## 1. Wittgenstein on Mathematics in the *Tractatus*



Wittgenstein's non-referential, *formalist* conception of
mathematical propositions and terms begins in the
 *Tractatus*.[1]
 Indeed, insofar as he sketches a rudimentary Philosophy of
Mathematics in the *Tractatus*, he does so by
*contrasting* mathematics and mathematical equations with
genuine (contingent) propositions, sense, thought, propositional signs
and their constituent names, and truth-by-correspondence.



In the *Tractatus*, Wittgenstein claims that a genuine
proposition, which rests upon conventions, is used by us to assert
that a state of affairs (i.e., an elementary or atomic fact;
'*Sachverhalt*') or fact (i.e., multiple states of
affairs; '*Tatsache*') obtain(s) in the one and
only real world. An elementary proposition is isomorphic to the
*possible* state of affairs it is used to represent: it must
contain as many names as there are objects in the possible state of
affairs. An elementary proposition is true *iff* its possible
state of affairs (i.e., its 'sense';
'*Sinn*') obtains. Wittgenstein clearly states this
Correspondence Theory of Truth at (4.25): 



> 
> 
> If an elementary proposition is true, the state of affairs exists; if
> an elementary proposition is false, the state of affairs does not
> exist. 
> 
> 
> 



But propositions and their linguistic components are, in and of
themselves, dead--a proposition only has sense because we human
beings have endowed it with a *conventional* sense (5.473).
Moreover, propositional signs may be *used* to do any number of
things (e.g., insult, catch someone's attention); in order to
*assert* that a state of affairs obtains, a person must
'project' the proposition's sense--its possible
state of affairs--by 'thinking' of (e.g., picturing)
its sense as one speaks, writes or thinks the proposition (3.11).
Wittgenstein connects *use*, *sense*,
*correspondence*, and *truth* by saying that "a
proposition is true if we *use* it to say that things stand in
a certain way, and they do" (4.062; italics added).



The *Tractarian* conceptions of genuine (contingent)
propositions and the (original and) core concept of truth are used to
construct theories of logical and mathematical
'propositions' *by contrast*. Stated boldly and
bluntly, tautologies, contradictions and mathematical propositions
(i.e., mathematical equations) are neither true nor false--we say
that they are true or false, but in doing so we use the words
'true' and 'false' in very different senses
from 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 the
world (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 limiting
cases--indeed the *disintegration*--of the
combination of signs" (4.466; italics added), where 



> 
> 
> the conditions of agreement with the world--the representational
> relations--cancel one another, so that [they] do[] not stand in
> any representational relation to reality, 
> 
> 
> 



tautologies and contradictions do not picture reality or possible
states of affairs and possible facts (4.462). Stated differently,
tautologies and contradictions do not have sense, which means we
cannot use them to make assertions, which means, in turn, that they
cannot be either true or false. Analogously, mathematical
pseudo-propositions are equations, which indicate or show that two
expressions are equivalent in meaning and therefore are
intersubstitutable. Indeed, we arrive at mathematical equations by
"the method of substitution": 



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



We prove mathematical 'propositions' 'true'
('correct') by 'seeing' that two expressions
have the same meaning, which "must be manifest in the two
expressions themselves" (6.23), and by substituting one
expression for another with the same meaning. Just as "one can
recognize that ['logical propositions'] are true from the
symbol alone" (6.113), "the possibility of proving"
mathematical propositions means that we can perceive their correctness
without having to compare "what they express" with facts
(6.2321; cf. *RFM* App. III, SS4).



The demarcation between contingent propositions, which can be used to
correctly or incorrectly represent parts of the world, and
mathematical propositions, which can be decided in a purely formal,
syntactical manner, is maintained by Wittgenstein until his death in
1951 (*Zettel* SS701, 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 of
how the world is, whereas the "truth-value" of a
mathematical proposition is entirely a function of its constituent
symbols and the formal system of which it is a part. Thus, a second,
closely related way of stating this demarcation is to say that
mathematical 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, by
determining whether or not a particular fact obtains (i.e., something
external to the proposition and the language in which it resides)
(2.223; 4.05).



The Tractarian formal theory of mathematics is, specifically, a theory
of *formal operations*. Over the past 20 years,
Wittgenstein's theory of operations has received considerable
examination (Frascolla 1994, 1997; Marion 1998; Potter 2000; and Floyd
2002), which has interestingly connected it and the Tractarian
equational theory of arithmetic with elements of Alonzo Church's
\(\lambda\)-calculus and with R. L. Goodstein's equational
calculus (Marion 1998: chapters 1, 2, and 4). Very briefly stated,
Wittgenstein presents: 


1. ... the sign '[\(a, x, O \spq x\)]' for the
general term of the series of forms \(a\), \(O \spq a\), \(O \spq O
\spq a\).... (5.2522)
2. ... the general form of an operation
\(\Omega\spq(\overline{\eta})\) [as] 

\[
[\overline{\xi}, N(\overline{\xi})]\spq (\overline{\eta}) (= [\overline{\eta}, \overline{\xi}, N(\overline{\xi})]). (6.01)
\]
3. ... the general form of a proposition
("truth-function") [as] \([\overline{p}, \overline{\xi},
N(\overline{\xi})]\). (6)
4. 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 form
of a number" (6.022). As Frascolla (and Marion after him) have
pointed out, "the general form of a proposition is a
*particular case* of the general form of an
'operation'" (Marion 1998: 21), and all three
general 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 relation
between the structures of its result and of its bases" (5.22),
Wittgenstein states that whereas "[a] function cannot be its own
argument,... an operation can take one of its own results as its
base" (5.251).



On Wittgenstein's (5.2522) account of '[\(a, x, O \spq
x\)]', 



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



Given that "[t]he concept of successive applications of an
operation is equivalent to the concept 'and so on'"
(5.2523), one can see how the natural numbers can be generated by
repeated iterations of the general form of a natural number, namely
'[\(0, \xi , \xi +1\)]'. Similarly, truth-functional
propositions can be generated, as Russell says in the Introduction to
the *Tractatus* (p. xv), from the general form of a proposition
'[\(\overline{p}\), \(\overline{\xi}\), \(N(\overline{\xi})\)]' by




> 
> 
> taking any selection of atomic propositions [where \(p\) "stands
> for all atomic propositions"; "the bar over the variable
> indicates that it is the representative of all its values"
> (5.501)], negating them all, then taking any selection of the set of
> propositions now obtained, together with any of the originals [where
> \(x\) "stands for any set of propositions"]--and so
> on indefinitely. 
> 
> 
> 



On Frascolla's (1994: 3ff) account, 



> 
> 
> a numerical identity "\(\mathbf{t} = \mathbf{s}\)" is an
> arithmetical theorem if and only if the corresponding equation
> "\(\Omega^t \spq x = \Omega^s \spq x\)", which is framed
> in the language of the general theory of logical operations, can be
> proven. 
> 
> 
> 



By proving 



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



Wittgenstein thereby outlines "a translation of numerical
arithmetic into a sort of general theory of operations"
(Frascolla 1998: 135).



Despite the fact that Wittgenstein clearly does *not* attempt
to reduce mathematics to logic in either Russell's manner or
Frege's manner, or to tautologies, and despite the fact that
Wittgenstein criticizes Russell's Logicism (e.g., the Theory of
Types, 3.31-3.32; the Axiom of Reducibility, 6.1232, etc.) and
Frege's Logicism (6.031, 4.1272,
 etc.),[2]
 quite a number of commentators, early and recent, have interpreted
Wittgenstein's Tractarian theory of mathematics as a variant of
Logicism (Quine 1940 [1981: 55]; Benacerraf & Putnam 1964a: 14;
Black 1964: 340; Savitt
1979 [1986: 34]; Frascolla 1994: 37; 1997: 354, 356-57, 361;
1998: 133; Marion 1998: 26 & 29; and Potter 2000: 164 and
182-183). There are at least four reasons proffered for this
interpretation.


1. Wittgenstein says that "[m]athematics is a method of
logic" (6.234).
2. Wittgenstein says that "[t]he logic of the world, which is
shown in tautologies by the propositions of logic, is shown in
equations by mathematics" (6.22).
3. According to Wittgenstein, we ascertain the *truth* of
*both* mathematical and logical propositions by the symbol
alone (i.e., by purely formal operations), without making any
('external', non-symbolic) observations of states of
affairs or facts in the world.
4. Wittgenstein's iterative (inductive) "interpretation
of numerals as exponents of an operation variable" is a
"reduction of arithmetic to operation theory", where
"operation" is construed as a "*logical*
operation" (italics added) (Frascolla 1994: 37), which shows
that "the label 'no-classes logicism' tallies with
the *Tractatus* view of arithmetic" (Frascolla 1998: 133;
1997: 354).



Though at least three Logicist interpretations of the
*Tractatus* have appeared within the last 20 years, the
following considerations (Rodych 1995; Wrigley 1998) indicate that
none of these reasons is particularly cogent.



For example, in saying that "[m]athematics is a method of
logic" perhaps Wittgenstein is only saying that since the
general form of a natural number and the general form of a proposition
are both instances of the general form of a (purely formal) operation,
just as truth-functional propositions can be constructed using the
general form of a proposition, (true) mathematical equations can be
constructed using the general form of a natural number. Alternatively,
Wittgenstein may mean that mathematical *inferences* (i.e., not
substitutions) 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" is
shown by tautologies and true mathematical equations (i.e., #2),
Wittgenstein may be saying that since mathematics was invented to help
us count and measure, insofar as it enables us to infer contingent
proposition(s) from contingent proposition(s) (see 6.211 below), it
thereby *reflects* contingent facts and "[t]he logic of
the world". Though logic--which is inherent in natural
('everyday') language (4.002, 4.003, 6.124) and which has
evolved to meet our communicative, exploratory, and survival
needs--is not *invented* in the same way, a valid logical
inference captures the relationship between possible facts and a
*sound* logical inference captures the relationship between
existent facts.



As regards #3, Black, Savitt, and Frascolla have argued that, since we
ascertain the truth of tautologies and mathematical equations without
any appeal to "states of affairs" or "facts",
true mathematical equations and tautologies are *so* analogous
that we can "aptly" describe "the philosophy of
arithmetic of the *Tractatus*... as a kind of
logicism" (Frascolla 1994: 37). The rejoinder to this is that
the similarity that Frascolla, Black and Savitt recognize does not
make Wittgenstein's theory a "kind of logicism" in
Frege's or Russell's sense, because Wittgenstein does not
define numbers "logically" in either Frege's way or
Russell's way, and the similarity (or analogy) between
tautologies and true mathematical equations is neither an identity nor
a relation of reducibility.



Finally, critics argue that the problem with #4 is that there is no
evidence for the claim that the relevant operation is *logical*
in Wittgenstein's or Russell's or Frege's sense of
the term--it seems a purely formal, syntactical operation. "Logical operations are performed with propositions,
arithmetical ones with numbers", says Wittgenstein (*WVC*
218); "[t]he result of a logical operation is a proposition, the
result of an arithmetical one is a number". In sum, critics of
the Logicist interpretation of the *Tractatus* argue that
##1-4 do not individually or collectively constitute cogent
grounds for a Logicist interpretation of the *Tractatus*.



Another crucial aspect of the *Tractarian* theory of
mathematics is captured in (6.211).



> 
> 
> Indeed in real life a mathematical proposition is never what we want.
> Rather, we make use of mathematical propositions *only* in
> inferences from propositions that do not belong to mathematics to
> others that likewise do not belong to mathematics. (In philosophy the
> question, 'What do we actually use this word or this proposition
> for?' repeatedly leads to valuable insights.)
> 
> 
> 



Though mathematics and mathematical activity are purely formal and
syntactical, in the *Tractatus* Wittgenstein tacitly
distinguishes between purely formal games with signs, which have no
application in contingent propositions, and mathematical propositions,
which are used to make inferences from contingent proposition(s) to
contingent proposition(s). Wittgenstein does not explicitly say,
however, *how* mathematical equations, which are *not*
genuine propositions, are used in inferences from genuine
proposition(s) to genuine proposition(s) (Floyd 2002: 309; Kremer
2002: 293-94). As we shall see in
 SS3.5,
 the later Wittgenstein returns to the importance of
extra-mathematical application and uses it to distinguish a mere
"sign-game" from a genuine, mathematical
language-game.



This, in brief, is Wittgenstein's Tractarian theory of
mathematics. In the Introduction to the *Tractatus*, Russell
wrote that Wittgenstein's "theory of number"
"stands in need of greater technical development",
primarily because Wittgenstein had not shown how it could deal with
transfinite numbers (Russell 1922[1974]: xx).
Similarly, in his review of
the *Tractatus*, Frank Ramsey wrote that Wittgenstein's
'account' does not cover all of mathematics partly because
Wittgenstein's theory of equations cannot explain inequalities
(Ramsey 1923: 475). Though it is doubtful that, in 1923, Wittgenstein
would have thought these issues problematic, it certainly is true that
the Tractarian theory of mathematics is essentially a sketch,
especially in comparison with what Wittgenstein begins to develop six
years later.



After the completion of the *Tractatus* in 1918, Wittgenstein
did virtually no philosophical work until February 2, 1929, eleven
months 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 attended
with F. Waismann and H. Feigl, but it is a gross overstatement to say
that he returned to Philosophy because of this lecture or that his
intermediate interest in the Philosophy of Mathematics issued
primarily from Brouwer's influence. In fact,
Wittgenstein's return to Philosophy and his intermediate work on
mathematics is also due to conversations with Ramsey and members of
the Vienna Circle, to Wittgenstein's disagreement with Ramsey
over identity, and several other factors.



Though Wittgenstein seems not to have read any Hilbert or Brouwer
prior to the completion of the *Tractatus*, by early 1929
Wittgenstein had certainly read work by Brouwer, Weyl, Skolem, Ramsey
(and possibly Hilbert) and, apparently, he had had one or more private
discussions with Brouwer in 1928 (Finch 1977: 260; Van Dalen 2005:
566-567). Thus, the rudimentary treatment of mathematics in the
*Tractatus*, 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 of
Brouwer, Weyl, Hilbert, and Skolem.


### 2.1 Wittgenstein's Intermediate Constructive Formalism



To best understand Wittgenstein's intermediate Philosophy of
Mathematics, one must fully appreciate his strong variant of
formalism, according to which "[w]e *make*
mathematics" (*WVC* 34, note 1; *PR* SS159) by
inventing purely formal mathematical calculi, with
'stipulated' axioms (*PR* SS202), syntactical
rules of transformation, and decision procedures that enable us to
invent "mathematical truth" and "mathematical
falsity" by algorithmically deciding so-called mathematical
'propositions' (*PR* SSSS122, 162).



The *core idea* of Wittgenstein's formalism from 1929 (if
not 1918) through 1944 is that mathematics is essentially syntactical,
devoid of reference and semantics. The most obvious aspect of this
view, which has been noted by numerous commentators who do not refer
to 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*, the
signs and propositions of a mathematical calculus do not
*refer* to anything. As Wittgenstein says at (*WVC* 34,
note 1), "[n]umbers are not represented by proxies; numbers
*are there*". This means not only that numbers are there
in the *use*, it means that the numerals *are* the
numbers, for "[a]rithmetic doesn't talk about numbers, it
works with numbers" (*PR* SS109).



> 
> 
> What arithmetic is concerned with is the schema \(||||\).--But
> does arithmetic talk about the lines I draw with pencil on
> paper?--Arithmetic doesn't talk about the lines, it
> *operates* 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 calculating
machine", which "works by means of strokes, numerals,
etc". The "justified side of formalism", according
to Wittgenstein (*WVC* 105), is that mathematical symbols
"lack a meaning" (i.e.,
*Bedeutung*)--they do not "go proxy
for" *things* which are "their
meaning[s]".



> 
> 
> You could say arithmetic is a kind of geometry; i.e. what in geometry
> are constructions on paper, in arithmetic are calculations (on
> paper).--You could say it is a more general kind of geometry.
> (*PR* SS109; *PR* SS111)
> 
> 
> 



This is the core of Wittgenstein's life-long formalism. When we
prove a theorem or decide a proposition, we operate in a *purely
formal*, syntactical manner. In *doing* mathematics, we do
not discover pre-existing truths that were "already there
without one knowing" (*PG* 481)--we *invent*
mathematics, bit-by-little-bit. "If you want to know what \(2 +
2 = 4\) means", says Wittgenstein, "you have to ask how we
work it out", because "we consider the process of
calculation as the essential thing" (*PG* 333). Hence,
the only meaning (i.e., sense) that a mathematical proposition has is
*intra-systemic* meaning, which is wholly determined by its
syntactical relations to other propositions of the calculus.



A second important aspect of the intermediate Wittgenstein's
strong formalism is his view that extra-mathematical application
(and/or reference) is *not* a necessary condition of a
mathematical calculus. Mathematical calculi *do not require*
extra-mathematical applications, Wittgenstein argues, since we
"can develop arithmetic completely autonomously and its
application takes care of itself since wherever it's applicable
we may also apply it" (*PR* SS109; cf. *PG*
308, *WVC* 104).



As we shall shortly see, the middle Wittgenstein is also drawn to
strong formalism by a new concern with questions of
*decidability*. Undoubtedly influenced by the writings of
Brouwer and David Hilbert, Wittgenstein uses strong formalism to forge
a new connection between mathematical meaningfulness and algorithmic
decidability.



> 
> 
> An equation is a rule of syntax. Doesn't that explain why we
> cannot have questions in mathematics that are in principle
> unanswerable? For if the rules of syntax cannot be grasped,
> they're of no use at all.... [This] makes intelligible the
> attempts of the formalist to see mathematics as a game with signs.
> (*PR* SS121)
> 
> 
> 



In
 Section 2.3,
 we shall see how Wittgenstein goes beyond both Hilbert and Brouwer by
*maintaining* the Law of the Excluded Middle in a way that
restricts mathematical propositions to expressions that are
algorithmically decidable.


### 2.2 Wittgenstein's Intermediate Finitism



The single most important difference between the Early and Middle
Wittgenstein is that, in the middle period, Wittgenstein rejects
quantification over an infinite mathematical domain, stating that,
contra his *Tractarian* view, such 'propositions'
are not infinite conjunctions and infinite disjunctions simply because
there are no such things.



Wittgenstein's *principal reasons* for developing a
finitistic Philosophy of Mathematics are as follows.


1. Mathematics as Human Invention: According to the middle
Wittgenstein, we invent mathematics, from which it follows that
mathematics and so-called mathematical objects do not exist
independently of our inventions. Whatever is mathematical is
fundamentally a product of human activity.
2. Mathematical Calculi Consist Exclusively of Intensions and
Extensions: 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 and
transformation, irrational numbers *as* rules), these
extensions and intensions, and the calculi in which they reside,
constitute the entirety of mathematics. (It should be noted that
Wittgenstein's usage of 'extension' and
'intension' as regards mathematics differs markedly from
standard contemporary usage, wherein the extension of a predicate is
the set of entities that satisfy the predicate and the intension of a
predicate is the meaning of, or expressed by, the predicate. Put
succinctly, Wittgenstein thinks that the extension of this notion of
concept-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 immediate *consequences*
for Wittgenstein's Philosophy of Mathematics.


1. Rejection of Infinite Mathematical Extensions: Given that a
mathematical extension is a symbol ('sign') or a finite
concatenation of symbols *extended* in space, there is a
categorical difference between mathematical intensions and (finite)
mathematical extensions, from which it follows that "the
mathematical infinite" resides only in recursive rules (i.e.,
intensions). An infinite mathematical extension (i.e., a
*completed*, infinite mathematical extension) is a
contradiction-in-terms
2. Rejection of Unbounded Quantification in Mathematics: Given that
the mathematical infinite can only be a recursive rule, and given that
a mathematical proposition must have sense, it follows that there
cannot be an infinite mathematical proposition (i.e., an infinite
logical product or an infinite logical sum).
3. Algorithmic Decidability vs. Undecidability: If mathematical
extensions of all kinds are necessarily *finite*, then, *in
principle*, all mathematical propositions are *algorithmically
decidable*, from which it follows that an "undecidable
mathematical proposition" is a contradiction-in-terms. Moreover,
since mathematics is essentially what we have and what we know,
Wittgenstein restricts algorithmic decidability to *knowing*
how to decide a proposition with a known decision procedure.
4. Anti-Foundationalist Account of Real Numbers: Since there are no
infinite mathematical extensions, irrational numbers are rules, not
extensions. Given that an infinite set is a recursive rule (or an
induction) and no such rule can generate all of the things
mathematicians call (or want to call) "real numbers", it
follows that there is no set of 'all' the real numbers and
no such thing as the mathematical continuum.
5. Rejection of Different Infinite Cardinalities: Given the
non-existence of infinite mathematical extensions, Wittgenstein
rejects the standard interpretation of Cantor's diagonal proof
as a proof of infinite sets of greater and lesser cardinalities.



Since we invent mathematics *in its entirety*, we do not
discover pre-existing mathematical objects or facts or that
mathematical objects have certain properties, for "one cannot
discover any connection between parts of mathematics or logic that was
already there without one knowing" (*PG* 481). In
examining mathematics as a purely human invention, Wittgenstein tries
to determine what exactly we have invented and why exactly, in his
opinion, we erroneously think that there are infinite mathematical
extensions.



If, first, we examine what we have invented, we see that we have
invented formal calculi consisting of finite extensions and
intensional rules. If, more importantly, we endeavour to determine
*why* we believe that infinite mathematical extensions exist
(e.g., why we believe that the actual infinite is intrinsic to
mathematics), we find that we conflate mathematical
*intensions* and mathematical *extensions*, erroneously
thinking that there is "a dualism" of "the law and
the infinite series obeying it" (*PR* SS180). For
instance, we think that because a real number "endlessly yields
the places of a decimal fraction" (*PR* SS186), it
*is* "a totality" (*WVC* 81-82, note
1), when, in reality, "[a]n irrational number isn't the
extension of an infinite decimal fraction,... it's a
law" (*PR* SS181) which "yields
extensions" (*PR* SS186). A law and a list are
fundamentally different; neither can 'give' what the other
gives (*WVC* 102-103). Indeed, "the mistake in the
set-theoretical approach consists time and again in treating laws and
enumerations (lists) as essentially the same kind of thing"
(*PG* 461).



Closely related with this conflation of intensions and extensions is
the fact that we mistakenly act as if the word 'infinite'
is a "number word", because in ordinary discourse we
answer the question "how many?" with both (*PG*
463; cf. *PR* SS142). But "'[i]nfinite'
is not a *quantity*", Wittgenstein insists (*WVC*
228); the word 'infinite' and a number word like
'five' do not have the same syntax. The words
'finite' and 'infinite' do not function as
adjectives on the words 'class' or 'set',
(*WVC* 102), for the terms "finite class" and
"infinite class" use 'class' in completely
different ways (*WVC* 228). An infinite class is a recursive
rule or "an induction", whereas the symbol for a finite
class is a list or extension (*PG* 461). It is because an
induction has much in common with the multiplicity of a finite class
that we erroneously call it an infinite class (*PR*
SS158).



In sum, because a mathematical extension is necessarily a finite
sequence of symbols, an infinite mathematical extension is a
contradiction-in-terms. This is the foundation of Wittgenstein's
finitism. Thus, when we say, e.g., that "there are infinitely
many even numbers", we are *not* saying "there are
an infinite number of even numbers" in *the same sense*
as we can say "there are 27 people in this house"; the
infinite series of natural numbers is nothing but "the infinite
possibility of finite series of numbers"--"[i]t is
senseless to speak of the *whole* infinite number series, as if
it, too, were an extension" (*PR* SS144). The
infinite is understood rightly when it is understood, not as a
quantity, but as an "infinite possibility" (*PR*
SS138).



Given Wittgenstein's rejection of infinite mathematical
extensions, he adopts finitistic, constructive views on mathematical
quantification, mathematical decidability, the nature of real numbers,
and Cantor's diagonal proof of the existence of infinite sets of
greater cardinalities.



Since a mathematical set is a finite extension, we cannot
*meaningfully* quantify over an infinite mathematical domain,
simply because there is no such thing as an infinite mathematical
domain (i.e., totality, set), and, derivatively, no such things as
infinite conjunctions or disjunctions (G.E. Moore 1955: 2-3; cf.
*AWL* 6; and *PG* 281).



> 
> 
> [I]t still looks now as if the quantifiers make no sense for numbers.
> I mean: you can't say '\((n) \phi n\)', precisely
> because 'all natural numbers' isn't a bounded
> concept. But then neither should one say a general proposition follows
> from a proposition about the nature of number.
> 
> 
> 
> But in that case it seems to me that we can't use
> generality--all, etc.--in mathematics at all. There's
> no such thing as 'all numbers', simply because there are
> infinitely many. (*PR* SS126; *PR* SS129)
> 
> 
> 



'Extensionalists' who assert that
"\(\varepsilon(0).\varepsilon(1).\varepsilon(2)\) and so
on" is an infinite logical product (*PG* 452) assume or
assert that finite and infinite conjunctions are close
cousins--that the fact that we cannot write down or enumerate all
of the conjuncts 'contained' in an infinite conjunction is
only a "human weakness", for God could surely do so and
God could surely survey such a conjunction in a single glance and
determine its truth-value. According to Wittgenstein, however, this is
*not* a matter of *human* limitation. Because we
mistakenly think that "an infinite conjunction" is similar
to "an enormous conjunction", we erroneously reason that
just as we cannot determine the truth-value of an enormous conjunction
because we don't have enough time, we similarly cannot, due to
human limitations, determine the truth-value of an infinite
conjunction (or disjunction). But the difference here is not one of
degree but of kind: "in the sense in which it is impossible to
check an infinite number of propositions it is also impossible to try
to do so" (*PG* 452). This applies, according to
Wittgenstein, to human beings, but more importantly, it applies also
to God (i.e., an omniscient being), for even God cannot write down or
survey infinitely many propositions because for him too the series is
never-ending or limitless and hence the 'task' is not a
genuine task because it cannot, *in principle*, be done (i.e.,
"infinitely many" is not a number word). As Wittgenstein
says at (*PR* 128; cf. *PG* 479): "'Can God
know all the places of the expansion of \(\pi\)?' would have
been a good question for the schoolmen to ask", for the question
is strictly 'senseless'. As we shall shortly see, on
Wittgenstein's account, "[a] statement about *all*
numbers is not represented by means of a proposition, but by means of
induction" (*WVC* 82).



Similarly, there is no such thing as a mathematical proposition about
*some* number--no such thing as a mathematical proposition
that existentially quantifies over an infinite domain (*PR*
SS173).



> 
> 
> What is the meaning of such a mathematical proposition as
> '\((\exists n) 4 + n = 7\)'? It might be a
> disjunction--\((4 + 0 = 7) \vee{}\) \((4 + 1 = 7) \vee {}\) etc. *ad
> inf.* But what does that mean? I can understand a proposition with
> a beginning and an end. But can one also understand a proposition with
> no end? (*PR* SS127)
> 
> 
> 



We are particularly seduced by the feeling or belief that an infinite
*mathematical* disjunction makes good sense in the case where
we can provide a recursive rule for generating each next member of an
infinite sequence. For example, when we say "There exists an odd
perfect number" we are asserting that, in the infinite sequence
of odd numbers, there is (at least) one odd number that is
perfect--we are asserting '\(\phi(1) \vee \phi(3) \vee
\phi(5) \vee{}\) and so on' and we know what would make it true
and 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'", which
doesn't provide the grammar of the former
'proposition', but only indicates an analogy in their
respective rules.



On Wittgenstein's intermediate finitism, an expression
quantifying over an infinite domain is *never* a meaningful
proposition, not even when we have proved, for instance, that a
particular 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 ought *not* to say '\((\exists x,
> y, z, n) (x^n + y^n = z^n)\)', since taken extensionally
> that's meaningless, and taken intensionally this doesn't
> provide a proof of it. No, in this case I ought to express only the
> first equation. (*PR* SS150)
> 
> 
> 



Thus, Wittgenstein adopts the radical position that *all*
expressions that quantify over an infinite domain, whether
'conjectures' (e.g., Goldbach's Conjecture, the Twin
Prime Conjecture) or "proved general theorems" (e.g.,
"Euclid's Prime Number Theorem", the Fundamental
Theorem of Algebra), are *meaningless* (i.e.,
'senseless'; '*sinnlos*') expressions
as opposed to "genuine mathematical
*proposition*[s]" (*PR* SS168). These
expressions are not (meaningful) mathematical propositions, according
to Wittgenstein, because the Law of the Excluded Middle does not
apply, which means that "we aren't dealing with
propositions of mathematics" (*PR* SS151). The
crucial question *why* and in exactly what sense the Law of the
Excluded Middle does not apply to such expressions will be answered in
the next section.


### 2.3 Wittgenstein's Intermediate Finitism and Algorithmic Decidability



The middle Wittgenstein has other grounds for rejecting unrestricted
quantification in mathematics, for on his idiosyncratic account, we
must distinguish between four categories of concatenations of
mathematical symbols.


1. Proved mathematical propositions in a particular mathematical
calculus (no need for "mathematical truth").
2. Refuted mathematical propositions in (or of) a particular
mathematical calculus (no need for "mathematical
falsity").
3. Mathematical propositions for which we know we have in hand an
applicable and effective decision procedure (i.e., we know
*how* to decide them).
4. Concatenations of symbols that are not part of any mathematical
calculus 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 ["possibilities
> for a proposition with respect to truth"]:
> 
> 
> 1. \(p\) has been experienced as true
> 2. \(p\) has been experienced as false
> 3. Neither 1 nor 2 has occurred yet, but we know a procedure to
> decide \(p\) (i.e., a procedure that will prove \(p\) or prove \(\neg
> p)\)
> 4. Neither 1 nor 2 has occurred yet, and we do not know a procedure
> to decide \(p\).
> 
> 
> 



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



As radical as the respective #3s are, Brouwer and Wittgenstein agree
that an undecided \(\phi\) is a mathematical proposition (for
Wittgenstein, *of* a particular mathematical calculus) if we
know of an applicable decision procedure. They also agree that until
\(\phi\) is decided, it is neither true nor false (though, for
Wittgenstein, 'true' means no more than "proved in
calculus \(\Gamma\)"). What they disagree about is the status of
an ordinary mathematical conjecture, such as Goldbach's
Conjecture. Brouwer admits it as a mathematical proposition, while
Wittgenstein rejects it because we do not know how to algorithmically
decide it. Like Brouwer (1948 [1983: 90]), Wittgenstein holds that
there are no "unknown truth[s]" in mathematics, but unlike
Brouwer he denies the existence of "undecidable
propositions" on the grounds that such a
'proposition' would have no 'sense',
"and the consequence of this is precisely that the propositions
of logic lose their validity for it" (*PR* SS173). In
particular, if there *are* undecidable mathematical
propositions (as Brouwer maintains), then at least some mathematical
propositions are not propositions of *any existent*
mathematical calculus. For Wittgenstein, however, it is a defining
feature of a mathematical proposition that it is either decided or
decidable by a known decision procedure *in a mathematical
calculus*. As Wittgenstein says at (*PR* SS151), 



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



The point here is *not* that we need truth and falsity in
mathematics--we don't--but rather that every
mathematical proposition (including ones for which an applicable
decision procedure is known) is *known* to be part of a
mathematical calculus.



To maintain this position, Wittgenstein distinguishes between
(meaningful, genuine) mathematical propositions, which have
mathematical sense, and meaningless, senseless
('*sinnlos*') expressions by stipulating that an
expression is a meaningful (genuine) proposition of a mathematical
calculus *iff* we *know* of a proof, a refutation, or an
applicable decision procedure (*PR* SS151; *PG* 452;
*PG* 366; *AWL* 199-200). "Only where
there's a method of solution [a 'logical method for
finding a solution'] is there a [mathematical] problem",
he tells us (*PR* SSSS149, 152; *PG* 393).
"We may only put a question in mathematics (or make a
conjecture)", he adds (*PR* SS151), "where the
answer runs: 'I must work it out'".



At (*PG* 468), Wittgenstein emphasizes the importance of
algorithmic decidability clearly and emphatically: 



> 
> 
> In mathematics *everything* is algorithm and *nothing*
> is meaning [*Bedeutung*]; even when it
> doesn't look like that because we seem to be using
> *words* to talk *about* mathematical things. Even these
> words 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, we
have altered the concept of proposition", he means that an
expression is only a meaningful mathematical proposition if we
*know* of an applicable decision procedure for deciding it
(*PG* 400). If a genuine mathematical proposition is
*undecided*, the Law of the Excluded Middle holds in the sense
that we *know* that we will *prove or refute* the
proposition by applying an applicable decision procedure (*PG*
379, 387).



For Wittgenstein, there simply is no distinction between syntax and
semantics in mathematics: everything is syntax. If we wish to
demarcate between "mathematical propositions" versus
"mathematical pseudo-propositions", as we do, then the
*only* way to ensure that there is no such thing as a
meaningful, but *undecidable* (e.g., independent), proposition
of a given calculus is to stipulate that an expression is only a
meaningful proposition *in* a given calculus (*PR*
SS153) if either it has been decided or we *know* of an
applicable decision procedure. In this manner, Wittgenstein defines
*both* a mathematical calculus *and* a mathematical
proposition in *epistemic* terms. A calculus is defined in
terms of stipulations (*PR* SS202; *PG* 369),
*known* rules of operation, and *known* decision
procedures, and an expression is only a mathematical proposition
*in* a given calculus (*PR* SS155), and only if that
calculus *contains* (*PG* 379) a known (and applicable)
decision procedure, for "you cannot have a logical plan of
search for a *sense* you don't know" (*PR*
SS148).



Thus, the middle Wittgenstein rejects undecidable mathematical
propositions on *two* grounds. First, number-theoretic
expressions that quantify over an infinite domain are not
algorithmically decidable, and hence are not meaningful mathematical
propositions.



> 
> 
> 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, this
> implies that '\((x)\)...' is meant extensionally and
> we 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 all
> and the '\((x)\)...' in arithmetic cannot be taken
> extensionally. (*PR* SS174)
> 
> 
> 



"Undecidability", says Wittgenstein (*PR*
SS174) "presupposes... that the bridge *cannot*
be made with symbols", when, in fact, "[a] connection
between symbols which exists but cannot be represented by symbolic
transformations is a thought that cannot be thought", for
"[i]f the connection is there,... it must be possible to
see it". Alluding to algorithmic decidability, Wittgenstein
stresses (*PR* SS174) that "[w]e can assert anything
which can be *checked in practice*", because
"it's a question of the *possibility of
checking*" (italics added).



Wittgenstein's second reason for rejecting an undecidable
mathematical proposition is that it is a
*contradiction-in-terms*. There cannot be "undecidable
propositions", Wittgenstein argues (*PR* SS173),
because an expression that is not decidable in some *actual*
calculus is simply not a *mathematical* proposition, since
"every proposition in mathematics must belong to a calculus of
mathematics" (*PG* 376).



This radical position on decidability results in various radical and
counter-intuitive statements about unrestricted mathematical
quantification, mathematical induction, and, especially, the
*sense* of a newly proved mathematical proposition. In
particular, Wittgenstein asserts that uncontroversial mathematical
conjectures, such as Goldbach's Conjecture (hereafter
'GC') and the erstwhile conjecture "Fermat's
Last Theorem" (hereafter 'FLT'), have no sense (or,
perhaps, no *determinate* sense) and that the
*unsystematic* proof of such a conjecture gives it a sense that
it didn't previously have (*PG* 374) because 



> 
> 
> it's unintelligible that I should admit, when I've got the
> proof, that it's a proof of precisely *this* proposition,
> or of the induction meant by this proposition. (*PR*
> SS155)
> 
> 
> 



> 
> 
> Thus Fermat's [Last Theorem] makes no *sense* until I can
> *search* 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 is
> no kind of search. (*PR* SS150)
> 
> 
> 
> I say: the so-called 'Fermat's Last Theorem'
> isn't a proposition. (Not even in the sense of a proposition of
> arithmetic.) Rather, it corresponds to an induction. (*PR*
> SS189)
> 
> 
> 



To see how Fermat's Last Theorem isn't a proposition and
how it *might* correspond to an induction, we need to examine
Wittgenstein'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, does *any*
number-theoretic proof by mathematical induction actually
*prove*?



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





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




If, however, "\(\forall n\phi(n)\)" is *not* a
meaningful (genuine) mathematical proposition, what are we to make of
this proof?



Wittgenstein's initial answer to this question is decidedly
enigmatic. "An induction is the expression for arithmetical
generality", but "induction isn't itself a
proposition" (*PR* SS129).



> 
> 
> We are not saying that when \(f(1)\) holds and when \(f(c + 1)\)
> follows from \(f(c)\), the proposition \(f(x)\) is *therefore*
> true of all cardinal numbers: but: "the proposition \(f(x)\)
> holds for all cardinal numbers" *means* "it holds
> for \(x = 1\), 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 is
customarily construed as the *conclusion* of the proof
(*PG* 406, 374; *PR* SS164), rather this
pseudo-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 "once
you've got the induction, it's all over"
(*PG* 407). Thus, on Wittgenstein's account, a particular
proof by mathematical induction should be understood in the following
way.





|  |  |
| --- | --- |
| **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* SS164)] uses
'\(m\)' rather than '\(n\)' to indicate that
'\(m\)' stands for any *particular* number, while
'\(n\)' stands for any *arbitrary* number. For
Wittgenstein, the *proxy statement* "\(\phi(m)\)"
is *not* a mathematical proposition that "assert[s] its
generality" (*PR* SS168), it is an
*eliminable* pseudo-proposition standing proxy for the proved
inductive base and inductive step. Though an inductive proof
*cannot prove* "the infinite possibility of
application" (*PR* SS163), it enables us "to
*perceive*" that a *direct* proof of any
*particular* proposition can be constructed (*PR*
SS165). For example, once we have proved "\(\phi(1)\)"
and "\(\phi(n) \rightarrow \phi(n + 1)\)", we need not
reiterate *modus ponens* \(m - 1\) times to prove the
particular proposition "\(\phi(m)\)" (*PR*
SS164). The direct proof of, say, "\(\phi\)(714)"
(i.e., without 713 iterations of *modus ponens*) "cannot
have a still better proof, say, by my carrying out the derivation as
far as this proposition itself" (*PR* SS165).



A second, very important impetus for Wittgenstein's radically
constructivist position on mathematical induction is his rejection of
an *undecidable* mathematical proposition.



> 
> 
> In discussions of the provability of mathematical propositions it is
> sometimes said that there are substantial propositions of mathematics
> whose truth or falsehood must remain undecided. What the people who
> say that don't realize is that such propositions, *if* we
> can use them and want to call them "propositions", are not
> at all the same as what are called "propositions" in other
> cases; because a proof alters the grammar of a proposition.
> (*PG* 367)
> 
> 
> 



In this passage, Wittgenstein is alluding to Brouwer, who, as early as
1907 and 1908, states, first, that "the question of the validity
of the principium tertii exclusi is equivalent to the question
*whether unsolvable mathematical problems exist*",
second, that "[t]here is not a shred of a proof for the
conviction... that there exist no unsolvable mathematical
problems", and, third, that there are meaningful
propositions/'questions', such as "*Do there
occur in the decimal expansion of \(\pi\) infinitely many pairs of
consecutive equal digits*?", to which the Law of the
Excluded Middle does not apply *because* "it must be
considered as uncertain whether problems like [this] are
solvable" (Brouwer, 1908 [1975: 109-110]). "A
fortiori it is not certain that any mathematical problem can either be
solved or proved to be unsolvable", Brouwer says (1907 [1975:
79]), "though HILBERT, in 'Mathematische Probleme',
believes that every mathematician is deeply convinced of
it".



Wittgenstein takes the same data and, in a way, draws the opposite
conclusion. If, as Brouwer says, we are *uncertain* whether all
or some "mathematical problems" are solvable, then we
*know* that we do *not* have in hand an applicable
decision procedure, which means that the alleged mathematical
propositions are *not decidable*, here and now. "What
'mathematical questions' share with genuine
questions", Wittgenstein says (*PR* SS151), "is
simply that they can be answered". This means that if we do not
know how to decide an expression, then we do not know how to
*make* it either proved (true) or refuted (false), which means
that the Law of the Excluded Middle "doesn't apply"
and, therefore, that our expression is *not* a mathematical
proposition.



Together, Wittgenstein's finitism and his criterion of
algorithmic decidability shed considerable light on his highly
controversial remarks about putatively *meaningful* conjectures
such as FLT and GC. GC is not a mathematical proposition because we do
not *know how* to decide it, and if someone like G. H. Hardy
says that he 'believes' GC is true (*PG* 381;
*LFM* 123; *PI* SS578), we must answer that s/he
only "has a hunch about the possibilities of extension of the
present system" (*LFM* 139)--that one can only
*believe* such an expression is 'correct' if one
knows *how* to prove it. The only sense in which GC (or FLT)
can be proved is that it can "correspond to a *proof* by
induction", which means that the unproved inductive step (e.g.,
"\(G(n) \rightarrow G(n + 1)\)") and the expression
"\(\forall nG(n)\)" are not mathematical propositions
because we have no algorithmic means of looking for an induction
(*PG* 367). A "general proposition" is senseless
prior to an inductive proof "because the question would only
have made sense if a general method of decision had been known
*before* the particular proof was discovered"
(*PG* 402). Unproved 'inductions' or inductive
steps are not meaningful propositions because the Law of the Excluded
Middle does not hold in the sense that we do not know of a decision
procedure 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' such
as GC. The intermediate Wittgenstein says only that "[a]
mathematician is... guided by... certain analogies with the
previous system" and that there is nothing "wrong or
illegitimate if anyone concerns himself with Fermat's Last
Theorem" (*WVC* 144).



> 
> 
> If e.g. I have a method for looking at integers that satisfy the
> equation \(x^2 + y^2 = z^2\), 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 a *stimulus*--but not a *question*.
> Mathematical problems are always such stimuli. (*WVC* 144, Jan.
> 1, 1931)
> 
> 
> 



More specifically, a mathematician may let a senseless conjecture such
as FLT stimulate her/him if s/he wishes to know whether a calculus can
be extended without altering its axioms or rules (*LFM*
139).



> 
> 
> What is here going [o]n [in an attempt to decide GC] is an
> unsystematic attempt at constructing a calculus. If the attempt is
> successful, I shall again have a calculus in front of me, *only a
> different 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., we
prove "\(G(1)\)" and "\(G(n) \rightarrow G(n +
1)\)"), we will then have a proof of the inductive step, but
since the inductive step was not algorithmically decidable beforehand
(*PR* SSSS148, 155, 157; *PG* 380), in
constructing the proof we have constructed a *new* calculus, a
new *calculating machine* (*WVC* 106) in which we
*now know how* to use this new "machine-part"
(*RFM* VI, SS13) (i.e., the unsystematically proved
inductive step). Before the proof, the inductive step is not a
mathematical proposition with sense (in a particular calculus),
whereas after the proof the inductive step *is* a mathematical
proposition, with a new, determinate sense, in a newly created
calculus. This demarcation of expressions without mathematical sense
and proved or refuted propositions, each with a determinate sense in a
particular calculus, is a view that Wittgenstein articulates in myriad
different ways from 1929 through 1944.



Whether or not it is ultimately defensible--and this is an
absolutely crucial question for Wittgenstein's Philosophy of
Mathematics--this strongly counter-intuitive aspect of
Wittgenstein's account of algorithmic decidability, proof, and
the *sense* of a mathematical proposition is a piece with his
rejection of *predeterminacy* in mathematics. Even in the case
where we algorithmically decide a mathematical proposition, the
connections thereby made do not pre-exist the algorithmic decision,
which means that even when we have a "mathematical
question" that we decide by decision procedure, the expression
only has a determinate sense *qua* proposition when it is
decided. On Wittgenstein's account, both middle and later,
"[a] new proof gives the proposition a place in a new
system" (*RFM* VI, SS13), it "locates it in the
whole system of calculations", though it "does not
mention, certainly does not describe, the whole system of calculation
that stands behind the proposition and gives it sense"
(*RFM* VI, SS11).



Wittgenstein's unorthodox position here is a type of
structuralism that partially results from his rejection of
mathematical semantics. We erroneously think, e.g., that GC has a
fully determinate sense because, given "the misleading way in
which the mode of expression of word-language represents the sense of
mathematical propositions" (*PG* 375), we call to mind
false pictures and mistaken, *referential* conceptions of
mathematical propositions whereby GC is *about* a mathematical
reality and so has just a determinate sense as "There exist
intelligent beings elsewhere in the universe" (i.e., a
proposition that *is* determinately true or false, whether or
not we ever know its truth-value). Wittgenstein breaks with this
tradition, in *all* of its forms, stressing that, in
mathematics, unlike the realm of contingent (or empirical)
propositions, "if I am to know what a proposition like
Fermat's last theorem says", I must know its
*criterion* of truth. Unlike the criterion of truth for an
empirical proposition, which can be known *before* the
proposition is decided, we cannot know the criterion of truth for an
undecided mathematical proposition, though we are "acquainted
with criteria for the truth of *similar* propositions"
(*RFM* VI, SS13).


### 2.5 Wittgenstein's Intermediate Account of Irrational Numbers



The intermediate Wittgenstein spends a great deal of time wrestling
with real and irrational numbers. There are two distinct reasons for
this.



First, the *real* reason many of us are unwilling to abandon
the notion of the actual infinite in mathematics is the prevalent
conception of an irrational number as a *necessarily* infinite
extension. "The confusion in the concept of the 'actual
infinite' *arises*" (italics added), says
Wittgenstein (*PG* 471), 



> 
> 
> from the unclear concept of irrational number, that is, from the fact
> that logically very different things are called 'irrational
> numbers' without any clear limit being given to the concept.
> 
> 
> 



Second, and more fundamentally, the intermediate Wittgenstein wrestles
with irrationals in such detail because he opposes foundationalism and
especially its concept of a "gapless *mathematical*
continuum", its concept of a *comprehensive* theory of
the real numbers (Han 2010), and set theoretical conceptions and
'proofs' as a foundation for arithmetic, real number
theory, and mathematics as a whole. Indeed, Wittgenstein's
discussion of irrationals is one with his critique of set theory, for,
as he says, "[m]athematics is ridden through and through with
the pernicious idioms of set theory", such as "the way
people speak of a line as composed of points", when, in fact,
"[a] line is a law and isn't composed of anything at
all" (*PR* SS173; *PR* SSSS181, 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 exclusively
of extensions and intensions (i.e., 'rules' or
'laws'), an irrational is only an extension insofar as it
is a sign (i.e., a 'numeral', such as
'\(\sqrt{2}\)' or '\(\pi\)'). Given that there
is no such thing as an infinite mathematical *extension*, it
follows that an irrational number is not a unique *infinite
expansion*, but rather a unique recursive rule or *law*
(*PR* SS181) that yields rational numbers (*PR*
SS186; *PR* SS180).



> 
> 
> The rule for working out places of \(\sqrt{2}\) is itself the numeral
> for the irrational number; and the reason I here speak of a
> 'number' is that I can calculate with these signs (certain
> rules for the construction of rational numbers) just as I can with
> rational numbers themselves. (*PG* 484)
> 
> 
> 



Due, however, to his anti-foundationalism, Wittgenstein takes the
radical position that not all recursive real numbers (i.e., computable
numbers) are genuine real numbers--a position that distinguishes
his view from even Brouwer's.



The problem, as Wittgenstein sees it, is that mathematicians,
especially foundationalists (e.g., set theorists), have sought to
accommodate physical continuity by a theory that
'describes' the mathematical continuum (*PR*
SS171). When, for example, we think of continuous motion and the
(mere) density of the rationals, we reason that if an object moves
continuously from A to B, and it travels *only* the distances
marked by "rational points", then it must *skip*
some distances (intervals, or points) *not* marked by rational
numbers. But if an object in continuous motion travels distances that
*cannot* be commensurately measured by rationals alone, there
must be 'gaps' between the rationals (*PG* 460),
and so we must fill them, first, with recursive irrationals, and then,
because "the set of *all* recursive irrationals"
still leaves gaps, with "lawless irrationals".



> 
> 
> [T]he enigma of the continuum arises because language misleads us into
> applying to it a picture that doesn't fit. Set theory preserves
> the inappropriate picture of something discontinuous, but makes
> statements about it that contradict the picture, under the impression
> that it is breaking with prejudices; whereas what should really have
> been done is to point out that the picture just doesn't
> fit... (*PG* 471)
> 
> 
> 



We add nothing that is needed to the differential and integral calculi
by 'completing' a theory of real numbers with
pseudo-irrationals and lawless irrationals, first because there are no
gaps on the number line (*PR* SSSS181, 183, & 191;
*PG* 373, 460, 461, & 473; *WVC* 35) and, second,
because these alleged irrational numbers are not needed for a theory
of the 'continuum' simply because there is no mathematical
continuum. As the later Wittgenstein says (*RFM* V, SS32),
"[t]he picture of the number line is an absolutely natural one
up to a certain point; that is to say so long as it is not used for a
general theory of real numbers". We have gone awry by
misconstruing the nature of the geometrical line as a continuous
collection of points, each with an associated real number, which has
taken us well beyond the 'natural' picture of the number
line in search of a "general theory of real numbers" (Han
2010).



Thus, the principal reason Wittgenstein rejects certain constructive
(computable) numbers is that they are unnecessary creations which
engender conceptual confusions in mathematics (especially set theory).
One of Wittgenstein's main aims in his lengthy discussions of
rational numbers and pseudo-irrationals is to show that
pseudo-irrationals, which are allegedly needed for the mathematical
continuum, 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, less
than, or equal to a rational number" (*PR* SS191))
and (b) that "[a] number must measure in and of itself"
and if a 'number' "leaves it to the rationals, we
have no need of it" (*PR* SS191) (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 genuine
real numbers because they fail to satisfy (a) and (b), Wittgenstein
defines 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* SS182); he similarly defines
\(\pi '\) as

 

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


(*PR* SS186) 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 to
Wittgenstein because, instead of using "the idioms of
arithmetic" (*PR* SS186), it is dependent upon the
particular 'incidental' notation of a particular system
(i.e., in some particular base) (*PR* SS188; *PR*
SS182; and *PG* 475). If we speak of various
base-notational systems, we might say that \(\pi\) belongs to
*all* systems, while \(\pi '\) belongs only to one, which shows
that \(\pi '\) is not a genuine irrational because "there
can't be irrational numbers of different types"
(*PR* SS180). Furthermore, pseudo-irrationals do
*not* measure because they are homeless, artificial
constructions parasitic upon numbers which have a natural place in a
calculus that can be used to measure. We simply do not need these
aberrations, because they are not sufficiently comparable to rationals
and genuine irrationals. They are *not* irrational numbers
according to Wittgenstein's criteria, which define, Wittgenstein
interestingly asserts, "precisely what has been meant or looked
for under the name 'irrational number'" (*PR*
SS191).



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



> 
> 
> [W]e cannot say that the decimal fractions developed in accordance
> with a law still need supplementing by an infinite set of irregular
> infinite decimal fractions that would be 'brushed under the
> carpet' if we were to *restrict* ourselves to those
> *generated by a law*, 
> 
> 
> 



Wittgenstein argues, for "[w]here is there such an infinite
decimal that is generated by no law" "[a]nd how would we
notice that it was missing?" (*PR* SS181; cf.
*PG* 473, 483-84). Similarly, a free-choice sequence,
like a recipe for "endless bisection" or "endless
dicing", is not an infinitely complicated *mathematical
law* (or rule), but rather no law at all, for after each
individual throw of a coin, the point remains "infinitely
indeterminate" (*PR* SS186). For closely related
reasons, Wittgenstein ridicules the Multiplicative Axiom (Axiom of
Choice) both in the middle period (*PR* SS146) and in the
latter period (*RFM* V, SS25; VII, SS33).


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



Superficially, at least, it seems as if Wittgenstein is offering an
essentialist argument for the conclusion that real number arithmetic
*should not* be extended in such-and-such a way. Such an
*essentialist* account of real and irrational numbers seems to
conflict with the actual freedom mathematicians have to extend and
invent, with Wittgenstein's intermediate claim (*PG* 334)
that "[f]or [him] one calculus is as good as another", and
with Wittgenstein's acceptance of complex and imaginary numbers.
Wittgenstein's foundationalist critic (e.g., set theorist) will
undoubtedly say that we have extended the term "irrational
number" to lawless and pseudo-irrationals because they are
needed for the mathematical continuum and because such
"conceivable numbers" are much more like rule-governed
irrationals than rationals.



Though Wittgenstein stresses differences where others see similarities
(*LFM* 15), in his intermediate attacks on pseudo-irrationals
and foundationalism, he is not just emphasizing differences, he is
attacking set theory's "pernicious idioms"
(*PR* SS173) and its "crudest imaginable
misinterpretation of its own calculus" (*PG*
469-70) in an attempt to dissolve "misunderstandings
without 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 within
mathematics, and they have proved their mettle in scientific
applications, but pseudo-irrationals are *inorganic* creations
invented solely for the sake of mistaken foundationalist aims.
Wittgenstein's main point is *not* that we cannot create
further recursive real numbers--indeed, we can create as many as
we want--his point is that we can only really speak of different
*systems* (sets) of real numbers (*RFM* II, SS33)
that are enumerable by a rule, and any attempt to speak of "the
set of all real numbers" or any piecemeal attempt to add or
consider new recursive reals (e.g., diagonal numbers) is a useless
and/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 not
included in *PR* or *PG*, Wittgenstein says: "The
concept 'irrational number' is a dangerous
pseudo-concept" (MS 108, 176; 1930; TS 210, 29; 1930). As we
shall see in the next section, on Wittgenstein's account, if we
do not understand irrationals rightly, we *cannot but* engender
the mistakes that constitute set theory.


### 2.6 Wittgenstein's Intermediate Critique of Set Theory



Wittgenstein's critique of set theory begins somewhat benignly
in the *Tractatus*, where he denounces Logicism and says
(6.031) that "[t]he theory of classes is completely superfluous
in mathematics" because, at least in part, "the generality
required in mathematics is not accidental generality". In his
middle period, Wittgenstein begins a full-out assault on set theory
that never abates. Set theory, he says, is "utter
nonsense" (*PR* SSSS145, 174; WVC 102;
*PG* 464, 470), 'wrong' (*PR* SS174),
and 'laughable' (*PG* 464); its "pernicious
idioms" (*PR* SS173) mislead us and the crudest
possible 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) his
discussion of the intension-extension distinction, and (2) his
criticism of non-denumerability *as cardinality*. Late in the
middle period, Wittgenstein seems to become more aware of the
unbearable conflict between his *strong formalism* (*PG*
334) and his denigration of set theory as a purely formal,
*non*-mathematical calculus (Rodych 1997: 217-219),
which, as we shall see in
 Section 3.5,
 leads to the use of an extra-mathematical application criterion to
demarcate 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 and
mathematical continuity has led to a "fictitious
symbolism" (*PR* SS174).



> 
> 
> Set theory attempts to grasp the infinite at a more general level than
> the investigation of the laws of the real numbers. It says that you
> can't grasp the actual infinite by means of mathematical
> symbolism at all and therefore it can only be described and not
> represented. ... One might say of this theory that it buys a pig
> in a poke. Let the infinite accommodate itself in this box as best it
> can. (*PG* 468; cf. *PR* SS170)
> 
> 
> 



As Wittgenstein puts it at (*PG* 461), 



> 
> 
> the mistake in the set-theoretical approach consists time and again in
> treating laws and enumerations (lists) as essentially the same kind of
> thing and arranging them in parallel series so that one fills in gaps
> left 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 a
description", for "[t]he one is not a substitute for the
other" (*WVC* 102; June 19, 1930); "there
isn't a dualism [of] the law and the infinite series obeying
it" (*PR* SS180).



"Set theory is wrong" and nonsensical (*PR*
SS174), says Wittgenstein, because it presupposes a fictitious
symbolism of infinite signs (*PG* 469) instead of an actual
symbolism with finite signs. The grand intimation of set theory, which
begins with "Dirichlet's concept of a function"
(*WVC* 102-03), is that we can *in principle*
represent an infinite set by an enumeration, but because of human or
physical limitations, we will instead *describe* it
intensionally. But, says Wittgenstein, "[t]here can't be
possibility and actuality in mathematics", for mathematics is an
*actual* calculus, which "is concerned only with the
signs with which it *actually* operates" (*PG*
469). As Wittgenstein puts it at (*PR* SS159), the fact
that "we can't describe mathematics, we can only do
it" in and "of itself abolishes every 'set
theory'".



Perhaps the best example of this phenomenon is Dedekind, who in giving
his 'definition' of an "infinite class" as
"a class which is similar to a proper subclass of itself"
(*PG* 464), "tried to *describe* an infinite
class" (*PG* 463). If, however, we try to apply this
'definition' to a particular class in order to ascertain
whether it is finite or infinite, the attempt is
'laughable' if we apply it to a *finite* class,
such as "a certain row of trees", and it is
'nonsense' if we apply it to "an infinite
class", for we cannot even attempt "to co-ordinate
it" (*PG* 464), because "the relation \(m = 2n\)
[does not] correlate the class of all numbers with one of its
subclasses" (*PR* SS141), it is an "infinite
process" which "correlates any arbitrary number with
another". So, although we *can* use \(m = 2n\) on the
*rule* for generating the naturals (i.e., our domain) and
thereby construct the pairs (2,1), (4,2), (6,3), (8,4), etc., in doing
so we do not correlate two *infinite* sets or extensions
(*WVC* 103). If we try to apply Dedekind's definition as
a *criterion* for determining whether a given set is infinite
by establishing a 1-1 correspondence between two inductive rules
for generating "infinite extensions", one of which is an
"extensional subset" of the other, we can't possibly
learn anything we didn't already know when we applied the
'criterion' to two inductive rules. If Dedekind or anyone
else insists on calling an inductive rule an "infinite
set", he and we must still mark the categorical difference
between such a set and a finite set with a determinate, finite
cardinality.



Indeed, on Wittgenstein's account, the failure to properly
distinguish mathematical extensions and intensions is the root cause
of the mistaken interpretation of Cantor's diagonal proof as a
proof of the existence of infinite sets of lesser and greater
cardinality.


#### 2.6.2 Against Non-Denumerability



Wittgenstein's criticism of non-denumerability is primarily
implicit during the middle period. Only after 1937 does he provide
concrete arguments purporting to show, e.g., that Cantor's
diagonal *cannot* prove that some infinite sets have greater
'multiplicity' than others.



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



> 
> 
> When people say 'The set of all transcendental numbers is
> greater 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*
> SS174)
> 
> 
> 



As with his intermediate views on genuine irrationals and the
Multiplicative Axiom, Wittgenstein here looks at the diagonal proof of
the non-denumerability of "the set of transcendental
numbers" as one that shows only that transcendental numbers
cannot be recursively enumerated. It is nonsense, he says, to go from
the warranted conclusion that these numbers are not, in principle,
enumerable to the conclusion that the *set* of transcendental
numbers is greater in cardinality than the set of algebraic numbers,
which is recursively enumerable. What we have here are two very
different conceptions of a number-type. In the case of algebraic
numbers, we have a decision procedure for determining of any given
number whether or not it is algebraic, *and* we have a method
of enumerating the algebraic numbers such that we can *see*
that '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 cannot
recursively enumerate each and every thing we would call a
"transcendental number".



At (*PG* 461), Wittgenstein similarly speaks of set
theory's "mathematical pseudo-concepts" leading to a
fundamental difficulty, which begins when we unconsciously presuppose
that there is sense to the idea of ordering the rationals by
size--"that the *attempt* is
thinkable"--and culminates in similarly thinking that it is
possible to enumerate *the real numbers*, which we then
discover is impossible.



Though the intermediate Wittgenstein certainly seems highly critical
of the alleged proof that some infinite sets (e.g., the reals) are
greater in cardinality than other infinite sets, and though he
discusses the "diagonal procedure" in February 1929 and in
June 1930 (MS 106, 266; MS 108, 180), along with a diagonal diagram,
these and other early-middle ruminations did not make it into the
typescripts for either *PR* or *PG*. As we shall see in
 Section 3.4,
 the later Wittgenstein analyzes Cantor's diagonal and claims of
non-denumerability in some detail.


## 3. The Later Wittgenstein on Mathematics: Some Preliminaries



The first and most important thing to note about Wittgenstein's
later Philosophy of Mathematics is that *RFM*, first published
in 1956, consists of *selections* taken from a number of
manuscripts (1937-1944), most of one large typescript (1938),
and three short typescripts (1938), each of which constitutes an
Appendix to (*RFM* I). For this reason and because some
manuscripts containing much material on mathematics (e.g., MS 123)
were not used at all for
*RFM*, philosophers have not been able to read
Wittgenstein's later remarks on mathematics as they were written
in the manscripts used for *RFM* and they have not had access
(until the 2000-2001 release of the *Nachlass* on CD-ROM)
to much of Wittgenstein's later work on mathematics. It must be
emphasized, therefore, that this *Encyclopedia* article is
being written during a transitional period. Until philosophers have
used the *Nachlass* to build a comprehensive picture of
Wittgenstein's complete and evolving Philosophy of Mathematics,
we will not be able to say definitively which views the later
Wittgenstein retained, which he changed, and which he dropped. In the
interim, this article will outline Wittgenstein's later
Philosophy of Mathematics, drawing primarily on *RFM*, to a
much lesser extent *LFM* (1939 Cambridge lectures), and, where
possible, previously unpublished material in Wittgenstein's
*Nachlass*.



It should also be noted at the outset that commentators disagree about
the continuity of Wittgenstein's middle and later Philosophies
of Mathematics. Some argue that the later views are significantly
different from the intermediate views (Frascolla 1994; Gerrard 1991:
127, 131-32; Floyd 2005: 105-106), while others argue
that, for the most part, Wittgenstein's Philosophy of
Mathematics evolves from the middle to the later period without
significant changes or renunciations (Wrigley 1993; Marion 1998). The remainder of this article adopts the
second interpretation, explicating Wittgenstein's later
Philosophy of Mathematics as largely continuous with his intermediate
views, except for the important introduction of an extra-mathematical
application criterion.


### 3.1 Mathematics as a Human Invention



Perhaps the most important constant in Wittgenstein's Philosophy
of Mathematics, middle and late, is that he consistently maintains
that mathematics is our, human invention, and that, indeed, everything
in mathematics is invented. Just as the middle Wittgenstein says that
"[w]e *make* mathematics", the later Wittgenstein
says that we 'invent' mathematics (*RFM* I,
SS168; II, SS38; V, SSSS5, 9 and 11; *PG*
469-70) and that "the mathematician is not a discoverer:
he is an inventor" (*RFM*, Appendix II, SS2;
(*LFM* 22, 82). Nothing *exists* mathematically unless
and until we have invented it.



In arguing against mathematical discovery, Wittgenstein is not just
rejecting Platonism, he is also rejecting a rather standard
philosophical view according to which human beings invent mathematical
calculi, but once a calculus has been invented, we thereafter discover
finitely many of its infinitely many provable and true theorems. As
Wittgenstein himself asks (*RFM* IV, SS48), "might it
not be said that the *rules* lead this way, even if no one went
it?" If "someone produced a proof [of
'Goldbach's theorem']",
"[c]ouldn't one say", Wittgenstein asks
(*LFM* 144), "that the *possibility* of this proof
was a fact in the realms of mathematical reality"--that
"[i]n order [to] find it, it must in some sense be
there"--"[i]t must be a possible
structure"?



Unlike many or most philosophers of mathematics, Wittgenstein resists
the 'Yes' answer that we discover truths about a
mathematical calculus that *come into existence* the moment we
invent the calculus (*PR* SS141; *PG* 283, 466;
*LFM* 139). Wittgenstein rejects the modal reification of
possibility as actuality--that provability and constructibility
are (actual) facts--by arguing that it is at the very least
wrong-headed to say with the Platonist that because "a straight
line *can* be drawn between any two points,... the line
already exists even if no one has drawn it"--to say
"[w]hat in the ordinary world we call a possibility is in the
geometrical world a reality" (*LFM* 144; *RFM* I,
SS21). One might as well say, Wittgenstein suggests (*PG*
374), that "chess only had to be *discovered*, it was
always there!"



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



> 
> 
> I want to get away from the formulation: "I now know more about
> the calculus", and replace it with "I now have a different
> calculus". The sense of this is always to keep before
> one's eyes the full scale of the gulf between a mathematical
> knowing 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: the
proof of the Fermat proposition is not to be discovered, but to be
*invented*".



> 
> 
> The difference between the 'anthropological' and the
> mathematical account is that in the first we are not tempted to speak
> of 'mathematical facts', but rather that in this account
> the *facts* are never mathematical ones, never make
> *mathematical* propositions true or false. (MS 117, 263; March
> 15, 1940)
> 
> 
> 



There are no mathematical facts just as there are no (genuine)
mathematical propositions. Repeating his intermediate view, the later
Wittgenstein says (MS 121, 71v; 27 Dec., 1938): "Mathematics
consists of [calculi | calculations], not of propositions". This
radical constructivist conception of mathematics prompts Wittgenstein
to make notorious remarks--remarks that virtually no one else
would make--such as the infamous (*RFM* V, SS9):
"However queer it sounds, the further expansion of an irrational
number 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 that
mathematics is essentially syntactical and non-referential, which, in
and of itself, makes Wittgenstein's philosophy of mathematics
anti-Platonist insofar as Platonism is the view that mathematical
terms and propositions *refer* to objects and/or facts and that
mathematical propositions are *true* by virtue of agreeing with
*mathematical facts*.



The later Wittgenstein, however, wishes to 'warn' us that
our thinking is saturated with the idea of "[a]rithmetic as the
natural history (mineralogy) of numbers" (*RFM* IV,
SS11). When, for instance, Wittgenstein discusses the claim that
fractions cannot be ordered by magnitude, he says that this sounds
'remarkable' in a way that a mundane proposition of the
differential calculus does not, for the latter proposition is
associated with an application in physics, 



> 
> 
> whereas *this proposition* ... seems to
> [solely] concern... the natural history of
> mathematical objects themselves. (*RFM* II, SS40) 
> 
> 
> 



Wittgenstein stresses that he is trying to 'warn' us
against this 'aspect'--the idea that the foregoing
proposition about fractions "introduces us to the mysteries of
the mathematical world", which exists somewhere as a completed
totality, awaiting our prodding and our discoveries. The fact that we
regard mathematical propositions as being about mathematical objects
and mathematical investigation "as the exploration of these
objects" is "already mathematical alchemy", claims
Wittgenstein (*RFM* V, SS16), since 



> 
> 
> it is not possible to appeal to the meaning
> [*Bedeutung*] of the signs in
> mathematics,... because it is only mathematics that gives them
> their meaning [*Bedeutung*]. 
> 
> 
> 



Platonism is *dangerously misleading*, according to
Wittgenstein, because it suggests a picture of *pre*-existence,
*pre*determination and discovery that is completely at odds
with what we find if we actually examine and describe mathematics and
mathematical activity. "I should like to be able to
describe", says Wittgenstein (*RFM* IV, SS13),
"how it comes about that mathematics appears to us now as the
natural history of the domain of numbers, now again as a collection of
rules".



Wittgenstein, however, does *not* endeavour to *refute*
Platonism. His aim, instead, is to clarify what Platonism is and what
it says, implicitly and explicitly (including variants of Platonism
that claim, e.g., that if a proposition is *provable* in an
axiom system, then there already exists a path [i.e., a proof] from
the axioms to that proposition (*RFM* I, SS21; Marion 1998:
13-14, 226; Steiner 2000:
334). Platonism is either "a mere truism" (*LFM*
239), Wittgenstein says, or it is a 'picture' consisting
of "an infinity of shadowy worlds" (*LFM* 145),
which, as such, lacks 'utility' (cf. *PI*
SS254) because it explains nothing and it misleads at every
turn.


### 3.2 Wittgenstein's Later Finitistic Constructivism



Though commentators and critics do not agree as to whether the later
Wittgenstein is still a finitist and whether, if he is, his finitism
is as radical as his intermediate rejection of unbounded mathematical
quantification (Maddy 1986: 300-301, 310), the overwhelming
evidence indicates that the later Wittgenstein still rejects the
actual infinite (*RFM* V, SS21; *Zettel* SS274,
1947) and infinite mathematical extensions.



The first, and perhaps most definitive, indication that the later
Wittgenstein maintains his finitism is his continued and consistent
insistence that irrational numbers are rules for constructing finite
expansions, *not* infinite mathematical extensions. "The
concepts of infinite decimals in mathematical propositions are not
concepts of series", says Wittgenstein (*RFM* V,
SS19), "but of the unlimited technique of expansion of
series". We are misled by "[t]he extensional definitions
of functions, of real numbers etc". (*RFM* V, SS35),
but once we recognize the Dedekind cut as "an extensional
*image*", we see that we are not "led to
\(\sqrt{2}\) by way of the concept of a cut" (*RFM* V,
SS34). On the later Wittgenstein's account, there simply is
no *property*, no *rule*, no *systematic means*
of defining each and every irrational number *intensionally*,
which means there is *no criterion* "for the irrational
numbers being *complete*" (*PR* SS181).



As in his intermediate position, the later Wittgenstein claims that
'\(\aleph\_0\)' and "infinite series" get their
mathematical uses from the use of 'infinity' in ordinary
language (*RFM* II, SS60). Although, in ordinary language,
we often use 'infinite' and "infinitely many"
as answers to the question "how many?", and though we
associate infinity with the enormously large, the principal
*use* we make of 'infinite' and
'infinity' is to speak of *the unlimited*
(*RFM* V, SS14) and unlimited *techniques*
(*RFM* II, SS45; *PI* SS218). This fact is
brought out by the fact "that the technique of learning
\(\aleph\_0\) numerals is different from the technique of learning
100,000 numerals" (*LFM* 31). When we say, e.g., that
"there are an infinite number of even numbers" we mean
that we have a mathematical technique or rule for generating even
numbers which is *limitless*, which is markedly different from
a limited technique or rule for generating a finite number of numbers,
such as 1-100,000,000. "We learn an endless
technique", says Wittgenstein (*RFM* V, SS19),
"but what is in question here is not some gigantic
extension".



An infinite sequence, for example, is not a gigantic extension because
it is not an extension, and '\(\aleph\_0\)' is not a
cardinal number, for "how is this picture connected with the
*calculus*", given that "its connexion is not that
of the picture | | | | with 4" (i.e., given that
'\(\aleph\_0\)' is not connected to a (finite) extension)?
This shows, says Wittgenstein (*RFM* II, SS58), that we
ought to avoid the word 'infinite' in mathematics wherever
it seems to give a meaning to the calculus, rather than acquiring its
meaning from the calculus and its use in the calculus. Once we see
that the calculus contains nothing infinite, we should not be
'disappointed' (*RFM* II, SS60), but simply
note (*RFM* II, SS59) that it is not "really
necessary... to conjure up the picture of the infinite (of the
enormously big)".



A second strong indication that the later Wittgenstein maintains his
finitism is his continued and consistent treatment of
'propositions' of the type "There are three
consecutive 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 7s
in the decimal expansion of \(\pi\)") is *not* a
meaningful mathematical "statement at all" (*WVC*
81-82: note 1). On Wittgenstein's intermediate view,
PIC--like FLT, GC, and the Fundamental Theorem of
Algebra--is *not* a mathematical proposition because we do
not have in hand an applicable decision procedure by which we can
decide it in a particular calculus. For this reason, we can only
meaningfully state *finitistic* propositions regarding the
expansion of \(\pi\), such as "There exist three consecutive 7s
in 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 in
*RFM* (Bernays 1959: 11-12). For example, to someone who says that since "the
rule of expansion *determine*[\(s\)] the series
completely", "it must implicitly determine *all*
questions about the structure of the series", Wittgenstein
replies: "Here you are thinking of finite series"
(*RFM* V, SS11). If PIC were a *mathematical*
question (or problem)--if it were finitistically
restricted--it would be algorithmically decidable, which it is
not (*RFM* V, SS21; *LFM* 31-32, 111, 170;
*WVC* 102-03). As Wittgenstein says at (*RFM* V,
SS9): "The question... changes its status, when it
becomes decidable", "[f]or a connexion is made then, which
formerly *was not there*". And if, moreover, one invokes
the Law of the Excluded Middle to establish that PIC is a mathematical
proposition--i.e., by saying that one of these "two
pictures... must correspond to the fact" (*RFM* V,
SS10)--one simply begs the question (*RFM* V,
SS12), 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, SS41; V, SS13).



Wittgenstein's finitism, constructivism, and conception of
mathematical decidability are interestingly connected at (*RFM*
VII, SS41, par. 2-5).



> 
> 
> What harm is done e.g. by saying that God knows *all*
> irrational numbers? Or: that they are already there, even though we
> only know certain of them? Why are these pictures not harmless?
> 
> 
> 
> For one thing, they hide certain problems.-- (MS 124: 139; March
> 16, 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 his
> *omniscience* decide whether they *would* have reached
> it after the end of the world? It cannot. I want to say: Even God can
> determine something mathematical only by mathematics. Even for him the
> mere rule of expansion cannot decide anything that it does not decide
> for us.
> 
> 
> 
> We might put it like this: if the rule for the expansion has been
> given us, a *calculation* can tell us that there is a
> '2' at the fifth place. Could God have known this, without
> the 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 omniscience
*might*, by calculation, find that '777' occurs at
the interval [\(n,n+2\)], but, on the other hand, God might go on
calculating forever without '777' ever turning up. Since
\(\pi\) is not a *completed* infinite extension that can be
completely surveyed by an omniscient being (i.e., it is not a fact
that 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, with
our modest minds, an omniscient mind (i.e., God) can only calculate
the expansion of \(\pi\) to some \(n\)th decimal
place--where our \(n\) is minute and God's \(n\) is
(relatively) enormous--and at no \(n\)th decimal place
could *any mind* rightly conclude that because
'777' has not turned up, it, therefore, will never turn
up.


### 3.3 The Later Wittgenstein on Decidability and Algorithmic Decidability



On one fairly standard interpretation, the later Wittgenstein says
that "true in calculus \(\Gamma\)" is identical to
"provable in calculus \(\Gamma\)" and, therefore, that a
mathematical proposition of calculus \(\Gamma\) is a concatenation of
signs that is either provable (in principle) or refutable (in
principle) in calculus \(\Gamma\) (Goodstein 1972: 279, 282; Anderson
1958: 487; Klenk 1976: 13; Frascolla 1994: 59). On this
interpretation, the later Wittgenstein precludes undecidable
mathematical propositions, but he allows that some *undecided*
expressions are propositions *of* a calculus because they are
decidable in principle (i.e., in the absence of a known, applicable
decision procedure).



There is considerable evidence, however, that the later Wittgenstein
maintains his intermediate position that an expression is a meaningful
mathematical proposition only *within* a given calculus and
*iff* we knowingly have in hand an applicable and effective
decision 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, SS6,
SS8), he does so in order to use the former to consider the
alleged conclusion of Godel's proof (i.e., that there exist
true but unprovable mathematical propositions), which he then rebuts
with his own identification of "true in calculus
\(\Gamma\)" with "*proved* in calculus
\(\Gamma\)" (i.e., *not* with "*provable* in
calculus \(\Gamma\)") (Wang 1991: 253; Rodych 1999a: 177). This
construal is corroborated by numerous passages in which Wittgenstein
rejects the received view that a prov*able* but unproved
proposition is true, as he does when he asserts that (*RFM*
III, SS31, 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, SS10, 1941) that "[a] new proof gives the
proposition a place in a new system". Furthermore, as we have
just seen, Wittgenstein rejects PIC as a non-proposition on the
grounds that it is not algorithmically decidable, while admitting
finitistic versions of PIC because they are algorithmically
decidable.



Perhaps the most compelling evidence that the later Wittgenstein
maintains algorithmic decidability as his criterion for a mathematical
proposition lies in the fact that, at (*RFM* V, SS9, 1942),
he says in two distinct ways that a mathematical
'question' can *become* decidable and that when
this *happens*, a new connexion is '*made*'
which previously did not exist. Indeed, Wittgenstein cautions us
against appearances by saying that "it *looks* as if a
ground for the decision were already there", when, in fact,
"it has yet to be invented". These passages strongly
militate against the claim that the later Wittgenstein grants that
proposition \(\phi\) is decidable in calculus \(\Gamma\) iff it is
provable or refutable *in principle*. Moreover, if Wittgenstein
held *this* position, he would claim, contra (*RFM* V,
SS9), that a question or proposition does not *become*
decidable since it simply (always) *is* decidable. If it is
provable, and we simply don't yet know this to be the case,
there *already is* a connection between, say, our axioms and
rules and the proposition in question. What Wittgenstein says,
however, is that the modalities *provable* and
*refutable* are shadowy forms of reality--that possibility
is not actuality in mathematics (*PR* SSSS141, 144,
172; *PG* 281, 283, 299, 371, 466, 469; *LFM* 139).
Thus, the later Wittgenstein agrees with the intermediate Wittgenstein
that the only sense in which an *undecided* mathematical
proposition (*RFM* VII, SS40, 1944) can be
*decidable* is in the sense that we *know* how to decide
it 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 his *anti-foundationalism* and his
criticism of the extension-intension conflation, Wittgenstein's
later critique of set theory is highly consonant with his intermediate
critique (*PR* SSSS109, 168; *PG* 334, 369, 469;
*LFM* 172, 224, 229; and *RFM* III, SS43, 46, 85,
90; VII, SS16). Given that mathematics is a "MOTLEY of
techniques of proof" (*RFM* III, SS46), it does not
require a foundation (*RFM* VII, SS16) and it cannot be
given a *self-evident* foundation (*PR* SS160;
*WVC* 34 & 62; *RFM* IV, SS3). Since set theory
was invented to provide mathematics with a foundation, it is,
minimally, unnecessary.



Even if set theory is unnecessary, it still might constitute a solid
foundation for mathematics. In his core criticism of set theory,
however, the later Wittgenstein denies this, saying that the diagonal
proof does not prove non-denumerability, for "[i]t means nothing
to say: '*Therefore* the X numbers are not
denumerable'" (*RFM* II, SS10). When the diagonal is
construed as a *proof* of greater and lesser infinite sets it
is a "puffed-up proof", which, as Poincare argued
(1913: 61-62), purports to prove or show more than "its
means allow it" (*RFM* II, SS21).



> 
> 
> If it were said: Consideration of the diagonal procedure shews you that the
> *concept* 'real number' has much less analogy with
> the concept 'cardinal number' than we, being misled by
> certain analogies, are inclined to believe, that would have a good and honest sense. But just the
> *opposite* happens: one pretends to compare the
> 'set' of real numbers in magnitude with that of cardinal
> numbers. The difference in kind between the two conceptions is
> represented, by a skew form of expression, as difference of extension.
> I believe, and hope, that a future generation will laugh at this hocus
> pocus. (*RFM* II, SS22)
> 
> 
> 
> The sickness of a time is cured by an alteration in the mode of life
> of human beings... (*RFM* II, SS23)
> 
> 
> 



The "hocus pocus" of the diagonal proof rests, as always
for Wittgenstein, on a conflation of extension and intension, on the
failure to properly distinguish sets *as rules* for generating
extensions and (finite) extensions. By way of this confusion "a
difference in kind" (i.e., unlimited rule vs. finite extension)
"is represented by a skew form of expression", namely as a
difference in the *cardinality* of two *infinite*
extensions. Not only can the diagonal *not* prove that one
infinite set is greater in cardinality than another infinite set,
according to Wittgenstein, *nothing* could prove this, simply
because "infinite sets" are not *extensions*, and
hence not *infinite* extensions. But instead of interpreting
Cantor'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 feeling
of paradox" (*LFM* 16-17)--a "giddiness
attacks us when we think of certain theorems in set
theory"--"when we are performing a piece of logical
sleight-of-hand" (*PI* SS412; SS426; 1945). This
giddiness and pleasant feeling of paradox, says Wittgenstein
(*LFM* 16), "may be the chief reason [set theory] was
invented".



Though Cantor's diagonal is not a *proof* of
non-denumerability, when it is expressed in a *constructive
manner*, as Wittgenstein himself expresses it at (*RFM* II,
SS1), "it gives sense to the mathematical proposition that
the number so-and-so is different from all those of the system"
(*RFM* II, SS29). That is, the proof proves
*non-enumerability*: it proves that for any given
*definite* real number concept (e.g., recursive real), one
cannot enumerate 'all' such numbers because one can always
construct a diagonal number, which falls under the same concept and is
not in the enumeration. "One might say", Wittgenstein
says, 



> 
> 
> I call number-concept X non-denumerable if it has been stipulated
> that, whatever numbers falling under this concept you arrange in a
> series, the diagonal number of this series is also to fall under that
> concept. (*RFM* II, SS10; cf. II, SSSS30, 31,
> 13)
> 
> 
> 



One lesson to be learned from this, according to Wittgenstein
(*RFM* II, SS33), is that "there are *diverse
systems* of irrational points to be found in the number
line", each of which can be given by a recursive rule, but
"no system of irrational numbers", and "also no
super-system, no 'set of irrational numbers' of
higher-order infinity". Cantor has shown that we can construct
"infinitely many" diverse systems of irrational numbers,
but we cannot construct an *exhaustive* system of *all*
the irrational numbers (*RFM* II, SS29). As Wittgenstein
says at (MS 121, 71r; Dec. 27, 1938), three pages after the passage
used for (*RFM* II, SS57): 



> 
> 
> If you now call the Cantorian procedure one for producing a new real
> number, you will now no longer be inclined to speak of *a system of
> all real numbers*. (italics added) 
> 
> 
> 



From Cantor's proof, however, set theorists erroneously conclude
that "the set of irrational numbers" is greater in
multiplicity than any enumeration of irrationals (or the set of
rationals), when the only conclusion to draw is that there is no such
thing as *the set of* all *the irrational numbers*. The
truly dangerous aspect to 'propositions' such as
"The real numbers cannot be arranged in a series" and
"The set... is not denumerable" is that they make
concept formation [i.e., our *invention*] "look like a
fact of nature" (i.e., something we *discover*)
(*RFM* II SSSS16, 37). At best, we have a vague idea of
the concept of "real number", but only if we restrict this
idea to "recursive real number" and only if we recognize
that *having* the concept does not mean *having* a set
of 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 later
writings on mathematics is Wittgenstein's (re-)introduction of
an extra-mathematical application criterion, which is used to
distinguish mere "sign-games" from mathematical
language-games. "[I]t is essential to mathematics that its signs
are also employed in *mufti*", Wittgenstein states, for




> 
> 
> [i]t is the use outside mathematics, and so the *meaning*
> [*Bedeutung*] of the signs, that makes the
> sign-game into mathematics. (i.e., a mathematical
> "language-game"; *RFM* V, SS2, 1942;
> *LFM* 140-141, 169-70) 
> 
> 
> 



As Wittgenstein says at (*RFM* V, SS41, 1943), 



> 
> 
> [c]oncepts which occur in 'necessary' propositions
> *must also* occur and have a meaning
> [*Bedeutung*] in non-necessary ones. (italics
> added) 
> 
> 
> 



If two proofs prove the same proposition, says Wittgenstein, this
means that "both demonstrate it as a suitable instrument for the
same purpose", which "is an allusion to *something
outside mathematics*" (*RFM* VII, SS10, 1941;
italics added).



As we have seen, this criterion was present in the *Tractatus*
(6.211), but noticeably absent in the middle period. The reason for
this absence is probably that the intermediate Wittgenstein wanted to
stress that in mathematics everything is syntax and nothing is
meaning. Hence, in his criticisms of Hilbert's
'contentual' mathematics (Hilbert 1925) and
Brouwer's reliance upon intuition to determine the meaningful
content of (especially undecidable) mathematical propositions,
Wittgenstein couched his finitistic constructivism in strong
formalism, emphasizing that a mathematical calculus does not need an
extra-mathematical application (*PR* SS109; *WVC*
105).



There seem to be two reasons why the later Wittgenstein reintroduces
extra-mathematical application as a necessary condition of a
*mathematical* language-game. First, the later Wittgenstein has
an even greater interest in the *use* of natural and formal
languages in diverse "forms of life" (*PI*
SS23), which prompts him to emphasize that, in many cases, a
mathematical 'proposition' functions as if it were an
empirical proposition "hardened into a rule" (*RFM*
VI, SS23) and that mathematics plays diverse applied roles in many
forms of human activity (e.g., science, technology, predictions).
Second, the extra-mathematical application criterion relieves the
tension between Wittgenstein's intermediate critique of set
theory and his strong formalism according to which "one calculus
is as good as another" (*PG* 334). By demarcating
mathematical language-games from non-mathematical sign-games,
Wittgenstein can now claim that, "for the time being", set
theory 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 can *make* the considerations lead us to
> that.
> 
> 
> 
> Or: we can say *this* and give *this* as our reason.
> 
> 
> 
> But if we do say it--what are we to do next? In what practice is
> this proposition anchored? It is for the time being a piece of
> mathematical architecture which hangs in the air, and looks as if it
> were, let us say, an architrave, but not supported by anything and
> supporting nothing. (*RFM* II, SS35)
> 
> 
> 



It is not that Wittgenstein's later criticisms of set theory
change, it is, rather, that once we see that set theory has no
extra-mathematical application, we will focus on its calculations,
proofs, and prose and "subject the *interest* of the
calculations to a *test*" (*RFM* II, SS62). By
means of *Wittgenstein's* "immensely
important" 'investigation' (*LFM* 103), we
will find, Wittgenstein expects, that set theory is uninteresting
(e.g., that the non-enumerability of "the reals" is
uninteresting and useless) and that our entire interest in it lies in
the 'charm' of the mistaken prose interpretation of its
proofs (*LFM* 16). More importantly, though there is "a
solid core to all [its] glistening concept-formations"
(*RFM* V, SS16), once we see it as "as a mistake of
ideas", we will see that propositions such as
"\(2^{\aleph\_0} \gt \aleph\_0\)" are not anchored in an
extra-mathematical practice, that "Cantor's
paradise" "is not a paradise", and we will
*then* leave "of [our] own accord" (*LFM*
103).



It must be emphasized, however, that the later Wittgenstein still
maintains that the operations within a mathematical calculus are
purely 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 really
> is 'playing a game'...[is] *acting* in
> accordance with certain rules. (*RFM* V, SS1)
> 
> 
> 
> To say mathematics is a game is supposed to mean: in proving, we need
> never appeal to the meaning [*Bedeutung*] of the
> signs, that is to their extra-mathematical application. (*RFM*
> V, SS4)
> 
> 
> 



Where, during the middle period, Wittgenstein speaks of
"arithmetic [as] a kind of geometry" at (*PR*
SS109 & SS111), the later Wittgenstein similarly speaks of
"the geometry of proofs" (*RFM* I, App. III,
SS14), the "geometrical cogency" of proofs
(*RFM* III, SS43), and a "geometrical
application" according to which the "transformation of
signs" in accordance with "transformation-rules"
(*RFM* VI, SS2, 1941) shows that "when mathematics is
divested of all content, it would remain that certain signs can be
constructed from others according to certain rules"
(*RFM* III, SS38). Hence, the question whether a
concatenation of signs is a proposition of a given
*mathematical* calculus (i.e., a calculus with an
extra-mathematical application) is still an internal, syntactical
question, which we can answer with knowledge of the proofs and
decision procedures of the calculus.


### 3.6 Wittgenstein on Godel 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]he
arguments are wild" (Kreisel 1958: 153), that the passages
"on Godel's theorem... are of poor quality or
contain definite errors" (Dummett 1959: 324), and that
(*RFM* App. III) "throws no light on Godel's
work" (Goodstein 1957: 551). "Wittgenstein seems to want
to legislate [[q]uestions about completeness] out of
existence", Anderson said, (1958: 486-87) when, in fact,
he certainly cannot dispose of Godel's demonstrations
"by confusing truth with provability". Additionally,
Bernays, Anderson (1958: 486), and Kreisel (1958: 153-54)
claimed that Wittgenstein failed to appreciate
"Godel's quite explicit premiss of the consistency of
the considered formal system" (Bernays 1959: 15), thereby
failing to appreciate the conditional nature of Godel's
First Incompleteness Theorem. On the reading of these four early
expert reviewers, Wittgenstein failed to understand Godel's
Theorem because he failed to understand the mechanics of
Godel's proof and he erroneously thought he could refute or
undermine Godel's proof simply by identifying "true
in *PM*" (i.e., *Principia Mathematica*) with
"proved/provable in *PM*".



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 reading *only* the informal,
'casual' (MS 126, 126-127; Dec. 13, 1942)
introduction of (Godel 1931) and that, therefore, his use of a
self-referential proposition as the "true but unprovable
proposition" may be based on Godel'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 is
not provable" (1931: 599). Perplexingly, only two of the four
famous reviewers even mentioned Wittgenstein's (*RFM*
VII, SSSS19, 21-22, 1941)) explicit remarks on
'Godel's' First Incompleteness Theorem (Bernays
1959: 2; Anderson 1958: 487), which, though flawed, capture the
number-theoretic nature of the Godelian proposition *and*
the functioning of Godel-numbering, probably because Wittgenstein
had by then read or skimmed the body of Godel's 1931 paper.



The first thing to note, therefore, about (*RFM* App. III) is
that Wittgenstein mistakenly thinks--again, perhaps because
Wittgenstein had read only Godel's Introduction--(a)
that Godel proves that there are true but unprovable propositions
of *PM* (when, in fact, Godel syntactically proves that if
*PM* is \(\omega\)-consistent, the Godelian proposition is
undecidable in *PM*) and (b) that Godel's proof uses
a self-referential proposition to semantically show that there are
true but unprovable propositions of *PM*.



For this reason, Wittgenstein has two main aims in (*RFM* App.
III): (1) to refute or undermine, *on its own terms*, the
alleged Godel proof of true but unprovable propositions of
*PM*, and (2) to show that, on his own terms, where "true
in calculus \(\Gamma\)" is identified with
"*proved* in calculus \(\Gamma\)", the very idea of
a true but unprovable proposition of calculus \(\Gamma\) is
meaningless.



Thus, at (*RFM* App. III, SS8) (hereafter simply
'SS8'), Wittgenstein begins his presentation of what
he takes to be Godel's proof by having someone say: 



> 
> 
> I have constructed a proposition (I will use '*P*' to
> designate it) in Russell's symbolism, and by means of certain
> definitions and transformations it can be so interpreted that it says:
> '*P* is not provable in Russell's system'. 
> 
> 
> 



That is, Wittgenstein's Godelian constructs a proposition
that is semantically *self-referential* and which specifically
says of itself that it is not provable in *PM*. With this
erroneous, self-referential proposition *P* [used also at
(SS10), (SS11), (SS17), (SS18)], Wittgenstein presents
a proof-sketch very similar to Godel's own
*informal* semantic proof 'sketch' in the
Introduction of his famous paper (1931: 598).



> 
> 
> Must I not say that this proposition on the one hand is true, and on
> the other hand is unprovable? For suppose it were false; then it is
> true that it is provable. And that surely cannot be! And if it is
> proved, then it is proved that it is not provable. Thus it can only be
> true, but unprovable. (SS8)
> 
> 
> 



The reasoning here is a double *reductio*. Assume (a) that
*P* must either be true or false in Russell's system, and (b)
that *P* must either be provable or unprovable in Russell's
system. If (a), *P* must be *true*, for if we suppose that
*P* is false, since *P* 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 what
*P* means or says, it is true that *P* is unprovable (which is a
contradiction). Second, if (b), *P* must be unprovable, for if *P*
"is proved, then it is proved that it is not provable",
which is a contradiction (i.e., *P* is provable *and* not
provable in *PM*). It follows that *P* "can only be
true, but unprovable".



To refute or undermine this 'proof', Wittgenstein says
that if you have proved \(\neg P\), you have proved that *P* is
provable (i.e., since you have proved that it is *not* the case
that *P* is not provable in Russell's system), and "you
will now presumably give up the interpretation that it is
unprovable" (i.e., '*P* is not provable in
Russell's system'), since the contradiction is only proved
if we use or retain this self-referential interpretation (SS8). On
the other hand, Wittgenstein argues (SS8), '[i]f you assume
that the proposition is provable in Russell's system, that means
it is true *in 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 interpretation
that engenders a contradiction. Thus, Wittgenstein's
'refutation' of "Godel's proof"
consists in showing that no contradiction arises if we do *not*
interpret '*P*' as '*P* is not provable in
Russell's system'--indeed, without this
interpretation, a proof of *P* does not yield a proof of \(\neg P\)
and a proof of \(\neg P\) does not yield a proof of *P*. In other
words, the mistake in the proof is the mistaken assumption that a
mathematical proposition '*P*' "can be so
interpreted that it says: '*P* is not provable in
Russell's system'". As Wittgenstein says at
(SS11), "[t]hat is what comes of making up such
sentences".



This 'refutation' of "Godel's
proof" is perfectly consistent with Wittgenstein's
syntactical conception of mathematics (i.e., wherein mathematical
propositions have no meaning and hence cannot have the
'requisite' self-referential meaning) and with what he
says before and after (SS8), 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 proposition
of calculus \(\Gamma\) is a contradiction-in-terms.



To show (2), Wittgenstein begins by asking (SS5), what he takes to
be, the central question, namely, "Are there true propositions
in Russell's system, which cannot be proved in his
system?". To address this question, he asks "What is
called a true proposition in Russell's system...?",
which he succinctly answers (SS6): "'*p*'
*is true = p*". Wittgenstein then clarifies this answer
by reformulating the second question of (SS5) as "Under what
circumstances is a proposition asserted in Russell's game [i.e.,
system]?", which he then answers by saying: "the answer
is: at the end of one of his proofs, or as a 'fundamental
law' (Pp.)" (SS6). This, in a nutshell, is
Wittgenstein's conception of "mathematical truth": a
true proposition of *PM* is an axiom or a proved proposition,
which means that "true in *PM*" is identical with,
and therefore can be supplanted by, "proved in
*PM*".



Having explicated, to his satisfaction at least, the only real,
non-illusory notion of "true in *PM*", Wittgenstein
answers the (SS8) question "Must I not say that this
proposition... is true, and... unprovable?"
*negatively* by (re)stating his own (SSSS5-6)
conception of "true in *PM*" as
"proved/provable in *PM*": 



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



This answer is given in a slightly different way at (SS7) where
Wittgenstein asks "may there not be true propositions which are
written in this [Russell's] symbolism, but are not provable in
Russell's system?", and then answers "'True
propositions', hence propositions which are true in
*another* system, i.e. can rightly be asserted in another
game". In light of what he says in (SSSS5, 6, and 8),
Wittgenstein's (SS7) point is that if a proposition is
'written' in "Russell's symbolism" and
it is true, *it must be proved/provable in another system*,
since that is what "mathematical truth" is. Analogously
(SS8), "if the proposition is supposed to be false in some
other than the Russell sense, then it does not contradict this for it
to be proved in Russell's sense", for "[w]hat is
called 'losing' in chess may constitute winning in another
game". This textual evidence certainly suggests, as Anderson
almost said, that Wittgenstein rejects a true but unprovable
mathematical proposition as a contradiction-in-terms on the grounds
that "true in calculus \(\Gamma\)" means nothing more (and
nothing less) than "proved in calculus \(\Gamma\)".



On this (natural) interpretation of (*RFM* App. III), the early
reviewers' conclusion that Wittgenstein fails to understand the
mechanics of Godel's argument seems reasonable. First,
Wittgenstein erroneously thinks that Godel's proof is
essentially semantical and that it uses and *requires* a
self-referential proposition. Second, Wittgenstein says (SS14)
that "[a] contradiction is unusable" for "a
prediction" that "that such-and-such construction is
impossible" (i.e., that *P* is unprovable in *PM*),
which, superficially at least, seems to
indicate that Wittgenstein fails to appreciate the "consistency
assumption" of Godel's proof (Kreisel, Bernays,
Anderson).



If, in fact, Wittgenstein did not read and/or failed to understand
Godel's proof through at least 1941, how would he have
responded if and when he understood it as (at least) a proof of the
undecidability of *P* in *PM* on the assumption of
*PM*'s consistency? Given his syntactical conception of
mathematics, even with the extra-mathematical application criterion,
he would simply say that *P*, *qua* expression syntactically
independent of *PM*, is not a proposition of *PM*, and
if it is syntactically independent of all existent mathematical
language-games, it is not a mathematical proposition. Moreover, there
seem to be no compelling non-semantical reasons--either
intra-systemic or extra-mathematical--for Wittgenstein to
accommodate *P* by including it in *PM* or by adopting a
non-syntactical conception of mathematical truth (such as Tarski-truth
(Steiner 2000)). Indeed, Wittgenstein questions the intra-systemic and
extra-mathematical *usability* of *P* in various discussions
of Godel in the *Nachlass* and, at
(SS19), he emphatically says that one cannot "make the truth
of the assertion ['*P*' or 'Therefore
*P*'] plausible to me, since you can make no use of it except
to do these bits of legerdemain".



After the initial, scathing reviews of *RFM*, very little
attention was paid to Wittgenstein's (*RFM* App. III and
*RFM* VII, SSSS21-22) discussions of
Godel's First Incompleteness Theorem (Klenk 1976: 13) until
Shanker's sympathetic (1988b). In the last 22 years, however,
commentators and critics have offered various interpretations of
Wittgenstein's remarks on Godel, some being largely
sympathetic (Floyd 1995, 2001) and others offering a more mixed
appraisal (Rodych 1999a, 2002, 2003; Steiner 2001; Priest 2004; Berto
2009a). Recently, and perhaps most interestingly, Floyd & Putnam
(2000) and Steiner (2001) have evoked new and interesting discussions
of Wittgenstein's ruminations on undecidability, mathematical
truth, and Godel's First Incompleteness Theorem (Rodych
2003, 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 following
passage seems to capture Wittgenstein's *attitude* to the
Philosophy of Mathematics and, in large part, the way in which he
viewed his own work on mathematics.



> 
> 
> What will distinguish the mathematicians of the future from those of
> today will really be a greater sensitivity, and *that*
> will--as it were--prune mathematics; since people will then
> be more intent on absolute clarity than on the discovery of new
> games.
> 
> 
> 
> Philosophical clarity will have the same effect on the growth of
> mathematics as sunlight has on the growth of potato shoots. (In a dark
> cellar 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 and
> doubts of the kind I develop. He has learned to regard them as
> something contemptible and... he has acquired a revulsion from
> them as infantile. That is to say, I trot out all the problems that a
> child learning arithmetic, etc., finds difficult, the problems that
> education 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 providing
philosophical clarity on aspects and parts of mathematics, on
mathematical conceptions, and on philosophical conceptions of
mathematics. Lacking such clarity and not aiming for absolute clarity,
mathematicians construct new games, sometimes because of a
misconception of the *meaning* of their mathematical
propositions and mathematical terms. Education and especially advanced
education in mathematics does not encourage clarity but rather
represses it--questions that deserve answers are either not asked
or are dismissed. Mathematicians of the future, however, will be more
sensitive and this will (repeatedly) prune mathematical extensions and
inventions, since mathematicians will come to recognize that new
extensions and creations (e.g., propositions of transfinite cardinal
arithmetic) are not well-connected with the solid core of mathematics
or with real-world applications. Philosophical clarity will,
eventually, enable mathematicians and philosophers to "get down
to brass tacks" (*PG* 467).