut he uses several others . Som etimes he says that in th e virtual

co nt inuum (plane of consistency) bodi es are charact erized by speeds and

affects (ibid .• p. 260).

Elsewhere , he says the virtual cont inuum (Aion) is 'the locus of incor­

poreal events and o f attributes which are distinct from qualities ' (Dc leuzc ,

Log ic eifSense, p. 165) . Here , ' events ' refers to singularit i s , whil e 'attributes '

are capacities to affect and be affected (to cut and to be cut, to us ' his

example).

N O T ES

47. lbid., p. 255. Rapidit y and slowness, however , should not be conceive d

as involving merely quantitative or exte nsive dilTer ences. Speed is an

int ensive property subject to critical thresholds, as in the case of fluids

which, below a critical speed, have one pattern of flow (laminar) but which ,

beyond the threshold , display a completel y dilTerent pattern (turbulence) .

See p. 371.

48 . lbid.; p. 258.49 . Th e term ' me chanisms of immanence ' does not , to my knowl edge, occ ur in

Deleuze , but he expresses himself in similar ways.

Many mo vem ents, with a fraai /e and delicate mechanism, intersect: that by

means of which bodies, states of alTairs, and mixtures, conside red in their

depth, succeed or fail in the production of ideal surfa ces [plane of

consiste ncy ]; and conversely , that by means of which the events of the

surfa ce are actuali zed in the present of bodi es (in accordance with

complex rul es) by imprisoning their singularities within the limit s of

worlds, individuals and persons. (Deleuze , Loaic ef Sense, P: 167; my

emphasis)

50. Co nnec tance is, in fact , controlled in food webs. Good evide nce suggests

that the number of connections in food webs is adjusted such that each

species maintains roughly a constant number of connections to othe r

species, regardl ess of the number of species in the web . . . [as displayed

inJ data on more than 100 food webs - terrestrial, freshwater , and

marine. A number of properties - such as length of food chains;

connectance ; ratios of top, intermediate and bottom species ; and ratios

of predators to prey - appear stable and scale invariant , both with respect

to the numbers of species in the web and with respect to the aggregation

of 'guilds ' of similar species int o single ' tro phic species' or the aggrega­

tion of similar species int o higher tax onom ic units. (KaulTman , The Oriai ns

C!f Order, p. 263)

51. Ibid . , p. 219.52. Although the famous Gaussian, or bell-shaped, distribution does represent

an important eme rgent property of widely dilTerent populations (that is,

ther e is someth ing recurrent or universal about it) it is nevertheless an

equilibrium distribution, and th e populations exhibiting this bell shap~ are

exa mples of distributions in extensity , fixed in their form and occ upymg a

metric, divisible space (much as sedentary cultures do). At the virt ual level ,

we must go beyond the e distributions, we mu st make a dilTerent use efhancc, Unlik e traditi onal games of chance (ro ulette, di c) in which fixed

rules lor rc th aleatory facto r to be r rained only at ce rtain points (the

NOTES

spin ning of the ro ulette, the throw of the dice) leaving the rest as a

mechanical development of the consequences , at the level of the virtual we

must allow the rul es to change with every throw and inject chance at eve ry

point, to yield truly nonmetric (or nomadic) distributions. In Deleuze' s

words

Each throw emits singular points . . . But the set of throws is includ ed in

the aleator y point [quasi-causal operato r ], a unique cast which is endlessly

displaced throughout the ser ies . . . Th ese thro ws are successive in

relation to one another , yet simultaneo us in relation to this point which

always changes the rul e , or coo rdinates and ram ifies the co rresponding

series as it insinuates chance ove r the entire length of the series . . . Each

thro w operates a distribution of singularities, a conste llatio n. But instead

of dividin g a closed space between fixed results which corre spond to

hypotheses [as in traditional treatments of probability], the mobile results

are distributed in the open space of the unique and undi vided cast. This

is a nomadic and non -sedentary distribution . (Dele uze , Loaic ef Sense,

pp. 59-60; emphasis in the original)

') l. Dcleuze olTers an alternative model for this task of the quasi-causal operator

which is based on the idea of entrainme nt, or more speci fically, the

phenome non of frequ ency entrainme nt. For tw o grandfather pendulum

clocks to entrain, weak siqnals must be transmit ted from one to the othe r to

couple them (in some cases , these are weak vibrations in the wo oden floor

on which the clocks are placed) . If the frequ encies of the two clocks are

close to each oth er they may resonate and the two clocks will lock into a

single frequ ency. Th e resulting entrainme nt of the two oscillators represents

a much stronaer linkaae (forced movem ent) between the tw o oscillators than

the weak signals which or iginally coupled them . In Deleuzc 's words:

A syste m must be constituted on the basis of two or mor e series , each

series be ing defined by the dilTerences between the terms which compose

it. If we suppose that the series communicate und er the impulse of a

force of some kind [e.g. the quasi -causal op erator), then it is apparent

that th is communication relates differences to other differences, constitu t­

ing differ ences between differ ences within the syste m. Th ese seco nd

d gre' dilTerences play the role of 'dilTere nciator' . . . Thi s state of alTairs

is adeq uately ex pressed by certa in physical concepts: coup/ina between

hct erogeneous syste ms, fro m whi h is derived an internal resonance within

the syste m, and fro m which in turn is deri ved a fo rced morcmcnr , the

am plitude of which exceeds that of the basic scric them selves. (Del uze ,

D!ffercn e and Repcu tion, p. 11 7)

N O TES

Deleu ze uses this ' re sonance ' model for th e action of the quasi-causal

ope rator in other places. For example,

Conce pts [multipliciti es], which have onl y consiste ncy or inten sive ordi­

nates outside of any coordinates , freely ente r int o relati onships of

nond iscursive resonan ce ... Concepts are centers of vibrations, each in

itself and everyone in relation to all others . Thi s is wh y th ey all resonate

rather than cohere or corres po nd to each other. (Dele uze and Guattari ,

What is Philosophyi; p. 23; my emphasis)

Clearly , if we interpreted th e term 'concept' as ' semant ic conte nt of a

term ' (or in any other lingui sti c way) this paragraph would be come

meaningless. Th e term ' inte nsive ordinates ' must be int erpreted in terms of

posit ive ordinal distances (which distinguishes it from any cardinal numerical

coordinate) and not as referring to on e of th e member s of the couple

'ordinates' and 'abscissas' which are simply th e nam es of two coo rdinates.

54 . De le uze , Loaic if Sense, p. 121. (My emphasis) Thi s is about th e specification

of the conditions of a problem, but problem s are , in Deleuzes onto logy ,

no thing but virtual multiplicities. I discu ss this relationship in Chapt er 4 .

55 . itewar t and Golubitsky, Fea1ul SymmetIJ', pp. 14-16.56 . Lawre nce Sklar, Space, Tim e, and Spa ce-Time (University of California Press,

B erkcley, 1977) , pp. 25 1- 86.57. T he reason why it is hard to find a physicist who would think of laws as

en ti ties in need of ontological analysis is that mo st of th em have an

instrumentalist or ope rationalist attitude toward th eoreti cal entit ies. Ever

since Ne wton refused to give me chani sms to explain the action of gravity

and settle d on describing how plan ets move, as opposed to explaining lYhy

they do so, man y physicist s have accepted a non-realist approach to laws, as

we ll as unobservable entit ies in gen eral. Thus, expe rime ntal laws (like

Boylc' s law) are defined as symbolic representations of laboratory regularities

or routines '?f experience, whil e fundamental laws become basic hypotheses

fro m which on e can derive experimental laws, and the validity of which is

not sett led empirically but through th e validity of their conse <j uence s. In

ncither case is the ontological status of the laws th emselves an issue . See

Lindsay and Margenau , Foundations if Physics, pp . 14-16 (for expe rime nta l

laws) and pp . 22-6 (for fundamental principles) .

While phil osophers can take this stance and argue that, if all speci fic

ex perime nta l laws may be deri ved from a set of fundamental ones, th en th e

latt e r may be see n as a set of axioms and treated as ete rn al truths, as in

Euclid's ax iomatic tr eatment of geome try . But as th e physicist Richard

I·c 'nman has argued, scientists cannot do this because they are awa re that ,

unlike essences, fund am ental laws may have seve ra l diffe r .nt forms. New-

N O TE S

ton 's laws of motion , for example , may be expressed in three ways wh ich

are, mathematica lly, completely different : the original force form, the field

form, and the var iational form . Th ese arc taken to express one and the same

law because they have th e same mathematical conse <juences and thu s we

canno t tell them apart expe rime ntally. But th e existe nce of a variet y of

forms docs eliminate th e temptation to adopt a Gr eek axiomatic approach,

forc ing physicists to adopt, as Feynman puts it, a Babylonian approach. See

Richard Feynman, The Character '?!. Physical Law (MIT Pr ess, Cambridge,

1995), pp . 50-3 .

Per haps th e only clear state me nt one can get from physicists as to what

funda me nta l laws are supposed to be co mes from the appli cation of gro up

theory to the law s th em selves , For example , th e well -known invariance of

cwton's laws under translation s in space and time impli es that

give n the same esse ntial initial conditions , th e result will be th e sam e no

matter when and where we realize th ese. Th is principle can be formulated

, . . as the state me nt that th e absolute po sition and th e absolute time are

never essential initi al conditions . . . If th e universe turned out to be

grossly inhom ogenous, th e laws of nature in th e fringes of th e universe

may be quite different from th ose we are studying .. . Th e po stulate of

invariance with respect of displacem ent in space and time disregards this

possibility, and its appli cati on on the cosmological scale virtually presupposes

" homoqeneo us and sta tionary uni verse. (W igne r , ln varian ce in Phy sical Theory ,

p. 4; my emphasis)

.lcarly, this is a more sophisticate d stance than naive essent ialism , since

1his post ulate of ln variance (which may imply that basic laws are simul­

t,' Ilt'ollsly valid every where , and have been so always) can , in turn, be treated

.1' .111 ap proxi rnate hypoth esis , I return to the <juestion of laws in Chapter 4.

, S "or if it is a qu estion of knowing, . . ' why water change s its state of

qu" lity at 00 centigrade', the <juestion is po orl y stated insofar as 00 is

cOllsidereel as an ordinary point in the thermomet er. But if it is

('ollsider xi, Oil the contrary, as a singular point, it is inseparable from

tlu- event occurring at that point, always being zero in relation to its

re.r lization on th e line of ordinary points, always Jorthcomina and already

/'" t . (Dcleuzc , Loaic if Sense, P: 80; my emphasis)

li lt" (·. ·.K! same formulation recurs thro ugho ut Dcleuzc 's work:

ion : Ihe inr h-Iin iu- time of the eve nt, the floatin g line that knows onl y

Iw,-ds and co nti nua lly divides th.lt wh ich tran spires into an alrea dy -t here

11..11 is .11 the sallie time no t -yc-t-h cre , a simultaneous too -Ian- and too -

N O TES

early, a something that is both go ing to happen and has just happened .

(Deleuze and Guattari, A Thousand Plateaus, p. 262)

T he meanwhile, the eve nt, is always a de ad tim e; it is th er e wh ere

nothing tak es place , an infinite awaiting that is alread y infinitely past ,

await ing and reserv e. (Dele uze and Guattari, What is Philosophy ?, p . 158)

"'J, Dclcuzc, D!lJerence and Repetition , p. 88. (My empha is)

Th e joint . . . is what ensures th e subordina tio n of time to those pr op erl y

cardinal points through which pass th e peri od ic movements wh ich it

measures [e .g . th e nest ed set of cyclic pr esents] . . . By contras t, tim e

out of joint means dem ented tim e . . . liberated from its overly simple

circular figure, freed from th e events that mad e up its conte nt ... in

short , time presenting itself as an empty and pure form. Time itself

unfold s . . . inst ead of things unfolding within it . . . It ceases to be

cardinal and becomes ordinal, a pure order of tim e .

1>0. I have said before that each cyclic present is a contrac tion of past and future

instants at a given temporal scale . Hence it is a veritable ' synthes is' of

present time, a syn thesis whi ch Del euze calls ' passive ' because it involves no

activity .ither by th e world or by th e subject.

Passive synthesis or co ntraction is essentially asymmetrical: it goes from

the past to the future in th e pr esent, thus from the particular to the

general, th er eby imparting direction to th e arrow of time . (Deleuz e,

D!fJcrence and Repetit ion, p. 71)

(,1 , The infinitely divisible event is always both at once. [future and past , acti ve

and passive] It is ete rn ally that which has just happ ened and that which is

about to happ en, but never that whi ch is happen ing .. . Th e eve nt , being

itself impassive, allows the act ive and the passive to be int erchanged more

easi ly, since it is neither the one nor the other, but rather their common

result. (De lcuzc, Loaic ifSense, p. 8)

1> 2. Dclcuzc, Loqic ifSense, PI" 94-5.(d . Ihid . , p. 147.

64 . lbid. , p. 165.

65 . Ibid . , p. 147.

h h. Ralph II . Abraham, ' Dynamics and el f-O rganizatio n ', in Se!f-Oraanizina

'stems. Th Emerqence if Order, ed. F. Euge ne Yates (Plenum Press, Ne w

York , 1987) , p . 606.

(,7 . 0 11 questi on s of simplicity and famili arity in th e foundat ion s of physics, sec

U nds . and Margellau , Foundations ifPhys ics, p. 18.

NOTES

68. Deleuze, Loa ic if Sense, p. 166.

69 . Ian Stewart, Does God Play Dice? The Mathematics if Chaos (Basil Blackw ell ,

Oxford, 1989), pp. 114- 21.

70 . Del euze and Guattari, A Thousand Plateaus, p. 25 1. (Emphasis in the

original)

71. lbid. , p. 9. Th e term 'line of flight' , referring to th e quasi-causal ope rato r,

is defined else where (I" 488) as a fractal line. Precisely because th e ope rator

and the plane it constr ucts mu st cut and pr eserv e N-dime nsions for eve ry

multiplicity, Del eu ze co nce ives of it as necessaril y having a fractal number

of dime nsio ns, a number wh ich is not a wh ole number but a fracti on. For

example , a flat piece of paper is a two-dimensional entity , but one fold ed

into a ball has a dimension between tw o and three , that is, it is a fractal

dim ens ion. 0 do es a one- dime nsional string so fold ed that it begins to fill a

plane. T he op erator itself would not be a transcendent agen cy ope rating in

N+ I dimens ions but on the contrary , it would work on N-I dimensi ons (a

line forming a plan e , or an aleatory poine cir culating through one- dime nsional

series) . O n th e fractal dim ensionality of the plan e, see also Del euze and

iua ttari, What is Pbilosophyi; pp. 36 - 8.

7) . I c1cuze uses th e term 'counte r-act ualization ' for the ex traction of ideal

events from actual ones in Deleuze, Loqic if Sense, pp. 150- 2. Il l' does not

II S l ' th e term 'pre-actualizat ion' hut thi s term do es capture th e meaning of

till' oth er task th e quasi -cause mu st perform .

In gene ral, as we have see n, a singularity may be grasped in tw o ways: in

its existence and distribution [in th e vector field), but also in its nature,

ill conform ity with which it exte nds and spre ads itself out in a det ermined

direction ove r a line of ordinary points. This second aspect alr ead y

represents a certa in stabilizatio n and a beainn ina if the actualizati on ifin,qu/ariti cs. ( Ibid., p. 109; my emphasis)

71 . . . th insta nt extracts singular points twice project ed - once into the

future and once int o the past - forming by this double equation the

const itutive clements of the pure event (in the manner of a pod which

r .lcases its spo res). (Ibid. , p. 166 )

/., [W Ill'1I ,1 multiplicit yI is grasped in its relation to th e quasi-cause which

prod uces it and distributes it at th e sur face , it inherits , participates in,

,lIId even enve lops and possesses th e force of thi s ideational cause. W e

h.ivc s e n that th is [qua si-jcau sc is nothing outside its effect, that it haunts

lhi, llll'( t , and that it maintains with the effect an immanent relation

whic h tu rn s th, product, the mom ent thai it is produced , int o sonl'thing

prod urtivr-. (Deh-uzc , 1.0lI'c of Sense, p. 95)

NOTES

This extract is about 'se nse' not 'a multiplicit y' but the two terms are

closely related .

75. Once communication between het erogeneous series is estab lished , all

sorts of consequences follow within the syste m. Some thing passes

betw een the borders, events explode, phenom ena lIash, like thunder and

lightning ... what is this agent, this for ce which ensures communicat ion?

Thunderbolts explode between differen t intensities, but they are pr eceded

by an invisible, imp erceptible , dark precursor, which determines their path

in advance but in reverse , as though intagliated. (Deleuzc, D!lJerence and

Repetiti on , pp . 118-1 9; emphasis in the or iginal)

76. Dcleuze does not speak of nonlin ear , no nequilibri um areas of the world , but

he does distinguish specia l pro cesses (such as th e spo ntaneo us formation of

metas table surfaces) fro m those character izing full equilibr ium structures .

O nly the former have the pow er to give rise to the virt ual.

When we say that bodi es and their mixtures produce [the virtual ], it is

not by virtue of an ind ividuati on which would presuppose it. Individua­

tion in bodies, the measure in their mixtures . . . pr esupposes . . . the

pre -indi vidual and impersona l neutral field within which it unfold s. It is

therefore in a differ ent way that [the virtual) is pr oduced by bodies. Th e

questio n is now about bodies taken in their undifferenti ated depth and in

their measureless pulsation . Thi s depth acts in an original way, by means

1!.f its power to orBanize suifaces and to envelop itsclf with in sUlfaces. (Deleuze,

LOBic 1!.f Sense, p. 124; emphasis in th e or iginal)

I have replaced refer ences to 'se nse' in this ext ract by ' the vir tual'. (The

term 'se nse' is closely related to 'v irtual mul tipli city' , but refers to th e

re lation between virtuality and language, a relat ion I do not ex plore at all in

this book.) The capacity of matt er to form sur faces, eve n surfaces at

equilibrium, constitu tes the most primitive form of self-organizat ion. The

surfaces of liqu id or solid bodies are , indeed , specia l or singu lar zones of

those bodies, very different fro m the ord inary bulk mat erial that they

enve lop . The bulk of a liqu id body, a lake or ocean, for instance , consists of

a populatio n of mo lecules on . which forces of attraction are exer ted in all

directions. At the surface of this body, on the othe r hand , there ex ists a

changing sub-population on which forces are exe rted inward but not

ou twa rd. This gives those surface molecules special prop ert ies not displayed

b thu bulk. In parti cular , they will possess a certa in amount of free energy

(·ncrgy available for do ing work) which acco unts for the surface's spon­

t,lIWOUS tendency to contract or minimize its ex te nsion (a 's ur face tension '

NOTES

which explains why dropl et s of water spon taneo usly acquire a round shape).

See Neil Kensington Adam, The Physics and Chemistry '?! Suifaces (Do ver, New

York, 1968), pp . 1-7.Even at equilibr ium, the surfaces of ind ividuated bodies are capable of

spo ntaneo usly giving r ise to asymmetr ical distributions of events, a distribu­

tio n which is the signature of the quasi-causal operator. This is part icularly

dear in the case of electrical phenomena occ ur ring at the surface of contact

between different phases of matt er .

When two conducti ng phases are in contact, a difference of electrica l

potential is generally established between them . The establishme nt of this

'phase boundary pot ential ' is int imately associate d with the formation of

an 'e lectrical doubl e layer', at the surface, i.e . an unsymmetrical distribution

f!I electrically charBed particles near the phase boundary, with an excess of

posit ive charges tow ards the phase which assum es a positive pot enti al and

of negative charges toward s the phase assuming negat ive pot enti al.

(p. 300; my emp hasis)

Here is Dele uze 's versio n of the same ideas,

Everything happ ens at the surface in a crystal which develops only on the

edges. Undoubtedly, an organism is not developed in the same manner

. . . But membranes arc no less important, for they carry potent ials and

reBcncrate polarities . Th ey place int ernal and ex te rna l spaces into contact

without rega rd to distance . Th e intern al and the exte rna l, depth and

height, have biological significance only through th is topoloqical surface '?!contact . Thus, even biologically it is necessary to und erstand that ' the

deepest is the skin'. The skin has at its disposal a vita l and properl y

superficial potential ene rgy . And just as [virtual) eve nts do not occ upy

the surface hut rather frequent it, superficia l ener8Y is not local ized at the

.<U~face but rather bound to its formation and riformation . (Deleuze, LOBie '?!Sense, p. J03; my emphasis)

77. The term 'line of flight ' is used in two ways, one to refer to relati ve , the

other to absol ute movements towards the virtual. A relative line of flight

refers to actual assemb lages, like those I described above when discussing

emhryogenesis and ecosys tems, defined by affects and relations of speed and

510 \\'n ss.

'omparntive rates of flow in these lines produce phe nomena of re lative

slowness or viscosity, or on the contrary, of acceleration and ru pture .

t\1I this, lines and measurable speeds, consti tute an assem blage. (I cleuzc

.uul Guattari , " Thousand Plateaus, p. 4)

NOTE S

I said that in these assemblages relative accelerations (neoteny, symbiosis)

allow an escape from rigid morphologies, the term 'relative line of flight '

refe rring to th ese ph enomena, among others. An absolute line of flight is a

further acce leration or boosting of th ese relative escapes whi ch allows th em

to leave th e extensive and intensive altogeth er.

Th ese relative movements should not be confused with th e possibility of

. .. an absolute line of flight ... The former are stratic or int erstratic [that

is, conce rned with exte nsities or intensities], wh ereas th e latter conce rn

the plane of consistency .. . There is no doubt that mad particles leave

minimal trace of their passage through th e strata as they accelerate, escaping

spatio-te m poral and even exi stential coo rdinate s as they tend towards ..

th e state of unformed matter of th e plane of consistency . (pp . 55-66)

And it is th ese absolute lines that creat e the heterogeneous virtual

continuum . 'Moreover, th e plan e of consistency does not preexist . . . the

lines of flight that draw it and cause it rise to the surface , th e becomings

that compose it' (p . 270) .

n . Philosophy is a const ructivism , but constructivism has two qualitatively

different complementary aspects : th e cre ation of concepts and th e laying

out of a plan e ... Concepts are absolute surfaces or volumes, formless

and fragm entary, whereas th e plan e is the formless, unlimited absolute,

neither sur face nor volume but alwa ys fractal ... Concepts are events

hut the plane is the horizon of events , the reservoir or reserve of purely

conceptual events .. . (Deleuze and Guattari, What is Philosophy ?, p . 36)

Here the term 'concept' does not refer to 'concepts of the understanding',

that is, to semantic or representational entitie s, but to virtual multiplicities :

'Every co nce pt . . . is a multiplicity although not every multiplicity is

conce ptual' (p. 15). Without this definition referen ce to conc epts as surfaces

or volumes (tha t is, as manifolds) would be meaningless. That virtual

mult ipliciti es cannot be conceived as int elle ctual con cepts is clear from the

following ex tract, wh ere th e term ' Idea ' gives a better rendering of what

'concept' mean s:

If the Idea elim inates variability, thi s is in favour of what mu st be called

var iety [a synonym of manifold I or multiplicity . The Idea as concre te

universal stands opposed to conce pts of the und erstanding. (Deleuze ,

[)!JJcrence and Repetiti on, p. 17 3)

7lJ. Dclcuzc and Guatta ri, What is Philosophy] , p. 126 .

HO . Ikl cuzl·. Lon ic r1 Sense, p. 148.

NOTES

4 VIRTUALITY AND THE LAWS OF PHYSICS

1. T he rejecti on of totalities and th e definition of social ontology as composed

en tirely of individuals op erating at differ ent scales needs to be defended in

detail. I am aware that th e way I present it here is rough and hardly

compelling. Moreover, a convincing case for thi s point of view needs of

lic e ssity to have a historical dim ension, that is, it need s to give the details

of .pccific individuation processes , for institutions, cities and nation states. I

have applied this ontology in th e conte xt of a historical analysis of W estern

history in Manuel DeLanda, A Thousand Years r1 Nonlinear History (Zone

Books, Ne w York 1997) .

lhcrc are man y approaches to th e question of the disunity of science. Some

p.rrticularly useful ar e John Dupree , The Disorder r1 Thinps. iHetaphysical

l-outulutions c1' the Disuni ty r1 Science (Harvard University Press , Cambridge,

199 5); Jerry Fodor, 'Special Scien ces, or Th e Disunity of Science as a

\ orking Hypothesis' , in The Philosophy c1'Science, eds. Richard Boyd, Philip

(;. Ispl·r and J. D. Trout (MIT Press, Cambridge , 199 3); Peter Galison,

' In t ro duct ion : Th e Context of Disunity ', in The Disunity ofScience, eds. Pet er

( ;.llisun and David J . Stump (Stanford Univ ersity Press, Stanford, 1996);

Vndrcw Pickering, The Alanale ifPractice. Time, AaenC)', and Science (University

ot Chicago Press, Chicago , 1995) .

lronical ly, some conte m porary socio logists of scien ce who are highly critical

01 the philosophers' s approach mak e the mistake thinking that a nov el

Pl'ro,ich to the study of science demands th e elimination of causal relations.

'I' I I. M. Co llins , Chanai na Order (University of Chicago Press, Chicago,

I'}')l), pp . 6-8 .

It is hard to tell wh ether Collins trunks causes do not exist , thus siding

\ IIh l lumc, or whether he thinks we should susp end belief in them as a

I II ' lhodological man oeu ver to highlight th e 'social ' aspects of scientific fields.

I II lat ter interpretation would avoid my criticism (that he is siding with the

old" ' 1 and most conse rvative philosophy of science) but it would sti ll be

" Jl" 11 to crit icism in a differ ent way: bringing 'society' as a totality into the

nalvsis.

I \II 11.ll'king , Represeminq and Interven inq (Cambridge University Press, Cam ­

11/1.1 'I ' , 1992) , P: 46. (My emphasis) In conte mporary philosophy th e re vival

II I 1 ,111. •Ility as a productiv e or gene tic relationship, one to be studied

II1Jlll'ic,d ly not mer ely conce ptually, was foreshadow ed by the philosopher

1.11 III Hungl' in 1959 , although the degr ee to which he has influenced

l U I n -ut aut ho rs is hard to evaluate . His key hook in this r 'spe t is Causality

"hi lIoJefTl S k nce (Dover, Ne w York. 1979). Her I adopt man y of Bunge 's

II \\ 1111 product ivity and depart onl y in the terminology. Il l' uses the term

NOTES

'd et ermination' for the general relation (including linear, nonlinear and

statistical causality) reserving th e term 'causality' for linear causality, so as

not to depart from tradition. I myself pr efer to speak of causal relations in

gene ral, taking the linear case as an untypical case , since th e point of my

discussion is to break with tradition in th ese matters.

5. Th e entire group of new philosophers that have tak en the 'causal turn ' are

unanimous in their rej ection of th e deductive-nomological model of expla­

nation (as w ell as related models which replace deduction by induction, and

exceptionless laws by statistical laws) for its emphasis on logico-linguistic

form at the expense of causal -productive processes. See Bunge, Causality and

Modern Science, PI" 290-1; Nan cy Cartwright , How the Laws rf Physics Lie

(C larendon Press, Oxford, 1983), PI'. 132- 3; W esley C. Salmon, Scientific

Explanation and the Causal Structure i!f the World (Prince ton University Pre ss,

Prin ceton , 1984), PI" 26-32; Dupree, The Disorder eif Thinas, Pl' 178- 9.Deleuze som etimes echoes th e philosophical mischaracterization repres­

ented by th e nomological -deductive model wh en he asserts that the object

of science is ' funct ions that are pr esented as propositions in discursive

syste ms' (Deleuze and Guattari, What is Philosophyi , p. 118).Alth ough in his early work Deleuze is very careful to differentiate

between mathematical functions which are close to lingui sti c statem ents

(such as algebraic functions) from those that ar e not (differe ntial functions),

in his last work where the differences between science and philosophy are

most dramatically stated, he lapses into a less car eful state me nt of the

question . Elsewhere (I" 128) he adds that '[TJhe fact that scien ce is

discursive in no way means that it is deductive ', but gives as an example of

non -deductiv e activity th e use of co mpute rs in the study of nonlinear

functions. I believe th e non -deductive aspe ct needs to be stressed mu ch

mo re and exte nde d to modelling pra ctices much older than com pute r -based

ex pe rime ntation . 1 have alread y arg ued that Delcuze' s main point, the

in.H!fficiency '?I J im ctions to capture the virtual , can be mad e without subordinat­

ing mathematical models to propositions, that is, by showing that function s

define indi vidu ation processes in such a way as to st res s th e direction

tow ard s the actual.

6 . Ron ald N . Giere, Explainino Science. A Coq nitive Approach (University of

'hicago Press, Chicago , 1988), 1'.82. (My em phasis)

7. ~omment i ng on a particular case of deri vation, that of th e mod el of th e

simple pendulum in one dim ension from the two-dim ensional case, Giere

says

Th e move from the mass-on -a-spring exam ple to the Simple pendulum

see ms lo me a clear case of what Kuhn called 'direct mod eling '. T he

NOTE S

two examples are not just special cases of a gene ral relationship . One

manages to reduce th e pendulum, a two-dimensional system , to the on e­

dim ensional case only by means of a judicious approximation that restricts

the pendulum to small angl es of swing . In particular, the ste p from th e

original application of Newton 's laws to the two-dimensi onal pendulum

to the one- dimensional version is not a matter of purely mathematical,

or logical, deduction . •Approximation' is a valid rul e of deduction only in

physicists ' jokes about mathematicians. (ibid ., 1'.71; sec also PI" 76-80)

" Ily.1 Prigogine , From Beino to Becoming (W. H. Freeman . New York, 1980),

p. 19." ( ',lTlwright, How the Laws rf Pby sics Lie, PI" 54-5 .

II I 'This fits better with my picture of a nature best described by a vast array

,.1 phenomeno logical [or causal] laws tailored to spe cific situations, than with

"'"' govern ed in an orderl y way from first principles, ' (ibid., p. 66).On Gie re's view see Giere , Explain ing Science, p . 85, and PI" 90-1 on

I", d ews on Car twright's work.

( ,,'twright, How the Laws eif Physics Lie, p. 107.I ),borah G. Mayo, Error and th e Growth eif Experimental Knowledge (University

," Chicago Press, Chicago, 1996), p. 128.

(" twright, How the Laws eif Physics Lie, PI" 96-7."'JI,is Kline, Mathematics and the Physical World (Do ver , New York , 1981),

I' 1-1 0 . (My emphasis)

\I, ,,ri s Kline, Ala thematical Thought from Ancient to Modern Times. Vol. 2

,I I li,rd l.lniversity Press, New York, 1972), p . 580. More generall y, on

II" lrist orv of variational techniques see Chapters 24 and 30.

l ;jvell appro priate variational principles each with an associated multiple

IIlk >ral and scalar int egrand, we can produce all the important partial

old ('rent ial equations in physics: the wave equation, th e diffu sion

l' I".ll ion, Po isson's equat ion , Shrodingcr 's equation, and each of Max ­

\\'(, 11'S equat ions . . . Such thinking bears fruit. General relativity and

' 1",lI1l UI11 mechan ics both originate d from variational principles . (Don . S.

1'1II01lS , Peifecl Form. Variat ional Principles, Meth ods and Applications in

I I /IIl'lIIary Php ics [Princeton Uni ver sity Press, Princeton , 1997J, p. 11 I)

I ,I, PI" 17 27. In a passage where Deleuze contrasts the propositional

II" ...HI. 10 the probl em atic one (o r what amo unts to the same thin g, an

1'1" ".J( h to though t in terms of its conditions as opposed to its producti ve

,". ,,) . II(' co mpares lhe Kanti an co nce ption of the con cpt of •shortes t

It 11I1l' ' (.IS a representational schema) to the conce ption made possihle hyIII I ,til II Ills of vari.u ious. Th e term 's hortes t ', as he S.l)'S,

N O T E S

may be understood in tw o ways: from the point of view of conditioning,

as a schema of the imagination which det ermines space in acco rdance

with the concept (the straight line defined as that which in all parts may

be supe rimpose d upon itsel f) - in this case the difference rem ains

ex ternal, incarn ated in a rul e of construction . . . Alte rna tively, fro m the

genetic point of view, the shortest may be understood as an Idea

[multiplicity] which . . . interi ori zes the difference between straight and

curved, and expresses this internal difference in the form of a reciprocal

determination [differ ential relati ons] and in the minimal condi tions C!f an

inteqral , (De leuze , D!fference and Repet it ion , p. 174)

J 8 . Leo nard Euler, quot ed in Stephen P. Tim oshenko, History c1 St renqtb C!fMaterials (Dover , New York , 1983), p . 31. (My emphasis)

19. Far from being conce rne d with so lutions, truth and falseho od primarily

alTect problems. A solution alwa ys has the truth it deserv es according to

the probl em to which it is a response , and a problem always has the

solution it deserves in pr oportion to its own truth and falsity - in other

word s, in proportion to its sense. (Deleuze, D!fference and Repetiti on,

p. 159)

In what follow s I will not speak of ' true problem s' but of 'correct ' or

'well-pose d problem s' but this constitutes, 1 believe , only a harml ess

terminological departure from Deleuze.

20. Kline , Mat hematics and th e Physical World , p. 441 . Within this traditi on , the

unifying power of Hamilton 's principle was almost inevitabl y inte rpre ted as

consisting in the gene rali ty of its truth , and axiomatic versions of classical

mechanics were produ ced in the nineteenth century (by Heinrich Hertz, for

example) to marry the unifying pow er of variati onal principles with the

concept of general truth . See Rob ert B. Lindsay and Henry Margenau ,

Foundations tifPhy sics (O x Bow Press, Woodbridge , 198 I), pp. 118- 20.

In a Deleuzian onto logy eliminating essentialism from physics involves

replacing clear and distinct truths (axioms and theorems) by problems, that

is, replacing dedu ctively connec te d linguistic propositions in the Euclidea n

geo metry mould by problem s defined by singu larities (events) and affects.

Greek geo metry has a general tend ency on the one hand to limit problem s

to the benefit of theorem s, on the othe r to subor dinate problem s to

theor ems themselves. The reason is that theorem s see m to ex press and

d rv .lop the prop rt ies of simple essences whereas problem s co n ern only

el'ents and Cf./ fections ... As a result , however , the Beneti c point of view is

fordbly rdegatcd to an inferio r rank : proof is given that so mething cannot

NOTES

be rather than that it is and why it is (hence the fre quency in Eucl id of

negative, indirect and [redu ctio ad absurdum I arguments .. .). Nor do

the essen tia l aspects of the situatio n change with the shift to an algebraic

and analytic point of view. Problem s are now traced fro m algebraic

equations . . . How ever just as in geo metry we imagine the probl em

solved, so in algeb ra we operat e upon unknown quanti ties as if they wer e

known: th is is how we pursue the hard work of reducing problem s to the

form of propositions capable of sen-ing as cases of solution. W e see this

clearly in Descartes. The Cartesian meth od (the search for the clear and

distinct) is a meth od for so lving suppose dly given probl em s, not a meth od

of inventi on appropriate to the const itution of problems or the und erstand­

ing of questions. (Deleuze , Difference and Repetuion, p. 160.)

Delcuzc, D!fference and Repetition, p. 189.' For Probl em s-Ideas are by nature unconscious: they arc extra-proposit ional

aru! sub-representati ve , and do not resembl e the propositions which represent

the affirmations to which they give rise' (I" 267; my emphasis).

I,HI I lacking, Bepresent lnq and ln terveninp , p. 41. (Emphasis in the original) In

.ldcl ition to ignoring causes and downplaying explanations, positi vist philo -

ophy holds a ' verificationist ' theory of meaning (if the truth of a stateme nt

,.II H1ot he tes ted the state me nt is meaningless) , a belief that verification

111\ olves comparison with raw data (data fro m the senses) and a disbelief in

theore tical (or uno bserva ble) ent ities. Hacking later on also expresses some

do ubts about the ro le of ex planations (PI" 52- 5) but this is, I believe ,

limited to thei r ro le as argume nts for realism . Hacking is well known for his

, h,lI11pioning of causal inte rventions in expe rimental reality as cri te ria for

I ••rlism , or for belief in unobservable entities .

\ II I(H'US on Whv questions is not meant to link these matter s to a specific

'.\ ut.ut ic: form, and is Simply a matter of case of ex position . Clearly, such

' 1" 1"\1 iOlls may be paraphrased in other ways : the requ est for a causal

I ' 1'1,11 1,11 ion ex pressed by the question ' Why did event X occ ur?' may be

' 1'I"l"\sed hy ' How was event X produced ?' or something like that. Though

1 1,, 1" 11"11' doe s not refer to Why questions he does differentiate between

'I" ' ,l illllS with simple propositions as answers (which subordinate the

' I" ' ,I jllil to a search for essences) from those mor e properly problematic .

IC,tioll,dism want ed to tie the fate of Ideas [multiplicities] to abst ract and

.1",111 ,'ssenc:es; and to the ex tent that the probl ematic form of Ideas was

1< ·' ·Il~lIi"led. it eve n wanted that form tied to the question of essences ­

11\ orlu-r words , to the ' W hat is X?' ... It should be noticed how few

I'flllm0l'lH"rs have placed their trust in the quest ion ' \ hat is X?' in orde r

til },.I\"I ' 1<i" ,ls. (' ..rtainly not risto t lc, O nce ti ll" diah-ct ir Itl ll' .irt of

NOTES

posi ng problem s] brew s up its matter instead of being applied to

propaedeutic ends , the qu est ion s ' How mu ch ' , 'How' , ' In what cases '

and 'Who ' abo und . . . These qu estion s are th ose of th e acciden t , the

event , the mult iplicit y. (Dc leuze, D!fference and Repetit ion , p . 188)

A more important omiss ion in my discussio n is that it does not include

Dclcuzc's distinction between problem s and questions. Problem s are th e

episte mological cou nterpar t to vir tual multiplicit ies, while questions (which

invo lve an imperative, a request or dem and for an exp lanatio n, for example)

are the sources of probl em s or the counte rpart of the quasi-causal operator.

Th er e are also episte mo logica l co unter parts to the inte nsive and th e act ual,

W e dist ingu ished four instances; imperati ve or onto logica l questions;

dia lectica l pr ob lem s or th e th em es that eme rge fro m them; symbo lic

fields of so lvability in which these problem s are 'scientifically ' ex pressed

in acco rdance to th eir conditions; th e so lutions given in th ese field s when

the probl em s are incarna ted in the actuality of cases. (I' . 200)

24. Alan Garfinkel , Forms if Explanation (Yale Uni versity Press, New Haven,

198 1), p . 21. O ther phil osoph ers have develope d similar approaches to Wh y

qu estions and the ir relat ion to the distributions of the relevant and the

irrelevan t . See , for ex ample, Salmon, Scientific Explanation and the Causal

. tru ctute '!.( the World, PI'. 1-6. See also Salmon's discussion of Van Frassen 's

approach to Wh y questions and contrast spaces (PI" 102- 6) which, unli keGarfinkel's, is complete ly lingui stic.

) 5. Alan Garfinkel, Forms ifExplanation, p. 40.

/6 . lbid.; p. 64 . Garfinke l takes this characterization of state space fro m Rene

Thorn, cre ator of catastro phe theory and of the conce pt of st ructural

stability . Here th e term 'crit ical point' may refer to both th e unstable

scparatrix tha t defines (as a repeller) th e border of a basin of attraction, or

to a bifurcation which defines the poin t of structural instability at which one

distribution of attractors changes into another.

27 . Deleuzc, DyJerence and Repetiti on , p. 159.

)H . Alan Garfin kel, Forms ifExplanation, PI" 53-8.)9. Ihid., Pl" 58-62.

lO. 11M, p. 168.

I I. Robert M. May, ' Chaos and the Dynamics of Biological Pop ulations ' , in

Dynami cal Chaos, ed. M. V. Berry (Lo ndon Royal Society, (987), PI" 31-2.

May's focus in this essay is chaotic at tractors, bu t he does m en tion periodic

at tractors , (T he latter are less controversial in populat ion st udies than the

Iormer .) I avoid discuss ion of 'chaos ' in the main t ext du e to thc excessive

hypc surrounding the subject, bu t more importantly, because on to logically

NO TES

the key notion is that of 'a ttractor ' no t the particular chao tic case . T hat is,

th e key is quasi-causality itself not anyone of its particular forms .

.~ 2 . Deleuze , Difference and Repetition , p . 212.

n . lbid. , p. 211.\4 . Dele uze views the so lving of a virtua l pr oblem by individuation processes as

an 'explanation ' or rather , an 'explication" , Th is term is used to refer to the

cancelling out of int en sive differences during a pr ocess of indivi duation, the

hid ing of int ensit y under the extensities and qua lities it gives rise to .

It is not surprising th at , strictly speaking, difference sho uld be ' inexplic­

able'. Difference is explicated, but in syste ms in which it tends to be

canceled ; this means on ly th at difference is essentially implicated, that its

being is impli cation .. . Intensity is developed and ex plicated by means

of an ex tension whi ch relates it to th e ex te nsity in which it appears

outside itse lf and hidden ben eath quality . (De leuze , D!fJerence and Repe­

tition, p. 228)

Some scien tists to day (C hr is Langto n, for instance) are begi nn ing to view

some processes of morphogen esis as involving th e solutio n to computational

problems.

A material near its critical transit ion point bet ween th e liqui d and th e gas

states , mu st, in effec t, come to a global decision abou t whether it must

settle down to a liqu id or to a gas . T his sounds almost anth ro po mo rphic,

hut th e results rep orted here sugges t tha t we must think abo ut such

syste ms as effectively computing thei r way to a minimum energy state.

( .hristopher G. Langton , ' Life at th e Edge of Chaos ', in Artificial Life II,·ds. Christopher G. Langton, Charles Tay lor, Doyne Farmer and Steen

Rasmussen (Addison- W esley, Redwood City, 1992), P: 82.

lh is, in fact, occurs in a differen t context. Deleuze never makes this point

n -l.u ivc to theoretical and ex pe rimental phYSiCS, but I believe his idea can be

I xtcnded in that direction . Th e act ual extract reads,

ot on ly do linguistic variables of expression enter into relations of

formal opposition or distinction favorab le for th e extraction of constants;

non .lin guisti c variables of content do also . As Hjelmslev notes, an

ex 1'1' .ssion is divided, for example , into phon ic units in the same way a

co nten t is divided into social, zoologica l, or phys ical uni ts . .. Th e

nvt work of hinaritics, or arborescences, is applicable to both sides. There

/I . however, no analytic resemblan ce, correspondence or co'!(ormity betll'ccn the

I II' '' planes . Rue their independence does not preclude isomorplusm . . . (DelCUZI'

,11 1< 1 Cuauarl, /1 Thousand Plateaus , p. 108; my emphasis)

NOT E S

\6 . Bunge, Causality and Modern Science, p . 175 . (My emphasis)

17. In a linear system the ultimate effect of the combined action of two

differ ent causes is merely th e superposition [e .g . addition] of th e effects

of each cause taken individually. But in a nonlinear syste m add ing a small

cause to one that is alre ady present can induce dramatic effects that have

no com mon m easure with th e amplitude of th e cause. (Gregoire Nicoli s

and lIya Prigogine, Exploring Complexity [W. H. Freeman, New York

1989J, p. 59)

\8 . Bunge , Causality and Modern Science, P: 127.

\9. lbid., p. 49 .

40 . Th is is W esley Salmon's character ization of statistical causality, meant to

replace pr evious versions state d in terms of high probabili ty. Th ese old er

vers ions, due to th c absoluten ess of th e probability value (near = 1), ar e

simply weak enings of nccessity (the case with probability = I) whereas

enhance d probability is not. Th e latt er demands that we know the prior

probabilities (th e probability of occurrence of an event without th e pr esen ce

of the cause) as well as th e posterior probabilities. Whether or not the value

uf the enhanced probability is near = 1 is not an issue in Salmon's version ,

hence it really br eaks with necessit y not just weak ens it. See Salmon,

Scient!fic Explanation and the Causal Structure rif the World, pp. 30-4 .41. lbid ., p. 203.

42. Bunge, Causality and Modern Science, Chapter 6 .

4\ . Ibid., Chapte r 8.

44 . Deleu ze and Guattari, A Thousand Plateaus, p. 408. (Emphasis in the original)

45 . Ian Hackin g , Representing and Interveninq, P: 158. (My em phasis) Hacking

explicitly compares expe r ime ntalists and artisans, both suffering a relatively

low er soc ial status due to th eir involvement with an active materiality, on e

that does not ob ey Sim ple theoretical laws or allow exte rn al forms to be

imposed on it as a command (p. 151). In classical m echanics perhaps th e

bcst examples of these tw o scient ific caste s are th e theorists Isaac Newton

or Rob ert Boyle, on one hand, and th e expe rime ntalist Rob ert Hooke, on

the othe r . As one scientist puts it, ' unlike Newton, Ho oke was inten sely

interested in what went on in kit chens, do ckyards, and buildings - the

mundane mechanical arenas of life .. . Nor did Hooke despise craftsme n,

and hc pr obabl y got the inspiration for at least som e of his ideas from his

friend the gre at Lond on clockmaker Th omas Tompion . .. ' Gam es Edward

Go rdon , The Science rif Structures and Mat ertals [Scientifi c Ameri can Library.

19881,p·18) .

46 . 'Phenomena accumulate . For example, Willis Lamb is trying to do optics

without photons. Lamb may kill off the photons [i.c. create a new theory or

NOTE S

a new paradigm for optics] but the photoelectric effect will sti ll be there '

(Hacking, Represenring and Intervening, P: 56. Also see pp. 155-62).,17. Ibid. , pp. 83-4.

IX. lbid., p . 265. (Emphasis in the original)

I (J. Pickering , The Mangle rif Practice, p . 70 .

r, O. Dclcuze, in fact, do es not refer to learning in a laboratory conte xt , but his

idea of lcarning as involving an intensiv e assemblage or a problematic.field is

d earl y applicable to th e case of expe rime ntal physics. Here's how Del euz e

expresses this idea,

For learning evolves entirely in th e comprehension of problem s as such

. . . Learning to swim or learning a foreign language means composing

the singular points of on e's own bod y or of on e's own language with

those of another shap e or cleme nt which tears us apart but also propels

us into a hitherto unknown or unh eard-of world of problems. (Diffe rence

and Repetiti on, p. 192)

And he adds that this composition of on e's singularities and affects with

tho se of water (in the case of swimming) or with thos e characterizing the

ounds and patterns of a language, forms a problematic field (p. 165). A

' problematic field' refers to a het erogeneous assemblagc since , as he says,

' le,lI'lling is the . . . structure which unites difference to differen ce, dissimi­

I.II·ity to dissimilarity, without mediating between them' (p . J66).

11>,.1., p. 164., Il.ll.:king, Representing and Intervening, P: 209.

IIII IIg c , Causality and Modern Science, p. 71. (Emphasis in th e original)

Ih leuz e , D!flerence and Repetition, p . 25.1/.,,1.• p. 177 .

t; " neralizing , we can say that a dynamical theory is approximately true

just if the modelin g geome tric structure approximates (in suitable

I "spects) to the structure to be mod eled: a basic case is where trajectories

III the model closely track traj ectories encoding physically rea l behaviors

(or. at least , track them for long enough) . (Pete r mith, Explaining Chaos

1C.llnhri dge Un iversit y Press, Cambr idge, 1998], p. 72)

r rhur " lberall , Towards a General Science I?!' Viable Systems (Mc Graw-Hill ,

• \\ York, 1972 ), p . 7 . (My emp hasis)

I I 1111 ( ;oodman, 'Seven tri ctures on Similarity' , in Problem and Projects

dIu),)" Merrill , Indianapolis, 1972 ), p . +45. Goodman 's att ack on the noti on

" uui l.rritv was as caustic as it was influential. Similarity, he said 'ever

.. I til so l\'\' philosophi cal pr obl em s and oven-orne obsta les, is .1 pn··

NOTES

tender, an impostor, a quack . It has, ind eed, its place and its uses, but is

more often found wh er e it do es not belong, professing powers it does not

possess' (p,437) . Tod ay' s generat ion of realist philosopher s who have

resuscitated this notion have learned Goodman' s lesson that any two things

are similar in some respel.t or another , and that therefore when ever valid

judgments of similarity are made the relevom respects in which things may be

said to he alike mu st be speci fied (p . 444) . But thi s, of course, simply

changes the task to on e of speci fying distributions of th e relevant and the

Irrelevant, and that is just wha t a problematic approach is supposed to do.

At this point the usual reply by defender s of similarity is to fall back on

subjectivism and say that questions of rel evan ce and irrelevance are int erest­

relative.

But far from sett ling the issue , to re latlvize r elevance to subjective

interests is fatal to realism . If there ' s one lesson to be learned from recent

sociology of scie nce it 's that , as a matter of empirical fact , the inter ests of

scientist s canno t be viewed as being purely episte mo logical, born from some

essent ial rational ity or a driving curiosity . If we are to relativ ize relevance

to inter ests th en we should bring th e full repertoire of interests here,

including not only selfish profession al and institutional interest s but also

those that may be deri ved from a sci entist' s membership in class or gende r

hierarchies, for example . Th e rampant relativi sm that thi s manoeuver has

sometimes given rise to should be a cautio nary lesson for any defender of

realism. Alan Garfinkel sometimes expresses him self as if the choice of

contrast space, that is, the choice of how to pose a problem, is relativ e to

human inter ests and values, as in th e different values held hy the pr iest and

the thief in his example, But qu esti ons of explanatory stability seem to point

to an object ivity of the distribution s of the rel evant and the irrelevant.

Whateve r relativity there may be in explanations it is an objecti ve one ,

depending on the existence of indi vidual s with their own eme rge nt causa l

capacities at many levels of scale. Human values would enter the picture in

thechoice of on e or another of these level s of scale as the level of interest,

but a correct ex p lanation, as Garfinkel says , ' will see k its own level'

(Garfi nkel , Forms l!I Explanat ion , P: 59) .

Gilles Dcleuze, D!fftrence and Repetition, p. 16 3.

There is nothing in the ordinary meaning of the words ' universal' and

' ~ i ngu lar ' that marks the philosophical distin cti on Deleuze is attempting to

draw here , In fact, ana lytical philosoph er s use the wo rds 'gen eral' and

'universal' alm ost inte rchangeably, and the terms ' part icular' and 's ingular '

JS rloscly relat ed . In D1ference and Repetition uni versality and singularity an'

(loth properties of object ive probl em s, the former defining their o nto logie.l l

-utus ,1S vir tual entit ies (capable or di vl.' rgt'n t act ualizat ion) the lat ter till'

NOTE S

status of that which defines th eir condition s (d istribut ions of the rel evant and

the irrelevant ). The very first page of thi s book state s 'Ge nerality, as

gt'nl'rality of the particular, thu s stands opposed to re peti tio n as unive rsality

of the singular ' (p. 1). Yet, Deleu ze is not consistent in his usage , and

else where he says th at the ' splendid ste rility or neutrality [of multipliciti es]

. . . is indifferent to the uni ve rsal and the singular , to the general and th e

particular, to th e personal and the co llec t ive' (Gilles Deleu ze , LOBic if Sense

[Col umbia University Press, Ne w York , 1990], P' 35) .

Ml. Dialect ic is the art of problems and qu estions ... Howev er , dialectic

loses its peculiar power when it remains conte nt to trace problems from

propo sitions : thus begin s th e history of th e long perver sion 'vhich places

it under the pO\\'er of the negative , Ari stotle writes: Th e differen ce

between a problem and a proposition is a differen ce in the turn of phrase .

{Dcleuze , Difference and Repetition, P' 158)

I . 1.1lI Stewart and Martin Golubitsky, Feaiful Symmetry (Blackwell, Oxford,

1')9 2), P: 4 2. (Emphasis in the ori ginal)

II } I Jcle uzc , D~fJerence and Repetit ion , P: 162.h' llu- impact of gro up th eory on physics is r evealed not onl y by the fact that

1111' ('hange fro m classical to relativistic physics can be described in group

tlu-un-t lc terms (Einste in replaced the old Ga lilean group of transformations

Ii ) ' another one , the Poin car e gro up) but also by th e fact that the switch to

n-Ia t lvi:..t ic mechanics involv ed a change of cognit ive st rate gy in which

mva rinnccs und er transformation s becam e more imp ortant than the physical

I.I\\"\ the mselves. As the physicist Eugen e Wi gner puts it ,

[Einste in's ] papers on speci al relativity .. . mark the reversal of a trend :

until then the prin cipl es of invariance were derived from the law s of

mo tion . . . It is natural for us now to deri ve the laws of natu re and to

tes t their validi ty hy means of the law s of invariancc, rath er than to

derive the laws of invarian ce fro m what we believe to be the laws of

o.uun-. The ge ne ral theory of relativity is the next mil eston e in th e

!.btury of invariance .. . It is the first attempt to derive a law of nature

Ii )' M'lt'ding the sim plest invari ant equat ion . .. (Euge ne P. Wi gner,

' Invartancc in Physical Theory ', in S/mmetries and Rtfteetiom , eds. W alter

Moon' and Michael Scr iven [O x Bow Press, W oodbridge , 1979], P' 7)

11111'b Kline, ,lIathemat ical Thouahr from Ancient to Modern Times, Vol. 2

I I ) lor d Llnivcrsity Pr ess, New York , 1972), P: 7 59. Thi s idea can he

• I'l. li lll' d hy .1Ilalogy with the use o f tra nsfo rmat ion groups to d assil)'

I t 1I1 1H·lrk.,1 figu n's. W hen un c s,lys thai a cube remains Invariant unde r a

NOTES

group of rotations (e .g. the set containing 0, 90, 180 and 270 degree

rotations) one means that, after performing one such transformation the

cube's appearance remains und1anged: an observer who did not witness the

transformation would not be able to tell that a change has in fact occ urred.

In a similar way, when Galoi s found a group of permutations that left

algebraic relations invariant he found a measure eif our isnorance eif the solutions,

since we cannot distinguish them from one another after they have been so

transformed.

65. Deleuze, D!iference and Repetiti on, PI" 180-1.

66 . Ibid . , PI" 179-80. That Deleuze views the progressive specification of a

problem as a kind of symmetry- breaking cascade (a term he never uses,

preferring Galois's idea of an 'adjunction of fields ') is clear from this ext ract:

On the contrary, 'solvability ' must depend upon an int ernal character ­

istic : it must be determined by the conditions of the problem, engendered

in and by the problem along with the real solutions. Without this

reversal, the famous Copernican Revolution amounts to nothing . More­

over , ther e is no revolution so long as we remain tied to Euclidian

geome try : we must move to . . . a Riemannian-like differential geometry

which tends to S hoe rise to discontinuity on the basis ifcont inuity , or to ground

solutions on the conditions of the problem. (I" 162; my emphasis)

67. Ian Ste wart , Does God PIa)' Dice? The Math emat ics if Chaos (Basil Blackw ell,

O xford , 1989), 1'1'.38- 9.

68 . onlinear equations , due to factors like the occurrence of higher pow ers of

the dependent variable, do not obey supe rp osition. On the differ ences

between the linear and the nonlinear , and on the (rare) conditio ns for the

exact solvability of nonlinear equations (auto nomy and separability) , see

David Acheson, From Calculus to Chaos: An Introduct ion to Dynamics (Oxford

University Press, New York 1997), Chapter 3.

On the superposit ion principle as crite rion to distinguish these tw o types

sec David K. Campbell, ' Nonlinear Science. From Paradigms to Practical ­

ities ' , in From Cardinals to Chaos, ed. Necia Grant Cooper (Cambridge

University Press, New York, 1989), p. 219.

69 . Stewart , Does God Pia)' Dice?, p. 83. (Emphasis in the original)

70 . June Barrow-Green , Poincare and the Three Body Problem (American Math­

ematica l Society, 1997), PI" 32-8 .

O n the history of this appro ach prior to the work by Poincare see Kline ,

Mathematica l Thou,qhtIrom Ancient to Modern Tim es, Pl" 72 1-5.

71 . We alwa ys find the tw o aspects of the illusion : the natural illusion wh ich

involves tracing pr oblem s from suppose dly preexistent propositions,

N OTES

logical OpiniOnS, geo metrical theorems, algebraic equatio ns , physical

hypoth eses or transcend ental judgments; and the philosophi cal illusion

which involves evaluating problem s according to their 'solvability' - in

other words, according to the extr insic and variable form of th e possibility

of their finding a solutio n. (Deleuze, D!ffirence and Repetit ion, p. 161)

7 1 . Bunge, Causality and Modern Science, PI" 203 - 4. (Emphasis in the original)

The fact that nonlinear theori es are rare is not so much a peculiarity of

natu re as a sign of the infancy of our science. Nonlinearity involves large

mathe matical diffi culti es; beside being math ematically clumsy, it affects

the very symbolic representation of physical entities. Thus forces that add

nonlin earl y (as gravitational forces do) cannot be exactly represented by

vecto rs since the addition of the latt er conforms to the superposition

' principle ' . From the moment it was realized that the laws of ferromag­

netism are nonlinear, it has been more or less clearly suspected that all

physical phenomena may turn out to be at least weakly nonlinear,

linearity being onl y an approximation which is exce llent in som e cases

but only rough in othe rs. (I' . 168; emphasis in the original)

l Iclcuzc, Difference and Repetition, p. 189.

Index

.IITects 62-5,69-70,75, 141,167,199 n.35, 218 n,46

assemblage 56-8, 62-4, 93, 136,142-3,236 n.50

.u tractor 15, 20, 31- 2, 36- 7, 50, 55,

79,90,109-1 1, 134, 148,214

n.19axiomatic 121, 154, 179

I" 'coming 84, 101-2, 107bifurca tion 19-20,32,50,79,86,

109-10, 134, 207 n.61

, .Iusality 75 , 119- 20, 126, 129,137-40, 142, 144-5, 228 nA

, mtinu urn 22-3,27,69,74-6, 107,

158, 161, onvcrgcnce

and communication betwe en virtualser ies 76 -7, 104, 160- 1,206

n.59, 208 n.67, 220 n.53

.uu l subjectivity 162- 3, 171

divergence.1IId ramification of virtual ser ies

74- 5, 104, 160, 169, 176, 205

n.58

.incl actualization 22, 28, 64 , 118

,. "'nc<' 3- 4 , 9, 16, 28, 39- 40 , 78,106, 119, 121, 128, 183 nn. IO &

11 , 231 n.20, 232 n.23t ' u-nsiv« 26 -7, 46, 51 3, 58, 64, 85,

144, 163

identity 4, 9, 22, 40, 42, 74, 86, 107,

118, 193 nn.59 & 60immanence 3,10,13,28,41,75,80,

103, 110-11 , 146, 177,2 18 n,49individuation 29,40,43,45-6,51 ,

71,84,97-8 , 101, 117, 142,145,1 6 1,1 64,1 71,1 95 nA, 202

n.5 1, 234 n. 34

intensive 4, 26-7,45-6, 50, 55, 58 ,

64,85,92-4,98, 135, 143-4,159,1 6 1- 2, 165, 167, 169,1 71,

173- 5, 188 n.28, 199 n.30invariant 18, 24, 69, 75, 83, 86- 7 ,

150, 210 nA, 238 n.63

law 83, 106, 118-19, 121, 123- 4 ,

141, 146, 150,180,210 n.4, 221

n.57

metric

space 24-6,51,56-7,69,73, 172,179,186 n.22, 187 n.26

time 84,88, 106,108-9,213 n. 17

multiplicities 9, 13, 22, 28, 32,40- 1,69,72-5, 105-8, 111- 13, 129,

135,148,1 59,1 66,1 70,1 74,

181 n. 1, 182 n.6

natural kind 9, 13, 43, 46

necessity 37-8, 192 n.54, 235 nA O

non-equilibrium 66-8nonlinear 37,5 2-3,66-7,87, 119,

123,131,136,140 1,144,153 ,

155, 235 n. 17, 239 n.6

I N 0 E X

ordinal series 70,73 -7,104,113 ,

159, 170,204 n.56

phase transition 19- 20, 27, 61, 78-80,103- 4, 107, 197 n.23, 209 n.76

plane of consistency 69, 77, 112-13,

115, 158, 166, 170,226 n.77

population 47-8, 122

possibility 10,13,29,33-5,37,40,

206 n.59

problems 5, I I, 102, 115, 125, 127,

129- 33, 135- 7,1 40,1 44 - 5,

146- 7, 149- 52, 154, 168,

176- 7, 23 1 nn.19 & 20, 239n. 34, 240 n.72

progressive differentiation 17, 25, 28,

54,69, 102, 105, 151, 164, 177,

185 n.20

quasi-causal

opera tor 75- 8, 80, 103, 108,

I l l-IS, 136, 160, 165-6, 168,

170, 174- 6, 219 n 52,220 n.53 ,

223 n.71, 224 nn.74 & 75

re lations 127,129,133-4,140,

146, 207 n.62

rates of change I I , 49, 53, 95-6,

99- 100

relevance 5, 13, 90, 130 - 2, 144 - 5,

148, 151

resemblance 4,9, 10,2 1,28,40-2,

60, 68, 136, 147- 8, 193 nn .59 &

60, 237 n. 58

singularity 5,15-16,31,36,64-5,72,77,92,94, 108, 125-7, 131,

134,137,141,146-8,154,167,

203 n.53state space 14-15,30-2,62, 130,

134, 136, 145-6, 154, 175

symmetry-breaking 18-19, 21, 24-6,

74,86-7,105,107,135, lSI,

185 n.20, 201 n.46, 239 n.66

time scale 87,90- 1, 108, 214 n. 19

topologica lfeature (forms, constraints) 15- 16,

29,31,72, 110-11, 147-8 , 183

n.S, 197 n. 19space 23, 52-3, 56, 69, 74, 179,

226 n.76

time 105-9,222 nn.58 & 59transcendence 3, 10, 13, 4 1, 80, 107,

113, 177

truth 5, 121,123, 134, 147, 177

typology 41-2,47-8,68, 177, 193

nn.59 & 60

unity 13, 113universal

as concrete entity 22, 28, 70, 112as mechanism-independence 16, 55,

75,79,92-3,127, 129, 133,207

n.6 1

as opposed to the general 148- 50,

237 n.59

Virtuality 33-8,44,65-8,78, 102,105- 9 , 127, 135, 147, 154, 165,

174- 5, 219 n.52, 225 n.76