FOM: "crises" and crises

[Lou van den Dries]

Dear Harvey,

By accident I addressed my last response only to you. So let me
respond again in roughly the same terms.

Okay, "crisis" is not quite the right expression, and i regret
having dropped that term. (It smells of those silly paradoxes,
and i certainly didn't have that in mind at all.) Perhaps a better
way of saying what was on my mind is this: At the time the now
prevalent way of thinking about mathematical objects and correct
way of reasoning about them had not yet consolidated itself, and
of course several of the leading mathematicians had strong ideas
about it and got involved. If I am not mistaken there wasn't even
a consensus yet what to make of the various non-euclidean geometries,
or the relation of geometry to physical reality. All sorts of
philosophical preconceptions were gradually put overboard (and perhaps
others were imported in the process), and this took some time to
play itself out, say from about 1870-1930.  And yes, I have read all
those books and articles people like Poincare, Hilbert, Weyl and so
on wrote on such "foundational matters". I must confess I considered it
as light reading, for the most part, and it never impressed me as
much as some other things they did. Some of what they said made a
lot of sense, and was taken over by those who came after them,
some of it is perhaps better forgotten.

Your idea of reviving foundations by creating a "permanent crisis"
is amusing. Well, nobody is stopping you from trying. But in my
opinion it is a grand delusion. All the best,
                                             Lou van den Dries

PS Leading mathematicians of our time also write about broad issues
concerning math and its relations to science and society. A good way
to find out is to take a look at Collected Works, things like that.
As I mentioned before (since Atiyah's name came up), volume 1 of
his Collected Works contains several quite thoughtful essays of
this kind. (I liked his Bakerian lecture where he sketches for a
general educated audience the preeminent role of geometric thinking
and alludes indirectly to cohomology when talking about the various
kinds of holes in mathematical spaces; he even manages to give a
very rough idea of the Weil conjectures in this connection. Maybe
Steve would like this, as he seems very eager to find out what all
this is about.)