FOM: "The Ignorance of Bourbaki"

[Stephen G Simpson]

Lou van den Dries writes:
 > Bourbaki needs no defense from me, but I cannot resist responding
 > to Simpson's latest production.

Lou, I knew you wouldn't be able to resist; that was the purpose of my
"production"!  But that's not to say I didn't mean every word of it.

 > (I'd be interested in getting Mathias' article though, predictable
 > as its contents are.)

I'll snail-mail it to you.  Does anybody else want a copy?

 > Bourbaki's little paper in the JSL of 1948 is forgettable 

Little?  It's 14 pages, long enough to do a lot of damage.
Forgettable?  Well, it's a truly rotten paper, one of the worst I've
ever seen, but I wouldn't call it forgettable.  I first saw it about
15 or 20 years ago, and it still amazes me to think that the author of
that paper can go around posing as an authority on lofty topics such
as "the architecture of mathematics".

Probably all of us have had the experience where non-mathematical
friends or relatives tell us they can't understand how anyone can do
research in mathematics, because "hasn't it all been done?"  This
attitude is readily excusable if it comes from a dentist or a plumber,
but we would be horrified if it came from scientists in neighboring
fields, e.g. physicists or statisticians.  Yet Bourbaki, the
archetypical pure mathematician, professes precisely this attitude
toward foundations of mathematics!  Have you ever encountered such
smug ignorance?

 > and has very little to do with the main ideas of Bourbaki
 > concerning mathematics.

What main ideas?  I've read some of Bourbaki's other articles, and
they seem to be internally quite harmonious with the JSL article.  You
could summarize Bourbaki this way: "Pure mathematics is my drug of
choice, and I resent anybody who tells me that I need to think about
anything else."

Lou finds it intellectually unrewarding to think and write about
issues and programs in foundations of mathematics.  Well, that's fine,
and I'm all for specialization and division of labor.  But to my mind
it's also interesting to ask: Why did the great mathematicians of the
past (Hilbert, Weyl, von Neumann, Poincare, Brouwer, ...) take such a
lively interest in these allegedly unrewarding topics?  Lou has a pat
answer:

 > A century ago there was something of a crisis in foundations
 > (although this has been exaggerated by those who write about
 > foundations), and that explains why these big names got into the
 > act.

Lou, I'm sorry, but that explanation won't do.  Profound developments
such as Weyl's program of predicativity, Brouwer's intuitionism, and
Hilbert's finitistic reductionism simply cannot be explained as
over-reaction to an alleged crisis brought on by the Russell paradox.
Bear in mind that, when faced with any sort of foundational issue,
people always have the option of yawning and turning away, so long as
it doesn't directly affect their current specialized problem.  That's
exactly what most pure mathematicians did, both then and now.  But
Hilbert and company didn't, and we have to ask why.  What were these
men getting at?

When we read what Hilbert and company wrote, it's obvious that they
were motivated by a keen interest in the nature of mathematical
objects, the logical structure of mathematics, and the relationship
between mathematics and the rest of human knowledge -- in short,
philosophy of mathematics, in the best sense.  They considered these
topics as part of their central concern as mathematicians.

Unfortunately, this kind of broad intellectual perspective is today
virtually extinct.  We have gone beyond specialization into the realm
of hermetically sealed compartmentalization.

 > There are large scale interactions going on, between fields of
 > mathematics, and mathematics and physics, mathematics and computer
 > science, etc.

Good, I'm glad you said this.  Let me comment on these three points in
order.

Interactions between fields of pure mathematics?  Sure, but what about
interactions between pure mathematics and everything else?  Rational
points on algebraic varieties may have a certain appeal to pure
mathematicians, but they are a far cry from foundations and a far cry
from what I would call general intellectual significance.

Physics?  There is string theory, but from what I understand of it,
it's nothing but pure mathematics.  It's not backed by one shred of
physical evidence.  (That's not to say that string theory is
uninteresting.  But I'd be very cautious about calling it physics.  It
seems to be a case where physical speculation has given rise to some
intricate mathematical questions, but from the physical standpoint,
where's the beef?)

Computer science?  Most pure mathematicians follow Bourbaki and
snobbishly reject all appliations.  That's why computer science split
off from mathematics decades ago.  We've declined a long way from von
Neumann, who wrote a technical report defining the architecture of the
stored-program digital computer.

Final comment:

I'm really glad that we got a good discussion going here.  Let's hear
from some new voices?  Neil?  

By the way Lou, I want to remind you that you promised Harvey a reply
to his long message on the thrill of foundations.  In particular, I'm
wondering how you will answer the following question from Harvey:

 > So now you know in some detail one of the biggest thrills for me in
 > FOM. The prospect of acheiving something that can only be compared
 > with the greatest intellectual events of all time. I simply don't
 > see how to do this in mathematics. Do you?

Best wishes,
-- Steve