Different forms of finitism. (English) Zbl 0838.03003

Czermak, Johannes (ed.), Philosophy of mathematics. Proceedings of the 15th international Wittgenstein-Symposium, 16-23 August 1992, Kirchberg am Wechsel, Austria. Part I. Wien: Hölder-Pichler-Tempsky. Schriftenreihe der Wittgenstein-Gesellschaft. 20/I, 185-194 (1993).
The author opposes the generally accepted opinion that the term “finite” in Hilbert’s programme is vague. He thus refutes attempts to create “generalized Hilbert’s programmes” by admitting constructive forms of infinite means – he particularly refers to Gentzen’s transfinite induction – as not being in Hilbert’s spirit. Regarding unpublished sources, the author argues for the universal epistemological character of Hilbert’s concept of finitism, giving evidence especially from his work of physics. According to this epistemology, the infinite exists only as an idea (a mere rule of thinking without any real meaning). Thinking itself, however, is taken as a real process, which is free of any trace of the infinite. Therefore every inference from the fact that no boundaries exist, to the infinite, is not an act on reasoning but of speculative invention. In order to be reliable, logical reasoning has to be finite, a condition which the author calls “reliability-condition of thinking”. In the light of Gödel’s second incompleteness result, the consistency proof demanded in Hilbert’s programme based on the reliability-condition looses much of its epistemological force. The author pleads for adopting Hermann Weyl’s standpoint of a “symbolic construction of the world” as a way out.
Reviewer’s comment: In advocating S. G. Simpson’s [J. Symb. Log. 53, No. 2, 349-363 (1988; Zbl 0654.03003); quote p. 353] harsh opinion that later proof-theoretic developments “are ingenious and have great scientific value, but they are not contributions to Hilbert’s program”, the author, as Simpson, underestimates the dynamical spirit of Hilbert’s programme.
For the entire collection see [Zbl 0836.00022].


03A05 Philosophical and critical aspects of logic and foundations
00A30 Philosophy of mathematics
01A60 History of mathematics in the 20th century
03-03 History of mathematical logic and foundations


Zbl 0654.03003