FOM: Kazhdan, Macpherson, f.o.m., 2050, 2100

[Harvey Friedman]

Reply to Steiner Fri, 14 Jan 2000 13:28 and Fri, 14 Jan 2000 13:34.

Before I begin,

a) g.i.i. = general intellectual interest;
b) are transcripts of these talks available in written form and/or in video
form?

>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.

I know the second, not the first. They are both very well known.

You wrote earlier:

>...one on
>the last 100 years in mathematics and the other on predicting the course
>of mathematics in the future (more precisely, whether it is possible to
>predict the future of mathematics).

Was Kazhdan on the 100 years and Macpherson on "predicting the future
course of math."? Looking at your account, I guess this was the case.

>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).

Nice to see a mathematical logician in such a position. Rabin also held
that position.

[One of the
>shapers of our mathematics department was Fraenkel (of ZF), thus logic
>has great prestige here.

I'm not sure that "thus" is appropriate here, but am aware of the
conclusion. It is in sharp contrast with the situation in the U.S.

>...the turnout for this conference was very large, filling a big chemistry
>lecture hall, ...

>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.

My own view is that Kazhdan is correct about existing mathematical
questions today. But that he is not correct about mathematical questions
generally. Specifically, there will be new mathematical investigations of
great mathematical and general interest, providing striking information
that is generally understandable about objects of everyday mathematics,
which simply cannot be developed without the demonstrable use of more
axioms than ZFC. I.e., can be developed with but not without use of axioms
going well beyond ZFC. This will be accepted by everyone by 2100.

In fact, there will emerge by 2100 a general kind of extra information that
one customarily seeks in virtually EVERY mathematical situation. And that
kind of extra information frequently can be obtained with but demonstrably
not without axioms going well beyond ZFC.

I can be more specific about these conjectures. Every interesting
mathematical theorem about everyday mathematical objects, T, has a certain
logical form that can be productively analyzed. In other words, one can
find an appropriate language tailor made to that theorem, and consider all
statements of that form. There may well be only finitely many statements of
that form - that finiteness in no way trivializes the situation.

The new kind of investigation is to determine which statements in this
language associated with T are true.

For many important standard theorems about everyday mathematical objects,
this process leads to new results that determine the truth or falsity of
all such related statements - but only if one uses new axioms that go well
beyond ZFC.

This cannot be done by routine construction of languages associated with
important standard theorems. A routine construction will lead to languages
with infinitely many statements where the set of true sentences is easily
seen to be not even recursively enumerable - so no new finite number of
axioms (that are consistent) could suffice to decide the set of true
sentences.

More specifically, here's how I think things will go. Given T, the first
languages associated with T for which the sentences can be solved in this
sense will be a somewhat weak, and easily handled without new axioms. But
then for any given T, one can persistently strengthen the languages and get
deeper and deeper analyses of the sentences, which require more and more
axioms.

We are still some distance from finding a single convincing example of
this, but some prototypes of this phenomena are being developed now.

> 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 am working with a number theorist and a computer science student now, and
we already know that all Diophantine problems of length at most 14 can be
solved in PA. We are well on our way to showing such results for all
formulas in PA of length at most 14. A full report will be made on the FOM
when this work matures.

Since you are a friend of Kazhdan, it would be interesting to see just what
thought was behind his answer "no."

>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.

Did the speakers touch on this topic?

>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.

What did Kazhdan say about this?

>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).

How did it contradict Macpherson's view?

>	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.

Well, apparently he did think that there was a good twentieth century
exmaple suitable for a broad audience - Godel's theorem(s). So he is
implicitly saying that Godel's theorem is unique in this respect?

>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.

Had no choice? What does that mean? Is he saying that if he is to give a
talk to a braod audience about mathematics in the last 100 years, then he
has no choice but to talk about philosophy and history of mathematics? A
most remarkable statement.

>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 am willing to agree that the great g.i.i. of foundations of mathematics
is, to some extent, dependent on the great g.i.i. of mathematics itself.
But what mathematics? There is no doubt that there is a substantial amount
of mathematics that is of great g.i.i. In fact, there is more than enough
mathematics of great g.i.i. before 1900 to support the great g.i.i. of
foundations of mathematics.

>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.

I would very much appreciate it if you could ask Kazhdan what exactly he
means by his "hope that mathematics would return to its roots in
philosophy." You may not be 100% sure of what that means, but I am
certainly not even 1% sure of what that means.

>	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."

This statement can be reconciled with views that look like they are
directly opposed. The real content of what he is saying is almost certainly
that there is an important distinction to be made between what has been
proved and what has been heuristically or probabilistically proved, and
that this distinction will always be of vital importance to mathematics.

Macpherson could get into trouble if he were to state that researchers
constructing heuristic or probabilistic proofs are crackpots. I know him
well enough to seriously doubt that he would put himself into that
position.

>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.

I tend to agree with Macpherson that the vital distinction between
different kinds of "proof" will remain of great importance - if that's what
Macpherson said!

> 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.)

I heard protests about giving the medal to Witten. But, alas, the Fields
Medal committee is a closed secret committee that does not have to explain
what they are doing and why. I wrote some time ago on the FOM that this is
bad for mathematics. It's very good for certain kinds of mathematicians,
but bad for the development of mathematics. The process should be opened up
with a moderated website in which, within limits, anybody can discuss what
they think of anybody. And also criticize what they think of what others
think of people. If this was done properly and orderly, there would not
even be a need for awarding the prize at all!!

See, the internet can change just about everything, including the most
venerable of institutions.

>	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.

Did Macpherson specifically comment in depth on Bourbaki?

>	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.

This is not surprising from one point of view: core mathematicians think
almost exclusively in terms of core mathematics then thinking about
mathematics.

Actually, in a way, I agree with them. At least in the following sense. I
don't think of f.o.m. as a branch of mathematics. It is a branch of
foundational studies. Of course, the practical problem comes when one
wishes to find a proper place for f.o.m. in contemporary University life.

>	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).

The mathematical community simply does not regard f.o.m. as a branch of
mathematics - which, in a sense, I kind of agree with. But here is where I
part company - they generally think that their research obligation to
University life is limited to supporting research in branches of
mathematics. They generally don't think that their research obligation to
University life extends to supporting other forms of mathematical research.

That is a generous interpretation. The less generous interpretation is that
they have a special aversion to f.o.m. and mathematical logic. That is
undoubtedly the case with many extremely influential core mathematicians.
Kazhdan and Macpherson? I don't know.

>	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).

And I repeat what I said earlier.

I am willing to agree that the great g.i.i. of foundations of mathematics
is, to some extent, dependent on the great g.i.i. of mathematics itself.
But what mathematics? There is no doubt that there is a substantial amount
of mathematics that is of great g.i.i. In fact, there is more than enough
mathematics of great g.i.i. before 1900 to support the great g.i.i. of
foundations of mathematics.

But, as I said elsewhere, f.o.m. also draws its great g.i.i. from its
seminal role in the development of foundational studies - a subject that
barely exists yet (smile).

>	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.

Which leads to the question of why these two very well known mathematicians
chose not to do it in the forum of this meeting.

>	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.)

Which leads nicely into my own view of mathematics/g.i.i.

Firstly, I believe that large portions of core mathematics would have
considerable g.i.i. -- if reworked and re-exposited along the lines of what
I call "foundational exposition." The usual way of presenting core
mathematics - although generally very efficient - simply does not make it
widely intelligible, or give it the general appearance of being important
science.

Of course, such a foundational exposition of core mathematics would
undoubtedly involve very substantial rearrangements of and rethinking of
the material, with different emphases. It would necessarily involve the
proving of new theorems and new kinds of theorems that would not naturally
fit into conventional approaches.