Re: How analogous are categorial and material set theories?

[Patrik Eklund <peklund@cs.umu.se>]

Is Category Theory a Theory? I think not. At least not in a logical
sense.

Is Logic a Theory. Of course not. Logic is a construction that embraces
ways of creating logical theories.

Can we describe Category in Logic? No, we cannot.

Can we describe Logic in Category. Yes we can. Topos, and all that, even
if I regret we are happy about quantifiers being adjoint functors to the
contra powerfunctor. Logic should be a bit more than just that,
shouldn't it?

---

Can we describe Logic over Category? Yes we can. This is less
recognized. This is the lative construction of logic from signatures,
through terms, to sentences, and so on, as Goguen (Institutions) and
Meseguer (Entailment Systems) did, without being explicit about
signatures. This is Category Theory as a construction embracing ways of
creating logics. We must be explicit about the starting point, the
underlying signature. Just think about it, we potentially have the
object of types in a monoidal category! It's just around the corner.
Let's go get it, and the world will never be the same!

---

Discussions under FOM now related to distinctions between first, second
and third order logic is really bizarre, or boring to say the least.
Much of what they try to say could be said more clearly if they would
understand to use category theory as a construction site to build what
they try to build.

Hilbert was not a bad person, if you ask me.

---

Best,

Patrik


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]