FOM: Picturing categorical set theory

Vaughan Pratt pratt at cs.Stanford.EDU
Thu Jan 22 19:12:05 EST 1998

From: Silver
>What I'm looking for is a development of category theory, or
>perhaps topos theory, that is based on some underlying *conception*.

From: Friedman
>Hopeless. As foundations of mathematics, it is conceptually completely
>incoherent. Proponents are zealots who want to hide this from you in the
>worst way. As technical mathematics, some of it, in moderate doses only,
>has limited and respected uses - it helps with the formulation of
>appropriate generalizations, and cleans up some proofs.

I didn't see this message from Harvey until after I'd posted my
cradle-to-grail collection-vs.-continuum message, otherwise I would have
organized my message as a refutation of his.

Category theory is conceptually simple both as a subject in its own right
like group theory and as a foundation for mathematics.  I presume Harvey
is not disputing the former (a category is after all just the common
generalization of a monoid and a poset, as well as being a quotient of
the free category consisting of all finite paths on its underlying graph).

I believe that the best way to relate to categories-as-foundations is
along the general lines of my juxtaposition of collection and continuum as
the respective conceptual starting points of the two styles of foundation.
This juxtaposition should not be controversial for the (regrettably few)
category theorists on this list, but I am fully resigned, based on the
lack of progress to date, to the rest remaining unconvinced that category
theoretic foundations can really be that simple.

One reason that natural transformations and adjunctions are as complicated
as they are is that their traditional treatment in category theoretic
texts is an awkward blend of categorical and set theoretic techniques.
Mixing the two while not unsound does create complexities of an order not
found in a purer treatment.  The 2-categorical treatment of these concepts
is much simpler, which is why I start from the 2-category perspective
in order to equip the traditional but difficult approach with simple and
elegant structure to serve as a sort of template for the eventual shape
of the traditional approach to natural transformations and adjunctions.

I have made this point about n-categories and geometry several times in
the past.  The essence of the point is that geometry and the continuum
constitute the proper conceptual starting point for categorical
foundations.  This starting point is neither incoherent nor complex.

What Harvey and I think many others on this particular list see in
categorical foundations is an incoherent mess being promoted by either
misguided zealots or charlatans.  What the category theorists see is a
simple and elegant yet rich and powerful basis for mathematical thought.
Both sides are well populated with technically brilliant mathematicians.
Both sides are notorious for their inability to convince each other
of their respective viewpoints.  I'd say we have here a breakdown in
communication approaching that in Ireland or the Middle East.

Name: Vaughan Pratt
Position: Professor of Computer Science
Institution: Stanford University
Research interests: Foundations of computation and mathematics
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