FOM: Does Mathematics Need New Axioms?

[Stephen G Simpson]

A reply to Steel's posting of today.

 > Simpson: What are the appropriate axioms for *all* of mathematics?

Steel:

 > We seem to agree that set theory deals with an age-old question of
 > great importance.

Yes.  Right now, set theory (moreso than other parts of f.o.m.) has
the best answers for the question about axioms for *all* of math,
because (1) the bulk of existing math is formalizable in ZFC, (2) this
kind of foundational scheme has achieved a certain degree of
acceptance among core mathematicians such as Bourbaki, (3) stronger
formal systems are interpretable in ZFC plus large cardinals, (4)
large cardinals have some applications to math.

For context, note that other parts of f.o.m. deal with similar or
related foundational questions.  For instance, reverse mathematics
deals (more effectively than set theory) with questions such as:

  What axioms are needed for *specific theorems* of mathematics?

and 

  What axioms are needed for *specific branches* of mathematics?

And intuitionism/proof theory deals (more effectively than set theory)
with questions such as:

  What are the appropriate axioms for constructive mathematics?

Etc etc.

Steel:

 > Steve wants to say that the question 
 [about axioms for *all* of mathematics]
 > is no longer of importance for "core" mathematics, 

New results, of the type that Harvey is looking for, could perhaps
make this question compelling for core mathematicians.  But right now
there is no strong reason for them to care about it.

 > "In order to avoid pointless wrangling about what is mathematics,
 > ...

The distinction between core math and f.o.m. is not pointless
wrangling.  Our discussion here on FOM has revealed several important
points, which should not be swept under the rug.  One of the points is
that, if set theory needs new axioms, that is very different from
saying that core math needs new axioms.

Steel:

 > Is the search for, and study of, new axioms worthwhile? Should people
 > be working in this direction?
 > This gets to the practical question."

The answer to this ``practical question'' is obviously yes.  But
pragmatism is the most impractical of philosophies, and the
``practical question'' is of little interest by itself.  We need to
supplement it with other questions.  What impact can this work be
expected to have on core mathematics?  Which kinds of mathematics can
be expected to need which kinds of new axioms, and for what purposes?
By what standards is this work to be evaluated?  Etc etc.

I think the original question for the panel, ``Does mathematics need
new axioms?'', is better (in that it covers more of the important
ground) than the above so-called practical question.

Steel:

 > If Steve answers the question above negatively, it must be because
 > he rejects set theory as a framework for answering the age-old
 > question.

Set theory has shown itself effective as a framework for studying
``the age-old question'', but not so effective for studying other,
similar questions that are of comparable importance and interest.

Steel:

 > The various constructivist schemes are fragments of ZFC.

Yes.  But constructivistic formal systems are best viewed as
stand-alone systems, not fragments of ZFC.  To view them as fragments
of ZFC is to miss the point.  The point is that, when you prove a
mathematical theorem in a constructivistic formal system, that is
supposed to reveal constructive content that is not revealed by
proving the theorem in ZFC.

 > They are not incompatible with ZFC unless one takes them as
 > outlawing the mathematics which they cannot support. 

Constructivists do not propose to outlaw ZFC, large cardinals, etc.
Constructivists know that proofs in systems such as ZFC can be carried
out constructively, just as set theorists know that proofs in
intuitionistic predicate calculus are also proofs in (classical)
predicate calculus, etc.

The incompatibility between constructivism and set theory is not at
the level of formal systems, but at the level of philosophy of
mathematics.  The underlying philosophical question (which Steel
apparently wants to short-circuit, because he thinks it leads to
``pointless wrangling'' and ``hot air'') is

  What is mathematics?

There are a number of possible answers to this question.
Constructivism represents a subjectivist position, according to which
mathematics consists of constructions in the mind of the
mathematician.  Set theory represents an intrinsicist position,
according to which mathematics is the study of a mathematical realm
remote from the world of entities and processes.  And there are other
coherent positions on the question ``What is mathematics?''.  These
positions are philosophically incompatible, but f.o.m. can sometimes
find common ground among them, at the level of formal systems.

 > Is this what you mean by an alternative scheme--the proposal to
 > abolish some part of ZFC or its large cardinal extensions?

This is not the right way to think about alternative foundational
schemes.  It misses the point.

-- Steve