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September 8, 2006

Category Theory and Philosophy

Posted by David Corfield

Bob Coecke wrote:

I strongly believe that the categorical approach provides the appropriate framework for a profound structural analysis for quantum theory, a currently lacking prerequisite for a decent philosophical analysis.

A couple of comments earlier, I had written:

Perhaps good things in the interpretation of Quantum mechanics are at last beginning to happen. It would be interesting to know what posterity had to say about the long period between its first appearance and its proper interpretation. There’s a philosopher of science, Michael Friedman, whose work I like very much, who sees advances in philosophy and mathematical science happening hand-in-hand. Whereas Newton led to Kant, and Einstein to the Vienna Circle, he feels that QM has never been philosophically worked over properly. For more on this see my paper.

Putting Bob and Friedman together, you could hardly miss the idea that the next revolution in philosophy should see category theory take a central place, just as logic did for the Vienna Circle in the 1920s and 30s.

But I’ve always been rather wary of the idea that some or other formal apparatus is going to act as a magic tool in the resolution of philosophical problems. Throughout my career as a philosopher I have frequently had the thought “I don’t see why you think a formal treatment of X is the right way to talk about X, but if you are going to do so why not use a proper formalism like category theory.” Presumably none of us here think category theory will help us much with a theory of justice or of history. On the other hand, as well as the ontology of physics, Mike Johnson’s category theoretic approach to databases give us a sense of what it can achieve at the level of everyday ontology. And I’d imagine it could help in statistics and probability. (For a recent attempt in statistics, see Peter McCullagh What is a statistical model? Ann. Stat. 30(5), 2002.)

So category theory is to feature prominently in the new philosophy, but, as Friedman’s examples make clear, it’s not just the taking on of a new mathematical language that constitutes a radical shift in philosophy. I’m not looking for an analytic philosophy Mark II.

Posted at September 8, 2006 8:33 AM UTC

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3 Comments & 1 Trackback

Re: Category Theory and Philosophy

It appears that there is now a Java application, Entity Attribute Sketch Implementation Kit, to help you with your database ontology.

If anyone knows how essential Censov’s use of category theory was to show that the Fisher information metric is the only sensible one to place on a statistical manifold, I’d be grateful to hear from them. This metric is the only Riemannian metric invariant under congruent embeddings by a Markov morphism. What this amounts to is requiring that the effect of re-partitioning an event space on a probability distribution be sensible. Presumably some kind of limits were used to get at the fine-graining.

Posted by: David Corfield on September 11, 2006 9:28 AM | Permalink | Reply to this

Re: Category Theory and Philosophy

To David:

Chentsov’s proof (the one in his book) doesn’t rely that heavily on category theory, it’s just phrased in category-theoretic language.

You can find a “bare bones” version of the proof in a paper by L. L. Campbell. I believe the title of the paper is “An extended Cencov characterization of the information metric.”

Not sure what you mean about limits.

Posted by: Travis Tambone on September 21, 2006 12:27 AM | Permalink | Reply to this

Re: Category Theory and Philosophy

Travis, thanks very much for that. By limits I meant them in the category theoretic sense.

If you’re interested in the next stage after Campbell, see Section 6 ‘Axiomatic Geometry for Conditional Models’ of Guy Lebanon’s thesis.

Posted by: David Corfield on September 21, 2006 8:54 AM | Permalink | Reply to this
Read the post What Can Category Theory Do For Philosophy?
Weblog: The n-Category Café
Excerpt: A possible philosophical meeting on category theory
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