FOM: category theory, cohomology, group theory, and f.o.m.

[Todd Wilson]

All quotes (indicated with ">" characters) are from the 21 Feb 2000
12:20:34 posting of Stephen G Simpson.

> This is a reply to two postings on category theory by Andrej Bauer
> yesterday (Feb 20, 2000).
> 
> The issue of so-called ``categorical foundations'' has already been
> discussed in great detail on FOM.  I suggest you have a look at the
> earlier discussion, so that we can perhaps avoid going over old
> ground.

As someone who read through all of that previous discussion (and
contributed a small part to it), I am not anxious to rehash all of
those issues here, and I concur with Simpson's suggestion concerning
"old ground".  However, it seems clear to me that Simpson and the
category theorists are once again talking past one another, and I
would like to see, this time around, if I can point out precisely
where this is happening.  I apologize in advance if, in doing so, I
misrepresent either side.

When confronted by statements to the effect that certain features of
category theory are not explained by set and class theory, Simpson
responds by pointing out how these concepts can indeed be represented
by the "more fundamental concepts" of set theory:

> For example, the concept ``group'' is fundamental with respect to
> group theory, but it is not fundamental with respect to mathematics as
> a whole, because in the context of mathematics as a whole, the concept
> ``group'' can be (and always is) explained in terms of more
> fundamental concepts such as ``set'' and ``operation''.
> 
> Similarly, the concept ``category'' is fundamental for category
> theory, but it is not fundamental for mathematics as a whole, because
> it can be (and always is) explained in terms of sets, classes, and
> operations.

However, just because certain features of a concept can be (or even
"always are") represented in terms set theory does not necessarily
mean that (1) *all* of the features of this concept have been thereby
"explained", and (2) set theory is therefore the more fundamental
theory.  As for (2), I'm sure that Simpson is well aware that many of
the conceptual systems we use to represent mathematics are mutually
interperatable.  One can try to bring in notions of "naturality" or
"elegance" of representation, and use them to argue the relative
merits of different representational schemes, but, although some of
these arguments are no doubt illuminating, they do not change the
basic sufficiency of the schemes involved.  Two bi-interpretable
schemes have the same inherent richness of concepts, even if the
translations between them are not symmetric in directness.  Is English
or Chinese the more fundamental natural language/conceptual scheme?

However, I think the more interesting point is (1).  What leads a
category theorist to say (as Andrej Bauer did in his post), "There are
concepts that are fundamental but are not at all exposed, clarified,
or easily studied within set theory", is the sense they have that, in
the process of "explaining" the fundamental ideas of category theory
in terms of set theory, something non-trivial is being lost and/or
something inessential is being added.  Category theorists have been
struggling for many years to make explicit just what it is that is
being lost or added, but, for the most part, they have not succeeded.
Simpson may view this failure as an indication that the large number
of category theorists who hold this view are suffering from a kind of
"mass hallucination" (my phrase, not Simpson's), but one can point to
many other instances where mathematicians have collectively held
strong intuitions for a long period of time that they weren't able to
make precise until much later.  One example that comes to mind is the
question of the consistency of infinitesimals.

    As an aside, I'd like to mention in this regard that, more
    recently than Robinson's work, a number of researchers in category
    theory have made an important advance in the explication of these
    intuitions through what is called "smooth infinitesimal analysis",
    a theory that is inconsistent with classical logic but consistent
    with and very fruitful under intuitionistic logic.  An excellent
    introduction to this work, written by someone who is both a set
    theorist and a category theorist, is the book

        J.L. Bell, A Primer of Infinitesimal Analysis, Cambridge
        University Press, 1998.

    (I especially recommend his Introduction.  Bell has also attempted
    to explain some of the intuitions of category theorists in the
    Epilog of his book "Toposes and Local Set Theories", Oxford Logic
    Guides 14, Clarendon, 1988.)

It is perhaps also worth mentioning another, more modern example of
intuition lagging strikingly behind results:  the almost universally
strongly held intuition that P is not equal to NP.

So, without taking up any of the particular matters that Simpson is
criticizing in Bauer's post (I see that Andrej has just written a
response), I would propose that we acknowledge that the intuitions of
category theorists concerning the fundamental nature of their subject,
even in the absence of tangible results vindicating these intuitions,
need not be a "mass hallucination", and instead make an honest attempt
to discover whether there really is anything to them.  For what it's
worth, my own intuition tells me that there is, but I am not any
closer to making this intuition explicit than the rest of the category
theorists.

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