Bühlmann - From Linear Algebra to Algebraic Invariances

8/18/2019 Bühlmann - From Linear Algebra to Algebraic Invariances

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in the light of the Master Argucomputability

PhD Kolloquium Wvera bühlmann, Dec


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Before the study of invariances, thestudy of linear equations was usuallysubsumed under that of determinants

transformations carried out on the variables („constants“) within algebraic forms

> not the solution of a system of

the solv ability of equations and th

from linear algebra to algebraic invariances propremaa systtrans

Quantics

In linalgebra,

determinana vaassociated w

a squmatrix . It cbe compu

from entries of

mat

a linear equation is an equation

in which each term is either aconstant or the product of aconstant and a single variable

(not raised to any power).

Linear Algebra studiessystems of such equations. No consideration was given to systems in wh

number of equations differed from the numbeunknowns . This changed with the interest in

2

http://en.wikipedia.org/wiki/Variable_(mathematics)http://en.wikipedia.org/wiki/Variable_(mathematics)http://en.wikipedia.org/wiki/Constant_termhttp://en.wikipedia.org/wiki/Constant_termhttp://en.wikipedia.org/wiki/Term_(mathematics)http://en.wikipedia.org/wiki/Term_(mathematics)http://en.wikipedia.org/wiki/Square_matrix#Square_matriceshttp://en.wikipedia.org/wiki/Square_matrix#Square_matriceshttp://en.wikipedia.org/wiki/Square_matrix#Square_matriceshttp://en.wikipedia.org/wiki/Square_matrix#Square_matriceshttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Linear_algebra


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what aboutspace and

geometry in

linear algebra ?

field - rational domain within the compelx numbers satifying certain con

vector - establishes the operability over a field, n-dimensionality, in ideal vectorspaces are taken as subsets in rings.

ring - establishes unique factorization, modularity, of a field

algebra - hypercomplex number system (a ring and a vector space over a

key terms in linearalgebra

they are dealing withinvariances - properties thatremain stable undertransformations operating onthe variables or coefficients

within algebraic forms .

> what is the nature of these invaProperties of what are we dealingThis is the main disconcertion as

with invariances from a non-techpoint of view.

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linear algebra (forms of homogenous transformations)- invariance and covariance, vector spaces and

matrices. no systematization, a cookbook.

Interest in the solvability of structures in abstractionfrom any specific „ground“ - through the conception ofthe ground in ideality, rings, fields, modules, groups,etc.

Interested in providing the conditions for solvability.

The ground is symbolic and engendered (not assuminga given „nature“ of numbers)

treats valences in physics and chemistry, according toprinciples of saturation (from the Latin word saturare ,meaning to fill ) - a notion of fullfillment based onreaching a maximum capacity.

algebra

quantics

abstract algebra

symbolic algebra

symbolico-physical

> Doping technologies.

ha nd l e t hi s w i t h s p e c i a l c a r e - t he dist inc t ions ar e assor t edow n r easoning ! no st andar d list of dist inc t ions!

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1 General Features of Invariance2 algebraic invariance3 the synthesis by transformation groups

4 codification of geometry by invariance5 intrinsic spatial invariance

E.T. Bell, in The Development of Math

chapter on Invarian"Invarian

changele

midst of

permane

world of

persistenfiguratio

remain t

despite t

stress of

hosts of c

transform

(E.T. BEL

5


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paragraph1 General Features of Invarian

along which path?

Quantics - the study of algebraic form

Abstract and Symbolic Algebra - an undiscovered continent

Relativism - the hypotheses at the foundation of geometry

Systematization - through seeking a unification of geometries

Physics - general theory of relativity (Conservation Laws)

invariant theory as aaddition to mathem

thought ?

Invariance as newunifying principle ,abstraction from thebook listing of calcuin the algebra of form

Riemann Space - a cimagination, againsabsolutist imaginati

encircling whic

George Boole

Bernhard Riemann

Albert Einstein

Felix Klein

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paragraph 2 Algebraic Invariance

Determinants - evolution of the symbolic me

from the Calculus of Quantics to the Algebra of Qua

from invariance as a property of quantities to invariance as the porperty of gro

settling in the undiscovered continent - devices of calculation bring about „new Provin

foundations of math

algebraic geometry (encoding of space and spac

chemico-algebraic theory (theory of chemical valence e.g. be

-> quantics was eventually absorbed into quantum

Problems of Quantics – is there a fundamental system, and a finite set of independent irreduci

for quantics YES, but for quantum theory cf the Hilbert - Brouwer/Weyl controversy . Intuitionism.7


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Bell speaks of an „army of algebraists and

projective geometers“ who storm into „the fertile

territory “ of abstract symbolic algebra. Those

who applied the symbolic method were

adventurers stigmatized as „illegitimate Kings“

striving for „profit“ in the domains of their

settlement: Bell writes how they were „recruting

masses of young mathematicians who mistake

the kingdom of quantics for the democracy of

mathematics“.

settling in theundiscovered

continent(Algebra) was

interpretedpolitically before

breaking abattlefield

betweenideological claims

for supremacy.

Are the realms of abstraction „terrundiscovered continent“ t

Are adventurers in abstraction supposed to be

Renaissance, obliged to sail in the name of the Queen > then the State turns into a monarch of the multitude, capable

L. conquirere for "to search for,procure by effort, win"

from com- for "together, together with, in combination,"

and quaerere "to seek, acquire"

conto susubmundanomaymat

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„Cayley's numerous successes, quikly followed by those o

Sylvester, unleashed one of the most ruthless campaignscalculation in mathematical history . [...] Such misdirectenot peculiar to the algebra of quantics in mathematics siaccompanying theory of groups, for example, especially groups, there was a similar panic. Once the means for raisupplies of a certain crop are available , it would seem to bcaution to keep on producing it till the storehouses burstcourse, the crop is to be consumed by somebody. There hconsumers for the calculations mentioned, and none foreasily digested. Nevertheless, the campaign of calculatioapparently, of mere calculation did at least hint at undiscin algebra, geometry, and analysis that were to retain thedecades after the modern higher algebra of the 1870's hato the dustier classics.“

(E.T.Bell p. 429/30)

Enlightenment – political seculariz

The Algebraists were accused of

totalitarian calulation: as

campaigning torecruit

mathematiciansfor theory with no

application

(useless)

science detached from metaphysics/theology

> fear of the potential applications that werBell‘s stance: criticizes the filling of the „storages“ with „intellectual nourishment“ that 9

b l f h L f h E l d d Middl


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Background to the Hilbert – Brouwer/Weyl controversy

Intuitionism

Gordan in 1868 proved the existence of finite fundamensystems of invariants and covariants for any binary quaand in 1870 did the same for systems of such forms. Hismethod was constructive, he demonstrated a procedur which can generate all the instances.

Hilbert extended the finite theorem significantly by givipurely formal proof without procedure to actually geneall instances. On purely logical grounds – by applying thLaw of the Excluded Middle to Cantor‘s infinite sets.

"This is nomathematis theology

„For this memorable victory overthe barbaric hordes of algebraicformulas, Gordan was crowned

"King of Invariants" [...] Heoccupied the throne exactly twenty-

two years, until Hilbert, a merestripling of twenty-eight, in 1890

snatched the crown from Gordan'sageing head and rammed it firmly

down on his own.“ (E.T.Bell)

Jan Brouwer and Herman Weyl (both students of Hilbercritizized the formalist (logical) method and instist ininfinitary methods when dealing with infinities.

Aristotle‘s privation-dynamicsis a prototype for infinitary method; the same

for Booles‘ Algebra of logics , and for Dedekind‘sand Noether‘s conceptual approaches .

contradictories for theaccidental properties,Contradiction for theessential properties.

non-absoluteness of the Law of the Excluded Middle

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paragraph3 The synthesis by transformatioStructural theory – inversion of problem-solving approaches: not is there a solu

problem, but which operations are sufficient and necessary, and wh

objects must be invented, to provide a solution for a problem of a

elimination through composition series (Liand algebraic characterization of the nature of irrationals (fields, domains, etc) dire

looking for integrability – The primary objectives (of algebraic structural theory) are

can be done rather than to do it , and to give criteria for wha

Distinguish between reasonable formulatio of problems and not reasonable formulation of problem

physical and chemical valences (doping) – construct manifestations of invarian(partial) equations whose conditions the abstract invar

Organic and non-organic ch

logistics, geodesics, analytical mechanics – application of invariants to kinem

(describe Euclidean Space kinematically (not static!) by working with differ

Lie secured linearization for differential equation like Newton did for infi

operat11


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paragraph4 The codification of geometry

Codification - from

Paradigms of langauge, informati

from kinematic space to quantum space - from equivalence transformations between group

commensurable by schemata (quantified), to transformations within groups that are iden

and need to be quantized

search for a unified field theory - abstracting from Einstein‘s gravitMaxwell‘s electromagnetic field – attempts developed by Weyl, leading to quan

point-set geometries – geodesics, Levi Civita parallel displacement (the sum of the angle

the earth is more than 180°), generalization to the geometry of path, eventually: break

connections (Dirac‘s Algebra of Q

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„A geometry is defined as the system of definitions

and theorems invariant under a given group of

transformations.

Two groups of transformations are to be considered in

connection with any geometrical relation: a group by

means of which the relation may be defined ; a group

under which the relation is left invariant . The more

restricted the group, the more figures will be distinctrelatively to it, and the more theorems will appear in

the geometry.

The extreme case is the group corresponding to the

identity [the transformation leaving everything

considered invariant], the geometry of which

is too large to be of consequence .“

codification of geometry

The Erlangen Program 1872 (Felix Klein 1872 )

note the indefinite article –a geometry, like a language

If at least one postulate of a mathematical (hypothsystem be suppressed , the sfrom the modified set of porestricted than the original modified system is more fumore general than the origi

group - structural solution space (can be expresse

all the properties of a spaceconsidered if an identity re

instead of equivalence rela13


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In the calculation of group characters, analytical considerations offeran alternative to finite summations of subsets, which are beingreplaced by integrations over the group manifold , the invariant

element of volume of a compact group having been suitably defined.

Klein‘s'Erlanger Programm' of 1872 for the codification of geometry as it existed at the time, essentially reduced geometry to the study of

equivalence under certain groups of transformations. In the modernizedpresentation of Klein's project (e.g. in Hilbert axiomatization of

mathematics), groups appear as groups of automorphisms and ofpreferred coordinate systems of the base space . The automorphisms leave

invariant the characteristic relations of the particular geometryconcerned. These automorphisms induce transformations on

the vector subspaces of the base space .

the characterization of groups throughinvariant integrals (not summation of subspaces)

(and Felix Klein) the definition of sets by code, working withrepresenting groups by summation of subspaces within coordinate systems

Weyl

Klein/Hilb

this needs a re

of groups as obassumes coordit cannot deal space, only wirepresentationtherein.

is purely oper works directlyspaces (non-c

spaces, manif14


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This brought to attention precisely those

Riemannian geometries about which the

Erlanger Programm said nothing, namely

those whose group is the identity. In such

spaces there is essentially only one figure,

namely the space structure as a whole.

„This [Klein's] point of view was the dominant one for

the first half century after it was enunciated. It was a

helpful guide in actual study and research. Geometers

felt that it was a correct general formulation of what

they were trying to do. For they were all thinking of

space as a locus in which figures were moved about

and compared [as was implied earlier in connection

with Lie's revision of Helmholtz' kinematic geometry].

The nature of this mobility was what distinguished

between geometries.“ (E.T.Bell)

space as alocus in whfigures we

moved aboand comp

With the advent of Relativity wbecame conscious that space

need not be looked at only as

"locus in which," but that it

may have a structure, a field-

theory , of its own. Einstein‘s Relativity Th> space-time.

„nature of this mobility“ > K

classical mechanics that des

bodies (objects) and systems

without consideration of th

No unified field theory for the

space of a geometry: Einstein‘s field of Gravitation andMaxwell‘s field Electromagnetism

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http://en.wikipedia.org/wiki/Classical_mechanicshttp://en.wikipedia.org/wiki/Classical_mechanics


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"A space is a set of objects with a definitesystem of properties, called the structure of

the space." (cited in Bell p. 448)

„A geometry is defined as the

system of definitions and theorems

invariant under a given group of

transformations“.

With relativity theory – matter itself baspect of spacetime . The measuremencurvature varies from point to point icorresponding to the amount of mat

In the absence of matter, spacetime i

Space is not a locus in which objects are placed and subjects act

(governed by whom/whatever).

metricsspace

matter

> this is what is so silly aboPhilosophy Movement! (O

tecthey are dealing withquantification, not with quphilosophy were game desi

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C di h


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A „wor

signs ovsigns a

com

(sy

{a, b, c .... x, y, z}

{0, 1, 2,.....8, 9} decimales alphabet

alphabet in letters

involves double discretization of amplitud

1) sampling (Abtastung)

2) quantization (Quantelung)

change of an alpe.g. analog/digital con

Coding – quantification vs quantization

formal Language vs Quantutransformation groups as alphabets

{0, 1} binary alphabet

... ...

quantization is wher

model of language-in

life streams and SOM

breeding clusters of d

be interpreted and fo

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in codification theory:

i i i i l i i


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paragraph5 intrinsic spatial invariance

topology - can it be interpreted as „the royal road „the devil of abstract algebra and the angel of topology“ (Herman Weyl): one striving for articulation and differentiatio

properties of a space - topology works with its own categories of how neighborhood, region, bounary, etc. These allow to predicate the properties of a

topological space - topology constructs its spaces according to the transformation gri.e. by the qualitative properties of space (independent of size, location, and shape): continous, homeomorphic (finite and con

knots – structural „objects“ (knots) in non-coordinated space (networks) can be a

aspectual space only (their appearances). They have relative dimensionality (topological invariances, c Analysis allows to enumerate and characterize all possible knots (Euler‘s bridge problem) relative t

topology illustrates properties of functions – appearances are constituted by their qualitat

topology in analysis - inversion of Descartes Accidential properties (dimensions in topology) for Descartes were series , starting from an absolute point. Construction had to build

Now we can postulate properties as invariants of groups of complex bodies, and then construct the spaces accordingly. In complex

imaginary numbers in the real numbers (Dedekind Cut). The r18


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an Aristotelian mindset

and contemporarymathematics ?recap the dynamics of privation

Everything is subject to privation in the events that can happen tothem, they gain their individuality thereby.

The space of quality is full – but never exhaustible!

He ascribed privationCan we ascribe it to

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„This garment, for example, may be cut in two and yet wibe cut in two, but will wear out first. In the same way, it mnot be cut, for it could not wear out first were it not poss

for it not to be cut in two. This holds for all other events a which are mentioned as having the same kind of potentia

Is this real or virtual ?

Aristotle‘s exampleof probabilistic

truth value based

on opinion:

Aristotle‘s future contingents, as in the example below

theory of the potentiality for contrar

A constradiction, if it concerns the accidoperationally and can be harvested in thin ethics .

abstracaccomodiversitbrings r

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to have a potentiality formeta-property for Aristo

For Aristotle, privation is if a thing is

hindered in fullfilling its potential.

To what is this kind of potentiality a proper potential (to whom or what does itbelong)? What could possible „fullfill“ it?Or is it not subject to privation at all?

Privation is what many (modern) theories hold asconstitutive for „the human“, and for ethics. The idea of

socalled Mangelontologien (ontologies of lack) is thatonly through affirming privation – the being hindered

through communality in fullfilling exhaustively ones

potential – can we live together socially.

excursion: privationthis is the kind of pro

abstract entities likeconstitution, plan, a generali

a form or a schema, etcpotentiality an abstract en

has, the higher its va

Abstraction allows to conse

myabstraction

the degree of(freedom) w

mastering of a govern its s

Not MANGELONTOLOGIE:

abstraction createsan excess of potential

properties for itssubjects. It

introduces aneconomy principle 21

b i i


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being-in-actbeing-in-potency

composed and divided yes, but

essential

accidential/probable

(the properties belong to a subje

no!

(the properties belong to an eve

no contradictions allowed in the essential

contradictions are operationalized within the dynamics of privation

as contradictories the can be

conjuncted and disjuncted

Contingency in the universe due todynamics of privation.

This is the Reality-Principle of A

Aristotle economized necessitarianist prede

for Aristotle,Truth had a Nature! „lending articulate voice to that with inarticulate eloquence“

A R T I C U L A T I O N

can algebraic invarianceplay the role of Aristotle‘

essential today?

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i i l f d


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[For Aristotle:]

„Accidental beings are not necessary butindeterminate, and their causes are unordered andinfinite.“

The principle of correspondence (at work in

language) transmits the properties of the things andtheir causes to the statements about them.

principle of correspondence And the Nature of Truth (within Realist Philosophy in Aristotle)

> correspondence is not representation but articul

how to te without coerci

for accidentials

the idea of a natural flow we ca

topologby estab

co

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being is not absolute for Aristotle

being-in potency can never be fully actualized.motioninfinity voidfullness

Reality actualizes from „linking up“ being(essences) and being-in-potency (accidencaught up in the dynamics of contrarity wdriven by potentiality, privation).

but numerouquantificationquantization because of the simultanousexistence of contraries

L. absolutufree, make "without renot relative

Aristotle‘s Reality is

analytical, ofdifferential make-up

which allows forgeneration and decay,

transformation,becoming

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th t hi h i ti l t ElAristotle‘s


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that which expresses inarticulate Eleganc

that which expresses the appearancesTarski‘sNature

Aristotle sNature

cannot be exhausted by language, the poetic principle of th

can be determined by formal language, the semantic principle of ththe domestication of Language by Algebra

the domestication of Nature by Language

What happens to theinarticulate elegance

of that whichappears without

being forced into

expression?

How can Algebra

domesticate Natuand not only coer form and consequthrough controlli(de dicto)?

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the beauty


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the beauty equation doe

appear if we solution, but from

promise of integradifferences wit

conif we can see inarticulate

promise in it! 26

Bühlmann - From Linear Algebra to Algebraic Invariances

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