Wittgenstein and Logic - Montgomery Link

Wittgenstein and LogicAuthor(s): Montgomery LinkReviewed work(s):Source: Synthese, Vol. 166, No. 1 (Jan., 2009), pp. 41-54Published by: SpringerStable URL: http://www.jstor.org/stable/40271156 .

Accessed: 25/01/2013 03:58

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Springer is collaborating with JSTOR to digitize, preserve and extend access to Synthese.

http://www.jstor.org

This content downloaded on Fri, 25 Jan 2013 03:58:45 AMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=springer

http://www.jstor.org/stable/40271156?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


Synthese (2009) 166:41-54 DOI 10.1007/sl 1229-007-9256-8

Wittgenstein and logic

Montgomery Link

Received: 28 January 2007 / Accepted: 5 September 2007 / Published online: 20 October 2007 © Springer Science+Business Media B.V. 2007

Abstract In his Tractatus Logico-Philosophicus Ludwig Wittgenstein ( 1 889-1 95 1 ) presents the concept of order in terms of a notational iteration that is completely logical but not part of logic. Logic for him is not the foundation of mathematical concepts but rather a purely formal way of reflecting the world that at the minimum adds absolutely no content. Order for him is not based on the concepts of logic but is instead revealed through an ideal notational series. He states that logic is "transcendental". As such it requires an ideal that his philosophical method eventually forces him to reject. I argue that Wittgenstein's philosophy is more dialectical than transcendental.

Keywords Wittgenstein • Logic • Order • Thought • McDowell • Operation • Number • Rule

Wittgenstein compares logic to a calculus with fixed rules, when an ordered sequence that determines each step of a procedure is specified ahead of time. It is predetermined in this case how to go on. Wittgenstein says that his philosophy changes over time. He began with questions about the status of the logical constants and the axioms of mathematical logic. His solution to those questions made tacit appeal to an ideal order, but that involved consequences for logic he had not anticipated. Later considerations about logic lead him away from rigid conceptualizations to the philosophical problem of the formation of concepts. This mainly internal development reveals a philosophy rather more dialectical than transcendental.

M. Link (El) Department of Philosophy, Suffolk University, One Bowdoin Square, Sixth Floor, Boston, MA 021 14, USA e-mail: [email protected]

<ö Springer



42 Synthese (2009) 166:41-54

1

When John McDowell in his incisive "Wittgenstein on Following a Rule" separates understanding from interpretation, his intent is to preclude the antirealist reading of the

Philosophical Investigations. !

According to McDowell Wittgenstein takes meaning as

normative, the "bedrock" being a "linguistic community". The ground on McDowell's

reading is the intuitive contractual conception.2 McDowell argues that it is the appeal to this bedrock that blocks the requirement for an antirealist reading of Wittgenstein. McDowell specifies that antirealism from the perspective of Wittgenstein would fail in its attempt to locate a "surface meaning" natural to philosophy since it cannot accom- modate the contractual conception that McDowell says is in the thoughts expressed by uttered propositions.

McDowell recognizes that, when Wittgenstein raises the question of the grounds for belief, the reasons given are not in every case "propositions which logically imply what is believed".3 Wittgenstein also remarks that "justification by experience comes to an end". But he is pointing out that there is no preset standard: "the standard has no

grounds". Late in the 1940s he will write that it must be possible to give a conceptual justification for both of two diametrically opposed remarks such as "But this isn't

seeing*" and "But this is seeing"* This is prima facie evidence that Wittgenstein's intellectual focus is the investigation of philosophical reasoning as a to and fro.

For instance in section 81 of the Philosophical Investigations Wittgenstein writes:

Ramsey once emphasized in conversation with me that logic was a "normative science". I do not know exactly what he had in mind but it was doubtless

closely related to what only dawned on me later: namely, that in philosophy we often compare the use of words with games and calculi which have fixed rules, but cannot say that someone who is using language must be playing such a

game. - But if you say that our languages only approximate to such calculi you are standing on the very brink of a misunderstanding. For then it may look as if

1 McDowell (1984, Sects. 7 and 14, in particular the last paragraph). The accepted practice is to divide Wittgenstein's career chronologically into three parts. Many direct attacks on realism occur in texts written during the middle stage, roughly 1929-1933, with the attack on platonism continuing later within the philosophy of mathematics. The realism of the natural sciences is not completely similar to the realism of set theory and the foundations of mathematics. 2 McDowell ( 1 984, Sects. 5,8, and 6). In the philosophy of mathematics the view that restricts mathematics to human processes is known as strict finitism. There are various readings of Wittgenstein as a strict finitist. McDowell (1984) is a response to the readings of Wright (1982) and Kripke (1982). Kripke connects strict finitism to his skeptical paradox. In the original Hao Wang (1958), and Bernays (1959) and Kreisel (1958), find anthwpologism in Wittgenstein (1956). Anthropologism is a core reading. A reviewer informs me that David Pears has a new book, Paradox and Platitude, in which the anthropologism attributed by Wang to Wittgenstein is taken as a starting point for interpreting many themes in his philosophy. I have not had the opportunity to consult the source, but Wittgenstein often comes close to anthropologism, perhaps never more so than in (1922). For the finitist reading: v. Marion (1995): cf. Link (2005, ch. 5). 3 McDowell (1984, Sect. 11); Wittgenstein (1953, Sects. 479ff., and v. 482-491). Cf. Wittgenstein (1922, 6.123). References to (1922) are by Nos. 4

Wittgenstein (1982, p. 81). Emphasis here and throughout follows the original.

Ö Springer



Synthese (2009) 166:41-54 43

what we were talking about were an ideal language. As if our logic were, so to

speak, a logic for a vacuum. - Whereas logic does not treat of language - or of

thought - in the sense in which a natural science treats of a natural phenomenon, and the most that can be said is that we construct ideal languages. But here the word "ideal" is liable to mislead, for it sounds as if these languages were better, more perfect, than our everyday language; and as if it took the logician to shew

people at last what a proper sentence looked like.

Logic is normative insofar as it is a standard for comparison; logic, however, is not related to language in the same way as natural science is related to the facts, for "we construct ideal languages". But an ideally constructed language may suggest that

something about the perfect essence of language has been revealed, whereas ordinary language has no general form, so that there are limits to logical form.

My reading is that language for Wittgenstein is wide open, although boundaries

provide clarity in local cases. I maintain there is a tension within Wittgenstein's later

philosophy that McDowell's reading glosses over: on the one hand Wittgenstein chal-

lenges the clarity of conceptual boundaries; yet, on the other hand he divides mathematical from philosophical investigations such that the two must be twain, even though they seem alike.5 For in philosophy concepts are wide open to reinterpretation, whereas in mathematics the properties of procedures are fixed and the concepts are formalized in First-Order Logic with Membership.

Without positive evidence of a resolution of this tension, I conclude that for

Wittgenstein thought does not stand necessarily on logical ground. Still there remain families of cases. Some of the family ties, for example in the case of the equations of elementary arithmetic, appear tight. But there is nothing in reality that corresponds to them. The truth of the arithmetical propositions is a mathematical requirement: philosophy would not really make sense of the propositions of mathematics, strictly speaking, even in a narrow and localized context such as a "language game".6 On this view philosophy remarks on but does not fall within the foundations of mathematics.

Well then, you might ask, is there a philosophy of logic here at all? I answer that there is: Wittgenstein's method already in the Tractatus is not to make positive asser- tions in philosophy but to question the metaphysical claims others make.7 He states that this would be the "right method of philosophy". That is early on, but throughout his career he draws contrasts for clarification, this method thereby becoming one

among many.

2

For criticism of his old philosophy Wittgenstein thanks Frank Ramsey in the preface to the Philosophical Investigations. Wittgenstein encourages the reader to compare

5 Cf. Dummett ( 1 959), Carnap ( 1 963, p. 25), and Maddy ( 1 986, p. 30 1 ). 6 Wittgenstein (1953, Sect. 136); v. (1953, Sect. 7 and p. 232). 7

Wittgenstein (1922, 6.53).

4ü Springer



44 Synthese (2009) 166:4 1->*

his new views with his former view in order to enhance the clarity of the description by contrast. But what exactly was the old philosophy?

Years before in one of the earliest records of his philosophical thought, a letter dated 22 June 1912, Wittgenstein wrote to his teacher at Cambridge University Bertrand Russell that the "props of Logic contain only apparent variables and whatever may turn out to be the proper explanation of apparent variables, its consequence must be that there are NO logical constants".8 He continues: "Logic must turn out to be of a TOTALLY different kind than any other science". At the heart of Witt- genstein's recasting of the place of logic within the body of science is his notational iteration. But the notational iteration for him is not part of logic per se, although this and the consequences thereof have been ignored in subsequent literature.

In the Tractatus Wittgenstein states ab initio that the "world is everything that is the case". The world as such is to be understood through the analysis of facts. One's own thoughts about these facts can be explicated as propositions. These thoughts and the thoughts of others might themselves become the objects of study in psychology. Wittgenstein, however, does not pursue a theory about the origin of these thoughts. Instead, he analyzes thoughts like other facts as propositions.

Propositions can be reduced to elementary propositions combined logically. An elementary proposition has no logical constants, identity, quantification, or truth- functional connectives. The general form of a proposition is [a, N, N'a ], where a is some series of elementary propositions, N is a truth-functional combination, and N'a is the result of applying that truth-functional combination to the series of elementary propositions. A mnemonic device for the general form of the proposition is Anna.

Next we shall see how Anna is an iteration on a. Let me make that precise in the vocabulary of the Tractatus, No. 6. The general form of the truth function is

[p, I #(!)]. Anna is the recursive variable going proxy for the operation Q on a proposition or series of propositions fj:

[I tf(f)] '(>?) = [ rç, |, tf(f)].

This is the "general form of transition from one proposition to another". This analysis indicates that the picture of logic can be perfected so that any factual situation can be

analyzed. To complete the consideration of Anna I must justify that Anna is a recursive

variable without presupposing the existence of a functional relation. "The general propositional form is a variable".9 Anna is the general propositional form, so Anna is a variable. Anna is recursive insofar as it is a general term for each element of a sequence. The general procedure connected to a formal series is determined by an operation O in that O'O'O'a results from three successive 0'£ iterations on a. Then the form of any element in the formal series

8 Wittgenstein (1995, pp. \4t)\ props are propositions. 9

Wittgenstein (1922, 4.53). V. (1922, 5.25-5.2522, and 5.4).

Ö Springer



Synthese (2009) 166:41-54 45

a, O'a, O'O'a, ...

can be given as the variable

[a, x, O'x].

The main idea is that a is the basis, x gives the form, and O'x shows how to go on. To a certain extent, then, no logical constants or objects would be required.

Tautologies are among the propositions produced by the combination of elemen-

tary propositions through the notational iteration.10 For Wittgenstein tautologies are different from other propositions in that they can be detected by inspecting the notation alone without looking at the world. The tautology is a purely formal mode for

presenting the world. It is a form of presentation signifying nothing, by which I mean a tautology is not true, strictly speaking, for there is no corresponding fact of the matter. Tautologies have no content. They are not "nonsense", but they are "sense-

less", to draw a critical technical distinction in the Tractatus. Wittgenstein says, "I know nothing about the weather, when I know that it rains or does not rain".

Logic under these circumstances does not provide for our knowledge of mathematics or of the world, not even along Russell's lines. Russell, for instance, reluctant to assume the axiom of infinity outright because of his universalism about logic, adopted it as a hypothesis.1

x This axiom of infinity is a precise formulation of the hypothesis in the Principia asserting that an infinite class of individuals exists.12 Wittgenstein rejects the position that any such hypothesis can be purely logical. In philosophy a

logical analysis at its barest need involve no more than a notational iteration, as in the case of Anna.

3

Wittgenstein's next step is to use the iteration to generate the natural numbers

0, 1, 2,....

10 Ibid., 6.1-6.13. V. Drehen and Floyd (1991). V. Wittgenstein (1922, 4.461ff). 1 ! For universalism in the thought of Russell v. Whitehead and Russell ( 1 9 1 0, p. 95) and ( 1 925, p. 9 1 ): "It is to some extent optional what ideas we take as undefined in mathematics We know no way of proving that such and such a system of undefined ideas contains as few as will give such and such results". Accordingly, the "chief reason in favour of any theory on the principles of mathematics must always be inductive" (White- head and Russell (1910 and 1925, p. v)). V. also van Heijenoort ( 1 967b). For the logocentric predicament v.Sheffer(1926, p. 228). 12 Whitehead and Russell (1912, * 120 • 03, p. 210): if a class a is an inductive cardinal, then there is at least one class of the right type that has a terms; cf. Wittgenstein (1922, 5.535). In (1922) Wittgenstein states that the theory of classes is superfluous in mathematics, a strong claim that depends on his notational iteration, just as his claim that Russell's type theory is superfluous for resolution of the paradoxes depends on his notational iteration (6.03ff. and 5.252f.). (1922) appears to make commitments Wittgenstein had not anticipated: v. Link (2005) for details. Russell's paradox then would vanish without requiring an appeal to type theory (3.33 Iff.); however, in (1922) Wittgenstein uses the ellipsis always to indicate a denumerable sequence, he will later write (1974, p. 268): cf. (1922, 6-6.01).

£j Springer



46 Synthese (2009) 1 66:4 1 -54

The general form of natural number is

[0, Ç, Ç + 1],

a variable symbolizing the recursive analysis of the finite ordinals, further evidence of the notational iteration, the formal sequence of which is

je, ß'jt, Œ'Œ'jc, ....

This Omega series displays the ordering concept: first, second, third, and so on. It is captured in the general form

[x, Ç, 0'*].

Here the variable presents the order: the initial element is x, £ expresses the difference, and ß'£ shows how to go on.

To explain, Wittgenstein gives a recursive definition of the natural numbers as exponents:

x = ß°'jc, n'nv'x = nv+l'x.

This stipulation allows him to rewrite the Omega series as the ordinal series

and to capture the concept of ordinality in the form

[nOfjc, nv'xt nv+l>*].

Before proceeding let's verify that last claim. For the basis step zero there is £2°'jc, the first in the ordinal series, identical with jc, the first in the Omega series. For the inductive step n + 1 introduce the further definitions:

0+1 = 1,

0+1 + 1=2,

and so on. One, for example, is the second in the ordinal series, £20+1'jc, which by the definition of the natural numbers would be Q'x, the second element in the Omega series. In general the n + lth element of the ordinal series will have as an exponent zero plus n applications of succession, equal by recursive stipulation to Q'ß"""1'*. But, if that is an element of the Omega series having n-fold applications of ß to *,

Ö Springer



Synthese (2009) 166:41-54 47

then QyQ'Qn~2'x is as well, and so on. Through this method each predecessor in the

sequence is manifested and can be verified, including the first.13 But isn't the Omega operation fundamentally truth-functional? Absolutely not. We

have already seen that an analysis differs according to its basis. When the basis is a series of elementary propositions rj, Q'(rj):

[I #(§)]'(»?) = [>;, lN(Ji)].

This is the form of transition from one proposition to another: here f gives the difference and N(ïj) tells how to go on. But when the basis is zero or £2°'*, £2'(0):

[£, $ + l]'(0) = [0, Ç, > + l].

This is the form of transition from one natural number to the next, where § gives the difference and the operation of succession tells how to go on. What these analyses have in common is the general term of the operation £2'(jc):

[ç, a** ]•(*) = [*. *. nf*L

The Omega series is the key to Wittgenstein's response to logicism. He does not provide a formalism nor need one. The natural numbers are exponents of an operation.

The ordinary minimal case for a logic-based metaphysics such as logicism requires that the propositions of elementary arithmetic be part of logic; for Wittgenstein, how-

ever, the equation '2 x 2 = 4' is not part of logic: a proof of this equation is not a proof in logic.14 The multiplication itself is provided for by the iteration. Further, equations like tautologies reflect the logic of the world, yet identity is not pure logic. In 1927

Wittgenstein will write a letter to Ramsey responding to Ramsey's suggestion that his mathematical explication of identity in terms of the tautology is close to that of the Tractatus}5 In the missive he insists that the concept of identity cannot be based on the tautology, not questioning Ramsey's work in the foundations of mathematics but its connection to the tautology of his own philosophy.

Gottlob Frege had discerned that the ancestral relation would provide the logical basis for the principle of mathematical induction. I involve Frege's analysis at this

juncture because that is the explicit context for Wittgenstein's analysis of succession

through the formal series: "We can determine the general term of the formal series by giving its first term and the general form of the operation, which generates the fol-

lowing term out of the preceding proposition". The consequence is that mathematical induction is not part of logic.

^ Wittgenstein (1922, 6.02). Wittgenstein's analysis is reminiscent of Richard Dedekind's essential research on number (1888, Sects. 9, 1 Iff.). 14 Wittgenstein (1922, 6.241). 1 5 TS 207: v. von Wright ( 1 993), Ramsey ( 1 925), Wittgenstein ( 1 995, pp. 2 1 6ff., a letter dated 2 July 1 927, and 219ff.). V. Frege (1879, Sect. 26), and cf. Wittgenstein (1922, 4.1273). For evidence that Wittgenstein could not have been completely committed to logical atomism v. Link (2007).

£l Springer



48 Synthese (2009) 166:41-54

4

In the last two sections I have described the essential form of logical analysis and the extent of the notational iteration into elementary arithmetic. But there are limits. No. 6 of the Tractatus begins with truth functions, then turns to the natural numbers; the

investigation of regularity, however, does not extend to natural science. Russell had reduced causation to a law of induction stated in terms of the probability calculus, a

general law, Russell says, that is a priori, if true.16 But for Wittgenstein logic is not universal and does not incorporate the natural sciences.

This leads to the concept of thought, although I still need to clarify the connection in order to throw new light on McDowell's reading, with which we began. In 1919

Russell, soon after first reading Wittgenstein's book, exchanges letters with the author. In a letter dated 1 3 August 1919 Russell asks: "What is the difference between Tatsache

[fact] and Sachverhalt [atomic fact]?"17 A thought in the Tractatus is a logical picture of the facts. Russell's illuminating question concerns the reality of the constituents of these facts.

Wittgenstein replies: "Sachverhalt is, what corresponds to an Elementarsatz

[elementary proposition] if it is true. Tatsache is, what corresponds to the logical product of elementary props when this product is true". Elementary propositions are not among those propositions composed through a notational iteration. If no thought or other fact corresponds to a certain logical product, the logical product cannot be true.

Wittgenstein is agnostic about a suggestion from Russell that a thought is composed of words, writing: "I don't know what the constituents of a thought are but I know that it must have such constituents which correspond to the words of Language. Again the kind of relation of the constituents of thought and of the pictured fact is irrelevant. It would be a matter of psychology to find out".

One suggestion for the notion of thought in Wittgenstein is that it is like Frege's notion of thoughts as significant propositions that different persons can share. Frege does write that by a thought he understands an "objective content, which is capable of being the common property of several thinkers".18 Frege offers the metaphor of the moon, the telescope, and the eye to differentiate between reference, sense, and

subjective idea. When different persons look through the same telescope at the moon, they are all looking toward the moon. The image of the moon as it is reflected in the

telescope also does not change between viewers. Frege preserves an appealing feature of thought not involved with truth. The reflection of the moon in the telescope does not vary between viewers, so the reflection is unlike the actual impingements on the

retina, those stimulations which vary from viewer to viewer.

My point is that for Frege a thought is an entity or, to be exact, an object. A thought is an object that realizes the sense of a sentence. Frege says that "for all the multiplicity of language" humans have "a common stock of thoughts". For "the task of logic can

hardly be performed without trying to recognize the thought in its manifold guises".

16 V.Russell (1914, pp. 225f.); cf. Wittgenstein (1922, 6.3-6.3751). 17

Wittgenstein (1995, pp. 121ff.) and v. (1922, 3) and (1995, pp. 124ff.). 18 Frege (1892a, p. 162n7).

£l Springer



Synthese (2009) 1 66:4 1-54 49

Wittgenstein is influenced by Russell and has avoided the influence of Frege's doctrine of sense in the respect that a proposition having sense for Wittgenstein is the same as a proposition being true or false. All three philosophers would count the sentence

'Odysseus fell asleep on the shores of Ithaca' as false, but Frege would countenance some additional reality to the thought expressed. He writes that "different expressions quite often have something in common, which I call the sense, or, in the special case of sentences, the thought".19

In the Tractatus thoughts are propositions having sense, which means being either true or false. This understanding of thought excludes the option that equations express thoughts as early as 1919, if not sooner. Wittgenstein again takes up the discussion of

thought in the winter of 1929.

5

Seven years after seeing his book published, Wittgenstein returns to Cambridge University. A manuscript entry dated 2 February 1929 records his first written philosophical remark: "Ist ein Raum denkbar de nur alle rationalen aber nicht die irrationalen Punkte enthält?"20 The question asks whether a space that contains the rational numbers and not the irrational numbers is "thinkable". What he means is that the rational numbers do not set a precedent for the irrational numbers necessarily. The rational numbers under certain assumptions fill a space without leaving a gap.

Were Wittgenstein's philosophy similar to Russell's at this juncture, one would

justly expect signs of a logical reduction to hypotheses; this passage, however, follows a variation of the "right method" in pursuing the question. Wittgenstein goes on to say that it would be misleading to conclude that the rational numbers are packed closely together on the number line: there is a constructive procedure for generating an irrational number from the rational numbers, so that the continuum can be thought of as a space of real numbers, as well. The space of the rational numbers, then, is thinkable with or without the irrational numbers. Wittgenstein in this case is using examples from mathematical logic to counter or curb philosophical arguments purporting to resolve the nature of mathematics.

Clearly Wittgenstein responds to the "scientific method" in philosophy, but this does not by itself mean he shares Russell's philosophy. A fair test would seem to be the concepts of addition and multiplication. It is conceivable that there is a narrower test. I have not yet considered the status of the truth tables. Perhaps one might say that

Wittgenstein had in mind some sort of logic-based metaphysics because he gave an

analysis through truth tables. The analysis is undeniable, but it is not an analysis to

objects of some logical reality: the truth table is a form of presentation manifesting the explication of the logical constants, these having been rendered unnecessary by the notational iteration.21

™ Frege (1892b, p. 185n7); cf. Wittgenstein (1922, 4). 20 MS 105, 1. This passage reappears six times in various manuscripts, finally being included in "On Set Theory" (1974, p. 460). For the Archimedean Axiom and the ray concept v. Hubert (1899). 21 Wittgenstein (1922, 4.27, 4.31, and 4.42). The presence of the ab notation (1922, 6.1203) shows that the truth tables are themselves inessential (1979, pp. 1 15, 124, 127, and 129). V. (1922, 6.121f. and 6.13).

£l Springer



50 Synthese (2009) 166:41-54

Even in this zero case logic is not a foundation. It is the iteration that generates any truth-functional combination of elementary propositions. The atoms requisite for a comprehensive analysis are themselves compiled by Anna, so that the analysis does not even require logical propositions. I conclude that Wittgenstein used the notational iteration to depict logic as a reflection of the world.

So far in this section I have presented evidence of a dialectical analysis of mathematical thought in writings from 1929. 1 have suggested that the use of an extended

asyndetic structure for contrasting philosophical arguments is a continuation of the

"right method". He will use other methods, as well. I have tried to distinguish Wittgen- stein's philosophical methods from the "scientific method" of Russell. I make room in

my reading of Wittgenstein's philosophy for the presence of traditional philosophical influences other than Frege and Russell but do not argue that there is an influence

greater than these two. Before I conclude this section let me summarize the evidence of the transcendental

philosophy in Wittgenstein's early career. Judgments on the transcendental under-

standing of formal logic in the Critique of Pure Reason are made up of concepts and

only concepts. Intuitions on this view are not included among the constituents of judgments within formal logic, the "science that exhaustively presents and strictly proves nothing but the formal rules of all thinking".22 When Wittgenstein writes that logic "is transcendental", part of what he means is that logic does not require intuitions.

For Immanuel Kant the "boundaries of logic" are "determined quite precisely".23 For Kant and Wittgenstein the logical is analytic. Further, logic for Kant is "com-

pletely a priori" and likewise for the early Wittgenstein. But Wittgenstein adds that the a priori is "purely logical". That needs to be explained: logic for the early Witt-

genstein is of pure crystal, form without content, and not really a region of science at all. To step back from logic for a moment, Wittgenstein might be said to differ from Kant in that mathematics for Wittgenstein does not require intuitions. These,

Wittgenstein says, arise in calculating. For him, though not for Kant, the a priori is all analytic.

6

In the previous two paragraphs I outlined a picture of logic that shows at least some

presence of the transcendental philosophy. Next I follow later developments to consider the degree of the persistence of this philosophy. By the middle of 1930 Wittgenstein updates Russell on his current research with a report in which he writes: "From the

very outset 'Realism', 'Idealism', etc., are names which belong to metaphysics. That

is, they indicate that their adherents believe they can say something specific about the

22 Kant (1787, Bviii-ix). The singularity of certain constituents in this case poses no difficulty for Kant, not requiring an appeal to something entirely non-conceptual. Immediacy and singularity are two indepen- dent conditions on Kant's intuition concept: v. Kant (1787, A19/B33 and A320/B377), (1900, Sect. 1), and the 1795 Wiener Logik (1900, 24:909), also Parsons (1984), Hintikka (1968), and Hintikka (1984). V. Wittgenstein (1922, 6.13, 6.126, and 6.233f.). 23 Kant (1787, Bviii); cf. Wittgenstein (1922, 5.4541). V. Kant (1787, A54/B78); cf. Wittgenstein (1922, 5.4731, 5.551f., and 6.3211).

£} Springer



Synthese (2009) 166:41-54 51

essence of the world".24 In the Philosophical Investigations Wittgenstein remarks on a passage about thought and logic from his earlier book:

Thought is surrounded by a halo. - Its essence, logic, presents an order, in fact the a priori order of the world: that is, the order of possibilities, which must be common to both world and thought. But this order, it seems, must be utterly simple. It is prior to all experience, must run through all experience; no empir- ical cloudiness or uncertainty can be allowed to affect it It must rather be of the purest crystal. But this crystal does not appear as an abstraction; but as

something concrete, indeed, as the most concrete, as it were the hardest thing there is (Tractatus Logico-Philosophicus No. 5.5563).

The once clear vision of a certain and concrete a priori order presented by the essence of thought now appears to be a fata morgana.

The problem is that the iteration was to suffice for the recursive definition of the natural numbers, and the denumerable sequence was to suffice for mathematical induction in place of Frege's logical analysis of the ancestral relation.25 Wittgenstein does not give up entirely on constructive procedures. He notes for instance the constructive nature of the unique "expansion" that Cantor's diagonal argument provides.26 Look-

ing back to Sect. 3, to finish the argument against Frege and Russell Wittgenstein proved that two times two equals four. That proof can be checked. The proof, then, throughout his career determines the equation and its use but not its truth beyond the mathematical requirement.

That thought itself is not determinative for the foundations of mathematics is con- sonant with what Wittgenstein says of his own method in a 1939 lecture: "I may occasionally produce new interpretations, not in order to suggest they are right, but in order to show that the old interpretation and the new are equally arbitrary. I will only invent a new interpretation to put side by side with an old one and say, 'Here, choose, take your pick' ".27 The "right method" has become more like a to and fro than like transcendental philosophy.

Logic remains separate from natural science: "logic does not treat of language - or of thought - in the sense in which a natural science treats of a natural phenomenon". Logic itself does not secure truth. Wittgenstein writes: "Our interest certainly includes the correspondence between concepts and very general facts of nature. (Such facts as mostly do not strike us because of their generality.) But our interest does not fall back upon these possible causes of the formation of concepts; we are not doing natural

science; nor yet natural history . . . ",28 That is a category distinction that he never obliterates.

24 Wittgenstein ( 1 975, p. 86); v. ( 1 953, Sect. 97). 25 I.e., Wittgenstein comes close to anthropologism in order to respond to logicismi. 26 Wittgenstein (1978, pp. 125-142, particularly p. 135): v. Kanamori (1996, p. 4). V. Wittgenstein (1978, pp. 116-1 19): cf. Floyd and Putnam (2006) and references therein. Cf. also Wrigley (1977) and Goldfarb (1983). 27 Wittgenstein (1976, pp. 13f.and 170). 28 Wittgenstein (1953, p. 230).

Ö Springer



52 Synthese (2009) 166:41-54

This leads to an aporia. How can there be a such sharp divide between logical investigations and natural investigations in his thought given the family resemblances that combine these disparate language games? It is well-known that he investigates the conceptual boundaries of metaphysics: "For I can give the concept 'number' rigid limits in this way, use the word 'number' for a rigidly limited concept, but I can also use it so that the extension of the concept is not closed by a frontier".29 He says that

concepts are elastic, which suggests they are fragile but not unusable in local cases. But that is an ad hoc measure that cannot resolve the tension about how to go

on, given the fragile boundaries of concepts. McDowell's idea of a normativity of

thoughts in understanding that is transcendental underplays the importance of this tension. Wittgenstein continues, in the passage from the Philosophical Investigations on natural history:

I am not saying: if such-and-such facts of nature were different people would have different concepts (in the sense of a hypothesis). But: if anyone believes that certain concepts are absolutely the correct ones, and that having different ones would mean not realizing something that we realize - then let him imagine certain very general facts of nature to be different from what we are used to, and the formation of concepts different from the usual ones will become intelligible to him.

Without idealism, without concepts {being in any way predetermined, Wittgenstein later in his career turns to the issue of concept formation.

The idea is that for an a posteriori investigation the propositions determine the

concepts, yet this, he says, is vague because '2 + 2 = 4' "forms a concept in a different sense from *p D p\\x).fx D fa\ orDedekind's Theorem. The point is, there is a family of cases".30 He comes to see the picture of an ideal a priori configuration that matches thought to the world to be air-drawn, like a dagger of the mind. The shift in his

philosophy is a shift away from an a priori requirement to a posteriori investigations, while the method of questioning without answering persists in multiple guises.

Acknowledgements Thanks to R. Jandovitz, A. Kanamori, and C. Mesa. Errors are my responsibility.

29 Wittgenstein (1953, Sect. 68); v. (1992, p. 24).

30 Wittgenstein (1978, p. 408) and v. (1929).

References

Bernays, P. (1959). Comments on Ludwig Wittgenstein's Remarks on the foundations of mathematics. Ratio, 2, 1-22.

Carnap, R. (1963). Intellectual autobiography. In L. E. Hahn & P. A. Schilpp (Eds.), The philosophy of Rudolf Carnap (pp. 3-86). La Salle: Open Court.

Dedekind, R. (1888). Was sind und was sollen die Zahlen? Brunswick: F. Vieweg. Sixth edition in Essays on the theory of numbers (Trans: Beman, W.W.) (pp. 31-115). Reprinted (New York: Dover, 1963).

Dreben, B., & Floyd, J. (1991). Tautology: How not to use a word. Synthese, 87, 23-49. Dummett, M. (1959). Wittgenstein's philosophy of mathematics. The Philosophical Review, 68, 324-348. Floyd, J., & Putnam, H. (2006). Bays, Steiner, and Wittgenstein's 'Notorious' paragraph about the Godei

Theorem. The Journal of Philosophy, 103, 101-110.

Ö Springer



Synthese (2009) 1 66:4 1-54 53

Frege, G. (1879). Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. (Trans: van Heijenoort (1967a, pp. 1-82)). Halle.

Frege, G. (1892a). On sense and meaning, in Frege (1984, pp. 157-177) (Trans: Black, M). From the original in Zeitschrift ßr Philosophie und philosophische Kritik, 100, 25-50.

Frege, G. (1892b). On concept and object, in Frege (1984, pp. 182-194) (Trans: Geach, P.). From the original in Vierteljahrsschrift fur wissenschaftliche Philosophie 16, 192-205.

Frege, G. (1984). Collected papers on mathematics, logic, and philosophy (B. McGumness, Ed.). Oxford and New York: Basil Blackwell.

Goldfarb, W. (1983). I want you to bring me a slab: Remarks on the opening sections of the Philosophical investigations. Synthese, 56, 265-282.

Hubert, D. (1899). Grundlagen der Geometrie. Festschrift zur Feier der Enthüllung des Gauss-Weber Denkmals in Göttingen. Leipzig: Teubner. In P. Bernays (Ed.), Foundations of geometry (10th ed.) (Stuttgart: Teubner, 1968) (Trans: Unger, L.) (La Salle: Open Court, 1971).

Hintikka, J. (1968). On Kant's notion of intuition (Anschauung). In T. Penelhum & J. J. Macintosh (Eds.), The first critique: Reflections on Kant's Critique of pure reason (pp. 38-53). Belmont: Wadsworth.

Hintikka, J. (1984). Kant's transcendental method and his theory of numbers. Topoi, 3, 99-108. Kanamori, A. (1996). The mathematical development of set theory from Cantor to Cohen. The Bulletin of

Symbolic Logic, 2, 1-71. Kant, I. (1787). Critique of pure reason (2nd ed.) (Trans: Guyer, P. & Wood, A.). Cambridge: Cambridge

University Press, 1998 (First edition published in 1781). Kant, I. (1800). Logik: ein Handbuch zu Vorlesungen (2nd ed.) (G. B. Jäsche, Ed.). Königsberg: Friedrich

Nicolovius. Kant, I. (1900). Kant's gesammelte Schriften. (The Deutsche Akademie der Wissenschaften, Ed.). Berlin:

Walter de Gruyter. Kreisel, G. (1958). Wittgenstein's Remarks on the foundations of mathematics. British Journal for the

Philosophy of Science, 9, 135-158. Kripke, S. (1982). Wittgenstein on rules and private language. An elementary exposition. Cambridge:

Harvard University Press. Link, M. (2005). Wittgenstein and infinity. Ph.D. thesis. Boston University. Link, M. (2007). Wittgenstein and logical analysis. In A. Pichler, H. Hrachovec, & J. Wang (Eds.),

Philosophie der Informationsgesellschaft. Kirchberg am Wechsel: Österreichische Ludwig Wittgen- stein Gesellschaft: 13 Iff.

Maddy, P. (1986). Mathematical alchemy. British Journal for the Philosophy of Science, 37, 279-314. Marion, M. (1995). Wittgenstein and finitism. Synthese, 105, 141-176. McDowell, J. (1984). Wittgenstein on following a rule. Synthese, 58, 325-363. Parsons, C. (1984). Arithmetic and the categories. Topoi, 3, 109-121. Ramsey, F. (1925). Foundations of mathematics, In Proceedings of the London Mathematical Society, s. 2,

25, 338-384. Russell, B. (1914). Our knowledge of the external world as a field for scientific method in philosophy. Open

Court. Reprinted (London and New York: Routledge, 2005). Sheffer, H. M. (1926). Review of Whitehead and Russell (1925). Isis, 8, 226-231. van Heijenoort, J. (Ed.). (1967a). From Frege to Godei. A source book in mathematical logic, 1879-1931.

Cambridge and London: Harvard University Press. van Heijenoort, J. (1967b). Logic as calculus and logic as language. Boston Studies in the Philosophy of

Science, 3, 440-446. von Wright, G. H. ( 1 993). The Wittgenstein papers. In L. Wittgenstein, Philosophical occasions. 1912-1951

(J. C. Klagge, & A. Nordmann, Eds.) (pp. 480-510). Indianapolis: Hackett. Wang, H. (1958). Eighty years of foundational studies. Dialectica, 12, 466-497. Reprinted as the second

chapter of A Survey of Mathematical Logic (Peking: Science Press, 1962). Whitehead, A. N., & Russell, B. (1910). Principia Mathematica (Vol. 1). Cambridge: Cambridge University

Press. Whitehead, A. N., & Russell, B. ( 19 1 2). Principia Mathematica (Vol. 2). Cambridge: Cambridge University

Press. Whitehead, A. N., & Russell, B. (1925). Principia Mathematica (Vol. 1, Second edition of Whitehead and

Russell (1910) with new introduction). Cambridge: Cambridge University Press. Wittgenstein, L. (1921). Logisch-philosophische Abhandlung. Annalen der Naturphilosophie, In Wilhelm

Ostwald (Ed.), Band XIV, Heft 3/4.

& Springer



54 Synthese (2009) 166:41-54

Wittgenstein, L. (1922). Tractatus Logico-Philosophicus. International library of psychology and scientific method (Trans: Ogden, C. K. with the assistance of F.P. Ramsey, with original German of Wittgen- stein (1921) included). London: Routledge and Kegan Paul. Corrected edition (New York and London: 1933) reprinted (Trowbridge: Redwood Burn, 1988).

Wittgenstein, L. (1929). Some remarks on logical form. In Proceedings of the Aristotelian Society, s.v. 9, pp. 162-171.

Wittgenstein, L. (1953). Philosophical investigations (G. E. M. Anscombe, Trans, with the original German text included). Oxford: Basil Blackwell; New York: Macmillan. (Second edition (Oxford: Basil Blackwell and Mott; New York: Macmillan, 1958)).

Wittgenstein, L. (1956). Remarks on the foundations of mathematics (G. H. von Wright, R. Rhees, & G. E. M. Anscombe, Eds., G. E. M. Anscombe, Trans.). Oxford: Blackwell.

Wittgenstein, L. (1974). Philosophical grammar (R. Rhees, Ed., A. Kenny, Trans.). Oxford: Basil Blackwell; Berkeley and Los Angeles: University of California Press.

Wittgenstein, L. (1975). Philosophical remarks (R. Rhees, Ed., R. Hargreaves & R. White, Trans.). Chicago: University of Chicago Press.

Wittgenstein, L. (1976). Lectures on the foundations of mathematics, 1939 (C. Diamond, Ed.). Ithaca: Cornell University Press. Reprinted (Chicago and London: University of Chicago Press, 1989).

Wittgenstein, L. (1978). Revised edition of Wittgenstein (1956). Cambridge: MIT Press. Wittgenstein, L. (1979). Notebooks 1914-1916 (2nd ed.) (G. H. von Wright & G. E. M. Anscombe, Eds.,

G. E. M. Anscombe, Trans.). Chicago: University of Chicago Press. Wittgenstein, L. (1982). Last writings on the philosophy of psychology (Vol. 1). Preliminary studies for

part II of philosophical investigations (G. H. von Wright & H. Nyman, Eds,, C. G. Luckhardt & M. A. E. Aue, Trans.). Chicago: University of Chicago Press. Midway reprint ( 1 990).

Wittgenstein, L. (1992). Last writings on the philosophy of psychology (Vol. 2). The inner and the outer. (G. H. von Wright & H. Nyman, Eds., CG. Luckhardt & M. A. E. Aue, Trans.). Oxford and Cambridge: Black well.

Wittgenstein, L. (1995). Cambridge letters. Correspondence with Russell, Keynes, Moore, Ramsey, and Sraffa (B. McGuinness & G. H. von Wright, Eds.). Oxford and Cambridge: Blackwell.

Wright, C. (1982). Strict finitism. Synthese, 57, 203-282. Wrigley, M. (1977). Wittgenstein's philosophy of mathematics. Philosophical Quarterly, 27, 50-59.

£l Springer



Documents

Wittgenstein and Logic - Montgomery Link