Comments for The Analogical Engine http://analogical-engine.com/wordpress A math and haskell blog Wed, 14 Apr 2010 16:46:25 -0400 http://wordpress.org/?v=2.8.6 hourly 1 Comment on Vandermonde Matrix at Roots of Unity by Mark Dukes http://analogical-engine.com/wordpress/?p=32&cpage=1#comment-217 Mark Dukes Wed, 14 Apr 2010 16:46:25 +0000 http://analogical-engine.com/?p=32#comment-217 Very interesting! Did you ever figure out the sign? I wanted to know this for something I was working on and this was the only place that I could find information on the determinant! Very interesting! Did you ever figure out the sign? I wanted to know this for something I was working on and this was the only place that I could find information on the determinant!

]]> Comment on Categorical Logic I by Dan Doel http://analogical-engine.com/wordpress/?p=98&cpage=1#comment-11 Dan Doel Wed, 09 Dec 2009 07:29:29 +0000 http://analogical-engine.com/?p=98#comment-11 I think it's also nice to note that for general Z, if you're thinking of your logic as a Hilbert system, then: f : Z x X -> Y => f^ : Z -> Y^X corresponds to the deduction theorem: Z, X |- Y => Z |- X -> Y which of course gets used all over the place. And naturally it corresponds to rules in natural deduction or sequent calculi where the deduction theorem is taken as a defining characteristic of the logic. I suppose that shows why the latter two are in some sense nicer than the former, as well. With a Hilbert system, you'd take your axioms and modus ponens, and define |- in terms of them, and show that the logic has this nice categorical structure. Natural deduction and sequent calculi just start with, "logics (are defined to) have this nice categorical structure," and go from there. I think it’s also nice to note that for general Z, if you’re thinking of your logic as a Hilbert system, then:

f : Z x X -> Y => f^ : Z -> Y^X

corresponds to the deduction theorem:

Z, X |- Y => Z |- X -> Y

which of course gets used all over the place. And naturally it corresponds to rules in natural deduction or sequent calculi where the deduction theorem is taken as a defining characteristic of the logic.

I suppose that shows why the latter two are in some sense nicer than the former, as well. With a Hilbert system, you’d take your axioms and modus ponens, and define |- in terms of them, and show that the logic has this nice categorical structure. Natural deduction and sequent calculi just start with, “logics (are defined to) have this nice categorical structure,” and go from there.

]]> Comment on Categorical Logic I by Brent Yorgey http://analogical-engine.com/wordpress/?p=98&cpage=1#comment-10 Brent Yorgey Tue, 08 Dec 2009 21:45:54 +0000 http://analogical-engine.com/?p=98#comment-10 Sweet! As much as I love category theory and logic, you'd think I would have thought of this or seen it presented as an example of a category before---but I haven't. I look forward to reading the rest. Sweet! As much as I love category theory and logic, you’d think I would have thought of this or seen it presented as an example of a category before—but I haven’t. I look forward to reading the rest.

]]> Comment on Permutations by Robin http://analogical-engine.com/wordpress/?p=6&cpage=1#comment-6 Robin Fri, 13 Nov 2009 12:11:19 +0000 http://analogical-engine.com/?p=6#comment-6 Thanks! This has been fixed. Its nice to know somebody is reading carefully! Thanks! This has been fixed. Its nice to know somebody is reading carefully!

]]> Comment on Permutations by Walt "BMeph" Rorie-Baety http://analogical-engine.com/wordpress/?p=6&cpage=1#comment-5 Walt "BMeph" Rorie-Baety Fri, 13 Nov 2009 05:14:32 +0000 http://analogical-engine.com/?p=6#comment-5 The bottom right string diagram of S3 should be flipped(over/upside-down). The bottom right string diagram of S3 should be flipped(over/upside-down).

]]> Comment on Vector spaces via monads part 1 by Dan Piponi http://analogical-engine.com/wordpress/?p=7&cpage=1#comment-4 Dan Piponi Tue, 10 Nov 2009 00:37:20 +0000 http://analogical-engine.com/?p=7#comment-4 Oh, and also check out <a href="http://www.cs.nott.ac.uk/~txa/publ/Relative_Monads.pdf" rel="nofollow">Monads Need Not Be Endofunctors</a>. Oh, and also check out Monads Need Not Be Endofunctors.

]]> Comment on Vector spaces via monads part 1 by Dan Piponi http://analogical-engine.com/wordpress/?p=7&cpage=1#comment-3 Dan Piponi Mon, 09 Nov 2009 19:41:14 +0000 http://analogical-engine.com/?p=7#comment-3 Cool! I think this ought to be better known. And it's a pity you have to resort to evil tricks to make it work nicely in Haskell. Cool! I think this ought to be better known. And it’s a pity you have to resort to evil tricks to make it work nicely in Haskell.

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