Comments on: Jenga mathematics http://www.johndcook.com/blog/2008/05/11/jenga-mathematics/ The blog of John D. Cook Thu, 21 Aug 2008 18:10:17 +0000 http://wordpress.org/?v=2.3.3 By: DD http://www.johndcook.com/blog/2008/05/11/jenga-mathematics/#comment-474 DD Tue, 13 May 2008 00:16:01 +0000 http://www.johndcook.com/blog/2008/05/11/jenga-mathematics/#comment-474 Oooo... How about the Riemann-Roch-Atiyah-Hirzebruch (did I leave anyone out?) theorem? Oooo… How about the Riemann-Roch-Atiyah-Hirzebruch (did I leave anyone out?) theorem?

]]> By: JPR http://www.johndcook.com/blog/2008/05/11/jenga-mathematics/#comment-466 JPR Mon, 12 May 2008 13:25:59 +0000 http://www.johndcook.com/blog/2008/05/11/jenga-mathematics/#comment-466 The axioms for homology theory come to mind, too. These are first verified as a theorem for a simple case such as simplicial homology but then become the defining axioms for any generic homology theory. Or much more simply, think of the axioms for a metric given that they are first known to hold in usual Euclidean space. The axioms for homology theory come to mind, too. These are first verified as a theorem for a simple case such as simplicial homology but then become the defining axioms for any generic homology theory. Or much more simply, think of the axioms for a metric given that they are first known to hold in usual Euclidean space.

]]> By: Jeremy Henty http://www.johndcook.com/blog/2008/05/11/jenga-mathematics/#comment-458 Jeremy Henty Mon, 12 May 2008 10:27:38 +0000 http://www.johndcook.com/blog/2008/05/11/jenga-mathematics/#comment-458 I think the notion of "semi-locally 1-connected" topological space is Jenga mathematics; it's exactly the condition on the base space of a covering map that makes homotopy lifting work. Unless it has other consequences I don't know about. I think the notion of “semi-locally 1-connected” topological space is Jenga mathematics; it’s exactly
the condition on the base space of a covering map that makes homotopy lifting work. Unless it has
other consequences I don’t know about.

]]> By: John http://www.johndcook.com/blog/2008/05/11/jenga-mathematics/#comment-457 John Mon, 12 May 2008 09:44:59 +0000 http://www.johndcook.com/blog/2008/05/11/jenga-mathematics/#comment-457 The Stone-Weierstrass theorem is an excellent example. Stone generalized the Weierstass approximation theorem, but the essential idea belonged to Weierstrass; Stone shouldn't get top billing. I agree with Thomas Nyberg that Stone's generalization is useful. Stone not only weakened the hypotheses, he simplified the proof. But the S-W theorem is starting to become inverted: some of the proof details are exposed in the hypothesis. Further generalizations are less original and more inverted. Many theorems from number theory were turned into abstract algebra theorems simply by applying vocabulary that didn't exist when the theorem was originally stated. That's valuable, but the theorem shouldn't be named after the person who updated the language. I've tried to think of egregious examples of inverted proofs, but such theorems are inherently unmemorable. The Stone-Weierstrass theorem is an excellent example. Stone generalized the Weierstass approximation theorem, but the essential idea belonged to Weierstrass; Stone shouldn’t get top billing.

I agree with Thomas Nyberg that Stone’s generalization is useful. Stone not only weakened the hypotheses, he simplified the proof. But the S-W theorem is starting to become inverted: some of the proof details are exposed in the hypothesis. Further generalizations are less original and more inverted.

Many theorems from number theory were turned into abstract algebra theorems simply by applying vocabulary that didn’t exist when the theorem was originally stated. That’s valuable, but the theorem shouldn’t be named after the person who updated the language.

I’ve tried to think of egregious examples of inverted proofs, but such theorems are inherently unmemorable.

]]> By: Thomas Nyberg http://www.johndcook.com/blog/2008/05/11/jenga-mathematics/#comment-454 Thomas Nyberg Mon, 12 May 2008 04:55:35 +0000 http://www.johndcook.com/blog/2008/05/11/jenga-mathematics/#comment-454 Stone-Weierstrass theorem is an example (though I wouldn't call the generalization useless...). Stone-Weierstrass theorem is an example (though I wouldn’t call the generalization useless…).

]]> By: jb http://www.johndcook.com/blog/2008/05/11/jenga-mathematics/#comment-444 jb Sun, 11 May 2008 19:53:06 +0000 http://www.johndcook.com/blog/2008/05/11/jenga-mathematics/#comment-444 Can you give some examples? Can you give some examples?

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