[...]
> Graham White is right that the terms involved all have techical
> meanings for Kant; but there remain two questions about this:
>
> a) What *did* Kant mean? I believe that there are rather different
> accounts of K's meaning of the term 'pure intuition'. The problem is
> that its supposed connection with empirical intuition (which is needed
> to support the applicability of geometry to space) is in conflict with its
>
> supposed role in mathematical demonstration. One obtains different
> views of K's idea as one attempts to take into account one of the
> other of these two poles.
>
> b) Did Hilbert use these terms with a Kantian meaning (as he
> understood K) in mind?
>
> It is so that there are passages (one in particular) in ``Ueber das
> Unendliche'', describing the finitist position, that sound very
> Kantian; but it can at least be questioned whether that should be
> read back into *Found Geom*.
>
> Bill Tait
The following article might provide some indirect help on the use of
'intuition' by Hilbert and Kant. There is a brief discussion relating
Goedel's views to aspects of each of theirs. This article is quite
interesting in itself as it discusses Goedel's work on a framework
(foundation) for mathematics prominently featuring the role of mathematical
intuition, based on Husserl's phenomenology and his notion of "intuiting
essences".
G\"odel's path from the incompleteness theorems (1931) to phenomenology
(1961), by Richard Tieszen, pages 181-203. The Bulletin of Symbolic Logic,
Volume 4, Issue 2, June 1998. http://www.math.ucla.edu/~asl/0402-toc.htm
Here is the text of one related passage on p. 189:
"...In addition, the abstract elements involved could not be given by
Hilbert's concrete intuition since concrete intuition is restricted to
finite sign-configurations. There must be, by the second incompleteness
theorem and the consistency proof for PA, a less restricted kind of
mathematical intuition or insight that accounts for our mathematical
knowledge. In Husserlian language, there must be an intuition of
mathematical essences."
And another related passage on p. 199:
"...Given the incompleteness theorems, there can for most mathematical
essences be no consistent machine that solves all of the well defined
yes-or-no questions that are left undecided by the original set of axioms
for those essences. Human reason, however, may be able to achieve such a
development by virtue of its ability to reflect on essences..... Thus, it is
through an adjusted philosophical viewpoint according to which we intuit
essences that Goedel seeks to make a place for Hilbert's optimism about
mathematical problem solving as well as Hilbert's idea that in mathematical
proofs we should strive for certainty.
Goedel says that the intuitive grasp of ever newer axioms that are
logically independent of earlier ones is necessary for the solvability of
all problems even within a very limited domain. He says that this agrees in
principle with the Kantian conception of mathematics. Goedel points out,
however, that Kant was wrong to think that for the derivation of geometrical
theorems we always need new intuitions, and that a logical derivation of
these theorems from a finite number of axioms was therefore impossible. But
in the case of mathematics in a more general sense, Kant's observation is
correct. Goedel says many of Kant's assertions are false if literally
understood, but that they contain deeper truths in a more general sense. It
is Husserl's (transcendental) phenomenology that for the first time does
justice to the core of Kantian thought. It avoids both the 'death-defying
leap of idealism into a new metaphysics as well as the positivistic
rejection of every metaphysics'..... It could be argued that the
phenomenological approach is not prey to the excesses and lack of balance
that characterized earlier..... viewpoints."
Possibly Bill's question regarding the clarity or lack thereof in "K's
meaning of the term 'pure intuition'" is reflected, if not explained, in the
report of Goedel's views above?
In any case, I found that Tieszen's article gave me renewed interest in
foundational questions, since I think it provides the best framework I know
for describing and representing how mathematics is actually done.
John Pais