FOM: attitudes of core mathematicians and applied modeltheoriststoward f.o.m.
Mark Steiner
marksa at vms.huji.ac.il
Fri Jan 28 04:58:02 EST 2000
Though my wife is starting to complain that I'm getting ADS from
staring at the Internet, I will answer something about number theory and
physics. (Jerusalem is snowed in, the worst snowstorm in 50 years, so I
can't go anywhere anyhow.)
First, however, I remark that though Lie Groups are indeed incredibly
useful in physics, this was not the view in the physics community till
the 1960's with the miraculous discovery of the omega minus particle
using group theory and the subsequent rise of gauge field theories,
quarks, etc. In 1913, James Jeans, a prominent physicist, recommended
that group theory be removed from the curriculum at Princeton, on the
grounds that "This stuff will never be useful in physics." Hermann
Weyl, whose work I think even Harvey will admit has supreme g.i.i. (look
at his book on Symmetry for the layman which is still in print, and his
books about Space Time and Matter), was called the "gruppenpest" by
physicists. Pauli's disdain for pure mathematics (and elementary rules
of rigor) is well documented. Gell-Mann himself, by his own admission,
should have predicted the omega minus a year or two before he did, but
his own arrogance about pure mathematics and, yes, Lie Groups, prevented
him from asking about them. Essentially he was trying to discover the
eight-dimensional irreducible representation of the group SU(3) by
himself by playing with matrices, but failed (these groups had long ago
been classified by mathematicians like Eli Cartan). His excuse was that
he was in Paris at the time, and in a constant state of
semi-drunkenness. It was only after he got back to California, and
unfermented fruit juice, that he thought to ask a mathematician. He
could have known about Lie groups earlier, since I believe he was
present at Guilio Racah's famous lectures on group theory and
spectroscopy at the Institute for Advanced Study, Princeton. I think
his excuse for this was he couldn't understand's Racah's "Yiddish
accent." (Racah, an Italian Jew, who didn't know Yiddish any more than
Gell-Man, was the head of the Hebrew University's department of
physics.)
So we have the intolerance chain: physicists are intolerant of pure
mathematicians who are intolerant of f.o.m. And philosophy often is at
the receiving end also (though almost invariably it's "the philosophy I
don't like" which is referred to by the word "philosophy").
It is remarkable that number theory, one of the earliest branches of
mathematics, has had very little application to physics, presumably
because the world is a spacetime manifold (or so it looks to us), and
the number system is discrete. Nevertheless, the Pythagoreans were able
to discover an amazing correspondence between numbers and music. They
attempted to extend this correspondence to other phenomena, even to
biology (gestation period of animals), but, by modern lights, did not
succeed. Since the 3rd century, Pythagoreanism has had an influence on
architecture (see Pythagorean Palaces, by I forget who), and culture,
but nothing of any substance in physics. You won't find the concept
"prime number" in
any physical laws to date.
Yet a leading mathematician recently predicted a change in this,
because of the quantum revolution, which makes nature more "discrete"
than before and which actually uses functions on thenatural numbers
(Balmer series, etc.). At the same time, Steven Weinberg became a great
fan of "string theory." String theory
applies modular forms which in fact is a topic in number theory.
In 1972, I believe, Freeman Dyson published his well known Gibbs
lecture "Missed Opportunities," in which he bemoaned the lack of
communication between
physicists and mathematicians. In fact, he complained, he doesn't even
speak to himself. I.e., when he did number theory he never thought of
physics. As such, he missed a possible discovery. Namely, he had
generalized a theorem T of Ramanujan about modular forms to the theorem
T(n), where n = 3, 8, 10, ..., 26, ..., and did not recall that these
numbers had to do with representations of Lie algebras, i.e. physics.
Writes Freeman, "except 26." He couldn't figure out what 26 was "doing"
in this series. Now, two weeks before I read this article (in the early
90's) I had heard about something called "strings." I knew exactly two
facts about strings: that they are related to modular forms, and that
dimension 26 has something to do with the matter. Excited, I actually
called up Freeman Dyson and asked him whether "strings" could be the
explanation for the strange "26" that appeared in his theorem. His
answer: "It has not escaped our attention."
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