FOM: General interest of mainstream mathematics?

[Josef Mattes]

First, my obligatory introduction:
 Name: Josef Mattes.
 PhD: 1995, University of California at Berkeley, under N.Reshetikhin.
 Position: Visiting Research Assistant Professor, UC Davis.
 Recent and current interests: Quantum groups, link invariants, moduli
  spaces of flat connections, related areas, teaching, photography,
  aikido.
 http://math.ucdavis.edu/~mattes

In this posting I will try to make a case for the general intellectual
interest of mainstream mathematics, inspired by the following posting by
Harvey Friedman.

>
>Date: Tue, 6 Jan 1998 05:02:36 +0100
>From: Harvey Friedman <friedman at math.ohio-state.edu>
>Subject: FOM: GII/clarification
>
>Often one has to reflect carefully on high level FOM in order to state the
>results in a way in which the general intellectual interest is most
>apparent and compelling. Let me mention a primary example. Let us consider
>Godel/Cohen to be the complex of results about the consistency and
>independence of the axiom of choice and of the continuum hypothesis (in the
>presence of the axiom of choice) from ZF (ZFC). One can on one extreme
>

>b) assert that there is a commonly accepted system of axioms and rules that
>is considered more than adequate to formalize all currently accepted
>mathematical proofs. Assert that this commonly accepted system is based on
>the single concept of set with membership, and is called standard axiomatic
>set theory. That they form the usual axioms and rules of inference for
>mathematics. Assert that the two most important problems in set theory -
>indeed the two main problems emphasized by the founder of set theory - were
>shown to be neither provable nor refutable within standard axiomatic set
>theory. More specifically, the first was shown to be neither provable nor
>refutable within standard axiomatic set theory, but has been regarded as
>having the flavor of an additional axiom. And that when added as an
>additional axiom - as is common today - the resulting system is not
>sufficient to prove or refute the second of the most important problems in
>set theory. This second of the most important problems in set theory is, in
>contrast, not of the flavor of an axiom.
>

Let's assume that the person you are talking to is a bit skeptical: "This
sounds to me like formalizing proofs is the main goal of mathematics. You
over there, aren't you a mathematician? Do you agree with this?"

And the mathematician says:

"Not really. As far as I can see, contemporary mathematicians are much
less interested in axiomatizing and formalizing than those 80 years ago.
Then, mathematicians were very much concerned with whether mathematical
reasoning was free of contradictions and it would have been a great feat 
if one could have shown that set theory is free from contradictions: Since
everything can be formalized in (some, there are several) set theory that
would have solved the problem for all of mathematics. Yet, it has been
shown in 1930 that this can not be done, which makes axiomatic set theory 
much less interesting (at least in my eyes). 

"But there was more to sets. For example, there were also basic problems
as to what is a function or a set of points on the line etc. In fact, this
is what the founder of set theory was working on when he was led to the
invention of set theory. Set theory provided a stable and reasonably clear 
framework of describing how general functions or geometric objects are
allowed to be which, I suppose, must have looked quite natural and
compelling at the time: What could be more obvious as saying that a circle
is just the corresponding set of points?

"Again surprisingly, it turned out that this is not always the right way
to think about geometry either. For example there is an area called
'noncommutative geometry' in which one studies objects that behave in many
ways as if they were geometric objects, but there is no underlying point
set. As one of the greatest living mathematicians put it [Alain Connes,
Noncommutative geometry, p.1]: '[There are] many natural spaces for which
the classical set-theoretic tools ... loose their pertinence.' 

"For functions, too, the set-theoretical view was not the final word. A
physicist came up with 'generalized functions' which it took
mathematicians about 20 years to squeeze into a formal framework
(which in turn can be done in several different ways, all that I
know of drawing on ideas from other areas of mainstream mathematics rather
than being simple extensions of the set theorettic idea of a function). 

"By the way, the influence of physics was also crucial in noncommutative
geometry which in turn fed back into physics, but also into what
previously were thought to be remote areas of the purest of pure
mathematics.  Why is this relevant? Because unification of mathematics was
included among the virtues of set theory, whereas here we see unification
without the help of set theory.

"I am not sure how much of the details of the above are of interest to  a 
nonmathematician,  but I think at least the following is an important
lesson: 
Mathematics has always been regarded as the prime example of the power of
human reasoning. But even mathematics is about meaning and understanding
rather than about formalisms (set theoretic, categorical or
otherwise), these are just tools. As another one of the great
mathematicians put it [Serge Lang, The beauty of doing mathematics, p.19]:
"Axiomatization is what one does last, it's rubbish. It's the hygiene of
mathematics, ...'  (A bit later he wrote 'The word rubbish is too
strong.').

"Human understanding starts and ends with human experience, and in my
view understanding is (or should be) the goal of science."

Josef Mattes