FOM: Kazhdan, Macpherson, f.o.m., 2050, 2100
Mark Steiner
marksa at vms.huji.ac.il
Sat Jan 15 14:05:12 EST 2000
I'll add some more information about the Conference.
I checked, and you can hear the lectures of the Millenium conference at
http://hug.phys.huji.ac.il/winterschool
I'm sure David Gross' lecture (which I didn't hear originally) will be
worth hearing.
I was not able to listen to Macpherson, through some technical fault in
the website, but Kazhdan is playing right now. I might be able to
supply more detail, or correct previous misstatements I made.
He claims that mathematics is a kind of thinking that cannot be captured
by formal logic. This is why mathematics has escaped the threat of
Goedel's theorem.
In particular, this has to do with the tendency of mathematicians to
seek generality. He mentioned Fermat's conjecture, which he and others
thought was probably unprovable, because the theorem for each n was a
different theorem. Once the conjecture was seen to be an easy corollary
of much more general hypotheses, it was immediately clear that it was
provable.
The reason that Kazhdan mentions Goedel's theorem is that it has
relevance to the question "Why is mathematics possible", i.e. to the
history and philosophy of mathematics. It is not typical,in its
content, of the rest of mathematics.
Kazhdan's talk is not "about" the philosophy of mathematics, it
expresses his own philosophy of mathematics, with occasional references
to Kant, some of which are not correct.
I had written:
>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).
The contradiction is that this relationship could change the status of
mathematical proof. After all, the status of mathematical proof as a
sine qua non of publication is relatively recent. Euler asserted that
the sum of the reciprocals of the squares IS pi^2/6 by a number of
arguments of "inductive" and "analogical" nature that he knew were not
proofs. (See my discussion of Polya on this matter in my *Mathematical
Knowledge*) Macpherson was talking not so much about mathematicians who
do probabilistic or "heuristic" mathematics, but of those who claim that
this kind of mathematics might partly displace the other kind (I'm
giving him the benefit of the doubt). Suppose the example cited by
Kazhdan, about the connection between mathematical theorems being
mediated by physics, in the absence of anything better remains.
Couldn't this lead to a changing of the role of proof? Granted that
mathematicians have succeeded in rigorizing such ideas as the Dirac
delta function; there is no a priori argument to say that they will
always be successful.
A final point about the role of f.o.m. as mathematics itself, where even
Harvey seems to buy the core mathematician's bias against f.o.m.
Actually, they are biased against all formal logic. I don't know
whether Harvey considers model theory f.o.m. or not, but it looks pretty
foundational to me, as an outsider. I believe that the original
motivation of that field was partly to explain mathematical phenomena in
ways that core mathematicians couldn't, because they don't pay attention
to the language of mathematics itself. Thus, the model theorist can
show that many different theorems of algebra are really special cases of
a theorem that the core mathematician can't even state. (I used to work
on the concept of explanation in mathematics and went through Chang and
Keisler to find examples) If I understand Harvey's program of generating
theorems along a gradient that can't be stated even by logicians today
and certainly not by mathematicians, there is no reason why such a
program should not be called "mathematics."
Aside from all this, model theory has produced some solid contributions
relevant to the decidability of actual mathematical conjectures (my
colleague Shelah has done work like this, as I don't have to tell the
readers). Since Kazhdan regards a result of undecidability as
mathematics, as long as the proposition itself is mathematical, there is
no reason not to consider Shelah for the Field's medal just because he
works in model theory.
I'd be interested in the actual contributions, to date, if any, of set
theory, meaning ZFC, to core mathematics, since the latter have a bias
against set theory also, so far as I can see. Do Harvey's remarks that
mathematicians have no awareness of ZFC as a foundation mean that there
are few such applications to date?
As for the relation between mathematics and philosophy, and Kazhdan's
views, maybe I'll just let you listen to the tape. (You need Real
whatever to hear it.)
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