[FOM] Re: John Steel on David Corfield's book
cxm7 at po.cwru.edu
Mon Oct 6 11:47:37 EDT 2003
At 12:01 05/10/2003 -0400, John Steel wrote:
That David Corfield has argued for philosophy of math to include questions
of why the latest mathematical advances are really advances. Then John writes
>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.).
But then one of two things must be true: Either, Connes' work is really
not important; or contrary to what John says the mathematicians have not
done the job of explaining it -- even to other mathematicians.
I think Connes' work is important, and Connes himself along with others has
done a great deal to explain it, and there is a great deal yet to do. It
is a genuinely huge new idea. There have been radical changes in the
understanding of geometry these past 50 years and they do not seem to be
slowing down. they have been driven by number theory as much as by quantum
Is this really ``geometry" or is it mere technicalities? That is a
philosophic question. And it is not a simple one. It surely has no
"yes/no" answer. It asks what we mean by geometry, and what we mean by
mere technicality, and what we mean by productive.
John well compares this to the Continuum Problem. But it is more a
convergence than a conflict. From Pen Maddy I learn that one strong
desideratum for a new axiom, which could settle the CP, is that it not be a
mere correct technicality. We already know many candidate axioms that
technically imply or refute the Continuum Hypothesis. But none yet strikes
people as a going far enough beyond the level of technicality to stand as
an axiom. And surely another goal for a new axiom is that it be
productive, not just proving new formal theorems, but giving new
insights. What is an insight? These are the very issues David wants to
discuss around Connes' geometry and other topics. They will certainly draw
on f.o.m. but I think not only on f.o.m.
As to the difference from math popularization, I'd say look at David's book
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