Jenga is a game where you start with a tower of wooden pegs and take turns removing pegs until someone makes the tower collapse. A style of mathematics analogous to Jenga reached the height of its popularity about 40 years ago and then fell out of fashion. I use the phrase “Jenga mathematics” to refer to generalizing a well-known theorem by weakening its hypotheses, seeing how many pegs you can pull out before it falls.
Many 20th century mathematicians spent their careers going over the work of 19th century mathematicians, removing every hypothesis they could. Sometimes a 20th century mathematician would get his name tacked on to a 19th century theorem due to his Jenga accomplishments.
Taken to extremes, Jenga mathematics turns theorems inside-out and proofs become hypotheses. Natural hypotheses are replaced with a laundry list of properties necessary to make the proof work. Start with some theorem of the form “Let X be a widget. Then X has a foozle.” Go back over the proof and see just what features of a widget are needed for the proof. Then restate the theorem as “Let X have the following apparently arbitrary list of properties necessary for my proof to work. Then X has a foozle.” Never mind whether anybody can think of anything other that a widget that satisfies the hypotheses of the new theorem.
Jenga mathematics is no longer fashionable. Mathematicians still value removing unneeded hypotheses, but they’re not as willing to go to extremes to do so. They are more interested in building new towers than in removing every piece possible from old towers.
{ 6 comments… read them below or add one }
jb 05.11.08 at 13:53
Can you give some examples?
Thomas Nyberg 05.11.08 at 22:55
Stone-Weierstrass theorem is an example (though I wouldn’t call the generalization useless…).
John 05.12.08 at 03:44
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.
Jeremy Henty 05.12.08 at 04:27
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.
JPR 05.12.08 at 07:25
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.
DD 05.12.08 at 18:16
Oooo… How about the Riemann-Roch-Atiyah-Hirzebruch (did I leave anyone out?) theorem?