Re: equality is beautiful

[Vaughan Pratt <pratt@cs.stanford.edu>]

Joyal wrote:
> Many people seem to distrust the equality
> relation between the objects of a (large) category.
> Is this a philosophical conundrum or a mathematical problem?

I don't even trust it for the *-autonomous category with two objects and
three morphisms.  This is the basis for the classical notion of truth as
normally understood when not trying to parse the sentences of
constructive mathematicians.

Some dark night someone could sneak into the basement where the
foundations of mathematics are stored and switch true and false around,
and in the morning we'd all wake up talking Doublespeak without even
realizing it.

Are you suggesting we should trust the equality relation between the
objects of that category?  Useful categories don't come much smaller.

It can't be a question of size because the free Heyting algebra on one
generator is infinite yet it doesn't have this structural ambiguity of
classical logic: we can trust its equality relation because we can tell
which element is the generator even if the arrows are accidentally
flipped---it's a sort of error-detection feature of that Heyting algebra
not possessed by any Boolean algebra.  This is a reason to prefer
intuitionistic logic over classical (not one I'd use myself mind you).

Best,
Vaughan


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