Re: Michael Healy's question on math and AI

[Todd Wilson <twilson@csufresno.edu>]

On Wed, 24 Jan 2001, F. William Lawvere wrote:
> (A)    The real issue is that for purposes of pure AND applied mathematics, 
> we need to be able to represent (without spurious ingredients) these 
> cohesive and variable sets (or `spaces') and their relationships.  The ZFvN 
> rigidification fails so miserably in doing this that even those geometers 
> and analysts who pay lip service to it as a `foundation' never in practice 
> use its formalism.
> 
> (B) [...]
> 
> Many people working in the new fields, striving to realize the dream of a 
> theoretical computer science, do not seem to be aware of points like  (A) 
> and (B). 

As someone who is "striving to realize the dream of a theoretical
computer science", I would better like to understand the point that
Lawvere is making here.  Am I right in assuming that, in using terms
such as "spurious ingredients" and "rigidification", Lawvere is
referring to the fact that (to use some computer science terminology)
set theory is too much implementation and not enough specification?
That the rigid epsilon-structure of set theory cannot represent
abstract mathematical structure faithfully, without introducing
unwanted details?

If so, can category theory really do better?  Can we give some
concrete examples in both "pure AND applied mathematics" that really
make the difference in representational ability clear?  (These
questions, like the ones below, are not rhetorical or deprecatory; I'd
really like to know some answers.)

To take the first example that comes to mind, consider the cartesian
product of two objects A and B.  The "implementation" of this in set
theory as a set of ordered pairs (which are themselves specific
doubleton sets) certainly introduces some "spurious ingredients", but
the category-theoretic version has its own idiosynchrasies as well:

- Although we constantly speak of "the" product, we really only have
  "a" product (at least if we take the category-theoretic perspective
  seriously).  What is really involved, formally, in making the move
  from "a" to "the"?  A formal language translation scheme?  Coherence
  theorems?  How much technical work is really involved here?

- Related to this, what about the fact that if

      (pi0: A x B -> A, pi1: A x B -> B)

  is a product, then so is (pi1, pi0), indistinguishable categorically
  from the other product?  Does the arbitrary choice between one of
  these products introduce a "spurious ingredient"?  If we find this
  particular "implementation detail" aesthetically displeasing, can we
  abstract away from it by defining an "unordered cartesian product"?
  (I couldn't see how to do it.)

- Is there anything to be made of the fact that the set-theoretic
  cartesian product is a local construction, involving only the sets A
  and B and certain small sets made up of their elements, whereas
  a/the category-theoretic product depends on the whole category
 (because of the quantification in the universal property)?

And if these idiosynchracies do carry any weight (and I'm not claiming
that they do), why are they "better" idiosynchracies than those of the
set-theoretic cartesian product?  And, finally, shouldn't "better"
really be "better for what"?  In other words, aren't the two
communities really just arguing past one another, like people arguing
over types of automobile?  What really is the issue here?

Sorry for all the questions (and all the "really"s).

-- 
Todd Wilson                               A smile is not an individual
Computer Science Department               product; it is a co-product.
California State University, Fresno                 -- Thich Nhat Hanh