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 h=
is
'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). Pau=
l
Kitcher, in his well-known article 'Hilbert's Epistemology' (Philosophy o=
f
Science 1976) takes Hilbert to be talking about *perceptual* intuition, b=
ut
his argument for this relies on quoting the Bauer-Mengelberg translation =
of
'Ueber das Unendliche' which simply introduces the word 'perceptual' thou=
gh
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 ju=
st
means 'intuitive' in the loose sense in which the word gets used today (c=
f
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 i=
ls
sont des intuitions d'une sort speciale, on peut continuer =E0 regarder l=
e
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