FOM: Ontology of Mathematics

[Todd Wilson]

On Mon, 10 Jul 2000, Jan Mycielski wrote concerning his "rational
philosophy" of mathematics, which I will attempt to summarize thus:

- mathematical objects are real in the sense that they exist as
  configurations in the brains of mathematicians, but differ from
  other "real" objects in not having external referents.

- only finitely many such mental configurations can exist at any one
  time (indeed, from the beginning of time up to the present), and
  they constitute the totality of the subject matter of (pure)
  mathematics.

I have myself often thought, naively (i.e., as someone not very well
read in the philosophy of mathematics), that this is the right way to
understand the ontology of mathematics.  But every time my thinking
gets very far along these lines, I run up against a problem that I
can't resolve.  Perhaps Professor Mycielski could comment on how this
problem is approached in his rational philosophy.

The problem has to do with how to explain the consistency of
mathematical communication across several mathematicians.  Ask any
group of mathematicians to count the number of non-isomorphic groups
of order 6, or to list the perfect numbers less than 100, say, and
they all (barring mistakes, of course) come up with the same answers.
If mathematics is about configurations in individual minds, what
accounts for this consistency?

Here is an analogy.  If a group of mathematicians are standing at the
entrance of a maze that has several exits, only one of which is
connected to the entrance, then, although they may each wander around
the maze quite differently, with different strategies and approaches,
they will all eventually come out the same exit.  It is the walls of
the maze that determine this unique outcome, despite the completely
different natures of the wanderers.

So:  What are the "walls" that create mathematics and how do they
operate?  In what sense do these walls "exist"?  Would the same walls
constrain non-human (say, Arcturan) mathematicians, even though their
brains may be constituted quite differently from ours?  Can these
walls change over the course of time?  In fact, does it even make
sense to talk about these walls over large spans of time and space, or 
might there be some kind of relativity involved that makes these
questions (like the question of which of two events happened first)
illusory?  

I would like to believe that mathematical ontology had a neurophysical 
explanation, but I don't see how to get beyond these questions without 
invoking some kind of extra-mental structure.

-- 
Todd Wilson
Computer Science Department
California State University, Fresno