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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 '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 $9 1-52. of vol. 1: translations from vol. 2 of 3156-67. 56- 137. i 39-44. 14 6-7 and Apprndix are included in eds. P. Gcach and M. Blxk Tr The purpose of this work is to answer the question. 'Did Wittgenstein intend bis Trm~irltsLogico-Pldusupliicr The problcm that îïrst xises concerning the question whether the Trticlcitirs was inrcndcd to support logicism is that of rxplaining what is to be undrrstood by the word 'Iogicism'. In itnswcr to the question whether Wittgenstein wüs a logicist disparate vicws Ii;iw bwn cidvanccd. Ths virus advanced on the question suggest that either (i) Wirgcnstcin ivas a logiçisr in the rame sense as were Fregc and Russell or iiil hti wos 3 ICI-icisi in ;i Jifferent sense or tiii) he was riot a logicist in any ssnse. Hençe. the views :iccording to which the Tra-tcitirs docs or does not contain a logicist philosophy of m:ittic.miitics ciin be disiinguished on the basis of different sènses of the word 'logicism'. Frcgc. 2nd Russell's vürious explmations of whüt is to be understood by rhc word 'Iogicisni' huggcsr diffcrent sensrs in which Wittgenstein crin be süid to be or not to be ;i Ioijcist (no[ 311 of whiçh are senscs in which borli Freg and Russell cm be said to br lugicists 1. Frcgc stlited that rtrithmetic is (i brlinch of logic (GA vol. l 1 ). In sontrast with R~isscll.tiis lugicism did riut include grometry. Russell stared that (i) pure mathematics ;inil lcgic arc identical (PrM v). (ii) the distinction of mathematics from losic is 'very ;irbirr:iry' iPrM 9). tiiii deduction in mathematics is sssentially the same as logicd dcdiiction ( Prbl 4) and (iv)mathematical concepts are definable by mcans of fundamental lu$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. it is rssential ro explain the sense in which Wittgenstein is snid to be or not to be a logicist. The views concerninp whether or not the Tracfatris was intended to support togicisrn have been various. Wittgenstein is said to have belirved that since mathematical propositions. like logical propositions, are pseitdo-propositions. mathematics is a pïirt of I«gid He is said to have maintained, following Frege and Russell. that mathematics is rcducibk to logic.2 He is also sciid. however. to have rejectrd logicisrn for the reason that tic Jid 1101 regard rnathematiciil propositions as propositions of logic.' But. on the sümr ;issunipiion. it hÿs brrn argued that Wittgenstein was ii logicist sincr. dthough hr rejectrd Fi-cgc's and Russrll's virw that mathematical propositions are logical propositions. hr rtiuught it essentiril to explaining logical mid mathematical propositions that they ciin be rccosnizcd to be correct without comparing them to actud facrs. Hrnce. hc is said to hnvc bclicwd the distinction between logic and mathematics to br somcwhüt arbitrary4 Fiirtlicr vicivs hrivs been ridvanced on the buis of' the assertions in the Trwtmls concçi-riing lopicd and rnathemnticiil coticepu Although the concept of number is inri-ocliiced in ihc Trtrcr~irit.~by means of a logical concept. i.r. the gencral form of a proposirion. Wittpstcin's understanding of the concept of nurnbcr hüs not alwliys been takcn into aççount in answers to the question whether the Trtrcrtirirs was intrnded to *upp«rt logicism. Yonetheless. hîs definition of numbrr has bren takrn to suggrst both iIirii tic did not helievr tha< mathematics is reducible to logic. and that he did not b~lirvr rtut it is rcduciblt: to logic in Frege and Russell's sens^.^ .-\ sccond problcm thac arises conccrnin_orhe question whcther the Trtrcrctnrs was intcn~ird iu support logicisrn is thüt of interpreting Wittgenstein's assenions on ' B. T: >lcGuinncsi;, 'Picturcs and Form in Wittgcnstcin's Trnctcrrirs', ESSLI!.~un \\'itf.qeri.sreirl 'S hcrm~. cd.;. 1. hl. Copi 2nd R. W.Bcard (London: Routlcdrc & Kegan Pwl. 1966) 137-3. -P. Bcnascrrrif ancf H. Putnam, ccls.. Philu.sop/~~of itl~~thrnrarics:Selei-rrd Readirigs. 2nd cd. i Cambridge: Cmibridgc Gnr\.crsity Prcss. I!JY 3 ) 16. 'F. P. Rattiscy, 'The Found 1029 ;ind 1931. in Waisrnann's 1930 lecture cit Konigsberg on Wittgenstein's philosopliy of rriiiihcniat ics and in the Pliilosophicd Renitrrks assume fundümental distinctions made in rlic Trt~~~ttitii.~.But it is difficult to Say. even in rqard to most of rhr assenions that wcni qiiitc clcarly to claborrite on what is said concemin; mathematics in the Trcrtrcrcrriis. whctticr or ncit rhcy can bt: taken as a basis for intrrpreting the Trmrciiiu. Hrnce. it is dift'icult to disccrn the reasons for Wittgenstein's assertions on mathematics in the Trlic.itr~ir.s.Having visited Wittgenstein in Puchbcrg during 1923. Ramsey wrote of him. 'Hc is. 1 anSCÇ. â littlr ilnnoyed that Russell is doin= a new edition of Principiu because hc thoughr hs hnd show Russell rhat it is so wrong that a new rdition would bc futile? Thus. thert. is relison to believe that the assertions on mathematics in the Trcictatrrs are intcndcd to stiow that Whitehead and Russell's logicist derivation of mathematics is fiind;~rrwntally misconceived. But what it is in particulrir that Wittgenstein intended to argue rigainsi is unclru. Hence. in evaluating virws concerning whether or not the "F. P. Rriniscy. '.-lppc.ndix of lctters by F. P. Ramsey 1923-1923'. Lertrrs ru C. K. Oydtcn rcirli. cûninienu uri rlitl Et~clisliTrmz.sk~tiort oj*tJre Tr~~ctatrrsLogico-Philosoplrictrs. cd. Ci. H. von Wright (London: Routledge k Kccg;m Paul. 1973 1 73. Tr1imiti1.s was mrant to support logicism. it is essentiai to explain what Wittgenstein intendeci to argue in asserting what he did on mathematics. Tkrc are disparate virws concerning what it is that Wittgenstein intended to xguc qiinst in regard to his assertions on mathematics in the Trcicttrfirs.His fundamental objwtion bas bern token to be against Whitehead snd Russell's conception of logical and rn~tIiern;ltimlpropositions: particulÿrly the Axiorn of Reducibility. the Axiom of Infinity arc IIUI Io~ic~llytrue. i.e. not rciirtologiccrl. and if indispensable for the derivation of tn~thzniatiçsthcn logicism must be rejected? But Wittgenstein's fundamental objection has dso bcen tcikcn to be ayainst Frege's and Russell's drfinition of nurnber. It h;is bern cl;tirricd thlit Wirigenstein's main purposc in assrrting what hs did on mathernarics was to rct'urc Frcge'r Pliitonist virw thar numbrrs are object~.~It h rcgiirdcd as soniething that Wittgenstein found fundamentally objectioncible in the logiçist dc!initiun ui nurnbcr.'') hlso. his drfinition of number is wpposed to elaborrite an objcciion to Frege's dcmonstrtiiion of the lo=ical charactrr of rnathrrnatical induction. i.e. hi., iindcrst:inding of foiiowing in the series of inductive cardinal nurnbrrs.'l Th~is.iherr are two questions in regard to whriher or not the Trmtcitris was inrcndcd to suppon logicism. (i) What is it that Wittgenstein intended to argue in assertin2 ivhat he did on mathematics'? (ii) What is to be understood by the word k~lclsrt~'~? I contcnd that Witt_oenstrin intended to clucidate in the Trncr~itiisa correct conceptuül notation for the concepts and propositions of mathematics. The conceptual -9. Ruswll. Itirtodr~~.rioriro .Mdrenicltical Pitilosopii~.(London: Crorgc Allen 9r Unwin. 19 19 1 194-706: F. P. Rriniacy. 'The Foundritivns of Mathematics', 3if. 'H.O. 1Iouncc. op. iir.. 60. "B. Ru~scII.Icttcr to Wittgenstein drtrcd 13 August 19 19. Lridwig \tïrtgensreirr: Cambridge Lerrerx. eds. B. F. ~Ic.Guinncssancf G. H. von Wright (Oxford: Basil Blrickwell, 1995) i2 Lff. Iu\1. Blrick. op. ci!.. 3 16-7. notation is supposed to üchieve two aims. (i) It is supposed to show thnt what Frege, and Whitehccid and Russell understood to be the similarity between logic and arithmetic, as rctleçtcd in the notations of the Grioidgeset:r der Ariflrtierik and Principici Mdiertinticci. is misconccived. Hrncr. it is intended to make perspicuous the tme sirnilarity between arithmctiç and logic. (ii) In addition. it is supposed to make clear differences between pure niathc.m:itics and natural science that are concenlsd by the notations of the G,-ir~iil,qt..u~r:eand Priticipici. The similuitirs between logic and lirithrnrtic. according to b'i t tgcnhtcin. arc twci. < i ) The fundamentai concept of logic. the concept of a proposition. is cinologou\ lo the fundrtmental concept of iirithmctic. thüt of a number. (ii) The mrthod of using iautolopies. as applied to propositions in logic. is analogous to the method of iising cquations. as lipplied to numbers in arithmrtic. The differences betwcen iii~itheniaticsand niitural science. Wittgenstein argued. are also two. (i) The propositions ot+niiitlieniatics. he belirved. are rquations. The propositions of naturril science are logical picturcs of fiictz. Fxts cari only bc depicted by propositions. Equations. howsvrir. are !~.s~~irrio-~~~*op~.~i~io~~.\+.i i i ) The concepts of mat hemarics. Wittgenstein bclicvcd. arc forma1 soncçpis. The concepts of natural science are concepts proper. .A concept proper is rcprcicntcd by mrüns of a function. A formal concept suçh as that of a numbcr is rcprficnied by an operation. i.r. a pseirdo-jiuiclio>i. According to the Trcicrcitlis. Frese. WiirchcaJ and Russell had not recognized that sentences used in mathematics express p,~"i~lo-pu)poxi~iott.s.and words used in mathematics express psriido-cortceprs. Hence. ils ;i rcsul t. Wiitgcnstcin beliwed. the notations of the Grrirrd,qest.t:r and Priricipia do not preciiidc dl phi losop hical confusion regarding mlitherniitics. including that hwing to do with thc similiirity betwetn logic and arithmrtic. In a correct conc~ptualnotation. according to Wittgenstein. the psrutlu-cottceppr cspresed by the word 'number' cannot be represrnted by a concept name. Vÿnous iirprncnts support this conciusion. For cxample. a conceptual notation must preclude as nonscnse whar ':is ii number' is supposed to say: for supposing 'jis a numbcr' had a rerise. since it aould otrly maks sense to substirute a number tor '5'. what the expression is iuppowd to sny would be completely supertluous. Hence. the concept of nurnber is not i IR. Rhccs. op. cir.. Y6. a concept proper. Wittgenstein's understanding of the concept of number as a formal concept. however. did not rest only on the preceding argument. Rather. it resultrd from ~irpmentswhich led to the complete dissolution of the cardinal ÿrithmrtics of the G')-wd,ycsrr:r der Aritlzrwrik and Pri~icipiclhf~iriiemoficcr, and also the relation arithmsiic of Priitcipi~i.In bis philosophical notebooks. Wittgenstein argues. Iürgely on the basis of his r-cjection of identity. thrit the definitions of the finite or inductive cardinal numbers. if malyzrd. art. shown to presuppose an understanding of what is supposcd to be defined. Hcncc. t hi: de tinit ions ore viciously circular. In addition. given his analysis. he concludes ~hitwhciher mg nurnber in the system of finite cardinal numbers can be dcfined drpends rzot on ;in iiirier iiccrssi~~of the concept of numbrr but on what cltisses. or funcrions tiaving stil'tïcicntly large extensions. Iicipprri to exist in the world. Hencc. in the Trlrc.tritrrs. Wittscnstein undertakes to Jefine rhc concept of number so thitt. given the mnccpt. rbc nurnber systrm cm be constnictcd (1 priori: and thus. hr rejects classes in the dcfinition of nurnbers. Accordingly. he also rejects Frege's. and Whitehead and Russell's. dcfinition of the concept of following in the series of nurnbers by means the inductive propcrtics of the finite cardinal numbers. i.e. by mrans of hercditiuy cliisstts. The number \crics. lie ;trsucb. is ;i hrrnal series. çonstructcd (1 priuri by rnerins of a formai Iriw. As the - - - corici.pt of s proposition is givcn by the (1 priori Iliw [ p. 5. N( 5 11. so ail propositions ;ire rcprcsented as the rrsuit of successive rippliciiiions of proposirions: the concept of a nurnber is givrn by the Priii~.ipici. lit. did indeed criticize the use of identity. the Throry of Cluses and the definition of th<: concept of an üncestral relation. the üppliciition of which to cardinal xirhmetic is supposed to explain following in the srries of inductive cardinal numbers by means of the principlt: of mathematical induction. However. none of thrse criticisms is whiir for \Vitt_nrnstein wu the fundamental criticism of Fre~e's.and Whitehead and Ruisell's conception of numbers. Rather ai1 elaborate on what he regardrd as the fundamental problem with their definitions of the numbers. i.e. the underlyin; confusion of the concepts of pure mathematics with concepts of nütural science. A similu confusion. according to Wittgenstein. underlies Whitehead and Russell's understanding of the propositions of pure mathematics. In a correct conceptual notation. he believcd. 'One and one is two' cannot be represented by a proposition. It is esmxilil to a proposition that what it says is either true or Mse. What the psrrtdo- propo.si[io/i 'One and one is two' is supposed to say, howsver. is not possibly trwe but cocs wi thout ;iying. and what it is supposed to drny is no: po.ssi& jirfsr but nonsense i PT 6.11 1 ). Wittgcnstrin did inderd criricize the Axiom of Reducibility. the Axiom of Iniïnity and thc .Clultiplicative Axiom for the reason thnt. as undcrstood in Prirtcipi~i .Ll(irl~~~rt~liric. \\*irigcnstcin ;irgued thiit. in contrüst with ri proposition of naturül science the general \.;ilidity of which. il' it hqprm to be (rue for (il! thinss. is ïrcc.idr:zrtii. the general vülidity 01' both thil propositions of logic and of mathematics is rssëmid (TLP 6.03 1. 6.123 1 >. Ho\icvcr. having shown that logical propositions are cssenticilly trut.. i.c. ttiutological. Wittgenhtein did not conclude that the propositions of arithmetic rnust be shown also to hc. csscntially [rue by the derivation of cirithmetic frorn logic. Rnthcr. he uyed thrit the esscntial validiry of arithmetical equations is shown by the mrrliod of using cquations in ;ipplic;ition to the number system: for the method is sirnilar to that of using tautolozies in ;ipplic;iiion ru systcnis of truth-funcrions such as that grnerareci from the totolity of clcmcntriry propositions. Açcording to the Trmtnrrls. Whitehcaci and Russell had iishimilatcd logicül and inathematical propositions to propositions of niltural science by taking the c haractrristic of such propositions to be general vrilidity. Hnving been concincçd by Wittgenstein that the propositions of logic are tautologies. Russell then ;iswmrd that the propositions of mathematics must also be tautological in the same sense that lugiciii proposirions are. Howsver. Wittgenstein believed that. given that assumption. Russcll \.vas unable to show that cven the propositions of e1ernrntu-y arithmctic cire r,s.sr)iiililI~valid. Funhermore. he believed that it is precisely this which musr be shown in LL correct conceptual notation for xithmetic; for such ri notation must show that. in contrüst wiih propositions of natural science. which say something, propositions of lirithrnctic go without saying. Thus. ~ivrnhis understanding of the concepts and propositions of rnathematics. did Wiit~enstcinintend the Trtrctcirlrs to suppon logicism? To answcr this question ir is ncc~ss;~r~~IO considrr whsther any of the explanations of loricism given by Frege and RiisscII suggcst a srnsc in which Wittgenstein could be raid to be a logicist. ti) Wittgcnstcin did not takr uithmetic to be a part of logic: and hencr. he did not conclude. 2s iiid Rii.;sell. that pure mathematics and logic are essrntially identical. (ii) He did not ;~cccprthLi[ whrit Russcil cailed 'an inference from ri to rr + 1' in the scries of numbers can hc c!i;ir~icicrizcdns a logicil drduction. c iii) He rcjrcted the assertion thiit the concepts of piii-c ni;itlieniatic:. ;ire reduciblr by definitions to lojicai concepts. and also the assertion that the pi.opositions of pure rnathematics are drrivabie by logicül drduction hmlogiclil propositions. Hcnce. he was ccrtainiy not a logicist in the sense that Frege. Whitehead md Riisicll wre. However. Wittgenstein did bclieve that rhcre arc sirnilarities betwern \»sic and nrithmetic. Did he tllke that to be sufficient grounds for the conclusion that thc dihicriori bciwccn logic and arithmetic is ÿrbitrary. and if SC. did he brlicve that hr was git ing in tlic T~-miirtc.va correct riccount of logicism? He cenciinly did >rot consider the distinction bctwccn logic and arithmetic to be rirbitrary. For logic depends on the concept i)i 2 propositicin. but clrithmetic depends on the concept of a numbrr. and hci did not rcgiird t hc distinc [ion bctwcrn numbers and propositions as orbitrq. That hr recognizrd \iniilnritics bctween Iogic and rnathematics. rnoreover. is not sufficient grounds for the conclusion thüt he supponed logicism. For example. Kant recognized that logic and ni;itlirm;ttics Iiavr certain essential chmcteristics in cornmon. but hr cirarly did not iindcrsix-d thc TI-mrscrirdrriralArsrlietic to suppon any kind of logicism. He regardrd Leibniz's efforts to esplicatr mothemntics in the manner of the rttetliorl~osyllugisric« as ü nierr subrlct);." Givrn his understanding of logii: and mathrmatics. evrn if the efforts lverc sucçcssful. he did not believe thnt that would explain how mathematicai knowledge is possible. If Wittgenstein regarded the very problern that logicism was supposed to solve as ii psrirtlo-problem. so that he understood Frege, Whitehead and Russell's efforts to bc /Iiiid~i~trriitd!\.confused, then thût is an evrn stronger reason for concluding that he entirclly rejectrd logicism. For if he belirved that Frege, Whitehead and Russell had misundcrstood the sirnilarity between logic and mathematics, and if his undenaking to ntiikr perspicuous the true sirnilarity was also intended to show that the purpose to be xhisved by the conceptual notations of the Gnoidgese~eand Priiicipin is fundnrnentally mi~conceivcd.then it is surely misleading to cd1 Wittgenstein a 'logicist'.lJ The fact that he niercly rccognizsd similari titis brtween logic and mothemiitiçs is not sufficirnt to slipport ihc claini that hr considered himself to bc ci logicist. Wtint Wittpstein regardrd as the fundamentcil problem of philosophy. and the iiiain piirposr of the Tructtrttis. is the distinction brtwern what cm be rxpressed by propositions and what ciinnot be expressrd but only shown (R37). Only facts crin br c.sprcssrd by propositions. i.e. the facts of naturd science (TLP 6.53). What Wittgenstein dcil 'the logiccil structure' of a Tact is reflecrrd in the rules of syntax in accordance with wliicli a propositioncil sipn must br used to be cible to espress rhat hçt (WWK 139j--.). .A rulc of synr;iu ciinnot be expresscd by a proposition. but is slioii*iiby the wop in which ihe propositi«nsl sign is used iTLP 3.334). It is ihr tiilure to understnnd the logical synta.x of Imguqe. Wittgenstein maintained. that results in the nonsensiccil psedo-propositions of vJiich philosopliy is full (TLP 3.374-5). Wittgenstein clairnrd that there is (i sense in whicti objccts and situations cm be said to have fornial propertirs and forma1 relations. ~indfiicts unbe s;iid to hnvc st~cturdproperties and structural re1iitior.s iTLP 4.132). To cnipli;isizc the contrat with extemiil properties and extemal relations. 1.e. proprnies and rcliitions propcr. Wittgenstein ülso used the words 'intemal property' instead of 'htriictural propcny': and 'interna1 relation' instead of 'structural relation'. It is imposhible. he stated. to express by mrans of a proposition the existence of an internai property of a possible situation: rather it s/ioir*sitself in the proposition representing the .;itucition. by means of an internai property of the proposition. Similarly. the existence of - - - - . ------. ''I. Kmt. 'The Dohna-WunJ1acken Logrç'. lnttriunuel Kmr: Lrcrurrs orr Logic. trans. and cd. J. ,Clichacl j'oung I Cxnbrid~c:Ccirnbridgt. University Press. 1992) Ak.780. ': Lcihni~cunsidercd Hçrlin's reduction uf the demonsrrarions of Euclid tu syllogisrns and prosyllogisms to wpport hi3 t.xplrinrit:on of geomctrical crminty. Kant acccptcd the intelIigibiIity of the purpose which chs rcductii~n~3s lo ~iih~evti for Lsibciz. i.e. to solve the problem of cxplrtiniq rnrithemritical ccminty. only an intrimiil relation between possible situations shows itself in language by means of an internai relation between the propositions representing them. A property, or relation. is internd if it is inconccivable that its objrct not possess it (TLP 4.123). Accordingiy, sincr thc Iogiç:il synttix of language is supposed to reflect the formal or interna1 properties of ohjcct.;. iind the stmçtur11 properties of hcts. i.r. the necrssary propenies without which ihcy would he inconceivable. philosophical nonsense involves the most fundamental conf~isionsconccniing the distinction between intemal and extemal propenies t TLP -1.122-4.123 i. Intemal propenies are representrd by formal concepts. Wittgenstein hdic~*c.dthat rhc confusion between formül concepts and concepts proprr pervtidrs pliilosophy md logic iTLP 4.172. 4.126). Morcovrr. the confusion nins through the n,liolc ul' Russsll's inlrrpretarion of mathematics 'like a red thrrad'I4. Russell did not u~dcrsrrindthat rhc concept of number is a forma1 concept: and tlius. represcnts it by niciins of functions or cltisses (TLP 4.1272). Hénce, he was Ied to assert nonsensical psciiiio-propositions such as 'There is only ont. 1' and 'There cire Hi, finitc numbers' r TLP 4.12711.Wittgenstein's conceptual notation for logic and mathematics wos intsndrd ro pïccliiclc such philosophical nonsense. blorcovcr. it was intended io preclude the fundamentiil epistemolo_oiçiilproblem tliiit Iogicism ~3ssupposed 10 solvr. i.e. the problrm of uncenainty in the foundations of iiiiitlicrnntics. Wittgenstein viewrd the problrrn as a pse~du-prohlrixIt is a problcm that ii~~uldnor wcn arise in a correct conceptual notation. Wittgenstein's insights regÿrding thz csscncc of logic wrre. Schlick argued. a 'decisive turning point'ls in philosophy. cnabiing hiin to dispose of the problrms of episternology üs previously understood in philoophy. Howver. whüt Schlick did not rrcognize is that Wittgenstein includrd the epistcm~lugic;llproblrm thrit ioyici'ini was intendrd to solve rirnong those thrit cm be ilispcnsed wirh. Ii is essential ro the logicism of Frege and Russell that the methods of no[ dit rnciins of xhicvrng it. (G. W. von Leibniz. NOL'ESSCI'S 011 Hrrrncln U~tdersrnndirig.trans. and ';II. Schlick. 'Thc Turninr Point in Philosophy'. Logicul Pusifivism. cd. A. J. Xyer t Nrw York: Frw Press. 1959 l 54-5. If' A. J. .-\lw.L~tr,qitq-e. Trtith cznd hgic. 2nd ed.. i New York: Daver Publications Inc.. 1952) Y 1-2. Furthermore. it ciin be said definitively that he rejected Frege's logicism regarding arithmetic. and Russell's logicism regardinz al1 pure mathematics. For Wittgenstein rejectcd the undertaking to demonstrate that what the propositions of arithmetic. or pure rn;itlirrnntics, sri. is expressçd by propositions of logic. ~Moreover.he rejected Frege's and Russell's hcliet' that logiçism can supply a solution to the problem of uncertainty in rnothsrncitics. Indeed. hr rejected the very problem of uncenainty in mathematics. and tiwing rcjrcrcd the problem. given that the propositions of logic are tautological. he rcg;irdcd the possibility of deriving mathematics from logic as inessential to elucidating rhc c!i;~rxt~rof matliernatical propositions. He did introduçe the concept of number in the Trirc,i<.iiit.sbu niccins of the prral form of a proposition: not ro satisfy !he conditions on rii;iilii'iiiaiicIiI definition necessitaled by the dernands of n rigorous dcrivation of ni;itticinntiçs irom logic. but to show that the fundamental concepts borli of logic and arithnietic cire forma1 concepts. He did introduce cirithmetical equations in the Trctcrmrs in a mnncr th~itemphasizes various similaritirs with the propositions of logic: nor to support the conclusion that thrre is no sssentinl difference between logiçai propositions md propositions of rnathemritics. but to show that neither the propositions of logic nor the proposirions of mathematics are proposirioris. Wittgenstein did not regard the simillirity bcnwen logic and müthenilitics to be that which Frege and Russell believrd it to be. and Iiis iiticnipt t« euplain the truc sirnilxity retlccts his rejection of the purposr that Frege's ;ind Russc.ll's logicism LUS supposed to achievc. The Analysis of the Concept of Number Wtiitchead and Russell werc the tlrst to evriluate the definitions of the numbers. md tliiit ut' rhc concept of nurnber. given in the Tr ;mt hiiic t ic of Pri~icipici.Cldrrmcitic~i. W ittgcnstrin's denial of classes in mathernatics. rcsulting llom his criticisms of the cardinal numbers. was triken by Whitehead and Russcll rncrcly to be the denial of classes (1s distim fhzjirnrrions. so that the cardinal iirithriicric of Priticipiïi remained. Accordingly. Russell rr,oardrd Wittgenstein's Jciinitions of ttic numbcrs rathrr 3s he regardeci the definitions given by Lcibniz and his i*ulloafcrs.Leibniz's rncthod of definition. Russell complained. is only applicable to fitiirr niirnbcrs. and sincc it necessitates that al1 other numbcrs be defined by means of the ncimbcr 1. it nilikes 'ri tircsome differcncc between 1 and other nurnbers' iPrM 1 12). The csplanation of ihe numbers given in the Trucrirria. according to Russell. ha the same dcic~t~.[t tus understood to be. not the result of a thorough analysis of the assumptions iindcrlying the cardinal arithmetic of Priricipi rr\.eals tiilit the definitions of the finite cardinal numbers presicppose an understanding of what is supposed to be defined. Thus. he concluded. the definitions are also nonsense. In hi.; 1922 .Introduction' to the Trcictarits. Russell clûims that he ûgrees with Wittgenstein thlit. givrn his ünalysis and resulting rejection of identity. it is impossible to assen, for cxtiinple. 'Therr are more than three objects in the world' or 'There are an infinite numbcr of objects in the world'. However. he believed incorrectly that Wittgenstein ihought it possible to assert the existence of numbers of objects having somr definite propcrcy. Hcncc. given that Wittgenstein's rejection of identity necessitates the use of mAi ;isscrtions in the definitions of the cardinal numbrrs. Russell's not hüving ~indersrood thlir Wittgenstein regardrd such assenions as nonsense resultrd in his not undcrstanding rhat Wittgenstein rcjectcd the definirions of the nurnbrrs in the cardinal xi t hnictic of Pri~icipitrMdtemntic~i. tnitially Russell was >ior convinced thlit the assertions regarding identity in the Tmcrirtirs provc that it is not possible to sciy that some number of objects enist: and tlius. as Wttpenstein belicved. supply the solution to 311 problcms nrising from the Axiom of Infinit!,. Witigcnïtein argueci in the Trcicftiriisthat whrrever the word 'objrct' is correctly ~iscil.it is c'rprcsxeci in a conceptual notation by a variable name. so that. for rxcirnplc. in rlic proposition 'There are Z objects which . . .' it is expresseci by '(3.r. y). . . '; and hence. II is inipossiblc io say 'Thsre ore objects' or 'There are Ki, objects' (TLP 4.1272). Hliving dho ;irgucd that the identity sign is not necessüry in a correct conceptual notation. Witigcnstcin concludes in the Tr~icmt~tsthat given such a notation pseudo-propositions Iikc *CI = i.Y.\-). .r = .r'. '(3s)..r = (1' and so on cannot even bc written down (TLP 5.533- 4 i. Hcncc. hc clliirned. riIl the problrms that have to do with such pscudo-propositions ciin bc dispenicd with. and a11 the problems rriised by Russell's Axiom of Infinity cm br iolvcd tTLP 5.535). in his lrtter IO Wittgenstein of 13 August 19 19. Russell stated that he cigreed with Wittgenstein's rejection of identity. and he supposed that Wittgenstein's views conceming what can be said about numbers of objects 'hiings togther with the rejecrion of identity'". He was not rntirely convinced. however. that it is not possible to '-B. Russcll. op. cir.. 112. (Russell. however. Inter explaincd thai he did not bclieve the use of identity in his 'Thcory of' Dcscriptmns could be dispenseci with (TLPxvi-~vri ).) say rhat some mmber of objects exist, arguing that even if nothing cm be said about the ~~t~ilnumber of objccts, i.e. Nc'V, One could till sny and 1 shoulti have thought that from such propositions one cculd obtain a meaning for 'Thcre cire at Ic~ist7 [objsctsl' . . . and sirnilx statements.lY .\cçorditigly. RusseLi did not belirve that Wittpstein's rejrction of identity proved that it is noi possihir to sriy thrit some numbrr of objects exist. and hc was unconvinced by the i~iggcs[ionin (lie Tmcrrrrits thnt the assertions concernins idrntity supply the solution to dl [tic problenis riliscd by the Axiom of 1nfin1ty.I'~ In his 1922 'Introduction' to the Tr~~c~rritris.however. Russell wüs inclinrd to iigrer avitti Wittgenstein's rinalysis of staternents of the ex'stence of numbers of objects. inçluding the Axioin of Infiniiy (TLP xvii). Wittgenstein's rejection of identity. he iligfmw~i.rcnioves any possibility of saying anything about the iotality of objrcts. For LVittzcnstein argued. according to Russell. that to drscribe the totality of what therc is in rhc tvorld. it i\ neccrsary to know of some property which musr belong to rvcqobjcct by losical ncccssity. and if self-identity is not such a proprny then nothiny is (TLP xvi-xvii). 1-lcncc. givcn thci rejrction of idrntity, Russell clairncd. the concept ot'objrcr is a pseudo- cunccpt. and to sriy '.r is an object' is to say nothing TLP 4.1272). It follows. according io Russell. that it is not possible to make such statements as 'There are more than tlircc objects in the world'. or 'There are an intinite number of objects in the world'. Houwcr. Russril supposed that it is possible. according to the Trac-rlints. to iay that thrrc Luc. numbcrs of objrçts having some dçtinitr propeny. Thus. for enample. hr claimed. it ciin be said. 'Thur Lire more than threr objects which are red' since in this statement the word 'ubjrcc' can be replaced by the variable x in the tunction '-Y is red'. But. if the \-ciriable .r in rhs iùnction 's = -Y' cannot be used in place of the word 'ohject'. no iubstitution cmbe made for the w~rdin 'There are more than three objects'; and hence. it is no: possible to sciy that there are more than three objects. Thus. it is not possible to say ivhot the Axiom of Infinity, and similar statrments, are supposed to say. Howrver. iilthough Russell argues. in his 'htroduction' to the Trncrunts. that what [hl: Auiom of Infinity is supposed to say cannot be said. his answer to Wittgenstein's ~riticisinsof the cixiom. in the 1925 edition of Principiri il.f~uhrrriciticcl,only dcds with the criticisiri thst Wiitgrnstrin raistd in November and Decernber 1913. initially ;icknowlc.Jipl in Riisssll's Irrrrod~~crioiztu Murltrniciriccil PI:ilosop,pli!. that the tixiorn is (i pr-opi.sizio)l and not rmrolog. (R22, 23). It doss not deal with the assertion in the Tm'.-i iri hi\ Introdiiction to the Tritcrms. that did not resuli in his revisin: Pri>icipi ibr Jcaling aith a11 txisting niathematics. Hence. in his 1922 'Introduction' to the #r nrcr Jcvclopniçni' rttquired of the Trwrcirrrs (c;c TLP txi. Thcy daim thüt. given Wittscnstein's. rejcction of the Axiorn of Reducibility and his assurnptions that functions of propositions iire always truth-functions and that a function can only occur in a proposition through its values. al1 the results of the first volume of Principk .Ildrermiiic~rclin be derived and the theory of inductive cardinals and ordinals remains. Hoivcver. they jtatr thnt 'ii aeerns that the tlieory of intinite Dedekindiiin and well- ordercd hcrirs liir~elycollapses. so that irrationiils. and real numbers grnrriilly. can no longcr bc xiequotely dealt with* (PM vol. 1 xiv). in addition. Cantor's proof that za > a fails unless u is finite. 'Perhaps some further auiom, less objectionable than the Axiom of Rcducibility. might give these results', they suggest. 'but we have not succeeded in finding such an axiom' (PM vo1.l xiv). Hence, the technical developmrnt that Russell reqliireJ of the Trilc~cttitr:vas based on the assumption thüt Wittgenstein had intendrd to cive a throry of inductive cardinals and ordinals. but had sirnply not deveioped the theory .;O LIS to dei11 tvith the cardinit1 and ordinal arithmetics of transfinite numbers: and hence, n~ithrtxl Lin;iiysis. Giveii ttiat he believed the views on numbrrs in the Trcicrms to be adequate for ciahoraiing the t heory of inductive cÿrdinals of Prirzcipicc rWzrho?icirini. Russell had to cspl;iin Wittgcnstein's rejjection of the Theory of Classes in mnthrmatics. Wittgenstein Ii;iJ clai nid. in hi5 1 3 Much 19 19 lctter to Russell. that the Trtrcrcims upsct his Theory of Clahsss. tiis Thcory of Nunibers and 'al! the rest' tR35). Russell initicilly undrrstood LVittgcrisic.in's rejeçtion of classes to mean that cardinal arithmeric is not possible. In his 13 .Aiigust 1919 leitcr to Wittgenstein. commenthg on the ,ocnerd form of a number giwn in thc Tr Yoii only ger tinite ordinals. You deny classes. so cardiniils collapse. What happens to K,.)If y~said cl~i~cswere supertluous in logk I would imagine that I undrirstood you. ti!. r~ipposiiiga distinction betwecn logic and mathrmntics: but when you sriy the? ore unncçcssriry in tmirlicni~iticsI am puzzlrd. E.,y. soi>rerhi>tgtmr is rxpressed by 'Is'Cl'u = ,S'"L - . Ho\\ do '011 restate this propositi~n'?:~) Howr.cvc.r. in the 1925 cdition of Pri~icipiu iM~irhrniciricn. Russell inrerpreted Wittgcnstein's denicil of classes to be the result of his assumption that funciions only IICCLII- throilgh their values. Givrn ihat assumption. Russell arped thnt al1 functions of fiincrion\ ;ire cstensionlil. i.r. @r y~r.3. fi~t ) = f(yt ): for sincr O can only occur in jo- > by ~ub~tiiutionof values of 9 in a truth-lunction of propositions p. q. r. . . .. and. if 0.r = ip-Y. then the substitution of 0.r for p in any truth-function of p gives the silmr truth- valiitt to rhe truth-iunction as the substitution of y.\..The consequence. he claimed. is that therc is no longer üny reason to distinguish betwcrn functions and classes: for it follows that or i, y-Y.3. ol = yri (PM vol.! xxxix). Hence. his initial interpretation of the Ti-~ict .ksioiii ot' Intïnity. hiid Idhim to rejcct cardinal cirithmetic cntirely. The exphnation of niinihcrs iri thc 7rtrcrmt.s \vas underïtood to br. not the result of a thcrough mcilysis of thc asbuniptions undrrlyinp the cardinal tirithmrtic of Prit~cipicr.Vdinuciric The assertions conceming nurnbers in Wittpmein's philosophicd notebooks wggcsi tti~ithc initially accepted Frege. Whitehead and Russell's view that numben tire propcrly definrd as classes of classes. His attempts to understand correctly Whitehead iind Russcll's Axiorn of [nfinity. however. resulted in Wittgenstein's rejection of the dcfinitions of the finite cardinal numbers given in Principici iM~itlientotic«.for the reason tliar the dctinitions if iinalyzrd. are found to prcsupposr an understanding of what thry rire .;upposcd to drfine. In the Tracfcif1i.s.Wittgenstein entirely rejects the cardinal cirithmetic ut' Priucipi .ind I it i rlic annlysis of the proposition. if it is a proposition. (3-Y).x = x. The rlucidütion of Imrh ncccssitatcd that Wittgenstein corne to a correct understanding rrguirding (i) whether .;i;i[c'mc'nts of SY istcnce th;~tare complrtrly gcnrrülized Lire propositions of natuml science or Iogiciil propositions. and tii) whsthrr the identity function .r = is a propositional fiinclion or a psrrrtlo-jiuicriori. W ittgenstein's concrrn with çompletrly jrneralized cx istsnw mtcnients involved him in difficulties rcgarding rlie pensrality of cornpletrly gcnc.ralizcd propositions. His analysis of generilized propositions and the idèntity iiinciiori 1c.J Wit!gcnsrein to the conclusion that stiiternents of existence of numbers of ttiingh. ;ind of nurnbcrs of things havine some Jefinitr property. are p.srir~io-pro~~osrio~~.s; and iicncc. to the rcjcction of the çardinül nurnbrrs. Russcll'c; definitions of the tlnite crirdind numbers and the transfinite cardinal No xc given in Tlw Prbicipks of' Mtirlirtircirics as follows. To define the finitr cardinal nunibers. Russell bqün with Peano's rixioms of uithmrtic. understanding him to have rc.duccd dl iiritlinictic to the following axioms containing only the concepts 0. ricrrctrtrl tlll~dii'r2nd stlc.ce.s.sor of t PrM 124-5). (,il O is a numkx (ii) If 11 is numbrr, the siiccchwr of 11 is a number. (iii) If the nunibers rr and trz have the sarne successor. rz = ru. i iw O is noi the succcssor of any nurnber. iv) If a is a class to which belongs O and also thc iuccessor of every nurnber belonging to o. then every number belongs to o. That is. the principlr of mathematical induction is applicable to the numbers. Assurning rhiit Pcano's asiorns rnust be sarisfird by any definition of the hite cardinal numbers. Russell initicilly detinrd 1i class using orrk Peano's axioms. i.r. the class that hr denotrd by 'Ku'. underriiking to dcfinr the class of cardinal numbers independently of Peano's axioms and then prove that it belongs to the class Ha (PrM 117). Russell defined a one-one relation as ;i relation such that if .Y and .x' both have the relation to then s = x'. and if .Y hiis the relation to both and J'. rhen y = y' (PrM 113). He defined No as the claïs of classes U. itssuiiiing cvery p is the domain of some one-one relation R. the relation of a member of p ro its successur. rvhich is such that there is at lest one mernber which succeeds no 0th nieriibr.~.. cvcry membrr which succeeds another mrmbsr hris a succrssor. and u is coiir:iincd in üny class o which contains a rnember of p having no predecessors. and also contains the succcssor of evrry member of p which bclongh to O (PrM 127). Given his dctinirion of K,,,Russsll ciefïned the finite cardinal numbers as follows (PrM 128). (i) O is the kiss oi' ~.lassrsthe 0n1y msmber of which is the nul1 çlass. (ii) A nunibrr is the class of al1 cllisscs sirnilar to riny jiven class. whrrr ri clüss is sirnilu to anothcr if there is a onc-ont. rc1;ition the domain of which is the one clriss and the convcrsc domriin of which ii~the othcr. ( iii) 1 is the clriss of al! çIrissc3s which rire not nul1 and lire such th. ifs is ri rnc.nibt.r i~t'riny one of these CL~SSCS. that çlass without -Y is the nul1 clriss: or such that, if .r ;id \. arc nicmbcrs of îny one of thesç cliissrs. rhen .r = J.. (iv) Assuming it has bern provcd rhrir if two classes arc similar. and to both classes a class having one mcmber is ddcd. the resultins çlüssrs are similar. then if fz is a number. n + 1 is defined as the nimber resulting (rom iidding a class having one msmber to ri class of 11 rnembers.?i (v) Hcncc. the finitc numbrrs rire those which are mrmbcirs of rvery class o to which brlongs O. ;ind to uhich ii + 1 belongs if n belongs. Having thus drfined the class of finitr cardinal nurnbers. Russell cloirned that it is possible to prove. as regards Peano's axioms. the following. (1 O is a number. (ii) If 'the successor of n' means 11 + 1. then if rt is a number. ~r + I is a nuniber. (iii) If rt + 1 = !ri + 1. then n = III. t iv) If ri is any numbcr. 11 + I is ilifkrcnt from O. i v If a is a clas. and O belongs to o. and if assuming 11 belongs CO o. n + 1 belongs to O. thrn (il1 finite numbrn belons to o. Thus. Russell clairnrd. given his 21~~~~~11dso dritincs rhc arithrnrticlil surn of the number di ri dus having ri membcrs and rhrir of 3 clss h:it.rng i mcmbcr in mother wriy. i.e. ris the numbcr of members of the clus the members of which bclong ro c'tw!, clm tn uhich both the clss hsving 1 mernber and the class having n members arc containeci. \i'!iichcvcr LicfÏnition of arirhrnctical addition is in question, Russell assumes thrit the member of the clriss havtng only I mcniber is rtor ri member of the class hriving n members. IPrM 117-23} definition of the finite numbers, al1 Peano's mioms are satisfied by the class of finite cardinal numbers. Hence, he concluded, the class of classes Ho has rnembers, rind the clriss filtire riwther is one definite member of Ku. Russell wris convinced that the class of finite numbers is one detlnite member of H,, \incc he belirvcd that he was able to prove. using the principle of mathematical induction. that the number of finite numbers is Ki). That is. using only the principle of matheni;iiical induction applird to finite nurnbrrs. hr belirved thnt he was able to prove rlic existçncr of Hi,. Given thai he regxded O as the number oî things satisfyins tiny coniiition th;it nothin2 satisfir's. to dernonstrate that thert: is ri number O, Russell brlieved tic niiihr show that there are such conditions. Sincr. for cxiimple. nothing is (i proposition whicti is both truc or falsr. he çoncludrd that the number of things that cire propositions wtiich arc both iruè rind t'alse is O. Thus. he clriimed, the ncinber O exists. Givcn that the nuniber of menibers in ri clriss is 1 if there is ri member in the ciriss such that. if it is taken ;iwy thc number of rnernbers remaining is O: to provc rhat slassrs hnving 1 rnembrr cxibt. RuwAl ürgued. it is necessary only to take the class of things identical with the nunibcr 0. The clriss of rhings identical with O contains oril? the number O. Thus. Russell cuncludcd. the number I exists. Similarly. to prow thlit clasïes htiving 2 members exist. it nt.cc.4sx-y only to takè the clrtsi, of things identical with either O or 1. Hence. Russell d;iinisti. thc number 2 can be show to cxist. Furthemore. using this method of constniçtin~numbcrs. he brlirvcd. it is possible to provr that if n is ciny finite number. thc nurnbcr of nurnbers tiom O to tt is the number rl + 1. Hence, if n exists. then ri + i c.xisi.\. Thus. accordin: to Russell. since O exists. it follows by mathematical induction chat riIl tinite numbcrs sxist. Since it can be proven that. if ni and n are finite numbcrs utlier than O. tu + tz is not identical with rithrr tn or ri. it follows that if tz is any finite riumbcr. ri is not the number of finite numbers; for the number of nurnbers from O to rr is ri + 1. and II + 1 is different liom rz. Thus. no finite number is the nurnber of finite nunibers. md since Russell defined ri cardinal number ris a class of similar classes iPrM 1 16 1. so thüt ii is possible to define a number intensionally. by a proprny that detemines 3 chs hatein2 that number: and hence. even infinitr numbers can br defined ( PrM 1 13). he wncluded that the number of finire nurnbers is infinite. Hence. Russell believed it possible to prove. without assuming the Axiom of Infinity. that the class of finite cardinal nurnbers has tilt: number Ko. Howcver. Russell argues. in Principiri Marlieni«ticn. that the Axiom of Infinity is nccesscin. if the existence of the infinite number Ku,or any other of Cantor's transfinite cardinîls. is to be provable within cardinal uithmetic (PM vol.? 203, 2608.). To explain Iiis rcclsoning. somr of the basic definitions of Russell's Theory of Cliisses and Throry of Relations. :ind the resulting typicd arnbiguity of his definitions of the cardinal numbers. nccd ducidririon. The class of classes drfinçd by cp is cls = aA[(3 '111~ncpion' of the universal class V = .i(.r = .r) Df.: whrre both 'il' and 'V' arc [>piililly .mibiguous. A descnptivr function. which describes a term .r by means of its rclaiiori to the iirpurncnt of a function cps. is clefincd by 'v(il\.) (PX)1. = : t jh) : cps . i,. x = h : \vh Df.'. wherc '(Ir) (9.r)' rnrtins 'the terni -r which satisfies cp-Y'. Russell givss the gcnsral &finition of descriptive function as R'y = (Ir)(sRy) Dt: and hencr. 'R'f means 'tlrr rcrni wtiich has the relation R to y'. The genrral drtinition of descriptive functions is rtie brihis of Riisscll's definition of the domriin of a relation R and the converse domsiin of R. Thc c.[,,ii.erise ai rio^^ is dcfined as Cnv =QP (.rQ? . o ,. ,. ?Px} Df.: and thus. Cnv*P = dQi{.t-(L)~. = ,. : . !*Px).Russell asserts that the class of tèms which have the relation R to is Rbv= .i:(.i-RF).whrre R = b! j{a = .t(.rR~))Df. The relation of the dornain of rttlatii~nR 10 X is D = cî R {a.= I((3y).xRy) J Df. and the relation of rhe converse domain of relation K to R is CD =pR{P = !((Zr). xR!)}Df. Given thrse definitions, Russell conclutles that thc domain of R is D'R = i((3y)..r&Jnnd the converse domain of R is CD'R = f ( r 3).-1-Q). The class of which the only member is .r is i'r = f (-Y = .y). where i = U .i ( u = f (J = .r>) Df. Given these dstinitions and resulting propositions. Russell dcfines the cardinal numbers: The cardinal number of a class a is defined as the clriss of al1 classes sirrlilar to a, two sIrtssss being similar when there is ri one-one relation between thern. The class of one-one relations is denoted by 141 and defined 3s follows: impossible to add s and y together to get a class of a and B members. Clearly, the same problem arises regarding any number of classes. If only a finite number of classes is concerncd. howrver. it is possible to construct a class having the cardinal number which is thc nrithmetical sum of the cardinal nurnbers of the clrisses of which it is the logical wrii: for ir is possible to raise the type of ti class without altering its cardinal nurnber. For csxnplc. siwn Liny cltiss x. the class L"X: that is. the dass the rnembers of which ut. the il;i\ils consisiing of' single members of .K. is of the samr cardinal numbrr but is of the ncst higher type ribove .r. Hence. given any finite number of çlassrs of different types. it is pussiblc tu raise ail of them to the type which is whüt Russell called 'the lowest miinion multiple' of al1 the types in question: and furthemore. it clin be done in such a ththe rcsulting classes have no common rnrmbers. Hrnce the lo_oical surn of a11 the ~-c'uitingcl;iliscs Gin bc cnnstructed. and its cardinal number is the arirhrnetical sum of the caidinril nurnbsrs of the givcn classes. But given an infinitr series of classes of higher and tiigticr types. tliis nietliocl clinnot be applied: hence it is appi~rentlynot possible &O provr thai thcrc miist bc intinitr classes. For suppose therc were only tz individuals ciltogethrr in 7 thc u-orld. whcrtl N is finite. There would then be 2" classes of individuals, and 2' ~.lasscsof çlasscs of individuals. and so on. Thus the cardinal number of classes of any ryc ~n-uuldbc finite. ;tnd drhough it would be possible to construct ;i number of classes grc;wr rhan ;in! givcn finitr nurnber. there would be no way of adding thrm to set an irltïilirc ncinibcr. Hcnct-. Russèll concluded. Lin uiorn is needed which States that no tinitc clw of individuals contains dl individuals. nnrnely. the Axiorn of Infiniry. Wittgenstein's notebooks indicate that the problems regarding rnathernatics that i~ccupicd him. during the Inter rnonths of 19 11. primanly concrmed the Axiom of Infinitu. In his notebook rntries. he suggests two strritegies for asessing the axiorn. (i) On 9 Octobcr 1914. he wrotr. NIthe problerns that go with he Axiom of infinity have drcad>-rd be suivcd in the proposition "&-).Y = s" ' (SB 10). t ii 1 On II October. he wrotc. 'It u.ould hr nrcessary to investigate the definitions of the cardinal numbrrs more cxcictly in order to understand the real sense of propositions like the Axiom of infinity' , XB 1 1 1. The Axiom of Infinity is defined in Principicr iMrtdiemurica by 'Infin ax . = : CG TC induçt . 2, . 3!a Df.' (PM vo1.2 183). The class denoted by 'NC induct' is the cliiss of irrclitctilv cardinals. i.r. dl cardinals but Cantor's transfinite cardinals (PM vol2 200). Ttius. @n the hypothesis of the dejî~iiensof Russell's definirion, Wittgenstein's second iuegestion for asscssing the Axiom of Infinity is understandable. His first suggestion. Iiorrwçr. rcquires rxplanation. The consequent of the ckfi~iirrisof Russell's definition is dciincd by '3!u= (3~).- -v E a DL': and hence. given Russell's definitions of the cardinal nuinbers. the Aaiorn of infinity amounts to the usumption that. in any givrn type. thrre is ;i cl;iss hwing riny givrn inductive numbtir of members. That is. regarding iriclii~idiitrls. ;in>, t'initc clw of individuais does not contain d1 individulils. Hence. to assume the .\siorri oi' Infiniry is to assume that. taking the type of the arguments of .r = .r to br indi\.illu;il~.[hcrc. csist ar least members of the class V = i(s= r).Thus Wittgenstein bclicved thnt al1 the essentilil problems having to do with the assurnption of the Axiom of lniïniiy cmbc ;issessc.d by considering '(31)..r = .r'. i.e. 'Thex rxists at least one tiii~~g'.-2 Tlic suggestion thnt al1 problems concçming the Axiom of Infinity can br solved b>- wniidcring '(3.r). .r = .r9 involved Wittgenstein in vanous diftïculties. He had iii;iiui:ii txd. in Iate 19 13. that (5). s = s is ri proposition of plr!sics. 2nd that the siirnr is tnic ~i tltc huiorn of Infinity: for whethrr therr exist Ki things is a matter for exprriencr io dccidc i R23 1. Howcvcr. in late 19 14. hc questioned whether that is so. On 13 October. If wc take 'i3.r)~= -Y' it might be understood to be wutolo_pical sincr it could not get rvriitcn dow Iir 311 if it were false. but here! This proposition can bc investigrited in place oi' ihc .A.uiorn of Infinity. . . Cm wt: 5pcitk of numbrrs if rherr rire only things*?1.r. if for rxarnplti the world only ionhisrcd of one rhing and of notliing else. could wt: say rhat thcre was ONE thing? Riisscll wouid probably say: If there is one thing then there is ülso a function (3.ri< = x. But!- If rtiis function does not do it then we can only talk of 1 if there is a material function wliich is salisfird only by one argument. How is it with propositions like: CIO).(3). @-Y 1 --:\icordin~ly, un the bsts of his discussions with Wittgensfc'in. Rrimsey xgucd that wherher ir is possible 10 .a> ?k'.r ir= .Y) 2 H,). thai 1s. to werr the Axiorn of Infiniiy. can be ;isscsscd by anïtyzing YC'X (.ï = CI i 1. t. tP. 131. .r = .r. (F. P. Rarnsey. op. cir.. 6 1.) Is one of these a tautology? Are these propositions of some science, i.e. are they p?-o[>o'i!io?lsat aH'? But Ict us remembrr that it is the varidhs and itor the sign of generality that is characteristic of logic. (NB 1 1) Con\.inced that (3-j. x = .Y can be considered in place of the Axiom of inlinity. Wiit~cnsteinasked whether it is a propusirion. as hr had belirved earlier. or n icrirrology: for if it wrr fdw then not èven objrcrs or imlividituls exist. iind it is not even possibi~ ri-il(). Tiir. possibility that (3).-r = .r is a tautology led Wittgenstein to question whethrr it a-crc to consist of only one thin;. there is seerningly no rnems of saying that it does. It onlv m~ikcswnsc to assrn something of one thing. Wittgenstein qued. aïsuming '(3). 5 = r' c;innot do it. il' that thing is the argument of ti maierial function 9-Ythat is satisfird by 01ili.ow argument. Thus. for cxrimplç. the proposition '(ZQ):.t 31): ~r: O?. 0:. 3,. . J = ,:' cmbr: taken to assen that therr is a property that only ow thins hüs: and assuming thx itic world consists of only one propeny ~pand two things. '(30).(3~). Q.V and (3~). i 3.r). - 0.r' unbe iakrn to assen thar only one thing has the propeny cp. But the question iipain arisw in regard to (34. x). Q iind (30. XI.- &r. whrther they are propositions of Iogic or naturcil x5r.ncr. Wittgenstein's understanding. in Decembcr 19 13. wcis that an r..;istential ~cnerüliziitionis 3 proposition of natural science: for it is for plr~misto say ii-lirrller mi!. rl,iiz,q r..risrs t R23). The propositions of logic. hc claimrd. m zenrralizlitions ai' citticr tci~itologirsor contradictions. as is. for enample. (.Y): .Y = .r. 2. (3~).= y 0322. 3 ): iind he clcxly intendrd urz(v universal generalizations. In his notebook, however. he suggests tliat it is not the sign of generality. but the use of irrriclblrs. in a grneralized iciutology ur contradiction. that is characteristic of Iogic. or of the generrility of Iogic. Ttiiis. Wittgenstein bçcrime involved in difficulties concrming whrthrr statements of csistcnce that are cornpletely generalized are propositions of natural science or logic. To solvc. the difficulties. Wittgenstein nad to dccide whrther a cornpletely -csneralizçd proposition is (1) n grneral description of facts or (ii) the genrraiizntion of a ra~iiology or contradiction. A proposition is a grnerczii;rd proposition. according to Russeil. if and only if it is either a universal or existential generalization (PM vol. 1 16 1). For cxample. (.y). Fx and (39).@z are generalized propositions. Hence. a proposition is co~upletelygenenilizrd if and only if it contains only variables and logical constants; and evcn variable must bs an cippcireiit variable. For examplr, (34 $). @Y is a cornplctely -zcncralized proposition. Wittgenstein considered various problems with the iissumption that ii çomplrtcly generalized proposition is a proposition of nütural science, but cilso with the ;issumption thiit it is a logical proposition (NB 12-3). (i) The assertion that there is n rcisnce of completely generalized propositions. he claimed. 'sounds extremely iniprobablc' (NB 1 1 ). If the completely general proposition contains only lugiccil cunst;ints. hm it cannot be cinything more than a logical structure. and cannot do ;inytliing more thnn show its own lo~jçiilpropenirs. Hencr. if it is concrived as a coinplctcly gcneralized proposition. rhen it can reprcsent the world ody by means of its own logical propcrries. It cmonly show logical propenies of the world. but cannot sriy mythin? that cm èither br true or false. Hrncr. it cannot br o completely generalizsd p-op.sjiioti ;ind inust bç a proposition of lu,+. (ii) But. according to Wittgenstein. if that i W. rhcn it must br possible to decide which of (30. .r). ox and - (36. r). ar is t~iuiolo~icalcind which coniradictory. Lt does not sccrn. howevsr. thiit the existence of the ijrm coniciincd in (30.1). 0.r could hy irsrlfdrtrrmine that it is rithrr trur or hisr: Ior it docs not sccrn EO be ~u~rliitlk~ihlethrit it be true or that it be faIse. Hence. it does not seem rliut ii is cither iautological or contradictory. Thus. it dors not seem to be a proposition of Io$ t iii) Houlever. Wittgenstein argurd. al1 complrtely generiilizrd propositions are ~iwnüs won as a lougiiigr is @en; and hence. it is hardly believriblr that a completely gcnçr;ilizc.d proposition says sornething about the ivorld. t ivi But. he suzgested. (30..ri. O-r cm also be arrived at by the successive grneralizotion of any given clementary proposition qt:and if rhr generalization of NB 6jj;). But. W ittgrnstcin argued. if a completely genrralized proposition has no clcriit.nts with which objects have bsen ürbitrarily correlated. thçn it is difficult to iiiidcrttincl how ir can represent anythinj outside itsclf (NB 12-3). He regÿrded the corrclations of the elements of a proposition with thrir mranings as something like iippcndagcs by mrans of which the proposition renches out to the world: and hcncr. he ~onsidcrcdthc grnertilizlition of ri proposition to be likr the drawing in of appendagcs. ~iniillïnally thc corripletelg genrral proposition dors not actudly roidi anything in the world i YB 13 1. Biit. Wittgenstein asked. is that what rdly happens iL instead of saying that (p(r. ii is wid that (3).cpx? On that understanding of gencrdization. hi: .wggestcd. it is cstrcrncly difricult to explüin propositions such as (30):ix). @Y: for it apprars almost scrttiin tlix it is ncither 3 tautology nor a contradiction. Thus. Wittgenstein concluded thnt i.\,cn if a complrtely gcnsral proposition is concrmcd only with the logical structure of the ~rx~rld.it niList bc a descripriori of the world: cind hence. thcre are cornpletely senerd /~t-~~>~strio>rst NB 14). The world. hr ciaimed. cün be completely described by mrüns of conipletely genrralized propositions. wirhout using my kinds of names or othrr denoting iicns.L and to arrive at piinicular descriptions it is necessary only to introducr namrs. or other signs. by sa@: after '(3~). . .'. for rxarnple. 'and that .r is (1' (NB 14). Howrvrr. hç could not esplain how a pruposirio~l cm br concemed only with tne logical structure of thc world (SB14-5). In his notebook entry of 23 October. Wittgenstein arsues thiit evrry proposition. in addirion to having the samc iogiccil propenies as that which it represents. mubt consisr of >iur?ies the rneanings of which are arbitrarily determînrd: for the possibi lity t hat a langage unambiguously represents different facts hoving the same logical proprrties depcnds on the giving of different names to the different objects of wtiicli the façts consist. cind that must take place by means of arbitrary stipulations. But. hr clainied. 'If one tries to cipply this to a completely gneralized proposition. it appeus th31 therr is some fundamentai rnistake in it' (NB 17). Nonetheless. Wittgenstein believed rlitit sincc a çoriipletely genrralized proposition concems al1 rhe things that there Itczppen to br. the generrility of a completely generaiized proposition is czccideritd generality, and a coniplr.tely prrtil proposition is a materinl proposition (NB 17). Accordingly. hr argued. If thc cornplrtrly grneriilizrd proposition is not cornplrtrly drmaterialized. thrn a pruposition does not set dematrrialized at al1 through genrralization, as Iused to think. Wicthcr 1 assert something of a piirticular thing or of ail the things that thrre are. the ;tl;scrtion is cqually material. (NB 17) Thus. Wittgenstein concludrd. a completely generdized proposition 1s neithrr a ~sncralizcdrnutology nor ;i generrilizrd contradiction: for it is m)t a &m~~~er~cilizd proposition. It is. hr belirved. the genrral description of ri hct. Hcnic. Wittgenstein had to explain how it is possible thrit a cornpiçtely gcncs;ilizcd proposition cïul drscribe a hct. I: was cssrntial to his understanding of a proposition thai in it things are cirriinged. as it were. e.~prrir~ie~itrilly.as they do not have to he in rcdity: for it is thus that a proposition can eithcr be tme or false (NB 7. 13). Hencr. the problcrn. as Wittgenstein undrrstood it. was to explain hzr is mangrci cspcrimentcilly in a complrtely genrralized proposition. In the "lotes on Logic'. he hd cl;iimiid rhat it is quitr possible to suppose that a symbol is complrx only if it contains names of objects: and hence. that the symbol '(I. O). ~r'.Tor example. is simple. to be rcprdcd as the name of a form (NL LW).Hence. on 25 October 1914. he wrote in his noiebook. 'Wc might also say that Our difficulty starts from the completely generalized propcisition's not appeiuing to be cornplen' (NB 18). A cornpletely genrralized proposition does na nppear. like a11 other propositions. Wittgenstein claimed. to consist of xbitrxily stipulated signifyins parts arranged in a logicd fom. It iippears not to /icii.e a torm. but to be a form completr in itself. Accordingly. in his notebook entry of 15 October. Wittgenstein asked. 'If there are quite general proposiiions - whot do we arrange cxperimrntlilly in them?' (NB 13). In answer to that question. he considered the following pozsibilitics. (i On 17 October. he stated. 'If there are quite general propositions, then it looks as if such propositions were experimental arrangements of "logical constants". (!)' (NB 13 1. That. liowrver. was not an answer consistent with Wittgenstein's understanding of logical constants. If a proposition contains ody lugicnl constants. then it is only a logical structure. But a proposifion cannot be concemed with the logical structure of the uvcirld: for the logic of the world is priur to di tmth and falsity (NB 14). (ii) Hence, ahsuniing rlirit 3 complrtely generalizrd proposition is not rin rxperimental arrangement of logicd constants. Wittgenstein concluded that it crin only be an arrangement of' apparent viirinblcs. tn his notebook entry of 3 1 October. he States thüt a proposition like '(3.r. $1. 0.is just 3s complrx lis an rlemcntary proposition. and that is shown in thrit it is nccchsary ro mcntion 'CI' and 'r' rxplicirly. as parts of the sign of scnerality: for both, indcpcndcntly. stand in signifying relations to the world. just as is the case in the clcrneniary proposition 'iy(rr)' (NB22). In his 3 Novembcr notebook entry. Wittgenstein cl~ibor:ttt's: TIi;ii srbiircii-y corrcliition of sign and ihing significd which 1s a condition of the pwsibility of the propositions. and which 1 found lacking in the completcly ~enrral propositions. occurs there by means of the penctrdity notarion. just as in the rlcmentq propohition it ocsurs by mrms of namrs. . . . Hence the constant feeling that genrrrility niakcs its ;ippcrirantx quite like an argument. t NB 35) If ;i complctçly generaiizcd proposition is a proposirimi. then the propositional sign must bc ;in csperimental arrangement of the signs that have been ürbitrarily correlatcd with the rtiings .;ignified in the proposition. It is by means of the variable names contained in the sign of sywiility. Witt~ensteinclaimeci. that the arbitrriry correlations brtween signs and things significd are given. Thus. he belirved. it is the variables that cire arranged ~spri~~i~rir~zll~~in a completely gcner~lized proposition. and that rnplains how it is pu\siblr: thrit a çornplstely _jenerdizcd proposition cmdescribe a hct. Howver. Wittgenstein's undenaking in his philosophical notebooks to zive an 2xpIanation of completely generalized propositions wns the result of his concem with n fiethrr or not i3.v~..r = -r is ri proposition. Havin; concluded that there are completely .r = 'i and so on. The solution to the difficulties, he believed, depended on whether it is possiblc to dispcnsc with the sign of identity in siving a description of the world. in wtiiçh case propositions like n = b and (31):(J). y = s would be shown to sa- nothing ~iboiitthc wxld. The dilfiçulties reglirding identity initially brcamr a concem for Wittgenstein giwn hi.; suggestion. in his notebook rntry of 17 October 1914. that a complstely genrrril proposition rnust br. a description of the world. even if it is concemcd ody with the Io-ic;il ~tniciurcof the world (NB 14. He rcgarded it as ubvious that a completely ;in:ii)mxi proposition. i.r. an elrrnentary proposition. contains just as rnany niimes as there ;ire rhinz.; contciinrd in its mctaning (NB II). Having cliiimed that the world çan be dscrihrid by nieans of completcly gcncralized propositions. without using ciny kinds of n;iiiics or othcr dcnoting signs. Wittgensrein hiid 10 decidc wheihcr the identity sign is nccchsary to give 3 complerr description of the world using completdy genrralizrd propositions. If the world can be so described. he clüimed. then it is possible to devise a pic~~ircof thc world without saying what is (i represrntation of what. Httnce. the identity bign semis to be essential to such a description: for to describs the world by mrans of ~oniplctclygcner;ilized propositions. without using drnoting signs. it secms necessary to idcntifv :il1 the differcnt objcicts of differrnt types that there are in the world. Wittgenstein cliibor;ited in his notebook entry of 17 October: Lct us suppose. r.g.. that the world consistsd of the things A and B and the property F. iind rh;it F( .A>werr the case and not F( B). This world could also be descnbed by mrans of And Iirre one iilso nrrds propositions of the type of the iast two. only in order to be able tu idcntify the objscts. From dl this. of course. it follows that tliere are conzpletr~yetrnrl proposirions! But isn't the first proposition above enough: (3.y. 0)Q.Y. - qy. x r y? The difficulty of identificcition can be done away with by describing the whole world in Hencc. Wittgenstein suggestrd arguments both for and ngainst the conclusion that the identity sign is necrssary in giving a complete description of the world using completely ~cnerdizedpropositions. (i) If the world consisis of only two things and one propeny then to describe the world using completely generalized propositions, Wittgenstein ~irgued.it is nccrssiiry to Say that therc is ot11y rhe property 0 and or14the things s and J. ;incl that .r is nut the scim~thing as y Thus. to give a co,qderr description of the world. hè concludcd. ir is nttcessary to use the sign of identity: for the description musc includr the propositions (30).(y). y = 0 and (3.y): .r # . (z).: = v z = y. (ii) Howrvrr. Wittgenstein also lirgurd thitt if the number of variable names for things. properties and rcl~iiioiis.giwn in the sign of generality of a completr!y generalized proposition w hich Jcscribc* tt~cW~O~C wurld is the sarnr as the number of things. propenies and relations in rhc wxki. thrn it is jior necessary to use the identity sign: for the number uf difkrent \-ririablc namcs of dit'fercnr types signifies the nurnber ut' different objcçts of different type\ thiit therc are in rhe world. Hencr. assuming thrit it is possible Io describe the world using «nly complrtely gencraiized propositions. Wittgenstein argued that ri complete dcscsiption of thc world cm br given without the identity sign. Thüt is. assuming that a cumplctely gcneralized proposition can drscribe the world. without using any kinds of namcs or other denoting signs. the identity sign cm be disprnsed with in giving such a Havin: later argurd that the occurrence of the sisn of generality in a generalized proposition such as (x). W. for enample. is similx to that of the argument in a proposition such 3s ai. Wittgenstein was able to conclude decisively thot the sign of identity is not necessa- to describe the world using completely general propositions (4NB 15. 90). It 1s not nerded in completely generalized propositions. he believed. for the same rerison thiit it is not nrrded in ungeneralized propositions. in the 'Notes on Logic'. Wittgenstein hdquestionrd whcther propositions are necessq in which the same argument occurs in dit'ferent argument places. but he claimed that it is impossible to dispeiise with such propositions since it is obviously useless to replace Q(u. a), for example. by @(a.b). n = b (NL101). He did not conclude that the sign of identity cm be dispensed with. since he belicved thnt '.Y = s' is a propositional function: and thus. that 'Socrates = Socrates'. for suample. is ii proposiriorl (NL 103). in the 'Notes dictated to G. E. Moore'. howcver. i\'ittgcnstr.in argues that the sign of identity signifies an irzfrrmrl relation betwern a funciion ;incl its Iirpnent. ir. that show by g Wittgenstein wrote in his notebook. ' "I = y" is tior ü propositiond fom'. and rirgued in qport O t' Iiis claim. Ir is clcar rhat 'trRlr' would have the sarne meaning as '(iRb. On 2 Sovriniber. he .;tated that 'tr = (4' is a tautology. although not in the samc sense as 'p 2 p' (SB 74). Thus. according to Wittgenstein. naithrr 'a = b' nor 'rr = tr' is a pt-~p~sirioiz.Hcncc. he believrd. '.Y = y' cannot be a proposi tionül function: and ;icc«rdingly. cannot bc used to say whrit. for example. is supposed to be sayble using the proposition '(30):. (5):~i.:@y. i)r. I,.. : . = z'. Thus. in his notebook entry of 29 ';~~vt.nibcr,Wirtycnstrin clnims thar hr brlirves it would be possible to ?ive a notation in u.liich thc sign of identity has bern dispenscd with rntirrly. so thot idrntity is indiciitcd mcrclv. bv- rhe idcntity of the signs (NB 34. In thiit case. he elaborated. 0( Hencr. ii rrsult of his ünalysis of both cornpletely genrralizrd propositions and thc idcntity function. Wittgenstein concluded that (3x1..r = -K is a psetdo-pruposiriori. If thc sig of identity can be dispensed with thrn in the resulting notation the idrntity of siens in a complrtely generniized proposition is indicated by the sign of generality (cf: NB 1 Y i. The sign of generality in a completely generalized proposition is not part of the piçture. triking part of the propositional sign to bc a picture (NB 25). Rather. according to Wittgenstein. in a completely generalized proposition, the sign of generality determines Iroii. r~ici~ixdistinct pans rire arranged in the part of the propositional sign that :ives a logical picture of ii situation. Hence. it is the sign of generality that determines the idcntity of the sigs in the part of the propositional sign that lo_oicrilly pictures the .;iruation. TIiiis. the apparent proposition '(3).x = .Y' signifies no more than does '(3).. .'. Wirtgcnstein states in his notebook entry of 12 May 1915 that if '- (33.-r = .r' is ~iippodr» sa? 'There rire no things' thrn it is n remarkable hct that in order to süy it wing the zynibuls of Prit~cipicihl~irlirnitrricrr. it is nrcessüry to lise the sign '=' with ivtiicfi thc proposition is rcally not supposed to be concerned at ail (NB 47). That is to s;i)-. the sign of idcntity is cornpletely unnccessiiry in ordsr to show whar is supposed to bc: sriid by 't3.r). .r = s'. Rüther. what '(3)..K = .Y' is supposed to say. Wittgenstein hclicvcd. is shown by the correct use of the sign '(31). . .' in a generalized proposition. Wittgcnsiein's understanding of (31).s = .r is sirnilarly explriincd in rin August 1917 ltlttcr [O R~mlsryin rrsponsr ro his writings conceming the assertions on the Axiom of ln finit? in the Tr~r . . . thc hiom of Infinity risserts merely that there are an infinite number of individuals. This appears . . . to be a mere question of fact; but the profound andysis of Wittgenstein ti~is shown that this is an illusion. and that. if it rneans anything, it must be rithrr a i;i~itolog>.or a contradiction. This will be much easier to rxpl;iin if we bepin not with int'init'. but witti some smaller number. Lst lis >tut wirh 'There is an individual'. or writing it ris simply as possible in logical ntmtion. Sow avhat is this proposition'? It is the logicd sum of the tautologies -Y = .Y for di values oi .y. and is rhsrefore a tautolo_oy. But suppose there were no individuals and therefore no vülues of .r. thrn the übovc formula is absolute nonsense. So if it means mything. it must be a ~iutology.~-' Wittgenstein answered that the symbol '(3~)..Y = r'. even if '=' is defined so as to make it tautolo=ical. does not say 'There is an individual'. He agreed that if there are no individuiils ihcn (3)..Y = s is absolute nonsense. But. he argued. if (3).x = r is to say 'Thcrr is an individual' then - (3).x = .Y says 'There is no individual': and hence. given tliat irmn - (3~)..Y = s it bllows that (3)..r = s is nonsense, - (3s)..r = .r must be nonsense i:self. and so mut (3.r). -Y = .r regxdless of whether it is assumrd that thrre cire no indi\~idulils.Witt~enstcin continued. 'Perhaps you will answrr: of course it dors not sa? "Therc is lin individual" but it sliuicls what we redly mran whrn we say "Thrre is an indi~.idual"'2.'. Howevcr. to that he respondéd that this is rior shown by (3s)..r = .Y. but simply by th<: correct use of the symbol (31).. . : and thus. (3-r)..Y = r can be said to show it only in [hi: sanie srnse that - (3s)..r = s cm be said to show it. Wittgenstein did not rakc i3.u. .r = r ro bc ;i tautology or - i3.r). .r = .r to be a contradiction. but regarded both ~1snunsensiciil pscudo-propositions: for both try to siiy usin: the idcntity siy something tti;it ciin onlv be show by the sign of generality. Ramsey claimeci that. given Wttgcnsitin's linalysis in the Trcictmts. the Axiom of Infinit? is either 3 tilutology or a contradiction. and hc assumed it to be tautol~gical.?~Howevcr. it was Wittgenstein's undmtnndinp that 'the Axiom of Infinity is nonsense if only because the possibility of txpressing ii would presuppose infiriitely müny things. i.r. whai it is tqing to assen' (PR 1141.Sirnilcirly. whüt '(3s)..r = .r' is intendrd to süy. htt believéd. presupposcs the cxisicncc of lit Iccist one thing. i.r. what it is supposed to asseri. In Iiis philosophiciil notebooks. Wittgenstein suggests two strritrgirs for lissessing rhe .\xioni of lntinity of which he says nothing in the Trmcinis. namcly. undenaking the mal>sis of (Li..r = r. and in addition. that of the definitions of the cardinal numbrrs. in thc Ttuc*t Iicir wpplied ri basis for giving the solution to 'al1 the problems that Russell's ".-\siom of Infinity" brings with it* (TLP 5.535). to the suggestion that he intended to assert that a11 -, I -'L.. \Yi t tgcnstcin. Ictter to Rrtmsey drtted 2 f uly 1927. Ludbr~igCVirr~qensstein: Cambridge Lerrers, t i 6-8. -< -. F. P. Riimscy. op. m.. 6 1. probkms having to do with the axiom are solved in the Trmronrs. he responded. 'This is~i'r ihir 1 tmwr IO fil!!'? Hence. Wittgenstein denied that the solution is jivrn in the Trrictirtrrs. Thus. sincr hr says nothing in the Trcrctcrrirs conceming the two strategies for assrssing the .biorn of hfinity to which he refers in hiç philosophical notrbooks. it clin rcasonably be cisksd whether Wittgenstein's solution to al1 the problems having to do wiih thc uiom is supposed to result from pursuin; the two strategies in question. The answer dspends on what basis Wittgenstein made the assenion in the Ti-trmrits. 'Al1 problems thrit Russell's "Axiom of Infinity" bring with it can br solvrd at thpoint' ITLP 5.535 i. Wittgenstein had claimeci. in the .?lotes dictritcd to G. E. Moore'. rhat tlic. fundamental "logical constants'' are generality. idrntity and the truth-funçtion a,tiich R~isscilcallcd 'incornpatibility' (Nb1 116). In the Trm-klt1r.s.howevrr. he iisserts that thc otil>. fundnmcnt;il logicül constant is incompatibiiity (TLP 5.5 j. In support of this awx-tton. hc argues i i ) rhat in a correct conceptual notation generality mus[ be dissociatcd t-rrim trurh-functions. i.e. from logiciil constants such as logical produçt and logiçal sum TL5.5. ; .vi. and (iii thit in such a notation identiry cm bi: rejcctcd altogerhcr iTLP 5.j;lï:i. It is against this background that. in the Tr~rcr ~11lvt.d. , . for hue they al1 lie iri rzrtce'? 1C'hiit Wittgenstein regarded as the problems that arise through Russell's Axiorn oi Infinit)!. which he believed Xe to br solvrd at the point indicated in the Trcictur~t.s.üre ~irg~iablythosc having to do with the analysis of (i) the proposition. or pseudo- proposition. Ch).,r = .Y' and (ii) the definitions of the finite cardinal numbrrs. It is clriir r li:it hc çonsidcred a correct understanding of both completel y gencralized propositions mJ rlie idcntity fiinction to be essential to the analysis of '(3s)..r = A-'. If ir crin be show rlut tic ;dso considcred it essential to the analysis OF the definitions of the finire cnrdincil riiiriibcrs. thcn it woiild serm thlit the problrms rirising rhrough the Axiom of Intïnity. to ushich Wiitgms&cin rckrs in the Trlicrtrrii.s. cire the problerns involved in giving an mîiysis of both '(3).= .Y' and the definitions of the finitr cardinal numbrrs on the b;i\is ol' 3 correct understandin_gof completely generdizrd propositions and the identity i'iinçticin. It hiis bern explained how Wittgenstein's understanding of ~eneralized priipositions and the identity function led him to concludr rhat '(3.r). s = .r' is a nonscnsic:il pscudo-proposition. It remains to br explainrd how Wittgenstein analyzed thc Jeiinitionc; of rhc t'inite cardinal numbers. LVitt~enstein's undenaking. in his philosophical notebooks. to give n propttr analysis ut' Russell's dcfinitions of the finite cardinal numbers. resultcd in his rejection of rhc conception of nurnbers undrrlying cardinal arithmetic. His understanding of both i ii thc idcniity function and (ii) assertions of the existence of numbers of things. Ird hirn ro concludc that the definitions of the finite cardinal numbers presupposr an und;.rst~inding of what is supposed to be definrd. Hence. given his iindysis of the i\.rrtch '( 3).Cr: - (3.J!. jr. fi' (TLP 5.3321 ). Henre. havinp apprirently dispcnsed with the sign of identity. \$-ittsc.nsrc.in clriirns thal in ri correct çoncepturil notation pseudo-propositions like 'u = a'. 'L; = b. b = c. 2.cr . = (- . (XI. .r = x'. '(Zr)..r = (1'. . . . ~mnotrven be wrrtten down tTLP 5.534). The Axiom of Intîniiy. ht: .;USCC~~S.1s ch ;1 PSCUJO-propt~s~t~on. =%.b'itrgcnstcin. op. cir., 50. 2'1f!7ici. detlnitions. Wittgenstein was Ied to conclude that numbers cannot be defined as classes of classes. That is. he was Içd to reject cardinal arithrnetic altogether. Giim Wittgenstein's rrjrction of rhe identity function, he concluded that if it is possible to defini: the tinite cardinal numbers, the definitions must contain completely gcncriil propositions thiit. as understood in Prittcipici Mdrerm~ic~i.assert that there txist soriic tïnitrr number of things having some definite property. If the identity function is a ~~.sc'rldo-jilricrior~.Wittgenstein believed. then it cannot be uséd ln the definitions of clsscs. tience. for eucirnple. Russell's universal class. and his nul1 class. must br rcjcciecl. In hinoiebook entry of 1 1 'lovember 19 1-1. Wittgenstein tirgued. Sincc '11 = b' is noi a proposition. nor '.Y = y' a function. a 'ciass .?(.Y = x)' is a chirnera. ;inJ so cquiilly is the so-called null class. (One did indrrd always have the feeling that ~i,ticrilvcrx = r. (1 = ci. rrc. were ussd in the construction of propositions. in a11 such cases oncl wiis only srtting out of a diftïculty by mcans of a swindlc; 3s though one said 'ci csists' mtxns Y%-)..\: = a'.) This i.s \iirotr,y:rince the tleji>ririorz uf ciasses irwy ,qr~lrcuitrrsrlir erisreticr of. rlw rc.ci! ]ilN ~.fiO?lS. 1Vticn 1 Lippcür to asscrt a function of the null class. I am sriyins that this function is true ut' di functionz that ;ire nul1 - and I crin say that cven if fia function is null. 1s .Y = .Y. = ,. 0.1-idcnticd with (.Y).-ar'! Crirtriinly! (NB 28-91 ;icç«rding to Wittgenstein. since neither is defined by means of a red function. What qpclir to bc assenions iibout rhr nul1 class. he believed. are really assenions about dl lunctions that are null: for it is not necessriry that there hr ri nul1 çlass in order ro r,:rike iiicli ~isxnions.eïen if rro function is null. Accordin@y. Wirtgcnstrin claimed that. for r.limplr.. ir is possible to write (x). - 0.r instead of .r F .r. =, Qr: so that. rather than say thtit a function is extcnsionally equivalrnt to the nul1 class. it is said that the function is itself niill. In that way. Wittgenstein believed. dl apparent references to the null class çan be zhown to be iinnecessriry. Thus. Russell's definition of the cardinal number O. i.c. O = L\ Df.. un be dispensed with: and for aimilar reasons. since I'X = = .Y). his crin dsu btt dispensed with. Hence. Wittgenstein concluded that if a nurnber is to be ddined as ii clnss of classes, as in cardinal arithmetic. and the identity function does not determine a class, then the definition of a finite cardinal number must contain the cornplrtrly generalized proposition which. as understood in Principiri Marhernciiiccl. says ihat for ronie property cp there exists rhat number of things having the propeny. Thus. @en Wittgenstein's understanding of completely generalized propositions. tir concluded that the definition of a finite cardinal number must shoiv what it is supposed to s 'Wliat the pseudo-proposition 'Thrre are ti things" tries to express is show in Ianguage hy ihc prcsencç of 11 proper names with different mcmings' (NB 20). For the same rc;ison. hc bclieved that it is not possible to scr\. ihere are rt things having somc definitr propcrry. Wittgenstein elaborated in a letter to Russell written in August 19 19: . . . what puwünt to siy by the apparent proposition 'There are 2 things' is slioirw by rherc king two names which have different meanings (or by there being one name which may have two merinings). A proposition cg. q(n. b)or (3p.-Y. y). q(r. y) doesn't say chat tlicrr are two tliings. it snys something quite different: hm irhctlirr ifs rrue ur fiilse. it SHOWS what you wünt to express by saying: 'There are 1 things'. (R37) Thc proposition dcp. .Y. -1. q(x.y) is true if .r 2nd J. thrn the proposition (39, r y). h'tthe Russtttlian dcfinition of nought nonsensical? Can wc speak of a class i(.r # r)at ;dl'! - Ciin wti speak of a class .?(.Y = x) eithrr? For is .Y + -Y or -Y = -Y a function of .va? - Musr not O be Jetincd by mrans of the huporllrsis (30):(r) -$Y'? And somerhing anrilogous uwld hold of 311 other numbers. Now this throws light on the whok question about the exisicnce of numbers of things. Ttic proposition must coizriiitr (and in this way show) the possibiiih of ifs rmth. But not niore than the possihiliiy. By my dcfinition of classes (x). - i(@)is the assertion that .?(@Y) is null and the definition of O is in thrit crise O = ii[(.\-). - a] Def. (NB 16) Thus. accordinp to Wittgenstein. the cardinal number O is defined as the class of dl classcs of mernbers rc such that Qii. for any ~.r^such that. for dl x. - is true. That is. the cardinal number O is the class of classes that are null. The cfefiriirns of Wittgenstein's Jefinition refm to soiiir function 0-r which happens to be null, rather than to Russell's mi11 clus: for the nul1 class is defined by the psr~tdo-jincriori.v = which as such cmnot determine ;t C~S.Hence. if there is no nul1 class. it seems that the definition of the cardinal number O must contain the hyp~thesis(30): (1). - +r. which presupposes that Waisrniinn's on Wittgenstein's views conceming mathematics. written during 1929 or 1930. it is explained that Russeli's definitions of the cardinal numbers are based on a confusion ofjbrrm and concepts (WWK 223flJ3* The numbers are forms. reflected in the rrr-ticrrrr-r of 3 proposition. Words signifying forms are not names of concepts. not possiblc constituents of propositions, but rules for the construction of propositions i WW K 120). Hencr. a number word symbolizes in a completrly different way from a concept ivord (WWK 26).This is shown. Wittgenstein brlisved. by the anülysis of rhr proposition 'There arc two things with propsnyj' by means of '(3-K.~).fi.b. - (3. y. 3. /:Y. /j-.ji'. It is the cciricibfrr contained in the proposition '(31.y). jk. fi. - (3.r. y. 3.j:~. fi. ' which signify that there are nvo things having propeny j: It is not signifird by prcdiuting a concept. i.r. the so-called concept of the numbrr 2. of things rhat have propcrty j. The same can be said. Wittgenstein believed. of the definitions of the finitr Russcll. too. had to rmploy . . . [the principle that numbers of thinzs tire repressnted by numbcrs o t' vari ables J when introducing the p~ticulitrnumbers. In order to introduce the nunibcr 7. hc ncrds to cmploy n symbolisrn which itself hos the rnultiplicity which it is Jcsigntxi io define. But then the multiplicity and not the drtïnition is the decisivi: thing. A dctlnition d~qïriessornething and zho\rs sornething. It is what a definition shows that wrrcspunds to a numbcr. IWWK 223-4) \iyittgcnstsin cir~uedthat ii fom cannot br defined as a concept is defined "'~j:F. IVaisniann, op. rir.. 63 alrcxiy be understood to understand the defniens of the definition. 'Lf a fom were det'inabis'. Wittgenstein ctaimed, 'we could not understand it without a definition' i WWK 224). .~ccordingly, he believed. as such definitions of logical forms are çoinplc.tely unnecrssary in logic. the definitions of the finite cardinal numbers in Pri~zcipi :id R~issèll'sdetlnitions of ihe numbers in cardinal arithmetic, or at least believed that the niinibcrï cire propsrly definrd as classes of cliisscs. until he undenook to iinalyze the Asiom of Infinity. The ÿnalysis of the Axiom of Intlnity required that hc mivr at correct understandine of the apparent proposition (h).r = .r and the definitions of thc finite clirdinnl numbers. Thus. he wu led to considrr problems concrming the senerality ol' c«mplctely zcneralized propositions and also the identity function r = y. His resulting vien*son (i the use of identity in the definition of classes such as the nuIl class and t ii) ;issertion.; of the existence of numbcrs of things having some definite property. led him to concludc ttiat ihê dcfinitions of the numbers as understood in cardinal xithmetic presuppoie iin understanding of whlit they are suppostid to define. Hence. he rejcçted ciirciind ririthmetic. Wittgcnstrin's introduction of the naturd numbers in the Trcicrcitits is. in a ssnse. hundcd on his rejection. in his philosophical notebooks. of the cardinal arithmetic of Pri~rcipi~i.bldiemiricci. If the numbers are defined as the extensions of c~cri~il pri>positionlil functions. as in Pri~icipiuitlutlie»iciticri. then how many numbers cm be dc1ïnc.d in cardinal cirithmetic depends on whether functions cxist that are satisfied by the ncçrssary nurnber of possible arguments. Hence. the possibility of defining any number rests. no[ un an ri priori law. but on accidental properties of the world. ie. on what there iilippriis to be in the world. Thus. the possibility of constructing the nurnber system rests on crctiicii propcnics of the world. However. Witt~ensteinargued that the concept of nurnbcr is a forma1 concept which is given n priori, not a posteriori. Wittgenstein wrote in the marzin of Ramsey's copy of the Tractatus. 'Number is [lie fundamental idea of cdçulus and must be introduced as s~ch'~!He later explained. in the Phifosophicd Rrwirks. 'Instsüd of a question of the definition of nurnber. it's only a question of the grlimmar [the logical syntax] of numerals'. claiminp thnt 'This is what 1 once rneant when 1 \;!id. it is with the calculus (system of calculation) that nurnbcrs enter into logic' (PR 9.By 'calculus' he apparently meut 'the number series' or 'the number system' i fhid. 1. Hcnce. Wittgenstein believed that the concept of number is fundamentally the concept of the nurnber srries (cf: PR 207): and that. given the chmcter of the number scric.3. the concept of number is not properly represrntrd by ÿ. funclion. signifying ü corninon propeny of the numbrrs. as in cardinal arithmetic. but gives the rule of logical \ynr;ix. thc forniri1 law. in accordance with which the number signs are constmctcd. The cswniial point of the Trumtits concrrning the charmer of the number serirs is. as Wirtgcnhiein statrd it. 'The order of the numbcr scries is not yovemrd by lin extemal rcliition but by an inicmal relation' (TLP 4.1352). Xccordingly. the Trcicicinis is intendrd to nikc it çlcar thlit < i) the concept of numbrr isjitndmreritïd(v the concept of the number wrics cind (ii) the nurnber series is ordered. not in riccordancc with an extemal relation, biit in xcordançe with an interna1 relation. It follows that the concept of number is ri furrti~ilconcept. the gencriil trrm of a formai srries: and hrnce. Wittgenstein believed. nn '1 psiori [al\. I-lcnçr. the Tnicrcitirs is intrnded to make it clear that not only cardinal arithmetic but ;ilm thc ordinal arithmetic of Primipiri iihfheniuticci must be rejected. Russell was ;ibic io cxplnin the ordinal properties of the numbers independently of cardinal uithrnstic by çonstruçting a relation withmetic of which he considered ordinal arithmetic a part1cul;ir application: for if the relations concemrd are restricted to well-ordered relations then relation arithmetic becomes ordinal arithmetic. i.r. the rtrithmetic of ordinal nurnbrrs Jcveloprd by Cantor (PM vol2 293ff.'). in his lrrtrr to Wittgenstein. wrirtrn 13 upst 1919 after he had received Wittgenstein's manuscript of the Trncrnnts. Russell wrott': 'Ic. Le~vy...A Note on the Tsxt of the Tractaus'. Mind July 1967: 122. 1 do not understand why you are content with a purely ordinal throry of number. nor why you use for the purpose an ancestral relation. when you object to ancestrd relations. This piirt ut' your work 1 want funher explained. hlso you do not state your reasons ajainst darses. l2 Tlic recimns for Wittgenstein's rejection of classes in the definitions of numbers. so far ÿs ciirdinal xithmétic is concemed. have becn cxplained by considering Wittgenstein's philosophical notebooks. However. in what sense the Trcrcfcrtiis can be said to contain an 'ordinal tIicon; of numbir'. and whether Wittgenstein used an ancestral relation as ihc basih fur riich ttieory. are questions that rernain to br answered. An ordinal numbrr. xcordirig to Russell. is ;i class of ordinally similür wrll-ordered series. that 1s. of relations gcncrating such mies i MLT 180). The relation of ordinal similiirity Russell defineci by .Siwr = P Q { (3s).S E 141. CD'S = C'Q. P = S 1 Q 1 CnvbS)Df.'. where C'Q is the field of te relation Q. i.r. .? ((3~):xQy. v. ~Qx).and s 1 Q 1 CnvS = S 1 i Q 1 Cnv.5). piven that S 1 Q = .? i ((3~).XSY. yQ:) Df. The chss of serial relations he dctïncd by -Sc.,- = P {.rPy. 2,. $,.. - LI. = y): xPy. -Pz. 3,.,., :. xPz: -r E C'P. 2,.. P *X u KY v P..v = CbP)DL'. cvhcre I'-.Y= j (.rPy} (cl:PM vol.:! 519). Russell defincd well-ordered wrid rcla~ionsby 32 = P {PE Ser: u c C-P. g!u.2, . 3!(a- CnvbPb«\J DI.' (cf PM ~~11.34). .;O that P is taken to generate a well-ordered serirs if P is srrial and 3ny class a cont;iincci in thc field of P and not nul1 hris ri tlrst term: for 'CnvgP'a' denotes the tcrrns comiiig after some term of a. Hence. drnoting the ordinal number of well-ordered rclation P by 'So'P'. and the class of ordinal numbers by NO'. Russell drfined No = ci P { P e R. a = S~iror.P 1 Df.. and NO = NobbRDI. MLT 18 1. cf. PM vo1.3 18). Thus. Riisscll rcgiirdrd an ordinal number as the çomrnon property of classes of serial relations wliiçh generate ordinally similar series. Assuming thlit al1 finirr ordinals cxist as indi~.idual-ordinals.i.e. ris thé ordinds of series of individuals. he bdieved that the tinite ordinds thsrnsclves torm a well-ordered series the ordinal number of which is the trrinsfinite ordinal denoted by 'o'.Russell claimed that 'only the serial or ordinal propcn.'ss of finitr numbers are used by ordinary mathematics. whrtt may be called the :--B. Russcl 1. Lrtd~igii'irtgcnsrein: Cambridge Lerrers. 1 2 1. logicil propenies [i.e. the properties characteristic of the cardinal nurnbers] being wholly irrelevant' (PrM 24 1). Wittgenstein would have agreed: and accordingly. the explanation of numbcrs given in the Tractcltus is. as Russell said, in a sense. 'purely ordinal'. But. it is clcar thar. having rejected the definitions of numbers in cardinal ürithrnetic. Wittgenstein did not sirnply accept Russell's relation arithmetic, or ordinal cinthmetic. Indeed. the staierticnt in the Trtrctritrrs that the numbrr series is ordered by an internul relation is inconsistent with the fundamental presupposition of the relation arithmetic of Principici .bld~emcrticii. hithough Wittgenstein undendces to givr a proper expianation of the ordinrii proprnirs of the natural numbers in the Tr~crrrrus.hr rejrctcd Russell's conception of rhr ordinal numbers. and the ordinal propenies of the cardinal numbers. Russell wu concerneci in Prutcipk il.luzlimuric ;mcrs~ralrelation t cf: PrM 199). A projression is explriined. in Principici iV~.ki~lirrncitic Accordingly. Whitehead and Russell give 1i direct definition of a progression by mrans of r hc propcnics in question. The characteristic that distinguishes progressions from other infinitr scrirs. Russell clliimed. is given by the principle of mathematical induction: or rlither. whnt hr regardecl as the essence of the principle (PrM 240. cf: 314). For m;ithcmritical induction he maintained. is the application to the number series of a conception which is applicable to al1 relations. namely, the conception of what he called .------.'-'.-kcordinslu. in The Pririciples of i~iarftem~tic~.Russell clriims thrit Peano's miorns detint: not the finitc numbcrs. ris Pcrino believcd. but rrither ri progrcssion (PrM xxv). Hence. Rusself's definition of the clriss K,,, to uhich hcrlongs the clrtssfktte nrrmber. is triken to detinc 3 progression. 'the ancestral rrliition with respect to a given relation' (PM vol.1 543). Thus. the de finition of a progression given in Priwipia Mdternaticn. whether generated by a one- one relation or a transitive relation. depends on the concept of an ancestral relation. The çlass of progressions generated by one-one relations. for example. is denoted in Principici .Clarlwilitic~iby .Prog7.and it is defined by 'Prog = (1 -t 1) n ~(D'R= R:B*R) Df.'. where B = .i' &.r E DBP- CD'P) Df.. Rm0xis called *the posterity of a term .Y' and R, is the ancesird relation with respect to R (PM vol.1 579. 607: vol2 2-15), The significancr oi' rhc concept of an ancestral relation. according io Russell. is rhat by means of it. sivcn a one-one rcliition R. which is n progression, it can be shown thai C'R is ordrred by a serial rclariun dcrived frorn R. Since Russell dcfined the concept of a serial relation by mrans of rhs conccpt of a tnillsitiw relation (PrM 2 17. PM vol2 -197fl.). to show that the field of a one-onc relation R has order ir is necessary to show that the ierms of the relation can be cli;iracterizeJ 3s standing to one another in a relation which is transitive. Russell brlievrd thiit rtic only iixy io do this without presupposing the finite cardinal. or ordinal. numbcrs 1s 10 Jcfinc the ircinsitive relation using the ancestral relation wirh respect to R: for again. rhc concept of ;ln ancesrra1 relation is obtained not hmthe finite cardinal nurnbers. but i'roni the principle of mathematical induction ( PrM 239-40. cf: 520). Hcnce. sivcn his conception of an ancestral relation. Russell was able to give an csplilnation of thc ordinal properties of the Rnitr cardinal numbcrs on the biisis of the principle of niathematical induction. Indeed. he dejirtetl the class of finite cardinal riunibcr~by means of mathematical induction (PM vol2 181-2). The clriss of finite cardinal numbcrs is drfined in Principiu Mdirrnntica by YC induct = C? ( a(+,1 ),O}. Thus. takine '+( 1' to drnotr the relation berween any two inductive cardinals v and L. + 1. if the ;\siom of Infinity is assurned. then it can be proven that the clss of finite cardinal numbrrs foms 3 series generated by a progression. For Russell miiintained that r is an iincestor of :with respect to R if and only if .r E C'R and :belongs to evrry hereditary dsss to which .r brlongs: which is to SV..rR,c. i:s E C'R: i; i (3-1. J E p. yRvJ c U. s E U. 3, . :E U. Hence. he de fined the ancestral relation with respect to R as R. = i i (1E CbR: ( (3.).j- E U. yRv} c p. .Y E u. 3,, . ;E p}.Thus. if the Axiom of Infinity is assumed then the class of finiie cardinal nurnbers is an infinite class. and sincr it is sssentidly defined as the posterity of O with respect to +, 1 (PM vol2 200). it follows from the definition of iin ancestral relation that a E NC induct. 1:. E p. 3: . C+,I E p: OE p: q.a E p. which wtcs [bat an inductive cardinal is one which possesses every propeny possessed by O and liy niin~hersrcii'!!ing frv~rdding ! :o nuEben pusbrhbing rne propenies - i.e. thrit the inductive cardiniils obry the principlr of mathcmaticd induction (PM vol2 18 1 ).J4 Thus. $cn tliat the relation +,l is a progression. the ordrr of the srries of inductive cardinal nuinbers is dctctrrtiinrd by the ancestral relation (+, l),. Hence, Russell's undenaking to ch;iractcrirr the ordinal propcnies of the cliiss of inductive cardinal numbers by defining thc clash by rnscins of the principle of mathematical induction depends. in the final iinal>.sis.on his definition of an ancestral relation. Witt_oenstcin*s vicws on the concept of an ancestral relation are given in the Tt- Iogic;~I structure 01 ;1 situation cannot br expressed by mrrins of ri proposition. the pn~pusition,slzow.s it (PT 4.1012 1). Accordingly. Wittgenstein claimed. there is a sense in w tiicli objects and situations can be said to have f~rmcilproperties and fin?itil rrlutions: 2nd facts crin be said to have stntcnoul properties and strtrcttird relritio~is(TLP 4.1 II). To enip tiasize the contriist ui th rxtrmal propenies and extemal relations. i.r. properties ~tndrcltiiions proper. Wittgenstein also used the words 'internai propsny' instccld of 'jlnictural proprrty': and 'interna1 relation' instead of 'structural relation'. It is impossible. he stared. to express by meons of a proposition the existence of an intemal propcrty of a possible situation; rather it shows itself in the proposition representing the c B. Russell. The Philosophical Implications of Mathematical Logic'. Essqs in Anniysis. ed. D. Lxktiy c Lundr~n:Gwrgt. Allen (Yr Unwin. 1973) 235. situation. by means of an internal property of the proposition (PT 1.10224. TLP 4.124). Similarly. the existence of an internal relation between possible situations shows itself in langurtge by means of an intemal relation between the propositions representing thrm (PT 4. i û2-5, 'îLF 4. i 25 j. A property, or relation. is intemal if it is inconceivable that its ohjcct not possrss it (TLP 4.123). Thus. it is nonsensicd either to assert or deny that an object ha3 :in internal propeny or an intemd relation. In a sense. Wittgenstein clairned. a propohition cün only say how a thing is. not dm it is (TLP 3.121 ). For logic is prior to thc question 'How'?'. not prior to the qucstlon 'What'!' (TLP 5-32). Wittgcnstcin explainrd his understanding of formal concepts. in the Trcicrcirlu. on rhc basis of his conception of forma1 propènies (TLP 4.126). Formal concepts. he bclieved. are nor to be confused with concepts proprr. If sornething PdIs under a forma1 conçcpt. one of its objects. as it were. thnt cannot be expressed by means of a proposition. Rxihcr. it is shown by the very sign for the object thar ftills undcr the formal concept. Tlius. a formai concept cannot be represented by means of a tùnction. as concept proper con. For the charticteristics of (i forma1 concept. i.r. its forma1 propcnieb. arc not txpressed by meüns of functions. The expression of a forma1 proprny. riccording to Wittgcnstéin. is an internai propeny of certain symbols. He cxplained in the 'Notes Ltictattxi to G. E. Moore': Tti:ii 11 is a tlii/,g ciin't be suid: ir is nonsense: but somrrlzblg is .slioror by the symbol 'W. In ihc sarne wciy. ihx a proposition is n subject-predicatr proposition can't be said: but it is sliown by the syrnboi. It is obi*ioiis.hc claimrd. that with a subject-predicate proposition. for rxcirnple. if it has Liny scnsc at AI. the form mut be known as soon as the proposition is understood. Hence. swn if thrre iwre a proposition of the fom ' "cpo" is a subject-predicate proposition'. it woüld br superlluous sincr what it tries to say is already known if 'cpd is undcrstood i >hl 105). Hencr. if thrre were a propositional function '< is a subject-predicate proposiiion'. it too would be suprfluous. It signifies no more than does the sign for the propositional variable S. Accordingly. Wittgenstein argurd in the Tr~icfrrtitsthat the sign for ;tic chmctrristics of a formal concept is not a propositional function. but a ~w-iïrble. The sign for the characteristics of a formai concept is the distinctive form of al1 symbols the meanings of which faIl under the concept (TLP 4.126). Thus, Wittgenstein claimed that the expression for a formai concept is a variable in which the distinctive fom alone is constant. The variable signifies the formai concept. and its values signify the objects that ftill under the concept (TLP 4.117. c$ PT 4.102271). Wittgenstein conclucird that cvq v:iri~iblc is the sign of a formal concept. for evsry variable represents ri constant forni tliiit iiII its values possess which can be regarded as a formal property of those values (TLP4.127 1 1. Hcncr. Wittgenstein argued that. in a correct conceptual notation. words used to rcprcscnt fornial concepts must be replaced by variable namrs. The variable name Y. for csiicriple. is the proper sign for the psrlido-concept signified by means of the word 'ohjcçt'. Whercvrr the word 'object' is correctly usrd. it is cxprrssed in conçeptu;ll notation by A vririable (TLP 4.1273. Wherevrr it is used üs the name of a proprr concept. W iitgenstcin claimed. nonsensical psctddo-propositions are the result. For it is nonsense. tic hclicvcd. ro say 'Therc cire objects'. as it might be said. 'There are books*.Sirnilarly. it 1s nonsense to sq. 'There are 100 objects'. or. 'There are Ho objrcts'. The same iippiies. according to the Trl~~.rtm.~.to the words 'complex'. 'fact'. 'function'. 'nurnber' and so on. To LIAihe question. 'Are there unanalyzablc subject-predicate propositions'?'. or 'Arc thcrc unanrilyzriblc relations hriving two terms'?'. is nonsense (TLP 4.1274). The propoïiiions ' 1 is a nurnbrr'. 'There is only one 0' and al1 similar expressions rire nonsilnhiciil (TLP 4.1272). The words 'complex'. 'fict'. 'furiction'. 'nurnber' and so on dl signi tj forniil concepts and are not represented in conceptual notation by jitnctions or cl~i.s.se.s iTLP 4.1177). 'Object'. 'cornplex'. 'fact'. 'number'. . . . are not narnes of concepts - ris Russell believed - but variables (PT 4.101373). Wittgenstein cxplnins his understanding in the Trmmriis of what ht: called 'the gcncrcil ierm of scries of forms' on the buis of his conception of a formal concept. A scries of forms. nccording to Wittpstein, is a senes that is ordered by an i~trenial reliition (TLP -1.1752). Hence. the concept 'rem of a series of foms' is a formal concept. Thc tsrms of a series of forrns are constructed in accordance with a tomal law (PT 4.10225 1 1 1. The srries of propositions '(1 Rb', '(31):aRr. xRb', '(3~.y):ciRr. xRy. ?Rb', is ;in example of such a srries (TLP 4.l252). if b stands in one of these relations to ci, \\:iiigcnstcin clilleci 11 'a succrssor' ot'tr. He furthcr rxplained: If u~ wcint to express in conceptual notation the general proposition. 'h is n successor of '1'. thcn wc rcquire an espression for the genenil term of the series of forms .... In ordrr tu express thc general term of a serirs of forms. we mut use 3 variable. becausr the cunccpt 'teriti of thni srries of forms' is a firrntrl concept. (This is whot Frege and Russell ovcrlooked: consequcntly the wciy in which they ivant to express gencral proposirions likc the one abovr is incorrect: it contains a vicious circle. WCcan dctcrniine the gcneral tcrm of a serirs of foms by giving its first term and the gcnenil form of the operation that produces the ncxt term out of the proposition that prcccdes it. (TLP 4.1773) Tliiis. açcording to the Trxfcints. to express in a conceptual notation the gcneral propirion 'b is ;i succrssor of d.it is necrssary to give an expression for the gçneral tcrm of [lie series of forms: oRb. (3~).tiRr. xRb. (Zr. F). ctRr. sRy. !.Rb. . . . . Howcvrr. rliat rcqiiircs rhe use of (i variable. because the concept 'term of a serirs of forms' is a iormnl concept. The variable. Wittgenstein suggests. must give the first terni of the senes and the senerd form of the operation that produces any term of the srries out of its iinnicditite predrtcessor. Thus. to understand Wittgenstein's conception of the generiil term of (i srries of forms. it is nrcessary to understand his conception of an operation. and ihs gsncral forrn of an operation. ln rlir Trcictatirs. Wittgenstein explains his understanding of an operation using ihc cscimplc. of operations on propositionul forms. Every proposition. he believed. can be construed as the result of an operation which has been applied to another proposition, i.r. the base of the operation. and which tums the latter into the former (TLP 5.2 1. PT 5.001). .An operation. however. can also have several bases (PT 5.001 1). Tnith-functions of clc.iiir.iitary propositions are results of operations with elementary proposirions as bases. namdy. the operations which Wittgenstein cülied 'wuth-operations' (TLP 5.231). Thus. logici11 addition and logicai multiplication. for example. are truth-operations (TLP 5.2341 ). An operiition is the expression of an intemal relation betwren the structures of propositions (TLP 5.2fl.). It is what has to be done to the one proposition in order thüt the 0thr.r rcsults (TLP 5.23). and that depends on their formai propenies, i.e. the intemul iimiiu-ity of tlieir forrns (TLP 5-231 ). Thus. the intemal relation by which a series of propositionA l'orms is orderrd is the operation that produces one term of the series irom ;inottit.r (TLP 5.232). Operations express the interna1 relations in which the structures of propositions stand to ont: another. .-\ççordingly. Wiitgrnsrein claimed that what an operaticn shows. i.e. how to get Ironi one iorm of proposition to another. must be shown in n variable (TLP 5.24). The v:iri;ihlc gives expression to the diffèrence bctween the forms (TLP 5.211). If the opcriition [hiit produccs '(1' from 'p. also produces 'r' from 'y' and so on. the only way rhat ihut is cxprcssiblr: is that 'p.. 'q'. 'r'. . . . be rcplaccd by variables that givr cxprcssion in a prrcil wiiy to certain forma1 relations (TLP 52-12).Thus. if 'Ob(p.q. r. . - . .)' is thc rcsult of an operation 'O' j' on 'p'. 'y'. Y. . . . . thcn the operation itself is Jcsiiyütcd by 'O*<<.q. j. . . .)' where ':'. 'q' and 'c' indicate argument places (PT j.OOlj), Howcver. Wirtgenstein rnaintüined that an operation is Ilor a hnction. 1.r. not a cunçcpt tTLP 5-25. cf: 5-44). A function cannot br its own argument. whrreas an opcr;ition crin tiike its own result as its base (TLP 5.3 1 ). In the Trcictntits. Wittgenstein calls ii 'the successive application' of an operntion. if an operation is repratedly applied ro its uwn results. i.r. if it is applied. as he says. 'and so on' (TLP 5.2521-5.2523). For - c.s;impk. '0'0'0'dis the result of threr successive applications of the operation '0' 5' to '~i'iTLP j.321). Accordingly. Wittgenstein introduces '[a. s. O'x]' for the general term of the series of fo,ms a. Oh. 0'0'~.. . . (TLP 5.2522). The bracketed expression is a uriiible. the firsr trrm of the bracketed expression is the bezinning of the series of forrns. thc second is the form of a term .t arbitrarily occurring in the series. and the third is the forrn of the term that imrnediately follows x in the series. Wittgenstein's understanding of the general form of an operation is explained in thc Trcictcitils using the gnenl form of a tmth-operation. The existence of the generÿl propositional form is proved. according to Wittgenstein. by the fact that. given al1 clcmcntary proposirions. there cannot be a proposition the form of which çannot be comtnicted (1 priori (TLP 4.5. cj: NB 75). A11 cruth-functions, he believed, are results of sucçrissivc. ;ippliccitions to elernentüry propositions of truth-operations (TLP 5.32). inderd. hc bc.1icvc.d that every truth-function is the result of successive applications to elcmenrÿry propositions of the operation N( 5 ) (TLP 5.5). N( 5 ) is the operation of ncgation of al1 - tlic v;ilucs of the propositional variable f (TLP 5.501).J5If 5 has only one value then - - Ni 5 = - p. if it has two values thsn N( 5 ) = - p . - q. and if it has üs its values al1 - \.aliir.s of ;i fiinction/i. for ai1 values ofx. then Fi( 5 ) = - (31).ji-. Hcncc. if evrry tnith- - iiiiiciion cmbe cunstructrd by succcssively ~ipplyingiI( < ) to elrrnentüry proposirions. it - - - 15 possible. to piw the general form of a truth-function as [ p. j. N( 5 )1, taking the wlucs of p to bc dcmentary propositions (TLP 6). According to Wittgenstein. the general forni of a truth-function is the general form of a proposition. So rhr general propositional korm i 3 v:riilble. a/i)nwl concept (TLP 4.53). The generril form of an operation R7 ------i, ri\-cn in the fiwrtirirs 11s [ 5. N( < )I'( q ) or [ q. 9. N( < )] (TLP 6.0 1 ). Hcnce. it is dm a variable. or (i fomd concept. The genrral form of an operation. Wittgenstein claincd. is the prneral form according to which propositions are constructed. for in it the ~ariciblc{ is not restricted to elernentq propositions (TLP 6.002). Hence. the general form of ;in operation is the most general form of transition from one proposition to mothrr (TLP 6.01 1. It is the general concept of ail foms of propositions thar are tr priori possible. Witrgcnstein's conception of an operation underlies the reason rhat he considered the way in which Frege and Russell expressed 'b is a successor of a' to contain a vicious sirçle. Frege gave the following explanation of following and preceding in a series. where the type of the series is detennined by the relation in which a rnernber of the series always stands to the member directly following (GA vo1.2 59-60. GIA 92-3. BS 56-60). He clainid rhltt if the proposition, 'if every object that stands in the relation T to A falls undsr ihc concept -F(j). and if from the proposition that E hlls under -F(j) it follows. for wery E. that evriry ubject which stands in the relation T to E also falls under -F(j): thcn CI> Mis under -F@' holds for every concept -F(j). then. by definition. follows 'r in the T-scries Frege. Russell cdlsd 11 'a heredituy clüss with respect to R' if j {(3x)..I E p. xRy] c p. i.e. if tliür w which p's stand in the relation R are p's. Ils is an ancestor of c then. if p is a hcrcclitq cl~i.\s10 which .r belongs. Russell claimed. then :dso belonjs to p. Conversely. if s bclongs to evc.N hrreditriry class to which x belongs. thrn -Y must be an ancestor of r. For hincc Russcll stipulates thlit .Y be includrd mong its luicestors. the cliiss of that which tus .Y ah an mxstor is ii hcreditary class to which .r belongs: and thus. by assurnption. to s,liich :hclongs. Hrncr. according to Russell. .r is an ancrstor of :with respect to R if md oniy if .r E C'R = (3~):iRy. v. yRv) and :belongs to every hereditary class to tr-liich .r hclongs. Thus. since sR.r. =: .r E CbR: 6 ((3~).I. E p. yRv} c p. .r E p. 3, . :E p. R~iswlldctines the ancestral relation with respect to R lis R. = {xE C'R: G((3y).y E p. ?Ri#)c u. . E u. 3 . :E p.Hence. Russell asserts that if .r e C'R thrn 'the dcsccndünri of .Y' are giwn by i (.rR.:) = ( ri ((3~).y E p. yRr} c p. -r E p. 2, . z E p}. ;\ccorcling to Wittgenstein. Frege and Russell both overlooked that the concept of iollowin- in 3 serieh is a fomal concept. Thus. bath tried to inrroduce it by mrans of an incorrect kind of generalizlition and. as ri result. presented the concept in a way which contains a vicious circle. In the Troctcttits. Wittgenstein distinguishes funml generalization resulting in a fomîl Iîw. from generalization resulting in a propositional function. Assuming an - expression i hiis propositions as its tems. the order of the tems being indifferent, he cliiimcd. the values of the variable j are the terms of the expression (TLP 5.501. PT -- - ''tt is this operiiion which Whitehead and Russell refrr to as incornpïuibili~in Principia .Llodtrrnaricli. 3.003'1.What the values of the variable are is stipulated. The stipulation is a description of the propositions that are values of the variable. Wittgenstein distinguished three kinds - of description: (i) direct enurneration of the propositions. (ii) giving a funciion j( .r ) the values of which for all values of rare the propositions to br drscribed and (iii) giving a fornial lwin accordlince with which the propositions are constructed. in which case the cxprcssiun has as its terms al1 the terms of a series of foms (TLP 5.501. PT 5.0053 1). In ttic second case. the values of the variable are al1 instances of a genrralized proposition u*hich describcs al1 objects Ming undrr a given concept (PT 5.00533). In the third case. r tic v;ilucs arc a11 propositions having cenain fhmul propsnies ( PT 5.00534). The interna1 rcliition by which the srries of forms is ordered is sxprrssed by an operation (PT 3.00534 1 ). Thus. for example. $1 propositions of the scries: oRb. (3).crRr. *Rb. (3s.-1. iiRr. .rR>-.!*Rh. . . . are chüracterized by a forma1 property: and the grneral form of thcse propositions çan bs presrntrd «rilr. by rneans of the form of a variable. i.e. (i forma1 law [PT 5.005342-5.00135). 'This second kind of gencritlization'. Wittgenstein assenrd. tiich cm bc cn1lr.d the jonmzl kind. was overlookrd by Russell and Frege* (PT 5.00534 1. cf TLP 1.1273). Wittgenstein elaborated on the distinction between the two kinds of genrrality Jc\~ribcLIin the Trricrcirirs in discussions with Waismünn during 1919 or 1930 i WWK 2 I3j":). Acçording to Wittgenstein. the distinction is that beiween a sysrrm and ri ror~rii~. .-\ toiality rssts on li proposiiio~idfitrictiorl. It is deteminrd by dl arguments for which the propositional funçtion is (me. However. a systrm rests not on a propositional funçtion. but ;in operririori. An operation can br applied to its own results. hence it can be iissd to construc[ a systrm of propositional forms. The system is determined by ail possi blc forms of proposition that result from the successive application of the operation. Opcrliiions occur whrn wr are dealing with propositional forrns that are ordered riccording to a tomal law. Thus the statements aRb (3x1aRr. .uRb t 3.3)czRr. .rRu. ?Rb ;ire orderrd according to a formal law. An operation is the transition from one propositional fom to another. It senerates one propositional forrn from another. If wr know the operation in question. we cm. starting from one propositional fom, generatr al1 otherr;. An operation is sompletely different from a function. A function cannot be its own argument. An operation. on the other hand. cm be applird CO its own results. In rnathenilitics we must always be dealing with systems. and not with totalities. Russt.ll's basic mistakc consists in not hriving recognized the essence of a system while reprsscntins empirical totalities and systems by rneans of the srime syrnbol - a propositional function - without drawing any distinctions. (WWK 2 16-7) Tlius. ~icçordingto Wittgenstein. the series of propositional forms: trRb. (31). clRr. .rRh. 13.1.1. &Y. IR?.. ?.Rh. . . .. is represrnted by the fomnl kind of generaliziition: for the \cric.; docs not form a totality. but rathcr a systcm. The terms of the srries cm bs cunstnictd systerniitically by succrssively applying to the first term of the series the c>pcr;ition which generates the series. Hencr. given the first term of the series. if we know ttic opcriition on which the series is based. then we know the entire serics (WWK 216). Thus. the series is rcprcsentrd in a genertil way by giving the grneral terni of the series. i.~.iiic gcncral hrm of thc operation in accordancc with whiçh it is construçtcd. Thc gcncrdity of the _ocnrrlil term of the srries. Wittgenstein belicvrd. is crseiirid gciieriilit).. Sincr. given the serirs of propositions: qucd. signifies thrit ün operütion can be applird to its own results: Thc conccpt 'and so on'. synibolized by '. . . .' is one of the most important of ail and like 211 the othrrs infinitely fundamental. . . . The concept 'and so on' and the concept of the operation are equivalcnt. Aftrr thc operation sign thcre follows the sien '. . . .' which signifies that the result of the uperation can in its turn be tciken as the base of the operation: 'and so on'. (NB 89-90, Sincs lin operation is the transition from a term to its immediate successor in a series of iornis. the concept of an operation is the concept of a form series (NB 8 1). The concept of an opcration is quite generally that according to which signs cmbe constructed according to ri nile (NB 90). Thus what is represented by the sign . . . is essential to any rcpresentation of ri formal series. 'When the general form of operations is found'. Wittgenstein stlitsd. 'we have ais0 found the general forrn of the occurrence of the conccpt "anci su on" ' (NB 90). Thus, he believed. the '. . .' in hRb. (3).ciRr. xRb. (3. y). clRt+.sR+v. ?.Rb. . . .' represents the rssr~rticilgenrrality of the operation in accordance with which the serirs is constructsd. The kind of generality involved in the representation of thc series: In PI-ir~cipi R' 1 R. . . . t PM vol. 1 216).'6 If S is ii power of R. thry que. then so is S 1 R. For since 1 R is dctïncd as the rclation of S 1 R to S (PM vol. 1 796). S 1 R is the relation 1 R'S. i.r. rlic rcl;itiun which hiis the reiütion 1 R ro S. Hrncr. Whitehead and Russell concludc. if R E p: S E K. zc . S 1 R E u: q,. P E p. thcn P is a power of R. for the class of powrrs of R is .i t.aluc. of which satisfies the hypothesis R E p: S E p. 25 . S 1 R E p. Conversely. xcording to Pri~rcipi 2, . P e p. Thus. drnoting the class of powers of R by 'PotbR'. Whitehead and Russell soncludc ihat P E PotmR.a: . R E p: S E p. 3s . S 1 R E p: q,. P E y. That. in turn. holds if' and only if P E PotbR. =. P( 1 R),R. Le. P occurs in the srries R. 1 R*R. Rb1 R'R. . . .. which is the series R. R'. R~.. . : for 1 R is the relation that Rn' ' has to Rn.Each of the rchtions R. RI.R'. . . .. has to its predecessor the relation 1 R. so that R' = 1 KR. R~ = "'~ussell wnsidcrcd rhe use of '. . .' io be inherently ambiguous. Thus. hc claimed. for ensmpk. [ha to prow thc cxistcncc of Ho ht: wris not content with apperils to ri vciguc 'and JO on'. Thc use of '. . .'. ris 1 R*/ R'. and so on. Thus every term of the series has to R the relation ( 1 R).. Hence, xcording to Whitehead and Russell. the powers of R cm be definrd as those relations ahich have to R the relation ( 1 R). (PM vol. 1 558). Thus. to say that the relation R 1 R, ho[& brttwecn 11 and b turns out to be equivalent to saying that one of the relations R, R', R'. . . .. hoids betwern [i and b. Thus. oR 1 Reb holds if and only if&b or (3).nRr. .rRb or i 3.~i. 'IRL .rRy ?.Rh or . . . . Hence. in Principio ~bldwmzricn.ciR 1 R.6 is tiiken to de fine tlic logicd sum uRb v (31).URÏ. xRb v (3.x. y). iiRr. rRy. -Rb v . . . . h'ittgenstcin cliiimed that the wciy in which Russell presents the prnerlil torm of the proporirion.; (rRb. t 3s).uRr. .[Rb. (31. y). dr.xRy. !,Rb. . . . is incorrect: it contains a vicious circle .t-R\,. iqRh. . . . presupposes what it tries to express. Russell clainis in Principitr .\ldrt~trricctrhat rhe powrrs of R. i.r. the class of relations R. R'. R'. . . .. unbr definrd ;is tiiose relations wliich hiive to R the relation ( (RI.. The brisis for his daim is his Jciiion\triition that P E PotmR.s. P( 1 R),R. He believed that rhe class of powcrs of R is to bc ciistinguishsd t'rom ihr srries generated by the relation ( 1 R). because the srries has a iictinitc order: but the clüss. although capable of various orders. has no order (4: PM ~1.7498 i. However. xcording to Wittgenstein. @en the definitions of the panicular powrs of R in Principio Mdterntrricri. the class of powers of R is ordered by an imenlcil rclaiion. The relations lire rlcfined such thüt R = .T j {.rRy). R' = R 1 R = .i. î. ((3:)..rk. .:RI-].R' = R' 1 R = .< f ((3:).XR~:. ZR-} = i f ((3:.11.). sk. :Rw. irRj). . . . . Hencr. rhc ordcr of rlie çlass of powers of R. Wittgenstein brlievrd. is givrn by the order of the propositional forrns xRy. (3:)..rk. ZR-. (3:. irp).IR;. rRiv. wRy. . . . . and the order of the pro posi t iunal forms .rRy. (3:). .r&. :RF. (3:. ii*). .xk. :Ritv. iv Ry. . . .. is an intemal proprny of thrse foms. The propositional form (3:).xk. :& sltorvs that it is constmcted from thc form .rR!. the form (31.):(3:). -&. :Riv. icRy slzuws that it is constmcted from thc r'orm i33. .rk. ZR!. and so on. Thus Russell's definition of the relation UR1 R.b pw.suppose.s the intemal structure of the srries: ciRb. (31). aRr. xRb. (3~.y). ~iRr.rRy. coinmtm as it ts ln marhernatiçs. he insisted. is not suictly acceptable. (B. Russcll. 'The Axiom of Infiniry'. >*Rb.. . . . For if the class of powers of R is given then the series of powers of R is given. Hcnçe. R 1 R. does not explain what it is to succeed in the series generated by R. it presupposcs it. Thus. ns an explmation of succeeding in the series. Russell's presentation oi the grncrtil form of the propositions ciRb. (3.r). aRr. xRb. (3s. y). ciRr. xRy. >-Rb. . . .' Hencc. Wittgenstein rejected Frege's. and Whitehead and Russell's. conception of ibllowing in a srries. that is. of a progression. ris drfined by means of the concept of an ;incc~iilrelation. Rrither. he introduces the series of natural numbers in the Tructutris as ri fornial srries. orderrd by an operation. .Accordingly. Wittgenstein usrd the general Corm of a logical operation to i.liicicl;ttc. rhc fomn1 characm of the numbcr series. and it is by mriins of this that the conccpt oi nurnbrir is introduccd in the Tmctctti Gi\,cri thc pwül form of an operation. Wittgenstein defines the natural numbers as . . . I givc thc following definitions .r = ~"'xDe f.. Q'QL'.y = Q' ' '.ï Def. So. in iiçcordüncc with thrse rules. which deal with sisns. 1 write the serirs .r. !Zr. Q'QkC2'n7n'.\-, . . . . in the iollowing way il{) 1 + 1 QO + 1 .. 1 + 1 ' T-~,fi() + 1 .r. . . . . Thereforr. instrad of '[x. <. R'S]'. 1 ii.ritt3 .[Q"-~,QV QL - 1 x] ' , .And I give the following definitions O + l = 1 Def., O+ 1 + 1 =2 Del'., 0+1+ i+1=3Def.. (and EÎ cm). (TLP 6.03) 75qf- - 1 "In tht: Pliriosophicai Rerru~rks.Wittgenstein gives 3 sirnilar critichm of Russdl's presentation of R,. Russcil'i; cxprcssm-i '~vrripsthe concept up in such ri way thrit its form Jisrtppears'. but conccpts which arc pickcd up in rhrs \va? Jcrivc [hcir mcrining (rom the definitions which package them (PR 306). Indced. Rur;sc.il", 'lirncirphous explrination' of R, scems to be nonsensical (PR 209-10. cf 153). On rhe hiisis of these definitions. Wittgenstein draws number of conclusions. The concept of number is what is cornmon to al1 numbers: it is the general forrn of a number cc$ PT 6.012). The concept of number is the variable number (TLP 6.022). The general iorm of ri naturd number is the variable [O. 6, j + 11 (TLP 6.03). Thus [O, 6. j+ l] slio\r*s thé inrèrnal relation in accord with which the number series is ordered (TLP 4.1352); for it si~niiirsthe operation that produces one number from another (TLP 5.732). Hence. Witigcnztein esplains. the Throry of Classes is çomplrtely supertluous in mathematics .id th;it rctlects thar thc gsnrrality required in mathematics is oot ~iccidr>irdgenerality i TLP 6.03 1 1. Wittgenstein clcarly intended to make thc samr point in assening that the wrd 'numbtir' signifies a formal concept. and it is represented in a conceptual notation by a \wiüble. not by a function or class as Frege and Russell brlievrd (TLP 4.1271). h jign for a nuniber .sliow.s that it signifies a number (TLP 4.136). Hence. it is nonsense to wert ' 1 is (i number': for there is no question of giving a general detinition of the cunçcpt oi nurnber fiiïirperirlrrirl~ of the definitions of the prinicular nurnbers iTLP 4.1171-4.1272 1 i. .4 formal concept is @en immediritely any object falling undrr ir is ~ivcn.S« ii is impossible to introduce independently both the concept of a number and part iculnr numbers (TLP 4. i 272 1 ). Thus. the assertion in rhe Trcicrciriu thrit the concept of nurnbcr is rhc gcncrnl form of ;i propositional funçtion: but by o formal law. the gcneral term of ;i senes of forms. Hcncc. Wittgenstein's bslief thrit it is essrrtrid generality which is required in inatlicmotics is explainrd by his understanding of a forma1 concept. that 1s. the grneral rcrni d a forrn .ieries. Wittgenstein believed that it musr be possible to write down riny iirbitrary number 14 priori by mrans of the system of numbers (NB 35). To =ive the ccneral form of a nurnber was. in his understanding. to give the number ~ystern.3~Thus. b he bclicved. in showing that it is possible to give the general fom of a number. he had >hou-nthat it is possible to construct any number a priori (cf NB 75. 89). Wittgenstein beIie\.ed. however. that it is not possible to construct the numbers [i priori as drfined in th< cardinal arirhmrtic of Priricipici .Cfutl~r»inficcr.Xssuming Wittgenstein's analysis of F. Waisrnnnn. Thc Neturc of Maihemîtics: Wirtgenstein's Standpoint'. 64. the finite cardinal numbers. whether any number can be defined as in cardinal arithmetic beçornrs ti question of whether there is a propositional function that is satisfied by that number of possible arguments. Wittgenstein argued that if the numbers were definable in ./'. cardinal arithnxtic. they could only be drfined as follows: O = fi(~~r)(- (3s).Q.~J Df.. 1 = /' /'- G(OW {(3).W-: - (3.J). @K. @y}Df., 2 = ~?(@il){(3, y). @Y. ~FI- (3~. y. :). Q.K. ~y.O:} Df. iind so on. The problem arising from the preceding definitions is rlaborated by Waismiinn in hi?; 1930 Kunigsberg lecture on Wittgenstein's views on mathematics." If thc numbers are defined 3s the extensions of trcticcii propositional functions. Waismann clairnsd. Iiow niciny numbers crin be defined in arithrneric depends on wlzcflicr tlinctions cxisi tharc ;itisfied by the necessary number of arguments. Hencr. the possibility of dctïning any number rests. tzot on the tr priori Iaw for constmcting the nurnber systcm. but on xcidental propenies of the world. i.e. on what there Iicippetis to be in the world. Hu~icvcr.cvcn if no class having >I members happenrd to exist in the world. i.e. no propositiond Ilinction were trur for only n arguments. it would still mnke srme to ionsidcr such dirsscs (WWK 214). Thus. the construction of the numbers in arithmetic airinot dcpend on ~icfirtrlpropenies (WWK 2 14-5). Xumbers express possibilities. i.r. the pmsibil it) of Irait. r?rw+~.not accidental proprrties of the world ( W W K 2 14. Yurnbers are loyicnl forrns: and thus. the niitural numbers do not form a totality. but riither ti ~ystern.~' Accordinply. Wittgenstein brlieved. the concept of number is a formal concept. givrn ci priori. as in the Trlicrcc tiis. not (i posteriori. as in Principiri iWif - -- -- '"~hi(!., 62-4. Cf: \Vii-K 2 l iff (Thc latter indudes what arc ckarly notes un which Waismnnn based pans O i' his Iccturc.. i 4'Jlhid.,64-5. 'l li \i.iitgcnstr.in's vie~vson number in thc Trucrririts ue aken to clarify grrlrruii~in mathernatics then thcl crin bc rcgarcicd as the starring point of his Iritcr thought on mathematics. Witt_gcnstcin's 1929 Iccturc for thc Joini Session c~t'the Aristott.lirin Society and the Mind Xssoci;ttidn. givcn instcad of 'Some Remxks tjn Logicd Form'. wris on genernlity and intinity in mathematics. t R5J) to express. Wittgenstein claimed in the Philosophicd Reninrks. would have io be slzo>vrz Dy a mathemarical induction. 'Grnercili~in arithmetic is indicated by [a mathematical] iiiciuction,' ( PR 150. c-200). '1 have always said'. Wittgenstein clüimed, 'you can't speak of 1111 nunibers. bccause there's no such thing ru: "al1 numbers" ' (Ibid.).The concept 'al1 nurnbers' is given by the mle for constnicting the number systern. i.~.[O, <, j+ 11. The riilcs for a number system. Wittgenstein clairnsd in the Pliilosoplricd Rrmrrb. contain cwryrtiing rtut is infinite about the nurnbers (PR 160- 1 ). Hencr. he maintaineci: Thc iniïiiitc nuinber srries is itself only . . . [the possibility contained in the niles for the uc of nuirirrals] - as ernerges clearly from the single syrnbol for it '[ t. j. 5 + 11'. This synihol is itsclf~inürrow with the first '1' as the tail of the mow and '< + 1' 3s its tip and irvhat is charcictrristic is thüt - just as length is inessential to an arrow - the variable 5 \liows herç that it is immaterial how tir the tip is from the tail. It ih pozsible tu .;pc'rik of things which lie in the direction of the arrow but nonsense to bpcak of ;il1 possible positions for things lying in the direction of the mow as an q~iivalcrrtfor this direction itself. (PR 162) For 'iiihitc possibility'. Liccording to Wittgenstein. does noi mean 'possibility of infinity' :is ag;iinst a1h;it is actuülly infinite (WWK 229). Rather ihe word 'infinitr' qualifies a p»\sihility ris cigainst othcr kinds of possibilities (WWK 228 1. Açcordingly. Wittgenstein clai nis i n the Plr ilo.soplii~dRe»r Gc.ricrcility in nlatliematics is a direction. Lin mow pointing dong the srries grneratrd by Lin oper;ition. And yuciin even say that the arrow points to infinity: but dors thlit mran ~ticrci.; sornething - infinity - cit which it points. as at a thing? Constmed in that way. it niiist O t' course Icrid to endless nonsense. it's iis though the mow designlites the possibility of u position in its direction. t PR 163) iicncc. Wittgenstein argued that the generality-sign rnakes no sense for nurnbers. The sonccpt 'dl nat~irrilnurnbers' is not a boundrd concept. Thus. wr cannot use genrrality. ~.LJ. the word '311'. in represrnting numbers. There is no such thing as 'dl numbers' jimply becausr there are infinitely many. and furthemore. because it is not a question hcre of the amorphous 'dl'. such as occurs in 'Al1 the apples are ripe'. where the class is given by Lin rxtcrnÿl description: it is a question of a system of stnictures, or forms. which niust bt: given precisely as sitcli (PR 148). The unboundedness of the concept of a number is not a problem of ambiguity which cm be dealt with by analyzing propositions that brtgiri with an arnorphous 'al1 the nurnbers . . .'; if ir is a problem, it is the problem of prtxiscly represrnting the essenrird unboundrdnçss of the numbers. i.r. rhe esscritial generiility of the concept of number. hccordingly. Wittgenstein explained to Waismann. in 1929 or 1930. it can ais0 be wid thlit the concepts of a finitr class and an infinite class are logiccrlly different (WWK 72s). 'Finite' and 'infinite' do not signify accidentcil deteminations of the concept of a clas%.It is not one and the samr concept that is quülifisd by the addition of 'finitr' or 'infinitc'. Russell promotcd this confusion. çlaimed Wittgenstein. bccausc hs used a \yribolim which represents both kinds of class in exactly the wme way. He was thus entircly prcvcnted from rccognizing the truc significance of the difference in question. A corrcct yiiholism has to reproducr an infinite class in a complrtely different way from a lïnlrc one. Finitcnrss and infinity of a claïs must be obvious from its syntax - as in the Triic[ in accordance with an iiirrrml relation. and accordingly the concept of number gives an (i III-iorr1w. Hrncrt. the numbers are not represented by means of a propositional function. but bu nisans of Lin induction (WWK 81).The generiility of induction is the generality of a niic of logical syntax (WWK 154). It is essential generalitv. no[ accidcniül senerality. Howcver. siyen his analysis of the definitions of the inductive cardinal nurnbers. whether rhc \ystern of cardinal numbers cm be constructrd cannot be decided u priori. Whether my nuniber crtn bc drfined in cardinal arithmeric drprnds on experirnce. Hcnce. \Vit tgenstcin rcjccts the cardinal arithmetic of Pri~icipkihldwrncuic«. and he undenakes in the T~LLC~CL~LISto introduce the numbers by means of an a priori liiw. The hnalysis of the Propositions of Mathematics Rxnsey wris the firsr to rvaluatr the virws on mathematical propositions c.xp1ainr.d in t hc Trrzc~ritirs.He iirgued that Wittgenstein', views involvrd insoluble Jifficultiçb. However. hr accepted the definition of pure mathematics given in Russell's Ir-oco ro ,hrlirninrictr[ Pliiloropliy Hence. hc believed Wittgznstein's undcrwnding of logicctl propositions to be no less esscntiül to the definition of inatlicn~aiiçs than the logicism of Pritlcipitr iMdirnz proctx of generalization lcads from 'If Socrates is a Greek. Socrates is a man' to 'If ci :ind h cire classcs. and (1 is contained in b. then "r is an a" implies ''.Y is a b" '. The rcsuiting proposition contains. according to Russell. the variables a. b and x. and the logical constants class. conrained in. and those involved in the concept of forma1 inipiiçation with variables. Thus. Russell concluded. it is a proposition of pure .':F.P. Rarnsc). 'Criiicril Notice: Tructu:!ts Lugico-Philosopliici~s',The Forrriclririons of :tIctthrnrurics und Utlwr Logtc-al ESSLI!.S( London: Routledsc Sr Krpn Paul. 193 1 ) 130. 182. *F.P. Rarnscy. 'The Foundxions of Mathcmatics'. 3. mathrrnatiçs. Ramsey argued. however, that mathematical propositions are not oniy ~ompl~td-vgeïirrd. but also necessariiy trrte. He believed. as did Wittgenstein. that there iiïc complctrly general propositions that are not necessuily true propositions. i.c. not tcittroIo,qies. but propositioris. which can be true or false; for rxamplr. the Axiom of Rcduci bilit!;. Thus. Rnmsry concluded. a de finition of mathematical propositions must includc not rnercly their completr genrrality but also their necrssity. The definition of piirc mathematics must explain not only (i) thüt the content of a proposition of iii;itlicrnatics is complerely generalizsd. but dso (ii) that the form is tautological. It is anothcr question. according to Ramsey. whether the form of a mathematical proposition is tautological in the sense that Wittgenstein defines in the Tr~icrczt~ts.Since a niathcmiitical proposition is necessarily trur. it must bç tiiutological in some srnse. Raniscy iuppossd. regnrdless of whether it is a tautology in the srnse drfined in the Trcic-rc~ttls.Wittgenstein regarded necessarily true propositions ris, in essence. mlcs of Iogiçai infcrcnce. Hcnce. thai 'QU. @z 3 ~[i:3 : va'. for example. is a tautology shows r li;it 'ipi cm btt logicdly infrrred from 'Qi' and 'oci (i yt~a'.But cquations. i.r. identities. ciin bc uscd in thc same way: for that '(1 = h' is AS triiitulugics. R~irnseyargued thnt Wittgenstein rejrcted io~icismfor the rerison that he was not ;iblc to rcduce mliihçmatics to tautologies in the sense defined in the Trcrcrcms. \\:ittgcnstsin slliboriites in the Tructatris a clearly defined sense in which logical propositions are tautological: and it is in Wittgenstein's sense thnt al1 axioms of Prirrcipic~ .Ildi~~rilirri~.trt.sct.pt the Axiom of Reducibility. Rarnsey claimed. can be undsrstood to be raurulo~iccil.He concludrd that consequently there ;ire two possibilities for the definition di rnlitlicmarics: either ii) the Axiorn of Rrducibility can be dispensed with in Pri~icipiri .Cl~idl~~~uciri~-~~and mathematics is reducible to tautologies in the sense defined in the T~-mmt.sor (ii) mathematics is tautological in some other sense. The difficulty with the first w_pptionconcems the possibility of reducing extensions: that is. classes of objects of the same logical type. to tmth-functions. Red analysis. riccording to Ramsey, is iùndarnenrülly c~remio~lnl;for its fundamental concepts. that of a real number and that of the funcrion of a reol variable. for example. must apparently be definrd txtensionally rrithcr than truth-tùnctionally. Hence. he believed that mathematical propositions 'assert rel~itions bctivt.cn extensions'-? But if mathematics is to consist of tautologies in Wittgt.nstr.in's sense. assertions of relations between extensions must be reduced to ;issrrtions of tnith-functions: for what Wittgenstein crillrd 'tautologies' Xe. in essence. initti-fiinctions that cigrce with al1 the truth-possibilities of the arguments of which thry arc ;~sscrtcd.R;lnlsry explnined the difficulty with such 11 reduction by an examplr: Lct ur tiike . . . the assertion that one class includes another. So long as the classes are defincd as tlic classcs of things having certain predicates @ and W. there is no difficulty. Ththe ci:iss of y's includes the class of @'smeans simply that cverything which is a m is .i \y. ~vliiçh. . . is a truth-function. But . . . mathematics hiis to dsd (at lcast apparently) ir,itti c1;issr.s whiçh are not given by defining predicütes. Let us take . . . the clus ((1.h. cl .irid thc cl~ihsclr, hl. Then that the class (a, b. c) includes the class (tt, I'l) is. in ri broad \cnhc. triiitological . . . but dors not seem to be a tautology in Wittgenstein's sense. [hot is ;i certain sort of tnith-function of rlcmentary propositions. The obvious way of tryins to makc it s rnith-lunçtion is to introducr idrntity and writc '(ci. h) is containcd in ((1.h. c)' as '1.1.):. .L = LI. V. .Y = Li: 2 : x = tr. v. .r = b. v. -Y = c'. This cenainly looks like a i;iiitologiçd truth-tùnction. . . . [the arguments of which are] propositions like 'ri = cd'. 'b = (1'. 'd = (1'. But these rire not real propositions at al1 . . . . When 'a'. '6' arc both namrs. tllc on1y ,ignific:ince which cm be plltced on 'a = b' is thot it indicrites that we use 'ri'. '6' ~ih. . . cqiiiv;tlcrit syrnbols.-'" Ir is the prccrdinp argument. Ramsey supposed. that Ird Wittgenstein to conclude rhat rixitlicnistics dors not consist of logical propositions: and hencr. to consider the posibility thrit nicithematics is tautologicd in some sense other than that defineci in the 7r~rc:litrr.s.Hcnce. it is for that rerison. Ramsey believed. thlit Wittgenstein rrjected the logic ism of Pri~icipicliI.1~lftientaticrr. Ramsey LUS convinced. however. that it is not possible that rnüthematics rests on identities. and that rixher it must consist of tautologies as drfined in the Trnctcitits. The ~.ic.\t.sun mathematics elaborated in the Trt~ctcituswere. hr supposed. restricted to e1r.mc.nta1-y arithmrtic. Hrnce, the evriluation of Wittgenstein's views depends on the possibility of cunstructing the rest of mathematics on his understanding of aithmetic. If it is supposed that the mosr fundamental propositions of mathematics are identities. then inotlicmatiçül propositions are. in some sense. tmth-functions of identities. The main cliificulty wirh this supposition concerns the occurrence of mathematical symbols in non- ~ii;iiliciiiaticrilpropositions. The proposition 'The square of the nurnbrr of Q'S is greÿter by rwo than the cube of the number of y's'. in which 'Q' and 'v' are tior logicd concepts. is ;i non-matheniatical proposition. Ramsey's analysis of rvhich is (1.1. î(@r) u~t , .îiy,ri €11 . w' = 11' + 2'. Hence. in the non-mathematical proposition there occurs ü 3 m;irheni;itical propositioii; namely. 'ur2 = tt + 2'. which cannot bt: made sense of. 3 t axuriiin~to Wittgenstein. unlrss it is takrn to be about the symbols 'nr-' and '11- + 2'. Thus. .iu2 = i; ' + 2' is not. as it srems to be. one of the arguments in .i3t~.tr 1. .i (QX) url . .ii iy.1-i £11 . t,r2 = tt ' + 2'. reiyded as the value of (i truth-function: but ratha- ptrrr of rhr rrurh-funcrion. Hcncr. it niust be analyzed trutli-functionally and. sinçr it is ü niatlicmatiç;il proposition. it is totitolugicd. in somr sense. for thc values of ttl and n that .;;iti.ify it. iind contradicrory for all others. Thus. for the values for which it is tautological. it niust be ;L t:~i~toIo~yin the sensr definrd in the Tmcrii~ris.Hrncr. Rnmsry rejccted \i,'itt~cnstcin's'idrntity throry of mathematics'. and sought a solution to the problrm of reclucing wcrtions of relations betwren rsre!isions to assenions of truth-functions. The importance of thc Tl~rcranrsfor the definition of mcirhemütiçs. he believed. is thrit it coni~iinsthe de finition of t ;LW nc'ccss3ri\y [rue. Raniscy believed that the views on mathematics elucidated in the Trcicrcinrs resultriti from an attempt to prove that mathematics is a part of logi~.~~He suggested that [lie rcasons rliiit Wittgenstein rejrcted logicisrn largely concemed difficultics revraled by his ind der standing thut logical propositions are tautologies. He was unconvincrd. ho we ver. thi~tthe reasons were de finit ive. in Principùi iI.liitltrmririnr. Whitehead and Russell h;id appÿrently bren able to deduce ail the concepts of mathematics from logical concepts by mcans of &finitions, and Ramsey found no rerison to reject their resuIts. He was Liware that the logical necessity of some of the uiorns of Principin iCiutlirnrciticri was qucstionahle: and iiccordingly, Wittgenstein's criticisms of Russell's conception of logiçlil propositions were acknowledged. indeed. Ramsey believed that Wittgenstein's conception of logiciil propositions as tautologies was indispensable for understanding the neccssity of logic and rnathematics. However. Wittgenstein's conception of mathematics w;is rcjtxted. For to understand Wittgenstein's conception of rnathcmatical propositions ;ih cqi~iiiionï.it is necessa- to understand the argument in the Tr Thc assertions conceming logicül and marhrmatical propositions in W ittgrnstein's c:~rIy writings ;id diçtations suggest that initially he did not question Whitehead 2nd Riiscli'.; vicw [hint the propositions of pure mathematics cire essenticilly logictil pr«positiims. The 'Notes on Lopic' indicütc that Wittgenstein likcly cigreed with iVhitclie;id and Russell that al1 mathematical propositions are derivable from the proposirions of logic. Howevrr. Wittgenstein's understanding. arriveci iit in latr 19 13. that ;di propositions of logic are either tautological or contradictory - and as a rrsult. his &finitive rcjection of the Axiom of Rçducibility. the Axiom of Infinity and the Zli~lriplic;iti\.cAxiom - put the possibility of deriving rnathemntical propositions from logiclil propositions in question. Accordin$y. whrthrr or nor mathematical propositions arc propositions of logic was also put in question. It remained in question until Wittpstein undenook. in his philosophical notebooks. to corne to a correct iinderstlinding of the .-\xiom of Infinity. It seems to have been only aiter he rejecied the definitions of the finite numbers in cardinal xithmetic that Wittgenstein concluded dc fin i t iwl y t hat the propositions of arithmetic are >toi logical propositions. In the a - " Hont.\cr. IL qpxm that by 1929. Wittgensfein rnay have succceded in explsiining his views to Ramscy. .ind convincing hm ut' their correctness: for Rrimsey claims in an essriy on philosophy. 'Logic issues in t;iurolo~ics.rnathcm;itics in identities. philosophy in definitions . . .' iF. P. Rmsey. 'Philosophy', Logical Posi~i~.isni.cd. A. J. Aycr i New York: Free Press. 1959) 322). Tr~rciti~s.hr rejects the assertion argued for in The Pririciples of Marliemntics. and which wtis supposed to have been proved in Principia Matlrernatica. that the propositions of pire mathematics are logical propositions. In The Prirrcipler of M~ithentatics.Russell's understandin: of the propositions of pure m;ithcmatics is rxplained by rnerins of his definition of pure miithemritics. He tlikes purc muthematics to consist of a11 propositions of the tom 'p implies q'. assuming that p ;ilid q .ire propositions contciining only variables. the same in sithrr proposition. and loyicd constants iPrM 3). The logicril constants çontained in propositions of pure miitlicm;itiçs are implication. relation. the relation of a terni to the ciass of which it is a iricnibcr and strc+ii hit. In addition. iiccording io Russell. mathematics iisrs the concept of irtirh. u-hich is nclt containcd in mathematical propositions but is containcd in logical propositions tPrb1 II). Russell sloimed that hr had arrived lit this definition of rn;itticni;itics cis ii result of an ündysis of rxisting pure mathematics; 2nd hence. that ~vhaicvcr ha?;. in the pnst, been regarded as pure mathematics. is included in the Jctinition. and whtitevcr is included has those chxricteristics by which mathematics is wrnnionly though vaycly distinguished from other subjects ( PrM 3).The purposr of The Pri~iciples of Jhirlret~iciiicswu to present Russell's analysis. and rhus justi fy his Jc finition of pure matlirlmatics. Rusdl brlievcd that the propositions of pure mathematics assert only iniplic:itions bccüu3t: pure mathematics concerns only whnt is possible. noi what is ;LCIU:I~.'~He Intcr explainrd thnt it was considcration of gcometry which led hirn to insist on thc iorm 'p irnplirs cf' in his definition of pure mathematics (PrM vii. C$ 5). Sincr Euclidean peomrtry and non-Euclidean geometries musi clrarly be included in pure iii;ittir.m:riçs. 2nd purc mathernatics cannot contain inconsistent propositions. Russell concludecl tlist it unuiily be asssrted in mathematics tlilit the axioms of ii given orometry imply rhe propositions of that geometry. not that the axioms ire [rue and the propositions ;ire [rue. Hencc. pure mathemûtics has no concem with the question whether or not the ii'rionih and propositions of Euclid hold of actid space: rather that is a question for iipplird rnathemütics to decide. Pure mathematics assens only that if the Euclidean 7 1 uioins are truc of actuill space then al1 the propositions of Euclid are true of actual space. In npplicd mathernatics, Russell argued. the consequent of an implication is assrned if the Iiypothesis is supported by rneans of actual facts. Indeed. the importance of an implication in ripplird mathematics is that it allows us to assert not only thiit the hypothesis implies thc conscquent. but also that. given that the hypothesis is true. since it is me. the consequent is truc However. in pure mathematics knowledge about the actual world. xhich is neçessxy to confirm the hypothcsis of a deduction in iipplied rnntherniitics. is nor ncedccl. In pure mathernatics. Russell claimed. it is assenrd that if an assertion p is truc ot' Iin entity .r. or a çlass of entities .K. y. :. . . . . then somr other assertion y is truc of thow cniitics: but ncithrr p nor q are rissened of the rntitirs tPrM 5). Hence. what is ~issertcdis 0ii1y a formal implication between the propositions p and q. Russcll br lieved that the propositions of pure mathernatics contain on1 y logiccil con3tanl.s bzçause. Iir supposrd. tliat explains the genrrd vriiidity of pure itititliematics. Purc ni;it\wrnatics is concerneci only with whrit Russell calkd 'the formal essence of ri propo\iiion' i PrM 7).The formal essence of o proposition. hc explaincd. is mivd cir by a procm 01' succcss ive genrralizaiion. i.e. hr successive substitution of variables for the cunstmts containrd in the proposition. Thus. br examplr. Russell argucd. givcn 'Soçrcitcs is a man*. the variable 's*cm be substituted for 'Soccites' siving '.Y is a man': dnd i:ikins as ri hypothesis 'r is Greek'. which implies the tmth of 'x is a man'. there rcwli.; ' ".r is Grerk" implirs ''A- is a man" * which is ti-ue for dl values of A-. Funhrr. the conccpts of Gwrk and miti cün be replaced by variables. The result is 'If CI and b are c.l;is~~and u is contained in b. then "x is an cl" irnplirs "x is a 6" '. which Russell clsimcd is a proposition of pure rnathrrnatics. containine only the variables (1. h and .Y. mJ the logicnl constants clciss. corirairird Ni and those involved in the concept of formai implicaiion (PM7). The resulting proposition cannot be funher genrralized. xcording io Russcll. for it is not possible to replace logical constants with variables (PrM 8-9). A Iogicâl constant. he explaineci. is a purely formal concept, a pure f~mi.~~Hence. if the Jeduction 'If a11 men Xe mortal. and Socrates is a mm. then Socrates is mortal* is 'h~~:B. Russcll. 'The Philosophicd hiplications of Mathomatical Logic'. 289-f. (The cssriy wris wittrn during ttic lime rhat Prirrcipia ~t.lurheniaricawrts being written. It wufirst published in 191 1.) '"lhrd., 237-f subrnittsd io a procrss of generalization, as explained in The Prirzciples of ibfutlieniatics. what results. Russell concluded. is the pure form of deduction on which depends the \,;ilidiiy of dl deduciions of the same fom. Given his analysis of pure rnathematics, Russell explains in ThPriricipks of .Iliiri~i~rdcs.that there are no esseriricil differences between the propositions of pure inathtmittics and the propositions of logic (PrM 9). The propositions of both logic and pure mathematics are propositions that contain o~lyvariables and logicnl constants. Logic coiisists of the axioms from whiçh al1 pure mathematics is derivablr and. in addition. al1 otlier propositions which contain only logical constants and variables but do not siitisfy thc dcîïiiition of the propositions of pure mlithemarics. Mathematics consists of dl propositions nohich Lissert fornitil implications and arc derivablc from the anioms of m;iitieriiatics. including those uioms which assen formal implications. Hrnce. according ro Riisscil. .;orne of the axioms of niathematics. for example. 'If p implics q and y implies i- ihcn 11 iriiplici r'. belong to mathemütics: and sorne. such as 'Implication is a relation'. bclong to logiç but nor to mathematics. Logic is thus more general than pure niiithemlirics (Pr.\l 1 1 I. Pure mathemütics is concemed with deductions involving the clüss of relations iia~~iiigttic iornid property of continuity. for exümple. but logiç concrms only the most gençral propositions in accordance with which deduction is possible. Howrvcir. iiccording to Riisscll. rhiit is not a fiindamental differcnce. Hence. he concludcd that therc are no csscntilil dil'fc.rençes betwern the propositions of pure rnathematics and the propositions of lugic. Howewr. the problems which Ied Russell to assume the Xxiom of Infinity. the .\Iuliipliciitii~rAniom and the Axiom of Reducibility in Principicr iCl«riiemiriccl raisrd \*iii.ious difficulties with his mülysis of pure mathernntics. During the writing of The Prir~c.jp/c..s O!- .Ifdirimtics. he had been unaware of the need to assume the uioms in question. The problems thût resultcd in Russell's undenaking to give the Theory of Loeical Types as the basis of his derivation of mathematics from logic. howevsr. sonvinccd him that the denvation necessitated the uioms. The problems hrtd to do with tlic occurrence of self-referential paradoxes within mathematics - for example, the paradox of the clliss of dl classes that are not members of thernselves which xises within cardinal arithrnetic. Supposing it is possible to define the class of al1 classes which are not rncnibsrs of thernselves, it can be asked whether or not this class is a member of itself. If it is. thcn it is one of thosr classes which are not members of thernselves. i.e. it is not a meniber of itself. But if it is not a member of itself, tnen it is not one of tiiose classes that rire not rncnibcrs of thrmselves. so it is a member of itself. Thus either supposition lads 10 a contrxiiction. Russrll's solution to this pruadox necessitated that he ordrr the proposiriunal kinctions of logic and matheniatics in a hierarchy of logical types. Howver. his solution resulted in further problems which required him to assume the .\siuni of Rcducibility. the Axiorn of infinity and the hlultiplicative Xxiom - and that ;iyin rcsultrd in difficulties. The difficulty with the Axiom of Reducibility. according to Russell. wstliat it is not self-eridrnr. Nonetheless. he belirved that he was able to justify ~iss~iniingthe axiom. The difficulty with the Axiom of Intïnity and the Multipliccitive Axiuiii. Iiuwvcr. was ihüt illthough thcy contain o~ilylogiccil constants. the axioms can oiily bc writicd by iicriid hcts. Thus. Russcll had to plxe çenüin rcstricrions on the use il I' c ir hcr :is iom w i thin the derivations of Principiri ;bZdimttrficc~.Hcncc. rlic logicist iicritxion of muthcmatics came io rrst on Russell's solutions for draling with the Axiom oi Inlinity. rhc .Multiplicative Axiom imd the Axiom of Reducibility. Riisscll 's ordrring of logical and müthernatical propositions in a hirrarchy of types i.; esplaincd in Pri~zcipiiiMdlrr~iaficcr as follows. The self-refrrentilil paradoxes ririsc. Riisscll bclirv~d.becausr it is assumrd that rnembers of n totality ciin br cietinrd by nicms of rhc totality. Thus. to rxcludr self-referentinl paradoxes. hr argued thut it ncccsïiiry to çonstruct logic and rnathrrnütics in accordance with what he called 'the s-clprncle. i.. 'If. provided ri certain class had a tord. it would have niernbsrs only dctinnble by means of that total. then it is nonsense to mkr stiitements about (il! irs rnernbcrs or 'Wharever contains an apparent variable must not be a possible valiic of tliat variable' (PM vol. 1 37). He qued that every propositional function has a cericiin muge ufsig~iijicci~ice.within which occur the arguments for which [hc function ha5 values. h function is said to be si_onificant for argument s if it has (i value for that iiripnent. The range of significmce of a function consists of dl the arguments for which the function is true. together with al1 the arguments for which the function is false. A Ïunstion is nonsense: that is. has no value, for arguments not within its range of si=nificance. Thus. every proposition containing dlassens that al1 values of the funcrion 3r~tnie. not that the function is true for dl arguments. since there are arguments for ~vliichiiny givrn function is nonsense. Hence. it rnakes srnse to refcr to al1 rnemben of a hss if and only if the class forms part or the whole of the range ofsign$cririce of some propositional function: the range of significancc being defined as the class of those Lirzuments for which the tùnction in question is significant. A gpe is defined as the range of significancc of a propositional function; that is. as the class of arguments for which the funciion lias viilues. Thus. if nothing containinj an apparent variable can be ii possible ~iilucof tliat variable thsn it must be of a differrnt type from the possible values of that uariiible. Russell says it is of o Iiigftcr type. The lowest type of objects are imiividids. the çon~titiicntsof whiit Russell callèd 'rlementary propositions': thüt is. propositions thrit contain no apparent variables. Elemenrriry propositions. togeiher with such as result from iipplying gcncralizotion to elcmentq propositions. arc collcd jirsr-ordrr propositions. ~vhicbform the wcond logicril type. Propositions in whichfirsr-order prtpoririom occur as ;ippurt.nt variables arc çdled srcotid-order propusiriorrs. which t'am the rhird logiçal ryc. Gc.neriilly. the (11 + 1 logical type consists of propositions of ordçr 12. Russell miiint~inedth;it it is meaningless to assen anyrhing of .L R~iscllcalled 'a predicative function'. Similarly. given a function q(î.T. . . .. t ). if it contains a variable in respect of which the hnction becomes prrdicative if qw~nents are bub\titutrd for al1 the orhrr variables. that function is also cülled 'a predicative function'. .hy tirsi-ordrr function the arguments of which cire individuals is a predicative function ol' :in individual. Russell denotrd such a function by. for rxarnple. 'cp!î' or 'cp!(f.j )'. Jcpcnding on ihe number of variables occurring in it. No tïrst-order function can contain function as apprirent variable: and hence. such functions form a totality and the Lippnrcnt variable and thcrr is no apparent variable of a higher type than cp is li second- ordcr proposrtion. If such a proposition contüins an individual x. it is not a predicative iunciion of .y: but if it conwins. for cxample. a first-order function Jcnotcd by 'j!( cp!: )' or y!( q!f. y!: )'. Thus. f is a second-order predicative function; thc possible values of!' again form a totality. iind /'cm be turned into an apparent variable. Tlius. third-ordcr predicarive funciions cm be drfined. which have third-ordrr propositions as volues iind second-ordcr prediccitive functions lis arguments. Higher ordrr prcdicat i vt: fiinct ions are de finrd similiuly in Priricipicz iI.l bc l ic\d it neccsscuy in Prùicipiu ;Mciihrninficci io assume the Axiom of Rzducibility. Since there arc various types of propositions and functions. and since pwdization can only br. applird within somr one type. the expressions 'dl propositions'. 'al1 functions'. 'suiiic propositions' and 'somr functions' are strictly meiiningless. Howcver. liccording to Rushell. in certain instances they can be given an unobjectionable interpretation. Furthcrmorc. he argued. if mathematics is to be possible at dl. it is absolutely necrssary that there be sorns method of making statements which. lis he put it. tvill usually be squivalent to what we have in mind when we inaccurately refer to 'dl propositionai functions sntisfied by s' (PM vol.1 166). Hence. Russell believed, it is necessary to finci zomr rnrrhod of reducing the order of a propositional function without affecting ~hetruth or frilsity of its values. This can be done by assuming the existence of classes. Given any propositioncil function qx. of whatever order. Russell claimed, this is ûssumed to be c.qui\,;ilent. for ail values of .Y to a statement of the form 'x belongs to the class a' - and assurning that therr is such an rntity as the class a. 's belongs to the class u' is a stmrnent contoining no apparent variable. and thus ir is a first-ordrr predicative function O . Plo. i 58 ). Howevrr. Russell maintaineci. the only advantagr thrit the function coilw-nirig u has over the function y.r is that it is first-ordrr. Thcrt. is no advanta=e. Ruïscll s1:iirncd. to wuming that theri: are such things as çlassrs: anci furthemore the pxidos conçerning the çIrissrs which are not members of themselvcs shows that. if there LUC clâsscs. thcy mut be somrthing very different from individuals. Thus what is really xhicved by assuming the existence of classes is the resulting method of reducing the order oi' 3 pr~p~siti~nillfunction. However. that can br achirvcd. without the resulting piirdos. bu aswiiiing the Axiom of Reducibiliry. The Xxiom of Reducibility is the ;issunipion that. siven iiny function. tlirre exists 3 t'ormally equivalcnt prrdiccit ive Icinction. Hcncr. for propositional functions of only one variable. the axiom states thüt giwn Lin? such propositional function gî. there is a function ~d such that. for evrry possible argument .r. 6.1 and VI( are either both tme or both falsr. and funher vi is a predic .A.~ioiiiof Rrducibility. Russell claimed. the inductive evidence for it is vcry strong. sincc rhc reasonin; which it dlows and the results to which it lc3ds are all such ris rippear valid. E-lcnw. Russcll concludrd that it seems very improbable thiit the axiorn would turn out to bc tilsc. Thus. altho~igh the uiom is not self-evident. Russell belièved that his ~i\wiiiptionof it LUS justified. Howcver. Russell was unable to similarly juscit'y his assumption of the Asiom of Infinit? and the !vlultiplicativc Axiom in Priticipiu Mdirnr~riicc~.He assumed the axioms ortl\. bccause tir was convinced that thry are necrssary for the derivation of pure ni:itlicmlitics: for hc considered both to be empirical propositions. Rrgarding the Axiorn of In finir'. Russell stated that 'if any one chooses to assume that the tod number of indi\.iduals in the universe is . . . 10. 367. therr seems no cr priori way of rehting his opinion' NLT 179). He ehborated: Tlis .\xiom of Iniïnity is purely ernpirical. Whritever v is. provided that it is a tïnits or rransfinite cardinal nurnber. it is possible LI priori that v be the number of individuals in the universe.'() "'B. RUSSCI~.ESS~VS in .-\naf~sis. ed. D. Lackey (London: George Allen & Unwin, 1973) 254. I-fencè. ivhether the number of individuals in the universe is finite or infinite is an cmpirictil question. Russell did not believe that. üccording to smpiricai evidence. it could be proven that the numbçr of individuals is nor finite: but he considered the cxisting empirical evidence sufficient to demonstnte that the hypothesis that the numbrr of indi~~idualsis finite involvrs rnany more difficulties than the contrary. Thus, he concluded on et~ipit-icalgounds that it is reasonable to assume the Axiom of hfinity. Accordingly, in Pri~icipi~i:Ilrirli~mcificci Russell statrs that it srrms clear that nothing in logic nccesit:itca the truth or falsity of the Axiom of Infinity. and that it cm only be believrd ur disbclic\+cdon cnipirical grounds iPM vo1.2 1 Y3 1. The same is true. he maintained. of tlic 4liiltiplicativr Asiom (PM vol. l 536). The Multiplicative Axiorn assens thüt givrn a cI:ic;s of classes having no members in common, none of which is null. there exists rit 1ecist onc class consisting of one mrmber of every class in the class of classes. Russell believed ill;it it is irnpossiblc to prove u priori that thrre are no possible universes in which it ii.oi!ld hc hlse (PrM viii). However. the iMultiplicative Axiom is ncrcrssary to prove. for cs;iniplc.. thi the two definitions of infinity cire cquivalcnt - i.r. thüt a cardinal numbtir is cal 1r.J 'intïnitc' if and only if its proprnies are non-inductive. and also if and only if it has propcr riibclasses ro which it is similar. Hcnce. dthough the Axiom of Lntinity and the Mu ltiplicutive ,+lxiornseemed necessary for the derivat ion of pure mathernatics. ne ithrr is il io:ic. holcis' (SL98 1. He did no&daborate on the rerisons for his assertion. but he seoms to have been con\-inced that. whrther or not it had bern achieved. the purpose of Pri~tcipitr .bfdremrric-ri.to prove that pure mathematics is a part of logic. is achievable. However. Wittgenstein's understanding of logical propositions in the 'Notes on Logic' led hirn into Jifficultics the solution to which was to result in his rejection of the Axiom of Rducibility; and hrnce. his qurstioning the possibility of deriving mathematics from hgc. Ii'ittgrnstrin'i understanding of logical propositions in the 'Notes on Logic' initi;illy led hirn into difficulties conccming how it is possible thiit a proposition of logic is LI pl-oposiriorz. His understanding was that logic consists of completely general propositions such üs '(p).p v - p' and 'p. p IJ q.z+,,, . ci'. Hrncr. Wittgenstein argues in tlic 'Notes on Logic' that reol variables have no proper use in logic: for if logic consists of pt-~p~.sirioris.the propositional functions in Priricipiri Mcrilit.»rclricci. which Russell süid miriii~zv;iriables. must be replaceci by gcner~lizedpropositions thüt have only qprrrit \.ciricibles i NL 103).Thus. for cxample. he regmirid '(x).-1 = .Y* and 'p.p 3 q. 3,., . y' lis propoiiticins uf lo@c. but not 'r = .r' or 'p v - p'. 'Siens of the form "'p v - p" are \cri\c.I~.as'. tir. claimrd. 'but not the proposition "(p).p v - 11" ' (NL 100). Hencr. Wittzrnstein's difficulty was to explain how a logiciil proposition such as '(p).p v - p'. n.hich swms to contain the nonsensical tûnction 'p v - p'. can have a srnsr. He iir_oued ht'(1)). p v - p' tus a mraning brcause it does tior contain the function 'p v - p'. but r;irhcs rhc function 'p v - y'. and he thought thar he could only explain how that is possible once he hnd given a correct aniilysis of identity. Wittgenstein did not regard '.Y = .r' as a nonsensiciil function. The reason that 'x = -Y' is not a proposition. hc believed. is that '.r' has no signification (NL 103). but he did not regard that as a reason to deny that '.Y = .Y' is 11 real function. Accordingly, iilthough he considered 'This rose is cither red or not rd to be senselcss. he clnimrd thrit 'Socrates = Socrates' is a proposition. Wittprnstcin mtes in (i lettrr written to Russell in early November 1913. in which he csp1;iins .;orne of ihr views rlaborated in the 'Notes on Logic'. '(p).p v - p is derived (rom rlie jioicliori p v - ri but the point will only become quite clear when identity is clcar' i R2O). To make it clex. he had been trying Thus. how (p).p v - p is derived from the function p v - q, Wittgenstein believed. can only he miide clear by makinj it clear how to express that there rire values of the funcrion II v - '1 such that p = q. Accordingly, he hiid to explüin why p v - p is nonsense but p v - LI. LI = p h;ls sense. and that, he believed, involved some fundamental difficulties. H~nvri\u-.Wittgenstein ncver solvrd the difficulties conccming rhe derivation of (pi. p v - p frorn p v - q: rnther hr carne to believe ihat. contrary to his view in the 'Yotes on Logic'. the logical proposition (p).p v - p dors contain the senseless tùnction p v - p. In ;I iertcr to RusseII. written on 30 October 1913. he supgests thiit in his attempts io wlve rht: problems conceming both idcntity and the occurrence of the sümr argument in cliffcircnt plrices of rhe same function. he had arrived at an important insight into the Tiicot-y ol' Deduction t RN). What his insight carne to. hc explainrd in tùnher lrttcrs \r.riticn ro Russcill in 'lovçrnber and Deçembrr 1913. was that al1 logical propositions are _zericrlilizarions of cither t Wittgenstein iirgurd for the conclusion that logical propositions rire gsncr;ilizations of rithcr tautolo_oiesor contradictions using the TF-notation introduced in ilici 'Norcs un Logic' INL 101-2).51 tn the 'Notes on Logic'. Wittgenstein insists ihat cwry proposition is cssentially 'tme-false' tNL 934). It is that which is reflected in his TF-notation. In the TF-notation. the proposition p is written as 'T-p-F'. tiiking 'T' to sisnify the possibility that it is true and 'F' to signify the possibility that it is fdse. Hence. hr ttxarnplc. - p is written as 'F-T-p-F-T'. for it is truc if p is frilsr and frilse if p is true: ;ind the proposition - - - p is written 'T-F-T-F-p-T-F-T-F' (R30). What signifies in the ''T~cnuimon 1s aiso cdlcd by Wittgenstein the 'ab-natation'. TF-notation. according to Wittgenstein, is the correlation of the outermost T and F with the innrrmost T and F, for by successively correltiting T's and F's, beginning with the innermost and endin: with the outermost. the outermost is thereby correlated with the inncrmost. Giiten the correlation of T's and F's. Wittgenstein believed that. with the TF- notlition. he could shoii that riIl logical propositions - or rathrr. the logical propositions of Pri~icipi idss. Il thc instances of a logical proposition are tautological. then the logicd proposition IS s3senti;iliy trur: if self-contrcidictory. thrn the logical proposition is rssçntially hlse. Wlicther ii proposition is tliurological or self-contradictory. Wittgenstein believed. cm be rtcoyi~edby the propositional sign as writren in the TF-notation (R22. cf. R73). Since. n~tiai.;ignitlt.s tn the TF-notation is the correlation of the outrrmost T's and F's with the inncrrnosr T'h and F's. 'F-T-F-p-T-F-T'. for example. çan be recognized to be the same \yrii bol :is ' F-p-T' . It cm bs recognized from the signs themselves that - - p is the same proposiiion ;is p. Thiis. p = - - p cnn bc shown to br ii tautology by writinr it as follou*~: It is (i triuiolop. Wittgenstein explainsd. brcause the outrrmost F is çonnectrd only with rhow pairs of innrrmost T's and F's that consist of ii T and F connected to the s~inic proposition p (R73). If the pair of innrrmost T's and F's consists only of T's or only of F'A. then the outrrmost truth-value is T. Thus. whatrver truth-value p has. the proposition (p).(p = - - p) is shown to be true. Hence. it is tautological. In the sarne way. \iritrgenstrin believsd he could show that dl logical propositions of the Theory of Dcduçiion of Pritxipin iM~zrizerncz~icaare tautological. and he had no doubt that he could do the süms in regard to the first-order propositions of the Theory of Apparent Variables O t' Priricipi~rM~~rhet~icrtica (R23). Howsvcr. although Wittgenstein belirved that he could show that al1 the propositions of logic in Principin Mathenmica are tautological. hr could not be so certain of the propositions of mathematics. Given that the propositions of logic. if tmr. ilri: esse~iritillytrue. Wittgenstein concluded that the Axiom of Infinity. the Multiplicntivr I\uiorn iind the Xxiorn of Rrducibility are not logicd propositions. These ~uioms.if true propositions. ;ire ciccitlt.milly trur (R17). Ewn if trur. it is possible that the axioms are Lilzc: so if truc t!ut would only be a fonunate accident. Regarding the Axiom of Intinity. W ittgcnstcin clairned that it is for pliyics to say whethcr any thing exists: and hence. the ;iuioiii. in that it States that there are H,, things. is not a logictrl proposition. The Axiom of Rcduçibility is not a proposition of logic. Wittgenstein argued. becausr it would tzot be Inic in a world in whiçh Ki) tlrings exist and. in addition to thcm. only ri siqle relation i.ai\is tioldiiig bewecn infinitrly rnany of the things and in suçh a wiiy that it dors not tiold betwt.cn caçh thing and every other thing and funhcr ncvcr held brtwern a finile nuniber of ttiinga.'' Whcther or not the world is of this kind. Wittgenstein claimeci. is not LL Ilutter for Iosiç LO decide. Russell had questioncd whether thçre was any wciy to Io,yic-~rfi!~prove or retùte the Axiom of [nfinity and the Multiplicative Xxiom. Wittgcnstein helieved that he had found a way to show that nrithcr is logically provable. ;inJ nor is the Axiom of Rrducibility: for none are propositions of logic. Hcnce. Wittgenstein's understanding of the propositions of lo,oic os :cncralizcitions of tautologies. or contradictions. hrid significant consequences for the iieriviiiion of mathematics from logic in Priricipiri Mritlicmciticti. Whitehead and Russell had rcglirded the Axiom of Infinity and the LMultiplicative hxiom as problematic. but bclievcd that they had found a way to deal with the auioms. The Axiom of Reducibility. however. was more difficult to deal with. 'If your hxiorn of Reducibility fails*. Wittgenstein wrotc to Russell in Deccmber 1913. 'thcn probably a lot of thinjs wiil have to be chrin_oeil8 iR73). Whitehead and Russell acknowledged the problem that I~~~~l.