FOM: NYC logic conference/Wide Perspective

[Harvey Friedman]

Simpson 10:55PM 12/26/99 wrote:

>The next day after the panel discussion, Gregory Cherlin in private
>conversation raised an objection to my point about Riemannian
>manifolds, Lebesgue measure, etc.  According to Cherlin, the
>development of these and many other mathematical concepts ought to be
>viewed as foundational work.  When I pressed him, he admitted that he
>is entertaining the following proposition: All high-level conceptual
>work in mathematics ought to be considered part of f.o.m. in the best
>sense.  I was also able to get Cherlin to admit that the
>Frege-Hilbert-G"odel line is foundational in a different sense of the
>word ``foundational''.  But according to Cherlin, core mathematicians
>view this ``traditional f.o.m.'' line as dull, passe, uninteresting,
>etc.  I said that this is a mistake on the part of the core
>mathematicians, as witness their shock over the fact G"odel and Turing
>are the only two mathematicians on the Time Magazine list of great
>20th century thinkers.
>
>Basically I think Cherlin is going back to the position of denying the
>interest of f.o.m.  This was of course the view taken by Cherlin's
>fellow applied model theorists (van den Dries et al) in the early days
>of FOM.

It is partly a matter of just how wide the perspective is that one is
concerned with. It is my impression that, for instance, physicists have
their own special view of what "foundations of physics" is which is quite
different from what mathematicians have in mind.

When mathematicians look at mathematical foundations of physics, they are
looking for general formulations that transcend particular conceptual
issues that physicists normally think of. Mathematicians look for exactness
where physicists do not - and in fact the physicists will question the
point of even having such exactness.

When f.o.m. researchers look at f.o.m., they are looking for general
formulations that transcend particular conceptual issues that
mathematicians normally think of. F.o.m. researchers look for exactness
where mathematicians do not - and in fact the mathematicians will question
the point of even having such exactness.

Take a look at Godel's second incompleteness theorem. It is a finding that
transcends any particular conceptual development in mathematics. And it has
to be exact where mathematicians aren't normally exact.

Take a look at the programs of concrete independence results, enormous
integers, reverse mathematics, etc. The issues and phenomena transcend any
particular conceptual issues in core mathematics. Yet the examples cut
across a huge variety of core mathematical topics. And more and more core
mathematical contexts are being treated from these perspectives.

These programs are clearly destined to eventually say something of clear
interest in every mathematical context whatsoever. Yet the overall messages
are of a character transcendent to any one of these contexts, and in fact
transcendent to all of mathematics. They fit into a wider framework - that
of foundational studies, which involves all subjects.

Simpson mentions "their shock over the fact G"odel and Turing are the only
two mathematicians on the Time Magazine list of great 20th century
thinkers".

There is nothing definitive about Time Magazine's list, of course. But it
is an indication that there is greater general interest in what Godel and
Turing did than what went on in core mathematics, regardless of how deep
and intricate it was. If the core mathematicians wish to compete with Godel
and Turing in the general intellectual culture of our times, they will want
to cast their subjects in more generally intellectually attractive and
generally understandable terms.