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Introductory remarks. (English) Zbl 0878.03004

Czermak, Johannes (ed.), Philosophy of mathematics. Proceedings of the 15th international Wittgenstein-Symposium, August 16-23, 1992, Kirchberg am Wechsel, Austria. Part I. Wien: Hölder-Pichler-Tempsky. Schriftenreihe der Wittgenstein-Gesellschaft. 20/I, 69-76 (1993).
In these “Introductory Remarks” to the special symposium on the occasion of the publication of his book: Frege: Philosophy of mathematics (Duckworth, London, 1991), the author gives what he regards as Frege’s essential motives for salvaging his attempts for a foundation of arithmetic being a consequence of his inability to solve Russell’s paradox. Dummett identifies as Frege’s eventual diagnosis that the failure of his earlier attempts “had been due to his attempt to combine logicism concerning number theory and analysis with platonism” (p. 69). Platonism as applied to any given mathematical theory may be regarded as comprising the following three tenets: “(i) what appear to be statement of the theory are genuine statements, characterizable as true or false, and possessing definite truth-conditions; (ii) the statements of the theory are to be interpreted at face-value, so that what look like individual constants or complex singular terms are construed as denoting objects, and what look like individual variables as ranging over a domain of objects; (iii) the statements are determinately either true or false – bivalence holds for them – and, in particular, the axioms and theorems of the theory are true” (ibid.). For Dummett the Fregean combination of logicism and platonism shows many affinities to Hartry Field’s neo-Hilbertian theory (p. 70).
One of Frege’s main aims in Grundlagen der Arithmetik (1884) was to give a justification for the existence of abstract objects. He distinguished between objects which are actual (wirklich) and those which are not. Frege’s main tool against nominalist objections to abstract objects, as Dummett points out, was the appeal to the context-principle that “it is only in the context of a sentence that a word means […] anything” (p. 71).
Dummett discusses the consequences of the context-principle and the limits of a realistic interpretation of mathematics.
For the entire collection see [Zbl 0836.00022].

MSC:

03A05 Philosophical and critical aspects of logic and foundations
03-03 History of mathematical logic and foundations
01A55 History of mathematics in the 19th century
00A30 Philosophy of mathematics

Biographic References:

Frege, Gottlob
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