Re: Intuitionism's (read "Philosophy's") Limits

[categories <cat-dist@mta.ca>]

Date: Wed, 5 Mar 1997 10:54:19 +0000
From: Steve Vickers <sjv@doc.ic.ac.uk>

> Constructive mathematics is a philosophy. Category theory is not.
> The question doesn't even type-check.
>
> Of course they're different.

Philosophy is the love of wisdom; type-checking is not.

Of course category theory has its philosophy. To me it's "all things are
connected" - you cannot fully describe anything purely in itself but only
by the way it connects with others. Category theory makes the connections
explicit (as morphisms) and then characterizes things by their universal
properties.

The philosophy plays a real role in categorical practice: for instance, in
the idea that isomorphism between objects is more important than equality,
which is not something that can be meaningful just in terms of the formal
mathematics.

The philosophy also yields a criterion for evaluating the theory: Is
categorical structure adequate for describing the connections that we
actually find? The strength of the categorical view of "connection
structure" is amply confirmed by the power of the universal properties it
can express (compare it with, say, graph theory); but if it does fail us
anywhere, how might it advance beyond its present formalization? (There is
already a plausible answer here: topology has a different way of describing
the connections between a point and its neighbours, and the categorical and
topological approaches combine to make topos theory.)

I hesitate to try to reduce the philosophy of constructive mathematics to a
single pithy phrase, not least because there are different schools of
constructivism with apparently different philosopies. I shall therefore
duck the question of comparing "the philosopies of constructive mathematics
and category theory", but I don't believe it's a meaningless one.

Steve Vickers.