The first Stage of the Idea of Mathematics, Pythagoreans, Plato, Aristotle

NIDOIT oss Pr. Au 3 The First Stage of the Idea of Mathematics: L- 3.185 Pythagoreans, Plato, Aristotle PETER H. NIDDITCH 1. INTRODUCTION 1.1. The sense to be carried by the frame ‘the idea of” in the title of this paper resembles the sense that frame has in Collingwood’s use, in The Idea of History (and elsewhere). Collingwood's practice, though with a different terminology, had a long line of precursors, including Whewell’s in his Philosophy of Discovery: Chapters Historical and Critical. Collingwood painted incisive portraits of a select succession of historical conceptions, i.e., of historiographies of historic moment; but his depictions were not passive replicas, for though he aimed at exhibiting the distinctive traits of his subjects, he vigorously engaged in revealing their deficiencies as perceived from his own perspectival center. The term ‘idea’, then, in the frame-phrase ‘the idea of’, in the context of philosophical history, may be employed for a notion considered extensionally and intensionally in conjunction, or even in combination, in critico-historical chapters (note the order within the epithet): extensionally, in reference to reflective articulations of the notion in question in their historical sequence and connection, and intensionally, in reference to the nature of that notion as conceived and critically utilized by the commentator from his or her own methodical and regulative viewpoint. 1.2. The treatment here of the initial, ancient phase of the idea of mathematics is selective and abbreviative in several ways. (i) It is restricted to so-called pure mathematics (arithmetic, geometry, etc.) and so far as possible disregards mathematical physics and other fields of so-called applied mathematics. Work on the idea of applied mathematics in antiquity would be a different inquiry. (ii) I do not claim to be covering all the features of the mathematical philosophies! of the Pythagoreans, AN |) MANDES STUDIES Suri rossa RY YOU. WN | Plato, and Aristotle, nor, consequently, do I claim to be providing a fully balanced account of these philosophies. On the other hand, broad notice of their major preoccupations and characteristics is an objective, within the limits of a concise,

a7bstracAtURgAroALuIpAs.dl.that4,14 thtehimsartehqeumiarteimceanlt.comunity.2.PYTHAGOREANS ptirtoup(tieh.c,g.sa,hnaldgoexwbisratfeonrctech,altachnuedlufstohlteoirwpoihrnysgaincfise)az.tautrieosn.s”.tdhireaeogsrouynpaalobsooiufttiaomsnaqstuhoaremeaafwthiiocramsaletsisaoisdnesaowhfhaotvlhee.“Puyntihta”golrengatnhs. egyoorcIcisbvwlpsrfthefuenxouanhirmtotipgraosrudel,pmsivttet,ndho)crhuuiasisdenetntbaacn|cislGooehlbu1Ttt(awn(staroe.clihrdebifvofp3ssyca)aq)tblcu.nrmt,eromgsusnMoaweeifrcAnTAbioiklptlueiacpsruethfwlaocomntreyspuchtemlfad,nlsoechoiuttayearpyibcdornmuf-tesrhiinel,ovatldwchxsmiysiefmtagohurt,nsn,himlthisduacpeeictoceso(trrmhppiaykp.atmdbechvsornatdoysue)ntalebihywoilsnnOctaelcrsgpottdomosgwiyyearnsvpuarin,fclgahwsonleuldcecoticirtpabsaeyoemfnafvuzyiittrs,cplpsrlkdn(,uoheethawatcramlln’ndiv,)ehfhyroatinsslb:aludel,ehcnr(fiHnpys,eofagecattgimlbGopdis,v.hlhraaeynfrfcogI.ctplltiwebusrfhtapdoswt.I,mildIeochhfaro-nlo,ty.csmrieudksee,plufiactrcgmmaponpeilefsighahtm;scnalpufob,gtxnehacaesiitdylrnobr-sdlsaeguralneiodtovtIm(sbrscvgenblaa.bfehmikiynrsicpnjcsgatimaoypso1eeuefp)nvttgnrwsocnoheliacilhgstdsrprfnnTyblaowo(chtsfcsiirdvpgwnoehceccfaolriritbfs(ga®emhltiooexcrtuunnaflvlcitameohbtsrtryeallcdfjaeht,sopbncmiuhmeptwtatr*,iahlfosk-ndrcictmhhnoetnsvmuhicplreaef.tmtacigtsnnoeihucilrvhednlgsatocsa|ua,oeurtbnnohlmauenfg1shduyctpirtnidwfkcaeolhinrfemcA:tcifornayeipayrxlnsidotwrmucda.io,hapnetwsrbnntiymodtfrliltslrcea¢ehyeuumms.i.ortcrtpngeagsfapyht.oandige)cipl)r,opIsuntnaees.itl;ogdymwt,hacfnublgrsaohodpdcuiyelktfsnedf:lhracuioyhiexttrlnovwas.u?ao.eeitxnlcwhnairurpfisOtlTveAldyhPoerrauwheo,ntmylbseixpnadhrdgagfeseiwciygt,auslrkdeeiowhdbclsaperennt,irydalftg,;srh.-gnof-prw2Zigslduftmcon.aeoSprgls1hetiunfmay.xrotcgdhbnamisruet.,toAbnsalgimedcrytfhuaoSi2yAn(ertsTufdv.awy2cpPbnei3(rse)htonm.adcwpliesh)xfoNtuWTnicl.ewvAiasuhtdobnelcvP'gamrdUsbicoep,ftyllnauo,ecmylrpithsftoagvednrAuyAlctekai.ga,oms)fhbctdrn;naoale.ieptgml,dyn;asiorhilgk,sdnaefocptsvmhdy,noairaltıs’eudlfohwin(toag'fceihd;lktunsmioba,cderftıir)oncsld1iuhpvg-stowdiencyhmuADasr,tyeodaGilnsbhLtuacpmoeslib,nrgpiafAdometeys.nPuprcGfa,ttioynmwdlskebhpaTEtirsylPbpchwason®eydhrPi,clvsUgf.nteohrtays—eupPl.foint,bsAyvrlanhgis4oetc,mda3ruiblhnoegLadsyu0alphtrxngoecdwauypeitrsmcPhgaofnBuet,hiymkla.wsodvhgtyafneCrciusobgh,atalmrdsefonucht~xsaidoug;¢crametp.nohynPa(,sgruciAfh.y,otslVeadr(mi')iksfa.bP-;aoc?ycegtfAjshPnaiw.mparoeuly,tsibh¿gdeanyrlcdimwuostheyrnxdliau,hmtsebgo'ry.ua?incmdslfeghtvrkaouHnrmiwfveachtspd.iaehosnrPctrvifmyeaIhnotiicfddeasshrtwonvpedi!f,cl(ra-tghnqpiydsaowmebaltru1hPpsaeoiyvkwracmbugsnedfoltatrmhgsxueovanrfkmtbpilabwuoeahynrdIlgpmcteanisoh)bmtkrlw,fvaygciueoh!otpdlPca%muntuvkfier>aoifsnh-e,gtdlrncs|-,

itrPo‘shefnydletaceuhtogmahitnegaobhsrnoet4sriah1etnriuae5dnpthrostauehntserl"ihearncldmtoeuawiicmnsatedgiyntdot;ensahfureoacysPUfprhyeFmtisaoenhlpogrIcemaDgoretpEmohareAbitcnlarhetinsticshdtinao.heltI.sh,ifusetmUiehrtruehlmabeStdsl:,iIeoaLIirnbtsghAarelpG.a,)wE.omtriasftl.re,lsFea,IIpAdtmK5rohaSli:etgsi4vInce%ptoain#rHh.teli#uscbteodhne4dpos:iaf3nwSge#todbrhtm.yao(hte3dgw:dea(2nI7rten)odhrbaeonladtoienv,deg sbtrauwdhacc“e1e.inynuaoolgtd-ndnsewr.mpehra:,icg’-stpaatesrgc,hqupts”a-aeuvttrunrmaafhoiwesodnarrefttvtdcinhlehwtioadsminoethbtvfiter,iweindlytseodtec,idfosnfhTcrmhc,abanles,wimtedintahawTmverrf.ailhidon1ipt.enistdtahtd.esihhrdetnt.mamTgsogyreleevahqfpe6rtoteaoune,siirsfmnagnc,oceredrt4,dsarle,het.leiacucwrslt2daoonepbyg,h,inuclsofeeaosnrcfiegtslfmtueroosrwdidmntmbM.uievsdtualbhcepeieltlvxgcidmnn'iyiroowpbgsbu,onnilrunofatdaneenk,rhalgdldoieravncstsruitaedqhaTotdoilsnnutehfiory,euiascnitoormmissoabthsnfealhn,eo,iusYagltmloi.ntneanaorsdainbtasgg.etudhnracioutrneedvhmpnadum.hoveitemamsiden.hovirbntnsbeptgwei5ehhrbhgtsac,rretteylh.s7mbsarhg,xeaut.seaocp,.tnilfnapoodtivadedo3fwnnnet,sdtismchetnider.:eont,gadrCtdsuiluaehnn3ehv1trmmesdtriesbnb,r4otfhT;eti,nmaeerrtWoahgiiveos2lafeinnpe,,smnanpr(wnom(oiteear“bssiuftdib)N)dticemmloohrenaebttmnirnTaehuptcdvtiigrrhnaonhltise$gdgreyhspart:ownhisforrelgautlearriittieexsp.raensdefdacPtloartsonAusmenriitcenaelso.f i(algorithms).proceduresnumericalitco.n”stirnutchtiinvgesdaelfianintdecaoresmtohesrwehroullee:tbhyeiebo“(ahsctsndnmuu3acfIouadtlwhmnemimn[fergttcwadnnibaifsmlhcs,-etn;uibrino]au.et-am—mhstadbnrudepfhccbwui1tieallyo,vymem)ofevansiiurpdt.ydh,algmre"lctrtviapnia.mfsWsdwcbehpth-"mna'uoesyei?bhc®fra"rmeaotailoh-yl®eapbnkfswcidtforasomlbngue“,vhbuapaeulsrtjinmasyorcoh,asehndttpbip.Alr,aveidptreolsmteuicatnchGyoiafy(hluvncrahsmsptre“poyrtieey)hldaesbd.unuof,trgywoacnatdiieT,rerv(hof.klfoc“sistq,hitaedrhe;.AoPswunid;etalsrycurn(tegohmpa,swaiitmbhnp;lsTnrat—aee)yipohutmt1rnncmfevaoinghyh,hty';oufimsneepiihntcnasorwif,dvdamyhgt)robiocesgpasAe:nritmealeod.onufrsycchwivprme*tdindpmacr?idtsooefuwasolcdvr“,ephimrntssiaelxprhm,oavlaietdrsinose)tipcsnh,fldmdyasheg,ltuibroaitSgw'pstduaoychcftlhki*dnrersf,Avayit*oFhandieu“nlnwa)lrdcciomsugeftg”hiareIonmpiudspwhtmsltePnfbcnyrdrehstyssfim,aaeuoIiPwhrcbtg,lusongbtcedmapda(hrithmcunltnosuseiaMohierh,yugmwrwtbdcn.ngi”aadrp,o,ih)fgLnets1ebibtnlahyoshotreiemuegmcslwldefoxs,rnre1hiioa1gmbnap.4ufecwqr"l(sokbrdtfaati2me”eyueamrnlht,s)o4ho"sreahudrtt1.avT?te1nlsob—e,prhsihrlno.4eahrypbatvn”hfoud(rnx1iiet.snbbheecpoydrdaisui)MatelfAbchnsAisNdporna)gwlbPalvcot—irse.uifapuhyinem,(tnrwtmnlpiqldoeSsfcsercalvAibhbouttoomfAdiTurTsaseoh,uannoeydNh(rihgasi.cf,martsinohbtnes)Rxlogehmdsrc“nprgpiebfiutl(derhosu(nN;WfAsc4i,faltgrnmugeO)e,nhoiho)aodumufcn,t(efso.mswbialnmgrSeykdtihipeobt)buTchliertdaecahcorswhlilehsynd,e”rs apropertyofaneffectmustbeapropertyofthecause. iasnpceotms.engesoumreatbrliec.alarheaslenvgetrh, qpuraiencltistyoe." tively:?! tsehceopnrdi,nctihpalte,bonththaesmoadtelroforhtuhmiangnsraenpdroadsufcotriomni,ngonbloythsutchehirampodliafricdautailointsyahnads pcoreplimurtoaniplvchmrtsinepmstiflaedvrtnse:g, sPfey(2tq34;run1ioht.aegnolmh,)rcatbseenyrs,prioctdn1vfFa.oucshreimotd.b,efrirdteni,mson poOTdahutlndielrmwihnueia.tcoited”r:s°p(robeftiOandvuhcornemtsfboe)dhrem, pesretahonmluvethrnmeselanbfi,tdtrs

LaÀ EN Eh, 1114144 (k) The Pythagoreans developed an arithmological epistemology conformable with their numerical ontology. (Wrongly, many scholars have bypassed or barely THE FIRST STAGE OF THE IDEA OF MATHEMATICS y — more precisely, arithmetized — the nature of things. and universally mathematicized A modern historian, keeping to Aristotle’s track but with a modern readiness to use glanced at this conception.) From the outset, starting with Pythagoras himself, they a more flamboyant vocabulary, might put it this way: the Pythagoreans, who at first laid special emphasis on mathematico-theoretical learning as a vital method for beengaged in the study of mathematics, were carried away by their successes in it and enthusiasm for it into supposing it to be the universal science. Aristotle's causal answer to the “why” question is too simple and one-sided. Like smoke, it obscures from view and even chokes off an inquiry into the intended coming “wise” and free. Thus, Pythagoras was early on described as a “polymath” and as one engaged in scientific inquiry. The Pythagoreans of the fifth century supposed that the natures and causes of things, as mathematical, are within human grasp: they can be exactly thought, understood, and expressed by human beings, though not by all human beings in the ordinary course of life. Exact wisdom, and the spiritual detachment that accompanies it, are attainable only by those who are specially gifted and purified, practicing the requisite detachment: for the Pythagoreans considerations likely to have been compelling in the minds of the Pythagoreans. Evidence of the grounds conceived —maybe no better than partially or blurrily, and far from articulatedly—by the early philosophical thinkers as warranting their re- “life is like a festival; just as some come to the festival to compete, some to ply their spective doctrines is, notoriously, mutilated and scarce. This general shortfall holds also for the Pythagoreans. Nevertheless, some definite and coherent indications surtrade, but the best people come as spectators, so in life the slavish men go hunting vive of what served as rational support for the Pythagoreans' mathematicism. for fame or gain, the philosophers for the truth.”?% (There is evidently a resonance Three branches of this rational support may be singled out of the historically more ramified complex. First, discoveries, some of them then recent, had been made of the remarkable and exciting applicability of mathematics to natural or manufactured systems/collections/items/modes, e.g., in astronomical predictions, in the correlation of numerical ratios with the principal intervals of the musical scale (a discovery attributed to Pythagoras himself), in the determination of distances by triangulation, and in the exact construction of the Egyptian pyramids. Second, at a more mundane level, there was the widespread recognition, established in the of this in Plato’s Republic.) Proof in the discipline of mathematics also had a general significance for them.” The epistemological side of the Pythagorean philosophy of mathematics is clearly attested by fragments ascribed to Philolaus;** it was a major key for them. “Actually, everything that can be known has a number; for it is impossible to grasp anything with the mind or to recognise it without this [number]”; “it would be impossible for any existing thing to be even recognised by us if there did not exist the basic Being of the things from which the universe was composed, [namely] both the Limiting and the Non-Limited”; and more fully in fr. 11: [Without] the power of the decad . . . all things are unlimited, obscure, and indiscernible. For the nature of number is the cause of recognition, able to give guidance and teaching to every man in what is puzzling and unknown. For none of existing things would be clear to anyone, unless there existed number and its essence. But in fact number, fitting all things into the soul through sense-perception, makes them recognisable and comparable with one another . . . .% (1) The Pythagoreans matched truth with, and wholly dissociated falsehood thought, language, and practice of humankind, of quantitative aspects of common things and the knowledge of how to count them or how to calculate some dimension of them. Aristotle, in a part of the Metaphysics other than the one cited at the beginning of this section, comes close to acknowledging this broader background to the formation of Pythagoreanism: “Again, the Pythagoreans, because they saw many attributes of numbers belonging to sensible bodies, supposed real things to be numbers —not separable numbers, however, but numbers of which real things consist. But why? Because the attributes of numbers are present in a musical scale and in the heavens and in many other things.””” Third, their cherishing mathematics was rationally self-supporting, for it was the supreme rational science through its proofs.” from, number. “Falsehood can in no way breathe on number; for falsehood is in- 2.5. The Pythagoreans scored a crucially original success in bringing matheimical and hostile to its nature, whereas truth is related to and in close natural union matics into distinct and fundamental intellectual prominence. Other early Greek with the race of number” (Philolaus, fr. 11). philosophers concentrated on qualitative conceptions, focusing mostly on certain 2.4. Why did the Pythagoreans produce a philosophy that gives a predominant natural sorts (e.g., water and air) or on abstracts from any natural types (e.g., role to the mathematical? Aristotle, in a sketch of his metaphysical predecessors, Anaximander's Infinite and Parmenides’ Being) and on quasi-passionate or quasiasserts chat che Pythagoreans’ study of mathematics itself was the genetic ground of intellectual cosmic agencies (e.g., Love or Reason). They did not pursue a positive and systematic reckoning with the numerical or the figured. their philosophical principles. Citing no reasons on their behalf, he says that “the socalled Pythagoreans, who were the first to take up mathematics, not only advanced Further, mathematical methods had been largely, in Greek states no less than this study, but also having been brought up in it they thought its principles were the in Mesopotamia and Egypt, motivated by practical needs —agricultural, astronomicoprinciples of all things.”?” This version of events is singular in suggesting that the and solely because mathematics was their frame of intellectual interest, that they religious, commercial, constructional, geographical, and medical. These methods had lacked any overt degree of theoretical organization or allure; still less had they received any philosophical conceptualization. The Pythagoreans initiated the latter turned to philosophizing, in which (he proceeds to elucidate) they fundamentally and made considerable contributions to the new development of mathematics as an Pythagoreans began their school as mathematicians only and that it was afterward,

abstract and liberal, as well as applicable, science. Perhaps, indeed, they were the founders of mathematics as an abstract, liberal science, which is what Proclus at- IHk FIKSI SILAGE Ut IME IDEA OF MATHEMATICS 11 2.8. Most of the Pythagoreans’ doctrines are lacking in clear and distinct ideas and in clear and distinct logical concatenation. Consider, for example, their propositributed to them in the person of Pythagoras: “Pythagoras transformed this study tion that numbers are the fundamental natures and causes of the bodies and modes into the form of a liberal education, examining its principles from the beginning in the cosmos. The Pythagorean adoption of mathematical-type notions and stanand tracking down the theorems immaterially (aulüs) and intellectually.”?! Perhaps dards by reason of their exactitude and plenitude is markedly at odds with their it was as such founders of “pure mathematics” that Aristotle had them in mind proposition's vagueness and inadequacy. How numbers are responsible for the when he remarked, in the historical introduction to the Metaphysics, that the properties of bodies (e.g., their mass, texture, color) and for such modes as justice, Pythagoreans “were the first to take up mathematics”; this possibility is strengthened reason, and opportunity (examples cited by Aristotle) is left obscure (see p. 14 by the fact that Aristotle’s own conception of mathematics was as an abstract, thebelow, however). This obscurity also surrounds the way in which numbers were oretical science. thought to be responsible for change, including motion and animate development. The intellectual prominence of mathematics that the Pythagoreans originated The Pythagorean analysis of bodies into geometrico-arithmetical figures does not by and promoted survived throughout subsequent antiquity; and even when not under itself have the capacity to explain change and nonextensional properties, and the their banner, yet it was under their radiating influence—via Plato and Aristotle**— invocation of geometrico-arithmetical transformations of one figure into another, as that it increasingly regained that status from the thirteenth century (when it was a kind of quantitative stereochemistry, brings with it the objection that it requires heralded anew by Roger Bacon)” until its full restoration in the seventeenth cenprinciples of transformation and that these principles must lie beyond the static— tury and after. because purely mathematical—concepts of their ontology. 2.6. A solid basis for the Pythagoreans’ historic success lay in their idea of the Aristotle goes further in criticizing the Pythagorean assertion that bodies are pervasively mathematical character of reality; through this idea they rough-hewed composed of numbers. In his characteristically concise and adversative style he says: the foundation stone of the later towering achievements of mathematical physics/ astronomy. Correspondingly, they pioneered the powerful methodological policy according to which one understands a species or a rhythm of fact if, and only if, one grasps it in appropriate quantitative terms. The seed of their mathematicism developed in its maturity into a threefold faith “justified” by many works. The first strand of the faith is that mathematics’s vitality depends on its renewing itself by seeking an ever more comprehensive description of nature in mathematical terms, thereby drawing on the inherently mathematical sources of nature and that in the [T]har bodies should be composed of numbers, and that this should be mathematical number, is impossible. For it is not true to speak of indivisible spatial magnitudes; and however much there might be magnitudes of this sort, units at least have not magnitude; and how can a magnitude be composed of indivisibles? But mathematical number, at least, consists of abstract units, while these thinkers identify number with real things; at any rate they apply their propositions to bodies as if they consisted of those numbers.** absence of such physical contacts mathematics degenerates into idle symbolism. The Aristotle here is pinpointing two objections: first, that bodies essentially possess second is that the adequate and clear expression of physical conceptions requires the spatial magnitude, which cannot be accounted for on the basis of numerical units, use of appropriate mathematical language, with its precision and abstractness and for these have no magnitude; second, that there is a categorial confusion in the Pywith its ratiocinative—and so precisely predictive—power. The third strand is that thagoreans’ identifying mathematical numbers, which are essentially abstract, with physical conceptions, when formulated in an appropriate mathematical way, exphysical realities as such. hibit a beautiful economy matching nature’s own. As already indicated, the Pythagoreans did not suppose number to be the absolutely fundamental ontological source of the quantitative; it stemmed from two 2.7. Unlike some of the Pythagoreans’ other teaching, e.g., that on the transmigration of souls, their philosophemes regarding numbers were no frenzied folly “principles” or “elements,” each consisting of a pair of contraries, namely, the and no crude relic of animistic or contagious-magical modes of thought. On the limit and the unlimited, and the odd and the even. It is the principle constituted by contrary: they made a decided intellectual advance and became the ancestors of the limit and the unlimit that is the absolutely fundamental ontological source of the quantitative, and of number specifically. Now, the sequence (1), 2, 3,4, ... ,°” which was recognized by the Pythagoreans, is an infinite sequence; it is a case, permany gifted intellectual offspring. Those philosophemes were not wholly sweetness and light, however. Besides having various other deficiencies and defects, the mathecore, just where it might seem, to an antirealist concerning abstract objects, to be haps the paradigm case, of the effect of the unlimited. Concurrently, each item in that infinite sequence is finite; this is a case, perhaps the paradigm case, of the effect solid and stable. This vagary occurs at the place—the central fireTM of the system— of the limit. The principle of the limit and the unlimited is, however, an unsatisfacwhere the Pythagoreans posited the inherence of numbers in things; the vapors of this realism concerning mathematical objects were inhaled and assimilated by the tory way of accounting for the infinite sequence of positive integers, because it cannot by itself determine the actual progression, which is in terms of a unit increase mathematical philosophies of Plato and Aristotle. at each step after the initial one. In other words, the law governing that sequence is matical philosophy of the Pythagoreans has a disturbing vagary at its metaphysical

AAA ALIA dl. award wal not covered, or at any rate is not precisely covered, by the principle of the limit and the unlimited. An analogous defect occurs in the Pythagoreans’ notion of the odd and the even as the “elements” of number, i.e., of 2, 3, 4, etc.; because although, obviously, the items in this sequence are odd or even, exclusively and exhaustively, the mentioned notion cannot by itself account for the alternation of odd and even numbers or for the successive unit increases in the actual progression. What is missing bite PAKS SAGE UF 111 tye bA Ut WA lieMALICS lo cognitive states and processes between which they discriminated are possible and have their distinctive natures. 2.10. The Pythagoreans’ naturalistic conception®' of mathematical entities and truths is unsatisfactory for the following reasons, among others. (i) Their conception would make mathematics a natural science, such as physics; whereas mathematics, including arithmetic, is not an empirical science, either in its subject matter is the logical equivalent of: 2=1+1,3=2+1;4= 3+ l;etc. 2.9. 1 will say a little about three further coordinates of the Pythagorean posior in its method of determining its assertibles. Patently, mathematical entities and tion before | recur to what I called the “disturbing vagary at its metaphysical core.” in order to avoid begging the question), e.g., in algebra, analysis, and topology, are mathematical “facts,” in general (1 leave aside the Pythagorean doctrine of monads First, with respect to the thesis that “the movement of the stars produces harnot given to us through sense perception; and the mathematician's demonstrations mony, the sounds which they make being in accord” cited in 2.3(c): this, taken prosaically, has long since been found empirically wanting. Critics from antiquity onward have noted the absence of relevant evidence supporting the claim, e.g., the stand or fall not by reference to the outcome of empirical verification/falsification, but by reference to his postulates and the rules of a deductive logic. The Pythagoreans' treatment of mathematics as physics is the naive origin of the persistent view of fact (conceded by the Pythagoreans) that one does not hear any musical sounds attributable to celestial causes (this unawareness they ingeniously explained as being due to the continuous presence of the sound, with no contrasting silence); and they the imaginative construction and pursuit of abstract types of possibility in mathehave noted observational facts incompatible with the Pythagorean astronomy or Pythagoreans’ view would require that the mathematician’s use of language be demusical theory. Therefore, in the prosaic sense the claim is a fantasy (not to say, scriptive/constative; on the contrary, his use, in general, is nondescriptive/nonconsta- Fantasia). Doubtless it is more precious when interpreted figuratively, but this lies tive, for typically his language is symbolical, geared to an abstract type of possibility mathematics asa body of “truths.” (ii) Their conception counterproductively debars matics; yet, it is this imagination that seeds the harvest of mathematics. (iii) The beyond my present province. rather than to depicting a definite actuality. (iv) Their conception does not make Second, with respect to the tenet that arithmetic, in geometrical form, is the reduction base of mathematics. This tenet is implied by the overt Pythagorean ideas clear, if indeed it allows at all, a distinction, which it is important to be able to about number. If their tenet were correct, all affirmations in mathematics should be reducible to affirmations in their arithmetic. Stretching the sense of “their arithmake firmly, between, e.g., the number 10 and this or that particular cluster of ten things. By the Pythagoreans’ ontology, “the number 10” can hardly be other than a(ny) decadal case. (For them, “the number 10 is thought to be perfect and to metic” as generously and sympathetically as possible, it still turns out that almost all mathematics is irreducible to theirs; this applies not only to later developments, comprise the whole nature of numbers”:*? 10 = 1 + 2 + 3 + 4.) (v) Although the e.g., in algebra and analysis, but also to already existing mathematical results. The concepts of elementary plane geometry, as established in the fourth and fifth cenlittle, if anything, to the theory of proof; and the “a priori” status of mathematical Pythagoreans recognized the significance of proof in mathematics, they contributed assertibles was not within their grasp. It was the Academy and the Lyceum that first turies B.C., are irreducible to ones in Pythagorean arithmetic; ‘angle’ and ‘area’ are examples. According to tradition, the Pythagoreans themselves were discomforted by the realization that their kind of arithmetic could not cope with irrational magnitudes, whose “existence” had recently been discovered—ironically, by one of their own members:*° thus, the square root of 2 cannot be expressed in terms of a regular and “rational” pattern of positive whole numbers in an exact and terminating way, so that here the limit and harmony are overruled by the unlimited and question and asking, as he used to do, ‘are we on the way from or to the first princidisorder. ples?’ "*? Plato’s mathematical philosophy is an ascent to the first Principles: to the philosophically deliberated on and articulated a methodology of mathematics. 3. PLATO 3.1. “Let us not fail to notice, however, that there is a difference between arguments from and those to the first principles. For Plato, too, was right in raising this Third, with respect to the Pythagorean (or, at least, Philolausian) theory of first principles of mathematics, and, cognately, of everything. Plato’s clambering knowledge: the theory’s insistence on number being the cause of all grasp, percepclimb to the first principles in his mathematical philosophy tallies, in its direction, tion, recognition, distinction, and comparison is understandable, given their numeriwith Russell’s characterization of mathematical philosophy. cal ontology whereby the characters, structure, and stability of things are a function of embodied number; but the dubiousness of this ontology proportionately threatens the tenability of their epistemology. Besides, the very plurality of cognitive terms used by them prompts an additional critical reflection: they failed to provide a philosophy of mind, conformable with their numerical ontology, that signals how the The more familiar direction [in the study of mathematics] is constructive, towards gradually increasing complexity: from integers to fractions, real numbers, complex numbers; from addition and multiplication to differentiation and integration, and on to higher mathematics. The other direction, which is less familiar, proceeds, by analysing, to greater and greater abstractness and

PRIER H. NIDDIACH logical simplicity; instead of asking what can be defined and deduced from what is assumed to begin with, we ask instead what more general ideas and principles can be found, in terms of which what was our starting-point can be defined or deduced. It is the fact of pursuing this opposite direction that characterises mathematical philosophy as opposed to ordinary mathematics.“ There is, nonetheless, a difference between Russell and Plato in this connection, which the common term ‘principles’ obfuscates. Stated simply it is this: Plato’s mathematical philosophy is wider than Russell’s constitutive characterization, inasmuch as it, unlike Russell’s, includes a broad epistemological concern. 3.2. Before I turn to an outline of some of the leading topics of Plato’s mathematical philosophy, 1 should draw attention to two of its traits. Each is a tension, or a self-conflict. And each is already apparent in Pythagoreanism. On the one hand, the Pythagoreans wanted to pinpoint the arithmetical constitution of bodies and modes, to show how, ina definite, physical way, a body or mode is actually “nothing but” a figured arrangement. This makes their mathematicism as concrete as possible: a body or mode is a complex of monads, On the other hand, the Pythagoreans were concerned about emphasizing the mathematical structures and regularities of things, rather than about claiming the identity (without any remainder) of the things with those mathematical aspects. Thus, they did not (I think) hold that musical concords are nothing but such and such numerical ratios, still less that the concords are series of monads; their conviction was that underlying the concords, acknowledged as sounds, lay definite mathematical structures and regularities. Take another example: justice. It is probable that the Pythagoreans numbered justice as four; they did so, perhaps, because justice was regarded as involving two parties and the proportionate distribution of two parts between them.** But this is not an identification of justice with four, either monadically or by way of maintaining that four and justice are identical; their idea was that justice has a four-term structure. Similarly, there is a tension, or a self-conflict, in Plato’s mathematical philosophy, especially (but not only) in his Unwritten Doctrine that the Ideas are numbers.*® This doctrine runs in two directions, one of these being toward the sheer identification of the ideas with — as nothing but—numbers, while the other direction is toward a number-type basis of the structures and regularities of the Ideas. The other tension, or self-conflict, common to Pythagorean and Platonic mathematical philosophy concerns the scope of the One. Trends toward dualism and infinitistic pluralism are quite marked in Pythagoreanism, which tended also to maximize the role of the One: “The One is the beginning of everything,” “Harmony is a Unity of many mixed (elements), and an agreement between disagreeing (elements).”*” Similarly, Plato was continually driven from pillar to post as between one and many. Each property and kind of the changing, sensible things amidst which we live and move, have, actually or potentially, many instances among these things (ourselves included); he found it necessary to posit an Idea, as the appropriate unity, accounting for each such instantial multiplicity of property or kind.*® Nevertheless, Plato’s quest for unity of this Ideal sort is, in his Dialogues, constrained and THE FIRST STAGE OF THE IDEA OF MATHEMATICS 15 qualified by his equal acceptance of the plurality of the Ideal sorts: there are really irreducible differences; and, consequently, the Ideal, intelligible realm cannot be assimilated without comprehending the relationships of its different members. This issue is tackled in, e.g., Republic (in connection with the Idea of the Good), Phaedrus, and Sophist. His reality becomes layered. The study of what the scholastics called “transcendental” terms, viz., ‘being’, ‘thing’, ‘something’, ‘one’, ‘true’, and ‘good’? was thus initiated. In seeking to determine the colligations and separations, the classifications and divisions, of Ideas, Plato found himself in difficulties with the status of One. He was disposed, and he was opposed, to a Parmenidean position, in the sense of the adoption of the absolute ontological primacy of the One. He may well have sensed the need for anultimate, all-embracing unity of reality. Also, oneness is the most basic transcendental, if only because the transcendentals and the Ideas are each of them one,TM as he supposed; whereas nothing else had for him that basic universal applicability. This disposition led him, in his Unwritten Doctrines, to allot a unique status to the One as the principle/origin of being. At the same time, he was steadily opposed to a Parmenidean position. Despite the attractiveness of a single principle/origin of reality, his reality is not uniform but inherently differentiated and layered: he is a thoroughgoing Ideal pluralist, who argued in the Parmenides that the One is not completely self-subsistent/coherent; and he is a pluralist concerning numbers. Plato's ambivalence concerning the One ran deep (and it ran long, through Neoplatonism and beyond), and the crosscurrent was not (I think) caused at all by his failure to discriminate the different connotations of the term ‘one’. (The term ‘the one’ (to ben) “can mean (1) Unity or Oneness in general; (2) the unity of anything that has unity or is one thing; (3) that which has unity, anything that is one; (4) the one thing we are speaking of, as opposed to ‘other ones,’ and so on.”*) 3.3. At first, and even at second, sight, Plato’s Dialogues give expression to a somewhat changing mathematical philosophy, which, however, in none of its phases, appears to be the same as the mathematical philosophy of his Unwritten Doctrines. It therefore has to be considered in a doubly divided way, one division attending to differences within the Dialogues, the other to differences between the Dialogues and the Unwritten Doctrines evidence. Despite the various differences, what is usually taken for granted as most characteristic of Platonism is indeed prevalent throughout all that is known of Plato’s thought in the philosophy of mathematics: his realist and constativist assumptions about mathematics and mathematicals. Some commentators have held that Plato adhered to the Unwritten Doctrines, with its arithmetization of the Ideas, as a programmatic project and/or an esoteric theory concurrently with his production of at least the middle, as well as the later, Dialogues.*? This is contrary to the usual view (with which I am inclined to agree) that the Unwritten Doctrines belong to Plato’s older age only, say from his sixties. Interesting as the question of choice between these two hypotheses is, 1 will here proceed in neutral detachment from both of them, in the absence of decisive enough reasons for choosing either. That 1 begin next with the Dialogues on their own, before discussing the Unwritten Doctrines, is to be taken as a procedural order, with procedural precedence to the philosopher’s own writings.

A GAL, Paria DALLA, a aan dad SARO thon baad AI st 3.4. Although the Dialogues | will consider expressly are not the only ones, they include most of the most important, for studying Plato's philosophy of mathematics; they date from the earlier part of the middle, to the final, period of his exists eternally and independently of us; in later Dialogues, he held that whatever is production as an author.TM In accordance with this chronology, | begin with Meno, of strict knowledge in contrasting this with “true” or “right” opinions. “[I]t is not, whose bearing on the philosophy of mathematics is exceptional among the Dialogues in being almost exclusively epistemological and methodological. (a) Recollection. A slave-boy —a prototype of ignorance —comes to recognize, through questions but (supposedly) without any information being conveyed to him by (the character) Socrates, what the correct answer is to the problem: Given a square each of whose sides is two feet long, what is the length of each side of a square having twice the area of the given square? (Answer: the length of the diagonal of the given square.) The process and success of the boy’s discovery is proffered as illustration and confirmation of the prior suggestion that Socrates had heard from priests and priestesses and from many divinely inspired poets that the human soul is immortal and undergoes reincarnations and so it “has seen all things both here and in the other world, [and] has learned everything that is.” Sliding down the rainbow of Plato's rhetoric to a plainer level of discourse, one may say that Plato was proknown is abstract (noëtos, “intellectual,” i.e., “falling within the province of” nous, was how helater expressed this status). In Meno he emphasized what is characteristic I am sure, a mere guess to say that right opinion and knowledge are different. There are few things that I should claim to know, but that at least is among them, whatever else is.”°? What is the difference? True opinions run away from a man's mind, so they are not worth much until you tether them by working out the reason [aitias logismö]. This process, my dear Meno, is recollection. . . . Once they are tied down, they become knowledge, and are stable. . . . What distinguishes one from the other is the tether [desmos].% According to a stock modern analysis or defining set of conditions, “A knows that p” means (i)A believes that p, (ii)A can “justify” his belief, and (iii) p is true. This modern definition, making ‘knowledge’ equal ‘justified true belief,’ is similar to Plato’s conception, whereby knowledge is true belief (or opinion) topped up by a justification, “the tether.” One might extrapolate from his remarks and infer that, jecting the notion that the attainment of conscious mathematical —more generally, thinking of mathematical knowledge, he took the tether to be like a mathematical a priori—knowledge is explicable only on the basis of direct learning previously. The demonstration and that what he called “working out the reason,” i.e., the excogitaperson must have latently possessed the relevant knowledge; his excogitation in response to suitable questioning resuscitates that latent knowledge into consciousness. His latent possession originated in an acquaintance, in a preceding state of life, with whatever is involved in that knowledge. Thus Plato, at a stroke, brought some major epistemological problems to the fore about the nature and genesis of a priori and, specifically, of mathematical tion of a rationale, was at least analogus to the mathematician's requirement and practice of a rationally demonstrative procedure in support of affirmed theorems. I would add that Plato, in assuming the truth condition for mathematical as for other knowledge, did not begin to discriminate between mathematical and any other sort of truth; for him, mathematical truth and its reality are on a par with all other truth and reality. Like the rest of the Greeks, Plato had no notion of mathematics knowledge. Unfortunately, the answers hovering on the surface of Meno are feeble as the disciplined exploration of imaginative possibilities or schemata. Mathematics fledglings that cannot get off the ground. What Plato tells us is itself, on the one was for him essentially “realistic,”* much as his conception of art was mimetic. hand, very problematical about soul and immortality and reincarnation, etc., and, on the other hand, very vague about “seen” and “all things” and “the other world” (c) Ideas? An insistent doctrine of the middle Dialogues in general is that the things known are ontologically altogether different from and superior to the things when he says of the soul that it “has seen all things both here and in the other believed or opined: the objects of knowledge are Ideal realities, whereas the objects world.” He has failed to indicate the nature of the things relevant to the slave-boy’s of belief or opinion are changeful vanities. In Meno, on the contrary, the different regained geometrical knowledge, e.g., whether they are objects or facts, and of what veridical cognitive states are not correlated with different objects; those states are sorts. Nor has Plato catered for the evident difficulty that the slave-boy, at the terdifferentiated only by virtue of the presence or absence of “the tether.” This fits minus of the inquiry, has apparently no remembrance of the occasion or acquainin with the fact that the Dialogue never mentions the term ‘Idea’ and says nothing tance originating his knowledge.“ implying the theory of Ideas. This is not to deny that, being wise after the event, Lurking inchoately at the back of Plato’s mind was the presumption, powerful one can discern a few green shoots from which the nascent theory will bud forth. in the rationalist breed of philosophies, that what is knowable and what is necessar- Those®' who have found the theory of Ideas in Meno have read the theory into it by ily9” or essentially the case coincide. Something like this presumption motivated and working backward from Phaedo and Phaedrus, where the doctrine of recollection is afforded warrant for his epistemological optimism, although it came to be modified indeed linked explicitly to an ontology of transcendent Ideas. Unlike these and later when Plato conjoined with it a sharply hierarchical view of humankind, whereother Dialogues written after Meno, in this Dialogue the objects of cognition are, it by only a select number of people have the firm°® native ability to gain an intelligence seems, unconfinedly heterogeneous, within “this world” and “the realm of Hades”; of truth and reality. not until later did Plato ascribe a purely celestial® location for the knowable—the (b) Characteristics of mathematical knowledge. Plato presupposed that mathematics is a matter of strict knowledge. All strict knowledge, for him, involves what Ideas. In Meno, then, Plato left undetermined and unrestricted the ontological status of cognitive objects, including those of mathematics.

b1m8eyntthoefsPaeEnmTseeEtshR®o)dH,o.lNogIyDofITaxCiHomaticmathematics. adinetcetasecrho(meradss).MvoafaIltstuhehceeumtraotvthsienuocgcrshpyattonhrfdsut—ihTnn.rto”uetnrhsBp.eucrttPolmhpaitaosonseyuixspttelodipcoiaacnItscdiaoeolannfs—cm.oeafpHteihte,ocmfsoanptcrifeoacipravolasesidpthhioeoflngoaatslvroueptthrohyun.tephr,IinawwcinaildsloscialwwpttohiPlbFtdcp“aoinabihfuaiohtywovglrsaelsjpfgahd:giotmlesameo,nncgthycmetsrfuua’irtin,.hetleymsdvlaEOsor3iewortenstqnidsv.,sgrf,h.nrokoew6ctthaftgdlfuo.iedenihacrethynlsuismxoa;nohfvI.heRmpsctcnwiddeguybinelatwcc,,SterlmpyetosianuoidlcowuuyhstntpmkieePadwbn:,reaattiTnvsghcilehlrlsno,[oetniaeHiyta,mslwhtynru,icae.hn¢uEoyoclmlar,.af"onevsbpfygpseonutitdo.kl,aesveFrsod”rnj]tfaehdbckughdaqrcwyartosiei4seudtIa,Robohlnfptcni;yhgm,iSdenvwamgattlesnlcgTksbouinheanmctrasn“rzgttlyiSsetorahkf,laiingnurdbsyTmrowolmtntu-sheigtaafelncdueAs,vmohcllitsoecbhxsuaanstlGeicritleopdpsiiounthpeEighfcludmgaoerlciphyelhsorte.irb,ttOameurDaliorsel,vtAb,migptnsaernpFodtleihya”ncebkuItuheo,ltrcanddseTpoiaePtinr,tmsdttciunoln(.iHooaiutg.tcahmvdunnn)roietfEshi—seHbtinreetdoofinbsndma,.mfoIe”nuobeoittrsasoacndpfrfDyoleiti—tnlhriernotEwi;mtptgdPtrine,shhfalouhlAvurtiabtenemagsiaoci.lotsoteexngfneOr.mcrtproa.pvwlbdecie,s'arhd(Famtltlbto.nsorfyrnauhee2osebsatcpm;mqnr(dMimsdtaybiuhntsoniliAnacaone)tefsdlvgilw,mhTgranfdPdemeitdohcoy"xlaHyhtiHqrts)frauw(lpese3heuoEbit.iarn"lieeiaaotugrs(eco)Mcpbxl,hvPneadetwtiomrepAacohanIhrrltasofdtedhTtirealntaognenreowsi)mtdIIpvtuacnotdo,unehfdIeir;Co,ctuesodatnzhrmrhibSedfhopntianousoatakeihsdsdo“mcbtsfrnpli;vfb,oqatsipogynreeahiurfnphhblvs,otodraaoecsiaq,pysikettfwlcuenartrnueoamiph.gepcettattsifrgnl1ahuselbhoheaayti.i.’9t,en.edyrsc--s 3ehigdouvs2amsjt“lpcRwtbfgtovnliIhruoieahscgrh,nfdtdpenyitmseeerahubgrstoausncihnbeeehs,vosfuttgc.ymnld(rhseobhtabapdre;i“3nIixps.dlrtyio)w5.he,acsnsbeha.5trwmn.h,dy)”riSt.aH(hptiisclwoo;aHhaehdrniyiwalsPnltesgcpegaomtoiesheltbMstndno,pofeaatihs“lrlpht,mr.yegauiedontciosn.osI?rerfwpheu,mgwis(gtb)aidonar1cdfynsmrthlatm,pHreiheoano”ibPyh‘ubgitcdcnrcuevoolsehrl,.a”tonayhfyademhnorcinotn:lepyvsrMaetiaihfcdnuomg.vtsanaeiealtoreho’fbyldreTdintsslorarImhbltlMpnfagoywm[nieycsonruvilutptrgfdgstheacnprebyntouskvdigfaargIw,hbe’n,eotseiulpcdyntx.mrhocdvsafeencdpsib“toyoisoadnwuucenalIh‘tDrnopssrdIlfmoe]yh[leindoytdsbafhn.tpwmaWie;cofgeesvdrtiluat]cyoHse(fcerttnrgIvpac,oid-oiystsemspghooncvena,.’ic.urnoylsaudfmns,hticloaetpuewrr-seHgiwaiTfhamtMsodcmnxde,leiype(hrtbuiutwndaidglnkpemcohcsmsroi,dltnzewaehnltpbccsieooiheyotlenarn,qMvpdwhctiduronosmftibhellclgtnamfsrf'ibouehaskittea,ero(dmdicsnl.)oxavrtreyncabgPabdetcmmpqhillrlk“eitzPehgrseumcwnda.yvohtnii.r,rom,lygcnaeeugaxsfivpsmotctllnru¢x.qnmeabVhi.todp"tugwschgeealearirhnylh.ysotnmiltf,deanpoecr;vfPsuutor”tsbitimlgefuarnisoewhk,ndrajtsrhpldolxetRufchnboii,oenrhac,tdsgurgnpoib’meeehtwh.mydlvogcssoeatduh*fr.wadtk;,mil“mounsslpobernihty,frmtapin(ltedSheb.mshgtiahnwriml)po.detItisFcanheraloenspPvtodiid.f,gIsylalreehcrmot(mpnndiisaky.eceo.vedrtnwgsanklsifape.,omhtiysdeBtnasos:)emnr'lIuw("rifpkool,tgoevsc’fdnielhnptmv/yfa’bfc.hzrolaigT“nuaeoeplt.itlpcr1Ishayrh,2owdlct,uigedtapfe1ansy)ihrIoeh”od'2mtl,-ewa;dfer,syotpitsnihefhosatepnitod,;h2fstdielfh“evoioe3xnbwrsdclgeielrrn.tseuxldwobe(aegdenmotesdpdirsl”Iotvstenhw;ihH),seiehbttiourenhptduvacdwemiaeonsdalaeirandnstokrdsoienp.tdao3hncretiaelxhuieptcdecamli.lpuvldurelHidyasethlestidshnryetaiochtbwlefiunc,otearneivsdhgoeedbnnsre.tspboiahufwtmertmefhanteIhlodumsiebduaser.snieotrq(apiountipme)yoinHsodayseinstdbmeiostpnfb,yadt.rueati(locvtpn)ihvwoieTIunhdagtecheat,dIrseitou.eabhgcuai.htns-,

a20totciotPnutdEeTmoEpflRmatiHen.deN.s”IeDncDeI.TCH mtSofts—ed“wurptaivebnolsnhdrpWtirutvefw[sdnphodAstatmrniTlaer.kehawc]sttiunenrfeldmayoP.eshntit)l]rdbnho(agceoihieaju;dusnw)ntbnbomiceplrmgvhjlsfogbMcykiwetuhan,yIsltraeihmnndohltca“sriiobeddekmh,tnuwcrab]soIbgvepnhtedscmndfjaml.io]thpacepde,sia”rgwnh.amocileItfhavdrlnsgslrhToytyaioRIeaulIfea)icy.msdnnd,rhrbPkitypcfwdgameosdliep,Atuvaomegstruhnbn;dntaflr:PsearioultnohlseegiraytdfIbemcnre,hidvtptuae.snslnrfgoqitahpptkhaysuedvnnul”oeeiwoTtsaIcibrlcddr)uhmla,ezniqet;gdolyhsiefbicehtbgwpdsidunarcightrealoecttvb,Iksagufcanpeiiulhwntlnrmselpdcoatyig,g-hhrufnotyietdv,pbewhfsamiithl”otmer[wshdupfnaatielqrep.oDmlchnasnhatw”ui..rtidllbvadehhnpakgti.edoociheS,vlasntfumriewymingtoachbeows,awgpgrnitmeu.rdhsintleo-urbate?cigdnSms.olIvohp,iaectecsdmnfoirAhs,slIya.taeeimrcbmotdrpsla(nti.ouep,tfcmohwrlnIratuns“pit.ebcydishaorotlnte,]cTsPrriapidfunlhvufgeossrxnaitlpotcstnemurherohsnlyttloca).dp,nerefii’ttamchl”h,uru—bds,eietpao'hnmjstcrvdeitsu‘”ebiwnhyoaw,msfchdremhaikpaeoyhl,dtvsniegnf”kdehatnoutpi,lvw(saie;pflnrhcudetonoue.miaihtpgnlrdwefdl,nyiaMoemhtsduolgcsfoapienwulgcrsditbaeo.oypi.nleg"t,yrfomuhpaidseu,rnoflTaTitrs.tn/mngu.yehdavpheas,rcn.itlg.ao(,utdIneilrm-.Tts,dsintPfmae(chszkfltvPrbeeaailshgnwzro-a)u,id.tens-, Whatpreventsitspractitionerfromattainingcompleteknowledgeoftruth/realityis eth.agi.ns,d.niu”smtbiIencrtsih,oentlhiegahstqutoafrebP,leatdnord'satwahnealdonigadigeoasnpableiaterdweurenlfeosrnteihdcetsoitsriantotaebarenmdscocnfovgiIncdietaidsv.eo”gfraidnecso,- ejdoThieodmccrnptstbdasctPifuahrnmsnhuonihsliivisiclertaoeaadtlaecnngrulp(niisatpshtn,ldaoe.irmeemotrpihyfgtaun)dpisoepmacwelrnnaiwttft.dp(BhoelbAscajogunticoi”sn.i/ebslfetubhrtsg)i,etidaweyednsosups?chigom,fs3aEttlivretraydhndIcesiAhrivemrbFnaahfptttsucwd.eh(eeoosmdi(nTvbniythrh,tmdnasnauhrebwedoeraiNin.satsttPhrcodemfcn;pPednhtliuacveifTbTlods,artaidoonlnhetacwssrohsbefHmtmfafrhoitdttaaljneon,heiobPuusElcsd.hmattr,tt)lligslenmieoadmmhhi1dabhCcytcu.poHnFltsaedettrohysra)oaohgdbtIemhien,sRr'p|tnlsibewisaatsRpuihshdolranmnec,ott,efspiShaengtavqlic’ihhlusoehssrReTfttuatmcnfmiowbh,emoaheiunttrgainsjseclapfSeycqrk'hhgt,nolisutipysuemwsesptohcaTyPfdiobhlo.nacnhefetrau,aleAhniMcpitsrnungegiisemoatatchTjmsGotthmriohainouaeeashcrbE,tfostsrlfhlmdnotletiuofkaionieuhedchmb,tlercsOatmm,osbgedfhaoinerituimdaacyvFeilkstumoasecotcuahnfhfnotiawdieatgo"dnTocbe'uhtaheceasoojistewmndlHxetlmrsgidcaoisovucta'quhse.obnntEerminemustfecl;cardosungaisObaerndPtticgmfprsiblehmnIcto(gnewtuschorspadcmraDeo(b,tieotuellrfnwsoycElaspkcsgnhieoe’iiatou,,prlntAsossiateuofbeirtlmremdncntenoxcthssdee—aOati,hbrhunt;ofreirshiatuoaestF)witn—meoiyw‘nrcegot.hoottaynx4rpgmhfgsishnebrvhwe.hM/nrg;aTcatereieXa,greodlloinhfhdAaonttnënrTmiyfepedenmesa.sxwouaTtn3ontrrlchgilInxaRt,uetmc.oerahsHidcmetrhn=,ehtea)fbwmstllecprmeedEnain,v‘eysocyaihceuystrpmoermsb1Msiiucbrh.tlasn2lanvdekocaidlmAnhtat’u,cgtagbeborToinrissrdouhfrpT,lt.jeauawydceeirhalmfyeycdt.vsbI,enoagobeaamhopisesemvh.lrasetgCarTIidoapmynnumhlvstdrhtoiiktSlktBneywbifadunnec(ohua,ficstedhbdrgsorsfmmteatfieilr,tsawuhvalmPecnhgboupaqlnellct’miasIioeeorhuPmegteprdatac,nrzclesdpniyeaibdxthtiaoctr'obvuf2rbnihgdlcitsae’ornelrein1,d)d.eostsf.re-r;-- ktshtesoainhsdgpifbsenl.g

THE FIRST STAGE OF THE IDEA OF MATHEMATICS 23 subject-matter,” on which thought and intelligence Plato’s attitude toward the mathematician was that the latter's knowledge is not self-supporting and cannot be left hanging in the air; nor can it be suitably grounded on empirical insights, for these are altogether too weak and unsteady a are engaged.*5 This Dialogue tells us (i) that mathematical objects and truths are “intellig ible,” objective, unchangingly existent, and exact; (ii) that no truly mathematical objects or perfect instances of basis. Further, Plato rejected the conception of mathematics as simply the working out consistently (bomologoumends) of its supposedly evident postulates (“hypothmathematical truths exist in the sensible world; and eses," i.e., axioms or axiomatic definitions); the alleged obviousness of the postuprehensiveness. None of this, any more than what was said in Republic, can rightly be read as implying that mathematicals are interme diate between sensibles and Ideas. Plato has, again, failed to clarify the relationship between mathematical truths and mathematical entities and to justify his existent ial assumptions about them. | 3.8. Before I leave Plato’s writings, reference should be made to his (if genuine) Seventh Letter. Every real being, he wrote, is knowable, and there are four relelectic, superior even to mathematics in its concern lates (including that of the very existence of their subject matter) and of the immanent logic of the mathematician was unsatisfactory, philosophically speaking, (iii) that there is a Science, Diawith truth/reality in all its combecause it relied on mere presumption or on sensible prompting and guidance. Plato insisted that mathematical knowledge required apposite vindication; for him that meant a complete ascent and descent, using Dialectic, through the realm of the impersonal Ideas, with all hypotheses being surmounted at the summit of the ascent vant things distinct from it: (1) its name, when the thinker “by thought itself” apprehends “the starting-point of all,” viz., variable; (2) its definition, e.g., in the case “the nature of the good itself,” which is the supreme ““confirmation.”*! Plato's obare everywhere equally distant from its centre,” and this too is not fixed jectually realist, and his constativist-veridicalist, presuppositions led him away and as it is composed of items of language that are e.g., “circle,” though this is arbitrary and of circle, “the figure whose extremities inasmuch arbitrary and variable; (3) an image, astray from the consistency conception of mathematics into the Dialectical labyrinth €.g., a so-called circle that we draw or erase; of his own making, where he was swallowed up by the Minotaur of his ambitious of (a) circle: “In the fourth place are knowle dge (epist®me), reason (nous), and right opinion (which are in our minds, not in words or bodily shapes and therefore obscurity. and (4) knowledge of the being, e.g., must be taken together as something distinct both from the circle itself and from the three things previously mentioned). . . . The same thing is true of straight-lined A continual factor of this obscurity was his failure to distinguish between the objectual and the propositional, to employ appropriately and clearly each of the as well as of circular figures; of colour; of the good, the beautiful, the just . . . .” And he added that “every circle that we make or draw in common life is full of two sides of this distinction, and to indicate how they are interrelated. He occasionally dropped hints of the relevance of the propositional: the mathematicians are characteristics that contradict the [real being itself], for it everywhere touches a only “dreaming about being, but the clear waking vision of it is impossible for them as long as they leave the assumptions which they employ undisturbed and cannot straight line, while the circle itself, we say, has in it not the give any account (logos) of them.”* But since he demanded that the account should longing to a contrary nature.” No doctrin e of the intermediate status of mathematicals can be detected in this letter; indeed, they are absolutely on a par with the real beings —the Ideas—that are the good, the beautiful, the just, and so on. be in terms of the Ideas, especially of the “starting-point” (arch €) of all” that is the (nature of the) Idea of the good, it is hard to discern in Republic a coherent expla- | nation of the foundations of mathematical knowledge. 3.9. The most convenient focal place for is Aristotle's Metaphysics, Book A, Chapte 3.7. Philebus illustrates the continuing importance, into his later period, that studying Plato's Unwritten Doctrines r 6. There, Aristotle cited no oral or writ- Plato attached to mathematical matters, partly the influence of Pythagoreanism, as ten sources, did not allude to this distinc tion, and gave in its deployment of the notion of the Jimir and the unlimited (cf. p. 6 above). In ophy underwent a change ( though Metaph this Dialogue Plato drew, and followed up for epistemological and other purposes, a a development) or that it was atall differe contrast between two kinds of arithmetic (and likewise of other branches of mathematics), viz., the “popular” and the “philosophical”; in the former one calculates of the myriad units under discussion is in any way different from any of the others" no hint that Plato’s philosysics, Book M, Chapter 4, recognizes such nt orally from what it was in the Dialogues. It looks as though all of Aristotle’s remark s in A6 were intended to apply to Plato’s oral teaching. Aristotle makes the followi ng statements. (a) The philosophy of Plato in most respects followed with unequal units, e.g., two armies or two oxen (each of which may be of different size), whereas the philosophical arithmeticians proceed ‘‘on the postulate that none slightest element be- “Italians,” i.e., the Pythagoreans (Aristotle the philosophy of the might just possibly have meant to include Parmenides, also). Plato said that the many things exist by “participation” in the —and analogously in regard to, e.g., practical mensuration versus philosophical geappropriate Idea; this is only a difference in termin ometry. Philosophical mathematics is superior to the popular most notably in its goreans' saying that things exist by “imitation” “precision.”®* Plato graded the ordinary arts and sciences according to their degree expression of the doctrine that things are numeri of such precision. He ranked Dialectic, though, even higher than philosophical nothing but numbers. See p. 14 above.) Aristot mathematics, for it alone being “concerned with the final truth [to on], the real for Plato the Ideas are, in some sense, number nature of things and unchanging reality is the most genuine [aléthestatos) knowl- | edge”; it has “as its province the clearest, most precise, and true [aléthestatos] jects of mathematics, which occupy an ology, however, from the Pythaof numbers. (This “imitation” is an cally structured, as against being le’s comparison clearly implies that s. This is an Unwritten Doctrine. (b) Plato held that, besides the sensible things and the Ideas, there are the obintermediate position. (See pp. 20-21 above.)

THE FIRST STAGE OF THE IDEA OF MATHEMATICS EpisI have argued that this Platonic view was not propounded in the Dialogues or there that n tles. Aristotle’s account does not answer the question: Did Plato maintai viz., one such are mathematical Ideas on top of the mathematical intermediates,yes yes Idea for each, many alike among the intermediates? The answer is and no:is not in that there are some mathematical Ideas, viz., Ideal numbers; no in that there an Idea for each sort of intermediates (e.g., there is no Idea of the circle, conne to the Unwritten Doctrines, unlike in the original form of the theory of Ideas). (c) For Plato, “the Ideas [are] the causes of all other things,” 1.e., — Num- 25 a layer of reality between Ideas and sensible things was unacceptable to Aristotle, basically because the very idea of the separate existence of abstract entities savored of nonsense to him. He sublimated his distaste in a series of arguments, including the following:” (i) The doctrine presupposed the reality of Ideas (distinct from the mathematical intermediates). He dismissed them because, inter alia, they are superfluous duplications of features of the world of our sense-perception and action, can have no causal efficacy, and are inherently useless and unknowable principles, which cannot be coherently defined nor consistently circumscribed. (ii) The doctrine the bers [= the Ideal numbers] are the causes of the reality of other things”: theallquesIdeas are Ideal numbers. This, I surmise, is why Aristotle did not deal with d, tion raised in (b): a general negative answer was implied by the doctrine he reporte ical chat all the Ideas are Ideal numbers; so, more specifically, there were no geometr jects, was repugnant to Aristotle because of its absurd reduplicative consequences, Ideas in Plato's Unwritten Doctrines. mathematical solids (e.g., tetrahedra) and likewise not at sensible but at mathematical | (d) Plato diverged from the Pythagoreans in two major respects, despite the resemblance described in (a). First, he introduced the Ideas. Second, it “is his view that the Numbers exist apart from sensible things, while they say that the things themselves are numbers, and do not place the objects of mathematics between Ideas of intermediates, with its commitment to the separate existence of mathematical obas he thought them, (1) According to the doctrine, the mathematician’s study is aimed not at sensible solids (e.g., pyramids) but at special —perfect and nonsensible — planes and lines supposed over and above sensible planes and lines. Those mathematical solids also contain mathematical planes and lines. There must, therefore (by consistency in appealing to what has priority, “for incomposites are prior to compounds”), be further mathematical planes and lines distinct from those in the and sensible things."”” initial mathematical solids. And each range of the so far mentioned mathematical tory. They derive from two elements/principles: one is limiting —1 is “essentialg reality, the One”; the other is the unlimited —it is the principle of plurality involvin the endless alternation of the lesser and greater, ¢.g., the Number 3 is less than the planes must contain distinct ranges of mathematical lines, additional to the initial (e) Plato accepted that the Ideal numbers are not self-sufficient/self-explana- | | Number 4 but is greater than the Number qe ized. ethereal reanism Pythago is chought On these points (d) and (e), Plato’s set of mathematical lines. “The accumulation becomes absurd. . . . With which of these [mathematical planes and lines], then, will the mathematical sciences deal?” Analogous difficulties of reduplication arise, he alleged, in regard to numbers. (2) If mathematical intermediates are required to be the objects of mathematics, then there should also, by parity of reasoning, be appropriate sorts of intermediates to be the objects of other sciences; so, for biology, there would have to be “intermedi- 4, ARISTOTLE 4.1 Plato's ontology gave pride of place to supersensible entities—primarily those universal in character—existing in their own right, although sensible particulars have merely a cavernous or shadowy subexistence. Aristotle came to be scornful of all this. He argued against Plato's transcendentalism in favor of his own metaphysical priority ascribed to individual sensible substances. (1 leave aside his theology, with its strained adoption of the reality of sheer form, perfect and partless—and living.) Aristotle's philosophy of mathematics”! was shaped by his critical preoccupation with Platonism, no less than by his constructive need to show how mathematicals have only a dependent reality, i.e., dependent on individual sensible substances and on our processes of abstraction and reason. I will skim the rooftops of certain terraces of Aristotle's ontology and methodology of mathematics: in his rejection of Plato's doctrines of intermediates and of numbers; his conception of mathematics in terms of the axiomatic method; his ontology of mathematicals generally and of geometricals, numbers, and infinity severally; and, associated with this ontology, his proposal of abstraction as the funda| mental procedure of mathematical concept of formation. in existing entities intelligible purely as cals mathemati of doctrine 4.2. Plato's ate” animals apart from ordinary animals: a reduction to the absurd. (iii) Neither, e.g., lines nor numbers have any primary existence as substances, forms, or material substrata in Aristotle's senses of these terms; since, accordingly, mathematicals have only a subsidiary mode of existence, they cannot exist in their own right as the doctrine of intermediates demands. Aristotle's irritation with Plato's theory of Ideas in its various versions spurred him to adduce a collection of criticisms against the theory of Ideal numbers, cach composed of units specific to it (see 2.3 (e) above).TM (i) In mathematics a number may derive, as a sum or product, from a plurality of numbers; but how can one Idea come from a plurality of Ideas? (ii) Plato was unable to explain how the unitary character of each of his Ideas is possible if these are composed of units. (iii) The relationship between an Ideal number (¢.g., Three) and the correlative mathematical number (e.g., three|s) )—the latter being the object(s) of the mathematician’s study — was left obscure. (iv) Plato was unable to explain how the two fives in the Ideal number Ten differ from the five units in the Ideal number Five. More generally, Aristotle charged to catch Plato on the horns of dilemma. Either all the units are alike or (as Plato had supposed) they are unlike specifically. (1) If all the units are alike, then the Ideas, i.¢., Ideal numbers, would be related as parts to wholes; e.g., Three would be a part of Four and of all succeeding Ideal numbers. This result is

deductionmerebynotsomethingunderstandtocomewethatemphasizedheand pAirnitser-uetsprbs.emiecspoeiusl.tdetmrwouhleiofcgrhioctamhlidnheigdsucbotybisoepnrevriev.cda.d,headevmewoasnhsotmuraltdhiwoenm,astuibycstoaonlfcyem,beumtahtloeidrnieosalvaoglfyidstoartloen'gs wotHgtTehthrheeeaosorptnyussnrhotbuaeesarensbltturhdsirbebatcesfpaananotslldaryticnrtttseuichyootwnseium,ouscecnleb,xohgnAoastasrtteokniahrbfgscnnehsltircareounslbtxgs(lhespa,tslaenh,inasfremapedehirsrnm,aoens,evammoiltceoavibrartlethomphsefanfherettoyorahhstrreo—ihseybmxcotmjiasvfkaeOttrcnouiaehtsfocrs.wnaseasmNlolufmunersebuafbaitdsocdsiotrhgtpchnaaehoennyrrmtsaocncs,laeeyiai—tsnhscntu.fiehd.e;sdbcomanssoartvacuetfihmeslncbsaiashtnlmetitrcrsaehlobhueitalfaetfpkisthebhio?mescsnitAaamfsbvnhritttadeebthimonsoeetivl,alnreciio)tnadtnamhtdishle,ulehsieaasfsm,oerlnocavsFptoiethoanrtiariiohfcltssnnHricpaoeeg4diylosra.deeci4ms,nwbnsi.tcahesHimevaiedetess, them,”ofouttogether aowitcpdq“rhiounmdifedobi,amTpbm-s,tvoonaeMHbhtiddprrtoegiwdenyixio-atertlbi;mnass.owycsenuap,qtlh,b.hiurjeolasythdnif.elcns.candrguatcitedmnsod,hcieta.rpmoabvhtieslijun.phmgiaefsnrbsnaepcthecyilsaiahspoet,vysriasuhcnefityadrodrcv“it]omalcfiwsuaoeo.bahftplsjbmldt"leu.ruaueio[ih”ltec”myfrdntasu-eacheilssgmxkq.ra,ptieseunldgtshcam,yaehoiorttbtis,enefhaajncwyhitesearensaochydsnod,mliveWt[tsasleimfitohbats,inrhxyasiudo;hmgltpoirtbukei/nlcucfrjtamdiodlnagehstarioischcdygfhxev]iwsttnoeeeilvsidsromcniyinofrfstiancnedhtpnluot.ixsegfwhaoynvri"ouethlfscstrniblhaiiadtrjetbocutosil“rtiaasnhw{nhA,dcidtolotasele.dfhepptmobyis-a.iehe;of,ncomasl,ydvtebfrtywiiseh,ttpfoHecnemsoua)erhildt,tpiwanrsesoiatccmdyhgrasohbp.tealconveuAhwsth-t,igs.rtmlynehJeowesrs.ianhsddviet,kdcrtGwxchmiehohiyrcaaencsfemrl.hyotgaeloaww",tmellr(ioOnkphhtmeuagbfreshismyttdunvoseaiorridcn“k.yitsbgeolyfics?hAteuuaebno”wIl.iTsmndfreg+hclihnul.tmtave,oacny+rhluswtnrdi,ar,le.auhtotsmehnehtai”r,dolen i26PETEilRingH.anNIinDfinIiTteCrHegres. aiawtsldehproWmntAig('cfedsrunlbhpiqoyc.etnalsuifPhpjcen.)amrltgis:P,udy.n-ohctmselgaoifFnesPurktp:yzh-bAifona,lweshgt;dq”krcbo’ifmltuyn—sw,xbeaif(m.Arlcoxitn,2erhdmoTaisnAewbfouhtkcrvlnypbds;ufiemgacwht)'xsnlorihecya.dk,nfoptrlihaPgcsbeKmtfirpn,clsAeohwgintPaprodlsce;fmtay,wdhnuoibs'retcaldnyoeqhvsgmdi,ry;fb':aeut.nico(pad,mtes.urlonihbewTgyutfscanid)(t.bIesrulonvjhcf,tise)dagfontsichpub.digycsntraewlmudhigcbeAyoartpns,fhmidu.aocenglrbtsicdfmoaeyrn‘lhbuptzsf.eDyilomctsapfdeihrultnNo,.scpamiItdexucifrahslometpnFbhriuoa“epAldvycfta,smnr2ipoehbcydt;mni,rfaep—oquhslgitxdcer,bomakniwtcdresubflmhgo¢(ay,tunipl.ebrqadcfhmtgsnwelou.iH'jtm’ars2phye,fvltirhcpe.awd4ngos¢f”h,imtylebA.a;p)Ii,dhgnrflvo.ust;iadsmenr,gotKibahnucmlskogvrtdeunmwpNlfisethavOrukbodcmfilwsrtnedgmaoh'fcetIylrndios)ac,bmthenpi.oudjarsblhimtesngwyuhocarf(len.i,dsbp2auchonfsr)t(mliuedvgco.athnsbvIfprmzoe)lcgw-y(vr.itbna,feTocsHiurphdgmFeoltyrnidhSAcsfmae,p_trohcu;.Psinebplvfocuamrsytowlicenhfgruatps+bd:cv—wionehslp)t'arIc,g—dsb-lietynw"oa.v-“xifIdsetnu(necdikpe“ntwaru.ssodnhaesnrtsoda.w;hienign,t)ldy” MSFTIO2AHDEHARM7EFGSTIAETCS

ahnstwierohnouclsefanetfmuitueB-lwbnOaoexipcdhelsnintlo:irseamdcsltncitbp?yneehsslf,nvo,btaie4ecrcn.atoriti5entumcn,ufedpatagb,inolaignaesqtoianapeut‘dcirnahytitohelc.4uiine.saagf,dfm’4lliost,iea,nfh,atpu§edinhtsometiyd4rsbimesihs.ymenaisat2enfscrtqh,iaof,iunselnciparmetnecliwhop“dtaetfioruhyhslfeemoi,nars,nh”cqoegiaunsHotncdthifeoewdyjsatneotds—utlicperndielovmnnoieasH,udyxileeonwinthfuwtsiePsyhugamnA;hbiieeoarstyl,no—netsies.aAudietngr.bicto,osesftHihutma,e.bcnsllai"hinngasfiateniyeidhnfincaeoitplcpnruehuholIgdy,intanusheecbi.U,duectennraihAtltsplnAciioeegatrksoniynatekhfsd,aiaoletmigto4ysnenahr.utfmgl6Hbefi.nseuaoiitfmnrstminwbtyunshetiedoarntirefs.dt so2htisslaPdAnicsdatfrfol9imi'yeehuarfsntavnaarwicnptendmogsgslii-lesahtoblnieupmtngmcitaylwmi-erbfdshntbgosyaib.reeeilsditannam)likeontib,tsrsecant.ientohwruoM,eso,jgmphfyiefltn"Aann;tev(morchs,mfudlefantTpie!haaitrdsoabeuaforwskinestHgdopilre?dmMnsuhidenmmaptEhrkosHbceec,snbaegtMth,hmteswmlfaniechtsa,dsdrtAloadcarioetrbithelosaTmilgbmewicwpfaseh“nili,amenscociIhs?xaenttnT;eptaodoimyClsusarxhiirltpcaietfsiootSceshhdrlnxratpzbstcnetoeyibiklaaatnevitslipnlccOkdecyssohfmtowreiantdmFiie:ssnhtoqfscanirnydh.)acbouisecukue'kglrIrldwrtocfnc"ineif(taye,prDH'htanHsnlohdnabmmf'ycduycEiglebwlq*.o.trnkiitsdrwAiusenhnroteaiemttHcssifu,hnqwjlnvahmhosiTpsuiories.gteyagasnaneHr,nmrvaystPtlgaetpilhnrEh,ayeowdclxsohitngfmeutnfioydaisercceamhhnOtrhse.aAsircmaabbpggyhtnrivfsnoiFne.tegias)geb,tgdwsnicntioso“h.ilfeulgeSctirn(roatts-oncugapnylkmicnrhTestpateonglae.efr;fcsAmadessnoulaetemiohinhmsGfdacdi,ntpabsindebtrhlioswhEaeodrAwiegcdtvtpuaahnueheonhnafcntsisdaiFr,potexfcylodtainIhpsahtontacgilsuieylfRbnouohcer:nmltcinmateqrntdyouSseamicilhmub“donotwstanTilreeGfssnomeohatditlsxiotipw,evrsamFThcenotndhcsrpnieafThtdreneHm,oapcuigiasdmyfrcdnpaoxoErnlrveetamtlesbuavitan“fPrhlPitdeaincrveoadtlysll;trotteein,qTadayvuibhafedP,irnntmthpdoaerefgywnoelosinthtbgaafldpuoynhiidhcehrm/ttamAivntooe4nkagogehuakyrbHt;ty.irgnlcidsetgWibae7btne:ocsgivhpns.ghldtareciaflessthoooPseblprh”nl..trtliu,eaweA"sa:g”lbughni!ieti'entlbadbohasyvhnn!'olailerftf'g?smes.;anesl osaaprtio‘eldbhscdirTupmtfjtisyeas(nh.hbvtsmr4tfeuceig’ian),dtoFrmsaeywpcn.fstlgeroicbiaT,hu4nhyotrldigmMcea,ienshtiboArtral5donusbothem.yl,akipairesdon,hnzstfQcu’mrwayedtoekonsp,phacilwuqtgldrfecshauomuleiitr,spanbmac‘vhftosue'eridrjynlitavms.to,cwbdkeigh'ntersac.if°yrdml’toew,srfho”n-cu,:aibg—esyfmtrlnotacg.hosiwftroduacegp,nnlhxdotlmArneivif/auetsdpob,,hngcaiidruoebtyseahnuam(l“rd.eiv,gbsncpq2lhd"oynbtd)"uIv.acimls(gote;uafith,tsvfbGd(lce)hi”nsoasi(rhmueb.evd,stnl3oarmepusy)itfcAehagrose.mliny,asmictbrGhqtsuaiyoldf‘etnhrmpiicwo.!syuvsnierba°lmo.tcawrg*nktuBeysihla.tvhuosenldilcrazeAty,nCeraipoxuh,lgdvs1tbgrimeanyoh%dojcbtmaiwurelootsgaemcebn.m,lvAsotalrbeemiuchnsaTg'‘,rothelyvpm.gcsrponeOti;adv(uiylc.egpbthrnsofacuIize,fk.rldob).nhaus’,ftyeomlbacis.gp,ta”emhiow;nsfyitreabplxohm’unsidtgjrpe.vahsafBuntdowlier“mpsycoan.rIthlewtisfdun=Rh“misvtpyaAsdeclonghtimsarifiohnxbeslmtidcn;fa,ghp.tesldis:ropuagnm,e—ti.ludro|yscmpgahe1,n.rflcd5tie,boamygsrhutwln,fac(etdigrbo.ulyhsne1wbtafvarmpikgyseh,ln)tbfuQoocadgwitlreyfunai¢hdshmrog,ln.eadbapsiycrntpvmae=zdhoF.,-ictrdOshybageo-nit,-“srga pmMewta(dieholrespjcotvuykwsn)fehlsi,rcoyse

THE FIRST STAGE OF THE IDEA OF MATHEMATICS learned things, while he, stressfully, accommodated to this confinement an insistence, —of all ction perfe and tness abstrac y—the idealit the from Plato's orientation, on | predecessors two other assumptions about/ me" took over from these veridica list nature of mathematical thought icals. mathematics. One is the constativist and discourse. The second is the intelligibility of mathematicals. For him, it may be sug-e gested, this feature was allied to the fact that mathematical concepts are an outcom of a certain kind of mental activity and attention by us; he made en cepts our internalized constructions."!5 This was indeed an advance. But it stil on this drawback, that it presupposed the priority and the limitations imposed by the sensible world as the matrix of mathematical concept formation. 31 14. Philebus, 16c-d, trans. J. Gosling (Oxford: Oxford University Press, 1975). 15. Aristotle, Metaphysics, AS, 9862-3; Aristotle, On the Heavens, ¡¡.9, 290b12-14, trans. T. L. Heath, Aristarchus of Samos (Oxford: Oxford University Press, 1913), p. 105. 16. System of Logic, I1.vi.2. J. Klein, Greek Mathematical Thought and the Origin of Algebra, trans. E. Brann (Cambridge, Mass., and London: M.I.T. Press, 1968), Part 1, stresses the ‘number of" sense of aritbmos. Cf. p. 28 below. 17. Metaphysics, M6, 1080a22-33. Italics in quotations are mine unless the contrary is noted. 18. Metaphysics, M6, 1080b16-21; M8, 1083b8-19, 19. My term ‘configuration’ covers the Pythagorean doctrine that solid bodies are formed of planes, planes of lines, and lines of monadic points. Relevant details are given in, e.g., W. K.C. Guthrie, History of Greek Philosophy (Cambridge: Cambridge University Press, 1962), Vol. 1, pp. 259ff. 20. Metaphysics, AS, 986a1 5-21. 21. Cf. W. D. Ross, Aristotie’s Physics (Oxford: Oxford University Press, 1936), pp. 542-45. 22. Philolaus, fr. 1, trans. Guthrie, op. cit., p. 330. Notes 1. In my usage the term ‘mathematical philosophy” has the forefront sense of and ical conof mathematics’; in appropriate cases, it can also carry *(a) philosophy with a mathemat | | tor style’ as a background sense. Cf. 83.1. 1900) { pre-Socratics and Plato n 2 Pd Milhaud, Les Philosophes-géomètres de la Grèce (Paris, ical Philosopby (New York: Alcan only), and E. A. Maziarz and T. Greenwood, Greek Mathemat Ungar, 1968), are cases in point. | LIM (1979), pp. 1-21. u ; Vol. "i Cf. my “Preface to the Grammar of Postulates," Aristotelian Society Supplement,u 4. For the Latin phrase, see Ockham, Philosophical Writings, ed. P. Bochner (London: Nel e E. M. Kleinberg, infinitesimal si 5. On ‘a system of hyperreal numbers, cf. J. M. Henle and and H. J. Keisler, Elementary . 1957), p. 41. Calcutus (Cambridge, Mass., and London: M.LT. Press, 1979), Calculus (Boston: Prindle, Weber and Schmidt, 1976). Austin, es 6. The term ‘illocutionary force’ originates on p. 100 (cf. pp. 98-99) of 3. L.index by P. H. with Sbisä, M, and Urmson O. J. ed. rev.. ed. 2nd Words, to Do Things with si | Nidditch (Oxford: Oxford University Press, 1980). 7. Metaphysics, AS, 985b23-24, following trans. by W. D. Ross. (A translator's name is cited in these notes only on the first relevant occasion.) 8. In Metaphysics, Physics, On the Heavens, and elsewhere; see W. T. Organ, An Index to ns. Py iversi y Press, 1949), s.v..v. Pythagorea Universit 3 seal (Princeto: n, N.J.: Princeton Aristotle a Ta 9. Translated in K. Freeman, Ancilla to the Pre-Socratic Philosophers (Oxford:c DE a«a of the Hellenisti 1947). Cf. H. Thesleff, An Introduction to the Pythagorean Writings of the Hellenistic Peno (Abo: Acta Academiae Aboensis, 1961), and The Pythagorean Texts Acta Academiae Aboensis, 1965). 10. E.g., by E. Frank, Plato und die sogenannten Pythagoreer (Halle: Niemeyer, 1923), pp. 313-15, footnote, and W. Burkert, Love and Science in Ancient Pythagoreanism, trans, E. L. . Minars (Cambridge: Harvard University Press, 1972), pp. 273-75. Cf. n.25. | 11. Cf. Burkert, op. cit., pp. 401ff.: B. L. van der Waerden, "Die Arithmetik der Pythagoreer, 23. Diogenes Laertius, viii.8, trans. G. S. Kirk and J. E. Raven, The Presocratic Pbilosopbers (Cambridge: Cambridge University Press, 1957), p. 228. Cf. Aristotle, Protrepticus, fr. 11. 24. Cf. lamblichus, Comm. math. sc., 25: "The Pythagoreans, having devoted themselves to mathematics, and admiring the rigour of its arguments, because it alone of the studies men undertake contains proofs . . . ," trans. }. Barnes, The Presocratic Philosophers (London: Koutledge, 1979), Vol. 2, p. 78. The text is given in Burkert, op. cit., p. SO, n.112, who argues that it is derived from a now lost work by Aristotle, On the Pythagoreans. Archytas, fr. 4, links Proof primarily with arithmetic, which is therefore superior to geometry. 25. The stock criticisms of the fragments’ authenticity, which stem mainly from Bywater and Frank and are conveniently adduced by Kirk and Raven (op. cit., pp. 308-11), are poorly reasoned, as was recognized by Guthrie (op. cit., pp. 331-32). Cf. G. de Santillana and W. Pitts, “Philolaus in Limbo,” Isis, Vol. 42 (1951), pp. 112-20. 26. Philolaus, frs. 4, 6, 11, trans. Freeman, op. cit. 27. Metapbysics, AS, 985b23-26. 28. See Burkert, op. cit., pp. 369ff. 29. Metapbysics, N3, 1090b20-25. 30. Cf. Philolaus (Diels-Kranz) A29, and n.24 above. A. Szabd, The Beginnings of Greek Mathematics (Dordrecht and Boston: Reidel, 1978) is interesting but is one-sided in stressing the role of dialectic in the rise of the deductive method. 31. I. Thomas, Greek Mathematical Works (London and Cambridge, Mass.: Harvard University Press, 1939), Vol. 1, pp. 148-49. 32. Aristotle treated quantity as an important category; mathematics has an important place in his scheme of the sciences; and in his Posterior Analytics it has, in some respects, a prototypical role among the sciences. 33. See, e.g., A. C. Crombie, Robert Grosseteste (Oxford: Oxford University Press, 1953), pp. 110ff., 139ff. 34. Cf. Aristotle, On the Heavens, 293a21-22. 35. Metaphysics, M8, 1083b11-19, 36. Aristotle uses both terms in his exposition in Metaphysics, AS, 986a. 37. Of course, the Pythagoreans had no cognizance of zero or of Negative Math. Annalen, Vol. 120 (1947-49), pp. 127-53, 676-700; B. L. van der Waesden, science Awakening, trans. A. Dresden (Groningen, Noordhoff, 1954), per index s.vv. "Pythagoras, "Pythagoreans;” W, R. Knorr, The Evolution of the Euclidean Elements (Dordrecht and Boston: Universe is too general and indefinite (cf. Philolaus, fr. 6) to provide a means of defense against Reidel, 1975). the criticism made in the paper. A 12. "Natures": Aristotle, Metapbysics, AS, 985b34; “causes”; A6, 987b24-25, and AB, 990a19-20. a 13. Essay Concerning Human Understanding, ed. P. H. Nidditch (Oxford: Oxford University Press, 1975), ILxvi.1: 205(8-10). My cross-quotations are meant to be taken as implying a subintegers. 38. "Harmony" as the agency of the combination of the limit and the unlimited in the 39. Cf. Aristotle, On the Heavens, ii.9. 40. Cf. the Euclid scholium quoted in Thomas, op. cit., (in n.31), pp. 214-17. 41. Cf. Aristotle, Metaphysics, AB, 989b33-34. 42. Aristotle, Metaphysics, AS, 986a8-9.

THE FIRST STAGE OF THE IDEA OF MATHEMATICS 44. Introduction to Mathematical Philosophy (London: Allen and Unwin, 1919), p. 1. 45. Cf. Aristotle, Nicomachean Ethics, 1131a20ff. 46. Cf., c.g., Aristotle, On the Soul, 404b24-25, and the Aristotelian references in W. D. Ross, Plato's Theory of Ideas (Oxford: Oxford University Press, 1951), p. 216 n. The term ‘Unwritten Doctrines’ comes from Aristotle, Physics, 209b15. A compendium of translations of source passages on the Unwritten Doctrines is in J. N. Findlay, Plato: The Written and Unwritten Doctrines (London: Routledge, 1974), Appendix |. 47. Philolaus, frs. 8, 10, trans. Freeman. The Parmenidean One should be kept in mind as a part of the background to Pythagoreans and Plato on One. 48. Cf., c.g., Republic, 507b, 596a; Parmenides, 130a-d. 49. Cf. G. Leff, William of Ockham (Manchester: Manchester University Press, 1975), pp. 164ff. Sameness and Difference are among the important higher-level kinds or Ideas in some later Dialogues, ‘Number’ is a “transcendental” at Sopbist, 238a-c. 72. Republic, 485b, 526a-b, 526d-e (cf. 508-9), 527a-b, trans. P. Shorey (Cambridge, Mass., and London: Harvard University Press, 1935). Cf. Sopbist, 2382. 73. Republic, 531d, 537c. 74, Metapbysics, A6, 987b14-17. 75. E.g.. J. Adam, The Republic of Plato (Cambridge: Cambridge University Press, 1902), Vol. 2, pp. 115 n., 159ff.; A. Wedberg, Plato's Philosophy of Mathematics (Stockholm: Almquist and Wiksell, 1955), p. 124; J. A. Brentlinger, “The Divided Line and Plato’s Theory of Intermediates,'” Phronesis, Vol. 8 (1963), pp. 146-66. (Brentlinger’s interpretation of Rep., 534a, which he regards as decisively showing that the objects of dianoia and noësis are on different ontological levels, is [1 think] unfounded.) 76. Republic, 526a; cf. Philebus, 56d-e. 77. Republic, SV11:cf. 533-34, 78. Republic, 510d, 525d-e. 50. Cf., e.g., Republic, 476a2-4. 79. Republic, 510a-511b, 525d. 51. Parmenides, 137aff. 80. Republic, 510-11. 52. F. M. Cornford, Plato and Parmenides (London: Routledge, 1939), p. 111. Locke, Essay, 81. Republic, 511, 532, 533. 82. Republic, $33b-c. Il.xvi.1, 2, slides from one to another of "one," “unity,” and “unit.” $3. Findlay, op. cit., is the fullest committed presentation in English. Guthrie, History of Greck Philosophy (Cambridge: Cambridge University Press, 1978), Vol. 5, Chapter 8, contains references to the relevant literature -and a sharp critique. 54. I follow the chronology in J. B. Skemp, Plato (Oxford: Oxford University Press, 1976), pp. 13ff., 52ff. Ross, Plato's Theory of Ideas, p. 10, dates Meno earlier, to before Plato’s first 83. In Timaeus (somewhat earlier among the later Dialogues than Philebus), numbers, proportions, and shapes play prominent roles cosmologically. The influence of Pythagoreanism on this is apparent. 84. Philebus, 55d-57e, trans. Gosling. 85. Philebus, 58a-59d. 86. Seventh Letter, 342-43, trans. G. R. Morrow, Plato's Epistles (Indianapolis and New Sicilian visit, 389-388 B.C. 55. Meno, 81a-d, trans. Guthrie (Harmondsworth: Penguin Books, 1956). On “the other world” (the realm of Hades), cf. Phaedo, 68a-b; and note “Hades under the earth" at Republic, 596c. 56. ‘Recollection’ is hardly re-collection: the slaveboy does not remember the original York: Bobbs Merrill, 1962). 87. Aristotle, Metapbysics, M4, discusses what it claims was the original form of Plato's theory of Ideas. 88. Aristotle, Metaphysics, A6, 987b18-25, Ross's trans. adapted; see also N2, 1090a4-6, and frag. 4 (Ross). Cf. Ross, Plato 's Theory of Ideas, Chapter 15. acquaintance or occasion. 57. Cf.. c.g., Republic, 458d. 89. Aristotle, Metaphysics, AG, 987b27-29, Ross's trans. adapted. 90. Cf, p. 12 above. For elucidations of Plato’s principles of generation of numbers, see 58. Cf. Republic, 491. 59. Meno, 90b. 60. Meno, 90a. 61. E.g., J. A. Stewart, Plato's Doctrine of Ideas (Oxford: Oxford University Press, 1909), Ross, Plato's Theory of Ideas, pp. 182ff.; and J. Annas, Aristotle's Metaphysics, Books M and N (Oxford: Oxford University Press, 1976), pp. 42ff. 91. Useful sources and references include: Aristotle, Metapbysics, Books p. 28. (The word eidos at Meno, 72c, does not mean “Idea.”) 62. Cf., e.g.. Pbaedo, 109e; Republic, 500c; Phaedrus, 247c. 63. Meno 81d. 64. This sense of ‘thesis’ was introduced by Lesniewski and made well known by Lukasiewicz. Cf. Lukasiewicz in S. McCall, ed., Polish Logic, 1920-1939 (Oxford: Oxford University Press, 1967), p. 44 n. 65. Meno, 86e-87a. On the whole matter, see R. S. Bluck, Plato's Meno (Cambridge: Cambridge University Press, 1961), pp. 75ff. 321 ff., 441ff.; and R. Robinson, Plato's Earlier A, È, Z, K, M, and N; T. Heath, Mathematics in Aristotle (Oxford: Oxford University Press, 1949); J. Barnes, Aristotle's Posterior Analytics (Oxford: Oxford University Press, 1975), with bibliography. 92. I select from Metaphysics, M2, 1076b11-1077436. 93. I select from Metaphysics, A9 and M6-8. 94, Metaphysics, A9, 991b31-992a1. 95. Metapbysics, M2, 1077a32-36. 96. Aristotle, Physics, ii.2, 193b24-194a12, trans. KR. P. Hardie and R. K. Gaye (Oxford: Oxford University Press, 1930). Dialectic, 2nd ed. (Oxford: Oxford University Press, 1953), Chapter 8. 66. Phaedo, 65c-e, 74a-7 5b, etc. 97. Metapbysics, Z10, 11. 67. See especially Phaedo, 101c, 104-5, and the commentary ad locc. in D. Gallop's transla- 98. Aristotle, Nicomacbean Ethics, vi.6, 1140b31-32, trans. Ross. tion (Oxford: Oxford University Press, 1975). (Gallop's criticism of Vlastos, on p. 186, is [1 99. Metaphysics, El, 1025b16-18. think} partly wrong.) 100, Metaphysics, M1, 1076236-37. 68. Phaedo, 74a-75b. 101. See Metaphysics, Z11, 1036a11-1036b3, 27-28, and K3, 1061a28-35. 69. Phaedo, 75c, 104a, 105c. 102. Aristotle, On Memory, 450a1-8. 70. Phaedo, 101b, c, trans. Gallop. My objection about specific differences at the end of (i) is made pace Cook Wilson, Ross, Gallop, et al. on 71. Republic, c.g., 382bff., 389b-c, 485c:; cf., c.g., Timaeus, 29c. | use ‘proposition’ in a broad sense, to contrast with *objectual”. 33 103. Cf. the notes on Chapter 6 of Aristotle's Categories in J. L. Ackrill's edition (Oxford: Oxford University Press, 1963). 104. Metapbysics, 413, 1020a7ff. 105. Metaphysics, A6, 1016b24ff.. and K2, 1060b10ff. 106. Cf. A.C. Lloyd, Form and Universal in Aristotle (Liverpool: Cairns, 1981), pp. 32ff.

107. Cf. Metaphysics, 428, Z10, 11. 108. Metaphysics, NS, 1092b19-20. 109. Metaphysics, A6, 1016b20ff.; Aristotle's whole chapter is a valuable discussion of uses of ‘one’. 110. Book Ill, Chapters 4-8. Valuable discussions are in Ross's editions, pp. 48ff., and in J. Hintikka, Time and Modality (Oxford: Oxford University Press, 1973), Chapter 6 (though its main purport is [I think] mistaken). 111. Physics, iii.6, 206a27-29. 112. Physics, iii.7, 207b10-13. 113. Physics, 111.7, 207b10. 114. Physics, iii.6, 207a7-8. 115. Aristotle, On the Soul, iii.4-7, is relevant in this connection.