Wittgenstein’s Philosophy of Mathematics

First published Fri Feb 23, 2007; substantive revision Wed Jan 31, 2018

Ludwig Wittgenstein’s Philosophy of Mathematics is undoubtedlythe most unknown and under-appreciated part of his philosophical opus.Indeed, more than half of Wittgenstein’s writings from 1929through 1944 are devoted to mathematics, a fact that Wittgensteinhimself emphasized in 1944 by writing that his “chiefcontribution has been in the philosophy of mathematics” (Monk1990: 466).

The core of Wittgenstein’s conception of mathematics is verymuch set by theTractatus Logico-Philosophicus (1922;hereafterTractatus), where his main aim is to work out thelanguage-reality connection by determining what is required forlanguage, or language usage, to beabout the world.Wittgenstein answers this question, in part, by asserting that theonly genuine propositions that we can use to make assertions aboutreality are contingent (‘empirical’) propositions, whichare true if they agree with reality and false otherwise (4.022, 4.25,4.062, 2.222). From this it follows that all other apparentpropositions are pseudo-propositions of various types and that allother uses of ‘true’ and ‘truth’ deviatemarkedly from the truth-by-correspondence (or agreement) thatcontingent propositions have in relation to reality. Thus, from theTractatus to at least 1944, Wittgenstein maintains that“mathematical propositions” are not real propositions andthat “mathematical truth” is essentially non-referentialand purely syntactical in nature. On Wittgenstein’s view, weinvent mathematical calculi and we expand mathematics by calculationand proof, and though we learn from a proof that a theoremcan be derived from axioms by means of certain rules in aparticular way, it isnot the case that this proof-pathpre-exists our construction of it.

As we shall see, Wittgenstein’s Philosophy of Mathematics beginsin a rudimentary way in theTractatus, develops into afinitistic constructivism in the middle period (PhilosophicalRemarks (1929–30) andPhilosophical Grammar(1931–33), respectively; hereafterPR andPG,respectively), and is further developed in new and old directions inthe manuscripts used forRemarks on the Foundations of Mathematics (1937–44;hereafterRFM). As Wittgenstein’s substantive views onmathematics evolve from 1918 through 1944, his writing andphilosophicalstyles evolve from the assertoric, aphoristicstyle of theTractatus to a clearer, argumentative style inthe middle period, to a dialectical, interlocutory style inRFM and thePhilosophical Investigations (hereafterPI).

1. Wittgenstein on Mathematics in theTractatus

Wittgenstein’s non-referential,formalist conception ofmathematical propositions and terms begins in theTractatus.^[1] Indeed, insofar as he sketches a rudimentary Philosophy ofMathematics in theTractatus, he does so bycontrasting mathematics and mathematical equations withgenuine (contingent) propositions, sense, thought, propositional signsand their constituent names, and truth-by-correspondence.

In theTractatus, Wittgenstein claims that a genuineproposition, which rests upon conventions, is used by us to assertthat a state of affairs (i.e., an elementary or atomic fact;‘Sachverhalt’) or fact (i.e., multiple states ofaffairs; ‘Tatsache’) obtain(s) in the one andonly real world. An elementary proposition is isomorphic to thepossible state of affairs it is used to represent: it mustcontain as many names as there are objects in the possible state ofaffairs. An elementary proposition is trueiff its possiblestate of affairs (i.e., its ‘sense’;‘Sinn’) obtains. Wittgenstein clearly states thisCorrespondence Theory of Truth at (4.25):

If an elementary proposition is true, the state of affairs exists; ifan elementary proposition is false, the state of affairs does notexist.

But propositions and their linguistic components are, in and ofthemselves, dead—a proposition only has sense because we humanbeings have endowed it with aconventional sense (5.473).Moreover, propositional signs may beused to do any number ofthings (e.g., insult, catch someone’s attention); in order toassert that a state of affairs obtains, a person must‘project’ the proposition’s sense—its possiblestate of affairs—by ‘thinking’ of (e.g., picturing)its sense as one speaks, writes or thinks the proposition (3.11).Wittgenstein connectsuse,sense,correspondence, andtruth by saying that “aproposition is true if weuse it to say that things stand ina certain way, and they do” (4.062; italics added).

TheTractarian conceptions of genuine (contingent)propositions and the (original and) core concept of truth are used toconstruct theories of logical and mathematical‘propositions’by contrast. Stated boldly andbluntly, tautologies, contradictions and mathematical propositions(i.e., mathematical equations) are neither true nor false—we saythat they are true or false, but in doing so we use the words‘true’ and ‘false’ in very different sensesfrom the sense in which a contingent proposition is true or false.Unlike genuine propositions, tautologies and contradictions“have no ‘subject-matter’” (6.124),“lack sense”, and “say nothing” about theworld (4.461), and, analogously, mathematical equations are“pseudo-propositions” (6.2) which, when ‘true’(‘correct’; ‘richtig’ (6.2321)),“merely mark … [the] equivalence of meaning [of‘two expressions’]” (6.2323). Given that“[t]autology and contradiction are the limitingcases—indeed thedisintegration—of thecombination of signs” (4.466; italics added), where

the conditions of agreement with the world—the representationalrelations—cancel one another, so that [they] do[] not stand inany representational relation to reality,

tautologies and contradictions do not picture reality or possiblestates of affairs and possible facts (4.462). Stated differently,tautologies and contradictions do not have sense, which means wecannot use them to make assertions, which means, in turn, that theycannot be either true or false. Analogously, mathematicalpseudo-propositions are equations, which indicate or show that twoexpressions are equivalent in meaning and therefore areintersubstitutable. Indeed, we arrive at mathematical equations by“the method of substitution”:

starting from a number of equations, we advance to new equations bysubstituting different expressions in accordance with the equations.(6.24)

We prove mathematical ‘propositions’ ‘true’(‘correct’) by ‘seeing’ that two expressionshave the same meaning, which “must be manifest in the twoexpressions themselves” (6.23), and by substituting oneexpression for another with the same meaning. Just as “one canrecognize that [‘logical propositions’] are true from thesymbol alone” (6.113), “the possibility of proving”mathematical propositions means that we can perceive their correctnesswithout having to compare “what they express” with facts(6.2321; cf.RFM App. III, §4).

The demarcation between contingent propositions, which can be used tocorrectly or incorrectly represent parts of the world, andmathematical propositions, which can be decided in a purely formal,syntactical manner, is maintained by Wittgenstein until his death in1951 (Zettel §701, 1947;PI II, 2001 edition,pp. 192–193e, 1949). Given linguistic and symbolic conventions,the truth-value of a contingent proposition is entirely a function ofhow the world is, whereas the “truth-value” of amathematical proposition is entirely a function of its constituentsymbols and the formal system of which it is a part. Thus, a second,closely related way of stating this demarcation is to say thatmathematical propositions are decidable by purely formal means (e.g.,calculations), while contingent propositions, being about the‘external’ world, can only be decided, if at all, bydetermining whether or not a particular fact obtains (i.e., somethingexternal to the proposition and the language in which it resides)(2.223; 4.05).

The Tractarian formal theory of mathematics is, specifically, a theoryofformal operations. Over the past 20 years,Wittgenstein’s theory of operations has received considerableexamination (Frascolla 1994, 1997; Marion 1998; Potter 2000; and Floyd2002), which has interestingly connected it and the Tractarianequational theory of arithmetic with elements of Alonzo Church’s\(\lambda\)-calculus and with R. L. Goodstein’s equationalcalculus (Marion 1998: chapters 1, 2, and 4). Very briefly stated,Wittgenstein presents:

… the sign ‘[\(a, x, O \spq x\)]’ for thegeneral term of the series of forms \(a\), \(O \spq a\), \(O \spq O\spq a\)…. (5.2522)
… the general form of an operation\(\Omega\spq(\overline{\eta})\) [as] \[[\overline{\xi}, N(\overline{\xi})]\spq (\overline{\eta}) (= [\overline{\eta}, \overline{\xi}, N(\overline{\xi})]). (6.01)\]
… the general form of a proposition(“truth-function”) [as] \([\overline{p}, \overline{\xi},N(\overline{\xi})]\). (6)
The general form of an integer [natural number] [as] \([0, \xi ,\xi + 1]\). (6.03)

adding that “[t]he concept of number is… the general formof a number” (6.022). As Frascolla (and Marion after him) havepointed out, “the general form of a proposition is aparticular case of the general form of an‘operation’” (Marion 1998: 21), and all threegeneral forms (i.e., of operation, proposition, and natural number)are modeled on the variable presented at (5.2522) (Marion 1998: 22).Defining “[a]n operation [as] the expression of a relationbetween the structures of its result and of its bases” (5.22),Wittgenstein states that whereas “[a] function cannot be its ownargument,… an operation can take one of its own results as itsbase” (5.251).

On Wittgenstein’s (5.2522) account of ‘[\(a, x, O \spqx\)]’,

the first term of the bracketed expression is the beginning of theseries of forms, the second is the form of a term \(x\) arbitrarilyselected from the series, and the third [\(O \spq x\)] is the form ofthe term that immediately follows \(x\) in the series.

Given that “[t]he concept of successive applications of anoperation is equivalent to the concept ‘and so on’”(5.2523), one can see how the natural numbers can be generated byrepeated iterations of the general form of a natural number, namely‘[\(0, \xi , \xi +1\)]’. Similarly, truth-functionalpropositions can be generated, as Russell says in the Introduction totheTractatus (p. xv), from the general form of a proposition‘[\(\overline{p}\), \(\overline{\xi}\), \(N(\overline{\xi})\)]’ by

taking any selection of atomic propositions [where \(p\) “standsfor all atomic propositions”; “the bar over the variableindicates that it is the representative of all its values”(5.501)], negating them all, then taking any selection of the set ofpropositions now obtained, together with any of the originals [where\(x\) “stands for any set of propositions”]—and soon indefinitely.

On Frascolla’s (1994: 3ff) account,

a numerical identity“\(\mathbf{t} = \mathbf{s}\)” is anarithmetical theorem if and only if the corresponding equation“\(\Omega^t \spq x = \Omega^s \spq x\)”, which is framedin the language of the general theory of logical operations, can beproven.

By proving

the equation“\(\Omega^{2 \times 2}\spq x = \Omega^{4}\spqx\)”, which translates the arithmetic identity“\(2 \times2 = 4\)” into the operational language (6.241),

Wittgenstein thereby outlines “a translation of numericalarithmetic into a sort of general theory of operations”(Frascolla 1998: 135).

Despite the fact that Wittgenstein clearly doesnot attemptto reduce mathematics to logic in either Russell’s manner orFrege’s manner, or to tautologies, and despite the fact thatWittgenstein criticizes Russell’s Logicism (e.g., the Theory ofTypes, 3.31–3.32; the Axiom of Reducibility, 6.1232, etc.) andFrege’s Logicism (6.031, 4.1272, etc.),^[2] quite a number of commentators, early and recent, have interpretedWittgenstein’s Tractarian theory of mathematics as a variant ofLogicism (Quine 1940 [1981: 55]; Benacerraf & Putnam 1964a: 14;Black 1964: 340; Savitt1979 [1986: 34]; Frascolla 1994: 37; 1997: 354, 356–57, 361;1998: 133; Marion 1998: 26 & 29; and Potter 2000: 164 and182–183). There are at least four reasons proffered for thisinterpretation.

Wittgenstein says that “[m]athematics is a method oflogic” (6.234).
Wittgenstein says that “[t]he logic of the world, which isshown in tautologies by the propositions of logic, is shown inequations by mathematics” (6.22).
According to Wittgenstein, we ascertain thetruth ofboth mathematical and logical propositions by the symbolalone (i.e., by purely formal operations), without making any(‘external’, non-symbolic) observations of states ofaffairs or facts in the world.
Wittgenstein’s iterative (inductive) “interpretationof numerals as exponents of an operation variable” is a“reduction of arithmetic to operation theory”, where“operation” is construed as a “logicaloperation” (italics added) (Frascolla 1994: 37), which showsthat “the label ‘no-classes logicism’ tallies withtheTractatus view of arithmetic” (Frascolla 1998: 133;1997: 354).

Though at least three Logicist interpretations of theTractatus have appeared within the last 20 years, thefollowing considerations (Rodych 1995; Wrigley 1998) indicate thatnone of these reasons is particularly cogent.

For example, in saying that “[m]athematics is a method oflogic” perhaps Wittgenstein is only saying that since thegeneral form of a natural number and the general form of a propositionare both instances of the general form of a (purely formal) operation,just as truth-functional propositions can be constructed using thegeneral form of a proposition, (true) mathematical equations can beconstructed using the general form of a natural number. Alternatively,Wittgenstein may mean that mathematicalinferences (i.e., notsubstitutions) are in accord with, or make use of, logical inferences,and insofar as mathematical reasoning is logical reasoning,mathematics is a method of logic.

Similarly, in saying that “[t]he logic of the world” isshown by tautologies and true mathematical equations (i.e., #2),Wittgenstein may be saying that since mathematics was invented to helpus count and measure, insofar as it enables us to infer contingentproposition(s) from contingent proposition(s) (see 6.211 below), ittherebyreflects contingent facts and “[t]he logic ofthe world”. Though logic—which is inherent in natural(‘everyday’) language (4.002, 4.003, 6.124) and which hasevolved to meet our communicative, exploratory, and survivalneeds—is notinvented in the same way, a valid logicalinference captures the relationship between possible facts and asound logical inference captures the relationship betweenexistent facts.

As regards #3, Black, Savitt, and Frascolla have argued that, since weascertain the truth of tautologies and mathematical equations withoutany appeal to “states of affairs” or “facts”,true mathematical equations and tautologies areso analogousthat we can “aptly” describe “the philosophy ofarithmetic of theTractatus… as a kind oflogicism” (Frascolla 1994: 37). The rejoinder to this is thatthe similarity that Frascolla, Black and Savitt recognize does notmake Wittgenstein’s theory a “kind of logicism” inFrege’s or Russell’s sense, because Wittgenstein does notdefine numbers “logically” in either Frege’s way orRussell’s way, and the similarity (or analogy) betweentautologies and true mathematical equations is neither an identity nora relation of reducibility.

Finally, critics argue that the problem with #4 is that there is noevidence for the claim that the relevant operation islogicalin Wittgenstein’s or Russell’s or Frege’s sense ofthe term—it seems a purely formal, syntactical operation. “Logical operations are performed with propositions,arithmetical ones with numbers”, says Wittgenstein (WVC218); “[t]he result of a logical operation is a proposition, theresult of an arithmetical one is a number”. In sum, critics ofthe Logicist interpretation of theTractatus argue that##1–4 do not individually or collectively constitute cogentgrounds for a Logicist interpretation of theTractatus.

Another crucial aspect of theTractarian theory ofmathematics is captured in (6.211).

Indeed in real life a mathematical proposition is never what we want.Rather, we make use of mathematical propositionsonly ininferences from propositions that do not belong to mathematics toothers that likewise do not belong to mathematics. (In philosophy thequestion, ‘What do we actually use this word or this propositionfor?’ repeatedly leads to valuable insights.)

Though mathematics and mathematical activity are purely formal andsyntactical, in theTractatus Wittgenstein tacitlydistinguishes between purely formal games with signs, which have noapplication in contingent propositions, and mathematical propositions,which are used to make inferences from contingent proposition(s) tocontingent proposition(s). Wittgenstein does not explicitly say,however,how mathematical equations, which arenotgenuine propositions, are used in inferences from genuineproposition(s) to genuine proposition(s) (Floyd 2002: 309; Kremer2002: 293–94). As we shall see in§3.5, the later Wittgenstein returns to the importance ofextra-mathematical application and uses it to distinguish a mere“sign-game” from a genuine, mathematicallanguage-game.

This, in brief, is Wittgenstein’s Tractarian theory ofmathematics. In the Introduction to theTractatus, Russellwrote that Wittgenstein’s “theory of number”“stands in need of greater technical development”,primarily because Wittgenstein had not shown how it could deal withtransfinite numbers (Russell 1922[1974]: xx).Similarly, in his review oftheTractatus, Frank Ramsey wrote that Wittgenstein’s‘account’ does not cover all of mathematics partly becauseWittgenstein’s theory of equations cannot explain inequalities(Ramsey 1923: 475). Though it is doubtful that, in 1923, Wittgensteinwould have thought these issues problematic, it certainly is true thatthe Tractarian theory of mathematics is essentially a sketch,especially in comparison with what Wittgenstein begins to develop sixyears later.

After the completion of theTractatus in 1918, Wittgensteindid virtually no philosophical work until February 2, 1929, elevenmonths after attending a lecture by the Dutch mathematician L.E.J.Brouwer.

2. The Middle Wittgenstein’s Finitistic Constructivism

There is little doubt that Wittgenstein was invigorated by L.E.J.Brouwer’s March 10, 1928 Vienna lecture “Science,Mathematics, and Language” (Brouwer 1929), which he attendedwith F. Waismann and H. Feigl, but it is a gross overstatement to saythat he returned to Philosophy because of this lecture or that hisintermediate interest in the Philosophy of Mathematics issuedprimarily from Brouwer’s influence. In fact,Wittgenstein’s return to Philosophy and his intermediate work onmathematics is also due to conversations with Ramsey and members ofthe Vienna Circle, to Wittgenstein’s disagreement with Ramseyover identity, and several other factors.

Though Wittgenstein seems not to have read any Hilbert or Brouwerprior to the completion of theTractatus, by early 1929Wittgenstein had certainly read work by Brouwer, Weyl, Skolem, Ramsey(and possibly Hilbert) and, apparently, he had had one or more privatediscussions with Brouwer in 1928 (Finch 1977: 260; Van Dalen 2005:566–567). Thus, the rudimentary treatment of mathematics in theTractatus, whose principal influences were Russell and Frege,was succeeded by detailed work on mathematics in the middle period(1929–1933), which was strongly influenced by the 1920s work ofBrouwer, Weyl, Hilbert, and Skolem.

2.1 Wittgenstein’s Intermediate Constructive Formalism

To best understand Wittgenstein’s intermediate Philosophy ofMathematics, one must fully appreciate his strong variant offormalism, according to which “[w]emakemathematics” (WVC 34, note 1;PR §159) byinventing purely formal mathematical calculi, with‘stipulated’ axioms (PR §202), syntacticalrules of transformation, and decision procedures that enable us toinvent “mathematical truth” and “mathematicalfalsity” by algorithmically deciding so-called mathematical‘propositions’ (PR §§122, 162).

Thecore idea of Wittgenstein’s formalism from 1929 (ifnot 1918) through 1944 is that mathematics is essentially syntactical,devoid of reference and semantics. The most obvious aspect of thisview, which has been noted by numerous commentators who do not referto Wittgenstein as a ‘formalist’ (Kielkopf 1970:360–38; Klenk 1976: 5, 8, 9; Fogelin 1968: 267; Frascolla 1994:40; Marion 1998: 13–14), is that,contra Platonism, thesigns and propositions of a mathematical calculus do notrefer to anything. As Wittgenstein says at (WVC 34,note 1), “[n]umbers are not represented by proxies; numbersare there”. This means not only that numbers are therein theuse, it means that the numeralsare thenumbers, for “[a]rithmetic doesn’t talk about numbers, itworks with numbers” (PR §109).

What arithmetic is concerned with is the schema \(||||\).—Butdoes arithmetic talk about the lines I draw with pencil onpaper?—Arithmetic doesn’t talk about the lines, itoperates with them. (PG 333)

In a similar vein, Wittgenstein says that (WVC 106)“mathematics is always a machine, a calculus” and“[a] calculus is an abacus, a calculator, a calculatingmachine”, which “works by means of strokes, numerals,etc”. The “justified side of formalism”, accordingto Wittgenstein (WVC 105), is that mathematical symbols“lack a meaning” (i.e.,Bedeutung)—they do not “go proxyfor”things which are “theirmeaning[s]”.

You could say arithmetic is a kind of geometry; i.e. what in geometryare constructions on paper, in arithmetic are calculations (onpaper).—You could say it is a more general kind of geometry.(PR §109;PR §111)

This is the core of Wittgenstein’s life-long formalism. When weprove a theorem or decide a proposition, we operate in apurelyformal, syntactical manner. Indoing mathematics, we donot discover pre-existing truths that were “already therewithout one knowing” (PG 481)—weinventmathematics, bit-by-little-bit. “If you want to know what \(2 +2 = 4\) means”, says Wittgenstein, “you have to ask how wework it out”, because “we consider the process ofcalculation as the essential thing” (PG 333). Hence,the only meaning (i.e., sense) that a mathematical proposition has isintra-systemic meaning, which is wholly determined by itssyntactical relations to other propositions of the calculus.

A second important aspect of the intermediate Wittgenstein’sstrong formalism is his view that extra-mathematical application(and/or reference) isnot a necessary condition of amathematical calculus. Mathematical calculido not requireextra-mathematical applications, Wittgenstein argues, since we“can develop arithmetic completely autonomously and itsapplication takes care of itself since wherever it’s applicablewe may also apply it” (PR §109; cf.PG308,WVC 104).

As we shall shortly see, the middle Wittgenstein is also drawn tostrong formalism by a new concern with questions ofdecidability. Undoubtedly influenced by the writings ofBrouwer and David Hilbert, Wittgenstein uses strong formalism to forgea new connection between mathematical meaningfulness and algorithmicdecidability.

An equation is a rule of syntax. Doesn’t that explain why wecannot have questions in mathematics that are in principleunanswerable? For if the rules of syntax cannot be grasped,they’re of no use at all…. [This] makes intelligible theattempts of the formalist to see mathematics as a game with signs.(PR §121)

InSection 2.3, we shall see how Wittgenstein goes beyond both Hilbert and Brouwer bymaintaining the Law of the Excluded Middle in a way thatrestricts mathematical propositions to expressions that arealgorithmically decidable.

2.2 Wittgenstein’s Intermediate Finitism

The single most important difference between the Early and MiddleWittgenstein is that, in the middle period, Wittgenstein rejectsquantification over an infinite mathematical domain, stating that,contra hisTractarian view, such ‘propositions’are not infinite conjunctions and infinite disjunctions simply becausethere are no such things.

Wittgenstein’sprincipal reasons for developing afinitistic Philosophy of Mathematics are as follows.

Mathematics as Human Invention: According to the middleWittgenstein, we invent mathematics, from which it follows thatmathematics and so-called mathematical objects do not existindependently of our inventions. Whatever is mathematical isfundamentally a product of human activity.
Mathematical Calculi Consist Exclusively of Intensions andExtensions: Given that we have invented only mathematical extensions(e.g., symbols, finite sets, finite sequences, propositions, axioms)and mathematical intensions (e.g., rules of inference andtransformation, irrational numbersas rules), theseextensions and intensions, and the calculi in which they reside,constitute the entirety of mathematics. (It should be noted thatWittgenstein’s usage of ‘extension’ and‘intension’ as regards mathematics differs markedly fromstandard contemporary usage, wherein the extension of a predicate isthe set of entities that satisfy the predicate and the intension of apredicate is the meaning of, or expressed by, the predicate. Putsuccinctly, Wittgenstein thinks that the extension of this notion ofconcept-and-extension from the domain of existent (i.e., physical)objects to the so-called domain of “mathematical objects”is based on a faulty analogy and engenders conceptual confusion. See#1 just below.)

These two reasons have at least five immediateconsequencesfor Wittgenstein’s Philosophy of Mathematics.

Rejection of Infinite Mathematical Extensions: Given that amathematical extension is a symbol (‘sign’) or a finiteconcatenation of symbolsextended in space, there is acategorical difference between mathematical intensions and (finite)mathematical extensions, from which it follows that “themathematical infinite” resides only in recursive rules (i.e.,intensions). An infinite mathematical extension (i.e., acompleted, infinite mathematical extension) is acontradiction-in-terms
Rejection of Unbounded Quantification in Mathematics: Given thatthe mathematical infinite can only be a recursive rule, and given thata mathematical proposition must have sense, it follows that therecannot be an infinite mathematical proposition (i.e., an infinitelogical product or an infinite logical sum).
Algorithmic Decidability vs. Undecidability: If mathematicalextensions of all kinds are necessarilyfinite, then,inprinciple, all mathematical propositions arealgorithmicallydecidable, from which it follows that an “undecidablemathematical proposition” is a contradiction-in-terms. Moreover,since mathematics is essentially what we have and what we know,Wittgenstein restricts algorithmic decidability toknowinghow to decide a proposition with a known decision procedure.
Anti-Foundationalist Account of Real Numbers: Since there are noinfinite mathematical extensions, irrational numbers are rules, notextensions. Given that an infinite set is a recursive rule (or aninduction) and no such rule can generate all of the thingsmathematicians call (or want to call) “real numbers”, itfollows that there is no set of ‘all’ the real numbers andno such thing as the mathematical continuum.
Rejection of Different Infinite Cardinalities: Given thenon-existence of infinite mathematical extensions, Wittgensteinrejects the standard interpretation of Cantor’s diagonal proofas a proof of infinite sets of greater and lesser cardinalities.

Since we invent mathematicsin its entirety, we do notdiscover pre-existing mathematical objects or facts or thatmathematical objects have certain properties, for “one cannotdiscover any connection between parts of mathematics or logic that wasalready there without one knowing” (PG 481). Inexamining mathematics as a purely human invention, Wittgenstein triesto determine what exactly we have invented and why exactly, in hisopinion, we erroneously think that there are infinite mathematicalextensions.

If, first, we examine what we have invented, we see that we haveinvented formal calculi consisting of finite extensions andintensional rules. If, more importantly, we endeavour to determinewhy we believe that infinite mathematical extensions exist(e.g., why we believe that the actual infinite is intrinsic tomathematics), we find that we conflate mathematicalintensions and mathematicalextensions, erroneouslythinking that there is “a dualism” of “the law andthe infinite series obeying it” (PR §180). Forinstance, we think that because a real number “endlessly yieldsthe places of a decimal fraction” (PR §186), itis “a totality” (WVC 81–82, note1), when, in reality, “[a]n irrational number isn’t theextension of an infinite decimal fraction,… it’s alaw” (PR §181) which “yieldsextensions” (PR §186). A law and a list arefundamentally different; neither can ‘give’ what the othergives (WVC 102–103). Indeed, “the mistake in theset-theoretical approach consists time and again in treating laws andenumerations (lists) as essentially the same kind of thing”(PG 461).

Closely related with this conflation of intensions and extensions isthe fact that we mistakenly act as if the word ‘infinite’is a “number word”, because in ordinary discourse weanswer the question “how many?” with both (PG463; cf.PR §142). But “‘[i]nfinite’is not aquantity”, Wittgenstein insists (WVC228); the word ‘infinite’ and a number word like‘five’ do not have the same syntax. The words‘finite’ and ‘infinite’ do not function asadjectives on the words ‘class’ or ‘set’,(WVC 102), for the terms “finite class” and“infinite class” use ‘class’ in completelydifferent ways (WVC 228). An infinite class is a recursiverule or “an induction”, whereas the symbol for a finiteclass is a list or extension (PG 461). It is because aninduction has much in common with the multiplicity of a finite classthat we erroneously call it an infinite class (PR§158).

In sum, because a mathematical extension is necessarily a finitesequence of symbols, an infinite mathematical extension is acontradiction-in-terms. This is the foundation of Wittgenstein’sfinitism. Thus, when we say, e.g., that “there are infinitelymany even numbers”, we arenot saying “there arean infinite number of even numbers” inthe same senseas we can say “there are 27 people in this house”; theinfinite series of natural numbers is nothing but “the infinitepossibility of finite series of numbers”—“[i]t issenseless to speak of thewhole infinite number series, as ifit, too, were an extension” (PR §144). Theinfinite is understood rightly when it is understood, not as aquantity, but as an “infinite possibility” (PR§138).

Given Wittgenstein’s rejection of infinite mathematicalextensions, he adopts finitistic, constructive views on mathematicalquantification, mathematical decidability, the nature of real numbers,and Cantor’s diagonal proof of the existence of infinite sets ofgreater cardinalities.

Since a mathematical set is a finite extension, we cannotmeaningfully quantify over an infinite mathematical domain,simply because there is no such thing as an infinite mathematicaldomain (i.e., totality, set), and, derivatively, no such things asinfinite conjunctions or disjunctions (G.E. Moore 1955: 2–3; cf.AWL 6; andPG 281).

[I]t still looks now as if the quantifiers make no sense for numbers.I mean: you can’t say‘\((n) \phi n\)’, preciselybecause ‘all natural numbers’ isn’t a boundedconcept. But then neither should one say a general proposition followsfrom a proposition about the nature of number.
But in that case it seems to me that we can’t usegenerality—all, etc.—in mathematics at all. There’sno such thing as ‘all numbers’, simply because there areinfinitely many. (PR §126;PR §129)

‘Extensionalists’ who assert that“\(\varepsilon(0).\varepsilon(1).\varepsilon(2)\) and soon” is an infinite logical product (PG 452) assume orassert that finite and infinite conjunctions are closecousins—that the fact that we cannot write down or enumerate allof the conjuncts ‘contained’ in an infinite conjunction isonly a “human weakness”, for God could surely do so andGod could surely survey such a conjunction in a single glance anddetermine its truth-value. According to Wittgenstein, however, this isnot a matter ofhuman limitation. Because wemistakenly think that “an infinite conjunction” is similarto “an enormous conjunction”, we erroneously reason thatjust as we cannot determine the truth-value of an enormous conjunctionbecause we don’t have enough time, we similarly cannot, due tohuman limitations, determine the truth-value of an infiniteconjunction (or disjunction). But the difference here is not one ofdegree but of kind: “in the sense in which it is impossible tocheck an infinite number of propositions it is also impossible to tryto do so” (PG 452). This applies, according toWittgenstein, to human beings, but more importantly, it applies alsoto God (i.e., an omniscient being), for even God cannot write down orsurvey infinitely many propositions because for him too the series isnever-ending or limitless and hence the ‘task’ is not agenuine task because it cannot,in principle, be done (i.e.,“infinitely many” is not a number word). As Wittgensteinsays at (PR 128; cf.PG 479): “‘Can Godknow all the places of the expansion of \(\pi\)?’ would havebeen a good question for the schoolmen to ask”, for the questionis strictly ‘senseless’. As we shall shortly see, onWittgenstein’s account, “[a] statement aboutallnumbers is not represented by means of a proposition, but by means ofinduction” (WVC 82).

Similarly, there is no such thing as a mathematical proposition aboutsome number—no such thing as a mathematical propositionthat existentially quantifies over an infinite domain (PR§173).

What is the meaning of such a mathematical proposition as‘\((\exists n) 4 + n = 7\)’? It might be adisjunction—\((4 + 0 = 7) \vee{}\) \((4 + 1 = 7) \vee {}\) etc.adinf. But what does that mean? I can understand a proposition witha beginning and an end. But can one also understand a proposition withno end? (PR §127)

We are particularly seduced by the feeling or belief that an infinitemathematical disjunction makes good sense in the case wherewe can provide a recursive rule for generating each next member of aninfinite sequence. For example, when we say “There exists an oddperfect number” we are asserting that, in the infinite sequenceof odd numbers, there is (at least) one odd number that isperfect—we are asserting‘\(\phi(1) \vee \phi(3) \vee\phi(5) \vee{}\) and so on’ and we know what would make it trueand what would make it false (PG 451). The mistake here made,according to Wittgenstein (PG 451), is that we are implicitly“comparing the proposition‘\((\exists n)\)…’ with the proposition…‘There are two foreign words on this page’”, whichdoesn’t provide the grammar of the former‘proposition’, but only indicates an analogy in theirrespective rules.

On Wittgenstein’s intermediate finitism, an expressionquantifying over an infinite domain isnever a meaningfulproposition, not even when we have proved, for instance, that aparticular number \(n\) has a particular property.

The important point is that, even in the case where I am given that\(3^2 + 4^2 = 5^2\), I oughtnot to say‘\((\exists x,y, z, n) (x^n + y^n = z^n)\)’, since taken extensionallythat’s meaningless, and taken intensionally this doesn’tprovide a proof of it. No, in this case I ought to express only thefirst equation. (PR §150)

Thus, Wittgenstein adopts the radical position thatallexpressions that quantify over an infinite domain, whether‘conjectures’ (e.g., Goldbach’s Conjecture, the TwinPrime Conjecture) or “proved general theorems” (e.g.,“Euclid’s Prime Number Theorem”, the FundamentalTheorem of Algebra), aremeaningless (i.e.,‘senseless’; ‘sinnlos’) expressionsas opposed to “genuine mathematicalproposition[s]” (PR §168). Theseexpressions are not (meaningful) mathematical propositions, accordingto Wittgenstein, because the Law of the Excluded Middle does notapply, which means that “we aren’t dealing withpropositions of mathematics” (PR §151). Thecrucial questionwhy and in exactly what sense the Law of theExcluded Middle does not apply to such expressions will be answered inthe next section.

2.3 Wittgenstein’s Intermediate Finitism and Algorithmic Decidability

The middle Wittgenstein has other grounds for rejecting unrestrictedquantification in mathematics, for on his idiosyncratic account, wemust distinguish between four categories of concatenations ofmathematical symbols.

Proved mathematical propositions in a particular mathematicalcalculus (no need for “mathematical truth”).
Refuted mathematical propositions in (or of) a particularmathematical calculus (no need for “mathematicalfalsity”).
Mathematical propositions for which we know we have in hand anapplicable and effective decision procedure (i.e., we knowhow to decide them).
Concatenations of symbols that are not part of any mathematicalcalculus and which, for that reason, are not mathematical propositions(i.e., are non-propositions).

In his 2004 (p. 18), Mark van Atten says that

… [i]ntuitionistically, there are four [“possibilitiesfor a proposition with respect to truth”]:
\(p\) has been experienced as true
\(p\) has been experienced as false
Neither 1 nor 2 has occurred yet, but we know a procedure todecide \(p\) (i.e., a procedure that will prove \(p\) or prove \(\negp)\)
Neither 1 nor 2 has occurred yet, and we do not know a procedureto decide \(p\).

What is immediately striking about Wittgenstein’s ##1–3and Brouwer’s ##1–3 (Brouwer 1955: 114; 1981: 92) is theenormous similarity. And yet, for all of the agreement, thedisagreement in #4 is absolutely crucial.

As radical as the respective #3s are, Brouwer and Wittgenstein agreethat an undecided \(\phi\) is a mathematical proposition (forWittgenstein,of a particular mathematical calculus) if weknow of an applicable decision procedure. They also agree that until\(\phi\) is decided, it is neither true nor false (though, forWittgenstein, ‘true’ means no more than “proved incalculus \(\Gamma\)”). What they disagree about is the status ofan ordinary mathematical conjecture, such as Goldbach’sConjecture. Brouwer admits it as a mathematical proposition, whileWittgenstein rejects it because we do not know how to algorithmicallydecide it. Like Brouwer (1948 [1983: 90]), Wittgenstein holds thatthere are no “unknown truth[s]” in mathematics, but unlikeBrouwer he denies the existence of “undecidablepropositions” on the grounds that such a‘proposition’ would have no ‘sense’,“and the consequence of this is precisely that the propositionsof logic lose their validity for it” (PR §173). Inparticular, if thereare undecidable mathematicalpropositions (as Brouwer maintains), then at least some mathematicalpropositions are not propositions ofany existentmathematical calculus. For Wittgenstein, however, it is a definingfeature of a mathematical proposition that it is either decided ordecidable by a known decision procedurein a mathematicalcalculus. As Wittgenstein says at (PR §151),

where the law of the excluded middle doesn’t apply, no other lawof logic applies either, because in that case we aren’t dealingwith propositions of mathematics. (Against Weyl and Brouwer).

The point here isnot that we need truth and falsity inmathematics—we don’t—but rather that everymathematical proposition (including ones for which an applicabledecision procedure is known) isknown to be part of amathematical calculus.

To maintain this position, Wittgenstein distinguishes between(meaningful, genuine) mathematical propositions, which havemathematical sense, and meaningless, senseless(‘sinnlos’) expressions by stipulating that anexpression is a meaningful (genuine) proposition of a mathematicalcalculusiff weknow of a proof, a refutation, or anapplicable decision procedure (PR §151;PG 452;PG 366;AWL 199–200). “Only wherethere’s a method of solution [a ‘logical method forfinding a solution’] is there a [mathematical] problem”,he tells us (PR §§149, 152;PG 393).“We may only put a question in mathematics (or make aconjecture)”, he adds (PR §151), “where theanswer runs: ‘I must work it out’”.

At (PG 468), Wittgenstein emphasizes the importance ofalgorithmic decidability clearly and emphatically:

In mathematicseverything is algorithm andnothingis meaning [Bedeutung]; even when itdoesn’t look like that because we seem to be usingwords to talkabout mathematical things. Even thesewords are used to construct an algorithm.

When, therefore, Wittgenstein says (PG 368) that if“[the Law of the Excluded Middle] is supposed not to hold, wehave altered the concept of proposition”, he means that anexpression is only a meaningful mathematical proposition if weknow of an applicable decision procedure for deciding it(PG 400). If a genuine mathematical proposition isundecided, the Law of the Excluded Middle holds in the sensethat weknow that we willprove or refute theproposition by applying an applicable decision procedure (PG379, 387).

For Wittgenstein, there simply is no distinction between syntax andsemantics in mathematics: everything is syntax. If we wish todemarcate between “mathematical propositions” versus“mathematical pseudo-propositions”, as we do, then theonly way to ensure that there is no such thing as ameaningful, butundecidable (e.g., independent), propositionof a given calculus is to stipulate that an expression is only ameaningful propositionin a given calculus (PR§153) if either it has been decided or weknow of anapplicable decision procedure. In this manner, Wittgenstein definesboth a mathematical calculusand a mathematicalproposition inepistemic terms. A calculus is defined interms of stipulations (PR §202;PG 369),known rules of operation, andknown decisionprocedures, and an expression is only a mathematical propositionin a given calculus (PR §155), and only if thatcalculuscontains (PG 379) a known (and applicable)decision procedure, for “you cannot have a logical plan ofsearch for asense you don’t know” (PR§148).

Thus, the middle Wittgenstein rejects undecidable mathematicalpropositions ontwo grounds. First, number-theoreticexpressions that quantify over an infinite domain are notalgorithmically decidable, and hence are not meaningful mathematicalpropositions.

If someone says (as Brouwer does) that for \((x) f_1 x = f_2 x\),there is, as well as yes and no, also the case of undecidability, thisimplies that‘\((x)\)…’ is meant extensionally andwe may talk of the case in which all \(x\) happen to have a property.In truth, however, it’s impossible to talk of such a case at alland the‘\((x)\)…’ in arithmetic cannot be takenextensionally. (PR §174)

“Undecidability”, says Wittgenstein (PR§174) “presupposes… that the bridgecannotbe made with symbols”, when, in fact, “[a] connectionbetween symbols which exists but cannot be represented by symbolictransformations is a thought that cannot be thought”, for“[i]f the connection is there,… it must be possible tosee it”. Alluding to algorithmic decidability, Wittgensteinstresses (PR §174) that “[w]e can assert anythingwhich can bechecked in practice”, because“it’s a question of thepossibility ofchecking” (italics added).

Wittgenstein’s second reason for rejecting an undecidablemathematical proposition is that it is acontradiction-in-terms. There cannot be “undecidablepropositions”, Wittgenstein argues (PR §173),because an expression that is not decidable in someactualcalculus is simply not amathematical proposition, since“every proposition in mathematics must belong to a calculus ofmathematics” (PG 376).

This radical position on decidability results in various radical andcounter-intuitive statements about unrestricted mathematicalquantification, mathematical induction, and, especially, thesense of a newly proved mathematical proposition. Inparticular, Wittgenstein asserts that uncontroversial mathematicalconjectures, such as Goldbach’s Conjecture (hereafter‘GC’) and the erstwhile conjecture “Fermat’sLast Theorem” (hereafter ‘FLT’), have no sense (or,perhaps, nodeterminate sense) and that theunsystematic proof of such a conjecture gives it a sense thatit didn’t previously have (PG 374) because

it’s unintelligible that I should admit, when I’ve got theproof, that it’s a proof of preciselythis proposition,or of the induction meant by this proposition. (PR§155)

Thus Fermat’s [Last Theorem] makes nosense until I cansearch for a solution to the equation in cardinal numbers.And ‘search’ must always mean: search systematically.Meandering about in infinite space on the look-out for a gold ring isno kind of search. (PR §150)
I say: the so-called ‘Fermat’s Last Theorem’isn’t a proposition. (Not even in the sense of a proposition ofarithmetic.) Rather, it corresponds to an induction. (PR§189)

To see how Fermat’s Last Theorem isn’t a proposition andhow itmight correspond to an induction, we need to examineWittgenstein’s account of mathematical induction.

2.4 Wittgenstein’s Intermediate Account of Mathematical Induction and Algorithmic Decidability

Given that one cannot quantify over an infinite mathematical domain,the question arises: What, if anything, doesanynumber-theoretic proof by mathematical induction actuallyprove?

On the standard view, a proof by mathematical induction has thefollowing paradigmatic form.

Inductive Base:	\(\phi(1)\)
Inductive Step:	\(\forall n(\phi(n) \rightarrow \phi(n + 1))\)
Conclusion:	\(\forall n\phi(n)\)

If, however,“\(\forall n\phi(n)\)” isnot ameaningful (genuine) mathematical proposition, what are we to make ofthis proof?

Wittgenstein’s initial answer to this question is decidedlyenigmatic. “An induction is the expression for arithmeticalgenerality”, but “induction isn’t itself aproposition” (PR §129).

We are not saying that when \(f(1)\) holds and when \(f(c + 1)\)follows from \(f(c)\), the proposition \(f(x)\) isthereforetrue of all cardinal numbers: but: “the proposition \(f(x)\)holds for all cardinal numbers”means “it holdsfor \(x = 1\), and \(f(c + 1)\) follows from \(f(c)\)”.(PG 406)

In a proof by mathematical induction, we do no actually prove the‘proposition’ [e.g., \(\forall n\phi(n)\)] that iscustomarily construed as theconclusion of the proof(PG 406, 374;PR §164), rather thispseudo-proposition or ‘statement’ stands‘proxy’ for the “infinite possibility” (i.e.,“the induction”) that we come to‘see’ by means of the proof (WVC 135).“I want to say”, Wittgenstein concludes, that “onceyou’ve got the induction, it’s all over”(PG 407). Thus, on Wittgenstein’s account, a particularproof by mathematical induction should be understood in the followingway.

Inductive Base:	\(\phi(1)\)
Inductive Step:	\(\phi(n) \rightarrow \phi(n + 1)\)
Proxy Statement:	\(\phi(m)\)

Here the ‘conclusion’ of an inductive proof [i.e.,“what is to be proved” (PR §164)] uses‘\(m\)’ rather than‘\(n\)’ to indicate that‘\(m\)’ stands for anyparticular number, while‘\(n\)’ stands for anyarbitrary number. ForWittgenstein, theproxy statement“\(\phi(m)\)”isnot a mathematical proposition that “assert[s] itsgenerality” (PR §168), it is aneliminable pseudo-proposition standing proxy for the provedinductive base and inductive step. Though an inductive proofcannot prove “the infinite possibility ofapplication” (PR §163), it enables us “toperceive” that adirect proof of anyparticular proposition can be constructed (PR§165). For example, once we have proved“\(\phi(1)\)”and“\(\phi(n) \rightarrow \phi(n + 1)\)”, we need notreiteratemodus ponens \(m - 1\) times to prove theparticular proposition“\(\phi(m)\)” (PR§164). The direct proof of, say,“\(\phi\)(714)”(i.e., without 713 iterations ofmodus ponens) “cannothave a still better proof, say, by my carrying out the derivation asfar as this proposition itself” (PR §165).

A second, very important impetus for Wittgenstein’s radicallyconstructivist position on mathematical induction is his rejection ofanundecidable mathematical proposition.

In discussions of the provability of mathematical propositions it issometimes said that there are substantial propositions of mathematicswhose truth or falsehood must remain undecided. What the people whosay that don’t realize is that such propositions,if wecan use them and want to call them “propositions”, are notat all the same as what are called “propositions” in othercases; because a proof alters the grammar of a proposition.(PG 367)

In this passage, Wittgenstein is alluding to Brouwer, who, as early as1907 and 1908, states, first, that “the question of the validityof the principium tertii exclusi is equivalent to the questionwhether unsolvable mathematical problems exist”,second, that “[t]here is not a shred of a proof for theconviction… that there exist no unsolvable mathematicalproblems”, and, third, that there are meaningfulpropositions/‘questions’, such as “Do thereoccur in the decimal expansion of \(\pi\) infinitely many pairs ofconsecutive equal digits?”, to which the Law of theExcluded Middle does not applybecause “it must beconsidered as uncertain whether problems like [this] aresolvable” (Brouwer, 1908 [1975: 109–110]). “Afortiori it is not certain that any mathematical problem can either besolved or proved to be unsolvable”, Brouwer says (1907 [1975:79]), “though HILBERT, in ‘Mathematische Probleme’,believes that every mathematician is deeply convinced ofit”.

Wittgenstein takes the same data and, in a way, draws the oppositeconclusion. If, as Brouwer says, we areuncertain whether allor some “mathematical problems” are solvable, then weknow that we donot have in hand an applicabledecision procedure, which means that the alleged mathematicalpropositions arenot decidable, here and now. “What‘mathematical questions’ share with genuinequestions”, Wittgenstein says (PR §151), “issimply that they can be answered”. This means that if we do notknow how to decide an expression, then we do not know how tomake it either proved (true) or refuted (false), which meansthat the Law of the Excluded Middle “doesn’t apply”and, therefore, that our expression isnot a mathematicalproposition.

Together, Wittgenstein’s finitism and his criterion ofalgorithmic decidability shed considerable light on his highlycontroversial remarks about putativelymeaningful conjecturessuch as FLT and GC. GC is not a mathematical proposition because we donotknow how to decide it, and if someone like G. H. Hardysays that he ‘believes’ GC is true (PG 381;LFM 123;PI §578), we must answer that s/heonly “has a hunch about the possibilities of extension of thepresent system” (LFM 139)—that one can onlybelieve such an expression is ‘correct’ if oneknowshow to prove it. The only sense in which GC (or FLT)can be proved is that it can “correspond to aproof byinduction”, which means that the unproved inductive step (e.g.,“\(G(n) \rightarrow G(n + 1)\)”) and the expression“\(\forall nG(n)\)” are not mathematical propositionsbecause we have no algorithmic means of looking for an induction(PG 367). A “general proposition” is senselessprior to an inductive proof “because the question would onlyhave made sense if a general method of decision had been knownbefore the particular proof was discovered”(PG 402). Unproved ‘inductions’ or inductivesteps are not meaningful propositions because the Law of the ExcludedMiddle does not hold in the sense that we do not know of a decisionprocedure by means of which we can prove or refute the expression(PG 400;WVC 82).

This position, however, seems to rob us of any reason to search for a‘decision’ of a meaningless ‘expression’ suchas GC. The intermediate Wittgenstein says only that “[a]mathematician is… guided by… certain analogies with theprevious system” and that there is nothing “wrong orillegitimate if anyone concerns himself with Fermat’s LastTheorem” (WVC 144).

If e.g. I have a method for looking at integers that satisfy theequation \(x^2 + y^2 = z^2\), then the formula \(x^{n} + y^n = z^{n}\)may stimulate me. I may let a formula stimulate me. Thus I shall say,Here there is astimulus—but not aquestion.Mathematical problems are always such stimuli. (WVC 144, Jan.1, 1931)

More specifically, a mathematician may let a senseless conjecture suchas FLT stimulate her/him if s/he wishes to know whether a calculus canbe extended without altering its axioms or rules (LFM139).

What is here going [o]n [in an attempt to decide GC] is anunsystematic attempt at constructing a calculus. If the attempt issuccessful, I shall again have a calculus in front of me,only adifferent one from the calculus I have been using so far.(WVC 174–75; Sept. 21, 1931; italics added)

If, e.g., we succeed in proving GC by mathematical induction (i.e., weprove“\(G(1)\)” and“\(G(n) \rightarrow G(n +1)\)”), we will then have a proof of the inductive step, butsince the inductive step was not algorithmically decidable beforehand(PR §§148, 155, 157;PG 380), inconstructing the proof we have constructed anew calculus, anewcalculating machine (WVC 106) in which wenow know how to use this new “machine-part”(RFM VI, §13) (i.e., the unsystematically provedinductive step). Before the proof, the inductive step is not amathematical proposition with sense (in a particular calculus),whereas after the proof the inductive stepis a mathematicalproposition, with a new, determinate sense, in a newly createdcalculus. This demarcation of expressions without mathematical senseand proved or refuted propositions, each with a determinate sense in aparticular calculus, is a view that Wittgenstein articulates in myriaddifferent ways from 1929 through 1944.

Whether or not it is ultimately defensible—and this is anabsolutely crucial question for Wittgenstein’s Philosophy ofMathematics—this strongly counter-intuitive aspect ofWittgenstein’s account of algorithmic decidability, proof, andthesense of a mathematical proposition is a piece with hisrejection ofpredeterminacy in mathematics. Even in the casewhere we algorithmically decide a mathematical proposition, theconnections thereby made do not pre-exist the algorithmic decision,which means that even when we have a “mathematicalquestion” that we decide by decision procedure, the expressiononly has a determinate sensequa proposition when it isdecided. On Wittgenstein’s account, both middle and later,“[a] new proof gives the proposition a place in a newsystem” (RFM VI, §13), it “locates it in thewhole system of calculations”, though it “does notmention, certainly does not describe, the whole system of calculationthat stands behind the proposition and gives it sense”(RFM VI, §11).

Wittgenstein’s unorthodox position here is a type ofstructuralism that partially results from his rejection ofmathematical semantics. We erroneously think, e.g., that GC has afully determinate sense because, given “the misleading way inwhich the mode of expression of word-language represents the sense ofmathematical propositions” (PG 375), we call to mindfalse pictures and mistaken,referential conceptions ofmathematical propositions whereby GC isabout a mathematicalreality and so has just a determinate sense as “There existintelligent beings elsewhere in the universe” (i.e., aproposition thatis determinately true or false, whether ornot we ever know its truth-value). Wittgenstein breaks with thistradition, inall of its forms, stressing that, inmathematics, unlike the realm of contingent (or empirical)propositions, “if I am to know what a proposition likeFermat’s last theorem says”, I must know itscriterion of truth. Unlike the criterion of truth for anempirical proposition, which can be knownbefore theproposition is decided, we cannot know the criterion of truth for anundecided mathematical proposition, though we are “acquaintedwith criteria for the truth ofsimilar propositions”(RFM VI, §13).

2.5 Wittgenstein’s Intermediate Account of Irrational Numbers

The intermediate Wittgenstein spends a great deal of time wrestlingwith real and irrational numbers. There are two distinct reasons forthis.

First, thereal reason many of us are unwilling to abandonthe notion of the actual infinite in mathematics is the prevalentconception of an irrational number as anecessarily infiniteextension. “The confusion in the concept of the ‘actualinfinite’arises” (italics added), saysWittgenstein (PG 471),

from the unclear concept of irrational number, that is, from the factthat logically very different things are called ‘irrationalnumbers’ without any clear limit being given to the concept.

Second, and more fundamentally, the intermediate Wittgenstein wrestleswith irrationals in such detail because he opposes foundationalism andespecially its concept of a “gaplessmathematicalcontinuum”, its concept of acomprehensive theory ofthe real numbers (Han 2010), and set theoretical conceptions and‘proofs’ as a foundation for arithmetic, real numbertheory, and mathematics as a whole. Indeed, Wittgenstein’sdiscussion of irrationals is one with his critique of set theory, for,as he says, “[m]athematics is ridden through and through withthe pernicious idioms of set theory”, such as “the waypeople speak of a line as composed of points”, when, in fact,“[a] line is a law and isn’t composed of anything atall” (PR §173;PR §§181, 183,& 191;PG 373, 460, 461, & 473).

2.5.1 Wittgenstein’s Anti-Foundationalism and Genuine Irrational Numbers

Since, on Wittgenstein’s terms, mathematics consists exclusivelyof extensions and intensions (i.e., ‘rules’ or‘laws’), an irrational is only an extension insofar as itis a sign (i.e., a ‘numeral’, such as‘\(\sqrt{2}\)’ or‘\(\pi\)’). Given that thereis no such thing as an infinite mathematicalextension, itfollows that an irrational number is not a uniqueinfiniteexpansion, but rather a unique recursive rule orlaw(PR §181) that yields rational numbers (PR§186;PR §180).

The rule for working out places of \(\sqrt{2}\) is itself the numeralfor the irrational number; and the reason I here speak of a‘number’ is that I can calculate with these signs (certainrules for the construction of rational numbers) just as I can withrational numbers themselves. (PG 484)

Due, however, to his anti-foundationalism, Wittgenstein takes theradical position that not all recursive real numbers (i.e., computablenumbers) are genuine real numbers—a position that distinguisheshis view from even Brouwer’s.

The problem, as Wittgenstein sees it, is that mathematicians,especially foundationalists (e.g., set theorists), have sought toaccommodate physical continuity by a theory that‘describes’ the mathematical continuum (PR§171). When, for example, we think of continuous motion and the(mere) density of the rationals, we reason that if an object movescontinuously from A to B, and it travelsonly the distancesmarked by “rational points”, then it mustskipsome distances (intervals, or points)not marked by rationalnumbers. But if an object in continuous motion travels distances thatcannot be commensurately measured by rationals alone, theremust be ‘gaps’ between the rationals (PG 460),and so we must fill them, first, with recursive irrationals, and then,because “the set ofall recursive irrationals”still leaves gaps, with “lawless irrationals”.

[T]he enigma of the continuum arises because language misleads us intoapplying to it a picture that doesn’t fit. Set theory preservesthe inappropriate picture of something discontinuous, but makesstatements about it that contradict the picture, under the impressionthat it is breaking with prejudices; whereas what should really havebeen done is to point out that the picture just doesn’tfit… (PG 471)

We add nothing that is needed to the differential and integral calculiby ‘completing’ a theory of real numbers withpseudo-irrationals and lawless irrationals, first because there are nogaps on the number line (PR §§181, 183, & 191;PG 373, 460, 461, & 473;WVC 35) and, second,because these alleged irrational numbers are not needed for a theoryof the ‘continuum’ simply because there is no mathematicalcontinuum. As the later Wittgenstein says (RFM V, §32),“[t]he picture of the number line is an absolutely natural oneup to a certain point; that is to say so long as it is not used for ageneral theory of real numbers”. We have gone awry bymisconstruing the nature of the geometrical line as a continuouscollection of points, each with an associated real number, which hastaken us well beyond the ‘natural’ picture of the numberline in search of a “general theory of real numbers” (Han2010).

Thus, the principal reason Wittgenstein rejects certain constructive(computable) numbers is that they are unnecessary creations whichengender conceptual confusions in mathematics (especially set theory).One of Wittgenstein’s main aims in his lengthy discussions ofrational numbers and pseudo-irrationals is to show thatpseudo-irrationals, which are allegedly needed for the mathematicalcontinuum, are not needed at all.

To this end, Wittgenstein demands (a) that a real number must be“compar[able] with any rational number taken at random”(i.e., “it can be established whether it is greater than, lessthan, or equal to a rational number” (PR §191))and (b) that “[a] number must measure in and of itself”and if a ‘number’ “leaves it to the rationals, wehave no need of it” (PR §191) (Frascolla 1980:242–243; Shanker 1987: 186–192; Da Silva 1993:93–94; Marion 1995a: 162, 164; Rodych 1999b, 281–291;Lampert 2009).

To demonstrate that some recursive (computable) reals are not genuinereal numbers because they fail to satisfy (a) and (b), Wittgensteindefines the putative recursive real number

\[\substack{5 \rightarrow 3 \\ \sqrt{2}}\]

as the rule “Construct the decimal expansion for \(\sqrt{2}\),replacing every occurrence of a ‘5’ with a‘3’” (PR §182); he similarly defines\(\pi '\) as

\[\substack{7 \rightarrow 3\\ \pi}\]

(PR §186) and, in a later work, redefines \(\pi '\)as

\[\substack{777 \rightarrow 000 \\ \pi}\]

(PG 475).

Although a pseudo-irrational such as \(\pi '\) (on either definition)is “as unambiguous as … \(\pi\) or \(\sqrt{2}\)”(PG 476), it is ‘homeless’ according toWittgenstein because, instead of using “the idioms ofarithmetic” (PR §186), it is dependent upon theparticular ‘incidental’ notation of a particular system(i.e., in some particular base) (PR §188;PR§182; andPG 475). If we speak of variousbase-notational systems, we might say that \(\pi\) belongs toall systems, while \(\pi '\) belongs only to one, which showsthat \(\pi '\) is not a genuine irrational because “therecan’t be irrational numbers of different types”(PR §180). Furthermore, pseudo-irrationals donot measure because they are homeless, artificialconstructions parasitic upon numbers which have a natural place in acalculus that can be used to measure. We simply do not need theseaberrations, because they are not sufficiently comparable to rationalsand genuine irrationals. They arenot irrational numbersaccording to Wittgenstein’s criteria, which define, Wittgensteininterestingly asserts, “precisely what has been meant or lookedfor under the name ‘irrational number’” (PR§191).

For exactly the same reason, if we define a “lawlessirrational” as either (a) anon-rule-governed,non-periodic, infinite expansion in some base, or (b) a“free-choice sequence”, Wittgenstein rejects“lawless irrationals” because, insofar as they are notrule-governed, they are not comparable to rationals (or irrationals)and they are not needed.

[W]e cannot say that the decimal fractions developed in accordancewith a law still need supplementing by an infinite set of irregularinfinite decimal fractions that would be ‘brushed under thecarpet’ if we were torestrict ourselves to thosegenerated by a law,

Wittgenstein argues, for “[w]here is there such an infinitedecimal that is generated by no law” “[a]nd how would wenotice that it was missing?” (PR §181; cf.PG 473, 483–84). Similarly, a free-choice sequence,like a recipe for “endless bisection” or “endlessdicing”, is not an infinitely complicatedmathematicallaw (or rule), but rather no law at all, for after eachindividual throw of a coin, the point remains “infinitelyindeterminate” (PR §186). For closely relatedreasons, Wittgenstein ridicules the Multiplicative Axiom (Axiom ofChoice) both in the middle period (PR §146) and in thelatter period (RFM V, §25; VII, §33).

2.5.2 Wittgenstein’s Real Number Essentialism and the Dangers of Set Theory

Superficially, at least, it seems as if Wittgenstein is offering anessentialist argument for the conclusion that real number arithmeticshould not be extended in such-and-such a way. Such anessentialist account of real and irrational numbers seems toconflict with the actual freedom mathematicians have to extend andinvent, with Wittgenstein’s intermediate claim (PG 334)that “[f]or [him] one calculus is as good as another”, andwith Wittgenstein’s acceptance of complex and imaginary numbers.Wittgenstein’s foundationalist critic (e.g., set theorist) willundoubtedly say that we have extended the term “irrationalnumber” to lawless and pseudo-irrationals because they areneeded for the mathematical continuum and because such“conceivable numbers” are much more like rule-governedirrationals than rationals.

Though Wittgenstein stresses differences where others see similarities(LFM 15), in his intermediate attacks on pseudo-irrationalsand foundationalism, he is not just emphasizing differences, he isattacking set theory’s “pernicious idioms”(PR §173) and its “crudest imaginablemisinterpretation of its own calculus” (PG469–70) in an attempt to dissolve “misunderstandingswithout which [set theory] would never have been invented”,since it is “of no other use” (LFM 16–17).Complex and imaginary numbers have grown organically withinmathematics, and they have proved their mettle in scientificapplications, but pseudo-irrationals areinorganic creationsinvented solely for the sake of mistaken foundationalist aims.Wittgenstein’s main point isnot that we cannot createfurther recursive real numbers—indeed, we can create as many aswe want—his point is that we can only really speak of differentsystems (sets) of real numbers (RFM II, §33)that are enumerable by a rule, and any attempt to speak of “theset of all real numbers” or any piecemeal attempt to add orconsider new recursive reals (e.g., diagonal numbers) is a uselessand/or futile endeavour based on foundational misconceptions. Indeed,in 1930 manuscript and typescript (hereafter MS and TS, respectively)passages on irrationals and Cantor’s diagonal, which were notincluded inPR orPG, Wittgenstein says: “Theconcept ‘irrational number’ is a dangerouspseudo-concept” (MS 108, 176; 1930; TS 210, 29; 1930). As weshall see in the next section, on Wittgenstein’s account, if wedo not understand irrationals rightly, wecannot but engenderthe mistakes that constitute set theory.

2.6 Wittgenstein’s Intermediate Critique of Set Theory

Wittgenstein’s critique of set theory begins somewhat benignlyin theTractatus, where he denounces Logicism and says(6.031) that “[t]he theory of classes is completely superfluousin mathematics” because, at least in part, “the generalityrequired in mathematics is not accidental generality”. In hismiddle period, Wittgenstein begins a full-out assault on set theorythat never abates. Set theory, he says, is “utternonsense” (PR §§145, 174; WVC 102;PG 464, 470), ‘wrong’ (PR §174),and ‘laughable’ (PG 464); its “perniciousidioms” (PR §173) mislead us and the crudestpossible misinterpretation is the very impetus of its invention(Hintikka 1993: 24, 27).

Wittgenstein’s intermediate critique of transfinite set theory(hereafter “set theory”) has two main components: (1) hisdiscussion of the intension-extension distinction, and (2) hiscriticism of non-denumerabilityas cardinality. Late in themiddle period, Wittgenstein seems to become more aware of theunbearable conflict between hisstrong formalism (PG334) and his denigration of set theory as a purely formal,non-mathematical calculus (Rodych 1997: 217–219),which, as we shall see inSection 3.5, leads to the use of an extra-mathematical application criterion todemarcate transfinite set theory (and other purely formal sign-games)from mathematical calculi.

2.6.1 Intensions, Extensions, and the Fictitious Symbolism of Set Theory

The search for a comprehensive theory of the real numbers andmathematical continuity has led to a “fictitioussymbolism” (PR §174).

Set theory attempts to grasp the infinite at a more general level thanthe investigation of the laws of the real numbers. It says that youcan’t grasp the actual infinite by means of mathematicalsymbolism at all and therefore it can only be described and notrepresented. … One might say of this theory that it buys a pigin a poke. Let the infinite accommodate itself in this box as best itcan. (PG 468; cf.PR §170)

As Wittgenstein puts it at (PG 461),

the mistake in the set-theoretical approach consists time and again intreating laws and enumerations (lists) as essentially the same kind ofthing and arranging them in parallel series so that one fills in gapsleft by the other.

This is a mistake because it is ‘nonsense’ to say“we cannot enumerate all the numbers of a set, but we can give adescription”, for “[t]he one is not a substitute for theother” (WVC 102; June 19, 1930); “thereisn’t a dualism [of] the law and the infinite series obeyingit” (PR §180).

“Set theory is wrong” and nonsensical (PR§174), says Wittgenstein, because it presupposes a fictitioussymbolism of infinite signs (PG 469) instead of an actualsymbolism with finite signs. The grand intimation of set theory, whichbegins with “Dirichlet’s concept of a function”(WVC 102–03), is that we canin principlerepresent an infinite set by an enumeration, but because of human orphysical limitations, we will insteaddescribe itintensionally. But, says Wittgenstein, “[t]here can’t bepossibility and actuality in mathematics”, for mathematics is anactual calculus, which “is concerned only with thesigns with which itactually operates” (PG469). As Wittgenstein puts it at (PR §159), the factthat “we can’t describe mathematics, we can only doit” in and “of itself abolishes every ‘settheory’”.

Perhaps the best example of this phenomenon is Dedekind, who in givinghis ‘definition’ of an “infinite class” as“a class which is similar to a proper subclass of itself”(PG 464), “tried todescribe an infiniteclass” (PG 463). If, however, we try to apply this‘definition’ to a particular class in order to ascertainwhether it is finite or infinite, the attempt is‘laughable’ if we apply it to afinite class,such as “a certain row of trees”, and it is‘nonsense’ if we apply it to “an infiniteclass”, for we cannot even attempt “to co-ordinateit” (PG 464), because “the relation \(m = 2n\)[does not] correlate the class of all numbers with one of itssubclasses” (PR §141), it is an “infiniteprocess” which “correlates any arbitrary number withanother”. So, although wecan use \(m = 2n\) on therule for generating the naturals (i.e., our domain) andthereby construct the pairs (2,1), (4,2), (6,3), (8,4), etc., in doingso we do not correlate twoinfinite sets or extensions(WVC 103). If we try to apply Dedekind’s definition asacriterion for determining whether a given set is infiniteby establishing a 1–1 correspondence between two inductive rulesfor generating “infinite extensions”, one of which is an“extensional subset” of the other, we can’t possiblylearn anything we didn’t already know when we applied the‘criterion’ to two inductive rules. If Dedekind or anyoneelse insists on calling an inductive rule an “infiniteset”, he and we must still mark the categorical differencebetween such a set and a finite set with a determinate, finitecardinality.

Indeed, on Wittgenstein’s account, the failure to properlydistinguish mathematical extensions and intensions is the root causeof the mistaken interpretation of Cantor’s diagonal proof as aproof of the existence of infinite sets of lesser and greatercardinality.

2.6.2 Against Non-Denumerability

Wittgenstein’s criticism of non-denumerability is primarilyimplicit during the middle period. Only after 1937 does he provideconcrete arguments purporting to show, e.g., that Cantor’sdiagonalcannot prove that some infinite sets have greater‘multiplicity’ than others.

Nonetheless, the intermediate Wittgenstein clearly rejects the notionthat a non-denumerably infinite set is greater in cardinality than adenumerably infinite set.

When people say ‘The set of all transcendental numbers isgreater than that of algebraic numbers’, that’s nonsense.The set is of a different kind. It isn’t ‘no longer’denumerable, it’s simply not denumerable! (PR§174)

As with his intermediate views on genuine irrationals and theMultiplicative Axiom, Wittgenstein here looks at the diagonal proof ofthe non-denumerability of “the set of transcendentalnumbers” as one that shows only that transcendental numberscannot be recursively enumerated. It is nonsense, he says, to go fromthe warranted conclusion that these numbers are not, in principle,enumerable to the conclusion that theset of transcendentalnumbers is greater in cardinality than the set of algebraic numbers,which is recursively enumerable. What we have here are two verydifferent conceptions of a number-type. In the case of algebraicnumbers, we have a decision procedure for determining of any givennumber whether or not it is algebraic,and we have a methodof enumerating the algebraic numbers such that we canseethat ‘each’ algebraic number “will be”enumerated. In the case of transcendental numbers, on the other hand,we have proofs that some numbers are transcendental (i.e.,non-algebraic),and we have a proof that we cannotrecursively enumerate each and every thing we would call a“transcendental number”.

At (PG 461), Wittgenstein similarly speaks of settheory’s “mathematical pseudo-concepts” leading to afundamental difficulty, which begins when we unconsciously presupposethat there is sense to the idea of ordering the rationals bysize—“that theattempt isthinkable”—and culminates in similarly thinking that it ispossible to enumeratethe real numbers, which we thendiscover is impossible.

Though the intermediate Wittgenstein certainly seems highly criticalof the alleged proof that some infinite sets (e.g., the reals) aregreater in cardinality than other infinite sets, and though hediscusses the “diagonal procedure” in February 1929 and inJune 1930 (MS 106, 266; MS 108, 180), along with a diagonal diagram,these and other early-middle ruminations did not make it into thetypescripts for eitherPR orPG. As we shall see inSection 3.4, the later Wittgenstein analyzes Cantor’s diagonal and claims ofnon-denumerability in some detail.

3. The Later Wittgenstein on Mathematics: Some Preliminaries

The first and most important thing to note about Wittgenstein’slater Philosophy of Mathematics is thatRFM, first publishedin 1956, consists ofselections taken from a number ofmanuscripts (1937–1944), most of one large typescript (1938),and three short typescripts (1938), each of which constitutes anAppendix to (RFM I). For this reason and because somemanuscripts containing much material on mathematics (e.g., MS 123)were not used at all forRFM, philosophers have not been able to readWittgenstein’s later remarks on mathematics as they were writtenin the manscripts used forRFM and they have not had access(until the 2000–2001 release of theNachlass on CD-ROM)to much of Wittgenstein’s later work on mathematics. It must beemphasized, therefore, that thisEncyclopedia article isbeing written during a transitional period. Until philosophers haveused theNachlass to build a comprehensive picture ofWittgenstein’s complete and evolving Philosophy of Mathematics,we will not be able to say definitively which views the laterWittgenstein retained, which he changed, and which he dropped. In theinterim, this article will outline Wittgenstein’s laterPhilosophy of Mathematics, drawing primarily onRFM, to amuch lesser extentLFM (1939 Cambridge lectures), and, wherepossible, previously unpublished material in Wittgenstein’sNachlass.

It should also be noted at the outset that commentators disagree aboutthe continuity of Wittgenstein’s middle and later Philosophiesof Mathematics. Some argue that the later views are significantlydifferent from the intermediate views (Frascolla 1994; Gerrard 1991:127, 131–32; Floyd 2005: 105–106), while others arguethat, for the most part, Wittgenstein’s Philosophy ofMathematics evolves from the middle to the later period withoutsignificant changes or renunciations (Wrigley 1993; Marion 1998). The remainder of this article adopts thesecond interpretation, explicating Wittgenstein’s laterPhilosophy of Mathematics as largely continuous with his intermediateviews, except for the important introduction of an extra-mathematicalapplication criterion.

3.1 Mathematics as a Human Invention

Perhaps the most important constant in Wittgenstein’s Philosophyof Mathematics, middle and late, is that he consistently maintainsthat mathematics is our, human invention, and that, indeed, everythingin mathematics is invented. Just as the middle Wittgenstein says that“[w]emake mathematics”, the later Wittgensteinsays that we ‘invent’ mathematics (RFM I,§168; II, §38; V, §§5, 9 and 11;PG469–70) and that “the mathematician is not a discoverer:he is an inventor” (RFM, Appendix II, §2;(LFM 22, 82). Nothingexists mathematically unlessand until we have invented it.

In arguing against mathematical discovery, Wittgenstein is not justrejecting Platonism, he is also rejecting a rather standardphilosophical view according to which human beings invent mathematicalcalculi, but once a calculus has been invented, we thereafter discoverfinitely many of its infinitely many provable and true theorems. AsWittgenstein himself asks (RFM IV, §48), “might itnot be said that therules lead this way, even if no one wentit?” If “someone produced a proof [of‘Goldbach’s theorem’]”,“[c]ouldn’t one say”, Wittgenstein asks(LFM 144), “that thepossibility of this proofwas a fact in the realms of mathematical reality”—that“[i]n order [to] find it, it must in some sense bethere”—“[i]t must be a possiblestructure”?

Unlike many or most philosophers of mathematics, Wittgenstein resiststhe ‘Yes’ answer that we discover truths about amathematical calculus thatcome into existence the moment weinvent the calculus (PR §141;PG 283, 466;LFM 139). Wittgenstein rejects the modal reification ofpossibility as actuality—that provability and constructibilityare (actual) facts—by arguing that it is at the very leastwrong-headed to say with the Platonist that because “a straightlinecan be drawn between any two points,… the linealready exists even if no one has drawn it”—to say“[w]hat in the ordinary world we call a possibility is in thegeometrical world a reality” (LFM 144;RFM I,§21). One might as well say, Wittgenstein suggests (PG374), that “chess only had to bediscovered, it wasalways there!”

At MS 122 (3v; Oct. 18, 1939), Wittgenstein once again emphasizes thedifference between illusory mathematical discovery and genuinemathematical invention.

I want to get away from the formulation: “I now know more aboutthe calculus”, and replace it with “I now have a differentcalculus”. The sense of this is always to keep beforeone’s eyes the full scale of the gulf between a mathematicalknowing and non-mathematical knowing.^[3]

And as with the middle period, the later Wittgenstein similarly says(MS 121, 27r; May 27, 1938) that “[i]t helps if one says: theproof of the Fermat proposition is not to be discovered, but to beinvented”.

The difference between the ‘anthropological’ and themathematical account is that in the first we are not tempted to speakof ‘mathematical facts’, but rather that in this accountthefacts are never mathematical ones, never makemathematical propositions true or false. (MS 117, 263; March15, 1940)

There are no mathematical facts just as there are no (genuine)mathematical propositions. Repeating his intermediate view, the laterWittgenstein says (MS 121, 71v; 27 Dec., 1938): “Mathematicsconsists of [calculi | calculations], not of propositions”. Thisradical constructivist conception of mathematics prompts Wittgensteinto make notorious remarks—remarks that virtually no one elsewould make—such as the infamous (RFM V, §9):“However queer it sounds, the further expansion of an irrationalnumber is a further expansion of mathematics”.

3.1.1 Wittgenstein’s Later Anti-Platonism: The Natural History of Numbers and the Vacuity of Platonism

As in the middle period, the later Wittgenstein maintains thatmathematics is essentially syntactical and non-referential, which, inand of itself, makes Wittgenstein’s philosophy of mathematicsanti-Platonist insofar as Platonism is the view that mathematicalterms and propositionsrefer to objects and/or facts and thatmathematical propositions aretrue by virtue of agreeing withmathematical facts.

The later Wittgenstein, however, wishes to ‘warn’ us thatour thinking is saturated with the idea of “[a]rithmetic as thenatural history (mineralogy) of numbers” (RFM IV,§11). When, for instance, Wittgenstein discusses the claim thatfractions cannot be ordered by magnitude, he says that this sounds‘remarkable’ in a way that a mundane proposition of thedifferential calculus does not, for the latter proposition isassociated with an application in physics,

whereasthis proposition … seems to[solely] concern… the natural history ofmathematical objects themselves. (RFM II, §40)

Wittgenstein stresses that he is trying to ‘warn’ usagainst this ‘aspect’—the idea that the foregoingproposition about fractions “introduces us to the mysteries ofthe mathematical world”, which exists somewhere as a completedtotality, awaiting our prodding and our discoveries. The fact that weregard mathematical propositions as being about mathematical objectsand mathematical investigation “as the exploration of theseobjects” is “already mathematical alchemy”, claimsWittgenstein (RFM V, §16), since

it is not possible to appeal to the meaning[Bedeutung] of the signs inmathematics,… because it is only mathematics that gives themtheir meaning [Bedeutung].

Platonism isdangerously misleading, according toWittgenstein, because it suggests a picture ofpre-existence,predetermination and discovery that is completely at oddswith what we find if we actually examine and describe mathematics andmathematical activity. “I should like to be able todescribe”, says Wittgenstein (RFM IV, §13),“how it comes about that mathematics appears to us now as thenatural history of the domain of numbers, now again as a collection ofrules”.

Wittgenstein, however, doesnot endeavour torefutePlatonism. His aim, instead, is to clarify what Platonism is and whatit says, implicitly and explicitly (including variants of Platonismthat claim, e.g., that if a proposition isprovable in anaxiom system, then there already exists a path [i.e., a proof] fromthe axioms to that proposition (RFM I, §21; Marion 1998:13–14, 226; Steiner 2000:334). Platonism is either “a mere truism” (LFM239), Wittgenstein says, or it is a ‘picture’ consistingof “an infinity of shadowy worlds” (LFM 145),which, as such, lacks ‘utility’ (cf.PI§254) because it explains nothing and it misleads at everyturn.

3.2 Wittgenstein’s Later Finitistic Constructivism

Though commentators and critics do not agree as to whether the laterWittgenstein is still a finitist and whether, if he is, his finitismis as radical as his intermediate rejection of unbounded mathematicalquantification (Maddy 1986: 300–301, 310), the overwhelmingevidence indicates that the later Wittgenstein still rejects theactual infinite (RFM V, §21;Zettel §274,1947) and infinite mathematical extensions.

The first, and perhaps most definitive, indication that the laterWittgenstein maintains his finitism is his continued and consistentinsistence that irrational numbers are rules for constructing finiteexpansions,not infinite mathematical extensions. “Theconcepts of infinite decimals in mathematical propositions are notconcepts of series”, says Wittgenstein (RFM V,§19), “but of the unlimited technique of expansion ofseries”. We are misled by “[t]he extensional definitionsof functions, of real numbers etc”. (RFM V, §35),but once we recognize the Dedekind cut as “an extensionalimage”, we see that we are not “led to\(\sqrt{2}\) by way of the concept of a cut” (RFM V,§34). On the later Wittgenstein’s account, there simply isnoproperty, norule, nosystematic meansof defining each and every irrational numberintensionally,which means there isno criterion “for the irrationalnumbers beingcomplete” (PR §181).

As in his intermediate position, the later Wittgenstein claims that‘\(\aleph_0\)’ and “infinite series” get theirmathematical uses from the use of ‘infinity’ in ordinarylanguage (RFM II, §60). Although, in ordinary language,we often use ‘infinite’ and “infinitely many”as answers to the question “how many?”, and though weassociate infinity with the enormously large, the principaluse we make of ‘infinite’ and‘infinity’ is to speak ofthe unlimited(RFM V, §14) and unlimitedtechniques(RFM II, §45;PI §218). This fact isbrought out by the fact “that the technique of learning\(\aleph_0\) numerals is different from the technique of learning100,000 numerals” (LFM 31). When we say, e.g., that“there are an infinite number of even numbers” we meanthat we have a mathematical technique or rule for generating evennumbers which islimitless, which is markedly different froma limited technique or rule for generating a finite number of numbers,such as 1–100,000,000. “We learn an endlesstechnique”, says Wittgenstein (RFM V, §19),“but what is in question here is not some giganticextension”.

An infinite sequence, for example, is not a gigantic extension becauseit is not an extension, and‘\(\aleph_0\)’ is not acardinal number, for “how is this picture connected with thecalculus”, given that “its connexion is not thatof the picture | | | | with 4” (i.e., given that‘\(\aleph_0\)’ is not connected to a (finite) extension)?This shows, says Wittgenstein (RFM II, §58), that weought to avoid the word ‘infinite’ in mathematics whereverit seems to give a meaning to the calculus, rather than acquiring itsmeaning from the calculus and its use in the calculus. Once we seethat the calculus contains nothing infinite, we should not be‘disappointed’ (RFM II, §60), but simplynote (RFM II, §59) that it is not “reallynecessary… to conjure up the picture of the infinite (of theenormously big)”.

A second strong indication that the later Wittgenstein maintains hisfinitism is his continued and consistent treatment of‘propositions’ of the type “There are threeconsecutive 7s in the decimal expansion of \(\pi\)” (hereafter ‘PIC’).^[4] In the middle period, PIC (and its putative negation, \(\neg\)PIC,namely, “It is not the case that there are three consecutive 7sin the decimal expansion of \(\pi\)”) isnot ameaningful mathematical “statement at all” (WVC81–82: note 1). On Wittgenstein’s intermediate view,PIC—like FLT, GC, and the Fundamental Theorem ofAlgebra—isnot a mathematical proposition because we donot have in hand an applicable decision procedure by which we candecide it in a particular calculus. For this reason, we can onlymeaningfully statefinitistic propositions regarding theexpansion of \(\pi\), such as “There exist three consecutive 7sin the first 10,000 places of the expansion of \(\pi\)”(WVC 71; 81–82, note 1).

The later Wittgenstein maintains this position in various passages inRFM (Bernays 1959: 11–12). For example, to someone who says that since “therule of expansiondetermine[\(s\)] the seriescompletely”, “it must implicitly determineallquestions about the structure of the series”, Wittgensteinreplies: “Here you are thinking of finite series”(RFM V, §11). If PIC were amathematicalquestion (or problem)—if it were finitisticallyrestricted—it would be algorithmically decidable, which it isnot (RFM V, §21;LFM 31–32, 111, 170;WVC 102–03). As Wittgenstein says at (RFM V,§9): “The question… changes its status, when itbecomes decidable”, “[f]or a connexion is made then, whichformerlywas not there”. And if, moreover, one invokesthe Law of the Excluded Middle to establish that PIC is a mathematicalproposition—i.e., by saying that one of these “twopictures… must correspond to the fact” (RFM V,§10)—one simply begs the question (RFM V,§12), for if we have doubts about the mathematical status of PIC,we will not be swayed by a person who asserts “PIC \(\vee\neg\)PIC” (RFM VII, §41; V, §13).

Wittgenstein’s finitism, constructivism, and conception ofmathematical decidability are interestingly connected at (RFMVII, §41, par. 2–5).

What harm is done e.g. by saying that God knowsallirrational numbers? Or: that they are already there, even though weonly know certain of them? Why are these pictures not harmless?
For one thing, they hide certain problems.— (MS 124: 139; March16, 1944)
Suppose that people go on and on calculating the expansion of \(\pi\).So God, who knows everything, knows whether they will have reached‘777’ by the end of the world. But can hisomniscience decide whether theywould have reachedit after the end of the world? It cannot. I want to say: Even God candetermine something mathematical only by mathematics. Even for him themere rule of expansion cannot decide anything that it does not decidefor us.
We might put it like this: if the rule for the expansion has beengiven us, acalculation can tell us that there is a‘2’ at the fifth place. Could God have known this, withoutthe calculation, purely from the rule of expansion? I want to say: No.(MS 124, pp. 175–176; March 23–24, 1944)

What Wittgenstein means here is that God’s omnisciencemight, by calculation, find that ‘777’ occurs atthe interval [\(n,n+2\)], but, on the other hand, God might go oncalculating forever without ‘777’ ever turning up. Since\(\pi\) is not acompleted infinite extension that can becompletely surveyed by an omniscient being (i.e., it is not a factthat can be known by an omniscient mind), even God has only the rule,and so God’s omniscience is no advantage in this case(LFM 103–04; cf. Weyl 1921 [1998: 97]). Like us, withour modest minds, an omniscient mind (i.e., God) can only calculatethe expansion of \(\pi\) to some \(n\)^th decimalplace—where our \(n\) is minute and God’s \(n\) is(relatively) enormous—and at no \(n\)^th decimal placecouldany mind rightly conclude that because‘777’ has not turned up, it, therefore, will never turnup.

3.3 The Later Wittgenstein on Decidability and Algorithmic Decidability

On one fairly standard interpretation, the later Wittgenstein saysthat “true in calculus \(\Gamma\)” is identical to“provable in calculus \(\Gamma\)” and, therefore, that amathematical proposition of calculus \(\Gamma\) is a concatenation ofsigns that is either provable (in principle) or refutable (inprinciple) in calculus \(\Gamma\) (Goodstein 1972: 279, 282; Anderson1958: 487; Klenk 1976: 13; Frascolla 1994: 59). On thisinterpretation, the later Wittgenstein precludes undecidablemathematical propositions, but he allows that someundecidedexpressions are propositionsof a calculus because they aredecidable in principle (i.e., in the absence of a known, applicabledecision procedure).

There is considerable evidence, however, that the later Wittgensteinmaintains his intermediate position that an expression is a meaningfulmathematical proposition onlywithin a given calculus andiff we knowingly have in hand an applicable and effectivedecision procedure by means of which we can decide it. For example,though Wittgenstein vacillates between “provable in PM”and “proved in PM” at (RFM App. III, §6,§8), he does so in order to use the former to consider thealleged conclusion of Gödel’s proof (i.e., that there existtrue but unprovable mathematical propositions), which he then rebutswith his own identification of “true in calculus\(\Gamma\)” with “proved in calculus\(\Gamma\)” (i.e.,not with “provable incalculus \(\Gamma\)”) (Wang 1991: 253; Rodych 1999a: 177). Thisconstrual is corroborated by numerous passages in which Wittgensteinrejects the received view that a provable but unprovedproposition is true, as he does when he asserts that (RFMIII, §31, 1939) a proof “makes new connexions”,“[i]t does not establish that they are there” because“they do not exist until it makes them”, and when he says(RFM VII, §10, 1941) that “[a] new proof gives theproposition a place in a new system”. Furthermore, as we havejust seen, Wittgenstein rejects PIC as a non-proposition on thegrounds that it is not algorithmically decidable, while admittingfinitistic versions of PIC because they are algorithmicallydecidable.

Perhaps the most compelling evidence that the later Wittgensteinmaintains algorithmic decidability as his criterion for a mathematicalproposition lies in the fact that, at (RFM V, §9, 1942),he says in two distinct ways that a mathematical‘question’ canbecome decidable and that whenthishappens, a new connexion is ‘made’which previously did not exist. Indeed, Wittgenstein cautions usagainst appearances by saying that “itlooks as if aground for the decision were already there”, when, in fact,“it has yet to be invented”. These passages stronglymilitate against the claim that the later Wittgenstein grants thatproposition \(\phi\) is decidable in calculus \(\Gamma\) iff it isprovable or refutablein principle. Moreover, if Wittgensteinheldthis position, he would claim, contra (RFM V,§9), that a question or proposition does notbecomedecidable since it simply (always)is decidable. If it isprovable, and we simply don’t yet know this to be the case,therealready is a connection between, say, our axioms andrules and the proposition in question. What Wittgenstein says,however, is that the modalitiesprovable andrefutable are shadowy forms of reality—that possibilityis not actuality in mathematics (PR §§141, 144,172;PG 281, 283, 299, 371, 466, 469;LFM 139).Thus, the later Wittgenstein agrees with the intermediate Wittgensteinthat the only sense in which anundecided mathematicalproposition (RFM VII, §40, 1944) can bedecidable is in the sense that weknow how to decideit by means of an applicable decision procedure.

3.4 Wittgenstein’s Later Critique of Set Theory: Non-Enumerability vs. Non-Denumerability

Largely a product of hisanti-foundationalism and hiscriticism of the extension-intension conflation, Wittgenstein’slater critique of set theory is highly consonant with his intermediatecritique (PR §§109, 168;PG 334, 369, 469;LFM 172, 224, 229; andRFM III, §43, 46, 85,90; VII, §16). Given that mathematics is a “MOTLEY oftechniques of proof” (RFM III, §46), it does notrequire a foundation (RFM VII, §16) and it cannot begiven aself-evident foundation (PR §160;WVC 34 & 62;RFM IV, §3). Since set theorywas invented to provide mathematics with a foundation, it is,minimally, unnecessary.

Even if set theory is unnecessary, it still might constitute a solidfoundation for mathematics. In his core criticism of set theory,however, the later Wittgenstein denies this, saying that the diagonalproof does not prove non-denumerability, for “[i]t means nothingto say: ‘Therefore the X numbers are notdenumerable’” (RFM II, §10). When the diagonal isconstrued as aproof of greater and lesser infinite sets itis a “puffed-up proof”, which, as Poincaré argued(1913: 61–62), purports to prove or show more than “itsmeans allow it” (RFM II, §21).

If it were said: Consideration of the diagonal procedure shews you that theconcept ‘real number’ has much less analogy withthe concept ‘cardinal number’ than we, being misled bycertain analogies, are inclined to believe, that would have a good and honest sense. But just theopposite happens: one pretends to compare the‘set’ of real numbers in magnitude with that of cardinalnumbers. The difference in kind between the two conceptions isrepresented, by a skew form of expression, as difference of extension.I believe, and hope, that a future generation will laugh at this hocuspocus. (RFM II, §22)
The sickness of a time is cured by an alteration in the mode of lifeof human beings… (RFM II, §23)

The “hocus pocus” of the diagonal proof rests, as alwaysfor Wittgenstein, on a conflation of extension and intension, on thefailure to properly distinguish setsas rules for generatingextensions and (finite) extensions. By way of this confusion “adifference in kind” (i.e., unlimited rule vs. finite extension)“is represented by a skew form of expression”, namely as adifference in thecardinality of twoinfiniteextensions. Not only can the diagonalnot prove that oneinfinite set is greater in cardinality than another infinite set,according to Wittgenstein,nothing could prove this, simplybecause “infinite sets” are notextensions, andhence notinfinite extensions. But instead of interpretingCantor’s diagonal proof honestly, we take the proof to“show there are numbers bigger than the infinite”, which“sets the whole mind in a whirl, and gives the pleasant feelingof paradox” (LFM 16–17)—a “giddinessattacks us when we think of certain theorems in settheory”—“when we are performing a piece of logicalsleight-of-hand” (PI §412; §426; 1945). Thisgiddiness and pleasant feeling of paradox, says Wittgenstein(LFM 16), “may be the chief reason [set theory] wasinvented”.

Though Cantor’s diagonal is not aproof ofnon-denumerability, when it is expressed in aconstructivemanner, as Wittgenstein himself expresses it at (RFM II,§1), “it gives sense to the mathematical proposition thatthe number so-and-so is different from all those of the system”(RFM II, §29). That is, the proof provesnon-enumerability: it proves that for any givendefinite real number concept (e.g., recursive real), onecannot enumerate ‘all’ such numbers because one can alwaysconstruct a diagonal number, which falls under the same concept and isnot in the enumeration. “One might say”, Wittgensteinsays,

I call number-concept X non-denumerable if it has been stipulatedthat, whatever numbers falling under this concept you arrange in aseries, the diagonal number of this series is also to fall under thatconcept. (RFM II, §10; cf. II, §§30, 31,13)

One lesson to be learned from this, according to Wittgenstein(RFM II, §33), is that “there arediversesystems of irrational points to be found in the numberline”, each of which can be given by a recursive rule, but“no system of irrational numbers”, and “also nosuper-system, no ‘set of irrational numbers’ ofhigher-order infinity”. Cantor has shown that we can construct“infinitely many” diverse systems of irrational numbers,but we cannot construct anexhaustive system ofallthe irrational numbers (RFM II, §29). As Wittgensteinsays at (MS 121, 71r; Dec. 27, 1938), three pages after the passageused for (RFM II, §57):

If you now call the Cantorian procedure one for producing a new realnumber, you will now no longer be inclined to speak ofa system ofall real numbers. (italics added)

From Cantor’s proof, however, set theorists erroneously concludethat “the set of irrational numbers” is greater inmultiplicity than any enumeration of irrationals (or the set ofrationals), when the only conclusion to draw is that there is no suchthing asthe set of allthe irrational numbers. Thetruly dangerous aspect to ‘propositions’ such as“The real numbers cannot be arranged in a series” and“The set… is not denumerable” is that they makeconcept formation [i.e., ourinvention] “look like afact of nature” (i.e., something wediscover)(RFM II §§16, 37). At best, we have a vague idea ofthe concept of “real number”, but only if we restrict thisidea to “recursive real number” and only if we recognizethathaving the concept does not meanhaving a setof all recursive real numbers.

3.5 Extra-Mathematical Application as a Necessary Condition of Mathematical Meaningfulness

The principal and most significant change from the middle to laterwritings on mathematics is Wittgenstein’s (re-)introduction ofan extra-mathematical application criterion, which is used todistinguish mere “sign-games” from mathematicallanguage-games. “[I]t is essential to mathematics that its signsare also employed inmufti”, Wittgenstein states, for

[i]t is the use outside mathematics, and so themeaning[Bedeutung] of the signs, that makes thesign-game into mathematics. (i.e., a mathematical“language-game”;RFM V, §2, 1942;LFM 140–141, 169–70)

As Wittgenstein says at (RFM V, §41, 1943),

[c]oncepts which occur in ‘necessary’ propositionsmust also occur and have a meaning[Bedeutung] in non-necessary ones. (italicsadded)

If two proofs prove the same proposition, says Wittgenstein, thismeans that “both demonstrate it as a suitable instrument for thesame purpose”, which “is an allusion tosomethingoutside mathematics” (RFM VII, §10, 1941;italics added).

As we have seen, this criterion was present in theTractatus(6.211), but noticeably absent in the middle period. The reason forthis absence is probably that the intermediate Wittgenstein wanted tostress that in mathematics everything is syntax and nothing ismeaning. Hence, in his criticisms of Hilbert’s‘contentual’ mathematics (Hilbert 1925) andBrouwer’s reliance upon intuition to determine the meaningfulcontent of (especially undecidable) mathematical propositions,Wittgenstein couched his finitistic constructivism in strongformalism, emphasizing that a mathematical calculus does not need anextra-mathematical application (PR §109;WVC105).

There seem to be two reasons why the later Wittgenstein reintroducesextra-mathematical application as a necessary condition of amathematical language-game. First, the later Wittgenstein hasan even greater interest in theuse of natural and formallanguages in diverse “forms of life” (PI§23), which prompts him to emphasize that, in many cases, amathematical ‘proposition’ functions as if it were anempirical proposition “hardened into a rule” (RFMVI, §23) and that mathematics plays diverse applied roles in manyforms of human activity (e.g., science, technology, predictions).Second, the extra-mathematical application criterion relieves thetension between Wittgenstein’s intermediate critique of settheory and his strong formalism according to which “one calculusis as good as another” (PG 334). By demarcatingmathematical language-games from non-mathematical sign-games,Wittgenstein can now claim that, “for the time being”, settheory is merely a formal sign-game.

These considerations may lead us to say that \(2^{\aleph_0} \gt\aleph_0\).
That is to say: we canmake the considerations lead us tothat.
Or: we can saythis and givethis as our reason.
But if we do say it—what are we to do next? In what practice isthis proposition anchored? It is for the time being a piece ofmathematical architecture which hangs in the air, and looks as if itwere, let us say, an architrave, but not supported by anything andsupporting nothing. (RFM II, §35)

It is not that Wittgenstein’s later criticisms of set theorychange, it is, rather, that once we see that set theory has noextra-mathematical application, we will focus on its calculations,proofs, and prose and “subject theinterest of thecalculations to atest” (RFM II, §62). Bymeans ofWittgenstein’s “immenselyimportant” ‘investigation’ (LFM 103), wewill find, Wittgenstein expects, that set theory is uninteresting(e.g., that the non-enumerability of “the reals” isuninteresting and useless) and that our entire interest in it lies inthe ‘charm’ of the mistaken prose interpretation of itsproofs (LFM 16). More importantly, though there is “asolid core to all [its] glistening concept-formations”(RFM V, §16), once we see it as “as a mistake ofideas”, we will see that propositions such as“\(2^{\aleph_0} \gt \aleph_0\)” are not anchored in anextra-mathematical practice, that “Cantor’sparadise” “is not a paradise”, and we willthen leave “of [our] own accord” (LFM103).

It must be emphasized, however, that the later Wittgenstein stillmaintains that the operations within a mathematical calculus arepurely formal, syntactical operations governed by rules of syntax(i.e., the solid core of formalism).

It is of course clear that the mathematician, in so far as he reallyis ‘playing a game’…[is]acting inaccordance with certain rules. (RFM V, §1)
To say mathematics is a game is supposed to mean: in proving, we neednever appeal to the meaning [Bedeutung] of thesigns, that is to their extra-mathematical application. (RFMV, §4)

Where, during the middle period, Wittgenstein speaks of“arithmetic [as] a kind of geometry” at (PR§109 & §111), the later Wittgenstein similarly speaks of“the geometry of proofs” (RFM I, App. III,§14), the “geometrical cogency” of proofs(RFM III, §43), and a “geometricalapplication” according to which the “transformation ofsigns” in accordance with “transformation-rules”(RFM VI, §2, 1941) shows that “when mathematics isdivested of all content, it would remain that certain signs can beconstructed from others according to certain rules”(RFM III, §38). Hence, the question whether aconcatenation of signs is a proposition of a givenmathematical calculus (i.e., a calculus with anextra-mathematical application) is still an internal, syntacticalquestion, which we can answer with knowledge of the proofs anddecision procedures of the calculus.

3.6 Wittgenstein on Gödel and Undecidable Mathematical Propositions

RFM is perhaps most (in)famous for Wittgenstein’s(RFM App. III) treatment of “true but unprovable”mathematical propositions. Early reviewers said that “[t]hearguments are wild” (Kreisel 1958: 153), that the passages“on Gödel’s theorem… are of poor quality orcontain definite errors” (Dummett 1959: 324), and that(RFM App. III) “throws no light on Gödel’swork” (Goodstein 1957: 551). “Wittgenstein seems to wantto legislate [[q]uestions about completeness] out ofexistence”, Anderson said, (1958: 486–87) when, in fact,he certainly cannot dispose of Gödel’s demonstrations“by confusing truth with provability”. Additionally,Bernays, Anderson (1958: 486), and Kreisel (1958: 153–54)claimed that Wittgenstein failed to appreciate“Gödel’s quite explicit premiss of the consistency ofthe considered formal system” (Bernays 1959: 15), therebyfailing to appreciate the conditional nature of Gödel’sFirst Incompleteness Theorem. On the reading of these four earlyexpert reviewers, Wittgenstein failed to understand Gödel’sTheorem because he failed to understand the mechanics ofGödel’s proof and he erroneously thought he could refute orundermine Gödel’s proof simply by identifying “trueinPM” (i.e.,Principia Mathematica) with“proved/provable inPM”.

Interestingly, we now have two pieces of evidence (Kreisel 1998: 119;Rodych 2003: 282, 307) that Wittgenstein wrote (RFM App. III)in 1937–38 after readingonly the informal,‘casual’ (MS 126, 126–127; Dec. 13, 1942)introduction of (Gödel 1931) and that, therefore, his use of aself-referential proposition as the “true but unprovableproposition” may be based on Gödel’s introductory,informal statements, namely that “the undecidable proposition[\(R(q);q\)] states… that [\(R(q);q\)] is not provable”(1931: 598) and that “[\(R(q);q\)] says about itself that it isnot provable” (1931: 599). Perplexingly, only two of the fourfamous reviewers even mentioned Wittgenstein’s (RFMVII, §§19, 21–22, 1941)) explicit remarks on‘Gödel’s’ First Incompleteness Theorem (Bernays1959: 2; Anderson 1958: 487), which, though flawed, capture thenumber-theoretic nature of the Gödelian propositionandthe functioning of Gödel-numbering, probably because Wittgensteinhad by then read or skimmed the body of Gödel’s 1931 paper.

The first thing to note, therefore, about (RFM App. III) isthat Wittgenstein mistakenly thinks—again, perhaps becauseWittgenstein had read only Gödel’s Introduction—(a)that Gödel proves that there are true but unprovable propositionsofPM (when, in fact, Gödel syntactically proves that ifPM is \(\omega\)-consistent, the Gödelian proposition isundecidable inPM) and (b) that Gödel’s proof usesa self-referential proposition to semantically show that there aretrue but unprovable propositions ofPM.

For this reason, Wittgenstein has two main aims in (RFM App.III): (1) to refute or undermine,on its own terms, thealleged Gödel proof of true but unprovable propositions ofPM, and (2) to show that, on his own terms, where “truein calculus \(\Gamma\)” is identified with“proved in calculus \(\Gamma\)”, the very idea ofa true but unprovable proposition of calculus \(\Gamma\) ismeaningless.

Thus, at (RFM App. III, §8) (hereafter simply‘§8’), Wittgenstein begins his presentation of whathe takes to be Gödel’s proof by having someone say:

I have constructed a proposition (I will use ‘P’ todesignate it) in Russell’s symbolism, and by means of certaindefinitions and transformations it can be so interpreted that it says:‘P is not provable in Russell’s system’.

That is, Wittgenstein’s Gödelian constructs a propositionthat is semanticallyself-referential and which specificallysays of itself that it is not provable inPM. With thiserroneous, self-referential propositionP [used also at(§10), (§11), (§17), (§18)], Wittgenstein presentsa proof-sketch very similar to Gödel’s owninformal semantic proof ‘sketch’ in theIntroduction of his famous paper (1931: 598).

Must I not say that this proposition on the one hand is true, and onthe other hand is unprovable? For suppose it were false; then it istrue that it is provable. And that surely cannot be! And if it isproved, then it is proved that it is not provable. Thus it can only betrue, but unprovable. (§8)

The reasoning here is a doublereductio. Assume (a) thatP must either be true or false in Russell’s system, and (b)thatP must either be provable or unprovable in Russell’ssystem. If (a),P must betrue, for if we suppose thatP is false, sinceP says of itself that it is unprovable,“it is true that it is provable”, and if it is provable,it must be true (which is a contradiction), and hence, given whatP means or says, it is true thatP is unprovable (which is acontradiction). Second, if (b),P must be unprovable, for ifP“is proved, then it is proved that it is not provable”,which is a contradiction (i.e.,P is provableand notprovable inPM). It follows thatP “can only betrue, but unprovable”.

To refute or undermine this ‘proof’, Wittgenstein saysthat if you have proved \(\neg P\), you have proved thatP isprovable (i.e., since you have proved that it isnot the casethatP is not provable in Russell’s system), and “youwill now presumably give up the interpretation that it isunprovable” (i.e., ‘P is not provable inRussell’s system’), since the contradiction is only provedif we use or retain this self-referential interpretation (§8). Onthe other hand, Wittgenstein argues (§8), ‘[i]f you assumethat the proposition is provable in Russell’s system, that meansit is truein the Russell sense, and the interpretation“P is not provable” again has to be given up’,because, once again, it is only the self-referential interpretationthat engenders a contradiction. Thus, Wittgenstein’s‘refutation’ of “Gödel’s proof”consists in showing that no contradiction arises if we donotinterpret ‘P’ as ‘P is not provable inRussell’s system’—indeed, without thisinterpretation, a proof ofP does not yield a proof of \(\neg P\)and a proof of \(\neg P\) does not yield a proof ofP. In otherwords, the mistake in the proof is the mistaken assumption that amathematical proposition ‘P’ “can be sointerpreted that it says: ‘P is not provable inRussell’s system’”. As Wittgenstein says at(§11), “[t]hat is what comes of making up suchsentences”.

This ‘refutation’ of “Gödel’sproof” is perfectly consistent with Wittgenstein’ssyntactical conception of mathematics (i.e., wherein mathematicalpropositions have no meaning and hence cannot have the‘requisite’ self-referential meaning) and with what hesays before and after (§8), where his main aim is to show (2)that,on his own terms, since “true in calculus\(\Gamma\)” is identical with “proved in calculus\(\Gamma\)”, the very idea of a true but unprovable propositionof calculus \(\Gamma\) is a contradiction-in-terms.

To show (2), Wittgenstein begins by asking (§5), what he takes tobe, the central question, namely, “Are there true propositionsin Russell’s system, which cannot be proved in hissystem?”. To address this question, he asks “What iscalled a true proposition in Russell’s system…?”,which he succinctly answers (§6): “‘p’is true = p”. Wittgenstein then clarifies this answerby reformulating the second question of (§5) as “Under whatcircumstances is a proposition asserted in Russell’s game [i.e.,system]?”, which he then answers by saying: “the answeris: at the end of one of his proofs, or as a ‘fundamentallaw’ (Pp.)” (§6). This, in a nutshell, isWittgenstein’s conception of “mathematical truth”: atrue proposition ofPM is an axiom or a proved proposition,which means that “true inPM” is identical with,and therefore can be supplanted by, “proved inPM”.

Having explicated, to his satisfaction at least, the only real,non-illusory notion of “true inPM”, Wittgensteinanswers the (§8) question “Must I not say that thisproposition… is true, and… unprovable?”negatively by (re)stating his own (§§5–6)conception of “true inPM” as“proved/provable inPM”:

‘True in Russell’s system’ means, as was said:proved in Russell’s system; and ‘false in Russell’ssystem’ means: the opposite has been proved in Russell’ssystem.

This answer is given in a slightly different way at (§7) whereWittgenstein asks “may there not be true propositions which arewritten in this [Russell’s] symbolism, but are not provable inRussell’s system?”, and then answers “‘Truepropositions’, hence propositions which are true inanother system, i.e. can rightly be asserted in anothergame”. In light of what he says in (§§5, 6, and 8),Wittgenstein’s (§7) point is that if a proposition is‘written’ in “Russell’s symbolism” andit is true,it must be proved/provable in another system,since that is what “mathematical truth” is. Analogously(§8), “if the proposition is supposed to be false in someother than the Russell sense, then it does not contradict this for itto be proved in Russell’s sense”, for “[w]hat iscalled ‘losing’ in chess may constitute winning in anothergame”. This textual evidence certainly suggests, as Andersonalmost said, that Wittgenstein rejects a true but unprovablemathematical proposition as a contradiction-in-terms on the groundsthat “true in calculus \(\Gamma\)” means nothing more (andnothing less) than “proved in calculus \(\Gamma\)”.

On this (natural) interpretation of (RFM App. III), the earlyreviewers’ conclusion that Wittgenstein fails to understand themechanics of Gödel’s argument seems reasonable. First,Wittgenstein erroneously thinks that Gödel’s proof isessentially semantical and that it uses andrequires aself-referential proposition. Second, Wittgenstein says (§14)that “[a] contradiction is unusable” for “aprediction” that “that such-and-such construction isimpossible” (i.e., thatP is unprovable inPM),which, superficially at least, seems toindicate that Wittgenstein fails to appreciate the “consistencyassumption” of Gödel’s proof (Kreisel, Bernays,Anderson).

If, in fact, Wittgenstein did not read and/or failed to understandGödel’s proof through at least 1941, how would he haveresponded if and when he understood it as (at least) a proof of theundecidability ofP inPM on the assumption ofPM’s consistency? Given his syntactical conception ofmathematics, even with the extra-mathematical application criterion,he would simply say thatP,qua expression syntacticallyindependent ofPM, is not a proposition ofPM, andif it is syntactically independent of all existent mathematicallanguage-games, it is not a mathematical proposition. Moreover, thereseem to be no compelling non-semantical reasons—eitherintra-systemic or extra-mathematical—for Wittgenstein toaccommodateP by including it inPM or by adopting anon-syntactical conception of mathematical truth (such as Tarski-truth(Steiner 2000)). Indeed, Wittgenstein questions the intra-systemic andextra-mathematicalusability ofP in various discussionsof Gödel in theNachlass and, at(§19), he emphatically says that one cannot “make the truthof the assertion [‘P’ or ‘ThereforeP’] plausible to me, since you can make no use of it exceptto do these bits of legerdemain”.

After the initial, scathing reviews ofRFM, very littleattention was paid to Wittgenstein’s (RFM App. III andRFM VII, §§21–22) discussions ofGödel’s First Incompleteness Theorem (Klenk 1976: 13) untilShanker’s sympathetic (1988b). In the last 22 years, however,commentators and critics have offered various interpretations ofWittgenstein’s remarks on Gödel, some being largelysympathetic (Floyd 1995, 2001) and others offering a more mixedappraisal (Rodych 1999a, 2002, 2003; Steiner 2001; Priest 2004; Berto2009a). Recently, and perhaps most interestingly, Floyd & Putnam(2000) and Steiner (2001) have evoked new and interesting discussionsof Wittgenstein’s ruminations on undecidability, mathematicaltruth, and Gödel’s First Incompleteness Theorem (Rodych2003, 2006; Bays 2004; Sayward 2005; and Floyd & Putnam 2006).

4. The Impact of Philosophy of Mathematics on Mathematics

Though it is doubtful that all commentators will agree (Wrigley 1977:51; Baker & Hacker 1985: 345; Floyd 1991: 145, 143; 1995: 376;2005: 80; Maddy 1993: 55; Steiner 1996: 202–204), the followingpassage seems to capture Wittgenstein’sattitude to thePhilosophy of Mathematics and, in large part, the way in which heviewed his own work on mathematics.

What will distinguish the mathematicians of the future from those oftoday will really be a greater sensitivity, andthatwill—as it were—prune mathematics; since people will thenbe more intent on absolute clarity than on the discovery of newgames.
Philosophical clarity will have the same effect on the growth ofmathematics as sunlight has on the growth of potato shoots. (In a darkcellar they grow yards long.)
A mathematician is bound to be horrified by my mathematical comments,since he has always been trained to avoid indulging in thoughts anddoubts of the kind I develop. He has learned to regard them assomething contemptible and… he has acquired a revulsion fromthem as infantile. That is to say, I trot out all the problems that achild learning arithmetic, etc., finds difficult, the problems thateducation represses without solving. I say to those repressed doubts:you are quite correct, go on asking, demand clarification!(PG 381, 1932)

In his middle and later periods, Wittgenstein believes he is providingphilosophical clarity on aspects and parts of mathematics, onmathematical conceptions, and on philosophical conceptions ofmathematics. Lacking such clarity and not aiming for absolute clarity,mathematicians construct new games, sometimes because of amisconception of themeaning of their mathematicalpropositions and mathematical terms. Education and especially advancededucation in mathematics does not encourage clarity but ratherrepresses it—questions that deserve answers are either not askedor are dismissed. Mathematicians of the future, however, will be moresensitive and this will (repeatedly) prune mathematical extensions andinventions, since mathematicians will come to recognize that newextensions and creations (e.g., propositions of transfinite cardinalarithmetic) are not well-connected with the solid core of mathematicsor with real-world applications. Philosophical clarity will,eventually, enable mathematicians and philosophers to “get downto brass tacks” (PG 467).

Bibliography

Wittgenstein’s Writings

•	1913, “On Logic and How Not to Do It”,TheCambridge Review 34 (1912–13), 351; reprinted in BrianMcGuinness,Wittgenstein: A Life, Berkeley & Los Angeles,University of California Press: 169–170.
Tractatus	1922,Tractatus Logico-Philosophicus, London: Routledge & Kegan Paul, 1974; translated by D.F. Pears and B.F. McGuinness.
•	1929, “Some Remarks on Logical Form”,Proceedings of the Aristotelian Society, Supplementary Vol.9: 162–171.
•	1929,MS 106, inWittgenstein’s Nachlass:The Bergen Electronic Edition.
•	1930,MS 108, inWittgenstein’s Nachlass:The Bergen Electronic Edition.
•	1930,TS 210, inWittgenstein’s Nachlass:The Bergen Electronic Edition.
•	1937,MS 117 (1937, 1938, 1940), inWittgenstein’s Nachlass:The Bergen Electronic Edition.
•	1938-1939,MS 121, inWittgenstein’s Nachlass:The Bergen Electronic Edition.
•	1939-1940,MS 122, inWittgenstein’s Nachlass:The Bergen Electronic Edition.
•	1940-41,MS 123, inWittgenstein’s Nachlass:The Bergen Electronic Edition.
•	1944,MS 124 (1941, 1944), inWittgenstein’s Nachlass:The Bergen Electronic Edition.
•	1942-1943,MS 126, inWittgenstein’s Nachlass:The Bergen Electronic Edition.
PI	1953 [2001],Philosophical Investigations,3^rd Edition, Oxford: Blackwell Publishing; translated by G.E. M. Anscombe.
RFM	1956 [1978],Remarks on the Foundations of Mathematics,Revised Edition, Oxford: Basil Blackwell, G.H. von Wright, R. Rheesand G.E.M. Anscombe (eds.); translated by G.E.M Anscombe.
•	1966 [1999],Lectures & Conversations on Aesthetics,Psychology and Religious Belief, Cyril Barrett, (ed.), Oxford:Blackwell Publishers Ltd.
Zettel	1967,Zettel, Berkeley: University of California Press;G.E.M Anscombe and G.H. von Wright (Eds.); translated by G.E.M.Anscombe.
PG	1974,Philosophical Grammar, Oxford: Basil Blackwell;Rush Rhees, (ed.); translated by Anthony Kenny.
PR	1975,Philosophical Remarks, Oxford: Basil Blackwell;Rush Rhees, (ed.); translated by Raymond Hargreaves and Roger White.
•	1979a,Notebooks 1914–1916, Second Edition, G.H.von Wright and G. E. M. Anscombe (eds.), Oxford: Basil Blackwell.
•	1979b, “Notes on Logic” (1913), inNotebooks1914–1916, G.H. von Wright and G.E.M. Anscombe (eds.),Oxford: Basil Blackwell.
•	1980,Remarks on the Philosophy of Psychology, Vol. I,Chicago: University of Chicago Press, G.E.M. Anscombe and G.H. vonWright, (eds.), translated by G.E.M. Anscombe.
•	2000,Wittgenstein’s Nachlass:The Bergen Electronic Edition, Oxford: Oxford University Press.

Notes on Wittgenstein’s Lectures and Recorded Conversations

AWL	Ambrose, Alice, (ed.), 1979,Wittgenstein’s Lectures, Cambridge 1932–35: From theNotes of Alice Ambrose and Margaret Macdonald, Oxford: BasilBlackwell.
LFM	Diamond, Cora, (ed.), 1976,Wittgenstein’s Lectures on the Foundations ofMathematics, Ithaca, N.Y.: Cornell University Press.
LWL	Lee, Desmond, (ed.), 1980,Wittgenstein’s Lectures,Cambridge 1930–32: From the Notes of John King and DesmondLee, Oxford: Basil Blackwell.
WVC	Waismann, Friedrich, 1979,Wittgenstein and theVienna Circle, Oxford: Basil Blackwell; edited by B.F.McGuinness; translated by Joachim Schulte and B.F. McGuinness.

Secondary Sources and Relevant Primary Literature

Ambrose, Alice, 1935a, “Finitism in Mathematics (I)”,Mind, 44(174): 186–203.doi:10.1093/mind/XLIV.174.186
–––, 1935b, “Finitism in Mathematics(II)”,Mind, 44(175): 317–340.doi:10.1093/mind/XLIV.175.317
–––, 1972, “MathematicalGenerality”, in Ambrose and Lazerowitz 1972: 287–318.
–––, 1982, “Wittgenstein on MathematicalProof”,Mind, 91(362): 264–372.doi:10.1093/mind/XCI.362.264
Ambrose, Alice and Morris Lazerowitz (eds.), 1972,LudwigWittgenstein: Philosophy and Language, London: George Allen andUnwin Ltd.
Anderson, Alan Ross, 1958, “Mathematics and the‘Language Game’”,The Review ofMetaphysics, 11(3): 446–458; reprinted in Benacerraf &Putnam 1964b: 481–490.
Baker, Gordon and P.M.S. Hacker, 1985,Wittgenstein: Rules,Grammar and Necessity, Volume 2 of an Analytical Commentary onthePhilosophical Investigations, Oxford: Blackwell.
Bays, Timothy, 2004, “On Floyd and Putnam on Wittgenstein onGödel”,Journal of Philosophy, 101(4):197–210. doi:10.5840/jphil2004101422
Benacerraf, Paul and Hilary Putnam, 1964a,“Introduction”, in Benacerraf & Putnam 1964b:1–27.
Benacerraf, Paul and Hilary Putnam (eds.), 1964b,Philosophyof Mathematics, Englewood Cliffs, NJ: Prentice-Hall.
––– (eds.), 1983,Philosophy ofMathematics, second edition, Cambridge: Cambridge UniversityPress. Note there are considerable differences between the first andsecond editions.
Bernays, Paul, 1959, “Comments on LudwigWittgenstein’sRemarks on the Foundations ofMathematics”,Ratio, 2(1): 1–22.
Berto, Fransesco, 2009a, “The Gödel Paradox andWittgenstein’s Reasons”,Philosophia Mathematica,17(2): 208–219. doi:10.1093/philmat/nkp001
–––, 2009b, “Gödel versusWittgenstein and the Paraconsistent Interpretation”, in hisThere’s Something About Gödel: The Complete Guide tothe Incompleteness Theorem, West Sussex, UK: Wiley-Blackwell,Chapter 12, 189–213. doi:10.1002/9781444315028.ch12
Black, Max, 1964,A Companion to Wittgenstein’sTractatus, Ithaca, NY: Cornell University Press.
Black, Max, 1969, “Verificationism and Wittgenstein’sReflections on Mathematics”,Revue Internationale dePhilosophie, 23(88/89): 284–95; reprinted in Shanker 1986:68–76.
Brouwer, L.E.J., 1907 [1975],Over de Grondslagen derWiskunde (On the Foundations of Mathematics), Ph.D. thesis,Universiteit van Amsterdam. English translation in Brouwer 1975:11–101.
–––, 1908 [1975], “De onbetrouwbaarheidder logische principes” (The Unreliability of the LogicalPrinciples),Tijdschrift voor Wijsbegeerte, 2: 152–158.English translation in Brouwer 1975: 107–111.doi:10.1016/B978-0-7204-2076-0.50009-X
–––, 1929 [1998], “Mathematik,Wissenschaft und Sprache” (Mathematics, Science, and Language),Monatshefte für Mathematik und Physik, 36:153–164. English translation in Mancosu 1998: 45–53.doi:10.1007/BF02307611
–––, 1948 [1983], “Consciousness,Philosophy and Mathematics”,Proceedings of the 10thInternational Congress of Philosophy, Amsterdam 1948, Amsterdam:North-Holland, 3: 1235–1249. Reprinted in Benacerraf &Putnam 1983: 90–96. doi:10.1017/CBO9781139171519.005
–––, 1955, “The Effect of Intuitionism onClassical Algebra of Logic”,Proceedings of the Royal IrishAcademy, 57: 113–116.
–––, 1981,Brouwer’s CambridgeLectures on Intuitionism, Cambridge: Cambridge UniversityPress.
–––, 1975,Collected Works. I: Philosophyand Foundations of Mathematics, Arend Heyting (ed.), Amsterdam:North-Holland.
Coliva, Annalisa and Eva Picardi (eds.), 2004,WittgensteinToday, Padova: Il Poligrafo.
Conant, James, 1997, “On Wittgenstein’s Philosophy ofMathematics”,Proceedings of the Aristotelian Society,97(1): 195–222. doi:10.1111/1467-9264.00013
Crary, Alice and Rupert Read (eds.), 2000,The NewWittgenstein, London and New York: Routledge.
Da Silva, Jairo Jose, 1993, “Wittgenstein on IrrationalNumbers”, in Puhl 1993: 93–99.
Dawson, Ryan, 2016a, “Wittgenstein on Set Theory and theEnormously Big”,Philosophical Investigations, 39(4):313–334. doi:10.1111/phin.12107
–––, 2016b, “Was Wittgenstein Really aConstructivist About Mathematics?”,WittgensteinStudien, 7(1): 81–104. doi: 10.1515/witt-2016-0107
Dreben, Burton and Juliet Floyd, 1991, “Tautology: How NotTo Use A Word”,Synthese, 87(1): 23–49.doi:10.1007/BF00485329
Dummett, Michael, 1959, “Wittgenstein’s Philosophy ofMathematics”,The Philosophical Review, 68:324–348.
–––, 1978, “Reckonings: Wittgenstein onMathematics”,Encounter, 50(3): 63–68; reprintedin Shanker 1986: 111–120.
–––, 1994, “Wittgenstein on Necessity:Some Reflections”, in Peter Clark and Bob Hale (eds.),Reading Putnam, Cambridge, MA: Blackwell Publishers:49–65.
Finch, Henry Le Roy, 1977,Wittgenstein: The LaterPhilosophy, Atlantic Highlands, NJ: Humanities Press.
Floyd, Juliet, 1991, “Wittgenstein on 2, 2, 2…: TheOpening ofRemarks on the Foundations of Mathematics”,Synthese, 87(1): 143–180. doi:10.1007/BF00485332
–––, 1995, “On Saying What You Really Wantto Say: Wittgenstein, Gödel, and the Trisection of theAngle”, in Hintikka 1995: 373–425.
–––, 2000, “Wittgenstein, Mathematics andPhilosophy”, in Crary and Read 2000: 232–261.
–––, 2001, “Prose versus Proof:Wittgenstein on Gödel, Tarski, and Truth”,PhilosophiaMathematica, 9(3): 280–307.doi:10.1093/philmat/9.3.280
–––, 2002, “Number and Ascriptions ofNumber in Wittgenstein’sTractatus”, in Reck2002: 308–352.
–––, 2005, “Wittgenstein on Philosophy ofLogic and Mathematics”, in Stewart Shapiro (ed.),The OxfordHandbook of Philosophy of Logic and Mathematics, Oxford: OxfordUniversity Press: 75–128.
–––, 2012, “Wittgenstein’s DiagonalArgument: A Variation on Cantor and Turing”, in P. Dybjer, S.Lindström, E. Palmgren, and B.G. Sundholm (eds.),Epistemology versus Ontology: Essays on the Philosophy andFoundations of Mathematics in Honour of Per Martin-Löf,Dordrecht: Springer, 25–44.
–––, 2015, “Review of FelixMühlhölzer’s Does Mathematics Need aFoundation?”,Philosophia Mathematica, 23(2):255–276. doi:10.1093/philmat/nku037
Floyd, Juliet and Hilary Putnam, 2000, “A Note onWittgenstein’s ‘Notorious Paragraph’ about theGödel Theorem”,The Journal of Philosophy, 97(11):624–632. doi:10.2307/2678455
–––, 2006, “Bays, Steiner, andWittgenstein’s ‘Notorious’ Paragraph about theGödel Theorem”,The Journal of Philosophy, 103(2):101–110. doi:10.5840/jphil2006103228
Fogelin, Robert J., 1968, “Wittgenstein andIntuitionism”,American Philosophical Quarterly, 5(4):267–274.
–––, 1987 [1976],Wittgenstein, SecondEdition, New York: Routledge & Kegan Paul.
Frascolla, Pasquale, 1980, “The Constructivist Model inWittgenstein’s Philosophy of Mathematics”,RevistaFilosofia, 71: 297–306; reprinted in Shanker 1986:242–249.
–––, 1994,Wittgenstein’s Philosophyof Mathematics, London and New York: Routledge.
–––, 1997, “TheTractatus Systemof Arithmetic”,Synthese, 112(3): 353–378.doi:10.1023/A:1004952810156
–––, 1998, “The Early Wittgenstein’sLogicism”,Acta Analytica, 13(21): 133–137.
–––, 2004, “Wittgenstein on MathematicalProof”, in Coliva & Picardi 2004: 167–184.
Frege, Gottlob, 1959 [1884],The Foundations ofArithmetic, translated by J.L. Austin, Oxford: BasilBlackwell.
Garavaso, Pieranna, 1988, “Wittgenstein’s Philosophyof Mathematics: A Reply to Two Objections”,Southern Journalof Philosophy, 26(2): 179–191.doi:10.1111/j.2041-6962.1988.tb00450.x
Gerrard, Steve, 1991, “Wittgenstein’s Philosophies ofMathematics”,Synthese, 87(1) 125–142.doi:10.1007/BF00485331
–––, 1996, “A Philosophy of MathematicsBetween Two Camps”, in Sluga & Stern 1996:171–197.
–––, 2002, “One Wittgenstein?”, inReck 2002: 52–71.
Glock, Hans-Johann, 1996, “Necessity and Normativity”,in Sluga & Stern 1996: 198–225.
Gödel, Kurt, 1931, “On Formally UndecidablePropositions ofPrincipia Mathematica and Related SystemsI”, in van Heijenoort 1967: 596–616.
–––, 1953–1957, “Is MathematicsSyntax of Language?”, Version III, in Gödel 1995:334–356.
–––, 1995,Collected Works, Vol. III,S. Feferman, J. Dawson, Jr., W. Goldfarb, C. Parsons, and R. Solovay(eds.), Oxford: Oxford University Press.
Goldstein, Laurence, 1986, “The Development ofWittgenstein’s Views on Contradiction”,History andPhilosophy of Logic, 7(1): 43–56.doi:10.1080/01445348608837090
–––, 1989, “Wittgenstein andParaconsistency”, in Graham Priest, Richard Routley, and JeanNorman (eds.),Paraconsistent Logic: Essays on theInconsistent, Munich: Philosophia Verlag: 540–562.
Goodstein, R.L., 1957, “Critical Notice of Remarks on theFoundations of Mathematics”,Mind, 66(264):549–553. doi:10.1093/mind/LXVI.264.549
–––, 1972, “Wittgenstein’sPhilosophy of Mathematics”, in Ambrose and Lazerowitz 1972:271–86.
Hacker, P.M.S., 1986,Insight & Illusion. Themes in thePhilosophy of Wittgenstein, revised edition, Oxford: ClarendonPress.
Han, Daesuk, 2010, “Wittgenstein and the RealNumbers”,History and Philosophy of Logic, 31(3):219–245. doi:10.1080/01445340.2010.485350
van Heijenoort, Jean, (ed.), 1967,From Frege to Gödel: ASourcebook in Mathematical Logic, Cambridge, MA: HarvardUniversity Press.
Hilbert, David, 1925, “On the Infinite”, in vanHeijenoort 1967: 369–392.
Hintikka, Jaakko, 1993, “The OriginalSinn ofWittgenstein’s Philosophy of Mathematics”, in Puhl 1993:24–51.
–––, (ed.), 1995,From Dedekind toGödel: Essays on the Development of Mathematics, Dordrecht:Kluwer Academic Publishers.
Hintikka, Jaakko and K. Puhl (eds.), 1994,The BritishTradition in 20^th Century Philosophy: Papers of the17^th International Wittgenstein Symposium 1994,Kirchberg-am-Wechsel: Austrian Ludwig Wittgenstein Society.
Kielkopf, Charles F., 1970,Strict Finitism, The Hague:Mouton.
Klenk, V.H., 1976,Wittgenstein’s Philosophy ofMathematics, The Hague: Martinus Nijhoff.
Kreisel, Georg, 1958, “Wittgenstein’s Remarks on theFoundations of Mathematics”,British Journal for thePhilosophy of Science, 9(34): 135–57.doi:10.1093/bjps/IX.34.135
–––, 1998, “Second Thoughts Around Some ofGödel’s Writings:A Non-Academic Option”,Synthese, 114(1): 99–160.doi:10.1023/A:1005006610269
Kremer, Michael, 2002, “Mathematics and Meaning in theTractatus”,Philosophical Investigations,25(3): 272–303. doi:10.1111/1467-9205.00175
Kripke, Saul A., 1982,Wittgenstein on Rules and PrivateLanguage, Cambridge, MA: Harvard University Press.
Lampert, Timm, 2008, “Wittgenstein on the Infinity ofPrimes”,History and Philosophy of Logic, 29(3):272–303.
–––, 2009, “Wittgenstein onPseudo-Irrationals, Diagonal-Numbers and Decidability”, inLogica Yearbook 2008, Michal Peliš, (ed.), London:College Publications: 95–110.
–––, forthcoming, “Wittgenstein and Gödel:An Attempt to Make ‘Wittgenstein’s Objection’Reasonable”,Philosophia Mathematica, firstonline 03 August 2017. doi:org/10.1093/philmat/nkx017
Maddy, Penelope, 1986, “Mathematical Alchemy”,British Journal for the Philosophy of Science, 37(3):279–312. doi:10.1093/bjps/37.3.279
–––, 1993, “Wittgenstein’sAnti-Philosophy of Mathematics”, in Puhl 1993: 52–72.
Mancosu, Paolo, 1998,From Brouwer to Hilbert: The Debate onthe Foundations of Mathematics in the 1920s, Oxford: OxfordUniversity Press.
Mancosu, Paolo & Mathieu Marion, 2002,“Wittgenstein’s Constructivization of Euler’s Proofof the Infinity of Primes”, in Friedrich Stadler (ed.),TheVienna Circle and Logical Empiricism: Re-Evaluation and FuturePerspectives, (Vienna Circle Institute Yearbook, 10), Dordrecht:Kluwer: 171–188. doi:10.1007/0-306-48214-2_15
Marconi, Diego, 1984, “Wittgenstein on Contradiction and thePhilosophy of Paraconsistent Logic”,History of PhilosophyQuarterly, 1(3): 333–352.
Marion, Mathieu, 1993, “Wittgenstein and the Dark Cellar ofPlatonism”, in Puhl 1993: 110–118.
–––, 1995a, “Wittgenstein andFinitism”,Synthese, 105(2): 143–65.doi:10.1007/BF01064216
–––, 1995b, “Wittgenstein and Ramsey onIdentity”, in Hintikka 1995: 343–371.
–––, 1995c, “Kronecker’s ‘SafeHaven of Real Mathematics’,” in Marion & CohenQuebec Studies in the Philosophy of Science, Part I,Dordrecht: Kluwer: 135–87.
–––, 1998,Wittgenstein, Finitism, and theFoundations of Mathematics, Oxford: Clarendon Press.doi:10.1093/acprof:oso/9780199550470.001.0001
–––, 2003, “Wittgenstein andBrouwer”,Synthese, 137(1–2): 103–127.doi:10.1023/A:1026230917748
–––, 2004, “Wittgenstein on Mathematics:Constructivism or Constructivity?”, in Coliva & Picardi2004: 201–222.
–––, 2008, “Brouwer on‘Hypotheses’ and the Middle Wittgenstein”, inOne Hundred Years of Intuitionism, Mark van Atten, PascalBoldini, Michel Bourdreau, and Gerhard Heinzmann (eds.), Basel:Birkhauser: 96–114. doi:10.1007/978-3-7643-8653-5_7
–––, 2009, “Radical Anti-Realism,Wittgenstein and the Length of Proofs”,Synthese,171(3): 419–432. doi:10.1007/s11229-008-9315-9
McGuinness, Brian, 1988,Wittgenstein: A Life—YoungLudwig 1889–1921, Berkeley & Los Angeles: University ofCalifornia Press.
McGuinness, Brian and G.H. von Wright (eds.), 1995,LudwigWittgenstein: Cambridge Letters: Correspondence with Russell, Keynes,Moore, Ramsey, and Sraffa, Oxford: Blackwell Publishers Ltd.
Monk, Ray, 1990,Ludwig Wittgenstein: The Duty of Genius,New York: The Free Press.
Moore, A.W., 2003, “On the Right Track”,Mind, 112(446): 307–321.doi:10.1093/mind/112.446.307
Moore, G.E., 1955, “Wittgenstein’s Lectures in1930–33”,Mind, 64(253): 1–27.doi:10.1093/mind/LXIV.253.1
Morton, Adam and Stephen P. Stich, 1996,Benacerraf and HisCritics, Oxford: Blackwell.
Mühlhölzer, Felix, 2010,Braucht die Mathematik eineGrundlegung? Eine Kommentar des Teils III von WittgensteinsBemerkungen über die Grundlagen der Mathematik [DoesMathematics Need a Foundation? A Commentary on Part III ofWittgenstein’s Remarks on the Foundations of Mathematics],Frankfurt: Vittorio Klostermann.
Poincaré, Henri, 1913 [1963], “The Logic ofInfinity”,Mathematics and Science: Last Essays(Dernières pensées), John W. Bolduc (trans.),New York: Dover: 45–64.
Potter, Michael, 2000,Reason’s Nearest Kin:Philosophies of Arithmetic from Kant to Carnap, Oxford: OxfordUniversity Press. doi:10.1093/acprof:oso/9780199252619.001.0001
Priest, Graham, 2004, “Wittgenstein’s Remarks onGödel’s Theorem”, in Max Kölbel, and BernhardWeiss (eds.),Wittgenstein’s Lasting Significance,London: Routledge: 206–225.
Puhl, Klaus, (ed.), 1993,Wittgenstein’s Philosophy ofMathematics, Vienna: Verlag Holder-Pichler-Tempsky.
Putnam, Hilary, 1994,Words and Life (James Conant(ed.)), Cambridge, MA: Harvard University Press.
–––, 1994, “Rethinking MathematicalNecessity”, in hisWords and Life, James Conant (ed.),Cambridge, MA: Harvard University Press: 245–263; reprinted inCrary and Read 2000: 218–231.
–––, 1996, “On Wittgenstein’sPhilosophy of Mathematics”,Proceedings of the AristotelianSociety (Supplement), 70: 243–264.doi:10.1093/aristoteliansupp/70.1.243
–––, 2001, “Was WittgensteinReally an Anti-realist about Mathematics?”, in TimothyMcCarthy and Sean Stidd (eds.),,Wittgenstein in America,Oxford: Clarendon Press: 140–194.
–––, 2007, “Wittgenstein and the RealNumbers”, inWittgenstein and the Moral Life: Essays inHonor of Cora Diamond, Alice Crary (ed.), Cambridge, MA: The MITPress: 235–250.
Putnam, Hilary and Juliet Floyd, 2000, “A Note onWittgenstein’s ‘Notorious Paragraph’ about theGödel Theorem”,The Journal of Philosophy, 97(11):624–632. doi:10.2307/2678455
Quine, W.V.O., 1940 [1981],Mathematical Logic,Cambridge, MA: Harvard University Press, 1981.
Ramharter, Esther, 2009, “Review of Redecker’sWittgensteins Philosophie der Mathematik”,Philosophia Mathematica, 17(3): 382–392.doi:10.1093/philmat/nkp008
Ramsey, Frank Plumpton, 1923, “Critical Notices: Review of‘Tractatus’”,Mind, 32(128): 465–478.doi:10.1093/mind/XXXII.128.465
–––, 1925, “The Foundations ofMathematics”, reprinted in 1990,Philosophical Papers,D.H. Mellor (ed.), Cambridge: Cambridge University Press:164–224.
–––, 1929, “The Formal Structure ofIntuitionist Mathematics”, reprinted in 1991,Notes onPhilosophy, Probability and Mathematics, M. Galavotti (ed.),Napoli: Bibliopolis: 203–220.
Reck, Erich H., (ed.), 2002,Perspectives on Early AnalyticPhilosophy: Frege, Russell, Wittgenstein, New York: OxfordUniversity Press. doi:10.1093/0195133269.001.0001
Redecker, Christine, 2006,Wittgensteins Philosophie derMathematik: Eine Neubewertung im Ausgang von der Kritik an CantorsBeweis der Überabzählbarkeit der reeleen Zahlen[Wittgenstein’s Philosophy of Mathematics: A ReassessmentStarting From the Critique of Cantor’s Proof of theUncountability of the Real Numbers], Frankfurt-Hausenstamm: OntosVerlag.
Rodych, Victor, 1995, “Pasquale Frascolla’sWittgenstein’s Philosophy of Mathematics”,Philosophia Mathematica, 3(3): 271–288.doi:10.1093/philmat/3.3.271
–––, 1997, “Wittgenstein on MathematicalMeaningfulness, Decidability, and Application”,Notre DameJournal of Formal Logic, 38(2): 195–224.
–––, 1999a, “Wittgenstein’sInversion of Gödel’s Theorem”,Erkenntnis,51(2–3): 173–206.
–––, 1999b, “Wittgenstein on Irrationalsand Algorithmic Decidability”,Synthese, 118(2):279–304. doi:10.1023/A:1005191706419
–––, 2000a, “Wittgenstein’s Critiqueof Set Theory”,Southern Journal of Philosophy, 38(2):281–319. doi:10.1111/j.2041-6962.2000.tb00902.x
–––, 2000b, “Wittgenstein’sAnti-Modal Finitism”,Logique et Analyse,43(171–172): 301–333.
–––, 2001, “Gödel’s‘Disproof’ of the Syntactical Viewpoint”,Southern Journal of Philosophy, 39(4): 527–555.doi:10.1111/j.2041-6962.2001.tb01832.x
–––, 2002, “Wittgenstein on Gödel:The Newly Published Remarks”,Erkenntnis, 56(3):379–397.
–––, 2003, “Misunderstanding Gödel:New Arguments about Wittgenstein and New Remarks byWittgenstein”,Dialectica, 57(3): 279–313.
–––, 2006, “Who Is Wittgenstein’sWorst Enemy?: Steiner on Wittgenstein on Gödel”,Logique et Analyse, 49(193): 55–84.
–––, 2008, “Mathematical Sense:Wittgenstein’s Syntactical Structuralism”, inWittgenstein and the Philosophy of Information, Alois Pichlerand Herbert Hrachovec (eds.), Proceedings of the 30th InternationalWittgenstein Symposium, Frankfurt: Ontos Verlag, vol. 1:81–103.
Russell, Bertrand, 1903,The Principles of Mathematics,London: Routledge, 1992; second edition, with a new Introduction,1937.
–––, 1914,Our Knowledge of the ExternalWorld, LaSalle, Ill.: Open Court Publishing Company.
–––, 1918, “The Philosophy of LogicalAtomism”,The Monist, 5(29): 32–63;190–222; 345–380; reprinted in R.C. Marsh, (ed.),Logic and Knowledge, London: Routledge, 1956):177–281.
–––, 1919,Introduction to MathematicalPhilosophy, London: Routledge, (1993 edition with a newIntroduction by John Slater).
–––, 1922[1974], Introduction toTractatusLogico-Philosophicus, London: Routledge & Kegan Paul,ix–xxii.
Savitt, Steven, 1979 [1986], “Wittgenstein’s EarlyPhilosophy of Mathematics”,Philosophy ResearchArchives, 5: 539–553; reprinted in Shanker 1986:26–35. doi:10.5840/pra1979528
Sayward, Charles, 2001, “On Some Much Maligned Remarks ofWittgenstein on Gödel”,PhilosophicalInvestigations 24(3): 262–270.doi:10.1111/1467-9205.00145
–––, 2005, “Steiner versus Wittgenstein:Remarks on Differing Views of Mathematical Truth”,Theoria, 20(2): 347–352. [Sayward 2005 available online]
Shanker, Stuart, (ed.), 1986,Ludwig Wittgenstein: CriticalAssessments, Vol. III, London: Croom Helm.
–––, 1987,Wittgenstein and the TurningPoint in the Philosophy of Mathematics, London: Croom Helm.
–––, 1988, “Wittgenstein’s Remarkson the Significance of Gödel’s Theorem”, in hisGödel’s Theorem in Focus, London: Routledge:155–256.
Skolem, Thoralf, 1923, “The Foundations of ElementaryArithmetic Established by means of the Recursive Mode of Thought,Without the use of Apparent Variables Ranging Over InfiniteDomains”, in van Heijenoort 1967: 303–333.
Sluga, Hans and David G. Stern, (eds.), 1996,The CambridgeCompanion to Wittgenstein, Cambridge: Cambridge UniversityPress.
Steiner, Mark, 1975,Mathematical Knowledge, Ithaca,N.Y.: Cornell University Press.
–––, 1996, “Wittgenstein: Mathematics,Regularities, Rules”, in Morton and Stich 1996:190–212.
–––, 2000, “Mathematical Intuition andPhysical Intuition in Wittgenstein’s Later Philosophy”,Synthese, 125(3): 333–340.doi:10.1023/A:1005118023258
–––, 2001, “Wittgenstein as His Own WorstEnemy: The Case of Gödel’s Theorem”,PhilosophiaMathematica, 9(3): 257–279.doi:10.1093/philmat/9.3.257
–––, 2009, “Empirical Regularities inWittgenstein’s Philosophy of Mathematics”,PhilosophiaMathematica, 17(1): 1–34.
Sullivan, Peter M., 1995, “Wittgenstein on ‘TheFoundations of Mathematics’, June 1927”,Theoria,61(2): 105–142. doi:10.1111/j.1755-2567.1995.tb00493.x
Tait, William W., 1986, “Truth and Proof: The Platonism ofMathematics”,Synthese, 69(3): 341–370.doi:10.1007/BF00413978
Van Atten, 2004,On Brouwer, Toronto: Wadsworth.
Van Dalen, Dirk, 2005,Mystic, Geometer, and Intuitionist: TheLife of L.E.J. Brouwer: Hope and Disillusion Vol. II, Oxford:Clarendon Press.
Waismann, Friedrich, 1930, “The Nature of Mathematics:Wittgenstein’s Standpoint”, in Shanker 1986:60–67.
Wang, Hao, 1958, “Eighty Years of Foundational Studies”,Dialectica, 12: 466–497.
–––, 1984, “Wittgenstein’s and OtherMathematical Philosophies”,Monist, 67:18–28.
–––, 1991, “To and FromPhilosophy—Discussions with Gödel and Wittgenstein”,Synthese, 88(2): 229–277. doi:10.1007/BF00567747
Watson, A.G.D., 1938, “Mathematics and ItsFoundations”,Mind, 47(188): 440–451.doi:10.1093/mind/XLVII.188.440
Weyl, Hermann, 1921 [1998], “Über die neueGrundlagenkrise der Mathematik” (On the New Foundational Crisisof Mathematics),Mathematische Zeitschrift, 10(1–2):37–79; reprinted in Mancosu 1998: 86–118; translated byBenito Müller. doi:10.1007/BF02102305
–––, 1925–1927, “The CurrentEpistemological Situation in Mathematics”,Symposion,1: 1–32; reprinted in Mancosu 1998: 123–142.
–––, 1949 [1927],Philosophy of Mathematicsand Natural Science, 2^nd edition, Princeton, NJ:Princeton University Press.
Whitehead, Alfred North and Bertrand Russell, 1910,PrincipiaMathematica, Volume I, Abridged, Cambridge: Cambridge UniversityPress; 1970.
von Wright, G.H., (ed.), 1973,Ludwig Wittgenstein: Letters toC.K. Ogden, Oxford: Basil Blackwell.
–––, 1982, “The WittgensteinPapers”, in hisWittgenstein, Oxford: Basil Blackwell:36–62.
Wright, Crispin, 1980,Wittgenstein on the Foundations ofMathematics, London: Duckworth.
–––, 1981, “Rule-following, Objectivityand the Theory of Meaning”, inWittgenstein: To Follow aRule, C. Leich and S. Holtzman, eds., London: Routledge:99–117.
–––, 1982, “Strict Finitism”,Synthese, 51(2): 203–82. doi:10.1007/BF00413828
–––, 1984, “Kripke’s Account of theArgument against Private Language”,Journal ofPhilosophy, 81(12): 759–778. doi:10.2307/2026031
–––, 1986, “Rule-following andConstructivism”, inMeaning and Interpretation C.Travis (ed.), Oxford: Blackwell: 271–297.
–––, 1991, “Wittgenstein on MathematicalProof”,Royal Institute of Philosophy Supplement, 28:79–99. Reprinted in his 2001Rails to Infinity,Cambridge, MA, Harvard University Press: 403–430.doi:10.1017/S1358246100005257
Wrigley, Michael, 1977, “Wittgenstein’s Philosophy ofMathematics”,Philosophical Quarterly, 27(106):50–9.
–––, 1980, “Wittgenstein onInconsistency”,Philosophy, 55(214): 471–84.doi:10.1017/S0031819100049494
–––, 1993, “The Continuity ofWittgenstein’s Philosophy of Mathematics”, in Puhl 1993:73–84.
–––, 1998, “A Note on Arithmetic and Logicin the Tractatus”,Acta Analytica, 21:129–131.

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