FOM: Does Mathematics Need New Axioms?

[Stephen G Simpson]

This is a quick reply to John Steel's posting of earlier today.

Steel said:

 > If any part of mathematics needs new axioms, then mathematics needs
 > new axioms. Of course, the importance of the need relates to the
 > importance of the areas in need.

and

 > If set theory needs new axioms, then mathematics needs new axioms.

Steel's argument seems to be that math *obviously* needs new axioms,
because set theory does, and set theory is (an important?) part of
math.

I would comment that there is a distinction between set theory qua
branch of math versus set theory qua f.o.m.  As a branch of
mathematics, set theory is relatively small and unimportant compared
to other branches such as geometry, number theory, and differential
equations.  Think of the history of these venerable branches.  Set
theory is relative newcomer.  The main importance of set theory
derives from its contemporary foundational role, as a proposed
foundation for *all* of mathematics.  In this sense, set theory, like
the rest of logic and f.o.m., is not really *part of* mathematics.  It
is a mathematical science, but not mathematics.  The term
``metamathematics'' comes to mind.  When core mathematicians hear that
*metamathematical* studies give rise to new axioms, they are not
unreasonable in thinking that such axioms may be irrelevant to their
central mathematical concerns.  

 > I don't think we need to be doing market research on the attitudes
 > of what Harvey calls "normal" or "core" mathematicians.

It is a mistake to interpret Harvey's view as a call for market
research or a shallow public relations campaign.  As Steel later
acknowledges, there are serious intellectual issues here.  It would be
wrong to ignore these issues or sweep them under the rug.

 > How can you know views which "began in earnest in the 1960's" are
 > fixed and fundamental? 

There are sound philosophical and historical reasons for thinking that
the shift back to concrete and computational math, which began
sometime in the the 1960's, is permanent and long overdue.  Of course
this statement is debatable, but it seems to me that the burden of
proof is on the proponents of highly abstract, non-computational math.

-- Steve