Arcadian Functor

occasional meanderings in physics' brave new world

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Marni D. Sheppeard

Wednesday, November 22, 2006

Need to Know Basics

Young OECD people must find it hard to imagine a world without computers and information technology. They have learnt the basics of logic in a way that even my generation cannot imagine. In my experience, they are not afraid of Category Theory, that is if they have any prediliction for mathematics. A Topos is a special kind of category which has a structure appropriate for doing intuitionistic logic. That funny word intuitionistic is actually a technical term, so don't worry too much about it. It includes ordinary classical (Boolean) logic, and other possibilities. All we need to keep in mind for now is that Topos Theory is the place where geometry meets logic.

There are now quite a few good books on topos theory. Rob Goldblatt's Topoi which is now available from Dover. For those with library access, there is the excellent book Sheaves in Geometry and Logic by the late Saunders Mac Lane and Ieke Moerdijk. Online texts include Triples, Toposes and Theories by Michael Barr and Charles Wells.


Blogger Mahndisa S. Rigmaiden said...

11 21 06

Other possibilities aside from the Boolean discrete case? Like fuzzy logic or what? I am curious. Thanks for this lesson. I think there is a tie in to language here, especially considering my finger posts when I discussed a linguistic tie in to three adics and ternery logic. There is also a fuzzy set tie in because of words like maybe or approximately.

OHHHHHH! After reading a bit, you are inviting us to explore MV logic. I see some of the connexions you are making here... How clever:)

November 22, 2006 1:14 PM  
Blogger Kea said...

Hi Mahndisa

You're one smart cookie. It is true that a 1-topos, although very powerful, can only do distributive logic. But even QM is non-distributive. Unfortunately one must have some idea about 1-topos theory before thinking about higher categorical generalisations. This is a very topical research area in mathematics. Ross Street was the first to develop a 2-categorical analogue, a 'cosmos'.

Fortunately, to do physics we can get away with understanding how the combinatorics of higher toposes should work. This is why Batanin's operads appear.

November 22, 2006 1:21 PM  
Blogger L. Riofrio said...

From the Wikipedia link, a topos behaves like sheaves on topological space. (Probably I missed a word there) This would seem to have many applications. Fascinating.

November 22, 2006 7:03 PM  

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