FOM: whither mathematics; set theory; cultural studies; historicism

[Stephen G Simpson]

David Corfield writes:
 > Does either side of this debate have ideas about where it wants to
 > see mathematics going, or the way it is taught?

Speaking only for myself, I'd like to see mathematics and the
philosophy of mathematics become more objective and reality-oriented.
I would hope for the following: (1) better integration of pure and
applied mathematics and various mathematical sciences; (2) healthy
skepticism about Bourbakian "mathematical architecture"; (3) greater
attention to and understanding of f.o.m. issues; (4) more emphasis on
mathematical rigor, precise thinking, etc.  I feel that all of these
would have a very positive impact on mathematics education as well as
philosophy of mathematics.

 > There have been consequences (possibly unintended) of the 'set
 > theory as foundation of math' picture. It has worked its way into
 > the minds of the larger part of Anglo-American philosophers. Are
 > FOM set theorists happy with that, or do they take their work to be
 > misappropriated? If the latter, should they not say so?

I think one of the bad consequences was the "new math" of the 1960's
and 1970's.  This was a US educational fad which consisted of teaching
set-theoretic f.o.m. to young school children in inappropriate ways.
See Morris Kline's book "Why Johnny Can't Add".

Though I see a lot of f.o.m. value in set theory, I'm not particularly
wedded to it.  In an ideal world, I would expect philosophers of
mathematics to explore objective alternatives to set-theoretic f.o.m.
In the real world that we live in, I'm alarmed about the subjectivist
or postmodernist trend in current academic philosophy, and I don't see
how anything good for mathematics can come of it.

 > David Corfield
 > School of Cultural Studies
 > Leeds Metropolitan University
 > U.K.
 > 
 > Interests: Historicist philosophy of mathematics, cognitive psychology

Hmmm, cultural studies.  Does this smack of the subjectivist turn that
I alluded to above?  And what about "historicist philosophy"?  Is it
Hegelian?  I'm not a big fan of Popper, but I think Popper's "The
Poverty of Historicism" made some good points.  An even better attack
on historicism (not widely known, unfortunately) is Ludwig von Mises,
"Theory and History".

-- Steve

Name: Stephen G. Simpson
Position: Professor of Mathematics
Institution: Penn State University
Research interest: foundations of mathematics
More information: www.math.psu.edu/simpson/