Synopsis of Jacques Bouveresse, ‘Philosophie des mathématiques et thérapeutique d’une maladie philosophique: Wittgenstein et la critique de l’apparence “ontologique” dans les mathématiques’
[‘Philosophy of mathematics and therapy for a philosophical illness: Wittgenstein and the critique of apparent “ontology” in mathematics’]
Jacques Bouveresse’s 1967 article on Wittgenstein’s philosophy of mathematics offers a reading of the later Wittgenstein’s polemic against attempts to ground mathematics in logical form. Wittgenstein’s philosophy was relatively unknown in France at the time, and the article is structured as a general introduction to Wittgenstein’s thought on mathematics. Bouveresse focuses on the posthumously published notes collected in Remarks on the Foundations of Mathematics, as well as Waismann’s transcriptions—published as an appendix to Philosophical Remarks—of conversations held with Wittgenstein in 1930-1931. It is in these remarks and conversations that Wittgenstein articulated his final stance on the question of the foundations of mathematics, and on the relationship of philosophy to mathematics.
Bouveresse begins by noting that at the end of the nineteenth century the field of mathematics suffered a crisis, as the appearances of ‘pathological phenomena’—the ‘monstrosities’ of irrational numbers, the paradoxes of, and non-Euclidian geometries—within the stable field of mathematics called the meaning of mathematical activity, and the status of mathematical truth, into question. This crisis led practicing mathematicians to make systematic inquiries into the foundations of mathematics. After his return to Philosophy in 1929, Wittgenstein responded to this crisis by arguing that not only does mathematics not need a foundation, but that mathematics and philosophy have nothing to say to each other. All a philosophy of mathematics can offer is a ‘clarification of the grammar of mathematical statements, just as they are’ (CpA 10.9:177). This ‘clarification’ shows that mathematics is not a preexisting field of universals that is unveiled by the discoveries of mathematicians, but that mathematics is a loose set of techniques—what Wittgenstein calls ‘the motley of mathematics’1 - that share a ‘family resemblance’. Each of these ‘mathematical games’ constructs, rather than reveals, a mathematical terrain. Through an analysis of the grammar of these mathematical games the supposed crisis in mathematical thought loses its urgency.
While Wittgenstein occasionally appeals to arguments that by turns appear intuitionist, logicist, or formalist arguments, Bouveresse writes that it is important to remember that the Remarks have ‘no doctrinal preoccupation’ (206). Rather, ‘the unity of [Wittgenstein’s] philosophical discourse on mathematics is first of all the polemical unity of a denunciation’ of the belief, held by Hilbert, Frege, Cantor, or Russell, that mathematics deals with essences, universals, or ideal objects. The goal of the Remarks is to heal philosophy of its search for the absolute, by destroying ‘the philosophical illusion of logico-mathematical realism’ (CpA 10.9:206).
Bertrand Russell provides one of the best examples of the realist position that Wittgenstein opposes. Russell argues that through a mathematical statement such as ‘two plus two equals four’ we discover a world of universals. For Wittgenstein, in contrast, ‘the mathematician does not contemplate essences, but creates them’ (CpA 10.9:183). Mathematics is an activity, not a doctrine, and the mathematician is an ‘inventor,’ not a ‘discoverer’ (182). Wittgenstein’s abiding metaphor for the individual techniques that form the field of mathematics is the game of chess. In the same manner that a chess piece can be exhaustively analyzed through the rules of its use, all we can know about a numeric symbol comes as we expose the techniques through which it is manipulated. As Bouveresse writes, ‘it is hardly more a question of truth and falsity in mathematics than in a game of chess: in the two cases one only deals with the configuration of symbols, where the transformations are regulated by a system of more or less arbitrary conventions. One can propose to describe exactly the functioning of a game, and rigorously discuss the relevance of the game with respect to other games, and one does not see where the supposed “problem of foundations” would be born, nourished, and eventually resolved’ (183). Just as the fact that checkers can be played with chess pieces does not lead us to believe that there is a unified logical space that describes both games, the fact that the same numeric symbols exist within different systems of calculation should not lead us to believe that these systems exist within the same metamathematical space. Wittgenstein fragments the unity of mathematics into ‘a certain number of insular domains which do not communicate with each other’ (178) to such a degree that when a student moves from counting with sticks to the decimal system, she learns an entirely new system of calculation, which has its ‘own life’ (188). The fact that the same numeric symbols appear in the two systems is an accident of notation, for the properties of numbers do not exist outside of specific techniques of calculation.
Since each new technique introduces a new mode of expression into language, ‘there is nothing more absurd than to try to understand the new method through the terms of older expressions’ (CpA 10.9:189). While Russell’s goal in the Principia Mathematica is to reconstruct mathematics by doubling each mathematical demonstration with a logical demonstration, Wittgenstein insists that a logical system does not have any hierarchical priority. Mathematical science does not have any theoretical character, but is ‘purely poetical’ (CpA 10.9:184); metamathematics itself is ‘another calculus, just like any other’2. Mathematics can thus never arrive at a theory of demonstration but only at new demonstrations that produce new concepts and new criteria, and that introduce new paradigms into language. As Wittgenstein writes, ‘I can also invent a game in which I play with the rules themselves…in that case I have yet another game and not a metagame’3
Comparisons between different mathematical games must therefore be stripped of any reductive intention, and the mysteries of mathematics resolved by an analysis of the specific techniques of calculation that are in play. Wittgenstein writes that ‘when, on hearing the proposition that the fractions cannot be arranged in a series in order of magnitude, I form the picture of a never ending row of things, and between each thing and its neighbour new things appear, and more new ones between each of these things and its neighbour, and so on without end, then certainly there is something here to make one dizzy’4). Yet when it becomes clear that it is only the technique of ‘fractions’ that is in question, ‘there is no longer anything queer’5.
Wittgenstein’s analysis of the diagonal argument that Cantor uses to construct a real number that is not part of the presumably complete enumeration of the set of real numbers follows similar finitist logic. The statement that ‘there is no biggest cardinal number’ means ‘that the authorization to “play” with cardinal numbers does not have an end’ (CpA 10.9:191). The mistake that Cantor makes is to suppose that this ‘technique without end’ can, in effect, be nominalized as an infinite set. Cantor confuses the technique through which the set ‘real numbers’ is constructed with the set itself. The clarification of the grammar of the calculus of real numbers resolves the ‘mystery’ of the transfinite, by revealing the ‘puffed up’ 6 trick that Cantor uses to show that there is a real number left out of the set of real numbers.
Wittgenstein circles this same problematic when he asks if the sequence ‘777’ appears in the decimal expansion of pi. The question, Wittgenstein insists, is meaningless, since the sequence is constructed through the technique by which pi is given a decimal expansion. ‘Could God have known this, without the calculation, purely from the rule of expansion? I want to say: no’7. There is no ‘mathematical space’ or ‘divine understanding’ (CpA 10.9:192) in which the decimal expansion of pi exists. The necessity that we encounter in writing the decimal expansion of pi comes from the fact that we employ a technique of calculation, which can no more be wrong than the metric system can be wrong. The force of the rule for writing the decimal expansion does not reside in the fact that we are obliged to follow it, but in the fact that we follow it. And yet while these techniques are regulated by convention, it would be a mistake to call Wittgenstein a ‘conventionalist’, for the fact that there is consensus about linguistic and mathematical use ‘does not require the juridical fiction of a original founding convention’ (CpA 10.9:205) with which we tacitly agree. For Bouveresse, Wittgenstein is more properly a ‘behaviourist,’ for he appeals to a model of stimulus-response in arguing that mathematics is a socially acquired technique, analogous to other systems of reactions that are conditioned by culture through education.
While mathematicians tend to think of non-contradiction as a fundamental law of thought, for Wittgenstein contradictions arise in specific technical situations when it is impossible to apply a rule. For example, the rule that a white piece can jump a black piece in a game of checkers results in a contradiction when the white piece is at the edge of a board. The possibility that such a contradiction might arise in no way diminishes the interest of the game, for ‘as long as the contradiction hasn’t arisen, it’s of no concern to me. So I can quite happily go on calculating. Would the calculations mathematicians have made through the centuries come to an end because a contradiction had been found in mathematics?’8 This would only show ‘that a contradictory arithmetic is enormously useful and that it would be better to modify our ideal of mathematics than to conclude that we have not yet had a real mathematics’ (CpA 10.9:197). The faith that we put in mathematics does not rest on demonstrations of non-contradiction, but on our experience of mathematics. Not only can we make a study of logic, with its contradictions, but contradictions might play an important role in human life. When a man says ‘I always lie’ he might mean that ‘what he says flickers; or nothing really comes from his heart’ 9. The pathological need to exclude contradictions only arises when mathematics, or language, is treated as more than a family of games.
There would seem to be some resemblance between Wittgenstein’s remarks on consistency and contradiction, and Gödel’s incompleteness theorem. Yet as Bouveresse remarks, Wittgenstein’s treatment of Gödel’s theorem is cavalier, and it seems that either Wittgenstein did not understand, or did not want to understand, the consequences of Gödel’s work. This is a surprising refusal, since it was in large part through the results of these formal inquiries into the limits of formalization that mathematics abandoned the ‘primitive logicist and formalist programs that Wittgenstein polemicized against’ (CpA 10.9:208). Bouveresse concludes that while the mathematical terrain has shifted away from the realist preoccupations of Russell’s Principia Mathematica, to which Wittgenstein opposed himself, at the very least a philosopher can learn, reading Wittgenstein, that even after twenty-five centuries of using mathematics, ‘we do not know, at bottom, what mathematics are’ (CpA 10.9:208).
References to this text in other articles in the Cahiers pour l’Analyse:
- Wittgenstein, Ludwig. Remarks on the Foundations of Mathematics, ed. G.H. von Wright, et al. trans. G.E.M. Anscombe. Cambridge: MIT Press 1978.
- ---. Philosophical Remarks, ed. Rush Rhees. trans. Raymond Hargreaves and Roger White. Chicago: University of Chicago Press, 1975.