FOM: total repudiation of McLarty
Vaughan Pratt
pratt at cs.Stanford.EDU
Wed Feb 25 18:15:20 EST 1998
From: Stephen G Simpson <simpson at math.psu.edu>
>My view is that "categorical
>foundations" is a sham, false, misleading, etc etc. When forced to
>discuss it, I refer to it as categorical mis-foundations, categorical
>dys-foundations, categorical non-foundations, etc.
Steve is like a pianist who has played only Beethoven all his life.
Confronted with a Scott Joplin score, his fingers tell him that this is
not even music.
In that sense Steve's position is understandable, if not entirely
reasonable. For those whose world view makes every object a set, there
can be no categorical foundations.
I have never met a category theorist who thought every object was a set.
Category theorists simply don't think that way. A category theorist
views sets as discrete entities, like sand (I think it was Poincare
who viewed sets in these terms, long before there was category theory).
Most objects encountered in practical mathematics are not sandlike, they
have some degree of coherence, of glue stiffening them up. The real line
is not just the set of its points, that would be just an uncountable set,
isomorphic to the set of continuous real functions. To be recognizable
as the real line it needs some glue to hold it together and give it
its shape.
Of course set theorists have to believe something like this too.
What distinguishes set theorists from category theorists is the order in
which they add the ingredients. Set theorists start with the points and
will add glue if persuaded that it is worthwhile. Category theorists
start with the glue and will add points if you ask nicely.
Sand and glue are pretty basic mathematical ingredients. Surely the
prejudice that all foundations *must* start from sand can be overcome just
as readily as can the various racist, sexist, and religious prejudices
that cloud our thoughts (which may not be saying much).
Vaughan Pratt
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