Friedrich Waismann’s Philosophy of Mathematics

Schroeder, Severin; Tomany, Harry

doi:10.1007/978-3-030-25008-9_4

Severin Schroeder⁴ &
Harry Tomany⁴

Part of the book series: History of Analytic Philosophy ((History of Analytic Philosophy))

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Abstract

In their paper Severin Schroeder and Harry Tomany parallel Waismann’s writings on existence in mathematics, the meaning of mathematical concepts, equations and tautologies as well as infinity in minute detail with Wittgenstein’s philosophy of mathematics. We learn that, aside from two substantial issues—conventionalism and conjectures in mathematics—Waismann would very much follow in Wittgenstein’s steps. As to conjectures, while Waismann’s criticism of Wittgenstein’s early views is well placed here, Wittgenstein would later amend his position in the 1940s. Schroeder and Tomany conclude that Waismann would remain a Wittgensteinian philosopher of mathematics.

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Notes

1.
Cp. Grassl (1982, 21–23), for a slightly different list of the main tenets of Waismann’s philosophy of mathematics.
2.
For more details about this project and its ultimate failure, see Engelmann (2013, 28–43).
3.
The seeds of this idea can already be seen in the early intermediate period when Wittgenstein talks of a calculus being ‘serious business’ because of its possible application(s) to ‘everyday life’ (WVC 170). This stands in contrast, however, to his more common discussion (at this time) of a mathematical calculus being an ‘application of itself’ (and seemingly fully meaningful) (PR 130–132).
4.
Wittgenstein similarly explains the Dedekind definition of an infinite class as ‘saying that it is a class which is similar to a proper subclass of itself’ (PG 464).
5.
It would seem that only Waismann employs the term ‘verification’ (as opposed to ‘method of verification’) in order to make this distinction. Wittgenstein employs the verification principle to make logical distinctions between types of propositions and does not use the term ‘verification’ to only refer to empirical verification. According to this understanding, mathematical propositions also have ‘verifications’ (i.e., different types of proofs). What is true: the method of verification of an empirical statement relates importantly to the world, whereas the method of verification of mathematical propositions does not. A large part of Wittgenstein’s work in the intermediate period and onwards is devoted to arguing for this point (often without any use of the term ‘verification’ at all—indeed the idea of ‘methods of verification’ at most complements the idea of categorially different methods employed by mathematics and the empirical disciplines). Despite the different use of terminology, Wittgenstein’s work obviously anticipates Waismann’s thought on this point also.
6.
For the purposes of this part of the chapter, we are ignoring various comments made by Wittgenstein in his intermediate period on the axiom of infinity (e.g. PR 124), since they are, at least at certain points, dependent on his earlier views. Instead, we base his views on the axiom here on his overall later philosophy of mathematics.
7.
Aside from a remarkable affinity between the wording of the two arguments, it also should be noted that the arguments are presented as arguments against the actual infinite (even though Waismann rejects this conclusion). Thus, it is virtually impossible that Waismann was not following Wittgenstein’s ideas in this context.
8.
Although Waismann would deny it, it is not clear to us that his argument cannot also serve as a refutation of the actual infinite. For if one imagines his example as one involving an algorithm constructed in the right way, it would appear that in order to speak of a completed infinite series one must have recourse to the infinite past. Yet this itself is unintelligible (it conflicts with other conceptual truths regarding calculation—e.g. our use of ‘begun’). And, in addition, the idea of an infinite series as a completed whole is undermined by numerous other comments made by Wittgenstein (e.g. PR 164; 167).
9.
For a more detailed account, see Schroeder (2012).

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