[FOM] Simpson on Tymoczkoism

[John Steel]

David Corfield has very eloquently argued that philosphers
of mathematics should broaden their perspective. Perhaps
oversimplifying his thesis, he suggests that rather than
focus (or focus only) on why the set theoretic foundations
erected in the early 1900's are justified, they should (also) focus on
why the latest mathematical advances are really advances.

Sounds good, but what does Philosophy have to contribute
here? At various points, it is hard to distinguish the
activity Corfield advocates from popular science writing.
That is a valuable activity, but it seems to be pretty
well taken care of by professional mathematicians. As to
judging the importance of Connes' latest advances, anyone
who has been on an appointments committee knows that is really
hard for mathematicians who do not work in his area. I don't
see what Philosophy can contribute, beyond abstracting
some platitudes (deep connections with other areas, etc.).

In contrast, in the foundational vein we have the Continuum
Problem, where it seems to me that Philosophy may make a real
contribution. A solution to the Continuum Problem will need some
accompanying analysis of what it is to be a solution to the
Continuum Problem. If the solution is that the language of set theory
is ambiguous, then it would seem that general issues in the
philosophy of language might be relevant. If Philosophy has
anything to say, it ought to have something to say regarding
justification and meaning.

Perhaps naively, I don't read Corfield as being as hostile
to foundations as does Steve Simpson. But as far as I can tell,
he is mistaken about where Philosophy has the most to contribute.

John Steel