Timeline for How do you present a non-existence theorem?
Current License: CC BY-SA 3.0
24 events
| when toggle format | what | by | license | comment | |
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| Feb 14, 2022 at 22:09 | comment | added | PrimeRibeyeDeal | "So you think there's an X? WRONG!" | |
| Feb 14, 2022 at 19:46 | review | Close votes | |||
| Feb 18, 2022 at 10:14 | |||||
| Jul 6, 2012 at 19:11 | answer | added | Terry Tao | timeline score: 14 | |
| Jul 6, 2012 at 18:03 | comment | added | Greg Marks | My first reaction on seeing the title of this question was to recall Rota's "Ten Lessons I Wish I Had Been Taught": completely erase the blackboard, begin writing in the upper left corner, .... | |
| Jul 6, 2012 at 13:22 | answer | added | Gerald Edgar | timeline score: 2 | |
| Jul 6, 2012 at 12:48 | history | edited | David White | CC BY-SA 3.0 |
Edited numerous typos and grammatical mistakes, since the question was on the front-page anyway
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| Jul 6, 2012 at 8:44 | history | edited | Federico Poloni |
http://meta.mathoverflow.net/discussion/34/5/tag-mergerename-requests/#Item_28
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| Oct 28, 2011 at 18:51 | comment | added | Colin Reid | I've always had the opposite prejudice. An isolated example can feel like just an amusing curiosity that people have stumbled across without understanding, or at best a warning that things are not as clear-cut as you might imagine. They feel more like questions than answers. Only a rare few examples are so important as to become the logical foundation of a theory (as opposed to being the historical inspiration). General statements (including classification results) on the other hand feel 'deeper' in setting out the laws of the land, spelling out what must happen and what cannot happen. | |
| Oct 28, 2011 at 14:07 | comment | added | Ramsey | This reminds me of a friend in grad school who would talk about proving amazing theorems about elliptic curves of large genus. | |
| Oct 28, 2011 at 7:22 | comment | added | Pietro Majer | My old dream is to make a short course on some non-existing objects X, say non commutative finite fields or so, stating and proving the non-existence result only at the last lesson, as the Classification Theorem for the objects $X$. | |
| Oct 27, 2011 at 23:19 | answer | added | Paul Siegel | timeline score: 16 | |
| Oct 27, 2011 at 22:36 | answer | added | fedja | timeline score: 29 | |
| Oct 27, 2011 at 22:15 | comment | added | Ramsey | I have to say, when I learned Liouville's Theorem, it was presented very much as a non-existence theorem. And it was shocking to me, which led me to come to regard complex-differentiability as something along the lines of an "insanely strong" hypothesis. I'm not sure what "a good one" means, but I can say that FLT feels very different to me because the hypothesis $x^{691}+y^{691} = z^{691}$ does feel so much strong as just kind of random (in the colloquial usage of the term, naturally...). I learned something from the statement of Liouville, not FLT. (the proof on the other hand...) | |
| Oct 27, 2011 at 20:49 | comment | added | MTS | I think that some of this is missing the point. Of course, we all know that there are logically equivalent ways of stating propositions using different quantifiers, etc. The question is about presentation. | |
| Oct 27, 2011 at 20:41 | answer | added | Will Jagy | timeline score: 15 | |
| Oct 27, 2011 at 20:30 | answer | added | MTS | timeline score: 5 | |
| Oct 27, 2011 at 20:29 | comment | added | Robert Israel | Any theorem of the form "all A such that B satisfy C" can be equivalently considered as non-existence theorems: "there is no A such that B and not-C". Whether the result is interesting depends, I guess, on whether people looking at things that satisfy some of the hypotheses might wonder whether they could satisfy the others. | |
| Oct 27, 2011 at 20:23 | comment | added | Thierry Zell | Off the top of my head, I don't see why you want to stick these non-existence theorems in a particular category (unlike, say, existence theorems). I mean, if you prove that no object can have property $p \wedge q$, you might as well say every object has property $\neg p \vee \neg q$, if you want to make the theorem more "positive". So Liouville's theorem says that a non-constant entire function must be unbounded, which, as far as I am concerned, is a positive and not-so-obvious statement about entire functions. | |
| Oct 27, 2011 at 20:21 | comment | added | user9072 | You mention Liouville's Theorem. In my impression it not typically, or at least not always, presented as a nonexistence theorem. One can say: If the function is nonconstant and entire, then it is unbounded. Or, if the function is bounded and entire, then it is constant. Perhaps, this is a half-way answer to your question. If possible, rephrase the nonexistence theorem and see whether the implication seems interesting. | |
| Oct 27, 2011 at 20:20 | comment | added | Will Jagy | It is a good deal of work, but one might try to present examples with one or more hypotheses weakened slightly. Not guaranteed to work, it may be another article to describe even one such example. For Liouville, I might discuss linear growth and polynomial growth. | |
| Oct 27, 2011 at 20:20 | comment | added | Cam McLeman | I'm not sure the concept is well-defined. In your specific example, you could just re-write this as "Bounded entire functions are constant," so in this sense, every theorem of the form "Blah always happens" is a non-existence result. Even among the most famous non-existence results, say the non-existence of a radical formula for solving the quintic, is just an existence result in the sense that it proves the existences of a non-solvable qunitic. Or is this a non-existence result, since it shows the non-existence of a sub-normal series of S5...you see my point. | |
| Oct 27, 2011 at 20:18 | comment | added | Felipe Voloch | Question is rather vague. There are no solutions to the equation $x^n+y^n=z^n$ in positive integers $x,y,z,n$ if $n>2$. Why do we care?The proof is kind of cool. | |
| Oct 27, 2011 at 20:13 | comment | added | Yemon Choi | Having recently proved something along these lines, I'd be interested in seeing people's answers | |
| Oct 27, 2011 at 20:09 | history | asked | Yuhao Huang | CC BY-SA 3.0 |