FOM: Kazhdan, f.o.m.

[Harvey Friedman]

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.

>...His [Kazhdan's] 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.

A part of the relevant program of mine is to show that every short sentence
*in primitive notation* in certain principal formal systems can be
decidable in those formal systems. An extreme extrapolation of this is to
prove such a result with abbreviations allowed - incomparably more
difficult, but perhaps not entirely impossible. There are other aspects of
this program concerning the canonical nature of certain principal formal
systems.

When I formulated this program, I did not really have the expectation of
entirely getting around the Godel incompleteness phenomena by saying
something like "well, all short statements in the language of ZFC with
abbreviation power (I call this practical ZFC) can be decided in ZFC."

I want to remind readers of this exchange that you brought this matter of
the decidability of short statements up with Kazhdan - while I was safely
in Ohio.

Of course, this program is valuable even if it does not entirely get around
the Godel incompleteness phenomena. In fact, I don't think for a moment
that one can get entirely get around the Godel incompleteness phenomena in
the sense that, perhaps, Kazhdan would like to.

But I think that something important has to be added to what you say that
Kazhdan said. The fact that long statements can be shortened by definitions
(abbreviation power) is not relevant here unless the hierarchy of
definitions is also short.

This leads to the following question. Take, say, PA or ZFC. Find an
impressively small presentation of a statement that cannot be decided,
where that presentation allows abbreviations. Just how small?

>> 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).

Can I surmise that Kazhdan is unfamiliar with work in this direction?

>>...By the way, was
>> ZFC mentioned in either talk as the preferred vehicle for the foundations
>> of mathematics?

>...  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.

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.

>...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?

>	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?

>I think Kazhdan's predictions (e.g.
>solvability of conjectures) are also inductive based on strong views
>about the "natural" nature of mathematical concepts.

What are these predications actually? Does Kazhdan predict that any
interesting mathematical conjecture can be resolved within the ZFC axioms?
We know more and more counterexamples already, and the range of
counterexamples is increasing, and the quality of those counterexamples is
improving. The counterexamples will be incomparably more convincing as the
century proceeds, and eventually it will become everyday phenomena.

Of course, any informed discussion of this must take into account the range
of counterexamples that exist at the moment. I am not sure what the Kazhdan
prediction really is.

>>...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.

>>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 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.

I claim that the appropriate notion now is "rigorous mathematics." And that
was not the appropriate notion then, but it is - quite appropriately - now.
Most of the mathematics that I think about is not rigorous - but when I
communicate results and write manuscripts, then it is rigorous. I use funny
images and dots and sketches on pads of papers, and mostly just
incommunicable things in my head. This is most of the work. But I require
that it gets in rigorous form before I call it a result.

Certainly Newton and Leibniz and many other greats could not get a full
time tenured position in any Y2K math dept on the basis of what they wrote
- maybe partly in physics/philosophy but not totally in math. And this
despite the fact that they invented a pillar of core mathematics. I know
that this doesn't make real sense since any Y2K math dept is so largely
based on the outgrowths of what they did. But imagine that only discrete
mathematics was developed, and with the current standards of rigor. Then
calculus comes about in the form of Newton/Leibniz. People would say - gee,
these people look like they have absolutely brilliant, seminal, fundamental
ideas with all kinds of applications. But the ideas are too ill formed, too
vague, too incoherent for us to accept them as professional mathematicians.
We have to wait until the ideas are in a proper form.

>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.

Not in a math dept. Logical clarity is an essential feature in mathematics.

OK, I will admit that I can imagine a strain of mathematical investigations
that do not neatly fit this model of mathematical practice. But that will
be based on some alternative notion of rigor that is regarded as totally
distinct. For example, mathematicians have been hired who do computational
work with no theorems. But that is a focused enterprise with its own rules
and standards. They are not "doing mathematics" but rather "engaging in
computational investigations." In Math depts, there is always the essential
element of absolute logical clarity - far higher than what is normal in
other Depts.

>Harvey (or anybody) what's the difference between f.o.m. as you
>understand and what used to be called "proof theory"?

Time for Simpson to answer this, since he has been so quiet.

>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 don't remember this rumor, which may predate my entering the profession
(1967). There are plenty of opinions attributed to Cohen that are very
negative about mathematical logic and f.o.m. I don't recall any opinions
attributed to Cohen that are positive about mathematical logic and/or
f.o.m.

>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.)

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?

Cantor originally presented basic set theoretic ideas in connection with
problems in Fourier series (not functional analysis). E.g., look at
Dauben's book on Cantor.

Set theory is a mathematical subject. Aspects of elementary set theory are
implicitly and explicitly used by core mathematicians in their work. I am
always careful to distinguish between

mathematics

and

mathematical subject.

I infrequently use the word "mathematics." I more often use the phrase
"core mathematics."

I think that I have a better idea of what core mathematics and mathematical
subjects are than what mathematics is.