wrigley - wittgenstein's philosophy of mathematics

7/29/2019 Wrigley - Wittgenstein's Philosophy of Mathematics

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Wittgenstein's Philosophy of MathematicsAuthor(s): Michael WrigleyReviewed work(s):Source: The Philosophical Quarterly, Vol. 27, No. 106 (Jan., 1977), pp. 50-59Published by: Blackwell Publishing for The Philosophical QuarterlyStable URL: http://www.jstor.org/stable/2218928 .Accessed: 23/07/2012 17:22

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50

WITTGENSTEIN'S PHILOSOPHY OF MATHEMATICSBY MICHAEL WRIGLEY

Wittgenstein'sontributionso thephilosophy fmathematics,n con-trastto his work n so manyotherareas ofphilosophy, ave oftenbeendismissed s of ittlevalue. GeorgKreisel,for xample,endshisreview fthe Remarksn theFoundations fMathematicsy saying hat "it seemstometo be a surprisinglynsignificantroduct f a sparklingmind".' I holdtheopposite pinion,ndmyaim nthispaper sto show hatWittgenstein'sphilosophyfmathematics as onlybeenheld n such owregardbecause thas beenmisunderstood,nd that nfact t is no lessoriginal nd importantthantherestofhis work.2Some writers ave takenWittgenstein'shilosophy fmathematicsobe an extreme orm fconstructivism,.e., he is said to holdthat theonlyvalid mathematicss thatwhich sesconstructiveroofmethods. Thistypeofphilosophy fmathematicss familiar romBrouwer, utWittgenstein'sconstructivisms supposed o be of a muchmorerestricted indthanBrou-wer's ntuitionism.Wittgensteins, we are told,a strict initist ho holdsthat theonlycomprehensiblend valid kind ofproofnmathematicsakestheform f ntuitivelylearmanipulationsfconcrete bjects. In hisarticleon the RemarksMichaelDummetthas attributeducha position o Witt-genstein.According o Dummett"Wittgensteindopts a versionof con-structivism"3 hich "is of a muchmoreextremekind than that of theintuitionists".4imilarlyaul Bernayswrites f"thefinitistndconstructiveattitude . . takenbyWittgensteinowards heproblemsfthefoundationsofmathematics"5nd he explicitly laimsthat "[Wittgenstein]maintainseverywhere standpoint fstrict initism".6 ut suchan interpretationscompletely t odds withWittgenstein'seneralconception f philosophy.In the nvestigationsereadthat"philosophymay n noway nterfere iththeactualuse of anguage; t can in the endonlydescribe t. For it cannotgive it a foundation ither. It leaves everythings it is. It also leaves

1G. Kreisel, "Wittgenstein's Remarks on the Foundations of Mathematics", BJPS,IX (1958-9), p. 158.2I have been much helped in writingthis paper by the discussions I have had onWittgenstein'sphilosophy of mathematics with Dr B. H. Slater. I must also thankDr Colin Radford, Mr AnthonyHodgetts and Mr Philip Welch fortheirhelpfulcom-ments on an earlier draft, and Mr Roger Picken and Mr. Michael Sissons for theirassistance in translating passages fromWittgensteinnd der Wiener Kreis.3Michael Dummett, "Wittgenstein's Philosophy of Mathematics", PR, 58 (1959),reprinted in Readings in the Philosophy of Mathematics,edd. P. Benacerraf and H.Putnam (Oxford, 1964) and in Wittgenstein: he Philosophical Investigations, d. G.Pitcher (London, 1968): Pitcher,p. 424.40p. cit.,note 3, p. 439.6p. Bernays, "Comments on Ludwig Wittgenstein'sRemarks on theFoundations ofMathematics",Ratio II (1959-60). Reprinted in Benaceraff and Putnam, p. 522.60p. cit.,note 5, p. 519.


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WITTGENSTEIN'S PHILOSOPHY OF MATHEMATICS 51mathematicss it is andno mathematical iscoveryanadvance t" (? 124).But an extreme onstructivisthilosophy fmathematicsnvolvesdrasticrevisions fmathematics nd by no means leaves it as it is. As Bernayspointsout, "for the strictly onstructivist iew a large part of classicalmathematicsimplydoes not exist".7 So it is quite clearthatDummett'sand Bernays' nterpretationfWittgensteinannotbe correct.A differentnterpretationfWittgenstein,ut one whichnonethelessmakeshima kindoffinitist,as beenputforward yKreisel and byKiel-kopf. Kreisel claimsthat "Wittgenstein'siewson mathematics re nearthoseofstrict initism" uthe adds the qualificationperhapsone shouldsay he concentrates n the strictly initisticspectsof mathematics . .[because]all the mathematicswhich[he] considers lear fitscomfortablywithin he frameworkf strict initism".8 ielkopf oncludeshis examina-tion ofWittgenstein'shilosophy f mathematics y claiming hat "[Witt-gensteins] an open-endedtrict initist",9hat s,he "acceptsstrict initismas an adequate philosophyfonlyas muchmathematicss can be donebystrictfinitisticmeans. However,he resolves o understand he remainderof mathematics y deviating s littleas possiblefrom trict initistichil-osophy".10Kreisel and Kielkopfare, I think,puttingforwardmuch thesame interpretationf Wittgenstein.Having graspedthe point that forWittgensteinhilosophys a descriptivectivity heyreconcile hiswiththeir laimthathe is a finitistyclaiminghat he is, at anyrateprimarily,interested nly in those veryelementaryreas of mathematics fwhichstrict initisms thecorrect escription.However, ven when hus modifiedtheclaimthatWittgensteins a finitistoes notsquarewithhisdescriptiveconception fphilosophy.Even for hosepartsofmathematicswhichcanbe re-done n a finitistic ay the finitistic ersion s a differentiece ofmathematics rom he original.We do not do any of our mathematicsna finitistic ayat present nd so ifWittgenstein'sim is to describemathe-matics s itnow s he couldnotacceptstrict initisms a correct escriptionofanypartof t. Nor s it truethatWittgensteins primarilyronly nter-ested nvery lementarymathematics.WhilstWittgenstein's athematicalexamples re predominantlyfan elementaryind, nd this s particularlytrueofthe Remarks n theFoundations fMathematics-theworkKreisel,Kielkopf,Dummett nd Bernaysbasedtheir nterpretationsn-if we con-siderWittgenstein's orkon the philosophy f mathematics s a whole,including he works whichhave been publishedsince the Remarks-thePhilosophicalGrammar, hilosophicalRemarks, nd Wittgensteinnd derWienerKreis-it becomesclear thathis interest n mathematicss by nomeans imited o itsrelatively lementaryreas. As we shallsee later, here

70p. cit.,note 5, p. 522.80p. cit.,note 2, pp. 147-8.9C. F. Kielkopf,Strict Finitism: An Examination of Wittgenstein'sRemarks on theFoundationsofMathematics" The Hague, 1970), p. 186.10Op.cit.,note 9, p. 182.


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52 MICHAEL WRIGLEYaregoodphilosophicaleasonswhyWittgensteinoesavoidvery omplicatedexamples, uttheyhavenothingo do withfinitism.I wantto devotetherestofthispaperto thepositive ask ofsetting utsomeof the themesnWittgenstein'shilosophyf mathematics hich eemto me to be the mostfundamental.Naturally have had to omitmanyimportant opicswhichWittgensteinonsiderst length, uchas theprob-lems concerninghe applicability f mathematics,he questionof whatconstitutes proof n mathematics,he infiniten mathematics,tc., butneverthelessthink hatthetopics do deal withgivea clear nough ictureofWittgenstein'shilosophyf mathematics or tshighly riginal haracterto emerge.

SincephilosophyorWittgensteins essentially descriptivectivity heaim ofthephilosopherfmathematicss tounderstandmathematicss it s,and it is no businessof his to criticize heway mathematicianso abouttheirwork. So it is no partofthephilosopher'sask to rewrite ll orpartof mathematicsnnewand supposedly etterways, s the ogicist, ormalistand intuitionist hilosophers f mathematicshad thought t was. Thephilosopherhould focushis attention n whatmathematiciansctuallydoand tryto give a correct escriptionfit. He shouldbe careful ot to bemisledby the descriptions hichmathematicianshemselves ive ofwhattheyare doing,forthesemay verywell be philosophicallyuite confused.Augustineknewwell enoughhow to use the word time' but could notanswer he question whatis time?". t is the same,Wittgensteinelieves,with mathematics. Mathematicians an discovermathematical acts but"what a mathematicians inclined o say abouttheobjectivitynd realityof mathematical acts s not a philosophy fmathematics, ut somethingforphilosophical reatment"PI 254).11 So if the philosopherwants todescribe mathematicshe must be careful o separatemathematicstselffrom he thingswhich mathematiciansnd philosophers ave said aboutmathematics. Time and again" Wittgensteinmphasizes I would iketosay: what I checkis the accountbooksof mathematicians;heir mentalprocesses, oys, depressions,nd instincts s they go about theirbusinessmaybe importantn otherconnections,ut theyare no concern fmine"(PG 295).Wittgensteinaw this distinctionetweenmathematicstself, nthe onehand,and what s said aboutmathematics,n theother, s of fundamentalphilosophicalmportance.He held that all the problemsof traditionalphilosophy fmathematicswhichappearedto be problemswithinmathe-maticswere in fact just confusionsn what was said aboutmathematicswhich esulted romn incorrectescriptionfmathematics.Likelanguage,mathematicsmustbe in order ust as it is, and the beliefthatphilosophical

llIn referencesto Wittgenstein'sworks I use the followingabbreviations: PI-Philosophical nvestigationsfollowed by sectionnumber); PG-Philosophical Grammar;PR-Philosophical Remarks;RFM-Remarks on theFoundations ofMathematics;andWWK-Wittgenstein und der WienerKreis (all followedby page number).


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WITTGENSTEIN S PHILOSOPHY OF MATHEMATICS 53investigationsf mathematicsmight ave repercussions ithinmathematicsand perhapsshow that certain areas of mathematicswere in some wayinvalidmust be mistaken.Philosophical roblems bout mathematics, sabout anything lse, are, in Wittgenstein'siew,simply matterofnotseeing hings learly.Whenwe achieve clearUbersichtf hesubjectmatterthephilosophical roblemshoulddisappear. In philosophyfmathematicsa firstteptowards chieving his s to separatethemathematicsrom hecomments,sides and explanationswhich ccompanyt-what Wittgensteincalls"prose". Theproof fa theoremsmathematics,ut a verbalexplana-tionof tssignificances "prose". As Wittgensteinxplained o Waismann,"it is very important o make the strictest ossibledistinction etweencalculus ndprose.Oncethisdistinction as been madeclear, hen ll thesequestions uch s consistency,ndependence,tc., renotraised" WWK 149).In thispassageWittgensteinontrastshe verbal "prose"which ccom-paniesmathematicswith the "calculi" whichmakeup mathematicstself.Wittgensteins usingthe term calculus' n a specialsense, nd in factthisconceptofa calculus ies at theheart ofWittgenstein'sccount ofmathe-matics. So I now wanttodiscuss hisconcept nd show howWittgenstein'sviewson someparticularopicsflow romt.Fregehad thought hat eithermathematicswas simply bout the pro-perties fmarks n paperor else thosemarks tood or omethingndwhattheystoodforwas thesubjectmatter f mathematics. ince,for xample,thesign 0' does not have thepropertyhatwhen ddedto thesign 1' thenit givesthe sign 1', Frege arguedthatmathematical ropositionsmust beabout certain ntities. But Wittgensteinhowed hatthere s a third lter-native,forconsider he question"whatis chessabout?". Plainly t is notabout thepropertiesfthepiecesused to playthegame,but on theotherhand thepiecesdo not stand oror meananythingn Frege'ssense. Witt-gensteingives the same answerto the question "what is mathematicsabout?". "I've beenasked . . . whether believe hatmathematics as todo with nkmarksn paper. To that answered:n exactly he samesenseas chess has to do withwoodenpieces" (WWK 104). Wittgensteins thussayingthatmathematical ormulae o not standforor meananythingnthemselves, ut have significancenly n so faras theyare manipulatedaccording o rules. A "calculus" is just a particular rocedure ormanipu-latingmathematical ormulaewhichis defined implyby its rules. InWittgenstein'ssage to "calculate"means to operatewithin calculus, othatanypieceofmathematics,e it a pieceofalgebraoranalysis relemen-tary rithmetic,s justas much pieceofcalculation.Wittgenstein'sccountof mathematics an now be stated very simply. "Mathematics onsistsentirely f calculations" PG 468), and so it is a mistake o think hat nmathematicswe are dealingwith propositions; ormathematicshas nosubjectmatter nd is not aboutanything.A mathematical proposition"such as 'Jexinx dx=- eX sinx - cosx)' or ei =-1', is not a proposition


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54 MICHAEL WRIGLEYat all, for t is just a piece in the symbolic game and has no sense. Mathe-matical "propositions" thus cannot be true or false, and so "proof" inmathematics is not a proofat all in the logical sense. Even mathematicalpropositionswhichcontainwords,such as 'fis differentiablend continuous',have no sense and do not express genuine propositions. Here too we arejust dealing with a formula n a calculus which is a mere piece in the gamealong with all the other signs. "In mathematics", Wittgenstein tells us,"everything is algorithm nd nothing meaning; even when it doesn't looklike that because we seem to be using words to talk about mathematicalthings. Even these words are used to construct an algorithm" (PG 468).Traditional philosophers of mathematics have all failed to realize this,and have puzzled over the status of mathematical propositions-are theytruths of logic (Frege and Russell), synthetic a priori propositions aboutourforms f sensible intuition Kant), ordescriptionsofmental constructions(Brouwer)? Wittgensteinshows that thereare no mathematical propositionsor truths,and so all the theories of traditional philosophy of mathematics,which flow fromthe assumption that there are, are so many Luftgebdude.In set theory the idea that Cantor was proving propositions about the"transfinite"generatedstormsofcontroversy. But both Cantor's supportersand his opponents were talking nonsense, for,as Wittgensteinremarked inanother context, "the decisive movement in the conjuring trick has beenmade, and it was the very one that we thought quite innocent" (PI 308).This view ofmathematical "propositions" leads to radical differences e-tween Wittgensteinand all the traditional philosophies ofmathematics. Inparticular it makes it quite clear that Wittgenstein could not have been afinitistof any sort, for even the finitistthinks that mathematics provespropositions about something.A picture which many traditional philosophers of mathematics haveworkedwith s that certainparts ofmathematics-set theoryand thevariousbranches of mathematical logic-are more fundamental than the rest.Consequently it has been thought that it is philosophically illuminatingtostudy those parts of mathematics, generallyknown as the "foundations ofmathematics". For example, an eminent contemporarymathematician andphilosopher, Jaakko Hintikka, has said that "it is not likely that anysubstantial progress can be made in the genuinely philosophical study ofmathematics without using the concepts and results of [symbolic logic andfoundational studies] to a much greaterextent than has happened so far".12Wittgenstein is completely opposed to such a view. Of course any philo-sophically valuable workin philosophy ofmathematics requires a knowledgeof more than elementary school mathematics, but Hintikka's suggestionthat the so-called "foundations of mathematics" are peculiarly relevant tothe philosopher is mistaken. Wittgenstein's position is that "the mathe-matical problems of what is called foundations are no more the foundations12K.J. J. Hintikka (ed.), The PhilosophyofMathematics London, 1969), p. 1.


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WITTGENSTEIN'S PHILOSOPHY OF MATHEMATICS 55of mathematics han a paintedrock is the supportof a paintedtower"(RFM 171). The "foundations f mathematics" rein no waymorefunda-mental hananyotherpartofmathematics.Theyhave merely een madeto look s ifthey re because oftheterminologyhichhas been ncorporatedintothesecalculi,and in thisway theyhave acquiredtheir aura ofphilo-sophical ignificance. ut appearances remisleading-"It is said thatthosenotions rehighly bstract nd wonderfullyeep. But theapparentdepthcomesfrom wrong magery.Theyare ust as pedestrians anycalculi".13The idea behind hetraditional onceptions "sincethepropositionsftherestofmathematicsan be analysed ntopropositionsboutsettheory ndmathematicalogic as the effortsf thelogicists howed) hen theproposi-tionsof set theory nd mathematicalogicare thosewhich upport herestofmathematicsnd on which hevalidityof otherbranches f mathematicsdepends".But forWittgenstein,ince here re nomathematicalropositions,thewholepicture s a false one. Results n algebra, opology nd analysis,etc.,do not logicallydependon propositionsbout "foundations" ecausethey renotpropositionsnddo not ogically ependonanything.fmathe-matics can be said to have foundations t all theyare ofquite a differentkindfrom hat usuallysupposed. Mathematics as its foundationsn theactivity fcalculating nd outsidethis t neither as norneedsany otherfoundations. What we have to do is to describe he calculus-say of thecardinalnumbers-thatis, we mustgive its rules and by doingso we laythefoundationsfarithmetic. each it to us and youhave laid its founda-tions" PG 297).An immediate onsequence fWittgenstein'sescriptionfmathematicsas consistingentirely f calculations" s that there an be no "metamathe-matics". Metamathematicss conceived f as expressing ropositionsboutmathematicsna formalmetalanguage,ut sinceforWittgensteinhecalculiof mathematics o not expressanything, formalmathematical meta-language" annot xpress ropositionsboutmathematics. heonly anguagewhichcan do this n thewaymetamathematicss supposedto is ordinarynon-formalizedanguage,but this is "prose" and not mathematics.ForWittgenstein,o-called metamathematics s just more mathematicsonall fourswith herest. It is nomore boutmathematics han chess s aboutdraughts. fthecalculiof"metamathematics"ppearto contain ropositionsaboutmathematicshen hat s justthemisleadingesult f their ontainingwords as well as othersigns,but hereas everywherelse in mathematics"everythings algorithm nd nothingmeaning". Wittgenstein sed hisfavourite nalogyto explain this point to Waismann: "I can play withchessmenccording o certain ules. But I can also invent game nwhichI playwiththe rules ofchess, nd the rulesofthegameare,say, the rulesof logic. In that ase I haveyet nother ameand not a metagame.What

13Wittgenstein, 939 lectures on mathematics, notes taken by Norman Malcolm(unpublished), pp. 65-6. (Cf. Wittgenstein'sremarks about the "charm" of Cantor'swork,Lectureson Aesthetics, sychology nd Religious Belief, p. 28.)


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56 MICHAEL WRIGLEYHilbertdoes is mathematics nd notmetamathematics.t is another al-culus just likeany other" PR 319). This radical attitude o metamathe-maticsmeans thatmetamathematicalheorems ikeG6del's ncompletenesstheorem, ecidability esults, tc., have onlyas muchphilosophical ignifi-cance as anyothermathematicalheorem,o wit,none.Hilbert'swhole motivationforinventing metamathematics"was ofcourseto deal withtheproblem fconsistency,nd, as we shouldexpect,Wittgenstein'sadical views on metamathematicso hand in hand withequally radical views on the questionof consistency.Wittgenstein's ayofdealingwith heproblemfconsistencystoshowuswhat n inconsistencyinmathematicseallys so that t becomes learthat traditionaldeas aboutinconsistencynd theneed forconsistencyroofshave arisenfrom n in-correct haracterizationf what it is for a mathematical alculus to beinconsistent.The root ofthe trouble s the mistakendea that in mathe-maticswe aredealingwithpropositions.t is thisthat makes hepossibilityof inconsistencyo alarmingbecause it appears that in an inconsistentsystemwe can provecontradictoryropositions.As Wittgensteinaid toWaismann: "the idea of inconsistency . . is contradiction,nd this canonlyarisein the rue/falseame, .e.,whenwe aremaking ssertions"PR321). Wittgenstein'sasic thesis s of coursethat "playingthe true/falsegame" is just what we are notdoingin mathematics. n mathematicalcalculi all we do is calculate, nd we do notprove nythingn the logicalsense,and so the possibility fproducing logicalcontradictionoes notarise. Since all we do is to calculateall thatcan go wrong s that we findourselves n a positionwhere we cannot calculatefurther,nd once thispointhas beengraspedtheneedfora consistency roofvanishes, or t isseen thatwhenan inconsistencyrises nmathematics,he situations pre-cisely nalogous o findinghattherulesof a gameconflict,nd is as easilydealt with. "If an inconsistencyere o arisebetween herulesofthegameofmathematics,t wouldbe the easiestthing n theworld o remedy.Allwehave to do is tomake a newstipulationo cover he case where he rulesconflict nd the matter'sresolved" PR 319). The discovery fan incon-sistency n a mathematical alculus is nothingmore than a momentaryhiatus n ourcalculatingwhich s overoncewe have laid downa new ruleto resolve heconflict. hephilosophicaleedfor onsistencyroofs anishesbecause all it means to say that a calculus is consistent s that we cancalculate n it and it needsno proof o tell us whether hat is so or not.Indeed,no proof ould ell us that,forno pieceofmathematicsouldprovethepropositionthis calculus s inconsistent";ecause no calculuscanproveanything.The formalist hilosophersfmathematicshinkthat the factthat we can go on calculatingn a calculus s not sufficiento showthatitis consistent, ormight here not be hidden nconsistencies aiting o bediscovered?But it isnonsense,Wittgenstein ouldreply, o talk of"hiddeninconsistencies",or "an inconsistencys only an inconsistencywhen it


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WITTGENSTEIN S PHILOSOPHY OF MATHEMATICS 57arises" (PR 319), a point he makes vividly with the followingexample:"Suppose two rules of a game were to contradict one another. I have sucha bad memory that I never notice this, but always forgetone of the tworules or alternatelyfollowone and then the other. Even in this case I wouldsay that everything's n order. The rules are instructionshow to play, andso long as I can play they must be all right. They only cease to be all rightthe moment I noticethey are inconsistent, nd the only sign forthat is thatI can't apply them any more!" (PR 321-2).In his article on the Remarks on the Foundations of MathematicsAlanRoss Anderson criticizedWittgenstein's views on consistency. AccordingtoAnderson"in trying o alterour attitude towards contradictionsWittgensteinsometimes seems to be recommending hat we stop playing the consistency-game altogether",14 nd he criticizesWittgensteinfordescribingthe formal-ists' attitude to inconsistencyas "the superstitiousfear and awe of mathe-maticians in the face of contradiction" (RFIM 53). But once we see in-consistencyforthe harmless thingit is we see the pointlessnessof Hilbert'sprogrammeofproving "the certitudeofall mathematical methods" bygivingconsistency proofs. Taken to its proper conclusion, Hilbert's view impliesthat there is no point in doing any furthermathematics until the calculusin question has been proved consistent. But if Wittgenstein is rightwemay well describe such an attitude as "superstitious fear and awe in theface of contradiction". Wittgenstein is not of course suggesting that wejust ignore nconsistencieswhentheyarise and stop "playing the consistency-game", as Anderson seems to think, but is simply showing us that incon-sistencies can only be dealt withwhen they arise and dealt withvery easilyat that.It mightappear that Wittgenstein'sposition impliesthat mathematiciansought to stop doing consistencyproofs,for ftheysee what an inconsistencyreally is then the need forconsistencyproofsvanishes. But Wittgenstein snot committedto such a view. He has nothingto say about what are called"consistency proofs" in mathematics qua mathematics, but he wishes topoint out that since they are not proofsof anything they do not prove any-thing about consistency. As Philip Welch has pointed out to me, proofsofconsistencyor nconsistency, .g., that 'V=L' is inconsistentwith the assump-tion that there exists a measurable cardinal, or that - GCH is consistentwith the axioms of ZF set theory, play an important role in set theory.But Wittgenstein's position in no way affectsthe status of such proofs asmathematics. They can be called "consistency proofs" on the groundsthatthe word 'consistent' appears in the calculus, but Wittgenstein wishes tomake it clear that the words consistent' and 'inconsistent' are just signs nthecalculus like any others, and this does not mean that those calculi haveanything to do with consistency.Just as everywhereelse in mathematics"everything s algorithmand nothing meaning".I now want to turn to a further spect of Wittgenstein's philosophy of14AlanRoss Anderson,"Mathematics and the "Language Game" ", ReviewofMeta-physics,XI (1958), reprinted n Benacerrafand Putnam, p. 489.


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58 MICHAEL WRIGLEYmathematicswhichshowsyet anotherway in which t differs rom hetraditionalviews. Apart fromsharingthe conceptionof philosophy fmathematicss a critical nd revisionaryctivity, he logicists, ormalistsand intuitionistslso shareda pictureof mathematics s a singlehomo-geneous nd monolithic tructure. n contrast o thisWittgensteinmpha-sizes thatthecalculiwhichmakeup mathematicsreextremelyiverse ndheterogeneous-whatecallsthe"motley" fmathematics. incephilosophyis a descriptivectivityt shoulddo justiceto this fact aboutmathematics.The philosopherfeelschanges n thestyleof a derivationwhich contem-porarymathematician asses over calmlywitha blank face" (PG 381).Thesechangesn style ndicatewhere ne calculus topsand another egins.It is one ofthephilosophicallymisleadingffects frewriting athematicsin an axiomaticway,as advocatedby the logicists nd formalists,hat itobliterateshis"motley"whichs suchan importantharacteristicfmathe-matics. The formalistnd logicist deal of presentingmathematics s asingleall-embracingxiomatic ystemwould "veil the important orms fproof o thepointofunrecognizability,s whena humanforms wrappedup ina lotofcloth" RFM 76),so that farfrom eing prerequisitef clearphilosophicalnderstandingfmathematics,chievementfthis dealwouldbe a majorobstacle o it. Here,as with onsistencyroofs,we shouldrealizethatWittgensteins not mpugningxiomatics ua mathematics.His objec-tionsare aimedat thephilosophical iewthat axiomatization ivesmathe-matics securefoundations nd revealsits fundamentaltructure.As wehave seen,Wittgenstein egards his programmes nonsensical, ut themathematicsroducedntheattempt o carryt out is just as valid as anyothermathematics.Onlywhen xiomatizedmathematicss seen as replacingnon-axiomatizedmathematics nd as beinga betterversionof the samethings the "motley"obscured.When t is recognizedhataxiomatizationjust producesmorenew and different athematics nd just adds to themotleycollection f calculi whichmakeup mathematics,hen all is well.So, for xample,Wittgensteinouldregard xiomatic ettheorynd "naive"settheory s just two differentalculi,oneno better r morerigoroushantheother.The traditional icture fmathematicss misleadingn anotherway,forit leads to thinking f morecomplex alculias extensions fsimpler nes:e.g.,therealnumbers rethought f as an extension ftherationalswhichare in turn hought fas an extension fthe ntegers.Thiswayofthinkingis, nWittgenstein'siew, totallymisguidedne. Each calculus,hestresses,is completeand self-contained ithno gaps whichneed to be filledby"extensions"of the calculus,and this is true no matterhow simplethecalculus s. Indeed,to talkof"extending" calculus s nonsense, or inceeach calculus s defined y itsrules, fwe add a newrulethenwehave notextended he old calculusbut invented newone. Thinking f onecalculusas anextensionf notherslike hinkingfchess s an extensionfdraughts.


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WITTGENSTEIN 'S PIIILOSOPHY OF MATHEMATICS 59Oneofthemost mportantonsequences f the factthat each calculus sautonomouss that all calculiare,qua calculi,on an equal footing romphilosophical ointof view. This is ofgreat mportance orphilosophy fmathematics,or f our aim is to understand he conceptcalculus nd theassociatedproblems,henone calculus s as good, .e., as interesting,s anyother-"None of them s moresublime hanany other" PG 334). It willtherefore e just as philosophicallylluminatingo considerverysimplecalculi s toconsidermore omplex nes. In fact here regoodphilosophicalreasonswhysimplecalculiare better uited to thephilosopher's urposes,forthemorecomplex calculusthegreater hedangerofcommittingheoriginal in oftraditional hilosophyfmathematics-confusinghe calculus

with"prose". Also withcomplexcalculi there s moredangerof gettinginvolvedwithpurelymathematical roblemswhichare none ofthephilo-sopher's usiness.Wittgensteinhoughthat n "foundations" hesedangerswereat theirgreatest; nd the factthattheyhave beenpreoccupiedwithjustthese reasofmathematicss no doubt the mainreasonwhy raditionalphilosophersf mathematics ave been led to say such absurdthings. Incontrast o thisWittgensteinocuses ur attention n simple alculi, o thatthephilosophical roblems an be seen themore learly nd it willbe easierto achievethatUbersicht hichwillresolve hem.Thisbrings s backto thetopic startedwith-Wittgenstein'supposedfinitism.t is easyto see that t is hisconcentrationnsimple alculiwhichhasmisledpeople ntothinkinghat he is a finitist. or example, he follow-ing passage fromthe Blue Book has been citedby Kielkopf15n supportofhis strict initistnterpretationfWittgenstein:If I wished o find utwhat sort ofthing rithmetics, I should be verycontent ndeed to haveinvestigated finite ardinal rithmetic. or (a) thiswould ead me on toall the morecomplicated ases, (b) a finite ardinal arithmetics not in-

complete,thas no gapswhicharefilled y therestofarithmetic"p. 20).I think t is perfectlylear that whatWittgensteins n factdoinghere ssettingut themethodologicalrinciplexplained bove andthishasnothingto do withwhat he himself eferredo as "theabsurditiesffinitism".16I have onlyscratched hesurface fthe wealthofmaterialwhichWitt-genstein as leftus onthephilosophyfmathematics,utI hope t is clearfromven uch brief nd selective urveyhatWittgensteins a philosopherofmathematics as much o offerhat shighly riginalnd ofgreat nterest.When t is morefully tudiedhisworkon mathematicsmaywellprovidemuch needed impetusfora genuinely hilosophical hilosophy f mathe-matics, n contrast o thedisquisitionsn the niceties f the latestresultsin mathematicalogicwhich o oftenpass forphilosophy fmathematics.At present, owever, verythings still o be doneto bring hisabout.UniversityfKent

150p. cit.,note 9, p. 177.161939 ectures see note 13 above), p. 23.

wrigley - wittgenstein's philosophy of mathematics

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