FOM: Millenium Conference
Mark Steiner
marksa at vms.huji.ac.il
Fri Jan 14 06:28:40 EST 2000
At Harvey's request, I'll supply more information about the millenium
conference.
The two mathematicians who spoke were Kazhdan of Harvard and Macpherson
of the Institute for Advanced Studies, Princeton. In the audience,
among the logicians, were at least Shelah and Magidor (who happens to be
now the president of Hebrew University--it is a great pleasure and honor
to have such a nice person and fine scholar as my boss). [One of the
shapers of our mathematics department was Fraenkel (of ZF), thus logic
has great prestige here. It is taught in the math department and in the
philosophy department--not only elementary logic, by the way.] Since
the turnout for this conference was very large, filling a big chemistry
lecture hall, I couldn't see for sure who else was there, but I tend to
assume there were other logicians there also.
Kazhdan marveled at the fact that Goedel's theorem has not hampered the
development of mathematics; that more or less mathematical questions can
be decided, with few exceptions. In the question period I asked whether
the speaker thought that one could put a bound of length on mathematical
theorems such that Goedel's theorem would not apply to theorems of that
length (I didn't want to misrepresent Harvey's ideas so didn't mention
his name in this connection) and was rewarded by a one word answer, to
the amusement of the crowd: no. Either Kazhdan doesn't have the concept
of a formalized mathematical theory like PA, or else(giving the benefit
of the doubt to a friend) he doesn't think it is a useful tool for
discussing questions like this.
I do think that the question "how is mathematics possible" is worth
discussing, and has great philosophical interest (or "general
intellectual interest," as Harvey puts it) though of course it needs
prior analysis to make sense. Naturally the question "What is
mathematics?" which has appeared on this list is quite relevant here.
Both philosophy and f.o.m. would be necessary.
Kazhdan gave a fascinating discussion of the phenomenology of
mathematics, i.e. when do mathematicians become convinced that an open
problem is solvable (such as Fermat's conjecture)even before it is
actually solved.
Most interest to me was his discussion of the relationship between
mathematics and physics (which, so far as I could see, contradicted
Macpherson's view). He stated that there is a new relationship here.
Whereas in the past mathematics was the connection between different
physical ideas, today it is often physics which connects different
mathematical ideas. For example, there are physical theories which
don't have any rigorous mathematical formulation at all. On the other
hand, for special cases (say, very high and very low energies)
mathematical formulations can be had. So that mathematicians are not
capable of "seeing" the connection between the two mathematical ideas
without looking at the physics of the system.
I asked Kazhdan privately why he mentioned no 20th century mathematical
theorems (he mentioned "category theory" as a major recent development
as a new way to look at the field of mathematics, but no detailed
achievements using category theory)and he said that he couldn't think of
any good twentieth century examples suitable for a broad audience. I
find this hard to believe. He admitted to me that his lecture was in
the philosophy and history of mathematics (and I pointed out to him
errors in the former) but said that he had no choice. As for Goedel's
theorem and general intellectual interest--as compared to that of other
mathematical results--I can't speak for him on this matter. Maybe he
would say that the g.i.i. of Goedel's theorem is what it is only because
the g.i.i. of mathematics is what it is. There are obvious
counterexamples to the thesis that the interest of Foundations of X
dependes on the interest of X, but I won't go on about this, and again,
I didn't ask him that question and don't have a view (about what his
view is). I should say in Kazhdan's favor that he was very open minded,
amenable to correction, and he concluded his talk with the hope that
mathematics would return to its roots in philosophy; I'm not 100% sure
what that means, but as a philosopher, I'm happy at the sentiment.
As for Macpherson, I found him somewhat dogmatic. Although he claimed
to believe that one cannot really predict the future of mathematics, he
stated dogmatically that anybody who claimed that proof would lose its
status as the criterion of truth in mathematics was a "crackpot." First
of all, the mathematicians he has in mind (and they can be named very
easily) don't strike me as crackpots. Second, he had just mentioned
Goedel's theorem as a landmark of mathematics, and I need not remind my
readers here that Goedel himself and many others saw the import of his
theorem as stating that truth and proof are two notions that should not
be confused. What he meant, probably, was that proof would continue to
be the criterion for the assertibility-as-true of a mathematical
proposition. But how does he know this? The role of computers in
mathematical knowledge cannot be predicted. Also, recall Kazhdan's
point that there are and may continue to be mathematical connections
that cannot be proved, but are seen by looking at physical models. If
I'm not mistaken (correct me if I'm wrong) Witten and even Mandelbrot
are considered by the mathematics community to have made real
contributions to mathematics without proving anything. They won prizes,
I think. (I recall there were protests in the mathematical community
about Mandelbrot. One mathematician wrote that to give Mandelbrot a
prize is equivalent to giving the Fields medal to a cirrus cloud, if it
inspired mathematicians to solve the Fermat conjecture.)
A point that Macpherson made that bears discussing: he drew two
historical curves with respect to time in the twentieth century: one
abstractness, and the other applications (or lack of them). It turned
out that they were the same curve: exactly during the years that the
Bourbakists reigned, applications to physics dropped.
An element that f.o.m. can justly take offense at is Macpherson's map
of interconnections among various fields of mathematics which is truly
mysterious. But he left out logic and f.o.m. from his map of
mathematics (I have seen omissions like this in other retrospective
books written by mathematicians). This after mentioning only a theorem
by Goedel in the entire lecture of one hour and fifteen minutes.
I think it is pretty clear that there is a bias in the so-called "core"
mathematical community against foundationalists. I think a number of
postings on f.o.m. expressed this idea, and I had thought that this was
a little exaggerated. However, the "chutzpah" of mentioning Goedel and
then not mentioning his field confirmed to me that there is indeed such
a bias, persisting even today (in my day at Columbia University
logicians could be hired only in the philosophy department; to teach
courses in logic, they brought people in as adjuncts).
As far as general intellectual interest of "core" mathematics, versus
philosophy, history, and foundations. Harvey asked me my view of their
view. There is no way really to know without an in-depth interview, but
my guess is (as I already said) that they would say that mathematics has
more intellectual interest than the history, philosophy, or foundations
of the field, if only because these other fields draw their intellectual
interest from core mathematics (of course, this is not an airtight
argument, as I pointed out above). As for twentieth century
mathematics, it has become to abstruse for general audiences, but it is
of a piece with earlier mathematics which can be appreciated by general
audiences. The fact that so many people came to hear core mathematicians
talk about their field indicates to me that lots of people feel that
mathematics HAS interest, except that THEY are incompetent to appreciate
it, much as a tone deaf person might come to realize that classic music
has general interest, which he unfortunately cannot appreciate. This is
different from the claim that certain fields have little general
intellectual interest even when they are understood.
I suspect, however, that the idea that, no matter how hard they try,
core mathematicians CANNOT explain even ONE theorem to a general
audience, not even the statement of the theorem to say nothing of the
general line of proof, is self-serving. I recall back in the '70's when
core mathematicians were worried about their funding, they organized
themselves into something called COSRIMS and put out a volume to explain
the importance of funding mathematics. They had no choice then but to
give expositions of some of the contents of twentieth century
mathematics, and I was able to learn something about the content from
the volume (or volumes) they produced.
In conclusion, I think that philosophers and f.o.m. people each in
their own way, can make a contribution to mathematics itself by making
clear the general intellectual interest of fields like mathematics,
since they may well be in a good position to make clear the content of
these fields to laypeople. (This is aside from the undeniable intrinsic
interest of philosophy and f.o.m. themselves.) I might add that I got
my first introduction to Mycielski's ideas from a book by a philosopher,
Understanding the Infinite, by Shaughan Lavine.
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