The Kantian character of Hilbert’s formalism. (English) Zbl 0849.00016

Detlefsen, Michael

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, 195-205 (1993).

Following Herman Weyl and Paul Bernays the author characterizes abstract axiomatic systems in the sense of Hilbert’s early formalism (up to 1904) as “a logical mold of possible sciences” dealing with “relational structures”. This gives an understanding for Hilbert’s regarding consistency as a criterion of truth: “The task of a mathematical theory is to define a ‘form’. It is ‘true’ just in case the ‘form’ it purports to define in fact exists. But since the form it purports to define exists just in case it is consistent, it follows that it is ‘true’ just in case it is consistent” (p. 196). Hilbert’s early formalism cannot easily be related to the trendy distinction between the realist and the idealist view on mathematics. The author recognizes, however, a turn into the direction of idealism in Hilbert’s later work. The author discusses Hilbert’s understanding of the infinite as a regulative device, and his abstraction, more radical than in his early writings, from meaning in order to preserve the classical logical status of mathematical reasoning and derived from his Kantian conception of the distinction between real and ideal propositions. The author shows some resemblances of these ideas with Kant’s critical philosophy.
For the entire collection see [Zbl 0836.00022].

Reviewer: V.Peckhaus (Erlangen)

MSC:

00A30	Philosophy of mathematics
01A60	History of mathematics in the 20th century
03-03	History of mathematical logic and foundations
03B05	Classical propositional logic

Keywords:

idealism; Hilbert’s formalism; Kantianism