FOM: Kazhdan, f.o.m.

[Mark Steiner]

Harvey Friedman wrote:
>
> Reply to Steiner 5:33PM 1/17/00. I think the discussion of Kazhdan's talk
> is winding down somewhat, and we are moving towards more general issues. I
> intend to listen to their talks on the web in the near future. We've got to
> get them to fix the link to Macpherson's talk.

So with your permission, I'll focus on the most important of the issues
that I have time for right now.

> >The latter.  He regards mathematical intuition as a developing
> >phenomenona (this was part of the lecture).
>
> Can I surmise that Kazhdan is unfamiliar with work in this direction?

	In philosophy?  What direction?
> >University's bad sense in hiring Fraenkel, while throwing Issai Schur to
> >the dogs.
>
> More evidence of the core mathematics community refusal to acknowledge
> their appropriate share of the responsibility for supporting research in
> f.o.m. and mathematical logic.

	Agreed, but to be honest, Fraenkel had a lot to do with throwing Schur
to the dogs.  His work has great general intellectual interest, even if
you restrict attention only to "Schur's Lemma".  See Sternberg, Group
Theory and Physics.  You could say the whole universe works according to
this lemma (if you believe quantum mechanics).  And I think a
philosopher could explain what this means to a general audience.
>
> >...Meir Buzaglo, who has just
> >finished a book on the subject of extending functions and concepts in
> >mathematics, using model theoretic techniques.
>
> I would like to hear more about this. Do you have any more info and/or
> references?
>

Yes, I'll get back with some more information.  His book has been
submitted to various publishers.  He distinguishes between "strong
forcing" and "forcing."  The former means that the uniqueness of the
extension is invariant over many choices of what axioms to preserve in
the extension.  He points out that the concept of "equinumerous" could
have been defined differently in such a way as to be equivalent to the
usual definition over the natural numbers, but such that there are no
noncountable cardinals (that's one way to solve CH!!).  Thus the
extension of this concept is not strongly forced.  Maybe I can get
Buzaglo to speak for himself.

> >       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.
>
> I don't think that my strong views about the logical structure of
> mathematics is mainly based on 20th century mathematics. Of course, it is
> true that mathematical practice conforms (in the appropriate sense) to this
> structure only in 20th and (up to now) the 21st century.
>
> >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."
>
> Not easy to answer. E.g., don't you think that "addition on the integers is
> commutative" is projectible? Or is it simply a temporary view of 20th
> century thinkers that is subject to radical change in the future?

	Believe it or not Frege discusses related questions.  He seems to argue
that his version of f.o.m. renders this a projectible hypothesis, or
more exactly, that without f.o.m. (meaning in his case the logicist
program) it is not projectible, since each number is unique.  This is
different, he says, from the situation in geometry, where the points of
space are indistinguishable.
>
> >>...Many people think that mathematics is about mathematical
> >> objects.
>
> >       I don't understand your answer.
>
> Do you mean that you don't understand what a mathematical object is? I can
> give you some examples to help.

The exchange went as follows:

> >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 what your answer has to do with my question.


>
> >>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.
>
> But I think that the key point is "related."
>
> I think you are referring to the possibility that an important concept of
> rigorous proof may emerge that involves using a physical theory that
> relates some measurable physical quantity with standard purely mathematical
> objects. Then the outcome of some experiment may rigorously imply that some
> mathematical statement is true - rigorously modulo the physical theory.
>
> But there are many weak links here which will always make this line of
> reasoning sharply distinct to the usual notion of rigorous proof in
> standard mathematics.
>
> Of course, there is a very special case of this, which is extremely
> distinctive, and which is now happening. And that is when the "physical
> system" is "merely" a digital computer running under a program. I claim
> that this case is quite different, and it is a defensible position to say
> that one can achieve absolute rigor this way. However, defending (or
> attacking) that position in a thoroughly convincing way is quite
> challenging -- and extremely interesting.
>
> I also believe (half know) that there are important setups where
> probabilistic reasoning is not merely heuristic, but has its own
> indisputable inherent absolute rigor. This rigor is demonstrably different
> from the usual absolute rigor. But there is an absolute rigor to it
> nontheless - in certain setups.
>

I agree with all this, but I had a different case in mind.  In studying
the application of mathematics to physics, spending lots of time talking
to physicists, it seemed that the manipulation of symbols which they do
is not only not rigorous, but they have no really persuasive argument
that the numbers they get should be correct (except that "it works",
which of course is a projection, cf. Goodman again).  E.g., there are
problems which can't be set up mathematically in three dimensions, so
they solve it in four dimensions, then analytically continue the
"solution" down to three dimensions, as if the problem where an analytic
function of dimension (of a magnet!).  The manipulation of symbols,
however, is behaviorially analogous to cases where there really is a
mathematical object (a Lagrangian, for example) to define, e.g. in the
nonrelativistic cases, where you can perform rigorous mathematical
reasoning.  On the other hand, there are special cases where you can
actually describe the system (say low or high energy) by a mathematical
object.  Then the fact that these objects describe a single system at
different energies might suggest (perhaps by the same mumbo jumbo that
isn't even what you would call "pre mathematics" but does answer to some
kind of physical intuition) some mathematical relation between the two
objects, special cases of which relation might be verified rigorously.
I think something like this is what Kazhdan had in mind, but he would
say it, of course, much better, since I think he has something specific
in mind.

> Close to W? Whoops! Tell me more. I have been trying to get a discussion
> going about W for a long time on the FOM. Isn't there a W cult? What's that
> cult all about?

	Maybe I'll start a discussion on W later.  For now, I'll just say that
you can translate his aversion to set theory into your language by
remarking that in his opinion set theory had no general interest.  No
application to physics nor to any branch of mathematics that has
application to physics. You are saying that in this he was right.  You
would disagree with his more general animus to set theory, because for
you ZFC is a handy way to codify mathematical reasoning, even if core
mathematicians don't use it.  It is useful in that an independence
result concerning ZFC means immediately that mathematicians using their
ordinary means cannot solve real problems.  There is no question (in my
mind) that W got Goedel's theorem seriously wrong, which prevented him
from seeing the above.  But my friend Juliet Floyd (BU) thinks he did
understand Goedel's theorem and would have appreciated your work
(projected?) on ZFC also.  And I agree with her that he SHOULD have
appreciated your work, given his GENERAL positions, so that in some
sense ideological blinders prevented him from understanding the logic of
his own position.  But perhaps I'll write more on this later, if there
is any interest in the community.

	I suppose there is a W cult, or more than one, but I'm not in it.
That's why I am hesitating about writing a book about W.  On the other
hand, there is a cult of W bashers, particularly concerning his
philosophy of mathematics.  E.g. Saunders MacLane bashes W by the banal
remark that W only discusses third grade arithmetic (not true, by the
way). Later, he asserts that there is no such thing as mathematical
knowlege--yet that was exactly W's conclusion!