FOM: speculations on the future of f.o.m.

[Stephen G Simpson]

This FOM posting is a follow-up to Sam Buss's FOM posting of 
Tue Dec 07 00:51:16 1999.

Buss formulates his main point as follows:

 > My basic assertion was that the real foundations of mathematics is
 > first-order logic, or the use of rigorous reasoning and
 > mathematical rigor that can be formalized in first logic.  This is
 > in contrast to the usual point of view that set theory is the
 > foundations of mathematics.

And I think I agree with the underlying sentiment, although I might
not formulate it exactly as Buss does.  

On the eve of the year 2000, let me try to restate the Buss/Simpson
point in terms of speculations/predictions about future history.

On the one hand, I speculate that the currently dominant, ZFC-style,
set-theoretic-foundational orthodoxy may well fall into disrepute
fairly soon.  This speculation is based on observation of recent
directions in f.o.m. research.  On the one hand, recent
set-theoretical research points to a need for large cardinal axioms
going far beyond ZFC, in a somewhat unpcontrollable way.  On the other
hand, recent research on subsystems of second-order arithmetic
indicates that the bulk of current mathematical practice is
formalizable in systems much weaker than ZFC.  My feeling is that both
of these research directions tend to put the ZFC orthodoxy into
question, and this could result in widespread rejection of the ZFC
orthodoxy within the next 100 years.

On the other hand, I predict that our current 20th century idea of
mathematical rigor, explicated in terms of formalization in
first-order logic, a.k.a. the predicate calculus, will surely remain
vital and of the essence in pure mathematics, for at least the next
500 to 1000 years.  To me these ideas seem so compelling that I can
see no viable alternative.

In sum, my feeling is that mathematical rigor and the predicate
calculus will endure much, much longer than ZFC set theory as an
f.o.m. orthodoxy.

FOMers, what are your thoughts on the future of f.o.m. over the next
100 or 1000 years?

-- Steve

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