FOM: Does Mathematics Need New Axioms? (response to Matthew Frank)

[Stephen G Simpson]

This is in response to Matthew Frank's posting of Mon, 22 May 2000
19:31:59 -0500 (CDT).

1. Frank says that the anti-foundation axiom might be a new axiom that
we need to look at in a set-theoretic context.  I am not sure why.
Could Frank please explain?

It seems to me that set theorists have good reasons for insisting on
the axiom of foundation.  Namely, by restricting attention to
well-founded sets, we simplify our picture of the set-theoretic
universe, and we lose very little, because the orthodox set-theoretic
foundational setup (natural numbers as finite von Neumann ordinals,
reals as Dedekind cuts, etc etc) takes place entirely within the
well-founded sets.

My impression is that the anti-foundation axiom has been of interest
not so much for set-theoretic f.o.m. but in somewhat different
contexts (Aczel's Frege structures, etc).

2. Frank sharpens Steel's ``practical question'' (whether research on
new axioms is worthwhile) to an even more practical issue:

 > "is this area of research worthy of government funding?"

But this immediately raises a question of political philosophy: ``Is
the funding of scientific research a proper function of government?''
And political philosophy is clearly off-topic for the FOM list.  In
order to avoid the political issue, why not put the question this way:

   Is this area of research likely to lead to new technology which can
   be expected to participate in improving the human standard of
   living?

I don't know whether anyone here on FOM would be willing to comment on
this version of the Steel/Frank question.  But at least this version
addresses the ``practical issue'' while avoiding some possibly
contentious political topics.

3. Regarding set-theoretic foundations, Frank says:

 > Simpson has pointed out that "the bulk of existing mathematics is
 > formalizable in ZFC", but I don't think that this carries much
 > weight for evaluating set theory against various other foundational
 > schemes.

I think my point carries a lot of weight, especially when combined
with other points that I made at the same time.  One of those points
was that ZFC-style set-theoretic foundations has achieved a certain
degree of acceptance among core mathematicians, which alternative
foundational schemes have not achieved.

By the way, Friedman has given a striking example of a mathematical
statement (involving symmetric Borel sets) which is straightforwardly
formalizable in the language of second order arithmetic, and provable
in ZFC, but not provable in second order arithmetic.  See Friedman's
posting of Fri, 12 May 2000 12:44:45 -0400.

4. Regarding constructivism, Frank says:

 > I recommend against using the word "constructivist" to describe
 > people, as Simpson does. ...

We need a term to describe a person who endorses a constructivist
philosophy of mathematics.  I think *constructivist* is the right
word, but Frank may want to introduce another term.  Of course Frank
is right in pointing out that there are people who ``work on''
constructivism (e.g., by proving metatheorems about
constructivistically-inspired formal systems) without endorsing the
underlying constructivist philosophy.  On the other hand, there are
also full-fledged constructivists like Bishop, Bridges, Richman, etc.
We could call them ``philosophical constructivists'', but that seems
redundant, since constructivism is nothing but a particular philosophy
of mathematics.

 > and certainly not of the subjectivist philosophy which Simpson
 > describes.

If subjectivism is not the essence of the constructivist philosophy,
what is?

Frank:

 > My own (pluralist) reasons for pursuing constructive mathematics
 > are now available in a paper on my web site, which might be of
 > interest to some fom-ers.

Yes, it is a nice essay.

One of the points in Frank's essay is that constructivism and other
alternative foundational schemes are often considered desirable simply
because they are unorthodox, i.e., their existence encourages people
to question authority.  This of course does not imply anything about
the strictly scientific merits of the schemes, but it is an
interesting point nonetheless.

Frank:

 > P.S.  An erratum to my post on Hilbert's second problem: I am now
 > convinced that, regardless of what I might consider a solution to
 > Hilbert's second problem, Hilbert would not have evaluated the
 > recent work of Friedman and Simpson in that way.

Why not?

-- Steve