FOM: CAT, SET and TOP

[Robert Black]

In the SET v. CAT debate the appeal of ZF for its proponents seems
basically to be that its axioms can be justified as being true to the
intuitive notion of forming a collection.  (More exactly, this seems
obviously right for hereditarily finite sets and there's a sensible project
of discussing just what notions about completable infinities, arbitrary
subcollections or whatever are required for the rest of the hierarchy.)
And the objection to CAT is that the axioms of topos theory (say) can't
easily be seen as supported by some comparably intuitive notion.

I'd like to compare here the relation between the axioms of general
topology and the intuitive notion of continuity.  We define a topological
space as a set of points, its topology as a collection of privileged
subsets, the 'open' sets (where 'open' is primitive) which contains the
whole set and the empty set and is closed under finite intersections and
arbitrary unions.  And we define a continuous mapping between topological
spaces as one where the inverse image of any open set is open.  Now just
what does this have to do with the intuitive notion of continuity?

For a start there's nothing intuitively obvious about the claim that
continuity has to do with mappings between sets of points at all.  It took
a couple of thousand years to develop the concept of the continuum as a set
of points (a view which, incidentally, was declared heretical by the
Council of Constance in the 15th century).  And even once we've decided
we're talking about mappings between sets of points, there's nothing about
'open sets' in the intuitive notion of continuity.  (You might try and say
that we can axiomatize in terms of neighbourhoods rather than open sets,
but I still don't think there's an intuitive notion which the axioms are
designed to capture; this is particularly obvious when one remembers that
the axioms allow the topology to be non-metrizable, a point at which my
intuitive notion of a neighbourhood, if I have one at all, gives out all
together.)

Rather, the story is something like this:  in reasoning about continuity,
at least in a set-theoretic context, we are led first to the notion of a
metric space, but then we realize that there is a structure in our
reasoning which floats partially free of the particular metric in use.
This structure turns out to have a simple axiomatization, giving us an
abstract *generalization* of our original ideas about continuity which
turns out to have amazingly wide applicability.

Now no-one would deny that the notions of general topology are of immense
conceptual (not merely technical) importance and that they have helped to
unify various branches of mathematics in an intellectually satisfying way.
But they are *not* justified as simply capturing with precision an
intuitive justifying idea.

It seems to me that category-theoretic ideas arise in a somewhat analogous
way, but this time as a generalization and reconceptualization of
*set-theoretic* ideas, namely composition of structure-preserving mappings.
We abstract from arguments with diagrams certain structures of thought
which can be separately axiomatized, the axiomatization then leading to
simple unifying ideas which can be used beyond the original realm from
which they were abstracted.  And it turns out (or so I am told - I'm no
category theorist) that elegant systems of axioms using only
category-theoretic ideas as primitive suffice for the development of real
analysis or indeed anything one can develop in ZF, while other systems lead
to interesting alternative structures.

Now the proponents of CAT might, I think, go along with all this and
proceed to say that just as once we have general topology we no longer need
a separate and more faithful mathematical analysis of the intuitive concept
of continuity to which it  gave rise, so once we have category-theoretic
'foundations' we can throw away the set-theoretic ladder by which we
reached them.  But that seems to me to be wrong.  The concepts of general
topology gave a generalizing reduction of thought about continuity *in
set-theoretic terms*, where the set-theory itself is still based pretty
directly on an intuitive concept, that of collection.  But just as I have
no intuitive understanding of 'open set' which directly leads to the axioms
of general topology, so I have no intuitive understanding of 'arrow' which
directly leads to the idea of a topos.  'Open set' is primitive in the
axiomatization of topology, but it's not really a foundational concept.
'Arrow' is primitive in category theory, but seems to me for similar
reasons not to be truly foundational.  While it's just as silly to be
generally rude about category theory as it would be to be rude about
topology, there still seems to be a pretty clear sense in which set theory
can claim to be foundational and category-theory can't.

Now of course a proponent of category-theoretic foundations might concede
this, while insisting that it doesn't matter.  But then my question would
be: doesn't matter *for what*?  I concede that it doesn't matter for
purposes of useful formal axiomatization; I'm happy to believe that *that*
can be done in category-theoretic terms, and that doing it leads to
interesting generalizations and problems.  But I do not concede that f.o.m.
is concerned only with useful (or even useful and fruitful) formal
axiomatizations.

Robert Black
Department of Philosophy
University of Nottingham