FOM: Kazhdan, f.o.m.
Mark Steiner
marksa at vms.huji.ac.il
Mon Jan 17 10:33:11 EST 2000
> Incidentally, you earlier said that one of the two misstated Godel's
> theorem(s). Which one was that, and what was the error?
>
I would like to withdraw this criticism. As I said earlier, Macpherson
stated that proof is a criterion for the truth of a mathematical
statement. I regarded this as evidence of misunderstanding of Goedel's
theorem, particularly if you regard the Theorem is philosophically
relevant (as Macpherson does) but it is not a misstatement. Since I
can't hear the lecture again, I can't ascertain whether he stated the
theorem correctly or not--for example, whether he prefaced the theorem,
as Kazhdan did, with the caveat that we are talking about consistent
systems.
> "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?
Definitely. His view on your program of proving that "short" theorems
can be decided is that long statements can be shortened by definitions
(accompanied by intuition), and thus "short" statements are subject to
Goedel's theorem. So your program would not shed light on what is
bothering him. (I am reporting a conversation, not endorsing the view.
As I said, I did not mention your name in this conversation, for fear of
misrepresentation. However, I did send him the issue of f.o.m. with one
of your propositions concerning length.
> 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?
The latter. He regards mathematical intuition as a developing
phenomenona (this was part of the lecture).
> >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?
I don't know the answer to any of these questions except the last: I
think not. As you have pointed out, core mathematicians ignore ZFC,
though usually force their students to take some set theory. In fact, a
well known core mathematician once berated ME for the Hebrew
University's bad sense in hiring Fraenkel, while throwing Issai Schur to
the dogs.
>
> >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?
Of course, the sixty-four thousand dollar word is is "can't"--a very
difficult word in mathematics, to say nothing of philosophy of
mathematics. Probably what you have in mind is that an f.o.m. expert
might construct a counterexample to the implied general rule that
Kazhdan believes in. However, I suspect the latter possibility, or
something a little more general, since I don't know whether "experts in
number theory" is the right term, given the thesis of the Unity of
Mathematics.
>
> 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?
I think it fits in with the part of the talk I didn't refer to, namely
about mathematical intuition, which of course Goedel refers to in his
paper, "What is Cantor's Continuum Hypothesis."
>
> 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.
>
I agree strongly with this, and add: there is also the question how is
it that mathematical concepts as such are so fruitful. Part of this
"fruitfulness" has to do with the fact that mathematicians tend [for
some reason] to choose concepts whose extensions to new domains are
"forced". (E.g., the exponential was not originally a function at all,
but a syntactic abbreviation for repeating the variable. The notation
suggested the idea that it could be understood as a function of two
variables, and it then turned out that rational, real, and even complex
exponentials could not be defined arbitrarily.) I believe it is a fact
that all the mathematically interesting functions on the reals turned
out to be analytic on almost all the complex plane, when the notion of
analytic continuation was invented/discovered (choose). This despite
the fact that we could easily define counterexamples and do. The notion
of "forcing" is that of my colleague, Meir Buzaglo, who has just
finished a book on the subject of extending functions and concepts in
mathematics, using model theoretic techniques.
> >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'll ask him.
>
> >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.
>
The complexity of this discussion is growing, but I'd like to point out
the remarkable nature of Harvey's expectation, which is based on
twentieth century experience in mathematics. But as Nelson Goodman and
others long ago pointed out, not every projection based on experience is
valid. For example, from 1840 on, every president of the US elected in
a year divisible by 20, died when in office. This happened seven times,
but nobody would predict on this, except superstitious people. We don't
regard the instances as "like" one another. Harvey's prediction is
based on the usual view of mathematicians that "mathematics" is what is
called by Goodman a "projectible predicate", and also that the
subcategories of "mathematics" are projectible. It would be interesting
to know the basis of this view--of course, we'd get into the question
again, "What Is Mathematics." I think Kazhdan's predictions (e.g.
solvability of conjectures) are also inductive based on strong views
about the "natural" nature of mathematical concepts.
> >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.
I don't understand your answer.
>By the way, what exactly does "mediated" mean here?
Suppose a physical system at high energies is related to a definable
mathematical object A and at low energies is related to definable
mathematical object B, but the description of the whole system cannot be
rigorized--on the other hand using standard physical reasoning we could
derive relations between A and B. Then I'd say that the system has
mediated. I'm not talking about computer modeling.
>
> >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.
Then you claim that the calculus wasn't mathematics till the 19th
century. That can't be right. And there is no relevant difference
between the cases. The kind of reasoning that goes on in physics today
is exactly like the case of the delta function and exactly like the case
of 17th and 18th century mathematics--the question is how to do calculus
on an infinite dimensional space.
I think, however, I didn't understand you here.
>
> 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.
>
Harvey (or anybody) what's the difference between f.o.m. as you
understand and what used to be called "proof 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.).
I had heard a rumor at the time that Cohen wanted to show that "anybody"
could do f.o.m. if they put their mind to it. Is there any truth to
this rumor? (For the record, I hope Cohen didn't say anything like
this. I believe strongly in the integrity of various intellectual
disciplines, and don't like it when scientists say that "anyone can do
philosophy.")
> > 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?
Of course.
>
> 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.
>
Isn't it the case that set theory originally made contributions to
functional analysis? Didn't Cantor introduce it for that purpose? If
the answer is "no", are you committed to the view that set theory is not
mathematics? (By the way, you are extremely close to Wittgenstein's
point of view on these matters.)
> 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?
I'm sure this is the case.
> 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.
It's amazing what mathematicians who think they're in one branch have to
learn. My friend, Sylvain Cappell, a topologist, has gotten involved in
combinatorics, class field theory, analysis, you name it (I can't).
He tells me that some of the stuff he has discovered accidentally is
patentable. (So he didn't tell me what it was.)
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