FOM: "crises" and crises

[Harvey Friedman]

I want to briefly respond to a statement of Lou's.

>A century ago there was something of a crisis in foundations
>(although this has been exaggerated by those who write about
>foundations), and that explains why these big names got into the act.

This is the full "explanation" Lou gives for the point that Hilbert, von
Neumann, Weyl, Brouwer, and Poincare were so interested in FOM and even
wrote extensively about it.

But this is not at all why these people were interested in FOM and wrote
about it so extensively. Just as an example, without going to the library to
make a thorough search for what these people wrote about FOM, and being at
my home library, I looked in Benacceraf and Putnam, where there is a reprint
of an article "On the nature of mathematical reasoning" was was excerpted
and reprinted from Henri Poincare, Science and Hypothesis, 1902. Russell's
paradox was made known to Frege in June, 1902, and so I doubt that Poincare
was influenced by this "crisis." Also in this particular excerpt, there is
absolutely no mention of nor trace of any kind of even remote crisis;
instead, Poincare talks about the nature and importance and necessary use of
induction. Of course, he does not prove any necessary use of induction - but
no one could doubt, after reading this excerpt, that he would have been
extremely excited about various results concerning the necessary use of
induction - or, for that matter, necessary uses of logical principles in
general. 

And Lou - this is just looking around my home office within a few feet of
reading your e-mail!! Haven't yet even looked at Poincare's books - there is
also one, I think, on "Foundations of Science." Nor Weyl's. Have you?

Just to give you an idea, Lou, of how closely connected Poincare's concerns
in this little excerpt are from state of the art FOM: At the end, Poincare says 

"... that this induction is only possible if the same operation can be
repeated indefinitely. That is why the theory of chess can never become a
science, for the different moves of the same piece are limited and do not
resemble each other."

In state of the art FOM, "chess cannot ever be a science" roughly
corresponds to questions about the practical algorithmic undecidability of
games, and various projects about the construction of simple games for
which, demonstrably, no formal determination can be made of who wins within
ZFC using a reasonable number of steps in the proof. Poincare - where are
you so we can tell you about modern FOM?

And then there's von Neumann. The later part of his intellectual life was
spent on various foundational topics, including FOM. Again, crisis was not
the guiding principle of his foundational work in computing, reliability of
components, self reproducing automata, game theory, etcetera. There is no
doubt from his writings and from people who knew him that he would have been
extremely fascinated by modern work in FOM.

Now as for crisis. For the working mathematician, there never was a crisis
since the time epsilon-deltas were put properly in analysis, and the
function concept clarified. (The Russell paradox was not a crisis for the
working mathematician, who simply ignored it.) After that, there has been no
practical issue as to whether a given proof is valid, or whether certain
principles are to be accepted. The closest thing to a crisis for the working
mathematician since the epsilon-deltas were straightened out and the
function concept clarified is probably Cohen's work in 1962 - but even that
it not really a crisis for various reasons I will get into later for the
working mathematician.

So the motiviation of Hilbert, Brouwer, von Neumann, Poincare, Weyl, etc.,
was definitely not crisis, but rather a genuine and deep interest in
foundational issues very similar to state of the art FOM. As much as you
wish to avoid that conclusion, you will be forced to admit it. And their
concerns are still very much alive today.  

Now, let's really talk about crisis. The program I outlined in my last
really big e-mail promises to create a genuine crisis for the working
mathematician, far more than anything since last century's problems with not
having epilson-delta and not being clear about the function concept. And it
will be a *permanent* crisis - so in some sense far more significant than
even the crisis from last century. I say *permanent* because it will only be
resolved by a permanent rethinking of the fundamental nature of the whole
mathematical enterprise - the complete overhauling and rejection of the idea
that mathematics can be divorced from the most profound philosophical
questions. By comparison, the problems solved with epsion-delta and the
function concept were rather simply resolved by thinking more carefully, and
mathematics retained its aloofness from profound philosophical questions.
This won't happen in this next crisis.

Lou - I am genuinely interested in hearing more about your views on this
matter.  - HMF.