FOM: Does Mathematics Need New Axioms?

[Stephen G Simpson]

This is a continuation of the dialogue with Steel.  I am replying to
Steel's message of Monday, 15 May 2000 18:44:17 -0700 (PDT).

There was the following exchange.  I said

 > > 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.

and Steel replied

 > Yes, although it is you who are emphasizing the importance of field
 > divisions within mathematics.  ...

The issue of field divisions is obviously crucial.  If large cardinal
axioms are needed in only a small and obscure corner of mathematics,
then that is a very different scenario from what Harvey envisions,
where the need for large cardinal axioms would be pervasive throughout
core mathematics (number theory, geometry, differential equations,
etc).

In any case, it seems to me that Steel himself had raised the issue of
field divisions, by arguing as follows:

  Set theory needs new axioms.

  Set theory is part of mathematics.

  Therefore, mathematics needs new axioms.

My contribution was to point out that this argument is somewhat weak,
because set theory is highly atypical when viewed as a part or branch
of mathematics.

For one thing, the main focus of set theory is foundational.  In other
words, set theory is better viewed as being part of f.o.m. rather than
part of mathematics proper.  Recall that the recent reorganization of
the Mathematics Subject Classification scheme abolished set theory as
a separate classification (04XX) and merged it into logic and
foundations (03XX).  The main importance of set theory is not so much
as a *branch* of mathematics but rather a proposed foundation for
*all* of mathematics, in full generality.  As Maddy has said, set
theorists are much more concerned than other mathematicians with
finding a framework for *all* of mathematics.  The goal of this work
is to answer the age-old foundational question:

   What are the appropriate axioms for *all* of mathematics?

Therefore, it is perfectly reasonable for core mathematicians to think
that, if new axioms are needed for set theory, then that doesn't
necessarily have any consequences for particular core math branches.

Steel spins this a little differently:

 > Yes, set theory has special importance because of its foundational
 > role.

My point is that set theory is *very different* from other branches of
mathematics, because of its foundational role.

Steel:

 > One should expect the need for new axioms, and the new axioms
 > needed, to show up here [in set theory] first.

Why ``first''?  Why not ``only''?  The core mathematician is perfectly
justified in raising this question.

Another relevant observation is that venerable core math branches such
as geometry, number theory, differential equations, etc., have very
many connections with each other, but very few connections to set
theory and the rest of f.o.m.  One way to partially capture this
observation in logical terms is to consider the complexity of the
statements involved.  The statements in core math tend to be
arithmetical (or at worst Pi^1_2), while the statements in set theory
tend to be Pi^1_3 or Sigma^2_1 or above, and non-absolute.

Considerations such as these highlight the crucial importance of
Harvey's effort (almost single-handedly) to bring large cardinals into
the realm of core mathematics.  The long-term viability of large
cardinals is at stake.

I characterized set theory as a proposed foundation for mathematics.
Steel replied:

 >  ("Proposed" understates the truth here.) 

If ``proposed'' is the wrong word, what is the right word?  In the
20th century, set-theoretic foundations achieved a certain level of
acceptance among core mathematicians, but that was a matter of
convenience rather than deep conviction.  There are alternative
foundational schemes that many people find attractive.

Here is another exchange.  I said:

 > > 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.

Steel replied:

 > ZFC and its large cardinal/determinacy extensions codify
 > mathematics, not metamathematics.

I concede this statement.  But Steel ought to concede that the
*process of codifying* mathematics within ZFC is metamathematics, not
mathematics.  And, the *discovery* of the ZFC axiom system and its
extensions and their uses is metamathematics, not mathematics.  And
the discovery of significant metatheorems concerning such formal
systems is metamathematics, not mathematics.  (I take
``metamathematics'' to be synonymous with ``f.o.m.'')

 > The objects of study are natural numbers, real numbers, sets of
 > real numbers,...,

Yes, but the way they are handled in set theory and the rest of
f.o.m. is very different from the way they are handled in core
mathematics.  

For example, arbitrary sets of reals are hardly ever considered in
geometry, number theory, differential equations, etc, but arbitrary
sets of reals are very often considered in set theory.  This is
another feature of set theory that makes it stand apart.

 > Steve seems to be saying that the axioms for all of mathematics do
 > not themselves belong to mathematics.

Yes, I am saying something like that.  Let me put it this way: The
discovery and study of such axiom systems and their use in the
formalization of mathematics is part of f.o.m.  It is not part of
mathematics proper.

 > If so, at what point in e.g. the proof that [0,1] is compact do we
 > enter mathematics?

``The proof'' that [0,1] is compact?  There are many proofs of this
well known theorem.  If we write down proofs with careful attention to
axioms, with an eye to formalization in various formal systems such as
ZFC, or Z_2, or WKL_0, then obviously we are engaged in
metamathematics, i.e., f.o.m.  We are not engaged in mathematics
proper.

Another exchange between me and Steel.  I said

 > > There are sound philosophical and historical reasons for thinking
 > > that the shift back to concrete and computational math, which
 > > began in the the 1960's, is permanent and long overdue.

and Steel replied

 > Permanent? How far ahead can you see?

It is important to try to see as far as possible.  The question that
we are discussing is, what is the long term future of abstract,
non-computational math as contrasted with concrete, computational
math?  This question is debatable, and I speak only for myself.

It seems to me that the history of mathematics (as expounded in books
such as Morris Kline's ``Mathematics from Ancient to Modern Times'')
is very much on the side of concrete and computational math: number
theory, geometry, differential equations, applied math, etc.

Philosophically, the idea of mathematics as the study of an abstract
mathematical realm remote from the real world of entities and
processes seems to be a dead end.  Applications of mathematics seem to
be where the action is.  Concrete, computational math has been the
core of math for thousands of years.

I said:

 > the burden of proof is on the proponents of abstract,
 > non-computational math.

Steel replied:

 > The main burden on the proponents of abstract, non-computational
 > math is to produce good mathematics of that sort.

What is ``good mathematics''?  ``Good'' by what standard?  ``Good'' as
judged by other large cardinal advocates?  It seems to me that Steel
is invoking the vague idea of ``good mathematics'' as a way of
short-circuiting the issue that we are discussing.

The real question is: How can the long-term viability of abstract,
non-computational math be promoted in the arena of core mathematical
and/or general intellectual interest?

-- Steve