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Over at Ars Mathematica there's a discussion about the merits of category theory. I mentioned in a comment there this site of preprints, of which my favourites are Lawvere's 1 and 8. I most enjoy the way category theory suggests that you transcribe pieces of reasoning into different 'keys', sometimes just to recover something you already knew, but hadn't viewed in this way, preferably to perform a new piece.

I'd like to use this kind of thinking in coming to understand information geometry. I should be able to learn from a series of papers by Chris Hillman, especially the ones on entropy and information. Another idea worth exploring is one which says that probability and optimization are in some sense dual. This is related to what I have posted about tropical or idempotent mathematics.

I'd like to use this kind of thinking in coming to understand information geometry. I should be able to learn from a series of papers by Chris Hillman, especially the ones on entropy and information. Another idea worth exploring is one which says that probability and optimization are in some sense dual. This is related to what I have posted about tropical or idempotent mathematics.

## 4 Comments:

It's a small world: I know Chris pretty well (though I haven't spoken to him recently). He and I used to discuss category theory frequently. In fact, when he was learning topos theory, he talked about it so much that it inspired me to learn it too.

Tranposing music into different keys is a very apposite metaphor here, particularly if you realize that under non-equal-temperament tunings, music played in different keys sounds decidedly different.

anonymous wrote:

Tranposing music into different keys is a very apposite metaphor here, particularly if you realize that under non-equal-temperament tunings, music played in different keys sounds decidedly different.It's also apposite because the transposition symmetry in equal-temperament tuning gives nice examples of torsors, and the category of G-torsors for a group G is equivalent to G itself, regarded as a category! Even better, this little portion of music theory is a branch of Klein's Erlangen program, which gave birth to category theory - as David's friend Jean-Pierre Marquis has explained.

I love your website. It has a lot of great pictures and is very informative.

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