FOM: reply to Graham White

[Stephen G Simpson]

Graham White writes:

 > It's a rant, and Mathias is no historian.

Are any points of history distorted in Mathias' account of "The
Ignorance of Bourbaki"?  Incidentally, Adrian Mathias is on this
mailing list, so he will be able to respond.

 > 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? ....

It's possible to do a lot in mathematics without knowing anything of
mathematical logic or f.o.m.  Bourbaki is a good example of this.
However, f.o.m. provides a certain perspective on mathematics and its
connections with the rest of human knowledge.  It does this by
focusing on and analyzing the most basic mathematical concepts.  This
posting is not the place for a full discussion of the goals and
achievements of f.o.m., but some of the relevant material has been
discussed on this mailing list.  See also the entire
f.o.m. literature.

 > 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,

Yes, mathematical practice has improved a lot: there is now a
generally accepted standard of mathematical rigor, namely formal
provability in Zermelo/Fraenkel set theory.  This hard-won consensus
is a result of foundational work by Dedekind, Frege, Zermelo, ...,
specifically set-theoretic foundations.  Bourbaki's paper "Foundations
for the Working Mathematician" relies on this earlier foundational
work.

 > .... So who *needs* foundations?

You do.  We all do.  Even if set-theoretic foundations is replaced by
something else, we will still need foundations in order to attain
rigor.

Your argument against foundations reminds me of the typical
self-contradictory argument against capitalism.  It runs as follows:
"Capitalism isn't needed for a good standard of living, because we can
get it more easily by expropriating large corporations."  (Never mind
how the corporations came into existence in the first place.)

 > I fail to see, in general, the equation you make between
 > "anti-foundationalism" and compartmentalisation

Part of the foundational outlook is a concern for the unity of human
knowledge.  From this it follows that compartmentalization (i.e. the
disintegration of human knowledge) necessarily entails
anti-foundationalism.

This issue has been discussed previously on the FOM list.  I stood up
in favor of the unity of human knowledge, and Sol Feferman stood up
against it.  See my posting of 13 Jan 1998 13:29:45.

-- Steve