[X-logic-www] [X-Logic] Re: OpenMind [10]

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Subject- Re: OpenMind
Url- http://www.rbjones.com/

You have to bear in mind that I am talking about foundation system in a technical sense here. i.e. a logic within which mathematics (inter alia) can be developed by conservative extension.

In this sense I know of no well-established categorical foundation system.

If you just mean would it be a good idea to use the concepts of category theory rather than those of set theory for giving semantics to programming languages, the answer is maybe.

I''ve done some work with monads and arrows, which are interesting ways of structuring a semantics.

Anyway, the use of a set theoretic foundation does not preclude giving a categorial semantics to a language, if that''s how you want to do it.

As to the unsuitability of set theory, I think you are mistaken. When people give a mathematical semantics to a language they are giving a set theoretic semantics, even if they don''t mention set theory.

Even when Dana Scott was doing domain theory, this was just mathematics, and the natural way to formalise it would be in classical set theory.

Actually there is a bit of work on a foundation system which is ontologically category theoretic on the X-Logic.org site at:
http://www.x-logic.org/pp/gst/pcf-defns.html

This is a first step towards constructing a rather eccentric categorial foundation system. If this were completed it would be logically equivalent to a set theoretic foundation but conceptually rather different.  Pragmatically, I suspect it would be no easier to work with than set theory, but there would be a lot of work to get to the point where one could make a judgement on that.

Roger Jones