FOM: Kazhdan, f.o.m.

[Harvey Friedman]

Reply to Sat, 15 Jan 2000 21:05 and 1/15/00 12:14.

The questions that I raise here about what Kazhdan said and/or meant may
not be answerable by just listening to his lecture. I noticed that you said
you knew him personally. I, and probably much of the FOM subscription list,
would be interested in having you contact him about his views at least to
the extent needed for clarification.

Incidentally, you earlier said that one of the two misstated Godel's
theorem(s). Which one was that, and what was the error?

>I checked, and you can hear the lectures of the Millenium conference at
>http://hug.phys.huji.ac.il/winterschool

I have written to my system administrator to set my Mac computers up so
that I can hear these lectures this coming week. In the meantime, I was
very curious about the further accounts that you have given us about
Kazhdan's talk.

>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.

I hope you have written to the appropriate people about this, so that they
know they must fix it.

>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.

"Cannot be captured by formal logic" has many different meanings, some of
which have something to do with "the threat of Godel's theorem" and others
have nothing to do with "the threat of Godel's theorem."

Does Kazhdan acknowledge that mathematics can be captured by formal logic
sufficiently clearly so as to establish the impossibility of deciding
particular mathematical statements by currently accepted mathematical
methods?

By "the threat of Godel's theorem" does he mean:

the threat that some statements mathematicians want to decide cannot be
decided in any mathematically legitimate way whatsoever?

or does he mean:

the threat that some statements mathematicians want to decide cannot be
decided by currently accepted mathematical methods?

>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.

That is a very remarkable thought on their part. Unprovable by what
methods? And was Kazhdan aware of just how recently we have been able to
obtain even a single really intelligible example of a statement in discrete
mathematics that is undecided in a strong system like ZFC? By the way, was
ZFC mentioned in either talk as the preferred vehicle for the foundations
of mathematics?

>Once the conjecture was seen to be an easy corollary
>of much more general hypotheses, it was immediately clear that it was
>provable.

Did Kazhdan explain why this would make any difference? Can't the more
general hypotheses still lead to a theorem whose known proofs at the time
are quite different for different choices of parameters? Or is this a very
murky situation that can only be properly perceived by experts in number
theory?

>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.

It is obviously true that it is not typical of the rest of mathematics in
its content. And I can see how Kazhdan can view Godel's theorem as relevant
to "why is mathematics possible?" Namely, Godel's theorem may suggest to
Kazhdan the possibility that mathematics may run up against undecidable
statements in everyday work, thereby getting stuck for reasons that are
"logical" rather than "mathematical."

But did he consider the other aspect of Godel's work - namely the
possibility that Godel himself suggested, that one can continually expand
the accepted axioms for mathematics in a nonending yet objective way, where
new mathematics becomes possible as one uses stronger and stronger axioms?
How does that Godelian idea fit into Kazhdan's talk?

In my own view, there are a lot of other important issues with regard to
"why is mathematics possible?" than the connection with Godel's
incompleteness theorems. For example, how is it that we can get away with
writing at most semi-formal proofs and not formal proofs, and have so much
agreement as to correctness? How is it that so few fundamental principles -
such as those codified by ZFC - go so far? The second question is of course
closely connected with some of what Kazhdan was saying.

>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.

Is Kazhdan interested in explaining and discussing his philosophy of
mathematics with f.o.m. or p.o.m. professionals, such as in this FOM e-mail
list? Should be invite him to join and discuss his views here? Would he be
interested in doing so?

>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*)

As I said earlier in this exchange, I think that a very clear distinction
will always be made between what we now call completely rigorous
mathematical proofs and "inductive", "analogical", "heuristic",
'probabilistic" proofs. That it will always be regarded as a very important
distinction, and that people will always be able to generate interest in
giving proofs of the completely rigorous kind. Part of my conviction is
based on the expectation that it will so often be possible to give such
completely rigorous mathematical proofs.

Yet I could imagine a situation where the distinction is regarded as not so
important if the normal mathematical contexts almost never allow the human
race to come up with completely rigorous proofs. But this seems very
unlikely sitting here in 2000.

A more tricky situation is the exact nature and importance of the
distinction between completely rigorous  mathematical proofs that do or do
not rely on "rigorous" computer searches. As is well known, there are a
variety of issues involved in turning such things into completely rigorous
proofs in various senses, and sometimes this is not only unfeasible, but
also in some way relies on physical principles. But that is a different and
long, interesting, story for another time.

>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).

I tend to side with Macpherson, but without the benefit of actually hearing
what they said.

>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?

Not really. Many people think that mathematics is about mathematical
objects. By the way, what exactly does "mediated" mean here?

>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.

But I don't see the connection because I don't think that murky talk about
"Dirac delta function" is mathematics. The closest it is to mathematics is
some sort of informal pre-mathematical idea that needs to be mathematically
formalized - such as informal talk of "deductive reasoning."

It is some sort of informal premathematical idea that is attempting to
serve as a mathematical model of physical phenomena.

>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.

I think that this is a somewhat misleading way to put my viewpoint. I think
that a strong conception of core mathematics has emerged which in no way,
shape, or form includes anything like -- all important mathematical
subjects.

This strong conception has emerged with great force because of the
continued interrelationships between certain kinds of mathematical
investigations whose roots can be traced through the classical themes of
measurement (including counting) and geometry. This core can also be
identified by considering the mathematical notions and constructions that
are used to effectively formulate physical theories. The latter does not
include the full core - e.g., does not include algebraic number theory.

Thus in particular, in this sense, f.o.m. is not core mathematics. But here
is where I part company with the core mathematicians:

1. The evaluation of f.o.m. in terms of general intellectual interest,
intellectual depth, etcetera.
2. The responsibility of the core mathematicians to play their part in the
support of f.o.m. research.

The support for f.o.m. research is by no means exclusively the
responsibility of the core mathematicians. But it is clearly partly their
responsibility.

>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.

Model theory has its roots in what is perhaps the most red blooded true
blue f.o.m. of all - Frege's development of predicate calculus, followed by
the Godel completeness theorem.

Model theory evolved into a subject which, as practiced in mathematics
departments, is not connected with f.o.m. at all. Rather it is becoming a
mathematical tool for dealing with certain kinds of mathematical situations
in a more productive way than appears to be possible without its use. More
productive means various things, including more general results, easier
proofs, different proofs, or earlier proofs, in core or close to core
mathematics. You are absolutely right in what you say in your last
sentence, but I regard that as applications to core mathematics and not
f.o.m.

There has been a move afoot among some applied model theorists to redefine
f.o.m. so as to include applied model theory as f.o.m. and also to regard
genuine f.o.m. as f.o.m. of a moot variety that is now obsolete. This is a
compeletely wrong point of view which has to be refuted with some care. The
bottom line is that applied model theory is not dealing directly with
fundamental issues of general intellectual interest, but rather with far
more focused issues that are motivated by the current development of core
mathematics.

This is perfectly consistent with the view that applied model theory is
interesting, valuable, and important. But its interest, value, and
importance is of a very different nature than that of genuine f.o.m.

Coming back to what is perhaps your point, it is certainly true that core
mathematicians are not normally interested in applied model theory, and
they have many reasons to be interested in it that are different than
reasons to be interested in f.o.m. However, some appreciable number of core
mathematicians are in fact interested in applied theory. For instance, look
at the last talk of Fall 99 at http://www.math.utah.edu/research/colloquia/

>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)

Yes, this is one important kind of contribution of contemporary applied
model theory.

>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."

But I would rather call it "mathematical research."

There is something seriously inappropriate about regarding f.o.m. as a
branch of mathematics. That commits one to having mathematicians evaluate
f.o.m. by the usual standards mathematicians use to evaluate branches of
mathematics. But that leads to absolutely absurd results. It is entirely
inappropriate to evaluate f.o.m. by the usual standards mathematicians use
to evaluate branches of mathematics - not much more appropriate than
evaluting chemistry by the usual standards mathematicians use to evaluate
branches of mathematics.

In particular, one of the usual standards mathematicians use to evaluate
branches of mathematics is in terms of its working  interconnections with
branches of core mathematics. Under this evaluation, Godel's first and
second incompleteness theorems are not outstanding. See what I mean by
absurd results?

>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).

This comes in various forms, shapes, and sizes, and needs more elaborate
discussion, at some other time. E.g., different kinds of actual
mathematical conjectures strike core mathematicians in entirely different
ways.

>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.

A handful of people specializing in mathematical logic have, over the
years, been considered for the Fields Medal without success - including
Shelah. (Cohen was not a specialist in mathematical logic, but won it
largely for work in mathematical logic - in fact, work in f.o.m.).

As I have said many times on the FOM, we don't get the benefit of seeing
how Fields Medal committees think since they operate in secret. I have
railed against this secrecy, proposing that it be replaced by a moderated
Internet website where anyone can, within reason, discuss what the
importance of other peoples' research is, and also discuss, within reason,
other people's discussion of what the importance of other peoples' research
is. If this is handled properly, perhaps there is no need for awarding the
Fields Medal at all.

The present setup is very good for some mathematicians, but bad for
mathematics.

>	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.

Since in some well known sense, all mathematics can be formalized in ZFC
(but see recent discussion on the FOM list regarding more refined senses),
all of core mathematics is an actual contribution of ZFC. But that can't be
what you are asking. Are you asking for peculiarly set theoretic methods?
Perhaps core mathematical theorems that require use of substantial portions
of ZFC?

Under a reasonably strict interpretation of core mathematics, a case can be
made that there aren't any -- yet. There is the question of how this comes
out when one adopts a very liberal interpreatation of core mathematics,
with regard to some things that lie on the periphery. This has to be
discussed in detail elsewhere.

>Do Harvey's remarks that
>mathematicians have no awareness of ZFC as a foundation mean that there
>are few such applications to date?

Not exactly. We need to clarify just what you are looking for.

>	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.)

We can still ask what you think of Kazhdan's views as to the relation
between mathematics and philosophy.

>Kazhdan touches tangentially on the question "What is mathematics" in
>his lecture, but in different ways than a f.o.m. specialist might.  The
>latter might tend to be interested in mathematical reasoning, and how it
>differs formally from other kinds.

>Kazhdan is interested in the subject
>matter of mathematics--what makes a concept mathematical, rather than,
>say, a game (like chess).

That spills over into the question of "how is mathematics evaluated?" or
"what is important mathematics?" or "what is the point of mathematics?" or
"what is mathematics trying to accomplish?"  I have seen very little
serious comments from core mathematicians on such questions.

>He doesn't deal with that question, but with
>a related one, why is mathematics one subject.  [I have always been
>interested in the reason that geometry and number theory were regarded
>as two parts of the same subject, and the historical question when did
>the same term "mathematics" start being used for both subjects.]

>Kazhdan says that there was no reason to believe that mathematics might
>not break up into different subjects in the twentieth century, but in
>fact this did not happen, for reasons he does not pretend to understand,
>but he thinks it's the same phenomenon that Wigner points to in his
>famous article about the relation of mathematics to physics.

I don't quite understand what the criteria are for seeing that one has one
subject instaed of two subjects. Is Kazhdan
referring to the fact that there are all of these working interconnections?
That when the working interconnections become minimal, then we have a
breakup into two or more subjects? Are branches of mathematics separate
subjects? I guess the idea is that that branches of mathematics are not
separate subjects.