Mueller, Euclid's Elements and axiomatic method

8/7/2019 Mueller, Euclid's Elements and axiomatic method

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29 0 ItIIIAlJtra xi om a ti cm e th o d, wh ic h. a cc o rd in g 't o E v es a nd NCW1 IOm( 19 5 8, p , 1 3) , i s'the very core of m odem m athem atics" D oth in the Elements Ilnd in ar ep re se n ta ti ve m o de rn wo rk l ik e H i lb er t's Grundlagen t ier Gtomelrie on efin ds s em e s en ten ce s p os tu la ted a s s ta rtin g p oin ts a nd th e re st d eriv edfro m the m. B ut ho w de ep is s uc h s im ila rity ? T he a ns we r to t he q u es t io nw ould seem to depend on the m eanings of the w ords 'pos tulated ' and'dcrh 'ed' in each ease , In the case o f the Grundlogell , th e m ea nin gs a rcr ela ti ve ly c le ar . 'D er iv e' m ea ns , o r c an b e t ak en to m ea n w it ho ut a ff ec tin gthe tenor o f the w ork, 'deduce using the principles o f m odem logic';'po stu la te ' is s om ew ha t m ore diffic ult bu t p ro ba bly m ea ns n o m ore th an'"T ite do wn an d use as prem isses in deductio ns'. In this appro xim ated ef in iti on o f 'p os tu la te ' I in te nd to b ri ng o ut th e S CH aI Ie d h yp ot he tic alc ha ra cte r o f m od ern a xio ma tic m athe ma tic s, M an y c om me nta to rs h av epointed o ut that, fo r the G rccb. m athem atics was n ot a h yp ot he tic als ci en ce i n t hi s 1 C n S C ; fo r them math emat ic a l a a ae r ti o ns were true and ofin te re st o nl y b ec au se th ey w er e tr ue . (Sec. f or e xam pl e. S ch o lz ( 19 3 0- 1) ,p . 2 ;6.) F or the m odem fo rm alist the question of truth as co nceiv ed bythe Grc :cks i s mathemati ca ll y i r re l evan t .T o s ay, ho we ve r, th at fo r the G rc cb 'p os tu la te ' m ea nt 'a ss um e a s tru e'i s t o o v er lo o k a v e ry s ig n if ic an t f ea tu re o f E u cl id 's Elements. T hree of the

f i" e p o stu la te s a re n ot e ve n c ap ab le o f b ei ng tr ue :I. L et it be postulated to draw a straight line from any po in t to any

point .2. A nd to e xte nd a lim ite d s tra ig ht lin e c on tin uo us ly in a s tra igh t lin e.3. A nd to draw a circle w ith an y cen tre an d distan ce.

G ramm ati ca lly , a t l ea st , th es e p os tu la te s a re n ot e xis te nc e a ss er ti on s l ik ethe ir m ode rn c ou nte rp ar ts (e .g . F or a ny tw o p oin ts th ere e xis ts e xa ctlyo ne straight line o n w hich they bo th lie).- N or are they descriptio ns ofp os sib ilities w hic h m ig ht in ra ct b e u nr ea lis ab le , the re by r en de rin g thed es cr ip tio ns r ais e. ~ ey a rc w ha t m ig ht b e cal le; l ic e nc e s t o p e rf o rm certaing eom et ri c o p er at io n s. T h at t hi s i s s o c an b e seen, 1th in k, f ro m E uc lid 's u seo f th em in -fo r ex am ple -the pro of o f pr op os itio n I, B oo k I :On ~ g iv e n l im i te d s tr a ig h t l in e t o con st ru c t a n e q ui la te r al t ri an g le .Le t AD be t he g iv e n l im it ed " ra ig ht l in e. T h us , i t i s n c cc sa ar y t o c o ns tr uc tan tq lOi l a te ra lt r iang l eon th e s t ra ight l i ne AD .Wi th cen te r A and d i s tance AD , l e t t h e c ir c le DC O be d rawn, a n d a ga in w it hc en te r 0 a n d d is ta n ce BA , l et t he c ir cl e ACE be d ra wn ; a nd fr om th e p oin t Ci n wh ich 1 I1ec ir c le s cu t o n e ano th e r, t o t he p o in t s A, D, l e t the Bt raigh tl i ne s CA ,CD, b e joined.

I I WU not diIcuu here H. G. Zcuthm'. (1196, pp. 119"116) 8lJUlllal1i that the poatuIates.re e" ..~ uscniona. Illoob to _ 88& h o v a h Zcuthcn .... 1_ dial lhe postu.lala r 1 a r lhe aole or uisl_ -.tiona in other rormul8liona of a-neh7.

&tliJ', E1c:tncntaa d 1M A xio m ,. ," M ellr oJ a9'N ow , s in ce th e po in t A is th e ce nte r o f th e c ird e C OD , A C is equal to AD'ag ai n , s in c e t he p o in t D i . I he c en te r o f t il e c ir cl e CAI~ II C is equlllto DA , an dC A w as p ro ve d e qu al to AD. t he re ro re , e ac h o f CA , CD is equal to A D. O utt hi ng s e qu al t o t he s am e t hi ng a re a ll O e qu al t o e ae h o ll lc r; a nd , t he re fo re , CA i se q ua l t o CD. t he r ef o re , t he t hr e e CA, AD, DCa r c e q u al t o o n e ano lh e r. Th e re for ethe t r iang l e ABC is equ il a te ra l; and it ha s b e e n con st ru c te d on t he g iv e n l im i te ds t ra ight l ine AD. Wh ic h w as to be done ._--~c~-~

D E

In w ha t s en se do cs E uc lid d eriv e p ro po sitio n J f rom f ir st p ri nc ip le s?C e rt ai nl y i n t he s e R S C th at c ac h s te p is s up po se d t o b e j us tif ie d b y r e fe re nc eto th es e p rin ciple s (a lth ou gh m ode rn a cc oun ts o f E uc lid c us to ma rily s aythat he canno t justify the usc of the po in t C in w hich the circles cut onea no th er , s in ce n on e o f h is f ir st p rin cip le s g ua ra nte es t he e xis te nc e o f S U clla po in t). B ut, in m an y c as es , j us tif ic atio n d oe s n ot m ea n s ho win g tha t a na ss ertio n is tr ue , bu t s ho win g th at a p erfo rm ed o pe ra tio n (fo r in sta nc e, tod ra w a c irc le w ith A a s c en tre a nd An as distance) is licens ed. H alf oft he p ro of o f p ro po si tio n J is g iv en to the p erfo rm an ce o f s uc h o pe ra tio ns( th e s o -c aU e d Ilallllkeut), re sultin g in th e c on stru ctio n o f a n e quila te ra ltrian gle o n the lin e A D . At the end of the proof it is described no t asshow ing som cthing to be true, but as do ing w hat w as to be done, T hec o nc lu si on s eem s i ne sc ap ab le t ha t p ro po s it io n J, whic h g ramm ati ca lly ist he s am e a s p os tu la te s J, :z , an d 3, is n ot an o ddly fo rm ulated assertio nbu t the de sc riptio n o f a ta sk w hic h E uc lid 'pr ov es ' b y do in g iL

The andents cal led p ro p os it io n s l ik e p ro po s it io n J 'problems' (pr06-ltrt,ata) a nd d is ti ng ui sh ed t hem f rom p ro v ab le g eom et ri c a ss er ti on s, w h ic llthey cal led ' thcorems' (lI,toremattl). Eu cl id m a rk s t he d is ti nc ti on b y w r it in g'w hic h w as to b e p ro ve d' a t th e e nd o f p ro ors o f the ore ms in stc ad o f 'w hic hwas t o b e d o ne ', Itmight be t ho u gb t t ha t E u cl id ea n d er iv a ti on s o f t he or em sa rc a t le as t n ot s ig nific an tly diffe re nt fr om m od em fo rm al de riv atio ns .B ut , in f ac t, th is is n ot tr ue , f or i n a lm o st e ve ry o ne o f E uc lid 's d er iv at io nst he c ar ry in g o u t o f c er ta in o p er at io n s, p re v io u sl y s hown p os si bl e, p re ce de sargum en tatio n in the usual sen se . T he characterisation o f a E udidcand er iv a ti on a s a 't ho u gh t e xp er im e nt ' i nv o lv in g a n i de al is ed p hy si ca l o b je ctw hi ch c an b e r ep re se nte d in a d ia gr am s ee ms c lc ar ly j us tif ie d. V er y o Cte n


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2 9 : & Ian Mutlkrthe diagram is inesaent ia l to the argument , s ince the words alone would' bea proof , and in no case is the diagram the subject of the proof. However ,it ha s often been pointed out (e.g. in Klein (1939), pp. 201-2) that in somecases the diagram does play an important role in Euclid's arguments.Such cases should not be looked on as mere bpscs from the axiomaticmethod but, rather, as indicative of the experimental character of Euclid'sgeometry. From this point of view the diagram is much more closelyrelated to the proof than the words which go along with it. One mightthink of Euclid 's proofs as standardised versions of the kind of discussionwhich Socrates has with the slave boy in the Meno (8zD-8SC) . Here it isquite obvious that the verbal argument is only an accompaniment todiagrammatic manipulation and that the diagram is both the source ofcon\;ction and the cour t of last resort in deciding the truth or falsity of ageometric assertion. The lIIeno discussion is often taken as representativeof the primitive kind of mathematical argument and contrasted withEuclidem proof (in, e.g, Szab6 (1964), pp. 34-40). The contrast seems tome &tylist ic rather than conceptual. In other words, Euclid's ' formalism'is much more like formalism in literature, which focuses on stylisticn icet ies , than l ike formalism in mathemat ics, which is motivated by aphilosophical conception of mathematics.

A Euclidean derivation, then, is a thought experiment of a certainkind-an apcr iment intended to show either that a certain operation canbe performed or that a cer tain kind of object ha a certain property. Thus,Eutl idean derivations are quite different from Hilbertian ones, which areusually uid to involve no usc of spat ia l intui tion (see, e .g. Poincare (1929),p. 467). The major difference bctwccn the mathematies of Euclid'sElelllents and more pr im itive predecessors is that the former contains anexplici t s ta tement of first principles, defin it ions. postubtes, and axioms.Ho\ \'C\ 'Cr, the historical explanation of this d ifference docs not requirepresupposing deep conceptual change. One need only refer to Szab6'splausible suggestion (in (11)60),among other places) t"'t the developnlentof the Greek axiomatic method is closely connected with the developmentof the dialectical method in Greek philosophy.' Socrates' description of hisdttlleros p l o w in the PIIt.edo (99D-IOJA) is perhaps the best indicationof the role which hypothesis came to play in philosophical diSctlssions, It isSocrates' claim that truth ( ti t on onlena) can b e reached only if one searcheson the basis or hypotheses accepted at least temporarily. Otherwise oneis li1tcIy to 'mix things up by disctassing a first princip le and i ts conse-quences at the same time, as the anti/ogikoi do'.ITh o con_ or my chanctaUatlon or Euclid'. lint prindpla doa nol, o r course.dcpcncl "1*1 & he uuth o r Saw'. hislOrical conjectwa.

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Hilt/ii", Elements a ll d t /~ A x io ll la ti cM . t/ ,a d 2 93There is g o o d reason to suppose that, in the fifth and rour th centuriesac., Grc:clt mathematies Buffered from as much confusion as did philo-

sophy. A passage in Aristotle (Prior A."alytit, II, 16 , 64bz8-65a9) makes itprobable that the treatment or parallels invoh-cd 'diseussing a firs t prin-ciple and its consequences at the same time' until someone made Euclid'sfifth postulate a hypothesis, We also know ortwo iacoudusive, and probablyfallacious, a ttempts to square the circle. (Sec Heath (1921), pp. 220-5 .)Some scholars (for instance, Hasse and Scholz) believe that the discoveryor incommensurability and perhaps also Zcno's paradoxes made thePythagorcans aware of the untenability of their whole conception orgeometry. As a final instance of the uncertaint) in Greek mathematics atthis period, I might mention the several solutions then known to theproblems of squaring the circle, duplicating the cube, and trisecting anangle. (Sec Heath (1921). pp. Z25-70') These solutions involved veryingenious operations, apparently much more complex than those involvedin, say, bisecting an angle. I t is not surpr ising that the question whetherthese solutions were satisfactory remained open; nor is it surprising thatsteps were taken to answer this question and those raised bydifficul ties l ikethe ones just refer red to. The steps takcn involved making assumptionsexplic it and standardising proof techniques. Taking these steps producedthe Euclidean axiomatic method-a method related to the difficultiesas Socratcs' method is related to the dif ficulties in philosophy created bythe Sophists. The evolution of the axiomatic method is explicable solelyin terma of the desire for clarity and order in geometry. Philosophicalconceptions of mathematics, such as those of Plato and Aristotle. weremore probably the rcsul t of phi losophically coloured reflect ion on mathe-mat ical practice than causes of that pract ice.

Euclid's f ir st principles , then, are things agrced upon for the sake of anorderly and unconfused development of mathematics. They arc of threekinds: (I) postulates concerning permissible constructions, (2) assumedassertions, (3) defin it ions. This division docs not coincide wi th Euclid'sown division IIf l int princil ,ICII in to IHlStulatCII,COUllllon notions, anddefini tions, for his postu la tcs and his defin itions include assumed tru ths,Although a great deal of work (notably in Von Fr itz (1955 has bcen doneon the ancients ' interpretation or first principles, no one has yet producedan adequate mathematical or philosophical cxpbnation of Euclid's division.There is no reason to discuss the attendant problems here, however,s ince I wish only to make some brief remarks about Euclid's assumptionsand definitions.

I said earlier that the Grccl ts took geometric assertions to be t rue. Theirdoing so is not incompatible with the view that f ir st principles arc things


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Z 9 4 laM"agreed upon i n o r de r t o m a k e po ss ib le r at io n al a rg umen ta ti o n, f o r this viewim plies on ly that the qucstion of the truth o f a geom etrical assertionca nn ot b e a part o f ratio na l arg um en tatio n in w hich it play s the ro le o ff ir st p rin cip le . T hu s, th e q uc stio n w he th er a g eome tr ic al f ir st p rin cip le istrue will not be a geom etrical question altho ugh a geom eter m ay v erywell be c on vin ce d o f its tru th an d p os tula te it o nly b ec aus e he be liev es itto be t ru e . P la to t ak es e xac tl y t hi s app roach t o ma th emat ic a l f ir st p ri nc ip le sin the Republ;c (5100 5I1D ) w here he claims they can be proved by amo re r ef in e d me th od t han t ha t o f t he ma th cma ti ci an - -d ia lc c ti c.

In the Postmor Analy t;u (I, 2 , 7Ib2 1-U ) A ristotle insists that thea ss um pt io ns o f a s ci en ce b e n ot m e re ly true. bu t also ' pr imary , immed ia te,a nd m ore kn ow n tha n, prio r to , an d ca us es o f th e c on clus io n d ra wn fro mth em '. I t is s om etim es a ss um ed th at E uc lid p ut t his A ris to te lia n d ic tu mi nt o p ra ct ic e i n t he Elements. Out, in10 f ar a s o ne c an te ll, th e f if th p os tu -la te w as a lw ay s a c on tr ov er sia l o ne , a nd it is u nlik ely th at E uc lid th ou gh ti t t o f :J lf il A r is to tl e's d es cr ip ti on o f a cc ep ta bl e a ss um p ti on s. M o re ov er , i no th er wo rk s o f G r ee k s ci en ce ( e.g . E uc li d's Opt i&l or Ari sta rchus' 0" tIltSiuJ alUl D ista nc a o j th e S un an d Moon) on e finds a ss um pt io ns wh ic h c anh ar dly b e c alle d imm ed ia te . A nd , o f c ou rs e. A rc him ed es ' r em ar ks in th eIctter accompanying Q ua dr atu re o f till PlJTa6014 s ho w th at h e c on sid er edth e s o ca Ilc d 'a xio m o f A rc him ed es ' o nly p la us ib le a nd r ea so na bly w ellc on fim le d. H ow ev er , e ve n A rc ltim ed es n ev er s ug ge sts th at th e q ue stio nof truth is irrelevan t or that one m ight investigate the co~ quences ofa ss um in g t he a xi om t o b e false.

E uclid 's de fin itio ns h av e freq ue ntly b ee n b elittle d by m od ern c om -m en la to rs o n th e Elenwru w ho lo ok at them from the perspectiv e of them od ern a xio matie m etho d. (S ec , fo r e xam ple , K lein (19 39 ), p. 1 96.) O fcour s e t he d e fi ni ti on s coul d n e ve r f ig u re ina fo rm al d eriv atio n, b ut that isj us t o n em o r e r ea so n f or d en yi ng t ha t E uc li d's p ro o fs a re f orma l d er iv at io ns .T he d ef in jtio ns s ho uld b e lo ok ed a t a s a tte mp ts to m ak e c le ar t he m ea nin gso f t he term s to b e u se d b ef ore a rg um en ta tio n b eg in s, th at is . t o m ak e c le arthe nature of the objects to be studied. That the most fundam entald ef in jt io ns ( e.g . o f p oi nt , l in e. s tr ai gh t l in e) s uc :c cc d o ~ w it h p cn ;o n s wh oalready ha ve so me id ea w ha t the o bjc c:ta in q ue stio n are do cs n ot re allym atte r if th es e d ef in itio ns a re ta ke n to r ep re se nt p re lim in ar y a gr ee me ntsa mo ng peo ple o f p res um ably n orm al in te Uigcn ce . (C f' Z cuthe n (I ).p. II,.)Out to say this is no t to say that the term s are really taken asprim itiv e be ca us e th e u nde rsta ndin g o f the n atu re o f the o bjects p lays aro le in E uc lid's pro ofs. In dee d, I a m in clin ed to b elie ve tha t m ost o f the'lo gic al g ap s' w hic h m od er n c omm en ta to rs f in d in th e Elements arc to beexplained b y reference to this understanding which for Euelid \vas I

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EudiJ' , Elc:mcnta"n d tilt Axi oma ti c Me th o d 2 C } Sd es cr ib ed a de qu ate ly b y th e d ef in itio ns : f or e xamp le , t he u niq ue ne ss o f th el in e o f p o st ul at e I b y th e d ef in itio n o f s tr aig ht lin e ( de fi nitio n 4 ); g ap s in w iv in g b ctw cc nn cs s b y r ef er en ce to d ef in itio ns lik e th at o f li ne ( de fin itio n2 ). (S ec Z cuthe n (189 6), p. 12 3.) I d o n ot m ea n to im ply, tho ugh. th at s uc hrefe re nc es rea lly do clo se the g aps . T he p oin t is tha t thes e g aps e xis t o nlyif Euclid is judged in terms of a m athematical m ethod w hich w as nothis.I conclude, then, that a E uclidcan derivation is an experim ent per-fo rm ed o n id ea lise d p hys ica l o bjec ts , the ex perim en t be in g lim ite d byp relim in ary agrc cm en lS (firs t p rin eip les ) c on cern itlg the n ature o f theo bje cts , s om e o f t h eir p ro pe nie s, a nd th e o pe ra tio ns th at m ay b e p erf orm edo n them . T hus , E uc lid's geo me try diffe rs rad ica lly fro m that o f H ilb ert,in w hich a deriv atio n is a d edu ctio n ac co rd in g to rrin ciplc s o f lo gic fro ma ce rtain g ro ul' o f s en ten ce s called p os tulates . T o 83y sim ply that b othHilbert's GrundlalD' and Eu cl id 's Elements em pl oy t he a xi om a ti c m e th odcan o nly o bsc ure the v ery s ign ific an t differe nce s w hic h I ha ve J UR t de -scribed.:I 1IAVING GlVIlN w ha t I rega rd as a n ac cura te de sc riptio n o f the app ea r-a n ce o f 1~ II c1 it l' lIli/tml'IIls, I w an t to c on side r o bje ctio ns w hic h hav e b ee nr ai se d t o i de nt if yi ng 't he r ea l E u cl id ' w it h t hi s a pp ea ra nc e. I 'r im a ri ly t he seo bje ctio ns dep en d o n II false dicho tom y: either the argum ents of theElements a rc in tu itiv e, e mp ir ic al, in du ctiv e, a nd in co nc lu siv e, o r th ey a rcfo rma l, l o gi ca l, d e du c ti v e. a nd conc lu s iv e . If t hi s d ic ho tom y b e a cc ep te d,it is clear that the ElDlJellts must be placed on the side of the formal,logical, ete, E qually, it ill c lear that w e can define term s in such a w ay asto in su re th at th e EIt",,"u a rc p la ce d o n th is s id c. S om e s ch ola rs s ee m, f orin sta nc e. to b elie ve th at th e u sc o f h yp oth es es (first p ri nc ip le s ) s u ff ic e s t os ho w th at a s ci en ce is d ed uc tiv e- he nc e lo gi ea l, h en ce f or ma l, h en ce c on -c1usive. Such assum ptio ns are liable' to m ake im possible an accurateco ncep tio n o f E uclid ea n argu me nt. T o attain s uc h a c on cep tio n w e m us tg iv e a e lc ar a nd r ea so na ble s en se to t erms l ik e 'i nd uc ti ve ' a nd ' de du ct iv c'an d th en d ec ide w he the r, in the !!C nICgiv en , the term s app ly to E uc lid'sEltmet.ts.C ha ra cte ris tic ally , a n in du ctiv c a rg um en t is d es cr ib ed a s o ne in w hi clla co nc lus io n is re ac hed a bo ut all o f the m em be rs o f IIc las s o n the ba sis o ran exam ination of som e of the m em bers (the in ductive sam ple). S inceEu cl id 's a rg um e nt s f or g en er al p ro po si ti on s i nv ol ve t he e xam in at io n o f o n lya s in gle c as e. to c all th em in du ctiv e w ou ld b e to c as t d ou bt o n th eir s ou nd -n es s. F or , o rd in ar ily , th e p la us ib ility o f in du ctiv e a rg um en t is th ou gh t toin c:rc as c: w ith the siz e o f the sa mple . U sin g the w ord 'in ductiv e' in this


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"i":':!'(:;'~~': ; i '/ T r ~ - ; ' i t , " ' 4, {.:f..aQ6}.!J~uPJui~I"'.. ' ,- ' - s . .~ 1 i S ~ '~ , t f . , i , :.. :~t~~~'":'1" ~ fq ~ :~ ,i~m.qldbc:~ng to call Euelid's arguments inductive; and in so far: ; " : : . : . ; u J ~ u: ~ 4 . ~ t ! f i c sthe inductive wi th the empirical, i t would b e wrong to;:;\ ,;-:c:alIj~cin:empirical also..",~:.;i~r~inc:;~~notion of the empirical seems to be involved in the standard, ::aiiilr.lstC~u~d in, e.g. Eves and Newsom (1958, pp. 5-9) or Reldemeister, .' '\l' "t. (.949, pp.,u-aa) o Greek mathematies with it s empirical Babylonian

" , :: ' '. ani l'Egyptian predecessors . The Babylonians and Egyptians arrived at, ",'~ (ormulu by mcuurcmcnts and ClCImination of cases, p~dtlrcs

, obviously foreign to Euclidean mathematics. The contrast here is of great. iinp,rtance, but it docs not tell the whole story and should not beover-emphasised. SzabO (1958), (or example uses it to call Greekmathemat ics anti-empirical and to connect Greek mathematics wi th the

. . , .anti-empirical epistemologies of Parmenides and Plato. Explaining the'non-empirical clJaracter of mathcmatics requires, however, no reference toexternal forces since this charactcris tic is intrinsic to mathemat ics. Whatdistinguishes mathematics from other sciences is that empirical obscrvationis seen to be indecisive in establishing certain general laws.

Consider, (or example, the question, 'What is the largest col lect ion ofobjects subject to combinatorial manipulations corresponding to addition,subtraction, etcJ ' There is no largest because the addition of one moreobject tosuch a col lection would not hamper a person 's manipul3ting theobjects mathematical ly. On the other hand, there arc col lect ions which nohuman being can in fact manipulate, even mentally. This ci rcumstanceis indicative of the di fficulty of formulating an ari thmet ic just in terms ofempirical observations. The desire for coherence more or less forces uponus the ideal ised extrapolat ion of the sequence of integers . And, al thoughinfini tely many members of th is sequence arc not even visualisable , usc ofthe sequence makes possible the formulat ion of general and simple lawsgoverning ordinary arithmetic manipUlations.

In some cases mathemat ical results are incompatib le wi th aU possib leobservation. Any direct measurement of the side",C a square will yield aresul t commensurable with the result of measuring the square's diagonal ,although it is an elementary mathematical truth that diagonal and sideare incommensurable. This mathematical truth and others like it wereundoubtedly important factors in the development of Plato's philosophy.But acceptance of the truth docs not depend on acceptance of an anti-empirical philosophy. For rel iance on observat ion makes i t impossib le tosay what the rat io of side to diagonal is , s ince the results of measurement...ary . Acceptance of a rat io given by just some observations would b e arbi-trary in a way that, given the proof, acceptance of incommensurability isnot. Moreover, the incommensur3bil ity of side and diagonal explains the

ElldiJ', Elements S l id ' /U I A x io l R ll '; e MltIJotl 29 7variations in measurement results and therefore helps to systematisecertain empirical facts.

The non-cmpirical character of Greek mathemat ics is , then, expl icablewithout refcrence to anti-empirical philosophical views. The very attemptto formulate mathematical generalisations leads to idealisation whichgoes,beyond empirical observation. However, there arc degrees of idcal-isation. Consider, for example, the gcol'l. 'etrie notion of a line. Euclid givesan abstract, apl ,arently I 'la tonic defini tion or l ine- 'A l ine is brcadthlesslength' (definition 2, Book I). But the only lines used in the E/II",,,II arestraight or circular. Aristot le seems to have bel ieved that al l l ines arc com-posed of ci rcular and stmight segmcnts ( D I C II I/ O , 268bI7). Later Greekmathematicians extended their working conception of lines only by addinga few other mechanical ly construct ible lines to thci r repertory. However,their conccption always remained within the bounds of the intuitivelyimaginable, hence, closely tied to the empirical. The separation of themathematical conception of a line from the intuitive one was a gradualprocess which began with the discovery of analytic geometry and thecalculus and ended, perhaps, with the general delinition of continuity.In terms of such definit ions one can reason about lines (e.g. space-fil lingcurves) in ways which completely transcend intuition.

I do not wish to claim that intuition has been banished from modernmathematics. It has not. Nevertheless, it is generally agreed that theelimination of intuitive notions ( rom the treatment of certain basic con-cepts is a-if not the- fundamental achievement or modern mathematics .I f ind no evidcnce of this approach to mathematies among the ancients.Indced, the character of Greek mathematical reasoning, which I havedescribed in the precOOing section, seems antithetical to such an approach.It therefore scems justified to say that Greek mathematics is, by mathe-matical standards, empirical. If, however, primitive mathematies orAr is totelian biology is taken as the standard, Greck mathematics is notempirical.The characterisat ion of the Elellle,,'s suggested in the preceding para-graph might seem incompatible with the obviously deductive nature ofthe work, but here again we must specify what sense we are giving to theword 'deductive'. The Elellll"ts is deductive in the sense that its pro-positions are derived from first principles; yet normally to call a workdeductive is to imply that the derivations arc logical in character. But, as Iargued in the previous sect ion , Euclid 's derivat ions arc not. In so arguingI do not mean to deny that mos t of Euclid's arguments presuppose certainprinciples of logic at one point or another. Itis a commonplace that pcopleusc logical principles unconsciously and correct ly . However, logically


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Z 9 8 1_ AI.,llercomet reasoning must be distinguished from reuoning based on logic.Euclid's reasoning is logically correct , but there is evidence that he had nounderstanding of the logic of his time. For, as DeMorgan remarks, 'Euclidmay have been ignorant of the identity of "Every X is Y" and "EverynOl-Y i s nol-X" for all that apppean in his writing; he makes the one tofollow from the other by a new proof every time' (quoted in Heath (19%5):n, p. %9) .t

Thus, the sense in which the EI,,,,,nll i s deductive does not precludeits being, in a sense, empir ical. However, it docs not seem correct to callthe Eltmtnl l iogical or formal in a sense approximating that given to thesenotions by modern logic. One might easily be misled by the fact that in alarge number of cascs Euclid's arguments can be transformed into oncswhich resemble arguments obviously sound by the principles of modernlogic . (How close the resemblance is wi ll depend on how sympathet ic thepenon making the transformations is.) However, th is possibi lity docs notshow that the transformations either preserve or rcvc:Il the 'inner workings'of the original argument. It can be shown, moreover, that the propertyof modern mathematics which makes logic directly applicable to it isabsent from ancient mathemat ics and, therefore , that the transformationof Euclid's arguments into modern logic is misleading.

It seems to be agreed generally that the idea of s tructure is essential inmodern mathematics. P. Dernays (1959, p. 2 % ) has described mathematicsas 'the theoretical phenomenology of structures' . My eontention is that thisconception of mathematics does not apply to ancient mathematics, inwhich structure plays no rolc. For the notion of structure has its roots inareas of mathematics which have no genuine analogue among the ancients:abstract algebra, analytic geometry, r e a l number theory, and mathematicallogic. It is not possible to describe here the evolution of the idea of struc-ture in the modern era. An important step in it is the realisation thatEuclidean geometry has an interpretation in the universe of the realnumbcrs-a realisation made possible by the c:xistcnce of analyticgcometry and making possible the unificat ion of two apparent ly diversesubjects, the study of space and the study of l umber. One effect of thisunification was the clear conception of spacc as a class of points and ofgeometry as the s tudy of the relations existing between them. Given theinterpretability of the class of points as the class of real numbers (pair s orITh~ diatinction bclWftn loaicall,.coned reuoninl and _inabaIccl on loaic a h o w sth e inconduslvcnal or suw'. (1964. pp. 4a08) arpmcnt thatlh1llllO or Indirect proofia a .ian or the non-lntuit i\'C character or Greek InIthanatia. One cannot haft anilnlac or ",hat Ia not the cue acomctrlcall" but ono can ... an I m a a e or dlqram Coshow that s o m e t h I n a Ia icwuclricall, impoAible, .. In Socrata' al)llllDllll In the M_Of in the proof or I, 6. -

Hucl id ' , Ulcmc:nl. am I ,Ill ,/briolllll,it M elh oJ 299triples of r e a l numbers) and of the geometric relations as relations betwccnrcal numbers, it is a short step to the view that geometry is nothing but thestudy of any class of objects and rel3tions on those objects isomorphicto the class of points and the or iginal geometric relations . In other wordsit is a short step to the view that gcometry is the study ofabst ract structure:Logic made poss ible the per fection of this view by providing a thcory ofstru~ure-prcacrving inference and a set of formal rules enabling one to~erive fro~ a set ~faxioms ~ll and only. the sentences true under everyIRterpretataon makmg the axioms true (m every structure in which theaxioms are t rue). I t i s important to realiss:, however, that these formal rulesdepend on the notion of structure, interpretation, or modcl for theirjustification.

I do not bclieve that the Greeks possessed the notion of mathematicalstructure in this sense. The descriptions of mathematics which havesurvived from antiquity never employ notions like that of structure. Themathematical practice of Euclid and other mathematicians of antiquitysuggests strongly the defini tion of geometry as the science of magnitudes(meg,lI,s). The gcometrical magnitudes which Euclid studics arc not' st ructural objects ', but, as I have argued, the in tui tively perecived spatia lobjects which arc cl,araetcrised in his defin it ions. There is no indicationthat he considered these objects (e.g. points ) as eonstituting a system orstructure. Had he so conceived them, he might equally have had the idc3of an isomorphic system; and had he had this idea, it is hard to sec whythe Ekmtlill should contain so many 'logical gaps'. For it is precisely thedevelopment of this idea which made it possible for the moderns to dis-cover thcse gaps . To attr ibute an understanding of abstract structure toEuclid or hi~ contemporarics is to obscure, i f not to obl iterate completely ,the revolutionary character of ninetccnth-ccntury mathematics . Theabsence of an understanding of mathematical structure among the ancientsmakes itmisleading and probably impossible to call Euclid's argumentationlogical or formaL For it is in terms of structure that the idea of logical orformal argumentat ion is given a precise sense . '

l'crhaps the major obs~cle to an aeceptance of the interpretation ofEuclid's arguments as thought experiments is the belief that such argu-ments cannot be conclusive proofs. In particular, one might as k howconsideration of a single object can establish a general assertion about allobjects of a given kind. Part of the difficulty is due, I think, to failure todistinguish two ways of in terpreting general sta tements l ike 'Al l isoscelest riangles have thei r base angles equal' . Under one interpretation the sta te-ment refers to (talks about, presupposes) a def inite totality- that is, theclass of al l i sosc:cJes t rianglca-and i t says something about cadl one of


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30 0 11mMil"them . U nder the o ther in terpretatio n no such definite to tality is pre-supposed, a nd th e s en te nc e ha s a m uc h m ore co nd itio na l c:h ara c:tcr-'U at ri an gl e i s i a os c: cI es ,i ts t wo b as e a ng le s a re e qu al .' A p er so n wh o i nt er pr et sa ge nera lis atio n in th e s ec on d w ay m ay h old tha t th e ph ras e 'th e dau ofall iso sce les trian gle s' is m ea nin glc ss be ea us e th e n um be r o f is osc elest r iang le s i s ab so lu tely i ndete rmina te .

I n m od er n m ath em atic s t he f irs t w ay o f in te rp re tin g g en er alis at io ns isc us to ma :y . T he r ea so n is that . in co nn ec tio n w ith a sy stem o r o bjec ts , it isn atu ra l :l nd a pp ro pr ia te to t re at a g en er alis at io n a s a n a ss er tio n a bo ut a llo bje cts in th e ay ate m. A lth ou gh m os t pro ofs o f g en era lis atio ns in vo lv ec st ab li sh in g t he a st lC rt io n f or a p ar ti cu la r b ut a rb it ra ry o bj ec t, n o p ro bl ema ris es in c on ne ctio n w ith th e q ue st io n w he th er th e o bje ct is a rb it ra ry . F ort hi s i s j us t t he q ue st io n wh et he r t he p ro o f i nv o lv es a tt ri bu ti ng t o t he o bj ec tan y p ro l,erties w hic h ha ve n ot be en pro ve d to be lo ng to all o bjec ts o f thelim e kin d. F orm alis atio n m ake s this qu es tio n e ffce tiv ely de cid able . Inc on ne ctio n w ith i ntu itiv e a rg um en t th e q ue st io n is m uc h m or e d if fic ul t,s in ce a ny i nl ui th -e Jy p er ce iv ed o bj ec t h as p ro pe rt ie s d is ti ng ui sh in g i t r romo th er s o f th e s am e k in d. T he I IC CO ndin te rp re ta tio n o f g en er alis at io nsp er ha ps c as cs th e d if fic ul ty s om ew ha t a nd is o bv io us ly m or e s uita ble f orm athe m;ltics in w hic h 5ya te ms o f o bje cts p lay n o ro le . U nde r it a In oo (c an b e lo ok ed o n a s g iv in g a p ro ce du re ( or v er ir yin g th at a g iv en a ss er tio nh old s in a ny p ar tic ula r c as e w hic h m ay a ris e. L oo ke d o n in th is w ay , a p ro ofo f a theorem is very like a proor o f a problem w hich gives a method ofc on st ru ctio n to a pp ly in a ny p ar tic ula r c as e. (Cr. t he a cc ou nt of finitaryrea so nin g in H ilbe rt a nd D crn ays (1934), pp . 32-45.) O ne ground forin te rp re tin g E ue lid ea n g en er alis atio ns in th e s ec on d w ay is th e la ct th atm ost o f the thco rem s in the Elemml l a re s ta te d a s c on di ti on al s en te nc es( e.g . p ro po sit io n 6 , B oo k I) . T he w or d 'a U' (pos ) occurs very r ar ely in th eEltmtnls a nd o ft en h a s to b e tr an sla te d 'a ny ' ( as in p os tu la te I). However,I thin k it w ould be w ro ng to p la ce m uc h re lia nc e o n s uc h lin guistic fa cts ,s in ce a lm os t c er ta in ly E uc lid 's t er min olo gy w as a m atte r o f c on ve nie nc er at he r t ha n o f a c on sc io us p hi lo so ph y o f m a th em a ti cs . T h e i nt er pr et at io no f g en er al a ss er tio ns in th e Elenunls m ust depend o "a gen eral accounto f th e w ho le w ork . T he ac co un t g iv en he re I C C 1 1 l S c omp at ib le o n ly w it ht he s e cond i n te rp r et at io n o f g e ne ra li sa ti on s .

E v en i f t h is i nt er pr et at io n o f E uc li de an g en er al is at io n s b e a cc ep te d, i t i ss till p os si ble to d en y th at a th ou gh t e xp er im en t c ou ld e st ab lis h t he m c on -dusively. For there is a sense in which one canno t be certain that ac on str uc tio n o r v er if ic atio n p ro ce du re c an b e c ar ri ed o ut in a ny c as e f ro ms eein g it c arrie d o ut in o ne ease, W hat needs to be said here seem s to beth at c on cl us iv cn cs s is a tim e- de pe nd en t n otio n. W e m ay if w e w is h i ns is t

Eudit/'. Eleanen.. ad 1MAxiDmtJlicM,tW 301th at o nly a rg um en ts m ee tin g th e s ta nd ar ds o f m od em m at hcma tic allo gi eare co ncluah 'C -in w hich case; E uclidean argum en ts are not. In fact,h ow ev er, ev ery on e a cce pte d E ud id's a rgu me nts a s s ou nd u ntil th e n in e-teen th century, and it is hard for me to make a e R S C of the claim thatthey w ere w rong. T he lim its o f in tuition arc reached at higher lev els o fm ath em lltics tha n the G ree ks a tta in ed-fo r in stan ce , in co nn ec tio n w iththe attem pt to define 'curve' or 'continuous'. To suppose that the in -c on clu siv co cs s a t th es e le ve ls s ho ws th at in tu itio n i s a lw ay s in co nc lu siv ei n m a th em a ti cs i s t o m a ke a n u nj us ti fi ed g en er al is at io n . 'C o nc lu si ve nc ss 'can be defined in term s of m athcm atieal practice as w cll as in term s o f ama th ema ti ca l i de a l ( c f. r cmark s 1-23 i n Wi t tgens te i n (196.. .3 IN JUSTIFYING h is c on ce pt io n o f G r ee k m a th em a ti cs , S za b6 r el ie s m u chm or e h ea vily o n G re ck a rith me tic th an o n g come tr y. I nd ee d, it i s p er ha psm ore ac cura te to s ay tha t h e in sis ts o n the m is lea din gn cs s o f g eo me try a sa b as is fo r in fere nc es ab out G ree k m athe ma tic s (s ee (1958), p. 123). Th etc na bility o f th illl,o si tio n is n ot a t a ll c le ar . F or t he g come tr y o f th e G re ek ss urp as se s the ir a rithm etic n ol o nly in q ua ntity-th c b ulk o f w ork d on e-b ut a ls o i n q u al it y. "Inc Gre ek s d ev el op ed o nl y a sm al l p or ti on o f a ri thm et ics ys tem at ic al ly ; a nd t he f ew c omp li ea te d r es ul ts wh ic h l he y o bt ai ne d, s uc ha s th e th eo re m o n p cr fe ct n um be rs ( pr op os itio n 36, nook IX ), ap pe ar asis ola te d in sig hts . I n c on tr as t, th e Elemell" c on ta in s a r el at iv el y e om p le ted ev el opm en t o f e lem en ta ry g com et ry , t o wh id , A rc hime dc s a nd Ap ol lo n iu sw e re a bl e t o a dd a l ar ge n umb cr oCc omp le x t he or em s , M o rc ov er , a s h a s o ft enb ec n p oi nt ed o ut , t he s ta nd ar di sa ti on o f m a th em a ti ca l t ec hn iq ue em bo di edin the Eleme,," is prim arily a ge om etrisa tio n o f m athe ma tic s. In m an ycaac:s, n um be r t he or etic r es ul ts h av e b ee n r cc ut a s g eome tr ic o ne s. W h at-e ve r th e e xp la na tio n C or t hi s f ac t, th e f ac t it se lr s ug ge sts th at, f or u nd er -s ta nd in g pro or in G ree k m athe matie s, g eo me try is m ore im po rta nt th anarithmetic.

On the other hand, it seems to have been IIg en er al ly a cc ep te d p hi lo -s op hi ca l b el ie f t ha t a ri thm et ic i s s om eh ow s up er io r t o g com et ry a nd , i nd ee d,to an y o th er sc ie nc e. I pre fe r to po stpo ne dis cus sio n o f this be lief u ntil Ih av e c om ple te d a d et aile d e xamin ati on o f S za b6 's a na ly sis o f a rith me ticproo f. Fo r I thin k it is clear that the analysis w ill n ot stand UPi and if it"ill n o t, e xp la na ti on o f o th er p hi lo so ph er s' c on ce pt io n s o C g eom et ry a nda rith me tic w ill d im in is h in im po rta nc e. I w ou ld l ik e to c on si de r S za b6 'sa na ly sis in terms o( the example which he has used on at least twooccasions (1958, pp . 118-21, an d 1964, pp . 3 9 -42 ) , p ro p os it io n s 21 an d22 o f B oo k IX .Ira ny n umb er oCe ve n n umb er s b e p la ce d t og et he r, t he wh ol e i s e ve n.


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30a 1_ MIIIIInF o r l et a n y n umbe r o r e v en n umb cr a AD ,DC , CD , DE , b e p ut t OBF th er ;I l Ayt ha t t he who le .AE , i . ev en . F o r a ince each 01 th e n um bc nl AD , DC , CD, DE , i,

A B oceven, each ha s a h al f p a rt ; t hu a t he wh ol e. AE , also has a h alf p ar t. D ut a n e ve nn umb er i , o n ewh ic h is d iv is ib le i n two ; t he re f or e , AE is even. Q.E.D.IfIn y even mu lt it ud e o f od d numbe r s b e p la c ed t o ge th e r, t he who le iteven.F o r l et an arbitrary even m ultitud e o f o dd n um be rs , A D, D C, C D, D E, b ep ut to gc thu ; I uy tha t the w ho le it CftD. Since each of AD , B C, CD , D E i.o dd , i f a u nit b e t ak en a wa y f r om each, each o f th e r em ain de R w il l b e e ve n.Thus, t he n umb er p ut t og et he r from t hem w il l b e e ve n .

A c EoBDut th e m ul titu de o f th e u ni ll is e ve n; th er ef or e, th e w ho le , A E , is also even.Q.E.D.S za bO c on tr as ts t he se p ro o fs w it h t he ir ' in tu it iv e a nd em pi ri ca l' o ri gi na ls ,r econ st ru c te d i n (1936) b y O . B ec ke r. T he se o rig in al p ro of s in vo lv ed th em an ip ula tio n o f c olo ur ed p eb ble s. D ec ke r's illu str at io n o f th e te ch niq uei n t erms o f p ro po si ti on ZI m ay be de scribe d as fo llo ws . W e im ag in e them athe ma tic ia n s ayin g, 'L et the n um be rs b e fo ur, s ix , ten , a nd tw o', an dl ay in g o ut p eb bl es i n t hi s p at te rn :

00 000 o.00000Movi ng t he p eb bl es g h'c s a r ep rc se nt at io n o f t he wh ol e:

00000000000.A nd rearranging show s that the sum in question is indeed even , i.e,d iv is ib le i n l \v o e q u a l parts:

00000000000T he pro cess described here is differen t in n o fundam ental w ay from

So cr at cs ' p ro o f i n t he Meno d on e w it h diagrams in the .. d. I have arguedthat the difF eren ce betw een So aatcs m ethod and a E udidean pro of isbasically that betw een spoken and w ritten argum en t. Is there a pro -f o und er d ifF er e nc e b c : t w e e n th e m an ipu latio n o f p ebb lcs jU lt d esc rib eda nd the E uc lid ean pro of de riv ed fro m itl It s ee ms th at pe rha ps the re is,f o r a lt hou gh Eu c li d d o cs d e sc ri be t he mO\ 'C l ll en t e o fl 'C l pond in g t o ch ang in gt he p eb bl es f rom the f i rs t pos it ion to th e s ec on d- i.e . p uttin g to ge th er th enum ben , he docs no t describe either dividing the numbers in tw o orre arran gin g th e ha lv es . It is s im ply s aid tha t th e e ve n n um be rs ha ve h alf

E"did' , Elements "IUI 'M AJdomati&MdW 303.p ar ts ( pr es um ab ly b y r ef er en ce to de fin itio n 6 , o oo k V II) a nd dircc tlyin fe rred th at th e w ho le, tile rc su lt o f pu ttin g th e n um be rs to ge th er, ba s ah alf pa rt-i.e . is ev en ,

O ut w ha t is th e n atu re o f this dire ct in fe ren ce? F ro m th e pe rs pe ctiv eo f m od ern a xio ma tic m ath em atic s the in fere nce is u njus tifie d. F or theinference rests o n the co mm utativity an d uso ciativity o f addition , an dEu cl id p ro v cs n o t he o rem s c on ce rn in g t he se p ro pe rt ie s. T h e t ac it usc o f t hep ro pe rt ie s s ho w s tM t E uc li de an a rit hm et ic r es ts o n in tu itio n as much asg eome tr y d oc s. A dd itio n is II pu tt in g t o ge th e r (1)'IIt/wil) o f c ol le ct io n s o fu ni ts , a nd it w as to E uc lid 'o b\'io us ' th at t he r es ult o f a dd iti on i s in de pe n-d en t o f th e o rd er in w hic h c olle ctio ns li re c om bi ne d. I n th e s am e w ay it was'obvious' to h im th at II lin e fro m a p oin t in stde a c irc le to a po in t o uts idein te rs ec ts t he c ir cle . T he re is n o g ro un d f or d is tin gu is hin g h is g eo me tr icin fe re nc es f ro m h is a rith me tic o ne s o n th e b ais o f m ath em ati ca l 'p ur ity '

S zab O pla ces g rea t e llll,h as is o n th e us c o f lin es in ste ad o f do ts to re -present num bers in the Eltnltlltl. Indeed, he ev en says that, w hen thism etho d o f re pres en ta tio n w as first us ed, 'II new era dawned upon theh is to ry o f ma th emat ic s ' (1964, 1' .p). The f ol low in g q uo ta ti on s i ll us tr at eh is r e aso n in g :T he : um e a c: ct in n sd en o te d b y t he um e l et te rs r ep n: ac :n t od d n umb er s i n t hedemonstra ti on of the fo ll owing the :o rem(Hltnrtll", IX , zz). Le t U8 m3ke i t c le a rth at th e dilT ere nc e be tw een the e ven an d o dd n um be rs c an by n o m ean s b ei ll us tr a te d w i th s e ct io n s o f a l in e , b e cau se any o f t he s c c; ti o ns can b e h a lv e d, y e to f the n um be rs o nly the ev en o ne s e a n be h alv ed. A nd in th e s pirit o f G re eka ri thme ti c n o t e v en t he :IIII;t can b e r ep r es e nt ed a s I IOmese ct io n o f a s tr a ig h t l in eb ec au se th e u ni t is i"rJivi,;6/, wh er ea s t he s ec ti on i s a lw a ys d iv is ib le . E u cl idd o cs no t c a re t o v i su a li se t he t ra n sfo rma ti on o f od d numbe r s i n to e v en numbe r se ith er. In th e de mo nstratio n o f the th eo re m IX , 2Z, f or in sta nc e, w e r ea d:'L et ua subtract on e fro m Iny o dd num ber, A D, D r, rIJ, an d IJE, Ind itbecomes an e ve n n umb er : T h is i . o rJ y a v er ba l s t at em e nt w it hi n t he d em o ns tr a-t io n , a nd no th in g i s u nde rt ak e n t o i ll us tr at e i t o n t he ab ov e -men ti o ne d s c tt io n so f a s tr a ig h t l in e (Il, p. 41).Ob vi ou sl y, b y u si ng p eb bl es , o n ly I IOm ec on cr et e e ve n o r o dd n umb er , s ay

';1',#WII, e te ., can be r cpr c sc :n tcd ,y e t n o i ll us tr a ti on c an b e g iv e n o f t he e v e: no ro dd n umb er i n g e n n td . A lCCt io no f II lin e, o n tile o th er h an d, c an a lw ay s b et he symbo l o f any a rbi tr a ry numbe r . Th i s me ans t ha t t he n ew wayo f r e pr e se n ta -tio n m igh t ha ve been d ev is ed -a mo ng o the r ras on a-by the e nde av ou r 10ach ieve :a h ighe r dcg rcc :o f gene ra l ity (Il, p. 42).SzabO seem s to say both that the diagram s play n o ro le in proo fs like

those under discussion an d that tbe usc o f the diagram s represents II" riv in g f or g re ate r g en er alit y. P re su ma bly , if th e G re ek s th ou gh t th at th ed ia gr am s w er e ir re le va nt. t he y w ou ld n ot h av e w o rr ie d a bo ut m ak in g th emm ore general. In any esse, the greater generality o r lines over dots,


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3~ 1l1liM"ellue mph as is ed b y S za b6 , is h ardly s ign ifica nt. F or m os t o f th e ar ith me ticp ro pn s it io n s i n t he Elements rcf 'er to an arbitrary n um ber of num bers o ro pe rl ltio ns a lth ou gh th e d ia gr am s a nd p ro of s d ea l w ith a s pe cif ic n um be r.Fo r e x amp le , p ro p o si ti on 21, Book IX. d eals w ith th e s um 01 an a rbi tr a rynum ber of even num bers, but diagram and proof m en tion four suchn um be rs . T he in fer en ce lro m the c on clu sio n tha t th e su m 01 lo ur e ve nn um be rs is C \'C nto the c on clu sio n tha t th e s am e is true o f a ny n um be r o fC\'CR numbers sc:cms to be the same in kind as the inference from 'thesum of four, six , ten , and two is even ' to 'the sum of any four evennumbers is e ve n'. D ot h \y ou id n orma lly be c aU ed in tu itiv e a nd c l a s s e dw ith ge om etric in fere nce s fro m a pr op erty's ho ld in g o f o ne fig ure to itsho ldin g of all figures 'o f the sam e kin d'. M OfC Over, e ven the greaterg en er alit y o f t he r cp ra cn ta tio n o f n um be rs b y lin es v an is he s w he n E uc lidr ep re se nts t he u nit ( as in VII, I,a nd I X, a 3, C or ex am ple ).

T he rep re sen ta tio n o f n um be rs b y lin es in ste ad o f do ts is s ufficien tlye xpla in ed by th e G rc :c k ge om ctris atio n o f m athe ma tic s. F or w ith th isre pres en ta tio n arith me tic o pe ra tio ns b ec om e n oth in g bu t g eo me trico pe r: :t io n s o n l in e s j us ti fi ed o r t ac it ly p re su pp os ed i n t he n o n ar it hm e ti cbecks, A dditio n o f on e num ber to an other is exten ding a line by a giv enl en gt h, m u lt ip li ca ti on b y a n umb er n e xte nd in g a lin e b y its o wn le ngth"ti me s, a nd d iv is io n b y tw o , b is ec ti on . P er ha ps th e m o st im po rta nt a ri th m etie notion w hich is given a geom etric guise w hen num bers are re-p re se nte d a s li ne s is m ea su re me nt. O f c ou rs e th e lin es h av e p ro pe rtic s-in fin ite d iv is ib ili ty , f or e xamp le , w hi ch n um be rs d o n ot h av e- bu t 8 0 d oc san y d ia gra m re pre se ntin g m athe ma tic al o bje cts. E ve n the 'e mpirica l'm ath em atic ia n p ay s a tte nti on o nly t o th e p ro pe rtie s o f a d ia gr am w hic h a rcm a th em a ti ca ll y r el ev a nt . M a ki ng i nf erCRc es a bo ut m a th em a ti ca l o bj ec tsu 8i nS d ia gr am s o r i ma ge s b ut i gn o ri ng t he ir i rr el ev a nt p ro pe rt ie s i s m a th e-m atk al a bs tr ac tio n a t its 1 0\ \'C Stl ev el. M a ki ng 8 uc h in fe re nc es is q uited if fe re nt f ro m m ak in g i nf er en ce s to w hic h n o im ag e c or re sp on d8 .

T here is , n ev erth cles s, a s en se in w hic h the a rithm etic d ia gram s d o n otiII us lra te the ir pro ofs a s fu lly a s g eo me tric diag ra ms do . 'l1 1e d cgrc c o fiIIuslrativeness varics am ong the arithm etic dia~ln the case of IX ,21, the d ia gram illu stra te s o nly the a dditio n .o f fo ur n um be rs , n ot th eirh ah i ng a n d r ea rr an gem en t. T h e f ai lu re to i ll us tr at e t he se o pe ra ti on s c anb e e xp la in ed b y a sk in g w ha t s ort o f d ia gram w ou ld illu stra te th e p ro of.T he answ er seem s to be that o nly a series of diagram s co rrespo ndin g toth os e in vo lv in g d ots , a s g iv en a bo ve , w ou ld . D es pite t he s im pli ci ty o f t hec omb in a to ri al a rg um e nt , i ts d ia gr amma ti c r ep re se n ta ti on i s c om p li ca te d.G co:netric argum en ts, on the other hand, are usually m uch easier' tor ep re se n t d ia gr amma ti ca ll y t ha n a re c om b in at or ia l a ri thm et ic a rg um e nt s.

...!:.":

H"t/id', 1~lc"'c:nl.""d tllf!lI~il"""tieMetW 305F er th is r ea so n o ne : m ig ht c all c om bin ato ria l r ea so nin g m o re a bs tr ac t th ang com et ri c a rg um e nt , a lt ho ug h b o th a re e qu al ly i nt ui ti ve a nd e xp er im e nt al .

T h e a ri thm et ic b o o k s of the Eklllellls furnish no go od evidence thatG re ek a rit hm etic w as a le ss in tu itiv e d is cip lin e th an g come tr y. It remainsto d is cu ss t he p hilo so ph ic al b clie f in 't he p rio rity o f a ri th me ti c'. M y m ainp ur po s e w il l be t o s ho w th e i ns uf fic ie nc y o f e vid en ce f or S za b6 's c la im th atth is b eli ef i s c o nn ec te d w ith a b eli cC i n th e a bs tr ac t c ha ra cte r o f a rith me ticre as on in g. S za bO lo ca tc s th e o rig in o f th e b elie f in th e prio rity o f arithm etic in a d is tin ctio n b etw ccn I,islorio, 'e mp ir ic al k no w le dg e o f v is ua lo rig in '. a nd a motlrnna, ' a d is ci pl in e oC tr ue m a th em a ti cs ', w h ic h h e a ll eg esthe P ytha go rc an s m ad e a s ea rly a s 50 0 D.C. (19 58 , p . 1 15 ). T he ev id en cef or s uc h a d is ti nc ti on i s , ir tu al ly n o n. cx is te nt . S z abO r ef er s t o l am b li ch us( 9 , s ec t. 8 c } ) and Ar is to t le (Melophysics, I , 5 , 9 8S ib J f T . ) to a rgu e th at thee ar ly P yth ag or ea ns c on sid er ed g eome tr y e mp ir ic al a nd a ri th me tic tr ulym athe ma tic al. Ia mb lieh us s ay s o nly th at P yth ago ra s ca lled gc om etryI,illoria a nd d oc s n ot giv e an e xplan atio n fo r th e n om en cla tu re . A ris to tlere fer s to th e s o-c alle d P yth ago re an s a 8 th e firs t to d ea l w ith an d ad van ceIIIOt/,tll,olo. H e docs no t suggest any special conno tations of the w ordIIIatlltJlla; thu s. o ne w ould 5UPI'OSC him to be using it in its custom arys en se t o r ef er to a lla cie nc ca in cl ud in g g eom et ry ( se c S ne ll ( 19 2 4) , p p. 7 7-81). n lis us e o f th c w ord is e ven a tte ste d fo r thc P ytha go re an s th em se lv es(D ic ls a nd K ra nz ( 19 56 ), 4 7.n .I). M ore ov er A ris to tle h im se lf giv cs w ha tis u sua lly a cc ep te d a s the co rre ct ex pla na tio n fo r th e P yth ago rc an b elie fin the prio rity of arithmetic: .

S in ce o f t he se p ri nc ip le s n umb er s a re f ir st b y n at ur e a nd t he y t ho ug ht I he y&aW m an y lik en es se s o f th e t hin gs t ha t a re a nd o f th e th in gs th at c om e t o b e inn umb er s r at he r t ha n i n f ir e, c ar th , o r w a te r 10t ha t t he y ca ll ed j us ti ce su ch ands uc h a q ua li ty o f n umb er s, l Ou l a n d r c: aa o na no th er , o pp o rt un it y a t hi rd ; a ndt he y t re at ed e ve ry th in g e la e s im il ar ly , a ce in g t he q ua li ti es a n d r at io s o f m u si cin n um be rs ; a nd s in ce o th er th in gs s ee me d to b e l ik e n umb er s i n t he ir w h ol en a tu r e and numbe rs t o be th e f i n t th ings o l fa1 ll 1 l1 lu rc ,Iheyassumed the c l emen tso f n umb er s t o be t he e lem en ts o f a ll t hi ng s a nd t he wh ol e h e a ve n t o b e a m u si ca ls ys tem a n d a n umb er ("'tinfoil,,;", I, S . I )8Sbz604)86az ) .

I c an no t s up po se A ris tu tl e to be w ro ng i n t his h is to ri ca l r ep or t, f or n oto nly is it p la us ib le b ut a ls o A ris to tle r ep ca ts i t m or e th an o nc e. M o re ov er ,n o go od e vide nce c on tr ad ic ts it. U A ris to tle is r ig ht. t he I 'y th ag or ca nco nce ptio n o r the p rio rity o f a rith metic re sts n ot o n a ny m ath em atica l o rlo gic al in sigh ts b ut o n the n aiv e n otio n tha t ev erythin g, p hys ic al o r c on -

. c e pt ua l, is m adc o ut o f n um be rs . T his n otio n s ee ms to be ne i thcr emp i ri ca ln or m at he ma tic al b ut, r ath er , w ild ly s pe cu la tiV c. I n th e c as e o f m an y la te rw r it er s t he a pp ea l o C s pe cu la tio n a nd o f t he a uth or ity s ee ms to h av e b ee nth e b as is f or p ro po un din g th e p rio rity o f a rith me tic .

u


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306 I_MwIInS ub O f in ds s up po rt f or h is ra dic al d is tin ctio n between a ri thm ct ic a ndg eome tr y in th e f am ou s r em ar k o f A n:h yta s: 'A nd log;l , i ," s ee ms to b e fa r

s up er io r to th e o th er a rts in w is dom a nd e sp ec ia lly to g eome tr y in tr ea tin gm ore clearly w hat it w ishes. A nd 10g;II; ," b rin gs p ro of s to e om ple tio nw he re geo metry lea ves the m o ut' (D ie ls an d K ran z (19 56) 4 7.B ",). S zaM(1 958, pp. 13 0-1) w rites a s if lop'. o bv io us ly m ea nt a rith me tic h er ea nd r ef er s f or in te rp re ta ti on to O. Ne ug eb au er ( 19 3 6) . N e ug eb au er , h ow -e ve r t ak es lop'i," to refer to an algebra very l ik e B ab yl on ia n a lg eb ra .H e a rg ue s c on vi nc in gly th at B oo k IIo f th e Eleme"" is a " tr an sla tio n o fB a by lo n ia n m e th od s i nt o t he l an gu ag e o f g eom et ri c a lg eb ra '. N e ug eb au err ea d~ t he f ra gm en t o f A r eh yt as a s a n e xp rc as io n o f t he d om in an ce o f a lg eb rain W es tern m athem atic s un til 4 00 B.C. T his p laus ible rc adin g m ake s itd if fi cu lt t o s up po se t ha t A r ch yt as m e an t b y p ro o fs s om e th in g l ik e Euc li de and er i' at io ns . F or th er e is n o s atis fa cto ry e vid en ce th at Iogu';," w as everan ythin g but a tec hn iqu e o f c ale ulatio n. In de ed, the ge om etrisa tio n o fa lg eb ra w a s u nd ou bt ed ly d ue t o G r ee k d iu at is fa ct io n w it h m e re ly s uc ce ss -f ul a lg eb ra ic c al cu la ti on . S uc h c al cu la ti on w a s r at io n al ly j us ti fi ed f or t hemo nly w hcn it had been r ed u ce d t o g e ome try.

The su pe ri or it y o f Jog;" ; ," t o g e ome tr is e d a lg eb ra i s p rimar il y t echn i ca l.lAgiJtikt not o nly prov ides so lution s to pro blem s m ore quickly thangeoR letry but also, unless the pro blcm is geom etrical, the geom etricr cp re se nt at io n o f t he p ro bl em i s c umb er som e a nd u nn at ur al . A r ch yt as m a ybe re ferrin g to the se fac ts w hen he s pe aks o f logis';kt b rin gin g p ro of s t oc om ple tio n w he re g eome tr y d oc s n ot. H ow ev er , h e m ay h av e r ea lis ed th atB a by lo n ia n a lg eb ra c ou ld be applied to equations of higher than thirdd eg rc c- cq ua tio ns w hic h c an no t b e t rea ted geome tr i ca l ly .

M o st o f t he c :x te ns h'C work which ha s been d on e o n P la to 's p hi lo so ph yo f m a th em at ic s h as b ee n b as ed o n A ri st ot le 's d is cu aa io n s o f t ha t p hi lo so ph yin the l t Ie tnpl ,ys;u . T he in eo nc lu siv cn ca a o f th is w or k a nd th e e om ple xityo f th e c vid en ce m ak e it im po ss ib le to h an dle th e s ub je ct in d eta il h er e. Its eems c le ar , h o\ \' CV e r, t ha t P la to i n h is l at er p hi lo so ph y a ss ig ne d a s pe ci all Ii gn il ic an cc t o n umbe fl l, b ll t i t h as n o t b ee n s at is fa et \l [i ly s hown h ow t hi ss ~l s ign ific an ce re lates to m athem atic al prac tice . ro r exa mple , n o o neh as b ee n a ble to g iv e p la us ib le m ath em at ic al s ig nif ic an ce to A ris to tle 'srefere nce s to P lato 's g en eratin g n um be rs fro m the o ne a nd th e great lindth e s ma ll. T he c ho ic e in in te rp re ta tio n s ee ms to b e b etw ee n m ak in g P la toa m athem atic al ge nius w ho se in sights w en : fa r ahe ad o f his tim e a nd n otu nd er sto od b y h is c on te mp or ar ie s a nd r eg ar din g h im a s a m ath em atic alam at eu r wh o i nd ul ge d in v agu e s pec ulatio ns . In n either ca se is o ne a bleto m ake a close connection betw een Platonic philosophy and G reekmathemat ica l p ract ice .

- ..)

, .

Hue/iJ'I aemcnta "" J 1MAxiom",ieMc~ 3 t ' /I n t he d ia lo gu es th em se lv es th er e is n o s atis fa cto ry e vid en ce th at P la tom ade a sign ifican t distinction betw een geo metry and arithm etic. H is

c ri ti ci sm s o f m a th em a ti es a t R~pub l k , 5 10 B- S , a n: o bv io us ly d ir ec te d a tbo th g eo metry a nd a rith metic. S za b6 fin ds sign ifica nc e in th e Ep;l Iomu(~ C-~~ w hen: 'Plato m en tioned geo metry in the seco nd place afteranthm ctlc (19 64, p. liS). H ow cver, in this paaaage, as in the s im ila rR e p u I J l i c S 2I D- s3 4A , g re ate r i mp or ta nc e s ee ms to b e a tta ch ed to s uc ce ed -in~ sciona;s- F ?r .in ea ch case. then : is a p ro gres sio n to wa rd a highestselencc-dlaleetic m the R~publ l t , I 18 tr o nomy i n t he Ep;IIomis.1

L ik e P la to , A ris to tle te nd s to tr ea t g eo me tr y a nd a rith me tic a s b as ic allyt he &_ am e .D oth s cien ces de al w ith o bje cts w hic h a re n ot ph ys ica l (Meta-P ! ' > : s t C S , III, 2 . 9 97b35-998al ; XIII, 3 , 1 07 8a z{ -z 6) b ut arc immovable(Ibid. I,. 8, 9 89 b3z-33); both proceed by proofs from first, unpro venassum ptio ns: and both an: co ntrasted w ith m ore physical aciences-h a,? ,o ?ie s in th c c as e o f a rith me tic (Po s t e r io r A"" l y ti a , I, 2 7, 87a33-34),opnes an the case of geome try (PI 'YI;u , 11, 2 , I 94 ll 7 -1 z ) . Howeve r, A r is to tl ed ~e s s ay . th ~t a rit hm etic is p rio r. to a nd m or e e xa ct th an g eome tr y b ec au searlthmeti~ IS b as ed o n fe we r thm gs ( P o s te r io r A"o ly t ;u , I, Z 7, 8 71134-37:M~tapl 'Yl lu , I. ' .z , 9 8 zaZ6 -z 8) : H e i ll us tr at es wh at h e m e an s b y s ay in g t ha t :I~lnt has position but an anthm etic un it docs not. W hat he says ean beI nte rp re te d in te nn s o f A ris to tle 's th co ry o f s cie ntif ic m eth od a nd in termso f ~ is ~ eta phys ics . A ~.rdin g to A risto tle, s cie ntific pro of is s yllo gis ticd ~ rav at lo~ f~om f ir s t p r i nC Ipl es ~p rc s se d a s c a t eg o ri ca l p ro p os it io n s . B e cau se~rathmetlc IS b as ed o n. f ew er. th mg s, th e f ir st p rin cip le s o f g eome tr y w illI n~ lv c t ~c t er :m s o f a ra thm et lc p lu s o th er s. e .g . ' ha vi ng p os it io n '. A lt ho ug ht ! ' 1 S ~o ct rl ne g lv ,: ,a cn sc t o t he n o ti on o f t he p ri or it y o f a ri thm et ic , i t b ea rs n oS Ign ifica nt rela tio n to th e m athe ma tica l fa cts . F or even t he s im pl es tm athem atic al a rg um en t o utrun s the p ow er o f the s yllo gis m. M ore ov er itis an aaa um ptio n o f A ris to tle's the ory o f pro of that pro oC s i n arithmetieand geom etry arc the sam e except for the particular term s w hich they~ ploy . A ristotle d~ distinguish, ~o w~er, between the objects of: l r&thmetlca~l t l the obJCCL~of matl i llmatu:s. Ih at i s t o s ay . t hc g cume tr ic ia npa~. atten~~~ .t? m o~ .pro perties or physical bodies-C or cxam ple,pOS it io n , d iV ISi bi li ty , s o ll dl ty - tl la n t he a ri thmet ic ia n (MdoPI 'Y1 i a , XUI3 , 1 07 8a 1l- 12 , z l- 26 ). T his d if fe re ntia tio n o r o bje cts is , h ow ev er a m atte ;o f. de gre e a nd is n ot r efle cte d in th e m eth od s o C reas on in g us ed i~ the tw oSCIences.

T he b clie r in th e p ri or ity o f a rit hm etic o ut ill8 ts t he f ou rth c en tu ry D.C. ~b6 (1960. pp. 86-93) ~ lCCta Plato'. calling IpKe tbe object or a baatan! rcuonillg(7'1_, saD) wl t~ ~ dupa"' aement or poIIICtry. In r lld. there ia 110 sisnllicant re-lation bet_ Euc lid. B"mc try AndtbeIlud)' or epace. See Jlmmer (1960), pp. 13....


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30 8 I t n I .1/"tlItrand is ret lffi rmed by many authors in the Christian era . However, theseauthors give no satisfactory explanation for their belief'. It seems very likelythat they hold the belief as an unquestioned dogma based on Pythagoreanand nco-I' la tonic principles. For examplc , Procius writes: 'The ancientssaid that geometry is a par t of the whole of mathematies and has secondplace afte~ arithmetic, being completed and determined by i t; for every-thing expressible and knowable in geometry is determined by ari thmet iclogo;' (1873, p, 48). Proclus seems to connect the pr iority of ar ithmeticwith the existence of incommensurable geometric magnitudes. Relationsbetween such magnitudes were inexpressible for the Greeks since theylacked satisfactory notation for them. But, given all the impressive resultsconcerning incommensurables obtained by Greek mathematicians, thereis RI) ground for saying that they are unknowable. Proclus writes as aperson aheady convinced of the superiority of arithmet ic on non-mathe-matical grounds.

Jlroclus also repeats Aristo tle 's explanation for the priori ty of arith-met ic, sh"wing no awareness of the connect ion oC the explanation withsyllogistic (.873, p. 59). In general, however, he treats arithmetic andgeometry 3S being very sim ilar. He repeats a standard contrast of arhh-1I11:Iiewith h:'lrlIIClnyand or geometry with spheric 1873), !lp. 59,35-36).Elsewhere (1873, pp. 8-9) he says that both arithmetic and geometry aresubordinate to universal mathemat ics. At another point 1873), p . 60) hesays that, of theorems common to both sciences, some arc transCerredfrom geometry to arithmetic, others from arithmetic to geometry. Proelus'sstating of these similarities further strengthens the impression that hisbelief in the priority of arithmetic is a matter of philosophical inheritance(or which he personally had no satisfactory grounds.

The Greek belief in the priority of arithmetic seems to b e based onphi losophical ideas and preconcept ions rather than on an examinat ion ofmathematical argument. It docs not provide any basis for Szab6's accountof the history and nature or Greek mathematical argument. The nownearly st:mdard account of Euclidean proof liS intuitive and empiricalmust be maintained. "TIl l U"i t 'ers il y o f C I, ic a go

REFERE:-;CESDECUR,O. (1936) Die Lelue \'011\ GaGen uad Unpnden im neunten Duch del' Euldi-dildlen Elemcnte. QUl I kn und S,tuJilll aur Gadlkht. tierM"rhcmG'ik, A.r_",k,

aI I4 Ph)'r i I t . 3, D, 533-53DERNAYS, P . ( 19 5 9) C om me nts o n Ludwig Wittacnatein'. n.",.,," on 'ht FOllnd,,'iolll 01lIf",M",,'iu. Rorio:a, I-U.

Euclid', Elements a nd ," . A Nio ma ,ic 1II"W 309DIIUI, H. lind KRANZ,W. (1956) Die P,"Plelf'. tic, V",MIIt,,,llk,, Vol. I,8th cd . Derlin.Ens. H. and NIWIiO." C. (1958) A n In ,ro Ja cIi"" ,,, 'h t F om ula rio n. aI I4 F tmtI " . . n t ' "ConaJl" o J M","-",iu. N ew Y or k.HASS I, H . IUldSelf 01.&, u. (19:18) Die Orundlaaen Krisis der griccbischen Mlllhmullik.KfnrI-SrIlt/im,33, 1-3...IISATII, T, (19:11) A I l i, , 1WJI 01 GrIJ!A 1I(""II",,,,i" \ '0 1. I , Oxford.HSAnI , T. (19Z5) TIll TIIi,tn Boolu o/lltldJ4'. B/~mt"", znd cd. Cambrid HILBERT,D. and DI!RNAYII, P. (1934) GrundlllgClIde r Mall,emn,iA Vol. I. Derlin.1 A . , D U a r u s . D t V i' a p ,l hD g or iC tl . ed. 1.. Deubne, (1937). Leip&ig.JA.I'IEII, M . ( 19 60 ) Conu,,, 0/ S/Hl". N ew Y or k.I(UIH, F. (1939) EleRlell'a,)' Ma"It_ria Ir on , a n AJL"""ttI S,,,,,tlpoi,,, Vol . a , NewYork.N'EUOEllAUEII, O. (1936) Zu, gcomet,i.chcn AI~br:l. QIle/ltll Imtl SIIIt/i,,, all' G"dJidl lede, ~1"""RI,, ' ;k, A " " , , , , , , , , ; , , 11,.. 1 PI , y JiA , 3, D, Z-f5-S9.PO INCARE , H . ( 19 :1 9 ) TIll Fou,t/a,ionr of Sdmu. If.O. D. 1I.llled. Ne... York.I ' R O C U I 8 . In P r j , , , , , , , , EuditlU EltRl'lllOrum U",m Co,"u,,"'a,ii, ed. O. Friedlein (1873).Leipzig.REiDIlMBlBI'Iln, K. (1949) D". e,''ac'e Den"e" de, G,i/"". Hamburg.ROS8, \Y. (1949) A,;'loIl., Prior a"d POI'~riorAlltlly,iu. Oxford.SalOl.&, H. (1930-I) Die AxiOltUltikdc. Atten. 814"., la,d~"',d,.PI ' i/O l 0 l' lI i e, . . , : 159- 78 .&ELL, D. (1914) Di t Au,drlitk./iJ,

Mueller, Euclid's Elements and axiomatic method

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Transcript of Mueller, Euclid's Elements and axiomatic method