FOM: General intellectual interest/challenges

[Harvey Friedman]

Lou writes:

>Well, P=NP is certainly highly interesting as a mathematical problem,
>and so is the abc-conjecture, and scores of other big problems,
>like the Lang conjectures.

It is easy to state clearly and concisely what the general intellectual
interest of P=NP is and its importance for lots of diverse contexts way
beyond mathematics. Furthermore, this can easily be done in such a way that
people from the following disciplines can readily grasp and relate to it,
and see its intellectual and other importance:

number theory, combinatorics, geometry, scientific philosophy, statistics,
finance, computer security, complex systems, complexity theory, database
theory, algorithm design, cryptography, logic, artificial intelligence,
robotics, networking, hardware design, control theory, operations research,
neural nets, quantum computing, expert systems, code verification, proof
checking, statistical mechanics, linguistics, dynamical systems, signal
processing, etcetera; and many, many more.

I repeat: it's not just that people from these diverse areas would grasp
the problem. It's that they would grasp its significance - although they
would be convinced that they pretty much know the answer. It would not look
like a frivolous puzzle to them.

Can you do this for the abc-conjecture and the Lang conjectures?

P=NP is something that I would not quite regard as mainstream FOM, although
it is clearly close in spirit, or at least far closer in spirit and actual
connection to FOM than any part of mathematics. It is definitely something
that emerges very very quickly in foundational studies. This is extremely
rare for a yes/no mathematical problem. I will save the big guns from
mainstream FOM in reserve for later.

> I am not inclined to compare them as to
>general intellectual interest.

I am, because otherwise you get to cite mathematical tradition without any
justification, in order to minimize the unique place of FOM in the history
of ideas with impunity. I use "general intellectual interest" to jolt you
into seeing that at least there is some sort of difference. Then maybe you
will cast the difference in your own terms, and we can proceed from there.

>Who, 200 years ago, would have
>predicted the central role of elliptic curves in so much that's
>going on? Or zeta functions? (Well, Euler had some inkling, presumably.)

Ah, the old needle in the haystack justification. It goes like this: "Since
no one can tell what is important in the long run, anything might be as
good as anything else. So we should just follow our instincts. Nothing
gained by being really critical. Let a thousand flowers bloom! Glory to our
instincts!! Those who waiver can't do!!!"

Well, this argument looks much less impressive when one injects a little
bit of probability theory in here. If you have a method by which you can
tell gigantic differences in probabilities of importance, you can't and
won't fail to use it. The real question is: can the method be taught?

>If these things are not of general intellectual interest, so much the
>worse for "general intellectual interests".

I have never met anybody yet who claims that eliptic curves and zeta
functions are of general intellectual interest. Of the people who care,
approximately 99.9.. percent are pure mathematicians (yes, they have had
uses - likely temporary - in some complexity theory situations). You don't
want to compare P=NP with eliptic curves and zeta functions once you step
out of the comfort of your math building - or do you?

>Of course, this is not
>to say I do not highly value good expositions on these things, and
>improvement in this direction is very desirable. But I don't see
>any reason for mathematicians to aim for approval by, for example,
>the big media, if this would subvert the intellectual standards
>we should be keeping up.

I would appreciate what you said if I thought you could delineate in a
convincing way any kind of fundamental underlying intellectual standards.
We know that mathematicians are addicted to complex and intricate
hierarchical strucutures, in which one can effectively use large complex
machineries - which have been built up over years. I have on several
occassions seen mathematicians, after a well known problem is solved, turn
a deaf ear because the solution did not employ big machinery. Which makes
me think that the mathematicians are not all that interested in
information, but rather in the process - one of the hallmarks of art.

What is unclear is to what extent mathematicians are driven by any wider
intellectual purposes, other than this special kind of process. To real
outsiders - and I am not really quite an outsider - it has all the
appearance of an intricate yet aimless art, which is admittedly very
"precious." Yet horrifically imbred, and very very snooty. To real
outsiders, it is supremely impenetrable. And when it is made to look
penetrable - on the surface - it merely looks like challenging puzzles
(coloring maps, FLT, etc.) - with the flavor of chess. A kind of climbing
of Mt. Everest in an age of airplanes.

I'll stop here for no good reason. One interesting question: suppose P=NP
is solved as an afterthought by people working on eliptic curves and zeta
functions? How would that affect my position? The answer is: not at all,
although given the way I am saying all of this, I could look real bad. Lou
- are you going to use your knowledge and experience with eliptic curves
and the zeta function to go try and solve P=NP? Or are you going to leave
it for the rest of us on the fom?