FOM: categorical non-foundations; three challenges for McLarty

Charles Silver csilver at
Sat Jan 31 07:48:24 EST 1998

> From: Stephen G Simpson <simpson at>

> >Challenge 3. McLarty needs to forthrightly concede that, as Harvey put it,
> >> there is no coherent conception of the mathematical universe that
> >> underlies categorical foundations in your sense.

On Sat, 24 Jan 1998, Vaughan Pratt wrote:
> While I believe there is such a conception, namely the geometric
> conception of morphism as a line segment, it troubles me that this isn't
> Colin's conception.  On the one hand Colin is an experienced category
> theorist while I'm barely an amateur.  On the other, if my elementary
> picture of morphism is wrong then I have *no* idea what other elementary
> concept, one that a 3-year-old could relate to, would improve on it.

	I for one would like to see more discussion of whether "line
segment" is a relatively coherent underlying concept of category theory,
and how important (vs. unimportant) it is to have such an underlying
concept to do f.o.m.  Several people (on both sides of the SET v. CAT
issue) seem to think that having "simple" formal axioms is the major
criterion (for f.o.m.).  I would like to see more discussion of this point
also.  For example, suppose I make up a number of "simple" axioms that are
clearly "about" nothing, but which turn out to be extremely useful.  Does
this qualify them as providing good foundations?  Suppose, after the fact
of laying down these simple axioms, it turns out that they have a nice
interpretation in terms of highly implausible types of objects.  For
example, suppose the nicest way to look at them is to suppose they
postulate the existence of an all-powerful, all-knowing God who is
interested in maximizing "existence"  but is bound by some version of
"freedom" for His subjects.  I know this is not only vague, but weird and
maybe ridiculous. The point I'm trying to bring out is that it seems
possible that helpful formal axioms could have a bizarre interpretation
that is helpful to think about when working in the field (or it could turn
out that the axioms have no "nice" interpretation at all). 

Charlie Silver
Smith College

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