Timeline for How do you present a non-existence theorem?
Current License: CC BY-SA 3.0
10 events
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| Feb 14, 2022 at 23:13 | comment | added | Kostya_I | @PaulSiegel, you are wrong about FTA. Depending on who you ask, the first correct proof was in 1799 by Gauss, in 1814 by Argand, in 1816 by Gauss again, or in 1819 by Argand. If there are "gaps" in these proofs, they are trivially fixed by nowadays standard analysis tools; e.g. the 1816 Gauss proof relies on the existence of a root of an odd-degree real polynomial, and Argand 1819 implicitly uses the existence of a maximum of a continuous function over a compact. But so does the proof by Liouville's theorem. Liouville was born in 1809. | |
| Jul 6, 2012 at 17:32 | comment | added | Lee Mosher | I took a little liberty here, correcting what I THINK was just a logical blooper in the first sentence. However, if I changed the intended meaning, feel free to revert. | |
| Jul 6, 2012 at 13:36 | comment | added | user2529 | I like this answer as it contrasts classification results with non-existence theorems. Classification results can be viewed as a kind of non-existence theorem. | |
| Jul 6, 2012 at 13:15 | history | edited | Lee Mosher | CC BY-SA 3.0 |
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| Oct 28, 2011 at 12:25 | comment | added | Paul Siegel | I'm not too confident about my knowledge of history, but I think Liouville's theorem led to one of the first correct proofs of the FTA and the non-existence of retractions was the basis of the original proof of the BFT. I do not claim that non-existence theorems are unavoidable, only that the philosophy of finding truths by whittling away falsehoods actually does work. | |
| Oct 28, 2011 at 7:34 | comment | added | Denis Serre | But a nice application of Liouville's Theorem is the description of the graded algebra of meromorphic functions in ${\mathbb C}$ with two given periods $1$ and $\omega$. | |
| Oct 28, 2011 at 7:27 | comment | added | Denis Serre | But we don't need Liouville's Theorem to prove the fundamental theorem of Algebra, and we don't need the non-existence of retractions from the disk to the circle to establish Brouwer's fixed point theorem. | |
| Oct 28, 2011 at 0:57 | comment | added | Yemon Choi | I think this answer comes closest to my own not-really-crystallised feelings. For instance, some problems in Banach and operator algebras ask if certain pathological things exist, not necessarily because such a putative example could be used for anything else, but because its existence would give us some kind of perspective on what we already know in the area, and its non-existence would give us hope that somehow all Boojums are of a form we feel we understand. | |
| Oct 28, 2011 at 0:00 | comment | added | David Roberts♦ | Speaking of the classification theorem, there is a huge difference between saying 'there are a finite number of sporadic groups', 'there are less than N sporadic groups, for some enormous N', and 'there are exactly 27 sporadic groups', and 'there are exactly 27 sporadic groups and we know what they are'. There are other examples in number theory where we can't even make the first statement, even though we only know a small number of examples (Fermat primes are my favourite). Making a contrast like this is handy. | |
| Oct 27, 2011 at 23:19 | history | answered | Paul Siegel | CC BY-SA 3.0 |