FOM: Do we need new axioms?

[Robert Swartz]

I come at this from a different perspective. One can look at axiom
systems as Gerry Sachs looked at groups -- as gadgets.  That is systems
of axioms are like card games.  We may apply a metric to the games to
decide which ones are better, but in the end it is a matter of taste.
Some games are clearly uninteresting and others are fascinating.  Why
shouldn't we look at axioms in the same way.

The above attitude will clearly upset those who believe that the goal of
axioms is to find a finite set of principles which will describe
intuitive concepts and decide questions about them.  This is an
extremely valuable approach, but what I am saying is that there is an
alternative. The nominalist approach of the preceding paragraph can be
justified by saying that axioms are really meaningless, because all we
have is syntax and symbol manipulation.  In this instance Turing
machines, lambda calculus, Post productions,  etc. are really the final
characterization of all logical thought, and everything else is simply
unnecessary complication.  We may enjoy working with axiomatic theories
as an intellectual exercise, but the semantic value we get out of it is
beside the point.  I am not saying that I agree with this viewpoint.  I
do not, but rather it is a valid way of approaching the question as to
whether or not we need new axioms. If you accept this nominalistic
approach, then you are like someone looking at an art museum determining
what is appealing  From this standpoint we need new axioms, the more the
better, let's be promiscuous about it because new theories can be so
much fun.

Another way to look at axioms is from the point of view that I believe
Godel had.  Here we are looking for those principles which correctly
reflect our intuitions about the nature of mathematical objects.  We
want to find those principles which are at once simple, intuitive,
clearly true, and simultaneously decide, in a way which makes sense, the
questions we have about mathematical objects.  In this regard the
Continuum Hypothesis is the poster child of the disease of mathematics.
Despite all efforts we have not been able to find any self evident
principle which decides CH.  It seems to me, and I think to many, that
this is what the axioms of set theory should do.  Similarly the halting
problem, or Hilbert's tenth problem are things which axioms should be
able to decide, but they can't. We have been playing around with various
axiom systems for thousands of years.  We have found, in general, that
finding new axioms is a slow process.  I don't hear anyone saying that
we are going to find new axioms for number theory which will solve
Hilbert's tenth problem.  As the discussion of this topic shows, the
mathematical foundations have serious problems.

I would like to consider another approach.  The axiomatic, formal
method  is not up to the task of providing a basis for mathematics. It
simply does not adequately reflect our intuitive notions. It is a
miracle that it does so well, but it is flawed.  In the end no amount of
work will find those stronger axioms that Godel was looking for  We may
find some new axioms which get us a little further, but we will always
fail in accomplishing our goal.  The work of Chaitin makes this clear.
Chaitin shows that the power of axioms is directly related to their
complexity.  The more information the axioms contain the more you can
deduce from them.  In the end we will have to find a new approach to
overcome the difficulties which we are encountering, and axiomatic,
formal systems are not the way.  I know this is a radical approach to
the problem, but in the end I believe that it is the only way out of the
difficulties that we have encountered.


Robert Swartz