Finitism and Intuitive Knowledge
|(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
|(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.