FOM: Orthogonal roles

[Todd Wilson]

As promised earlier, here is my response to the thoughtful posting of
A.R.D.Mathias concerning my posting of 27 Feb 2000, entitled "A dual
view of foundations".  In short, I find myself in agreement with most
of Mathias's points.

On Fri, 3 Mar 2000, Andrian-Richard-David Mathias wrote:
> 2. Start from Wilson's remark that 
> 
> W4    topos theory can provide an ontology that is just as
>       abundant as the one set theory provides. 
> 
> To a set-theorist that is plainly untrue: taking topos theory to be a 
> decorated version of Mac Lane set theory, it cannot, for example, prove the 
> existence of an infinite set of infinite cardinals. 
>
> [Mac Lane, be it noted, does not see the issue between topos theory and set 
> theory as an ontological competition.]  

The intent of my claim (numbered W4 above) was that topos theory *plus
additional existence axioms* is able to match ZFC in the "ontological
competition" -- in particular, note that I said "can provide" rather
than "does provide".  Some evidence for this claim, in the form of
references to recent work in type and topos theory, was kindly
provided by Erik Palmgren in his FOM posting of 18 Mar 2000.  There is
no doubt, however, that, in the ontological race, set theory (with its
hierarchy of large cardinals) is still several lengths ahead.

But, like Mac Lane, I don't view this as the main issue, as I hope my
follow-up FOM posting of 15 Mar 2000 (a response to Stephen Simpson)
made clear.  The ontological capabilities of topos theory are
important to the extent that worlds are available in the universe of
toposes where various large sets may find a home, thereby assuaging
(at least some of) the concerns of critics that topos theory is too
limited.  But the real foundational interest of topos theory lies, I
believe, more in its "relativity" and "geometric coherence" (as
proposed ibidem and in the references cited therein), about which I
would enjoy reading some comments from FOM readers.

> 3. But let us turn Wilson's remark around and take it to mean that
> the objects supplied by a system of the strength of topos theory
> meet Wilson's ontological hunger --- as he says,
> 
> W5    What matters is that there are enough objects available, and enough 
>       relations between these objects representable, that we can map
>       whatever we wish to speak about onto what's provided.
> 
> [Aside: that smacks slightly of the idea of a once-for-all foundational job 
> being done, an idea I distrust as I believe foundational questions permeate 
> mathematics. I think of particular superstructures of abstract ideas being 
> called into being to solve particular problems; different superstructures 
> being invoked at different times.]  

I agree that my remarks and questions in the post to which Mathias is
responding were made and asked concerning single systems that purport
to be "foundations for mathematics" -- drawing the appropriate charge
of "once-for-all foundational job".  I was interested, in that post,
in inquiring into the relative importance of what I saw as two
different foundational roles that such "once-for-all" systems might
have.  But, as I tried to indicate (at least indirectly) in my
follow-up post, I believe that topos theory may have something to
offer Mathias in his distrust of such monolithic systems.

> 5. Personally I define set theory as the study of well-foundedness, and
> regard its foundational successes as occurring when it meets a need for a
> new framework for a "recursive" construction (in a suitably abstract
> sense).  I don't think it succeeds at all in accounting for geometric
> intuition.  [In line with Wilson's final point, that failure should not be
> allowed to obscure its successes; but nor should its successes be judged a
> reason for sweeping its failures under the foundational carpet.]

I am happy to agree with Mathias on this point, which he states very
well.

> 9.  Wilson writes: 
> 
> W7    These axioms can be seen as an attempt to capture everything about the 
>       notion of collection that can be subjected to rigorous examination.
> 
> Everything ? surely incompleteness means that there is no hope of doing
> that.  Ideas slowly evolve. 

Of course, but it seems nevertheless that the promoters of set theory
as ultimate foundation -- especially those with a strong conviction in
the independent existence and complete determinateness of the
cumulative hierarchy -- view ZFC as a partial axiomatization of the
associated notion of collection, and efforts to extend ZFC are driven
by the goal of discovering and formalizing that which the current
axioms have yet to capture about this picture.  I personally have less
faith in the ultimate determinateness of the picture, despite how far
it has already taken us.

-- 
Todd Wilson
Computer Science Department
California State University, Fresno