Abstract
According to Mark Steiner, Wittgenstein’s intense work in the philosophy of mathematics during the early 1930s brought about a distinct turning point in his philosophy. The crux of this transition, Steiner contends, is that Wittgenstein came to see mathematical truths as originating in empirical regularities that in the course of time have been hardened into rules. This interpretation, which construes Wittgenstein’s later philosophy of mathematics as more realist than his earlier philosophy, challenges another influential interpretation which reads Wittgenstein as moving in the opposite direction, from a more realist toward a less realist position. Both of these readings, I argue here, tend to overlook the crucial role of conventionalism in shaping Wittgenstein’s later philosophy. I show that during the transition period, Wittgenstein was first strongly attracted to conventionalism, but then, upon discovery of the rule following paradox, came to realize its weaknesses. Did this realization mark the end of the liaison with conventionalism and a wholehearted acceptance of the empirical regularities account? Illustrating Wittgenstein’s ongoing ambivalence toward conventionalism and pointing to his novel understanding of this position, I answer this question in the negative.
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Notes
- 1.
One example of a disputed issue regarding Wittgenstein’s later philosophy is verificationism; some interpretations ascribe to him a verificationist semantics whereas others deny this verificationist reading.
- 2.
It could be objected that unless there are empirical regularities ‘in the world,’ there could be no regularities in human behavior and that the view that Steiner ascribes to Wittgenstein must therefore boil down to anchoring mathematics in empirical regularities in nature. It seems to me, however, that Wittgenstein was open to the possibility that the same natural world would give rise to different kinds of regularities in human behavior. Hence, he would not be willing to reduce regularities in human behavior to natural regularities and laws. I therefore distinguish Wittgenstein’s position (and Steiner’s reading of Wittgenstein) from any empiricist account of mathematics.
- 3.
In her paper in this volume as well as her comments on a draft of this paper, Juliet Floyd urges a more nuanced periodization of Wittgenstein’s later philosophy. She agrees that rules were central to Wittgenstein during the middle period (roughly1929–1935), but maintains that in his later philosophy, the far richer notion of form of life moved to center stage. She therefore could accept my description of Wittgenstein as torn between conflicting tendencies towards mathematical truth when applied to the middle years, but not to his later philosophy in its entirety. Floyd is right about the importance of forms of life and I concede that my account is not sufficiently fine grained with respect to the difference between the middle and late positions. Having said that, I am not convinced that the specific tension I point to (and which can be detected even in Wittgenstein latest passages on the relation between language and reality) is completely resolved in the framework in which rules are replaced with forms of life. I realize, of course, that the ascription of such an unresolved tension might be taken to speak against the interpretation I offer, but happen to believe that deep tensions do not detract from the profoundness of a philosophical position.
- 4.
Wittgenstein’s familiarity with conventionalist arguments is attested to—to mention but one example—by his repeated comparison of the adoption of a convention or a rule with the choice of a unit of measurement, a comparison that runs through conventionalist writings from Poincaré onwards.
- 5.
“We use the perceptible sign (spoken or written sign, etc.) of the proposition as a projection of a possible situation… I call the sign with which we express a thought a propositional sign.—And a proposition is a propositional sign in its projective relation to the world” 3.11–3.12.
- 6.
“There is a general rule by means of which the musician can obtain the symphony from the score, and which makes it possible to derive the symphony from the groove on the gramophone record, and, using the first rule, to derive the score again. … And that rule is the law of projection which projects the symphony into the language of musical notation. It is the rule for translating this language into the language of gramophone records” 4.0141.
- 7.
On the origins and history of the ‘free creation’ idiom see Ben-Menahem (2006) p 143ff.
- 8.
The Tractatus does not endorse conventionalism, then, but notably it does not critique it either. I see it as noteworthy because both Frege and Russell did engaged in explicit critique of conventionalism.
- 9.
The earlier post-Tractatus writings are more conventionalist while the later ones, are less so. At the moment, I ignore these chronological differences. See also note no. 3 above.
- 10.
Similarly ([1956], 1978 RFM I:168): “The mathematician is an inventor, not a discoverer.”
- 11.
For example, members of the first group are not always committed to a platonist ontology, but they do uphold the objective but nonempirical nature of the truths in question.
- 12.
In the Tractatus, logic and mathematics are clearly distinguished; e.g., mathematical theorems are not tautological. These differences need not concern us here.
- 13.
Wittgenstein makes extensive use of a related idea in On Certainty, where he argues against Moore that what we cannot doubt, cannot be said to be known. See, e.g., §56–58, 155, 203.
- 14.
He compares mathematical proofs to schematic pictures of experiments several times in RFM I.
- 15.
Putnam (1994) draws attention to the Kantian roots of this position, and the tension between the platonist and Kantian strands in Frege’s writings.
- 16.
See, e.g., Wittgenstein ([1956] 1978, RFM I:132).
- 17.
I am not suggesting that Wittgenstein likens mathematics to games—indeed, he stresses that games, unlike mathematics, have no application. Yet even the inapplicable rules of chess are, in this sense of ‘arbitary,’ nonarbitary.
- 18.
For example, Engelmann (2005) cites the following lines (his translation): “I don’t need another model that shows me how the depiction goes and, therefore how the first model has to be used, for otherwise I would need a model to show me the use/application of the second, and so on ad infinitum. That is, another model is of no use for me, I have to act at some point without a model (MS 109, p. 86).
- 19.
This conception of laws of nature, although still quite common, is not the only one in the field. David Lewis is one of its opponents.
- 20.
See Tractatus 4. 122–4.1251. There, however, the internal relation is not purely linguistic. The above noted parallelism of representation and what it represents recurs: “The existence of an internal relation between possible situations expresses itself in language by means of an internal relation between the propositions representing them” (4.125). Even though the term ‘internal relation’ is not used by Wittgenstein when describing the connection between the rule and its application, the terms that he does use, such as the rule and its application making contact in language, convey the same idea.
- 21.
Floyd would object that “purely linguistic” is misleading for linguistic expressions are rooted in forms of life. But couldn’t we raise the same question about forms of life? Are they dictated by nature or responsible to it?
- 22.
I am saying it is less intuitive because the relation between a picture or an image and what is depicted usually appears to us more objective than the relations figuring in the previous examples. As for mathematics, simple rules like ‘add 1’ naturally seem to satisfy the linguistic account better than more complex examples. Whether and how it is possible to go from the simple to the complex in a Wittgensteinian framework is a difficult problem I will not address.
References
Barrett, R. B., & Gibson, R. F. (Eds.). (1990). Perspectives on Quine. Blackwell.
Ben-Menahem, Y. (2006). Conventionalism. Cambridge University Press.
Coffa, A. (1991). The semantic tradition from Kant to Carnap. L. Wessels (Ed.). Cambridge University Press.
Engelmann, M. L. (2005). The origins of the so-called ‘Rule-following paradox’ from the ALWS archives: A selection of papers from the International Wittgenstein Symposia in Kirchberg am Wechsel. Papers of the 28th IWS F. Stadler, M. Stöltzner) (Eds.).
James, W. (1955) [1907–9]. Pragmatism. Meridian.
Poincare, H. (1952) [1902]. Science and hypothesis. Dover.
Putnam, H. (1994). Rethinking mathematical necessity. In Words and life (pp. 245–263). Harvard University Press.
Quine, W. V. (1969). Ontological relativity and other essays. Columbia University Press.
Quine, W. V. (1990). Comment on Hintikka. In Barrett and Gibson (Eds.). (pp. 176–177).
Steiner, M. (2009). Empirical regularities in Wittgenstein’s philosophy of mathematics. Philosophia Mathematica (III), 17, 1–34.
Wittgenstein, L. (1971) [1921]. Tractatus logico-philosophicus (D. F. Pears, & B. F. McGuinness, Trans.). Routledge and Kegan Paul.
Wittgenstein, L. (1953). Philosophical investigations (PI); Trans. G. E. M. Anscombe, G. E. M. Anscombe and R. Rhees, (Eds.). Basil Blackwell.
Wittgenstein, L. (1978) [1956]. Remarks on the foundations of mathematics (RFM), Trans. G. E. M. Anscombe, G. H. von Wright, R. Rhees, and G. E. M. Anscombe (Eds.). 3rd Rev edn. Basil Blackwell.
Wittgenstein, L. (1974). Philosophical grammar (PG), Trans. A. Kenny, R. Rhees (Ed.). Basil Blackwell.
Wittgenstein, L. (1976). In C. Diamond (Ed.), Lectures on the foundations of mathematics (LFM). Cornell University Press.
Wittgenstein, L. (1977). On certainty (OC), Trans. D. Paul and G. E. M. Anscombe, G. E. H. Anscombe and G. H. von Wright (Eds.). Basil Blackwell.
Wittgenstein, L. (1993a). In J. Klagge & A. Nordmann (Eds.), Philosophical occasions (PO). Hackett.
Wittgenstein, L. (1993b) Notes for the ‘philosophical lecture’ In Wittgenstein (1993a). (pp. 447–58).
Acknowledgement
I am deeply grateful to Juliet Floyd and to Carl Posy for their very helpful comments on earlier versions of this paper.
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Ben-Menahem, Y. (2023). The Turning Point in Wittgenstein’s Philosophy of Mathematics: Another Turn. In: Posy, C., Ben-Menahem, Y. (eds) Mathematical Knowledge, Objects and Applications. Jerusalem Studies in Philosophy and History of Science. Springer, Cham. https://doi.org/10.1007/978-3-031-21655-8_16
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