FOM: reply to Graham White

[Graham White]

>I meant it in the following sense: the writings of Bourbaki exhibit no
>interest in or understanding of f.o.m. or other interfaces between
>mathematics and the rest of human knowledge.  In this sense Bourbaki
>is a paradigm of anti-foundationalism and compartmentalization.  Read
>Mathias's article "The Ignorance of Bourbaki".
>
Well, I have read it. It's a rant, and Mathias is no historian. S.L. Segal
(who's a competent historian of mathematics) writes, in his review of
Mathias' paper in the Zentralblatt "In this paper polemics have precedence
over history -- and it does not seem worth much as the latter".

Apart from that, suppose we lay aside Mathias' remarks about Bourbaki's
motivation (which are, as Segal remarks, a bit confused, because he
simultaneously accuses the Bourbakists of having gone wrong because
i) they were influenced by Hilbert, and ii) because they were French
chauvinists)). Mathias' main point seems to be that the Bourbakists
had an inadequate grasp of logic; suppose we grant this. Well, you could
argue, what difference does it make? Their mathematics seems to be
extremely reliable; they don't make mistakes, and they succeed in doing the
usual stuff.

In this respect the situation now seems to be different from
that of the turn of the century, when (apart from the set-theoretical
paradoxes) there were large areas of mathematics that just seemed
unreliable (algebraic geometry, statistical mechanics, calculus of
variations...). At the turn of the century, it could be argued,
mathematics looked as if it needed foundations. But now there just isn't
the significant disagreement about the validity of mathematical results
that there was then; mathematical practice has evidently improved, to
the extent to which you can get it right even if you know as little
logic as the Bourbakists. So who *needs* foundations?

But, of course, reverse mathematics remains interesting. But it's not
foundations of mathematics: it's metamathematics, that is, it is
the mathematical reflection on mathematical practice. Now I can't see
that this sort of position would be an instance of "compartmentalisation",
because, since it treats mathematics as an intellectual practice, it
leaves lots of room for connections between mathematics and other
human activities. I fail to see, in general, the equation you make between
"anti-foundationalism" and compartmentalisation: there are, of course,
non-foundationalist positions which compartmentalise a lot, but there are
also non-foundationalist positions which emphasise the interconnection
of different areas of intellectual activity. Nancy Cartwright's philosophy
of science, for example, is a good modern activity (and builds on
Duhem's work, if I can be forgiven for mentioning another Frenchman...)

Yours,
Graham