Re: Michael Healy's question on math and AI

[Michael Barr <barr@barrs.org>]

At the risk of offering my head on a tray, I will make a stab at some of
this.

On Thu, 25 Jan 2001, Todd Wilson wrote:

> 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?
> 

Any two products are uniquely isomorphic in a way that preserves the
projections in the obvious way.  That is all that need be said.  There are
coherence statements that can be made, but they follow from the above and
are unnecessary.


> - 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.)
> 

The product is the product of the set {A,B}, which is equal to the set
{B,A}, but our orthography forces us to write one or the other.  Of
course, a product is really defined for {A_i|i in I} and is inherently
unordered.  In set theory, the usual A x B is quite a different set from B
x A and in category theory they are indistinguishable.  You seem to
consider that a disadvantage to category theory, but I consider it an
advantage.

> - 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)?
> 

Our familiar categories have regular generators and for them the product
condition can be reduced to the universal mapping condition when the
domain is/are the generator(s), which is local.  On the other hand, check
out the product in the category of affine schemes that is really
comprehensible only in terms of the categorical definition.

> 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).
> 


For another example, consider the traditional definition of Z as the set
{0,{0},{0,{0}},{0,{0}{0,{0}}},...}
and contrast that to the categorical specification.

Michael Barr

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