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Hilbert’s twentyfourth problem
 American Mathematical Monthly
, 2001
"... 1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Cong ..."
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1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Congress of Mathematicians (ICM) in Paris has tremendous importance for all mathematicians. Moreover, a substantial part of
Remarks On Finitism
 Reflections on the Foundations of Mathematics. Essays in Honor of Solomon Feferman, LNL 15. Association for Symbolic Logic
, 2000
"... representability in intuition. (See [2, p. 40].) But our problem is, of course, not the finiteness of a number, but the infinity of numbers. There is, I think, a di#culty with Bernays' notion of formal object, where this is intended to extend to numbers so large as, not only to be beyond processing ..."
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representability in intuition. (See [2, p. 40].) But our problem is, of course, not the finiteness of a number, but the infinity of numbers. There is, I think, a di#culty with Bernays' notion of formal object, where this is intended to extend to numbers so large as, not only to be beyond processing by the human mind, but possibly to be beyond representablity in the physical world. [2, p. 39]. This di#culty ought to be discussed more adequately then + This paper is based on a talk that I was very pleased to give at the conference Reflections, December 1315, 1998, in honor of Solomon Feferman on his seventieth birthday. The choice of topic is especially appropriate for the conference in view of recent discussions we had had about finitism. I profited from the discussion following my talk and, in particular, from the remarks of Richard Zach. I have since had the advantage of further discussions with Zach and of reading his paper 1998; and I use his scholarshi