National Library Bibliothèque nationale af Canada du Canada Acquisitions and Acquisitions et Bibliographie Services services bibliographiques 395 Wellington Street 395, me Wellington Ottawa ON Ki A ON4 Ottawa ON K 1 A ON4 Canada Canada The author has granted a non- L'auteur a accordé une licence non exclusive licence allowing the exclusive permettant à la National Library of Canada to Bibliothèque nationale du Canada de reproduce, loan, distribute or seil reproduire, prêter, distribuer ou copies of th~sthesis in microform, vendre des copies de cette thèse sous paper or electronic formats. la forme de microfiche/filrn, de reproduction sur papier ou sur format électronique. The author retains ownership of the L'auteur conserve la propneté du copyright in ths thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts &om it Ni la thèse ni des extraits substantiels may be printed or othenvise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation. Canada The Philosophy of Mathematics of Wittgenstein's Tractatus Logico-Philosophicus Ihplication and Analysis of Wittgenstein's Early Philosophy of ma the ma tics by Eric A. Snyder Graduüte Progriim in Philosophy Submitted in partial fulfilmrnr of the requiremrnts for the degree of Doctor of Philosophy Faculty of Graduatc Studies The University of Western Ontario London. Ontario Xugust 2000 O Eric A. Snyder 2000 W hether Wittgenstein intended his Tractatus Logico-Pliilosopliic~tsto support logicism is a question on which disparate views have been advanced. The views ~idvancrdon the question are that either (i) Wittsrnstein was a logicist in the same sense as werc Frege and Russell or (ii) he was a logicist in a different sense or (iii) he was not 3 logicisi in Iiny scnss. Thrre are two reasons for the displtrity of views. (il What W itrywcin intcnded to say in the Tr~~crrrmsregarding rnathematics has nrver been ;idi.i1~1at~'1~explained. (ii) Wittgenstein's understanding of logicism. panicularly his iinJcr.;taiicling of rhr purpose that he believrd logicism wüs supposcd to achirve. has nt.\qt.r brcn cvctluated. To explain what Witt~ensteinintended to wy in the Trcictritirs rc~cirding niatherniitics. 1 have tried ro elucidatr various iuguments concrrning ni;iihcmiitics borh from his eulier writings and from thosr latçr writings which explain ~.ieu.shc hcld won aftrr his retum to philosophy in 1929. The writings on which 1 have hascd my conclusions lire found in Wittgenstein's eiuly letters. dictations. notebooks and rhc Pror~~r>-~ic.rrrr,~s.;ind also in rransçripts made of Wittgenstein's conversations with Schlick cind hisniann during 1929 and 1930. Waisrnann's 1930 lecture nt Konipbrr: on Wiitgcnstcin's philosophy of mathematics and the Pliib).sopliicd Rrmirks. On the bibis ut' thcse writings. I have been able io givr an intrrpretrition of the philosophy of mat heniatics of t htt Trc~crurirswhich explains Wittgenstein's assertions on series of forms. ihc ancestral relation. the nntural number srrics. the concept of numbrr. arithmeticcil sqiiritions. clahses in rnathematics and the penerality of rnathematics. Furthemore, the rcsulting intcrprctation of the philosophy of mathematics of the Trmtcirils includes an esplmation of what Wittgenstein undrrstood to be the simililrity between logic and srirhmetiç. niimely. that the respective methods of using taurologies and equcitions are andopus. Hence. given this explanation. 1 have been able io decide whether \\,'ittgenstcin intended the Tracrurits to support logicism. I contend that he did not. Not only did he believe that Frege's, and Whitehead and Russell's logicism misrepresent the siniilarity between logic and anthmetic. he regarded the very problem which logicism LWS qqmzd io solve as a pseudo-problem. Key Words: logicism. Wittgenstein.C cardinal numbrr. concept of number. mathematical induction. iinccstriil relation. formal senes. ÿnthmetical rquation. tautology For unly alter we have lramt . that the intuitions or perceptions of space and time are qilitc diifercnt t'rom ernpirical perception. entirely independent of any impression on the jcnsc.. condi tioning this and not conditioned by it. Le.. are (1 priori. and hsnce not in any w+ cxposed ro wnx-clecrption - only then can we see that Euclid's logical method of ircaiing riicitliciiiatics is it useless precaution. ri cmtçh for sound legs. Schopenhauer TABLE OF CONTENTS Cert ificritt. of Examinrition Abstrrict Thlc of Contents List of .Abbreviations t ntroduction 1. The Analysis of the Concept of Number 1. Whitehead and Russell's views on the Trtrcr~ltrison number 1. Wittgenstein's txly views on the cardinal nurnbers 3. \Vittgc'nstein's virws on nurnbers in the Trcrcturiis II. The :inrrlysis of the Propositions of Mathematics 1 . Ranisc y ' s vicws on the Trcictcitiis on mat hematical propositions 1. Wirrgcnhtein's etirly views on mathemarical propositions 3. M'itigcnstein's views on arithmrticnl propositions in the Trcrcrcrrrrs Conclusion LIST OF ABBREVIATIONS 1. .-lhhre~~intiotis~brrrorks by Wittgemtrir2 R Liuiwig IVirrgrristri~i:Cmbriclgr Lrnrrs. correspondence ivih Ritssell. Keynes .Moore. Ro~~isey~irid Sriiflci. sds. B. F. McGuinness and G. H. von Wright i Oxford: Basil Blrickwell, 1995). Nt 'Xotes on Logic'. ~Vmbooks1914-1916. eds. O. H. von Wright and G. E. W. .Anscombe t Oxford: Basil Blrickwell, 196 1 1. SM 'Sotiis dictatecl 10 G. E. Moore in Norwciy. April 19 14'. iVoreL>ooksI Y 14-1Y 16. cds. G. H. von Wright and G. E. M. Anscombe (Oxford: Basil Biackwell. 196 1). YB .Vorrhooks 1914-IY 16. trans. G. E. .M. Anscombe. ed. G. H. von Wright and G. E. M.Anscurnbe iOxford: Basil Blrickwell. 196 1 ). PT Prololj-trclcrtrl.\,. trms. D. F. Perirs and B. F. .McGtiinness. eds. B. F. bIcGuinness. T. Sybcrg and G. H. von Wright (London:Routlcdgt: & Kcgm Paul. 197 1 ). TLP Tr~~ct~zritsLogico-Plitlo.sopI~ic~~~~~. trcins. D. F. Pears and B. F. McGuinncss ( London: Ruutleclgr & Kegm Paul. 1972). WWK IVittyrmteiii ri>idtlrr Virnnci Circlr. shonhand notes recorded by F. Waisrnann. cd. and rrrins. B. F. hkGuinness (Oxford: Basil Blackwell, 1979). PR Plriioropili~.cilRemirks. trans. R. Hargreaves and R. White. rd. R. Rhees iOxford: Brisil Blackwell. 1975j. PrXl The Priiiciples of iM(itlzemitics. 2nd ed. (revised) (London: George Allen & Unwin. 1937). 1 ILT 'blathematical Logic as based on the Theory of Types'. From Frrgr to Godel: A So~~rccBo~k il1 .Cl~itlwnrclticcilLogic. 1379- 193 1. cd. J. van Hrijcnoort i Cmibrid_oe: Harvard University Press. 1977 1. PM PrOrctpi~irbf(irliori(itic<i, vols. 1-3 (with A. Y. Whitehead), 2nd ed. (Cambridge: Cmbridgr University Press. 1975-7). 'Conçcptual Notiition. a formula Ianguagci of pure thoüght modelled upon the iorniiilii Ianguage of iirithrnetic'. trans. S. Bauèr-Mrngelberg in Front Frege to Giidc~i:A Soio-ce Book in Mdir>ricitictrl Logic. 1879- 193 1 . cd. J. van Hrijenoon i Cambridge: Harvard University Press. 1977). ThBmic Lmqs of'rlrirltrrieric:exposition of rjte qstnti. rd. and trans. .M. Funh i Berkeley and Los Angeles: University of Colifornia Press. 1964) - a translation of Introduction and $9 1-52. of vol. 1: translations from vol. 2 of 3156-67. 56- 137. i 39-44. 14 6-7 and Apprndix are included in eds. P. Gcach and M. Blxk Tr<i~isiariot~s/kutrirltr Plzilosophi~ulI.Vritingr of*Gutrlub Frege ( Oxford: Basil Blrickwll. 1977). Introduction The purpose of this work is to answer the question. 'Did Wittgenstein intend bis Trm~irltsLogico-Pldusupliicr<s to support logicism'?'. 1 contend that he did not. I do not ;u-ye. Iiowevcr. as others have. that Wittgenstein believed logicism to be Mse. Rathcr. 1 ciryr. rliat he regarded the very problem which logicism was supposed to solve as a /~.sc'"tio-p''"I>I~~~~l. The problcm that îïrst xises concerning the question whether the Trticlcitirs was inrcndcd to support logicism is that of rxplaining what is to be undrrstood by the word 'Iogicism'. In itnswcr to the question whether Wittgenstein wüs a logicist disparate vicws Ii;iw bwn cidvanccd. Ths virus advanced on the question suggest that either (i) Wirgcnstcin ivas a logiçisr in the rame sense as were Fregc and Russell or iiil hti wos 3 ICI-icisi in ;i Jifferent sense or tiii) he was riot a logicist in any ssnse. Hençe. the views :iccording to which the Tra-tcitirs docs or does not contain a logicist philosophy of m:ittic.miitics ciin be disiinguished on the basis of different sènses of the word 'logicism'. Frcgc. 2nd Russell's vürious explmations of whüt is to be understood by rhc word 'Iogicisni' huggcsr diffcrent sensrs in which Wittgenstein crin be süid to be or not to be ;i Ioijcist (no[ 311 of whiçh are senscs in which borli Freg and Russell cm be said to br lugicists 1. Frcgc stlited that rtrithmetic is (i brlinch of logic (GA vol. l 1 ). In sontrast with R~isscll.tiis lugicism did riut include grometry. Russell stared that (i) pure mathematics ;inil lcgic arc identical (PrM v). (ii) the distinction of mathematics from losic is 'very ;irbirr:iry' iPrM 9). tiiii deduction in mathematics is sssentially the same as logicd dcdiiction ( Prbl 4) and (iv)mathematical concepts are definable by mcans of fundamental lu$xl concepts md mathematical propositions ;ire dcduciblr from fundamental logicd principles tPrM IV). Ir çan br iirgued. for rxamplr. that Wittgenstein believed the distinction of mathematics tiom logic to be somewhat xbitrary but did not beiirve that ni:ittit.nilitics is a part ut' logic. and it cm be çoncludrd rirher that Wittgenstein was not ii logicist in the same sense as were Frege and Russell or that he was nor a logicist in any ww. Hcnce. in zvaiuating an); particular view conceming whether or not the 7i«crunis uss intended to support logicism.