FOM: set vs category as foundation

[Harvey Friedman]

This is a reply to Martin Davis, 6:27PM 1/22/98.

>Just want to suggest that the right question is not which leads most
>appropriately into real analysis or algebraic geometry or whatever. The
>right question about a foundational approach is: what sort of insight does
>it yield into the outstanding foundational problems of our time? Personally
>I believe these center about undecidability/independence results. We know
>that any fixed foundational scheme (leading to an r.e. set of theorems) will
>fail to yield proofs for some true statements asserting that certain
>Diophantine equations have no integer solutions. We know that easy questions
>to ask about cardinal arithmetic are left undecided by existing foundations.
>
>Which of the proposed foundational approaches is more likely to help in
>understanding/resolving this?
>
Set theory, of course. Paul Cohen is well known to have absolutely no
patience for general nonsense, and it was he who invented forcing in order
to prove the independence of the continuum hypothesis. The method was
refined by others to obtained many many other independence results, as well
as crafted into a tool for use in other aspects of f.o.m. other than
independence results. Even for proving outright theorems in descriptive set
theory. E.g., Harrington's proof that a Borel measurable equivalence
relation on the reals has either countably many equivalence classes or
continuumly many equivalence classes.

Meanwhile, there was the realization that one could give an alternative
general nonsense proof of some of Cohen's original independence results
from the topos theoretic point of view. However, this alternative way of
looking at things never produced any new independence result or any new
descriptive set theory, despite the fact that many important questions were
respected open problems. E.g., the status of the Lebesgue measurability of
all sets of real numbers.

Another point. Cohen got the 1966 Fields Medal* without any hesitation by
the power brokers in mathematics at that time - for a result whose interest
completely and utterly depended on ZFC and the absolute significance of ZFC
as THE foundations of mathematics. I.e., he got it for showing that the
continuum hypothesis could not be proved nor refuted by the accepted
methods of mathematics. If there had been any question about the
suitability of, or the robustness of, or the appropriateness of, ZFC, then
he would not have gotten the Medal. Because, if there were any questions
raised about ZFC, then one could not really claim that his work showed that
CH "could not be proved nor refuted by the accepted methods of
mathematics." And to this day, the reason virtually everybody in
mathematics is still impressed with this acheivment is because it is
virtually universally accepted that ZFC has this status.

Contrast this with the following: Cohen showed that in some topoi with some
additional properties, the translation of CH into topoi language comes out
sometimes with truth value 0. How boring!! He wouldn't have even got
appointed to a decent math dept with that result!!

So here is one GIANT difference between topos theory and set theory. You
can impress everybody by showing that a suitable statement is independent
of the axioms of set theory, but you cannot impress anybody by showing that
a suitable statement is independent of the axioms of (extended) topos
theory - except maybe other topos theorists. You could impress more people,
however, if you then showed that a suitable statement is independent of the
axioms of set theory, via a translation result. The other side: ponder
carefully why this asymmetry exists.

*The Fields Medal - in 1966 it was awarded to Michael Atiyah, Paul J.
Cohen, Alexander Grothendieck, and Stephen Smale. It has had a sacred
reputation among pure mathematicians (not applied) for a very long time,
until recently, when I have heard lots of complaints from all sorts of pure
mathematicians. Back then, it was beyond reproach. I was always suspicious
of it since it is conducted in a great deal of secrecy where there is no
systematic open discussion of the merits of fields of study, and of what
directions mathematics should go in, and what mathematics should be aiming
at. But, nevertheless, there does appear to be some sort of minimal quality
of work needed in order to get this award. And it has never been clear
whether f.o.m. counts as mathematics for this Medal. Cohen managed to
overcome this hurdle, and he has always had my sincere congratulations.