Timeline for Theorems which say "such and such method cannot possibly prove FLT"
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 15, 2010 at 20:30 | comment | added | T.. | A catalogue of "10 ways to not prove FLT" would be publishable if it went beyond the well known familiar observations such as congruences not sufficing. | |
| Jul 14, 2010 at 10:18 | comment | added | Charles Matthews | I can see a narrow use for such results, in deterring ex nihilo efforts to reprove FLT with methods that are bound to fail. I can't get excited about them. Would they even be publishable? | |
| Jul 13, 2010 at 19:30 | comment | added | T.. | Well, considering FLT itself, "a greater danger of spending too much time on futile advancement of methods that are provably insufficient" seems a pretty accurate diagnosis. I thought you were making a more general point, however. | |
| Jul 13, 2010 at 18:06 | comment | added | Charles Matthews | @T. Well, consider also that the concept of "depth" has a longer history in analytic number theory, and proof-theoretic analyses as well as genuine advances in elementary techniques (I'm thinking of the Prime Number Theorem) have had quite a thorough discussion in this context. I don't see that P =? NP is really comparable. | |
| Jul 13, 2010 at 17:39 | comment | added | T.. | Charles, consider your comments in the context of P=NP. There, a proof is unknown, but an enormous amount of effort has been invested in showing that certain lines of attack are not powerful enough to solve the problem. Limitative results may be dangerous if they inhibit certain lines of thought but there seems to be a greater danger of spending too much time on futile advancement of methods that are provably insufficient. See the P=NP discussion at the end of my answer. | |
| Jul 13, 2010 at 13:47 | comment | added | Bill Dubuque | Simple examples of such "elementary" unwindings are ubiquitous in elementary number theory proofs, where higher-order concepts such as (principal) ideals or modules are hidden in the background. For example, consider the well-known Zermelo-Lindenmann elementary proof [1] that Z is a UFD - which directly inlines a proof that an ideal is principal by Euclidean descent. Similarly [2] for irrationality proofs, which essentially inline a proof that a conductor ideal is principal (so cancellable) - so PIDs are integrally closed - a 1-line proof! [1] bit.ly/cIcw8c [2] bit.ly/aeKDlR | |
| Jul 13, 2010 at 13:03 | comment | added | JBL | I certainly hope it's not true both that such a proof exists and that we can prove it doesn't! ;) | |
| Jul 13, 2010 at 11:13 | comment | added | Pete L. Clark | +1. Phrases like "elementary proof" are being thrown around like they have agreed upon and well-understood meanings, but I don't believe this is the case. In this instance, an unstated implicit condition seems to be "proof that a number-theoretic amateur can understand, in contrast to the Taylor-Wiles proof". I find it very unlikely both that such a proof exists and that we will be able to prove that it does not exist. I could however well imagine an "elementarization" of the T-W proof which resulted in a 50,000 page truly unreadable mess. | |
| Jul 13, 2010 at 11:04 | history | answered | Charles Matthews | CC BY-SA 2.5 |