FOM: Boolean rings; the quandary of categorical mis-foundations
Stephen G Simpson
simpson at math.psu.edu
Thu Mar 12 13:18:41 EST 1998
Regarding the Boolean rings debate, and the larger debate concerning
categorical mis-foundations:
Here's the quandary that we are in, as I see it. Back in my posting
of 31 Jan 1998 16:04:27 (see also Sol Feferman's posting of 19 Nov
1997 23:46:02), I tried to make a point about the different
motivations behind Boolean algebras and Boolean rings. Boolean
algebras: George Boole, propositional logic, {and,or,not}. Boolean
rings: a special kind of commutative ring, a.a=a. These questions of
motivation are very important from my perspective as an
f.o.m. researcher. I wanted to discuss these important questions of
motivation in a simple context, Boolean algebras versus Boolean rings,
where presumably everybody is familiar with the underlying
mathematical notions. However, Vaughan Pratt and other proponents of
categorical mis-foundations have stymied this discussion by
obstinately denying the underlying mathematical notions, e.g. by
refusing to admit that there is any distinction whatsoever between
Boolean rings and Boolean algebras. (By categorical mis-foundations I
mean the absurd attempt to claim that category theory is f.o.m.) And
in my opinion this perverse obliteration of technically and
motivationally significant mathematical distinctions, e.g. signature,
is a large part of the problem that arises in connection with
categorical pseudo-foundations. (For example, a similar issue came up
in connection with McLarty's absurd claim that topos theory is the
same as intuitionistic higher order logic. Such a claim can only be
made if one ignores significant distinctions. This is explained in,
for instance, my posting of 3 Feb 1998 14:24:1.) In order to expose
the obstinacy of Vaughan Pratt and others, I am forced to drag Pratt
and the entire FOM list through a lot of very elementary mathematics.
And undoubtedly some people are finding their patience stretched thin.
Unfortunately, I don't know of any other strategy to establish a basis
for rational discussion with Pratt and others.
Why then do I continue this discussion? Well, it would be easy and
not at all inappropriate to dismiss categorical pseudo-foundations as
total nonsense. (Indeed, many mathematicians dismiss the entire
subject of category theory as "abstract nonsense".) However, I have
not taken that easy course. I may be wrong, but to me it seems
potentially worthwhile to engage in a dialogue with proponents of
categorical pseudo-foundations here in the light of day provided by
the FOM list. The purpose of such a dialogue is to get to the bottom
of exactly why the claims of categorical dis-foundations are mistaken,
and in the process hopefully to illuminate some genuine f.o.m. issues.
I think this can work, provided everybody proceeds with patience and
intellectual integrity.
Vaughan, what do you say?
Sincerely,
-- Steve
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