Re: Categories ridiculously abstract

[Todd Wilson <twilson@csufresno.edu>]

On Wed, 29 Nov 2000, Michael Barr wrote:
> And here is a question: are categories more abstract or less
> abstract than sets?

There is a trap lurking in this question, and it has to do with
understanding the term "abstract":  different notions of "abstract"
can lead to different answers to the question.  In the case of sets
and categories, since these are of different similarity types,
something other than inclusion of classes of models is meant.  For
example "abstract", applied to sets and categories, might mean:

1. Having wider applicability.  In this case, we can observe that the
   theorems of category theory (e.g., products are unique up to unique
   isomorphism) are generally more widely applicable than theorems of
   set theory (e.g., the powerset of a set has greater cardinality
   than the set itself), and so we would be inclined to say that
   categories are more abstract than sets on this criterion.

2. Having more general conditions for being an instance.  In order to
   specify a set, we need only give (list, characterize) its members.
   To specify a category we need to do the same thing for both the
   collection of objects and the collection of arrows, and then we
   need to specify the composition law.  (Even in an arrows-only
   formulation of category theory, we still need to specify both the
   collection of arrows and the composition law.)  So, on this
   criterion, sets come out as more abstract.

Some time ago, on the Foundations of Mathematics mailing list (FOM),
there was a long and sometimes heated debate on alternative
foundations of mathematics (where alternative meant non-set-theoretic)
-- in particular on the viability of some kind of category-theoretic
foundation for mathematics (e.g., elementary topos theory + some
additional axioms) -- and the majority view seemed to be that

- Set theory is more all-encompassing.  The standard arguments about
  the bi-interpretability of category theory and set theory were met
  with the challenge (unanswered, as far as I know) to produce, in a
  category-theoretic foundation, a natural linearly-ordered sequence
  of axioms of higher infinity that can be used to "calibrate" the
  existential commitments of extensions to the basic axioms comparable
  to the large cardinal axioms of set theory, where the naturality
  requirement supposedly precludes the slavish translation of these
  large cardinal axioms into the language of category theory.  (Recall
  that all known large cardinal axioms for set theory fall into a very
  nice linear hierarchy that can be used to gauge the consistency
  strength of a theory.)

- Set theory is conceptually simpler.  Set theory axiomatizes a
  single, very basic concept (membership), expressed using a single
  binary relation, and posits a natural set of axioms for this
  relation that are (more or less) neatly justified in terms of a
  fairly (some would say perfectly) clear semantic conception, the
  cumulative hierarchy.  Category theory, the view goes, could only
  approach the scope of set theory, if at all, by adding many axioms
  that are unnatural and quite complicated to state and work with
  without the aid of multiple layers of definitions and definitional
  theorems (for products, exponentials, power-objects/subobject
  classifier, higher replacement-like closure conditions on the
  category, etc.).

The arguments put forward in support of these views were very similar
to those that are implicit in the labeling of category theory as
"ridiculously abstract", and there are no doubt many readers of this
list who would disagree with part or all of these views (me, for one).
However, my intention in reporting them here is *not* to start another
set-theory vs category theory thread, but rather to point out that,
although category theorists have yet to make a convincing case -- at
least I haven't seen one -- that category theory is more fundamental
or foundational in any important sense (sorry, Paul), recent research
in cognitive science on the embodied and metaphorical nature of our
thinking indicates that category theory may well be able to make such
a claim after all.  See the books

    G. Lakoff and M. Johnson.  Philosophy in the Flesh: The Embodied
    Mind and Its Challenge to Western Thought.  Basic Books, 1999.

    G. Lakoff and R. Nuñez.  Where Mathematics Comes From: How the
    Embodied Mind Brings Mathematics into Being.  Basic Books, 2000.

for a popular account of this research.  I should mention, of course,
that, closer to home, the book

    F.W. Lawvere and S.H. Schanuel, Conceptual Mathematics:  A First
    Introduction to Category Theory.  Cambridge University Press, 1997. 

is certainly a step in this direction.

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