Table Of Content.WITTGENSTEIN
and the Turning-point
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in the Philosophy
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of Mathematics
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S.G.SHANKER
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State University of New York Press
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CONTENTS
Preface vii
© Stuart Shanker 1987
ListofAbbreviations x
All rights reserved. No part ofthis publication .
may bereproduced or transmitted. in any form or by any means. without 1. Wiltgenslein's Tnrning-Pninl
permission. 1. Philosophy'sDebt to Schlick I
First published in U.S.A. by 2. ScepticalConfusionsabout Rule-following 13
State University of New York Press, Albany 3. Dispersingthe Cloudsof Epistemological
Confusion 25
For information, address State University ofNew York Press,
State University Plata, Albany, N.Y. 12246 2. The Strains in the Realist/Anti-realist Framework
I. Wittgenstein'sVerificationism 39
Printed in Great Britain 2. Wittgenstein's 'Conversion' 48
3. TheObjectivityofMathematicalKnowledge 59
Library of Congress Cataloging-in-Publication Data
3. The Nature of Proof
Shanker, Stuart.
Wittgenstein and the turning point in the philosophy 1. The Burden ofProof 75
ofmathematics. 2. Observingthe Law ofExcluded Middle 88
3. Mathematical 'Stimuli' 104
Bibliography: p.
Includes index. 4. Surveyabilily
I. Mathematics - Philosophy. 2. Wittgenstein,
1. TheBoundsofPerspicuity 120
Ludwig, 1889-1951. I. Title.
QA8.4.S53 1987 510'.1 86·23053 2. 'ProbabilisticProofs' 130
ISBN 0·88706·482·5 3. TheAppel-HakenSolution of the Four-Colour
ISBN 0·88706·483·3 (pbk.) Problem 143
5. The Perils of Prose
1. The Nature ofInfinity 161
2. Comingto Gripswith the Irrational 175
3. 'Prose':The Meeting-pointBetween Mathematics
and Philosophy 198
6. Consistency
I. Hilbert's Programme 220
2. MathematicalTotemsand Tabus 231
ToMy Parents
3. The Application ofan 'Inconsistent System' 246
7. The Recovery of Cerlainly
1. The Foundationsof the Foundationsof
Mathematics 259
2. EquationsareRulesofSyntax 274
3. UnconventionalConventionalism 288
8. Freedom and ecessity
1. Discovering,Creating, Inventing 303 PREFACE
2. AnarchyversusAutonomy 315
3. The Interrelatedness ofMathematicalTruths 329
Bibliography 342
354
Index
On the first page of Culture and Value we read: 'There is no
religious denomination in which the misuse of metaphysical
expressions has been responsible for so much sin as it has in
mathematics.' Here is a charge that must either be exploded or
primed; on no account can itbe ignored, whatever one's attitude
towardsWittgenstein'sconceptionofphilosophy.Unfortunately,the
widespread obloquy that Wittgenstein's work in the philosophyof
mathematics has provoked tends to dampen enthusiasm for such
aninitiative.Evensympatheticadmirersarecowedintosubmission
by such disparaging assessments as Dummett's reproach that
'Many of the thoughts [in Remarkson theFoundations ofMathe
maticsJare expressed ina mannerwhichthe author recognised as
inaccurate or obscure;some passages contradict others;some are
quite inconclusive; some raise objections to ideas which
Wittgenstein held or had held which are not themselves stated
clearly in the volume; other passages again, particularly those on
consistency andonGodel'stheorem,areofpoorqualityorcontain
definite errors." It is the spectre of technical mistakeswhich must
be particularly haunting for the aspiring Wittgensteinian; but this
issuemust besquarelyconfronted, ifonlyto certifywhetheror not
Remarks on the Foundations of Mathematics is an area which
would best beleftundisturbed bytheprudentWillgensteinian.
The origin of this workwasacommission to undertake the first
stepof thisunenviabletaskbyidentifyingthespecificmistakesthat
critics were alluding to in their passing asides on Wittgenstein's
failure to grasp the mechanics of Godel's second incompleteness
theorem, It quickly became manifest, however, that far more was
involved here than was immediatelyapparent. It was obvious that
Wittgenstein's remarks on Godel's theorem could not be grasped
without a prior understanding of his attack on meta-mathematics
and Hilbert's Programme, yet these latter issues could not be
broached before Wittgenstein'sdiscussionsofthenature ofmathe
maticalpropositionsand proofhad beenaddressed. But howcould
the lattertopics be understood withoutplacingthem inthe context
of Wittgenstein's attack on the use of 'prose' in the interpretation
ofmathematics, hismanyexamplesufthephilosophicalconfusions
vii
viii Preface Preface ix
that had resulted from the indiscretions of prose, and most approach to Wittgenstein's remarks on the philosopby of mathe
importantly of all, his striking oew approach to the character of matics by concentrating on the material of the early 1930s. For
mathematical necessity and the propriety of scepticism in the unlesswe carefullyretrace the stepswhich Wittgenstein took from
philosophy ofmathematics? hisreturn to Cambridgeand philosophyin 1929 (tbe timeat wbich
Running through all this was a growing awareness that the above quotation from Culture and Value was written) tbe
Wittgenstein's attack on Godel's interpretation of his second obstacles to understanding his mature writings on philosophy
incompleteness theorem could not be dissociated from his pro particularly in the pbilosophy of mathematics- are formidable, if
posed resolution/dissolution of the 'foundations crisis'. If any notinsuperable.ThegreatimportanceofPhilosophicalRemarksand
thing, the critiqueof Godel's theorem was merely a by-product of Philosophical Grammar for our understanding of Wittgenstein's
the much larger investigation into what Wittgenstein regarded as later investigations in the pbilosopby of mathematics is tbat
the conceptual confusions inspiring the foundations dispute.Thus, Wittgenstein discussed here in considerable detail the various
what began as a short paper on Wittgenstein's attitude to Godel's technical themes in bigher mathematics that are generally only
theorem had soon blossomed into a full-scale monograph on alluded to in Remarks on the Foundations of Mathematics. It
Wittgenstein's extensive involvement in the philosophy of mathe is tbus by focusing on these works that we can best expose the
matics. For one thing had become all too apparent: Wirtgenstein fallacies underlying tbe currently prevailing interpretations that
was assailed by the earlyreviewers of Remarkson theFoundations Wittgensteio was intent on some form of 'Anti-realist' attack on
of Mathematics for the mistakes wbich purportedly riddle the the foundations of mathematics, or that he was interested in a
book, yet invariably these 'errors' were only listed, never actually species of 'full-blooded conventionalist' or 'radical constructivist'
substantiatedassuch.Obviouslytbegreat appealofsuch a polemic critique. Freed from these critical incubuses, we shall then be in a
is that it is much easier to dismiss an argument on technical position to grasp the full implications of Wittgenstein's anti
grounds than to refuteit philosopbically.But what were presented metaphysicalexertions in tbe philosophyof matbematics.
as corrections were, infact,covert philosophicalobjectionswhich,
because of the prior assumption, were developed without any It gives me great pleasure to thank Gordon Baker and Peter
effort to clarify, let alone challenge, the philosophical background Hacker here for their unstinting help and encouragement. I am
on which Wittgenstein had based his approach to the foundations especially grateful to Peter Hacker for bis showing me the details
dispute. Whetheror not Wittgenstein's criticisms hit their mark is of an immense landscape which I could not possibly have known
obviouslyanissue whichwe cannot hope to consideruntilwe have mywayaround.I would also like to thank Brian McGuinness and
first 'established the grounds for the points which he raised; and Bede Rundlefortheir valuablecomments.[amdeeplyindebted to
whether these grounds are warranted will depend on whether tbe generous funding which 1 have received from the Canada
or not Wittgenstein's criticisms hit their mark. It was all too clear Council, first as a Doctoral and then as a Postdoctoral Fellow.
that any satisfactory treatment of Wittgenstein's writings in the Finally, 1 would like to thank Richard Stoneman and Mark
philosophy of mathematics would have to satisfy both of these Barragry for services rendered far above and beyond the call of
demands. duty.To mywife ...
Ironically, there is no discussion of Godel's theorem in what
S.G.S.
follows; tbat remains to be pursued in a subsequent work which Christ Cburch,Oxford
will be devoted solely to the clarification of Wittgenstein's attack
on the standard - meta-mathematical - conception of Godel's
second theorem in light of bisphilosophical scrutinyof the frame Note
work which underpins Godel's interpretation of his proof. The
1. MichaelDummett.'Wiugenstein'sPhilosophyofMathematics',inS.O.
present book might be seen as a prolegomenon to this subsequent
Shanker(00.),Ludwig Wittgenstein': CriticalAssessments,vol. III(London.Croom
exercise; its primary goal is to establish the outlines for a fresh Helm, 1986),p. 121.
Abbreviations xi
ABBREVIATIONS
(trans.), amended 2nd edn (Oxford, Basil Blackwell,
1980)
RPP RemarksonthePhilosophyofPsychology/Bemerkungen
iiber die Philosophie der Psychologie, vol. 1, G.E.M.
Works by Wiltgenslein Anscombe and G.H. von Wright (eds), G.E.M.
Anscombe [trans.](Oxford, BasilBlackwell, 1980)
NB Notebooks 1914-1916, G.E.M. Anscombe and G.H. Remarks on the Philosophy of Psychology/Bemer
von Wright (eds), G.E.M. Anscombe (trans.) (Oxford, kungen iiber die Philosophie der Psychologie; vol. II,
G.H.vonWright,G.H.andH.Nyman(eds), e.G.Luck
BasilBlackwell, 1961)
TLP Tractatus Logico-Philosophicus, D.F. Pears and B.F. hardtandM.A.E.Aue (trans.)(Oxford,BasilBlackwell,
1980)
McGuinness, (trans.) (London, Routledge & Kegan
Paul, 1961)
RLF 'Some Remarks on Logical Form', Proceedings of the
Lectures and Conversation Notes
AristotelianSocietySupplement,vol. 9 (1929), 162-71
WWK Ludwig Wittgenstein and {he Vienna Circle: Conver
LWL Wittgenstein'sLectures, Cambridge1930-1932,D.Lee
sations recorded by Friedrich lVaismann, B.F.
(ed.)(Oxford, BasilBlackwell, 1980)
McGuinness (ed.), J. Schulte and B.F. McGuinness
AWL Wittgenstein's Lectures; Cambridge 1932-1935, A.
(trans.) (Oxford, Basil Blackwell, 1979)
PR Philosophical Remarks, R. Rhees (ed.), 2nd edn, R. Ambrose (ed.) (Oxford,BasilBlackwell, 1979)
LA Lectures and Conversations on Aesthetics, Psychology
Hargreaves and R. White (trans.)(Oxford,BasilBlack
well, 1975) and Religious Belief, e. Barrett (ed.) (Oxford, Basil
PG Philosophical Grammar, R. Rhees(ed.), AJ.P. Kenny Blackwell, 1966)
(trans.) (Oxford, BasilBlackwell, 1974) LFM Wittgenstein's Lectures on the Foundations of Mathe
BB The BlueandBrown Books: Preliminary Studiesfor the matics,1939, C.Diamond(ed.) (Hassocks,Sussex,The
'Philosophical Investigations', 2nd edn (Oxford, Basil Harvester Press, 1976)
Blackwell, 1969)
RFM Remarkson theFoundations ofMathematics,G.H. von
Wright,R. Rhees andG.E.M. Anscombe(eds), G.E.M. Derivative.Primary Sources
Anscombe (trans.) 3rd edn (Oxford, Basil Blackwell,
IMT Friedrich Waismann, Introduction to Mathematical
1978)
PI PhilosophicalInvestigations,G.E.M.AnscombeandR. Thinking:TheFormationof ConceptsillModernMathe
Rhees (eds), G.E.M. Anscombe (trans.), 2nd edn matics:TJ.Benac(trans.)(Lon~on, Hafner, 1951)
PLP FriedrichWaismann, ThePrinciplesofLinguisticPhilo
(Oxford, BasilBlackwell, 1958)
z Zettl, G.E.M. Anscombe and G.H. von Wright (eds), sophy,R. Harre (trans.) (London, Macmillan, 1965)
G.E.M. Anscombe (trans.), 2nd edn (Oxford, Basil
Blackwell, 1981)
OC On Certainty,G.E.M.Anscombeand G.H.von Wright
(eds), D. Paul and G.E.M. Anscombe (trans.) (Oxford,
BasilBlackwell, 1977)
cv Culture and Value, G.H. von Wright (ed.), P. Winch
x
1 WITTGENSTEIN'S TURNING-POINT
I am persuaded that we are at present in the midst of an alto
gether final change in philosophy, and are justly entitled to
consider the fruitless conflict of systems at an end. The present
age, Imaintain,isalreadyinpossession ofthe meansto make all
such conflict essentiaUy unnecessary; it isonlya matter of reso
lutely using them. ... The methods proceed from logic. Their
beginnings were obscurely perceived by Leibniz; in recent
decades important stretches have been opened up by Gottlob
Frege and Bertrand Russell; but the decisive turning-point was
firstreached byLudwigWittgenstein.
MoritzSchlick,'TheTurning-PointinPhilosophy'
Philosophy'sDebttoSchlick
For allthe accolades,Schlickhas receivedrather uncharitable press
in the memoirs of his colleagues from the Vienna Circle, largely
because of the sentiments which are expressed in the above quo
tation. Herbert Feiglseems to have been particularly disturbed by
<theenormouseffectofWittgenstein,whichsetaquitenewscampon
Scblick's thinkingduringthelast tenyearsofhislife'.' In'No Potof
Message' he recalled: 'To my chagrin Schlick ascribed to
Wittgenstein philosophical ideas that he (Schlick) had already
expounded much more lucidlyin his 1918 book on epistemology.I
was also disappointed with Schlick's compromise with positivism
(phenomenalistic version) - and the abandonment of his critical
realismas"metaphysicallysuspect"."Whetherornotthiscensureis
regarded as unjust willdepend,ofcourse,on one's attitude towards
Wittgenstein'slater philosophy.What would beunpardonable,how
ever, would be to sympathise with Schlick's appraisaland yet do
nothing to defend his reputation. For the fact of the matter is that
Wittgensteinstudiesowea tremendousdebt to Schlick.
We shall never know just how much Schlick contributed to
Wittgenstein's decision to return to philosophy, nor howmuch the
rapid evolution of Wittgenstein's thought during what Alice
Ambrosehassoaptlydescribedasthe'Olympian years'of1932-5
owes to Schlick's success in persuading Wittgenstein to undertake
the joint project withWaismann of writing Logik, Sprache, Philo-
I
2 wiugenstein's Turning-Point Wittgenstein'sTurning-Point 3
sophie.' Finally,there isthequestionofhowmuchSchlickwasable sort of questions as those which have COme to the fore in the
to influence Wittgenstein's thought during the conversations on 'sceptical' interpretation, and Wittgenstein's answers provide a
philosophywhich took placefrom 1929-32.That Wittgensteinhad perspicnous indication of the anti-sceptical direction in which his
anenormousimpactonSchlick'sthought- and throughSchlick,on thought was moving. Schlick repeatedly responded to the inno
such younger membersof the Circleas Waismann and Juhos - has vations which Wittgenstein was proposing to introduce into the
beenamplyrecorded byvariousmembersofthe Vienna Circle.But fabric of the Tmctatuswith the anxietythat Wittgenstein had failed
there is some evidence to suggestthat, if only by the nature of the to clarify the epistemological orientation of his new argument,
questions and objections which he persistently expressed - which, thereby threatening to undermine the great service which the
not surprisingly, all demonstrate his pre-established 'logical Tractatushad performedfor philosophybyprovidingan 'insightinto
empiricist' bias- Schlickforced Wittgenstein to clarifythemes that the nature of the logical itself,' Just as persistently, however,
wereto becomecentralto hisdevelopment duringthe 1930s. Wittgenstein responded that epistemology had nothing whatsoever
This is an issue of more than passing relevance to our inter todowiththecurrentissue:thathewassolelyconcernedwithclarify
pretation of Wittgenstein's writings on the philosophy of ing what at the time he described as the logical syntax of those
mathematics. There is a widespread feelingtoday that Wittgenstein expressionswhichhad resulted in philosophicalproblems.Indeed, it
was engaged in a sceptical assault on the foundations of mathe will be argued that, ina fundamentalsense, Wittgensteinperceived
matics; a subject that had already been rocked by serious his essential task in the philosophy of mathematics as that of
epistemological doubts. Yet this is the very opposite of removing the epistemologicalframework from the foundations dis
Wittgenstein's basic objectives in the philosophy of mathematics. pute altogether, without whichthe 'foundations crisis' would simply
Rather, Wingenstein was intent on demonstrating that, like all collapse. Before commencing these investigations, however, some
philosophical sceptical issues, the principal problems in the found explanation is perhaps called for to account for why this theme
ations dispute stem from conceptual confusion, and thus call for should havebeen solargelymisunderstood orunappreciated.
logicalclarification asopposed to epistemologicalrefutation.This is In large part these misconceptions would seem to be due to the
obviouslya crucial - indeed, perhaps the single most important manner in which Wittgenstein's Nachlass has been published,
issue in the interpretation of Wittgenstein's remarks on the philo together with the philiosophicaltrends that have been dominant at
sophy of mathematics. If we start - as some influential the time of their appearance. Confronted with Remarks on the
commentators have recently suggested - with the premise that Foundationsof Mathematics- aworkwhichbyanystandardsmust
Wittgenstein wasengagedin a speciesof'rule-followingscepticism', be judged unusually opaque - philosophers not unnaturally
we shall inevitably be led to view Wittgenstein's arguments as attempted to penetrate Wittgenstein's highly enigmatic remarks in
intended to take us everfurther down a sceptical path whose pur terms of the problems and approaches that current!y preoccupied
pose was to dislodge us from the prevailing truth-conditional them. In particular, recent analytic philosophy had become
conception of semantics. If, on the other hand, we accept that increasingly interested in the so-called 'sceptical' consequences
Wittgenstein never wavered from his fundamental belief that whichcould bedrawn from the paradoxes inset theory and natural
'Scepticismisnotirrefutable,butobviouslynonsensical,whenittries language. Since Wittgenstein was demonstrably interested in
to raise doubts where no questions can be asked' (TLP 6.51; NB developing paradoxes of his own, it seemed conceivable that his
44), then weshallsee theseargumentsas attacks,eachofwhichwas, writingscould becitedasan importantauthority- ifnot thesource
intended to dissolvesome philosophical problem in the foundations - of the various sceptical dilemmas that had captured the imagin
ofmathematics. ation of philosophers of mathematics and language. That
Fortunately, we have the precedent of the discussions which Wittgenstein was primarilyinterestedinconstructingparadoxes asa
resulted from Schlick's own confusion on thisscore, as recorded in method of deflatingmetaphysicscouldnotevenbeconsidered;fora
wiugenstein and the Vienna Circle, to corroborate the latter inter reductioadabsurdum that wasintended to undermine philosophical
pretation. For, as we shall see, Schlick posed very much the same scepticism could certainly have no place in a sceptical
4 Wittgenstein'sTurning-Point Witrgenstein'sTurning-Point 5
Weltanschauung. The very question from whence this 'sceptical' be unfair; disposing ofit bymeansof another question isnot.') We
interpretation springs could not allow for this possibility; for if we are thus left with a series of questions, metaphors and analogies
beginwiththe premisethateveryphilosophymustfallIDtOeitherthe which,althoughtheycanbestrikinglyevocative,oftenseemtofailto
category of 'Realism' or 'Anti-realism' in Dummett's sense (cf. advancematters significantly,oreven to clarify wbatconstitutesthe
Chapter 2), we soon find ourselves grappling with the thorny centralpointatissue.Indeed,theeffectachievedisoftenthereverse
question of which camp can rightly claim Wittgenstein as an from that which Wittgenstein intended, only serving to confuse or
adherent. But if the premise iswrong - if Wittgenstein belongsto even undermine the invariably logical point of clarification that he
neither school ofthought,for the veryreason that he had embarked wasstrivingtoestablish.
on a course which would undermine the very foundation of the Unlike the obscure discussions presented in Lectures on the
Realist/Anti-realist distinction - the 'sceptical' interpretation of Foundations ofMathematics and Remarks on the Foundations of
Remarkson the FoundationsofMathematicsisitselfundermined at Mathematics, however, the arguments developed in Philosophical
a stroke. Thus, the first task of this book willbe to establish that Remarks and Philosophical Grammarare noticeably detailed and
Wittgenstein's arguments are in a crucial sense fundament~lly explicit, and thus provide an invaluable introduction to the later
opposed to the basicpremiseunderlying thisapproach:a fact which work. They also furnish the much-needed explanation for why
accountsformuchofthestrain inrecentexegesis. Wittgenstein should have concentrated to such an inordinate extent
To makemattersworse,forstylisticreasons- nottomentionthe on problems drawn from elementary arithmetic in Remarks on the
method whereby Wittgenstein compiled his manuscripts from his Foundations of Mathematics. This feature of Wittgenstein's
notebooks - thesethemes can be exceptionallydifficult to compre approachwasquicklyseizedonbyhiscriticsasproofofhistechnical
hend solelyon the basisofreading Remarks on the Foundations of limitationsinthe philosophyofmathematics.But no suchaccusation
Mathematics. In his 1931paper 'The Future of Philosophy' Schlick can be levelled against Philosophical Remarks and Philosophical
compared Wittgenstein to Socrates on the grounds that wecan dis Grammar.Itisonlywhenweread throughtheseearlyworksthatwe
cern an important parallel between their similar approaches to the can begin to understand the important srylistic and thematic
solution of philosophical problems.' Wiltgenstein must have been development whichWittgensteinexperienced duringthe 1930..The
fullyawareofthiscomparison,and~rhapsevensancti~ned suchan more he addressed the fundamental confusions underlying the
allusion (if only by his silence); certainlythe style of his arg~me~ts 'foundations crisis' the more strongly he began to feel that the
becameincreasinglySocraticoverthenexttwodecades.Inthisvern, philosophicalproblemswhichsurfaceinthevarious realmsof higher
one of the most notable themes, frequently repeated in the manu mathematics are merelymorecomplex versions of the same issues
scripts and the lectures, is Wittgenstein's insist~n~e that he ~oul? which arise inelementaryarithmetic.Forexample, the typeofprob
refrain from the dogmatic assertion of any opuuon or thesis; his lemsthat emergedwith theconstruction oftheTransfiniteCardinals
route would be one which proceeded to philosophical clarification areessentiallythesarneasthosethatcharacterisetheconstructionof
via completely banal assertions. And yet this is hardly what hap anynewDumbersystem.Hence, Wittgensteinsoughtto gain inper
pened: at least in the realm of the philosophyof mathematics. For spicuity what he lost in detailed application by presenting his
here Wittgenstein's comments often seem anything but uncon criticismsofthequestionswhichprefigureinhighermathematicsin
troversial,andsomeofhismore notoriousremarks ~ suchasthose the context ofthe problemswhichoccurinelementaryarithmetic.
onconsistency- have struckhiscriticsaspositivelybizarre,ifnot The benefits of adopting a criticalapproach which takes Philo
proof of his technicalincompetence in the field.Furthermore, by sophical Remarks and Philosophical Grammaras its starting-point
the end of the 1930., Wittgenstein had purged his style of almost are principallytwofold, therefore: the first is simplythat the taskof
any remaining elements of a direct approach. (In Book 111 §6 of clarifying exactlywhat themes Wittgenstein was objecting to in the
Remarks on the Foundations ofMathematics he insisted that 'In philosophy of mathematics and what position he was arguing for
philosophy it is alwaysgood to put a question instead of an answer becomes far morestraightforward. But there isafurther reason why
to aquestion. For ananswertothe philosophicalquestion mayeasily it isessential to study thismaterial ifwe are to appreciate fully the
6 Wittgenstein'sTurning-Point Wittgenstein'sTurning-Point 7
argumentsofthe later work.Wittgenstein'sdevelopment asa mature seen as comprising a complex network of interlockingcalculi:auto
philosopher was in many ways remarkablyself-contained. Thus,the nomous 'propositional systems' eachof whichconstitutesa distinct
argumentswhich evolved over the nextdecade were based asmuch 'logical space'. This would enable Wittgenstein to reconcile the
on criticisms of his own earlier views as on those of other philo principlethatcolour-statements'exclude'oneanotherwiththe Trac
sophers. For exegetical purposes the significance of this fact is that tatus'insistence that'allnecessityislogicalnecessity'.Suchachange
we must try to enter into the concrete details of Wittgenstein's necessitated, however,a radical shiftfromthe Tractatusconception
development ashe himselfperceived them. Inotherwords, wemust of inference. Thus, as Wittgenstein explained to Waismann and
try to follow as closely as possible the actual steps which Schlick: '[When I wrote the TractatusJ I believed that elementary
Wittgenstein took followinghis return to active philosophy in 1929. propositions mustbeindependentofoneanother, thatyoucouldnot
Otherwise,the resultislikelyto be,asWittgensteinsorightlyfeared, infer the non-existence of one state of affairs from the existence of
that hisargumentswould be largelymisunderstood. another. But if mypresent conception ofa system ofpropositions is
The origin of Wittgenstein's approach to the foundations of correct,itwillactuallybethe rule thatfromtheexistenceofonestate
mathematics lies in the rapid development of his thought following ofaffairsthe non-existenceofallthe otherstates ofaffairsdescribed
his return to Cambridge and philosophy in 1929. Wittgeostein's by this system of propositions can be inferred.' (WWK 64) That is,
immediateproblem wasto remedythe glaringerror in the Tractatus 'A is red' does indeed entailthat 'A isnot blue, green, yellow ...'.
account oflogical necessity(which Ramseyio particular was press But Wittgenstein nowaccepted tbat 'A isred' is an atomicproposi
ing home). In his curt remarks on the colour-exclusioo problem at tion; i.e. not all entailments are to be accounted for by inner
6.375-6.3751 Wittgenstein had argued that: (i) it is 'logically propositional complexity. Hence the truth-tabular definition of the
impossible' for twocolours tobe inthe sameplaceat thesame time, logicalconnectives isdemonstrablyinadequate as a means of deter
and that 'the statement that a point in the visual field has two dif mining all theformsofinference that arepermissiblein a language.
ferent colours at the same time is a contradiction'; (li) that this For, in the case of a determinate exclusion, we can only be said to
exemplifies the fundamental principle that 'the only necessity that bave grasped the meaningofthe determinateifwe bavegrasped the
existsislogical necessity';and (iii) thatitfollowsfrom thistbat 'red' entire Satzsystem; i.e. to grasp tbe meaning of a determinate is to
cannot be the name ofa simple, and thus that 'A isred' isnot fully grasp how it logically excludesthe possibility of the combination of
analysed.But,asRamseyobjected inbisreviewofthe Tractatus,this theother determinatesinthat system. Asweshallsee, thisnotion of
argument only serves to push the problem back to a deeper level: logicalexclusioncameto playanevermoreprominentroleinWitt
'even supposing that the physicist [can provideIan analysis of what genstein's approachto the'foundationscrisis'.
wemeanby"red",Mr.Wittgensteinisonlyreducingthedifficultyto The price whichWirtgenstein wasforced to payforthisproposed
that of the necessary properties of space, time, and matter or the resolution of the colour-exclusion problem. therefore, was to
ether." Ramsey would undoubtedly have pressed home this point abandon the hallmark of the 'absolute' logical atomism which char
when Wittgenstein returned to Cambridge, and it thus seems likely acterisesthe Tractatus:the premisethat elementarypropositions are
that 'Some Remarks on Logical Form' reflects Ramsey's successin logically independent. But how was Wittgenstein going to reconcile
persuading Wittgenstein that the argument at 6.375-6.3751 needed the introduction of Satzsysteme with another of the Tractatus's
emendation," major themes:theargument that, contraFrege,names bavemeaning
In orderto correctthisdefect,Wittgenstein introduced two major (Bedeutung) but notsense(Sinn)? The answerhere wastoabandon
innovations which be believed would enable him to resolve the the referential conception of meaning: the Tractatusargument that
colour-exclusion problem insuch a wayas to preserve the Traaa the meaning ofa nameisthe objectwhichitdenotes.Inplaceoftbis
tus'srigiddemarcation between logical and empiricaltrutb.Hisfirst Wittgenstein now argued that a word only has meaning in the con
step was to abandon the Traaatus's; sweeping model of a single textofitspropositionalsystem,andthat themeaning ofawordisthe
amorphouscalculusunderlyingnaturallanguage,and toshiftto what totalityof rulesgoverningitsuseinthatsystem.Whenawordisused
he described asa Satzsysteme' conception, inwhich language was outside the context of its legitimate system it becomes a 'wheel
I
8 wittgenstein'sTurning-Point Wittgenstein'sTurning-Point 9
turning idly'. Generally we can simply ignore the obvious Satz briefsummaryofWittgenstein's new Satzsystemeconceptionof lan
systeme confusions which proliferate in language (in as much as guage on a note ofpersonal dismay:TheTheoriescontained in this
these belong to the province of the grammarian), but occasionally newwork of Wiltgenstein's [PhilosophicalRemarks) are novel,very
Sattsystemeconfusions arise which from the grammarian's point of original,and indubitably important, Whether theyare true, Ido not
view are perfectlywell-constructed,but which none the less seem to know. As a logician who likes simplicity, Ishould wishto think that
generate a profound 'ontological' or 'epistemological' perplexity. theyare not."
Such, Wittgenstein now wanted to argue, are the unique source of There thus arose from the ashes of the colour-exclusion problem
philosophical problems: they result from the transgression of the a bold new argument which more than anything else heralded the
rules,notof ordinarygrammar,butrather,of logicalsyntax.Inorder turning-point in Wittgenstein's approach to the philosophy of
to understand the confusion underlying this type of 'idly turning language, and thence,the philosophyofmathematics.Itwasthispic
wheel', we must clarify the logico-syntactical system in which the lure of language as compartmentalised into distinct, autonomous
word properly belongs, a task which can often he extremelytaxing, systems, each of which operates on its own specific internal rules,
and which we accomplish by considering the use of the term in which was to lead him ever funher from the Tractatussconception
question. This theme led Wittgenstein to articulate the principlethat of the formal isomorphism between language and reality. Ina short
'the sense ofa proposition is the method of its verification' (WWK timethe last vestigesofthecalculusmodel wouldalsodisappear,and
66). The method of verification manifests to which Satzsystem a from these Satzsysteme would evolve 'language games': artificial
proposition belongs, and hence revealswhen words which belongto microcosms of language whose sole purpose was to clarify various
multiple or different systems are being used meaningfully or illicitly aspects of actual linguistic practice. Perhaps the most important
(ct.Chapter 2). point to cometo termswith here, however,isthe fact that this thesis
On the basisofthese developmentsin hisconception of language wasintended solelyas a matter ofclarifying logicalsyntax,which as
Wittgenstein set about the task of salvaging what remained of the such was devoid of any epistemological overtones. This is brought
Tractatus's grand design. When he first introduced Satzsysteme in out particularly forcefully in Wittgenstein's exchanges with Schlick
1929 it was not with the intention of repudiating the Tractatuss on the question of whether there is'nothing that canbe said inreply
calculus theoryof meaning perse; rather Wittgenstein wasarguing to the question, "How can I know that one syntax is rightwhile
that the Tractatusconception of a global calculus underlying natural another isnot. ... In what relation does empirical knowledge stand
language failed to account for the complex 'multiplicity' of logical to syntax?'" (WWK 65).
syntax, and what was needed was a 'network' conception of over The argument which developed between the two on this point is
lapping 'logical spaces'. The price that Wittgcnstein was thus somewhatcurious,insofaraseachseemsto havebeen intenton a
prepared to pay in 1929 in order to shore up the crumbling foun different issue. Schlick's main concern was to ensure that
dations of the Tractatus edifice was to abandon the pristine Wittgenstein did not allow the dreaded ' synthetic aprioritruth' to
simplicity of the structure which he had earlier envisaged.Thus he slip back into philosophy,thistime under the auspices of the 'logical
repudiated the Frege-Russellnotationsin PhilosophicalRemarksas exclusions' which characterise Satzsystemeinference.Thus he asked
far too oversimplified to serve as concept-scripts for natural Wittgenstein, 'What answer can one give to a philosopher who
languages. He mayhavestillbeen committed to the universal calcu believes that the statements of phenomenology are syntheticzrpriori
lus model at this time,but itwaswiththe important proviso thatthis judgements?' To which Wittgenstein responded: 'Let us take the
turns out to be enormously more complicated than he had antic statement,"Anobject isnotred andgreen atthesametime." IsallI
ipated when writing the Tractatus. It is hardly surprising, therefore, want to say by this that I have not yet seen such an object?
that Russellshould havebeen so alarmed bythe direction thatWitt Obviouslynot. What Imeanis,"I callnotsee suchan object," "Red
gcnstein'sthoughtwastaking in 1930.Inhisreporton Wittgenstein's and green cannot be in the same place." Here I would ask, What
most recent work to Trinity College (made on behalf of doestheword"can"meanhere?Theword"canU isobviouslyagram
Wittgenstein'sapplicationfor a researchgrant) Russellconcluded his matical (logical) concept, not a material one' (WWK 67). But this
