Timeline for An example of a proof that is explanatory but not beautiful? (or vice versa)
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| Aug 22, 2023 at 16:04 | comment | added | Timothy Chow | Concrete examples of what Andrés E. Caicedo is referring to include Helfgott's proof of the ternary Goldbach conjecture and Catalan's conjecture. In both cases, reducing the size of the finite computation involved significant conceptual advances. Arguably, therefore, the earlier arguments were not entirely "explanatory." | |
| Jul 25, 2013 at 6:48 | comment | added | Timothy Chow | I think that in your number-theoretic example, it's still reasonable to hold out hope that the computational residuum can be eliminated by a conceptual advance. I intentionally picked an example where there are heuristic reasons, based on computational complexity, for believing that there is no purely conceptual argument, if for no other reason than that any proof must be too long to be conceptual. | |
| Jul 25, 2013 at 6:38 | comment | added | Andrés E. Caicedo | (A different issue is whether such a proof is not beautiful. I am not addressing this here.) | |
| Jul 25, 2013 at 6:37 | comment | added | Andrés E. Caicedo | A proof is produced that very large $n$s have the required property. The proof explains why this is so, but only applies to large values of $n$, say $n>M$. We then complete the proof by extending our numerical computations, until we reach $M$. But a proof like this, though explanatory, seems lacking, as a "true explanation" should perhaps address the small numbers (with something other than a table), since they were what made us suspect the conjecture to begin with. Yes, what is lacking is a more conceptual argument. Somehow it seems unsatisfactory to call the current proof explanatory. | |
| Jul 25, 2013 at 6:34 | comment | added | Andrés E. Caicedo | Though I agree, there is something that bothers me here. Let me try to illustrate it using a different example: Many results in, say, analytic number theory, assert all $n$ have some property. Perhaps this was conjectured based on copious numerical evidence. (Cont.) | |
| S Jul 25, 2013 at 6:24 | history | answered | Timothy Chow | CC BY-SA 3.0 | |
| S Jul 25, 2013 at 6:24 | history | made wiki | Post Made Community Wiki by Timothy Chow |