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Robert Black

Posts: 1
Registered: 12/3/04

[HM] Hilbert's epigraph from Kant
Posted: Jun 23, 1998 8:46 PM

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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