WITTGENSTEIN’S philosophy was Kantian from beginning to end. In other ways, too, he never changed. He always wrote in that maddeningly distinctive, bare bones, dramatically epigrammatic style, flashing like some powerful stroboscope, bewildering the intelligence with alternating brilliance and darkness and often causing a complete blur. The foreword to Philosophische Bemerkungen is an intense expression of his persistent view that what matters in the world of the mind must be clear and simple (see also Tractatus, 5.4541), and his tragedy was that he knew everything in his own life was less clear and simple than he thought it had to be. The progressive spirit of so-called empirical science and the methodological programmes of its philosophy were always alien to his temper (see Tractatus 6.372). He favoured the transcendental ‘other world’ of metaphysics and logic and mathematics over the valueless inexplicable contingencies of science and of common sense (see Tractatus 6.13, 6.421, 6.4312). The only kind of explanation that wouldn’t prematurely stop short while still giving the appearance of explaining everything would reveal the deepest constant features of our ways of thinking about the world, and he never arrested the Kantian thrust to demonstrate the fundamental categories of thought and experience. But such experiments could be conducted only in the mind itself. So, in his later work, after he had abandoned the ideal of a unified language, he left to philosophy only the task of describing the ‘motley’ of human thought.
This view of Wittgenstein’s thought brings into perspective his revisitations to the philosophy of mathematics, which kept his interest until at least 1944, in the Tractatus, Philosophische Bemerkungen and in those writings collected together under the title, ‘Remarks on the Foundation of Mathematics’ (which, for ease of reference, I hereafter shall refer to respectively as ‘T’, ‘B’, and ‘R’). As is well known, he came to Cambridge to study the foundations of mathematics with Russell, and the writings of Frege and Russell on the relationship between mathematics and the use of language were the first and the main continuing influence on his thought, later also affected in conversations with Brouwer and Ramsey and by the writings of Hilbert and Goedel. Wittgenstein’s results in the philosophy of mathematics were always inconclusive, inadequate to the issues, technically terrible and sometimes even silly, in R as well as in T—in Kreisel’s words, ‘… a surprisingly insignificant product of a sparkling mind’.1 In a later review article on The Blue and Brown Books, Kreisel complains that Wittgenstein characteristically failed, when considering mathematics or almost anything, to go beyond fun-poking to a more satisfying, fuller analysis of theoretical problems.2 But still there is something fascinating about it all, for reasons revealed perhaps by its limited relevance to logic and to the most concrete parts of classical mathematics—geometry, mechanics and arithmetic. I shall argue that according to Wittgenstein successful mathematizing would expose and confirm the conceptual connexions which obtain in our everyday and scientific ways of thinking and talking. Mathematics, like metaphysics, is, according to this, the conceptual investigation of necessary connexions. Despite his occasional declarations to the contrary, Wittgenstein should have seen mathematics and metaphysics as Platonic cousins, both methods for uncramping the mind (see R page 17). He believed in geometry and mechanics because here the relationship of mathematics to our ordinary ways of thinking is immediately visible. He thought axiomatics and the use of other heavy methodological machinery pernicious. Pure mathematics and speculative metaphysics alike are apt to be swamped in images and in their own meaningless abstractions, and nowhere, he thought, is this more evident than in the development of set theory from the time of Cantor.
In this paper, I wish to present Wittgenstein’s philosophy of mathematics, as I read it, with the primary intention of indirectly establishing the fact of a substantial carryover from T to R, a continuity which I also believe is typical of his whole philosophy. I shall hold that Wittgenstein always thought of mathematics as a method or assortment of methods that aims to demonstrate conceptual connexions latent in or imposed upon our ordinary and scientific uses of language. What gives mathematics its meaning and accounts for its necessity is the civil nonmathematical rôles of the concepts investigated. Wittgenstein not surprisingly lays stress on calculation and on what Hilbert-Cohn-Vossen called ‘Intuitive Mathematics’, on that kind of non-systematic thinking well illustrated by classical thought-experiments typically conducted outside of mathematical theory by methods which are indeed methods of demonstration and not logical derivation. Wittgenstein, just as expectedly, deprecates the highly but artificially structured theories of ‘pure mathematics’, which are mainly directed towards other parts of mathematics itself.
I shall try to break Wittgenstein’s position down into a number of connected ‘themes’, which I shall develop and elaborate with observations and with references to T, B, and R. My indirect argument for the indicated conclusion is just the references which, however, have been collected pretty haphazardly and are meant to be only illustrative and are certainly incomplete. Sometimes the citation of a reference will also constitute implicit interpretation.1
My secondary intention is to offer an alternative to Mr. Dummett’s well-known but widely disbelieved interpretation of R.1 I agree with Dummett that mathematics cannot tolerate anyone’s assumed privilege to stipulate when an assertion is justified, but that is not a criticism of Wittgenstein. My colleagues, Chihara, and Stroud, are right to take Dummett to task for making Wittgenstein out to be a conventionalist in that silly sense.2 They see that Wittgenstein was out to make logical compulsion intelligible and not to advance claims for some high Carnapian game of language relativity. Kreisel observed that Wittgenstein’s disbelief in mathematical objects is not a disbelief in mathematical objectivity ([1], page 138, n. 1). Chihara goes so far as to suggest that Wittgenstein’s ‘constructive view of mathematics’ is consistent with the ‘realist’s discovery view’ (op. cit., page 34), and if forced to a choice I would rather call Wittgenstein a ‘Platonist’ than a ‘conventionalist’ My main theme below will be that Wittgenstein held that mathematics is derivative from the civil use of language and (as Chihara points out, op. cit., page 26, n. 17) that he agreed with Frege that it is this civil application which gives mathematics its meaning. Mathematics unfolds the properties of familiar notions. I would understand if someone described Wittgenstein’s philosophy of mathematics as ‘transcendental Platonism’ or (if you wish) ‘conceptualism’, for Wittgenstein thought that mathematics at its most characteristic is the conceptual investigation of (other) ‘language games’. Something like this is his alternative to the other standard ‘philosophies’. Formalism wrongly strips mathematical concepts of their civil rôle, but is right in recognizing that ‘mathematics must fend for itself’ [T: 5.473; R: 67 twice]; ‘psychologism’ and ‘empiricism’ preserve meaningfulness at cost of replacing the necessities of mathematics with the contingent background of thought; and Platonism rescues necessity only by indulging in the worst kind of theology and alchemy. But ‘transcendental Platonism’ and ‘conceptualism’ are wrong also, because Wittgenstein throughout stood firm in his opinion that the kind of mathematics that matters is not a body of doctrine or a theory but rather a record of man’s successful efforts to make explicit what is essential in our forms of thought. Mathematics is a difficult kind of reflective, intuitive ‘nonobservational’ knowledge. Conventional fiats are as much, and no more, part of this as self-interpretation is part of our knowledge of our own intentions.
In presenting and arguing for this interpretation, I shall mostly limit myself to what Kreisel distinguished as questions in ‘general philosophy’ in contrast with questions in the ‘philosophy of mathematics’, viz., to questions concerning the relation of mathematics to ‘life’ in contrast with questions raised by specific mathematical investigations, for which my incompetence is enormous. My policy goes well with Wittgenstein’s reluctance to deny anything and with his declared desire to leave mathematical results as they are [R: 104, 157, 174; see also Dummett, op. cit., page 325], if not with his conclusions about the nature of mathematics which I already have suggested intimate a resemblance with metaphysics. For all that, Wittgenstein’s early use of truth-tables was a small but historically consequential contribution to mathematical logic itself, and he certainly did often appear to be denying things. I gladly defer to the authority of the mathematicians who say that Wittgenstein didn’t grasp the mathematical significance of the results he interpreted and that his specific observations are useful only to those who, in their mathematical innocence, are apt to make horrendous misconstructions. My own judgement is that the useful part of what Wittgenstein has to say about true but unprovable propositions in R [50 et. seq.] has since been better said by others and surely does not show much sensitivity to the subtleties of the mathematical questions at issue. I shall, however, say something about Wittgenstein’s views on real numbers and consistency.
The bulk of what follows will be a review of recurrent themes in Wittgenstein’s philosophy of mathematics. I begin with some general historical remarks and finish with a perfunctory assessment and with an attempt to impart a sense of the plausibility of Wittgenstein’s ideas.
Some Background
Perhaps the most distinctive and revolutionary feature of Frege’s and Russell’s logistical programmes for pure mathematics was the declared aim of anchoring mathematics to the bedrock of nonmathematical thought and language. Logic and mathematics are, according to this view, responsible to the familiar activities of inferring, counting, measuring, and ultimately are theories of the propositions which formulate everyday facts. Both thinkers found it necessary to lay down a number of original and still influential theories about language by which their logical theories were secured. Frege put special emphasis on the distinction between the alleged Bedeutungen of proper names and function names (so-called). Of his other key distinctions, that between Sinn and Bedeutung was essential but only occasionally visible in the system of the Grundgesetze, and that between ‘content’ and ‘judgement’, though highly visible was insufficiently elaborated. Russell, with greater abandon and less coherence, purported to erect the whole structure of logic and mathematics on the so-called Vicious Circle Principle which first and foremost was a principle about language in its ‘civil’, nonmathematical appearances.
Wittgenstein found much to provoke him in these theories of language taken simply in themselves. More importantly for our purposes, he thought that the whole logistical programme foundered on a misconception about the relation between mathematics and language. The misconception involved the assumption that mathematics with logic could be arranged as an autonomous theory eligible for systematic recasting as a deductive discipline. The systems of the Grundgesetze and PM seemed to Wittgenstein like Muscovite monuments to this mistake, against which he polemicized at length across the pages of T. Among the more dramatic symptoms that something was wrong were the contradiction and the ultimate incoherence of the theory of types as a theory of language (see esp. vol. II of PM, prefatory statement), and the occurrence among the fundamentals of poorly understood and apparently contingent propositions. Wittgenstein exploited all these difficulties to the full, together with such borrowed positive points of doctrine as Russell’s affirmed belief that the theory of types is a theory of symbolism and Frege’s argument that we cannot formulate the crucial distinction between function and object. But the main difficulty simply was that neither Frege nor Russell were able to say precisely what they thought the ultimate relation between mathematics and language really was. The logistical method frustrated the logistical intent. Both men finally were driven to the idea that the laws of logic are extremely general propositions, perhaps about the world at large (Russell), perhaps about what we ought to take as true (Frege), or (mysteriously) about a domain of things preferable to the denizens of traditional mathematics only for being more evidently factitious. Wittgenstein for a time concurred with the thesis that logical propositions are very general propositions, but later, taking his cue from the way Frege explained and justified his axioms, argued that the truths of logic are not universal propositions about language or about anything at all, but rather reflections of how we use language. Proof, in this conception, becomes ‘demonstration’ in the original sense—we exhibit how things are in our language—and not the logical derivation of propositions. Notoriously, the resulting doctrine was excessively rigid in its exaggerated demand that demonstration in the sense of ‘showing how it is’ is the only recourse available to one who would wish to comprehend better the foundations of thought and the nature of necessity.
This theory of mathematics carried over almost intact into Wittgenstein’s ‘middle’, ‘verificationist’ period, of which B is up to now the main published record. The extreme position on ‘showing’ and even the details are preserved. There were some additions and some changes of interest. In general philosophy, he was showing an advancing interest in issues of scepticism and in the problem of privacy in particular. The 1929 article on logical form witnessed a significant alteration of detail though not in principle in the T idea that language is a unified structure with a unified logic which could be comprehended at a shot. His reputed conversations with Brouwer may have confirmed his tendencies towards ‘intuitionism’ in mathematics and encouraged him to ask us to consider how one could verify statements about (e.g.) real numbers, where only arithmetic and recursive methods generally were acceptable procedures. (See B: 174ff.) In B, Wittgenstein showed a much greater interest than appears in T in the actual ideas of modern mathematics, chiefly the modern conceptions of the infinite and of real numbers and in the theology of Mengenlehre. He makes continual but novel application of the idea of ‘rules for going on’, only dimly prefigured in the Tractarian appeal to operations. In the second Anhang, which is a record of conversations with Schlick and Waismann, he turned his argumentative weapons against the philosophy of Hilbert and the emerging doctrine of metamathematics, polemicizing stridently against the philosophy which underlay the concern with consistency and against the alleged importance of Goedel’s incompleteness theorem.
These interests and critical tendencies persist into the writings collected together as R in which (I argue) Wittgenstein’s Tractarian doctrine also survives, softened somewhat by his dawning recognition of the varieties of language and consequent stress on the ‘motley of mathematics’. The chief change, I believe, is that finally he relaxed his rigid insistence that necessities in no way can depend upon the mere contingencies of the human situation. Wittgenstein was finally able to find something like a foundation for mathematics in the natural history of man.
Theme 1.1 WE MAY BEGIN BY OBSERVING THAT WITTGENSTEIN TOOK
AS HIS PRIMARY EXAMPLES OF AUTHENTIC LIVING MATHEMATICS THEORY OF INFERENCE [obvious in T, B, and R alike, but notice that he became progressively less permissive about logic and more relentless in his criticisms of its claims],
CALCULATION,
Kreisel observed that Wittgenstein was concerned with the neglected question of elementary computations. [Also see: Bernays, op. cit., p. 11.] In The called his method of arriving at tautologies calculation, thus assimilating the theory of inference and logical truth to arithmetic [T: 6.1203, 6.126]. He thought that mathematics was nothing else but the method of substituting into equations, also described as calculation [T: 6.2331, 6.24].
One can suppose that he had something like trigonometry in mind. [B: 177, 180f.]
(The theory as presented is probably incoherent if only because the equations in question would not involve mathematical symbols but rather ordinary names.)
His only example contains numerals [6.231, 6.241], and his theory of numbers charitably can be seen as equivalent to the Peano scheme for introducing arithmetical operations.
Stress on calculation and on number systems continues into b [throughout],
It is significant that he appears to adhere still to the theory of the T [132f., 142f.].
and R [e.g., 88f.], where he observes the connexion with counting [5f., 28]. It also is evident that his maxim that process and result are equivalent is nowhere more plausible than for calculation.
GEOMETRY,
which he likens to calculation [T: 2.0131, 3.032, 6.35; B: 152, 216f.; R: 16ff., 77],
TOPOLOGY [R: 174F.; also see knot-untying example, B: 184],
KINEMATICS,
Wittgenstein was fascinated by gears and simple mechanisms. These are among his favourite examples of proofs by diagrams, where the picture defines the connexions and conveys the sense of rigidity and compulsion [R: 35–9, 119f., 127f.], and also excellent illustrations of the way in which an actual movement can be taken as a ‘demonstration of what is essential’ [R: 25f., 139, 195f.];
AND, DERIVATIVELY, MEASUREMENT,
In both B and R, Wittgenstein frequently appeals to measurement to illustrate internal relations among concepts and the imposition of controls [e.g., R: 27f.; 159f.; 173f., 194];
MECHANICS,
Especially in the T [4.041, 6.321–6.3611, 6.3751]. There also are a few stray examples in b, especially in the first Anhang and in R [e.g., 136f.].
AND, FINALLY, IN LATER WRITINGS, THE KIND OF DEMONSTRATION OR GEDANKENEXPERIMENT BY WHICH PHYSICISTS GET US TO COMPREHEND THE PRINCIPLES OF PHYSICS. [See references above under ‘kinematics’ where a diagram or an actual working mechanism is used as a proof.]
WITTGENSTEIN CONSPICUOUSLY NEGLECTS CLASSICAL DEMONSTRATIONS IN ANALYSIS WHICH CANNOT BE ACHIEVED EITHER BY MERELY EXHIBITING A FIGURE OR BY MAKING A FORMAL CALCULATION. [See Bernays, op. cit., p. 2.]
Theme 2. LOGIC AND MATHEMATICS ARE METHODS FOR THE ‘TRANSCENDENTAL’ DEMONSTRATION OF THE LOGICAL, ESSENTIAL PROPERTIES OF OUR ‘CIVIL’ (NONMATHEMATICAL) WAYS OF THINKING AND TALKING ABOUT THE WORLD.
(For the kind of thing he first had in mind, see T : 6.12, 6.121; B: 142f. He uses ‘logical’ throughout T in this sense, effectively equivalent to ‘internal’, ‘formal’, ‘necessary’, and ‘essential’, all of which, especially ‘internal’, recur frequently in B and R. He employed the highly Kantian ‘transcendental’ at T: 6.13. For interpretation of what it means, see below.)
The task of mathematics is to demonstrate what it makes sense to say,
by showing how to apply a rule; by laying down tracks in language [R: 12, 77f., 80];
by putting ordinary concepts characteristically having a nonmathematical use into memorable relations [R: 25f.].
‘What makes people accept a proof is that they use words as language’ [R: 44]; ‘It is essential to mathematics that its signs are also employed in mufti’ [R: 133]; ‘Concepts which occur in “necessary” propositions must also occur in non-necessary ones’ [R: 153]. This idea of the ‘civil’ occurrence of mathematical concepts, which is a recurrent one in R [also 8, 41, 79], echoes the Tractarian thought that logic and mathematics must be ‘in touch’ with reality via their ‘application’ [T : 2.15121, 5.5 5 7]. What makes the game mathematics is the ‘application’ to ordinary propositions [B: 131, 135, 143; R: 133] ; ‘Logic gets its whole sense simply from its presumed application to propositions’ [R: 118, 133], e.g., through the ordinary sentences which occur in tautologies,
(Tautologies while senseless are not nonsense, but rather limiting, degenerate propositions—not schemata [T : 4.4611, 4.466, 5.143, 6.112, 6.121; also R: 79])
and the ordinary names which occur in the equations of mathematics [T: 6.22, 6.23, 6.2341; B: 143];
(But Wittgenstein himself later observed that, since he left no place for identity statements, these equations are not even degenerate propositions [B: 142].)
it is seen in Wittgenstein’s insistence that ideas like those of the numbers which are submitted to mathematical investigation are dependent on concepts [B: 123; R: 150],
(In the T numbers were connected essentially with the rules by which propositions were generated, but the idea was not worked out [T: 6.02, 6.03].)
and is part of what he sometimes had in mind when he spoke of ‘use’ [R: 3f.]. The general conception of language is the only mathematical primitive [T: 5.472], language itself is the reality beyond [R: 6, 39, 80] and supplies the necessary intuition upon which mathematics depends [T: 5.4731, 6.233]. The manner in which logical properties of language are demonstratively reflected is well illustrated by Wittgenstein’s use of truth-functional analysis, borrowed from Frege [T: 4.431], and which suggests what has since come to be called ‘semantics’.
Proof thus conceived serves to reveal the essential ‘internal’ properties and relations of concepts, e.g., relations of logical consequence or the dependence of measurement on counting. The enterprise presupposes conventions to be sure and perhaps other facts as well, but these are facts which enable our civil nonmathematical ways of talking,
The need for conventions and the dependence of mathematics on ‘forms of life’ is a persistent theme in R [e.g., 94 et. seq.] but was anticipated already in T [3.342, 4.1121]— the employment of ‘thought’ broadly hints that logic presupposes psychical facts.
from which the mathematics is derivative—mathematics is ‘postulated’ with the language, and not conversely [T: 6.1233; R: 43f., 159].
Here is where Wittgenstein parted company with Frege and Russell. While agreeing with them in insisting on the fact of a relation between mathematics and civil language and in the thought that application is what gives mathematics its meaning, he thought that they had failed to apprehend this dependence with sufficient clarity or constancy [B: 137; R: 41, 78], in supposing that logic could supply laws of truth to language [T: 5.132, 5.4733; see Frege, Grundgesettze v. I, xvi; also in his late article ‘Der Gedanke’, translated and reprinted in Mind, 1956, pp. 289–311, esp. p. 289] with resulting confusions about truth and meaning [T: 4.431, 6.111].
(Actually, Wittgenstein’s later unsettled way of speaking about mathematical proofs changing concepts [for which see pp. 8 1f. below] sometimes sounds suspiciously like what Frege had proposed.)
Following Wittgenstein’s own usage, call these ‘civil’ conceptual activities the ‘application’ of mathematics; to repeat, the application is what gives ‘meaning’ to mathematics [B: 201 ; R: 118, 147f., 172, 186].
(Also [B: 229], where he says that application is the criterion of reality in mathematics. Only here ‘application’ may mean something like ‘effectively computable’, and illustrates Wittgenstein’s occasional willingness to count in ‘internal applications’, within mathematics itself. He is very careless about this crucial distinction upon which the distinction between pure and applied mathematics depends.)
An immediate consequence of this is that the traditional distinction between pure and applied mathematics is not valid.
The logic of T is like the plain formal ‘nonmathematical’ logic of the earlier teaching tradition imparted in direct connexion with examples and thought to be evident in the very understanding of language, [See T: 5.13, 6.12, 6.1221.]
(Also [R: 186], which I interpret to mean that concepts which are introduced for exclusive application within mathematics itself, such as 2ℵo, are apt to be footless and very easily thought about in the wrong way. Their meaning too depends upon distant relations to non-mathematical activities like measurement.)
We cannot ‘give’ mathematics an application or interpretation either by appending explanations on the side [T: 5.452] or, more formally, by assigning objects to names and classes to predicates or by ‘interpreting’ variables,
(See B: 327 where he appears to challenge the Hilbertian distinction between game and theory.)
for the application cannot be called into question, hence cannot be stipulated or anything of the kind. The application must fend for itself [B: 130f.; R: 67 and T: 5.473, but here observe without the important ‘Anwendung’]. Mathematical truths are essentially self-applicable [e.g. B: 130ff., R: 176].
The ‘internal’ features of language and thought are typically demonstrated in arguments of a kind which have been called ‘transcendental’.
The truth of what is shown is guaranteed by the sense of the formulation, and the proof is less an indication of truth than of what it makes sense to say [T: 3.04, 6.2322; B: 144, 170, 200f.;R: 77, 80]. These internal, formal features are given with the objects to which they apply, and their presence cannot meaningfully be doubted or denied because they are presupposed in the very formulation of the doubt.
(This is a familiar theme in philosophy. Wittgenstein probably found his precedents in the turn of the century furor over the contradiction and in Russell’s unavailing attempt to formulate the Vicious Circle Principle, which still left him in the situation where he could not make meaningful misascriptions or true denials of types, as well as in the Moore-Russell doctrine of ‘undefinables’, explained as ideas that would be presupposed in any attempted definition.)
In the T, Wittgenstein also held, ‘It would be as nonsensical to ascribe a formal property to a proposition as to deny it a formal property’ [4.124].
(For which, again, he had the precedent of Russell’s admission that we cannot say an object is not of a type other than it is and also of Frege’s conclusion that we cannot say that a function is a function.)
Wittgenstein put that by saying it would involve us in an illicit attempt to transcend the limits of thought and language [T: 2.174, 4.041, 4.121, 5.61, 6.45–7]. The same idea carries over into B where Wittgenstein held that you can neither deny nor assert fundamental principles and presuppositions or definitions [B: 172, 120, 193, 330]. Though Wittgenstein later repudiated the ‘transcendental metaphysics’ of ‘possibilities’ and of the ‘ideal’ which so naturally attends these arguments [e.g., Investigations, I, §§ 89–105; R: 6, 22f.], something similar remains in the important thought that the fundamental agreements in the ways in which we act and the fundamental facts of human nature which constitute the environment in which our concepts are formed and upon which our mathematics reposes are also the limits of empiricism’ [R: 121, 171, 176].
Reductio proofs conceived of as deductions are especially suspect on this view, for they would appear to depart from meaningless assumptions [B: 190; R: 147, 177]. They perhaps could be redeemed were we able to rid ourselves of the idea that mathematical demonstration is logical derivation.
It always was part of Wittgensteins view of the dependence of mathematics on civil language that mathematical results should not be thought of as empirical or empirical-like statements [for development of this, see theme 4 below], and that we should resist every inclination to suppose that mathematical propositions have their own subject-matter— a domain of ‘mathematical objects’—for this can only be an obscuring and pernicious pretence. Hence, the Tractarian polemic against ‘logical objects’ [4.0312, 5.4 et. seq.], later extended against the claims of Mengenlehre. [See Theme 6 below.] The only logical primitive, again, is language itself. Also, logistical systematisation, if ever appropriate, would be appropriate only to empirical theories. [For development, see Theme 8 below.] Indeed, mathematical results are not properly to be regarded as propositions at all for, like the laws of logic, they ‘… show what we do with propositions, as opposed to expressing opinions and convictions’ [‘Math Notes’], and what we do with propositions is brought out in the mathematical demonstrations themselves and not in the supposed conclusions of logical derivations. [For development, see Themes 8 and 9 below.]
Theme 3. MATHEMATICAL PROOF IS AN INSTRUMENT OF CONCEPTUAL CONTROL
An ‘instrument of language’ [R: 78, 80, 165]
USED TO DETERMINE WHAT IT MAKES SENSE TO SAY OR WHAT IS POSSIBLE. [See above and B: 140; R: 116.] WHEN SUCCESSFULLY EMPLOYED, IT RESULTS IN THE UNFOLDING [R: 24, 30] OR FIXATION [B: 201], 249; R: 80, 127, 195f.] OF WHAT WE TAKE TO BE ESSENTIAL [R: 12f., 30, 163].
In explaining how this comes about, Wittgenstein appeals to a bewildering variety of analogies and technical notions.
(He always was careless about terminology, and sometimes regretted it [R : 16 3, 18 8, 195, in connexion with ‘concept’].)
‘logic’
(Especially prominent in T, where the word usually was employed to cover everything having to do with the a priori determination of the essential ‘internal’ features of language, but sometimes more narrowly confined to the method of tautologies and contradictions [e.g., 6.22].)
‘internal’ [throughout T, B and R]
‘grammar’ [B: 129f, 135, 186, 188, 309; R: 40, 77f., 119]
‘syntax’ [T: 3.327, 3.33, 3.344, 6.124; B: 143, 178, 189, 216] ‘forms’
(of language, thought, often identified with ‘possibilities’ [T throughout; B: 178])
‘dictionary’, ‘definitions’ and ‘what we call’ [B: 135, 194; R: 28, 76, 174]
‘rules’ [T: 4.0141, 5.476; B: 143, 178, 216, 311, et. seq.; R: 21, 26, 32, 47, 77, 81, 115ff, 127, 159, 163, 196]
(Notably ‘rules of inference’ [B: 134; R: 178ff., 185])
‘convention’ [R: 6, 159]
‘decision’ [R: 77]
‘moves’ and ‘positions in games’ [R: 94f.]
‘pointers’, ‘paths’, ‘channels’, ‘handrails’ [R: 82, 116, 122, 193]
‘pictures’, ‘patterns’, ‘models’ and descriptions thereof [R : 11f., 29, 75, 117]
(N.B.: In R, unlike in T, pictures and models are not themselves propositions, but rather instruments of conceptual control used to regulate what is to count as a proposition and to demonstrate connexions among propositions.)
‘methods of experiment’ and ‘of prediction’ and formulations thereof [R: 13, 65, 187]
‘frameworks of description’ [R: 160; also T: 6.341]
‘impress a procedure’ [R: 14]
‘paradigms’ [R: 45f., 82, 193]
‘measures’, ‘standards’, ‘norms’, ‘controls’ [B: 212; R: 47, 76, 99, 199, 194]; and the general idea of a criterion of identity [R: 96, 196], among others.
Proof, by bringing us into position to see things in a certain way [R: 13, 18], makes visible and enables us better to comprehend what previously may have been only implicit and latent in the civil practice, e.g., which thing or what kind of thing we are talking about [T : 6.232 ; R: 27], or the ‘possibilities’ (which are the ‘actualities’ of mathematics) admitted by thought and language [T: 2.0121, 6.361; B: 138, 157, 161, 164, 217, 253; R: 39, 116] and to control those civil operations, [B: 212; R: 117], e.g. by guiding inferences [T: 6.211], or by supplying a criterion for mistakes in counting [R: 27, 76]. Coming into this position is no more a matter of discovery than is coming to know of the existence of the north pole [T: 6.1251, 6.1261; B: 182, 189f.; R: 127].
Or, rather, the making of an expedition to the pole, which is something to be done, is like proof, in that facts are not in dispute in either case. What is important is not so much where one gets as getting there.
The illusion of discovery is due to an inadmissible switching back and forth between empirical and conceptual questions [R: 26, 126f.].
It is to be likened rather to ‘nonobservational’ reflective knowledge of one’s own actions.
(Could one argue that all ‘transcendental knowledge’ is of this kind?)
I think here of Wittgenstein’s use of ‘reflection’ [T: 6.13] and of his later stress on ‘doing’, ‘knowing how to go on’, foreshadowed in the T idea of an ‘operation’ [T from 5.51; B: 191, 199; R: 3, 7, 117f., 123, 176, 179]. I find his example of untying knots particularly trenchant [B: 182, 184f.].
Wittgenstein sometimes put this in an exaggerated way by speaking of ‘invention’ [B: 186; R: 47, 59, 140]. When Wittgenstein gave up his ‘static’ conception of language as a unified system governed by the ‘postulate of the determinateness of sense’ with a single general form of proposition whose whole logic could be comprehended all at once [T: 2.0124, 3.23, 4.5, 4.53, 5.47, 5.476, 5.55, 5.557; B: 177f., 187f.], when the unity of language gave way to the variety of language games, a monolithic logic to the ‘motley of mathematics’ [R: 84, 194],
(The seeds already were sown in the T discussion of mechanics [esp. 6.34–6.35]. He tried to hush up the incoherence at 6.3431. His use of ‘Forderung’ in the sense of ‘demand’ at 3.23 and 6.1223 also anticipated the future. Also in B [170], where Wittgenstein worries how proof is possible if proof depends only on sense which must be comprehended before proof can be attempted—this naturally leads to the thought that proof also alters or creates sense.)
and mathematical demonstration became in its applications comparatively unstable, like a four-legged table [R: 115, 180], he was faced with the problem of saying to what extent the demonstration of conceptual connexions also changes and creates concepts.
Wittgenstein was uneasy with this way of talking about concept alteration and innovation to which some of his commentators have attached so much importance [R: 126, 154]. The old concept, at all events, always is in the background [R: 121]. The fact is he had a number of different more or less acceptable things in mind. The most important was that of establishing new connexions between [old] concepts, and creating thereby the concept of a connexion [R: 79, 154, 188, 195].
(But, he asks, is the conceptual apparatus a concept, is a ‘conceptual path a concept’ [R: 154, 188]? In this regard, it is worth noting that the mathematical concepts in question, e.g., prime number, typically have no civil use.)
But also : Turning things around so that they look different [R: 18, 122, 192];
extending old paradigms and rules to cover new cases [R: 47, 193];
changing rules and introducing new ones [R: 124];
introducing new paradigms [R: 78, 82];
introducing new paradigms for internal application within mathematics itself, e.g., recasting arithmetic in an algebraic mould [B: 202f.];
(Wittgenstein’s way of talking about creating concepts is, for obvious reasons, more appropriate for these ‘internal’ applications which Wittgenstein’s general philosophy is calculated to play down and overlook. One could rewrite the mufti theme to say that mathematics is ‘ultimately’ created for increasing our reflective comprehension of unconscripted civil concepts which must be ‘there’ in advance of mathematizing.)
fixing criteria of identity, e.g., making it explicit that a cardinal number is unaffected by the direction from which an assemblage is counted.
Our description of the motley of mathematics should, at any rate, capture the natural conceptual order of the different techniques, an order which duplicates the relations of dependence within language [B: 244f; R: 7]. This is an important instrument in Wittgenstein’s criticism of attempts to reduce one part of mathematics to another, e.g., the theory of numbers to logic. [For development, see theme 7 below.]
Theme 4. MATHEMATICAL RESULTS, WHATEVER ELSE THEY ARE, ARE ABOVE ALL CONTINGENCY AND MUST BE NECESSARY AND RIGID [T: throughout, but see esp. 2.012, 5.55 et. seq., 6.111, 6.1222, 6.1233].
(In B and R, Wittgenstein repeatedly contrasts calculations, pictures, paradigms and other assorted mathematical instruments with causality, experiment, prediction [B: 125, 133, 152, 187, 209f., 213, 235, 238, 240, 313; R: 19, 28f., 32, 69, 81£, 91, 94 et. seq., 113f., 119, 124f., 171, 186f., 189ff; also T: 6.2331].
Axioms and consequences alike [R: 79, 114].
For this reason he approved of Frege’s and Russell’s campaign against ‘psychologism’ and ‘empiricism’ [T: 4.1121]. In T, the Axiom of Reducibility and the whole of set theory were held to be mathematically spurious because empirical [T: 6.031, 6.1232, 6.1233], and the classification of propositions according to form forbidden for the same reason [T: 5.553–5.5542]. The rule of necessity abetted Wittgenstein’s suspicions about the deductive theories of mathematics which appealed to such empirical paradigms as the coordination of objects, and commonly treated possibilities as realities [B: 140, 164f., 212], and in which demonstration was assimilated to a pattern of deducing empirical statements from other empirical statements leading back finally to axioms which simply are self-evident or obvious [T: 5.4731, 6.1232, 6.127].
A necessity, according to the early Wittgenstein, is something whose contrary cannot coherently be conceived and which, therefore, cannot meaningfully be doubted [T: 3.03–3.0321, 5.4731]. Transcendental arguments must establish necessities which are, so to speak, the other side of paradox. The truth of such propositions is determined with their sense [T: 3.04, 3.05; B: 144], and they have no justification except comprehension itself. He later would say that these propositions are not so much true as proven by use [R : 4]. They are a priori and known before the fact, from language alone [T: 3.04; B: 143]. The requirement of necessity for mathematical results is companion to Wittgenstein’s early idea that the actualities of mathematics are the possibilities of daily life, coordinate with what we can say [T: 2.0121; B: 135, 140, 153, 161, 164, 253 ; R: 116], whose existence cannot be meaningfully questioned, because proven by their essence [B: 124] and guaranteed by language [T: 5.525, 3.04]. While Wittgenstein later sloughed off this way of talking he never forsook the thought that necessity is a kind of dependence on the use of language [B: 13 5 ; R : 4, 20, 153], though the sharpness of the dictum was dulled by his emendation that we sometimes put forward necessities in order to fix a sense previously underdetermined [R: 113 et. seq., 121]. In T, Wittgenstein staunchly maintained that what is necessarily so is utterly without contingent consequences and in no way can depend upon the facts [T: 2.0211, 5.551, 5.552, 5.5542, 5.634, 6.1222]. This thesis, though still rather fashionable, probably is incoherent, and Wittgenstein himself gave it up in his later writings,
(For an important anticipation, see T [6.342, 6.343]. The old view lingers on in B [212]. In this connexion, it is notable that Wittgenstein began his Tractarian journey by drawing our attention to the actual world of actual facts, suggesting thereby that logic presupposes a world. A similar interpretation may be put on the extremely obscure T: 5.5521, though he appeared to withdraw the thought in B [164 n].)
where he allowed that the sense of what we say, hence what our language requires and permits, does indeed presuppose and is grounded upon unquestioned facts about the human situation—upon our actual ways of living and acting [R: 20f., 36, 43, 98, 124, 159, 173] and upon fundamental agreements among men [R: 13, 34, 94, 97, 164]. He did not think this in any way endangered the distinction between the contingent and the necessary.
(Though he now allowed for synthetic a priori truths [B: 129, 178; R: 125f.].)
Necessities are, in any case, not ‘about’ their contingent presuppositions [R: 159ff., 170 et. seq., 187]. And, however multi-form [R: 125], still function as controlling paradigms, rules, etc., and hence are not empirical statements of fact [R: 32, 46f., 81, 159, 174]. Though we indeed might have thought otherwise than we do, thinking as we do limits what is thinkable and determines what must be. What are recognized as necessary propositions would simply be unsayable in another world. If the world were very different from what it is, our actual concepts could not be made available, and some of what we now can comprehend wouldn’t be intelligible; obversely, we cannot now intelligently conceive all of what would then make sense. Wittgenstein exclaims ‘How can we describe the foundations of our language with empirical propositions?’ [R: 120; see also 4f., 14, 96 et. seq., 120, and Stroud’s development of the theme, op. cit.] These fundamental presuppositions are the contingent ‘limits of empiricism’ [R: 96, 171, 176] implied by what we cannot meaningfully question.
Theme 5. MATHEMATICAL RESULTS ARE NOT GENERAL CONCLUSIONS BUT PERSPICUOUS PROOFS.
(From the beginning, Wittgenstein worried about the apparent generality of the propositions of logic. In letters to Russell and in the ‘Notes on Logic’, he proposed that the propositions of logic were complete closures. [See Wittgenstein’s Notebooks 1914–16, pp. 119, 103, 126.] He turned away from this already in ‘Moore’s Notebooks’ [ibid., p. 107], and, in T, his dictum, ‘The mark of logical propositions is not their general validity’ [6.1231, 6.1232], became one of the foci of his criticism of Frege and Russell. His sense of the importance of the thought that mathematical truth is not universal validity is evident throughout B [138, 144, 148, 150] and continues into R [156].)
The apparent generality of mathematical results is not that of a universal proposition about mathematical objects but consists in the open applicability of a particular rule, paradigm or control [B: 150, 195, 312; R: 156].
It is a ‘direction’ (a vector?) [B: 163] ; it is knowing how to go on in a certain specific way, an ‘induction’ [B: 150, 250, 328], which does not establish a proposition about ‘all numbers’ but validates the use of a particular rule, e.g., that of binomial expansion, to arbitrary special cases.
(It is significant that many theorems which nowadays commonly are formalized according to recursive rubrics could just as convincingly have been established by consideration of an arbitrary special case, and indeed the latter kind of proof usually has greater explanatory power than the routine application of mathematical induction to a given formula. Wittgenstein also observed that in geometry we often begin with, e.g., ‘Take a triangle’ [B: 152]. He once put this by saying that in mathematics the general and the special cases must be mutually intersubstitutable [B: 207; see also 214 where he discusses incompatibility between the general and the special in mathematics]. This recalls Kant’s view that in mathematics we come to know synthetic necessities by ‘intuition’. Both Wittgenstein and Kant thought that mathematics deals with general concepts. Kant put this by saying that mathemathics uses ‘Anschauungen’, Wittgenstein by saying that particular and general coincide in mathematics. [For a nontechnical explanation of the Kantian theory see J. Hintikka’s ‘Kant’s “New Method of Thought” and his Theory of Mathematics’, Ajatus, 1965, pp. 37–47].)
The apparent infinite extension can be grasped in one step [B: 146f., 149], and the indefinitely applicable rule finitely formulated [B: 149, 314, 329]. Mathematics demonstrates the particular essentials of our conceptions—singular necessities, if you wish [B: 152, 182, 200].
(At B 182, he observed that the contradictory of ‘It is necessary for all’ is not ‘It is necessary for some not’, but rather, ‘It is not necessary for all’. Working on the assumption that all mathematical results are implicitly to the effect that something is necessary, the upshot would seem to be that the apparent universal statements of mathematics are not what they seem [also B: 249 f.].)
An example would be that a particular form of proof is valid [B: 198f.]. On this view, mathematical demonstrations do not establish that all the members of a certain totality have a certain property. [See theme 6 below.]
Wittgenstein’s alternative to ‘logical proofs in mathematics’ [R: 84] was perspicuous, memorable, reproducible [these three words occur throughout R; see the index of that book], geometrically cogent [R: 83] demonstrations, which reflect [T: 6.13] the essentials of civil language.
(This is part of what is meant in calling Wittgenstein an ‘intuitionist’. It goes well with his earlier concentration on verification and recursive procedures.)
Initially, recursive rules or operations for ‘going on’ were his favourite examples.
In T, the general form of proposition was given by exhibiting a putatively recursive rule [5.21–5.32, 5.5–5.503, 6.0–6.031]and everything in the domain of formal logic and mathematics was reduced to formal series and operations [4.1252, 4.1273, 5.1–5.150, 5.252, 5.2523, 6.0, 6.031] and calculation [6.126, 6.2331]. This trend of thought carried into B, where Wittgenstein continually returns to mathematical induction and formal sequence [throughout, but see 150, 182, 187, 202f., 250, 313 for some notable passages], holding, among other things, that the essence of a real number is an induction [234; for development see theme 10 below]. Induction still appears on the pages of R [e.g., 90], and, of course, Wittgenstein in this period was fascinated with ‘knowing how to go on’ [see early pages of R and parts of Philosophical Investigations].
(Kreisel traces Wittgenstein’s impatience with Goedelian results to an alleged dissatisfaction with the appeal to some general but still indefinite idea of recursiveness [[2] pp. 245f.] I doubt this because of his own uncritical fascination with ‘Knowing how to go on’.)
But in R, ‘perspicuity’ seems to cover almost any kind of ‘knowing how to’ [R: 3] and anything ‘plain to view’ [R: 83].
(There is little variety among Wittgenstein’s examples— arithmetical and stick representations of reckoning operations, simple geometric constructions, topological problems, kinematical diagrams, conversions of units— and almost no detail.)
Perspicuous proof apparently may make use of most anything which is not brought into question at all [R: 45] because illustrative of what is essential [65], hence above all contingency [124] and about which the understanding cannot be deceived [75, 81, 90f].
The demand for perspicuity was employed powerfully by Wittgenstein in his criticism of ‘reductionism’ in mathematics: alleged alternative proofs (e.g., logistical proof of arithmetical identities) lack the necessary perspicuity—what is proven rather, and proven perspicuously, is a general correspondence between two systems [R: 5 et. seq.; for development, see theme 7 below].
Wittgenstein has been styled a ‘finitist’ [Kreisel, [1], p. 148, and Bernays, op. cit., p. 11 ; but note Wittgenstein’s own implicit denials at R: 63, 150]. I think part of what is meant has something to do with his demand for perspicuity. A proof must exhibit or demonstrate essential connexions. That can be done by making a ‘construction’ [e.g., B: 132], or by exhibiting an apparently concrete representation, or by allowing the machine to symbolize its own possible movement [e.g., R: 37f.]. Such representations are taken as self-applicable and hence have incontrovertible geometric cogency [see B: 132f.].
Consider the following demonstration of a negative answer to the question whether one can with seven 2 × 1 tiles finish tiling a bathroom floor already having 1 × 1 tiles in two opposite corners:
or the following ‘shortest’ proof of the Pythagorean Theorem
Observe that the area of ABC is the sum of the areas of ADB and BDC, all of which are similar.
Of course, one must recognize what is essential in the representation to be able to see them as proofs, and much preparation and prompting may be needed for that.
(Though I am perfectly certain that the ‘shortest proof’ above must be a proof, I’m not so sure of the proof.)
Theme 6. MATHEMATICS IS NOT THE ABSTRACT STUDY OF THE INFINITE.
Wittgenstein keenly appreciated that the use of a concept of infinity was both peculiar to and the hallmark of mathematics.
(In his ‘middle’ period especially, Wittgenstein was preoccupied with gaining a correct understanding of this concept of the infinite. See especially Part XII and first Anhang, both of which contain reminders of his use in T of the idea of an operation for going on.)
That is because mathematics is concerned with the unlimited possibility of applying rules which are implicit in the use of language. The mathematical concept of the infinite clearly is not an idea which belongs to civil language, but it can be comprehended only as part of the use of mathematical demonstration to control our understanding of civil language. Infinitude is not a feature of anything we talk about or conceptualize in civil language, but, so to speak, a feature of our forms of conceptualization themselves [a highly Kantian theme; see esp. B: 155–61].
Not something known by ‘experience’ [B: 154f., 157f., 304 et. seq.]. The appearance of ‘infinite’ in this usage shows that we are dealing with ‘possibilities’ [B: 153, 155, 159, 164, 313], with ‘syntax’, ‘grammar’, and ‘rule’ [B: 160f., 309, 313f.], and different kinds of mathematical infinity are features of different conceptions and not of different realities [R: 57].
In imputing infinitude in this sense one registers confidence that one has perspicuously apprehended the unlimited full range of application of a (single) concept (rule, form, etc.) [T: 2.0131; B: 153, 157, 313f.].
The assertion that a sure event will occur sometime in the infinitude of time is like a tautology [B: 153, 311].
But one is apt to misconceive a finitely formulated rule to be an unverifiable universal statement about a large aggregate of things—to take each possible application of a single rule as an actual mathematical object [B: 314; for ‘unverifiability’, see 149].
One regards the singular possibilities latent in our forms of thought as plural actualities. But an infinite possibility is not the possibility of an actual infinity [B: 164f., 159, 219, 312f.].
(In supposing, for example, that the possibilities of pairing things off—the natural numbers—can themselves be paired off like apples and pears [B: 140, 162].)
The mistake is in part a result of mixing up the material, temporal ‘can’ of capability and opportunity with the ‘adverbial’ timeless ‘can’ of logical possibility [B: 161f., 219, 311ff.; also R: 38f.]. It is a confusion of ‘den Elementen der Er-kenntnis’ with physical questions [B: 168].
(This is an instance of the familiar theme of grammatical misassimilations.)
The possibilities are singular, and their alleged extensions cannot have independent existence.
(The existence of an infinite extension is proved by essence [B: 124, 221f.], e.g., the language has an unlimited number of names [T: 5.535].)
Supposed infinite sets always presuppose (logical) concepts and must be ‘constructed’ [B: 155, 221, 244].
The spuriously resulting world of ‘sets’ may present itself as the proper and autonomous subject-matter of mathematics to claim the titles of the discredited philosophies of empiricism and psychologism.
(There are precedents for this kind of ‘Platonism’ in the ancient idea that knowable, necessary, propositions must have their own special subject-matter—perhaps universals. Russell had advanced this kind of proposal in Problems of Philosophy.)
That has happened, in fact, with Mengenlehre towards which Wittgenstein’s early criticism of Frege’s ‘logical objects’ was redirected [B: 206f., 211].
(We already have seen that earlier he had held the subject was straightforwardly empirical [T: 6.031].)
Though this investigation can be pursued with exactness and precision, the claim that it provides a subject-matter and a foundation for mathematics is at best a pretence, based only on false pictures, which obscures essentials and unfailingly leads to mystery and paradox.
The set theorist seems to know what he is talking about because he uses borrowed pictures [B: 162, 218, 221; R: 62, 144ff., 149ff.] and imposes apparently unexceptionable principles like Excluded Middle [B: 176; R: 140, 149], but mostly because his concepts are introduced in application to familiar and compelling examples which unquestionably do make sense [B: 208f.: R: 60, 137, 148 et. seq.].
(I believe that Cantor’s point of departure was the theory of Fourier Expansions.)
But he leaps beyond these to fanciful and underdetermined illustrations [B: 224, 232; R: 9f., 55, 148, 180].
(Where there is no way of perspicuously surveying the supposed infinite extensions or making ‘selections’ [B: 167, 224].)
And his operations become rootless [B: 211; R: 149f.]. The result is mystery, glitter and darkness which widens the eyes and makes us gasp and reel [R: 142, 148].
(A kind of legerdemain, ceremony and incantation [B: 229; R: 60, 136f., and 53 for a different but similar instance], full of meaningless problems [B: 175f.], which Wittgenstein sometimes called ‘alchemy’ [R: 142]. He was an occasional and unimpressive practitioner in his own early days, e.g., in a letter to Russell in which he purported to prove that the Axiom of Reducibility was empirical and contingent.)
And the mathematician appears like nothing so much as a guardian of a cult, like the ancient priests and astrologers.
Set theory, whose general employment would result only in covering the distinctive features of different parts of mathematics with a uniform formal structure, least of all could be a foundation for mathematics [B: 206; R: 150: for development, see theme 7 below]. Set theory (whose credentials, as one part of the motley, are not in question) can be redeemed philosophically only by deflating the theology of ‘pure mathematics’ with its pantheon of free floating objects [R: 142] and its fanciful images [R: 60f., 180f.] and scholastic questions [B: 149; R: 59], by returning the subject to its concrete illustrations and applications [R: 62f., 133ff., 146, 152f.], steering away from gratuitous abstractions [see Kreisel [2]]. Mathematicians in general, and set theorists in particular, are not giving general descriptions of amorphous aggregations but providing general schemes for dealing with particular cases. These applications to cases are of the essence, and we must always attend to the civil and mathematical rôles of mathematical concepts in their order of dependence.
Regarding dependence, the real numbers presuppose the natural numbers and must be comparable with the rationals [B: 231f., 236ff.; see theme 7 and theme 10 below]. Our understanding of continuity and other such notions is built upon our familiarity with numbers [B: 207f.] and has important connexions with geometry [R: 148, 151]. Set theory should not try to reverse the dependencies or suppose it can make them disappear [B: 211], nor try to hide the distinctiveness of the various parts of mathematics with an amorphous, uniform elucidation [B: 206, 209; R: 146; for further development, see theme 7 below]. Civil rôles should dominate mathematics, Wittgenstein thought, but it is significant that set theoretical concepts are applied almost exclusively within mathematics itself [R: 186]. Writing about the compactness of the rationals, he says:
‘ “Fractions cannot be arranged in an order of magnitude.” First and foremost, this sounds extremely interesting and remarkable.
‘It sounds interesting in a quite different way from, say, a proposition of the differential calculus. The difference, I think, resides in the fact that such a proposition is easily associated with an application to physics, whereas this proposition belongs simply and solely to mathematics, seems to concern the natural history of mathematical objects themselves.
‘One would like to say of it, e.g., it introduces us to the mysteries of the mathematical world. This is the aspect against which I want to give a warning.’ [R: 60.]
It remains that the concepts of pure mathematics are in danger of losing their feet [R: 186]. Much of the so-called ‘foundations of mathematics’ seems to be dedicated to this possibility. Kreisel reproves Wittgenstein for not taking sufficiently well into account that logic ‘ … provided concepts necessary for the description of mathematics, just as, according to Wittgenstein, mathematics provides the concepts necessary in the description of nature’ [[1] p. 143]. Wittgenstein could have accepted that [see R: 145f.] and thrown in set theory too, and then have argued that for just that reason these subjects are ill-suited to serve as foundations.
Theme 7. NO ONE PART OF MATHEMATICS IS A FOUNDATION FOR ALL THE OTHERS.
The desire for foundations is in part due to a mistaken hankering for justification [R: 8f., 76, 82],
But, mathematics is the measure not the measured [R: 99]. Every part must fend for itself [T: 5.473; B: 131; R: 67], and must show in itself and in its application that it is true [B: 143f.]. We cannot explain the application with remarks in the margin [T: 5.452], nor can it be secured by any other part of mathematics [R: 67]. This is a kind of generalization of Brouwer’s reliance upon ‘basal intuition’. Ultimately, we just calculate as we do [R: 98], and the only justification for that is what we do outside of mathematics and in how we talk [T: 5.47, 5.472, 6.233; R: 9, 72, 82].
and to a methodology which demands a uniform presentation.
We feel that this insures comprehensibility and controls derivations, but logical notation is no better than prose [R: 155], and logical and set theoretic formulations hide important differences and defects in conception under an amorphous presentation [B: 206, 221; R: 76f., 89, 145f.].
From the beginning Wittgenstein objected to unified logistical formulations and doubted the claims of formal logic to be a foundation [R: 72f., 83, 145f.].
In T mathematics was described as a logical method of calculation, of making substitutions into equations [6.2, 6.23 3, 6.234, 6.2341, 6.24], and contrasted with the method of tautologies and contradictions used in formal logic for purposes of revealing relations of logical consequence [6.22]. Mathematics, so regarded, is neither part of logic (in the narrow sense of ‘logic’), nor can it be deductively derived from logic.
(Russell’s proposed definition of ‘=’ in purely logical terms was banned on the grounds that objects are only contingently indiscernible [T: 2.0233, 2.02331, 3.221, 5.5302]. The essential, necessary ‘pure numerical difference’ needed in mathematics cannot be captured in this way [4.1272, 5.5303].)
But it is like logic in being a method of calculation which is derived from nothing at all, but reflected in the use of language. Logic and mathematics are true together in so far as, ‘If a word [sic., “God”?] creates a world so that in it the principles of logic are true, it thereby creates a world in which the whole of mathematics holds’ [‘Notes on Logic’], viz. by being together implied by the fact that we use language as we do, and by the consideration that totalities of elementary propositions with which logic operates and of objects whose mutual distinctness is presupposed by mathematics impose the same limits on thought and reality [T: 5.5561].
(There is in T, from 5.11 on, a strong tendency operating to reduce the concepts of logic and mathematics together to the idea of a putative recursive rule for going on. [See also 4.1252, 4.1273.])
Wittgenstein’s whole unsatisfying doctrine about identity and mathematics persisted into B, where he found new arguments to support his view that equations could not be reduced to tautologies [B: 141ff., also 126, which anticipates R].
(This appears to be a response to Ramsey’s interpretation of T: 5.535 [see Foundations of Mathematics, pp. 60f.].)
He goes so far as to suggest that arithmetic, in its independence from logic and reliance upon its own form of autonomous insight, is an example of Kant’s synthetic a priori [B: 129].
These objections and doubts later were directed against reductions of every kind.
First, because reduction would destroy the essential perspicuity of proof [B: 125f.; R: 62f., 68, 70, 81, 83, 91]. Second, it would not get us what we had and wish to retain, e.g., a logistical reduction would not teach us how to calculate or to solve differential equations [B: 127; R: 66, 71, 89]. Third, the result justifies the attempted reduction, and not vice versa—the shorter, original proof tells us how the longer ought to come out [B: 127; R: 73f., 81, 83, 91, 171]. And, finally, re-ductionism systematically confuses a representation of one theory by another with identification [R: 66, 72, 84, 89f., 91]. Wittgenstein places an exaggerated but interesting stress on difference and autonomy.
For instance, on the differences in mathematics among apparent existential statements and between existential statements and truth functions [B: 149; R: 141, 144], small and large numbers [R: 67, 74], equations and inequations [B: 249; R: 1].
His criteria for mutual independence of theories and for mathematical autonomy seemed to be these:
(1) Could one theory (technique, etc.) have been learned independently of the other? [R: 86.]
(2) Does the subject have its own characteristic techniques [R: 85ff., 145];
(3) its own characteristic application, e.g., in surveying? [R: 88, 190.]
(4) Does purported reduction utilize or otherwise presuppose the analysed concepts? [B: 125ff.; R: 66f., 71f., 83, 85.]
(5) Finally, are the concepts in question immediately self-applicable, e.g., as when we count the numbers or use a geometric construction to illustrate a proof? [B: 132f.]
Self-application guarantees perspicuity and independence of contingency. If the proof is an illustration of how it is, one cannot deny that that is how it is, anymore than one who is honestly screaming can fail to know that this is pain [B: 130, 132].
Wittgenstein’s alternative to reductionism of any kind—whether to numbers, geometry, logic or sets—was attention to the varieties of language eligible for mathematical elucidation and to the consequent motley of mathematics.
(Though the ‘motley’ is not a theme in T, where Wittgenstein seemed to demand a single unified language, the idea is foreshadowed in his interesting but unsatisfying remarks about mechanics and the principles of physics [T: 6.3211, 6.33, 6.34 et seq.], which are made to appear to have a certain autonomy, even though they ‘speak of the objects of the world’ [6.3431]. On the other hand, [6.3751] intimates the idea that logical distinctions of every kind can be presented within the formalism of mathematical analysis.)
Theme 8. MATHEMATICS IS NOT SO MUCH DOCTRINE AS METHOD.
A ‘method of logic’ [T: 6.2, 6.234].
For exhibiting essential features of the civil use of language [T: 6.12, 6.1201, 6.121, 6.1221, 6.124, 6.22; R: 79]. In T, Wittgenstein was particularly exercised to combat the idea that logic and mathematics could be regarded as a corpus of propositions, ideally to be deductively presented as a unified theory à la Frege and Russell.
Apparent logical derivations can be replaced by calculations in which the distinction between proof and conclusion cannot easily be drawn [see below; T: 6.126, 6.2331]. Wittgenstein’s method of truth tables abolished the artificially imposed distinction in rank as between alleged primitive propositions and their consequences [T: 6.126, 6.127]. He argued that there are no fundamental concepts in logic and mathematics [T: 5.45–5.4541], and that the classification of ideas which is so much a part of any kind of theorizing is here a symptom of error [5.554–5.555]. The logistical presentation falsely mis-assimilates the necessities of logic to contingent truths [6.113, 6.1263], and the formal concepts of mathematics to material concepts [4.1272, 4.1274, 6.1231–6.1233]. Construction of theories of logic and mathematics of this kind must presuppose anyway the mathematical methods Wittgenstein thought were fundamental [T: 6.123, 6.1263].
(This position is strongly fortified by the argument Lewis Carroll advanced in his famous paper, ‘What the Tortoise Said to Achilles’ [Mind, 1895, pp. 278–80; T: 5.132].)
Wittgenstein was not concerned with special forms that might be defined in this or that alleged theory of logic, but rather with ‘what makes it possible to invent such things’ at all [T: 5.555].
The supposed conclusions of mathematical arguments are not really propositions which could be true or false, but integral parts of the whole logical demonstration which itself cannot be asserted or denied, only exhibited [B: 192, 198f.].
In T, Wittgenstein’s ban was codified in the exaggerated dictum that what could thus be shown could not be said. He stuck to this in B, where he continued to look upon language as an untranscendable unity all of whose propositions have a perfectly determinate sense [B: 123, 139, 143f., 152, 168, 178, 198, 203, 208, 234].
(At [208] he turns the principle that certain sets can only be described and not presented squarely on its head.)
The thesis that the results of at least some parts of mathematics cannot be formulated propositionally was additionally and incoherently supported by use of the verification principle [B: 172, 174f., 190, 336, 338]. A faint echo of ‘showing’ is still distantly heard in R [79], though by then Wittgenstein had broken out of this indefensibly tight position.
Wittgenstein always was chary about the conception of a mathematical proposition because, he argued,
(1) they have no proper subject-matter [T: 6.111, 6.211; see theme 6],
(2) they convey no information [T: 2.225, 5.142, 6.11, 6.122, 6.2321–6.2323; R: 31, 53f.];
(3) they do not admit of meaningful alternatives [T: 4.463, 6.1222];
(4) they presuppose their own correctness [T: 6.123, 6.1261, 6.1264, 6.1265, 6.23, 6.232–6.2322].
Their sense presupposes their truth [B: 144], and they are no more assertable than their counterparts among the self-referential paradoxes.
Wittgenstein expectedly had hard words for the so-called conjectures and hypotheses in mathematics, e.g., the Riemann Hypothesis [B: 190f., 338]. An interesting and not entirely implausible consequence of his view is that the apparent propositions of mathematics do not have negations; rather, apparent negations (e.g., inequations) are independent determinations [B: 247–251]. In B and R, Wittgenstein was apt to classify these nonpropositions as commands [R: 49, 142ff.], rules and schematic applications of rules [B: 143, 194, 322f.; R: 47, 77, 118, 120], definitions [B: 198] or simply as techniques [R. 43]. He pointed out the possibility of imparting mathematical techniques without benefit of formulated propositions or apparent propositions, e.g., we teach a person to count or to integrate without bothering to impart such ‘facts’ as the fundamental theorems of arithmetic and the calculus [R: 49, 118]. Wittgenstein’s thesis that mathematical results are not propositions holds especially well for his favourite examples of calculating [B: 172; R: 32, 76, 115]
where the distinction between proof and conclusion cannot be drawn easily [for development, see theme 9],
and knot-unravelling [B: 184f.], and for the kind of demonstration one witnesses in the physics lecture hall. But it is not entirely evident that these are what mathematicians are wont to call ‘proofs’; certainly 13 × 14 = 182 is not a ‘theorem’.
Theme 9. IN MATHEMATICS IT IS ALWAYS THE PROOF THAT MATTERS.
(Here there is a curious partial agreement with Frege who defined analytic necessity in terms of derivability from the axioms of logic.)
Mathematical results are proof constructions [B: 183; R: 92], for the proof is what shows the conceptual connexions [R: 25f., 75f., 80].
The ‘real’ mathematical proposition is the proof itself [B: 184].
(But that is no proposition.)
So-called mathematical propositions essentially are proof conclusions [B: 192; also Dummett, op. cit., p. 327].
If one may believe mathematical propositions at all, then that is to believe that one has a proof [B: 204; R: 32].
(I think the thought would be true if read, ‘… that there is a proof’.)
To know a mathematical proposition is to know how it can be proved, and to know that is to have proved it [B: 199].
They cannot be understood when cut off from the proof [B: 183; R: 26f., 52, 77].
(Like the surface of a body [B: 192].)
With reference to Wittgenstein’s favourite example of calculation, there is here no clear distinction between proof and conclusion [T: 6.126–6.1265; B: 130; R: 26, 32f.; and see Kreisel [1], p. 140].
(Discussing Goedel’s Incompleteness Theorem and the common way of speaking of it in terms of asserting something of itself, he says, ‘In this sense the proposition “625 = 25 × 25” also asserts something about itself: namely that the left-hand number is got by the multiplication of the numbers on the right’ [R: 176].)
Wittgenstein plausibly claims that the alleged conclusion is itself a form or an indication of a form of proof in the ‘application’, e.g., it is a form of Modus Ponens [T: 6.1264] or a rule telling how many objects there should be if we have arithmetically melded two groups of things [B: 145; R: 77].
The conclusion so regarded is given its sense by the mathematical demonstration [B: 180f.; R: 5 2 (an example where the application is within mathematics), 77] by schematically showing how the alleged conclusion is applied as a rule of inference in civil life: the proof of the ‘conclusion’ shows how the conclusion may be used as a rule of proof. The conclusion in its turn tells us how to read the demonstration. Its sense is to tell us how to use the proof of which it is the ‘conclusion’ [R: 76]. The sense of the alleged conclusion is that this has been proved [B: 181, 192].
(He also says—if I understand the passage—that in inductive proof—the kind he liked best—the conclusion is to proof as a sign is to signified [B: 328f.].)
From this Wittgenstein concludes that the sense of the conclusion is its proof [T: 6.1265; B: 192].
(And that is why it must be so, on Wittgenstein’s theory of necessity. See theme 4 above.)
Apparently Wittgenstein believed that this way of thinking about proof and conclusion is supported by the fact that the alleged conclusion is often self-applying [B: 130, 132]. He summed it all up in the maxim that in mathematics process and result are the same [T: 6.1261; R: 26].
(But this maxim bears other interpretations too, e.g., that there are no processes in mathematics.)
An obvious and strong objection is that Wittgenstein’s thesis, if true, makes it meaningless to speak of establishing the same conclusion in two different ways [R: 92f.].
Sometimes he appears to accept the conclusion that there cannot be two independent proofs of the same mathematical proposition [B: 184, also 193].
(But note the word ‘independent’.)
Sometimes he incoherently allows that we can be brought to accept the same rule in different ways [R: 92f.], through new connexions, but one way dominates, e.g., that defined by multiplication [R: 93]; again, he allows that we might get to the same place by two routes [R: 92, 165], or that we are working in distinct systems [R: 165], always with the suggestion that the application furnishes the connecting fabric. Wittgenstein also says that alternate proofs supply equally suitable instruments for the same purpose [R: 165] and persuade us to ‘stake the same thing’ on the truth of the proposition [R: 186].
These replies do not meet the objection and reveal what I feel may be the weakest place in Wittgenstein’s whole philosophy of mathematics, his failure to say how the different proofs for all sorts of different things, throughout the many parts of mathematics, can be related.
Theme 10. IN HIS MIDDLE AND LATE PERIODS WITTGENSTEIN SHOWED A WAKENING INTEREST IN THE APPLICATION OF MATHEMATICAL CONCEPTS WITHIN MATHEMATICS ITSELF, ESPECIALLY IN CONNEXION WITH THE CONSISTENCY PROBLEM (WHICH WE SHALL CONSIDER NEXT BELOW) AND IN SET-THEORETIC INTERPRETATIONS OF THE CLASSICAL REPRESENTATION OF REAL NUMBERS BY NON-TERMINATING DECIMALS. [For the latter, see B throughout, but especially parts XII, XV–XVII and first Anhang; R: APPENDIX II.]
Wittgenstein was gravely suspicious of the idea that real numbers could be regarded as arbitrary infinite sets of nested intervals of rationals or as arbitrary Dedekind ‘Cuts’, conceived of as existing outside our conceptions without need of rule or specification. This is the worst form of ‘extensionalism’, where we only seem to know what we are talking about, based on fanciful images and full of all sorts of spurious problems and misapplications [see themes 2 and 6 above].
We think of a real number as a definite but infinitely long string of things which we could systematically tick off and, after an infinite time, look back upon as a job done, a state we might now already be in had we lived from the beginning of time [B: 149, 164 et. seq., 236f.]. We are apt to think in this way about infinite strings and infinite processes because we mistake and project accidental features of the representation as essentials of the conception [B: 231f.; R: 251].
(The powerful dialectic Wittgenstein directed against the easy assumption that the representation of a real number simply either does or does not contain a certain pattern of digits carries the germ of a genuine mathematical point, viz., the distinction between generally recursive and merely recursively enumerable sequences. See esp. R [139ff.] where he makes the supporting observation that the denial of ‘There exists a law that p’ is not ‘There exists a law that ~p’ [R: 141; also B: 228f.].)
He objected particularly to the idea of arbitrary ‘free choice sequences’ speciously thought of as generated by some mechanical temporal process, such as flipping a coin [B: 165ff., 218ff., 233].
Wittgenstein also had doubts about Cantor’s ‘Diagonal Argument’,
In apparently presupposing familiarity with a real number still not defined, the proof seems to require us to act in ignorance and without concrete comprehension [B: 226]. The proof assimilates the introduction of a new concept to a deep and mysterious discovery. But the depth is an illusion and the mystery due to the fact that, even after the argument has been understood, it remains unclear where and how the concept applies, and we try to fix its sense in terms elsewhere appropriate, e.g., in terms of comparisons of magnitude [R: 54 et. seq.].
and about Dedekind’s theorem that the real numbers are closed over additional cuts [B: 224f.; R: 148ff.].
He objected particularly to the image of ‘fitting in’ reals among the rationals [B: 223, 339; R: 151].
Apparently Wittgenstein did not object to the classical conception of a real number as the limit of a sequence of partial sums, represented perhaps by assigning an argument to a well-defined power series expansion. Here we still can discern the relation of real numbers to the civil institution of measurement [B: 230, 235; my perhaps incorrect interpretation of ‘messen’], and are less apt than with the more abstract conception to obliterate clear lines of dependence of the reals upon other mathematical theories and especially upon the system of rational numbers [B: 228; R: 148].
A particularly insistent theme in B was Wittgenstein’s demand that particular real numbers should effectively and uniformly be comparable with the rationals upon which they depend [B: 227, 236 et. seq.].
(The rule of development of a real is the method of comparison with rationals [B: 236–44].)
The introduction or definition of a real number should make clear from the outset what those relations are and not leave it as something to be discovered later on [B: 238f.].
Wittgenstein sometimes appeared even to allow that the general idea of an infinite decimal had a useful, non-mysterious sense, capable of covering a variety of different systems [R: 58],
(perhaps still other forms of numbers different from [e.g.] 1/3, √2 and π, themselves first introduced in quite different connexions)
all of which he once was inclined to bring under the umbrella conception of a rule for going on—an ‘induction’ (a ‘recursive rule’) [B: 223f., 234ff.]. Positively, Wittgenstein identified particular real numbers with particular such rules [B: 227–34, 308f.; R: 144].
Theme 11. THE NUB OF WITTGENSTEIN’S OBJECTIONS TO THE CONSISTENCY PROBLEM SEEMS TO HAVE BEEN THAT THIS IS JUST ONE KIND OF MATHEMATICAL QUESTION WHICH HAS BEEN GIVEN AN EXAGGERATED IMPORTANCE BY THE FASHIONS OF CONTEMPORARY THEORIES OF THE FOUNDATIONS OF MATHEMATICS [R: 52, 107].
The earliest extensive examination of the matter by Wittgenstein of which we have a printed record is the transcription of conversations with Schlick and Waismann incorporated as the second Anhang in B, although there is a clear anticipation at B: 189ff. He returned to the topic of consistency and consistency proofs frequently in R, esp. in parts II and V. It is clear, however, that Wittgenstein’s thinking about the question was conditioned largely by the turn of the century concern with the logical paradoxes, which he regarded merely as confusions to be resolved by analysis and not by proofs [B: 320].
(But note his suggestion that mathematical demonstration is never anything else but ‘analysis’ [B: 192]. Usually Wittgenstein illustrated his observations with Hetero-logical and kindred paradoxes [R: 51, 102, 104f., 150f., 166, 170, 175, 182]. This may explain why he so thoroughly misunderstood the aims and results of metamathematics.)
Wittgenstein’s polemic squared at all points with his negative attitude towards ‘foundations’ and with mathematical theories about mathematics [B: 320, 327, 330, 336; R: 109].
He had just as little patience with the related metamathematical concern with the questions of independence and completeness [B: 189f, 319, 324, 335ff.].
He held that the fear of an hitherto undisclosed contradiction was either a pretence or neurotic [B: 318f., 323, 325, 332, 338, 345f.; R: 181], first, because the appearance of a contradiction isn’t the only thing that can go wrong in mathematics [B: 325; R: 105, 130, 196]; second, because the demonstration of a contradiction itself would be just another mathematical result, although in a system other than the one in which the supposed inconsistency would be found [B: 189, 320, 328, 330, 335, 341; R: 167f.];
He argued that only formalized, derivational mathematics could pretend to view a contradiction as a disaster. But in fact a formal contradiction would be interesting only if it were also an inconsistency, thus presupposing that the system had truth, meaning and application [B: 321ff., 333, 337, 339; R: 104, 166].
(A contradiction is just another piece in the imagined game of formal mathematics [B: 318f., 326, 331f.].)
Wittgenstein believed that his own views about the meaning and application of mathematics left no place for significant philosophical questions about provability and consistency [B: 189, 322, 329ff., 339; R: 104, 109, 166ff., 178, 181].
(This claim was sometimes unsatisfyingly supported by use of Tractarian principles about what can only be shown and not said, and about the impossibility of meaningfully crossing the limits of language [B: 326, 330, 336].)
third, contradictions can always be taken in stride [R: 51, 101, 141, 15 of., 166, 168, 170, 181F.], and even be used [R: 150F., 166, 171, 183]. Systems with contradictions can at the very worst always be patched up with more or less ad hoc repairs [B: 319, 333, 345; R: 102, 181]. At all events, a consistency proof would not give us the controls we would want and the confidence we would lack if we really were sceptical about mathematics [B: 330, 345; R: 104, 106f., 109f, 181].
Conclusion
I hope to have created a convincing sense of constancy and continuity in Wittgenstein’s thinking about mathematics, a constancy in basics which swells under a continuity of surface changes in emphasis, and interests and occasionally in doctrine. Certainly the hard lines of T were softened in later days, but a recognizable doctrine survived to be given a broader and looser application.
Constructively, the two dominant continuing themes in Wittgenstein’s thinking are that mathematics is an assorted kit of instruments for the conceptual control of civil language and life at large, and the conception of mathematical proof as the perspicuous demonstration of essentials, viz. necessities. The doctrine was illustrated for the most part by examples of simple arithmetical calculations and the calculation of tautologies with some occasional attention to other parts of ‘intuitive mathematics’. Critically, we find a persisting scepticism about the deductive paradigm of mathematical demonstration, a stiff resistance to the conception of mathematics as an autonomous subject-matter which is best codified in a growing corpus of propositions about mathematical objects; Wittgenstein never really became reconciled to mathematical propositions and never accepted the historically recurrent thesis that mathematics can be unified into a single theory. His distrust of the claims of ‘pure mathematics’ was sustained by a serious inattention to the actual contemporary conduct of mathematical theory, only partially compensated for by his waxing and then waning concerns with the idea of a real number and the conception of the mathematical infinite and his still later critical reaction to metamathematics (Wittgenstein apparently was pretty ignorant about recent work in algebra, analysis and geometry, from which he might have received some moral support). This late and grudging and lazy interest in pure mathematics did not have entirely happy consequences, for it seems to have abetted his occasional indifference to the distinction between civil and mathematical applications of mathematical concepts, which throws a darkening shadow of suspicion over his whole philosophy of mathematics. Something at once more interesting and more compelling might have emerged from an investigation of how internal applications (e.g., of probability concepts to the theory of numbers) finally reach down into ‘life’. Wittgenstein would have done well to take seriously the thesis that nondenumerability is an inescapable implication of the physical and technological applications of mathematical analysis. [See Bernays, op. cit., p. 14.]
The most obvious surface change was that the unified language of T was fragmented into a welter of language games regarded as forms of behaviour, and mathematics in train became dependent upon conventions and upon agreements among men, on the ways we act and forms of life. He became ever more apt to formulate his thoughts in the vocabulary of ‘rules’, ‘pathways’, ‘paradigms’ and ‘norms’ and to pay perhaps exaggerated attention to the conceptual transformations effected by mathematical demonstrations. The original thesis that formal logic and mathematics are distinct logical methods was softened into the ‘motley’; the apparently crystalline conception of ‘and so on’ was diffused into the general notion of perspicuity. Wittgenstein expressed uneasiness about sharp lines [see R: 155, 163, 186] and imputed an over-rigid theory about language to his own earlier self [R: 182].
This naturally was attended by a generally more relaxed attitude towards language at large. Most notably, Wittgenstein, after putting up a staunch resistance, finally abandoned the idea that even propositional language was a unified system analytically resolvable into a ‘totality’ of elementary judgements, themselves anchored to the ‘totality’ of objects to which reference ultimately is made—a view which issues in what I have elsewhere called the ‘Absolute Spielraum Principle’ [Inquiry, 1964, pp. 411f.] The rigid distinction between saying and showing is broken and therewith is destroyed the ‘transcendental metaphysics’ of the other ‘world’ on the other side of the limits, the world of ‘possibilities’ which must have the perfect structure of an ‘ideal’ to hold language inflexible inside.1 It seems to me that the most important change of all, deeper and less visible and more consequential than the others, was in his view of necessity. What was proposed rhetorically in the T question, ‘What must be the case in order that something can be the case?’ [5.5542], finally gave way in the concession that ‘There correspond to our laws of logic very general facts of daily experience’ [R: 36]. Otherwise, the ideal world of logic of which he speaks so eloquently in the Philosophical Investigations would have remained starkly separated from the contingency of fact. But now the benchmarks of contingency, the ‘limits of empiricism’ are themselves posted in other contingent facts.
I think Wittgenstein was right but not right enough. Mathematics is built on the presuppositions of everyday life and therefore has contingent implications. But I doubt that Wittgenstein saw this with sufficient clarity or perspicuity in R, nor did he follow up with enough detail. Surely the facts that we walk on two legs and speak a babel of tongues—important in the human situation—are not among the bourning stones of empiricism. But why not? We need many cases of at least as much detail as Wittgenstein devoted to his lumber merchants [R: 43f.]. Then perhaps we could begin to understand what is so important, that every part of the motley of mathematics has a motley of applications [see e.g., R: 152].
Assessment. We have already observed a number of defects in Wittgenstein’s presentation and unresolved problems for his theory of mathematics.
Wittgenstein simply didn’t know what to do about pure mathematics, where the ‘civil application’ is already mathematics. It is important here to see what Wittgenstein himself sometimes failed to notice, that mathematical concepts are not themselves civil concepts. The natural numbers are not the numbers of ‘how many?’ or ‘which one?’, but simply the numbers which we can calculate with and prove theorems about, doubtless often for civil purposes of regularizing our ideas of ‘how many?’ etc. The number 6 is just as much a mathematical notion as perfect number or ℵo. Perhaps Wittgenstein already was bumping unwittingly against the difficulty in B with his frequent application of the verification principle to the supposed propositions of number theory. The application is, I believe, entirely reasonable regarded by itself, but hardly coherent with Wittgenstein’s ban on mathematical propositions. The problem stayed to haunt him in R, e.g., where he found himself unable to find a civil technique for 2ℵo to be a property of [R: 186]. He seems to have felt deceived by the two-facedness of mathematics, which looks outwards to its civil application and inwards to its own theory. [See R: 117ff. That is one way of taking the ‘twofold character of the mathematical proposition as law and as rule’; R: 120.] Since he didn’t want to deny anything, Wittgenstein had to find some adjustment to the fact that mathematicians work within their own walls and establish all sorts of interesting things like the irrationality of √2 and the transcendence of π. That dulls the sting of Wittgsenstein’s criticism of metamathematics which is, in its operations, not all that different from Galois theory or any other part of pure mathematics in which men have reasoned successfully about mathematical objects. Wittgenstein allowed that the demonstration of a contradiction would show that we don’t know our way about [R: 104]. Agreed, and that is an important result, which may resemble the discovery of the irrationality of √2; but it is also a problem, which may have a similar though doubtfully as grand a significance for mathematics as the discovery of irrationality; we don’t see immediately what is wrong with our intuitive idea of class abstraction in the way most of us are able to see why division by o is forbidden. I don’t think Wittgenstein was wrong in stressing civil applications—on the contrary; but I doubt whether he could have resolved the difficulty without going into unwonted detail with the aim of explaining how internal applications of mathematical concepts also, in a perhaps refracted and diffused way, reflect the essentials of civil, nonmathematical language.
This first difficulty is tied up with the unsatisfactorily indefinite state of Wittgenstein’s ideas about conceptual change. The most obvious alterations occur in connexion with internal applications, e.g., the closing up of the projective plane by the introduction of a point at infinity, the extension of the real number field, the algebraic reworking of arithmetic and geometry, and the systematic use of set theory as a formulary.
We have also noticed that Wittgenstein was never really able to say how different proofs could have the same conclusions. Otherwise put, he had no place in his framework of thought for the idea of a theorem. The availability of alternative proofs goes together with the common assumption that one can understand a theorem one has never seen proved, which Wittgenstein apparently also wanted to deny, not without conceding the ring of paradox [see B: 183; R: 92]. Here again I think Wittgenstein could have dispelled the difficulty only if, contrary to his every inclination, he would have looked at many different particular cases in considerable detail. I don’t know how to do this; but I believe that what we must come to understand better is the obvious, open, multivalent applicability of interesting mathematics to cases of every kind, both inside and outside; this perhaps would better reveal how one thing can be looked upon as a model for another kind of thing and (to speak figuratively) how the same case can be rotated in different directions into the same position.
Another major problem which Wittgenstein himself recognized is to explain how mistakes in calculation specifically and in demonstration generally are possible. Either you do or you do not know how to calculate; but if you do, then your calculation gives the right result. So far the problem sounds a bit like accounting for mispelling. But it is given special consequence by Wittgenstein’s maxim that process and result are the same in mathematics and poignancy by the consideration that if proof were (as Wittgenstein held) the revelation of sense, then it would be hard to explain how we could, with understanding, set about to prove something if we didn’t know the result in advance [see B: 170]. We can agree up to a point with Wittgenstein’s confessional declaration, ‘I have not yet made the rôle of miscalculating clear. The rôle of the proposition: “I must have miscalculated.” It is really the key to an understanding of the “foundations of mathematics”.’ [R: 111; see also 33, 95, 120.]
There are a number of other, smaller unresolved problems. Wittgenstein was ill-prepared to cope with the obvious, if only occasionally relevant, distinction between axioms and theorems [see R: 79]. Why, for instance, were the ancients so chary about the parallel axiom—it generally was thought to be true? (And see Wittgenstein’s own remarks at R: 113f.) Again, how does Wittgenstein account for the fact that mathematicians do ‘discover’ proofs if not theorems? Where do trial and error methods fit into his scheme? Though I can believe that something I don’t see led Euler to his calculated disproof of the Fermatian conjecture, that all numbers of the form 
are prime, most of us see 
as a concrete counterexample [for partial recognition, see B: 134 and R: 188]. All these are obvious, still unresolved difficulties for Wittgenstein’s philosophy of mathematics.
Switching onto another critical track, I think most commentators would agree that Wittgenstein’s style and manner were inappropriate to the subject. The flickering images of his variable terminology are regrettable; and he uses it in a careless analogical way, with little discrimination. Just think how many different things have been and could be called ‘rules’; think how very different are the rules of English from the rules of calculating. In no other part of philosophy is detail more in demand. Wittgenstein draws our attention to the ‘motley’; but where is it in his book? He works with a few jejune examples, notable chiefly for their vague similarity to more important ones. (Wouldn’t the Koenigsberg Bridge Problem have done better as an illustration than the joints in the wall [R: 174]?) Wittgenstein was inexcusably lax about technicalities, and sometimes they matter. Everyone has felt, with Wittgenstein, that there must be an important formal difference between the sequence of primes and (say) the sequence of even numbers [see B: 251]. I believe that mathematicians, some of whom are of a like mind with Wittgenstein about the real numbers, have tried unsuccessfully to say what that difference is.
The otherwise finely wrought T is unbelievably bad on important technical questions. The main such defect is that Wittgenstein’s operation for generating propositions was not, as he later could have put it, generally recursive, as it had to be. I have racked my brains trying to figure out how a theory of descriptions could be fitted into the T theory of language, as Wittgenstein intimated it could be at T: 3.24. But, more to our point, the T theory of mathematics per se is simply a mess. First, the equations in question are not even degenerate sentences, so it is not clear how they can be sinnlos but not unsinnig singular points of logical revelation. Second, these equations are described as containing names and not (e.g.) numbers, though Wittgenstein’s examples are framed for number. I suspect Wittgenstein had something like trigonometry in mind; but for the reason just given, his theory is inadequate even to simple calculations. He briefly sketches out a theory of number, which may be congruent with the Peano scheme; but then one wonders how this can be applied to apples and oranges. It would take more than two applications of joint denial to get ‘apple1 is in the bowl and apple2 is in the bowl’. There are ways out of this, but they lead into still deeper difficulties.
Yet, after all this has been noted down, I think that Wittgenstein was substantially right in his proposed if fragmented alternative to the other ‘philosophies’ of mathematics. In the course of tracking through these texts, I found myself becoming more and more agreeably disposed to the thought that the formal derivational method of presenting mathematical proof against which Wittgenstein polemicizes is only a method of presentation, and a rather artificial one at that, a kind of uniform which I had been taught to respect from training by the professors of logic. I am completely convinced by what he intimates regarding the ‘meaning’ of mathematics. I should now like to try to convey this conviction.
Many share Wittgenstein’s distaste for the theology of pure mathematics and his wish to dispel the arcane mysteries of the cult. But perhaps the mysteries are preferable to saying that mathematics is something it patently is not (who would exchange the mathematics we know for psychology?), and to the naturalism of the ‘meaningless game’. Wittgenstein’s view of mathematics as a kit of instruments for conceptual control is at once a refuge from empiricism and an alternative to Platonism. We must try to see how mathematical objects are creatures of human conception in a way in which coconuts and thoughts are not. Wittgenstein’s guiding thought is that they are features of our originally non-mathematical ways of thinking about the world, and they owe their apparent superpalpability to the fact that they may be presented ‘nonobservationally’, in this most immediate manner. We explain the meaning of mathematics by attending to those applications, to the dominating illustrations, and not by appeal to borrowed images. Wittgenstein also fortifies us against those obviously false but still dangerously inviting ways of thinking of mathematical sequences as like actual physical processes which go on perhaps for a terribly long time. The infinitudes of mathematics are not processional, but features of the open if regular application of single rules. I believe that many practising mathematicians share Wittgenstein’s idea that real numbers must answer to ‘rules’.
Evidence for the correctness of Wittgenstein’s way of thinking about mathematics appears in many places. Consider that until quite recently only in geometry among the many mathematical disciplines were proofs supposed to proceed from stipulated axioms and postulates. Why? Because elsewhere demonstration began with the mathematician drawing our attention to what everyone could see had to be the case, e.g., one could see that the numbers were unlimited just because one had learned a rule of civil language, to count thus, permitting one always to count higher than ‘this’. Even in geometry, axioms were formulated as reminders that geometric proportions and not (e.g.) sensation intensities were the subject-matter, and postulates were stipulated where a kind of generality was needed for which civil practice did not provide immediately validating construction. Geometric demonstration remained for the most part a matter of getting one to ‘look and see’, where what one looked at was the ‘application’. Outside of geometry, with the growth of pure mathematics, this application was commonly already something mathematical (e.g., the order and the number of the exponents of polynomial functions).
There is something right in Wittgenstein’s ideas about proof ‘creating’ concepts. One first thinks of the important if hackneyed examples from the history of the extension of the concept of number, which I shall not dwell on except to observe the differences. We get to ℵo by generalizing upon one application of the numbers and ω by generalizing upon another; the complex numbers, by way of contrast, arose out of a demand for algebraic roundness. The final resolution of the classical geometry problems affords a different kind of illustration. Wittgenstein once saw these as establishing connexions between previously separate ‘language systems’ [B: 177]. Suppose, what is historically inconceivable, that Archimedes knew what he did about geometric magnitudes and was also somehow master of the complete algebraic theory of equations, the concept of a derivative function, and the analytic theory of power series expansion. He, like Lindemann, might have been able to prove that e was the root of no polynomial equation; and hence not πi; and hence not π; but not even Archimedes could have seen immediately that therefore the circle could not be squared. Lindemann’s proof would have been incomprehensible to the ancients and would not have answered their question. The sense of ‘π’ has since undergone a continuous but marked change, mostly as a result of applying a growing body of analytical technique to problems that originally arose elsewhere. The original concept of a geometric ratio was, however, always there, as Wittgenstein might have said, in the background. One may tendentiously interpret Lindemann’s demonstration as an explanation of why the geometric problem could not have been (hence never was) solved.
This case description, which I hope is acceptable, is meant to illustrate how application makes meaning; here the main ‘application’ was mathematical, but the mathematical application itself got its meaning from the nonmathematical superimpositional procedures of comparing areas. Now we really don’t want to deny anything, and the question arises whether this way of thinking would make any useful sense if directed to more recondite and abstract parts of mathematics which begin with long enlisted, highly regimented applications. I hazard the suggestion that it might, though here my ignorance may betray me. I have been bemused to observe the excitement generated by Cohen’s recent proof of the independence of the Axiom of Choice. The great question is what will the mathematicians do now. The consensus seems to be that they will go on as before with a sense that their previous confidence in using the Axiom has been vindicated.1 I feel that one could ‘philosophically’ explain that ‘decision’ along the following, Wittgensteinian lines. Start with Wittgenstein’s thought that mathematical conceptions of the infinite are introduced for purposes of fixing in a nonarbitrary manner the limits of already available concepts. A proof of de-numerability exhibits a rule by which that can be effected. A proof of nondenumerability shows that there is no such procedure. Still, in order that anything should be limited, a concept with an allegedly nondenumerable extension must be pinned down to something we can comprehend. Proofs employed in the stratosphere of set theory get their sense from their connexions with closer, gravitating, ‘constructive’ things. The limits are drawn to enable us to survey those cases in a nonarbitrary way. Now, if we are sure that a principle holds for any such finite or otherwise ‘constructive’ case, it behooves us to adopt it, if it is also known to be consistent with all other such principles. I believe that that is now the situation with the Axiom of Choice. I’m not entirely confident that what I say makes sense; but it is, at all events, an example of Wittgensteinian thinking.1
It must be evident already that I am sympathetic with Wittgenstein’s thesis that mathematical proof is perspicuous demonstraation and not logical derivation; or, better, that logical derivation is only one kind of demonstration. I believe that this in fact would have been the traditional sentiment, and Wittgenstein’s philosophy represents a significant return from recent fashion. I am inclined to think that Hilbert was wrong and Euclid right, that postulates are not to be laid down as assumed truths, but are rather to be posted here and there to tell you what you can do. Anyway, I agree with Wittgenstein that it is the proof that matters most and not the conclusion. Mathematicians discover proofs, not theorems. Mathematical propositions are indeed essentially last lines of demonstrations. One could not properly claim to know a theorem if one thought it had not been demonstrated, and for one to believe a mathematical proposition would imply one believed it could be demonstrated.
Wittgenstein’s maxim that process and result are the same is an extreme formulation of the position. Even that has a certain initial plausibility for examples of nonsystematic demonstrations, such as Cauchy’s proof that the sum of the numbers of the faces and the vertices less the number of the edges of a polyhedron is always 2, and the solution of the Koenigsberg Bridge problem. One could argue for the thesis by concentrating (as Wittgenstein does) on calculations, where the calculation seems to be a proof that the calculation can be done. I don’t like this defence, since I doubt whether calculation is itself ever demonstration, not-withstanding the fact that Goedelian exercises seem finally to culminate in an elaborate but still elementary computation. I should prefer saying that the theorem is that there is a calculation that can be so regarded. Generalizing upon this, one can preserve something of Wittgenstein’s doctrine. A demonstration shows that a certain construction exists, and that cannot be better effected than by exhibiting such a construction. One thinks here of the classical precedent of Euclidean geometry, where constructions were exhibited with proofs appended. But the doctrine applies best to nonsystematic examples. I recall the experience of a mathematician friend (the same one who showed me the ‘shortest’ proof of the Pythagorean Theorem) who was engaged to present TV lessons in elementary geometry to classes of second graders. Axiomatics was out. One week he taught them to bisect a line. The next week he showed them how to draw a circle, and then he asked how one might find the centre of a circle. Two children were reported to have seen the solution immediately. What I find interesting about this and also about our two examples given on pp. 87f. is the near impossibility of seeing the demonstrations that attend the conclusions as deductions from axioms. Where indeed are the axioms? Well, perhaps they are ‘implicit’. Perhaps, but how do they figure as premisses? I do not deny that we could find derivational proofs from axioms for the claimed conclusions. But the proofs I have mentioned are not of that kind. I say that the proof gets us to see the construction in a certain way which may flick off and on and, indeed, we may actually have something rather more like a duck-rabbit situation than a derivation.1
I have read somewhere that compound interest was invented by the Genoese bankers of the fifteenth century. I can imagine that they could have supported their introduction of this new form of merchandising with the following demonstration: if liquidity has its price, then, in an ideal frictionless economy, interest should be compounded continuously, for a perfectly rational investor could continuously withdraw and reinvest his principal with accrued interest. This kind of ‘proof’ must, I think, resemble the early and consequential discovery by the Babylonians of an explanation of why fields with equal perimeters do not produce equal yields, and the explanation of why Del Cano’s log-book was off a day when he put in at a Portuguese port in Africa towards the end of the first circumnavigation of the world. I think these are the kinds of demonstrations Wittgenstein had most in mind. They do not proceed deductively from axioms; they take on their meaning from their direct connexions with an application which is immediate and palpable. At the same time, it also is evident that nature doesn’t always behave as the demonstration seems to demand. The economy is not frictionless; it is too much trouble to be ‘perfectly rational’; ‘fertility’ as well as area may affect the crop. But again the proof puts us into position to spot these other factors, and thus may be used as an instrument of conceptual control.
In everyday usage words like ‘demonstration’, ‘proof’, and ‘inference’, in company with ‘explanation’, ‘elucidation’ etc. cover activities which while essentially linguistic also go beyond the use of language to what can be gained in the way of creating conviction and organizing knowledge. The modern formal conception of proof as logical derivation, like the currently popular ‘covering law’ conception of explanation, would lop off the non-linguistic factors as something extraneously psychological and reduce the notions of demonstration and explanation to their purely linguistic components. Indeed, one of Goedel’s achievements was to show how formal proofs could be regarded as purely linguistic structures. Still, subsumption is not enough to explain, and formal proof is demonstration only under important if implicit caveats.
Now, subsumption under general laws is a form of logical derivation: in reversal of that, I’m also inclined to think that mathematical demonstration is at its best a kind of explanation.
That thought taken together with the examples considered just above brings me back to ‘concept formation’. The nonsystematic demonstrations we have been considering are successful only if they get us to see something in a new way, or if they reveal distinctions previously overlooked. They do that the more effectively, the more they also ‘explain’; they are ‘perspicuous’ if not immediately convincing. I am, after reading Wittgenstein, frankly puzzled by the fact that there can be different proofs of the same theorems. But now, with an eye to examples selected from the little mathematics I learned at school, I am impressed by the fact that some proofs do explain better than others. Go back to the ‘shortest’ proof of the Pythagorean Theorem cited earlier which makes use of this construction.
Compare this with another very convincing, ‘short’ proof using the construction
(a + b)2 = a2+ b2 + 2ab, where, of course, the four corner triangles have a combined area of 2ab. I confess that I am still not sure that I see why the first shortest proof is a proof; I am perfectly convinced by the other. Nonetheless, I believe that the first proof is probably the ‘better’ one because it intimates an explanatory connexion with the fundamental principle that corresponding parts of similar figures have proportional magnitudes.1
One last example. All of us have seen the classical proof for the irrationality of √2 ascribed to Pythagoras. We begin by assuming that 2 = m2/n2, where m and n are relatively prime. I once complained to a mathematician friend that, although I found this perfectly convincing, I felt that I didn’t really understand what was happening. He sympathized and told me to look at the demonstration as a special case of the Unique Factorization Theorem. Now I understand (as I had ‘implicitly’ in school when I ‘saw’ that √5, 
and 
are all of them irrational). The Unique Factorization Theorem itself seems or once seemed to me perfectly obvious, though I confess that the standard proof, ascribed to Euclid, strikes me as ‘unperspicuous’. I understand or would understand things better from some proofs of a given theorem than from others because the ‘better’ proof relates the conclusion more closely to the civil applications with which my thinking began—to the operations of adding and multiplying which support the theory of numbers or to the construction and use of Euclidean triangles for measuring areas.
1 G. Kreisel, Review of ‘R’ in British Journal of the Philosophy of Science, 1958, pp. 135–58. Hereafter ‘Kreisel [1]’.
2 ‘Wittgenstein’s Theory and Practice of Philosophy’, BJPS, 1960, pp. 238–52, esp. pp. 239ff. I came onto this paper (hereafter ‘Kreisel [2]’) only after I had written the substance of this paper. From it I got a lead to P. Bernays’ ‘Comments on Ludwig Wittgenstein’s Remarks on the Foundation of Mathematics’ (Ratio, 1959, pp. 1–22) which comes closer to my own views about Wittgenstein’s philosophy of mathematics than anything else I have read.
3 I confine myself pretty much to the three books mentioned. When drafting this paper, I did not have the unauthorized ‘Math Notes’ which circulated widely in the 1950s, though I have been glad to find useful quotations from them in my thesis on T. I believe those notes have been superseded by R. Also, because of unavailability, I made no systematic use of the ‘Notes on Logic’, ‘Moore’s Notebooks’ and Wittgenstein’s journals. So far as the Philosophical Investigations is concerned, I have relied on my retained general impression occasionally confirmed by notes on my thesis. References to B and R will be given simply with those initials followed by page numbers.
4 ‘Wittgenstein’s Philosophy of Mathematics’, Phil. Rev., 1959, pp. 324–48.
5 See C. Chihara, ‘Mathematical Discovery and Concept Formation’, Phil. Rev., 1963, pp. 17–34; B. Stroud, ‘Wittgenstein and Logical Necessity’, Phil. Rev., 1965, pp. 504–18.
6 I shall employ this format:
MAJOR STATEMENT OF THEME
Development
Discussion [References]
(Further discussion)
Rungs sometimes will be skipped.
7 The first break came with his abandonment of the requirement that the elementary judgements be mutually independent. This is documented in the 1929 paper on logical form and in B, parts VIII and XXI and on p. 317, where he rewrites T: 2.1512 to say that not single sentences (as he thought before) but the whole, still rigid system of language is applied to reality.
8 There may be some holdouts, chiefly because the Axiom of Choice has some counterintuitive consequences—observe the immediate appeal to the non-mathematical application!—notably the Tarski-Banach theorem. But that kind of intuition is always on the defensive anyway. Recall Wittgenstein’s own favourite example of the distance a band around the earth would stand off the surface if its length was increased half a foot.
9 Similar reasoning might lead us to adopt the Axiom of Reducibility were it shown to be independent of the other axioms of the Ramified Theory of Types. The Axiom obviously holds for finite models, for there we can easily form predicative functions which define the extensions of impredicative functions, e.g., x bas all the properties of a great general has the same extension as some function or other rather like x = Epaminondas … v x = Alexander … v x = Hannibal… v x = Caesar … v x = Sabutai… v x = Marlborough … v x = Napoleon … v x = Rommel.
10 I once was set and solved the problem of showing that a triangle with two equal bisectors is isosceles. I have not been able to recapture the proof though I do recall that the conclusion came when I saw a relationship which, like the faces in the trees, has now disappeared into a jumble of lines.
11 Perhaps the other proof also intimates a deep explanation connected with consideration of symmetry. What I don’t see at all are the connexions between these proofs, and with the others which idle minds have produced over more than two millennia.