Hilbert is quoting Kant correctly, but it's not A298/B355, it's A702/B730. I'm not sure what Bill means about his question re what Hilbert meant by 'Ideen' - I'd have thought this was just (as it is for Kant) idealizing concepts going beyond the bounds of possible experience (in particular for Hilbert infinitistic mathematics). I feel there's more of a question about just what Hilbert meant by 'Anschauung' - perhaps not in *this* passage, where it's pretty clear that he's just saying that geometry starts from perceptual data, conceptualizes them, and then moves to infinitistic idealization, but in later Hilbert where it's crucial that the methods of proof theory be 'anschaulich' or 'konkret', and that's supposed to distinguish finitary methods from intuitionist ones. Bill's own classic article on the subject ('Finitism', J.Phil. 1981) would be the starting-point for any such discussion at a philosophical level, but herewith some historical trivia:
The neo-Friesian philosopher Leonard Nelson, Hilbert's friend and protege, took Hilbertian proof theory to depend on a priori geometrical intuition - essentially the geometry of dried ink - (see the last item reprinted in his 'Beitraege zur Philosophie und Mathematik), but this can't be right (Paul Bernays in his 1923 Math. Annalen reply to Mueller mentions that we could use a series of noises in time rather than marks on paper as signs). Paul Kitcher, in his well-known article 'Hilbert's Epistemology' (Philosophy of Science 1976) takes Hilbert to be talking about *perceptual* intuition, but his argument for this relies on quoting the Bauer-Mengelberg translation of 'Ueber das Unendliche' which simply introduces the word 'perceptual' though it's not there in the original German. It's at least arguable that for Hilbert the key notion is something like self-evident truth of combinatorics (going back to Dedekind's 'Was sind und was sollen die Zahlen'), having nothing essentially to do with space, time or perception. It's also interesting to look at Zermelo's use of 'anschaulich' - e.g. in his 'Neuer Beweis fuer die Moeglichkeit einer Wohlordnung' (Math. Annalen 1908) he claims the axiom of choice is 'anschaulich evident', where 'anschaulich' here seems to have lost all its Kantian associations and just means 'intuitive' in the loose sense in which the word gets used today (cf also the title of the Hilbert-Cohn-Vossen volume 'Anschauliche Geometrie'). Volker Peckhaus ('Hilbertprogramm und Kritische Philosophie' pp.97-8) quotes a letter from Nelson to Hessenberg where Nelson reports much expenditure of time and effort making the concept of pure intuition clear to Zermelo. In Goettingen of the time the word seems to have floated around between philosophers using it in a pretty strict Kantian sense and mathematicians using it more loosely. Among the philosophers things were complicated by the fact that the Marburg neo-Kantians (Cohen, Natorp, Cassirer) had rejected Kant's distinction between Anschauungen as passive and thought as active - even Anschauungen were the result of an original synthesis. (However, Nelson didn't think much of the Marburg school, and may well have persuaded Hilbert that they weren't to be taken seriously.) Another person to mention here is of course Poincare, who was Kantian in thinking that mathematical induction depended on a priori intuition, but doesn't thereby seem to have meant anything more than that the ultimate ground of knowledge was synthetic rather than analytic. One gets this use in Zermelo as well: 'Si ces axiomes ne sont que des principes purement logiques, le principe de l'induction le sera egalement; si au contraire ils sont des intuitions d'une sort speciale, on peut continuer ÃÂ regarder le principe d'induction comme un effet de l'intuition ou comme un "jugement synthetique a priori"' (Acta Mathematica 1909).
Robert Black Dept of Philosophy University of Nottingham