The Philosophy of Mathematics Today

By Matthias Schirn | Go to book overview
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Finitism and Intuitive Knowledge
In the introduction to his paper of 1958 on an extension of the finitary method in proof theory, Kurt Gödel remarks that 'finitary mathematics is defined as that of intuitive evidence',1 by which he evidently means:
(1.1) If a proposition has been proved by the finitary method, then it is intuitively evident.
(1.2) If a proposition is intuitively evident, it can be given a finitary proof.
(1.1) and (1.2) formulate in a succinct way a thesis about the significance of the finitary method that did, in my view, belong to the outlook of the Hilbert school although it is not stated quite so directly in their writing. I have stated it in the way I have because I think it useful to consider (1.1) and (1.2) separately. It is also useful to consider them in connection with two theses that together constitute a mathematical characterization of finitism:
(1.3) Proofs in primitive recursive arithmetic (PRA) are finitist; hence any theorem of PRA is finitistically provable.
(1.4) If a proposition in the language of primitive recursive arithmetic is finitistically provable, then it is a theorem of PRA.
(1.3) is clearly expressed in writings of Hilbert and Bernays; an analysis of finitism that did not yield it would be hard put to it to show that it was

I am grateful to the participants in the Munich conference for their comments, especially to Geoffrey Hellman for subsequent correspondence. Since then the paper has been presented to other audiences, which I also wish to thank; in some cases I am conscious of not having done justice to the comments. I am grateful to Jaakko Hintikka for the invitation to speak at a symposium on Hilbert's Philosophy of Mathematics at Boston University, which led me to focus the paper on finitism.

"'Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes'", Dialectica, 12 ( 1958), 280-7, at 280. The paper is reprinted with an English trans, in Collected Works, ii: Publications 1938-1974, Solomon Fefermanet al. (eds), Oxford University Press, 1990. However, translations of quotations are my own. In the 1972 English version of the paper, what corresponds to the quotation in the text is the remark that 'finitary mathematics is defined as the mathematics of concrete intuition' (ibid. 272, emphasis Gödel's).


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