[FOM] Re: John Steel on David Corfield's book

[Colin McLarty]

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 
gravity.

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 
to see.

best, Colin