Timeline for The resolution of which conjecture/problem would advance Mathematics the most? [closed]
Current License: CC BY-SA 3.0
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| Mar 17, 2017 at 10:13 | history | edited | CommunityBot |
replaced http://meta.mathoverflow.net/ with https://meta.mathoverflow.net/
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| Dec 24, 2014 at 3:04 | review | Reopen votes | |||
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| Dec 17, 2014 at 3:00 | review | Reopen votes | |||
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| Dec 6, 2014 at 21:40 | review | Reopen votes | |||
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| Dec 2, 2014 at 22:12 | comment | added | Andreas Thom | Right, I didn't say it was any different. I understand what you mean but I don't understand why you doubt the legitimacy of the problem and the potential progress in case of a solution. Anyhow, I guess we can end this discussion. | |
| Dec 2, 2014 at 16:39 | comment | added | Emil Jeřábek | A proof of anything in ZF is only meaningful if ZF is consistent. The consistency of PA is not any different in this respect from any other theorem. | |
| Dec 2, 2014 at 13:39 | comment | added | Andreas Thom | My initial problem did not involve the word "consistency", I was talking about the existence of a concrete contradiction (if you want, we can call that a meta-question about mathematics). It seems that this is not a mathematical problem in your sense, and your mathematical formulation does not answer the legitimate meta-question. A proof in ZF that PA is consistent is only meaningful if ZF is consistent. In particular, it cannot have any meaning if PA is inconsistent. But certainly, this meta-question is a problem about mathematics with potential to advance our understanding of it. | |
| Dec 2, 2014 at 12:13 | comment | added | Emil Jeřábek | ... the Peano representation theorem happens to talk about the consistency of some formal system is irrelevant, all these issues arise for any result with extra-mathematical applications. | |
| Dec 2, 2014 at 12:13 | comment | added | Emil Jeřábek | ... level of elementary algebra. (2) The theorem can be interpreted as saying something about reality outside mathematics. This is a philosophical problem, which is not what this question asks about. As such, it is subject to endless debate whether applying the theorem to reality in this way makes sense in the first place, and if so, whether the reality validates the theorem (if not, mathematics may be in serious trouble), and so on. There is nothing mathematics can do to "resolve" this debate, as it is neither mathematical nor conclusively resolvable by its very nature. The fact that ... | |
| Dec 2, 2014 at 12:12 | comment | added | Emil Jeřábek | On second thought, the problem seems to be that you are constantly confusing theory and metatheory. It's probably unhelpful that the result we are talking about involves the word "consistency", which has the curious psychological effect on many mathematicians to treat it as special, and pretend they suddenly became die-hard finitists. So let's call it the Peano representation theorem instead. Now, just like many other theorems, it has two aspects. (1) It is a mathematical problem; and this is the aspect this question asks about. The problem has been solved some time ago, and it's on the ... | |
| Dec 2, 2014 at 10:52 | comment | added | Emil Jeřábek | Oh really? Show me your proof of consistency of Cantor's theory, I'm all ears. | |
| Dec 2, 2014 at 8:29 | comment | added | Andreas Thom | @EmilJeřábek: I am sure you can prove consistency of Cantor's set theory pretty much the same way as consistency of PA, using only "daily basis" naive set/model theory. And this exactly shows how useful such a proof can be (in any situation) when it comes to the question of existence of a contradiction - but you know that. Your point can merely be that this seems rather unlikely to happen (given all our positive experience with basic reasoning) and I agree (but not based on any mathematical reasoning/argument, rather some sort of belief/platonism.). | |
| Dec 1, 2014 at 23:05 | comment | added | Emil Jeřábek | Cantor’s set theory was a huge leap into an uncharted territory full of new unfamiliar concepts and pitfalls. This is not comparable to the proof of consistency of PA using the satisfaction predicate for the ring of integers, which only employs simple standard techniques that are used on daily basis all over mathematics (the most complicated being the construction of a sequence of objects $\{a_n:n\in\mathbb N\}$ by induction on $n$.) | |
| Dec 1, 2014 at 22:55 | comment | added | Andreas Thom | @EmilJeřábek: Your way to argue applies just as well to naive set theory, before Russell's paradox. Our vision was advanced considerably by the need to rethink how we set up mathematics in a way that avoids the paradox. You can "mathematically solve" these problems ten times, but a concrete contradiction would force you to start again from scratch. Call that progress/advancement or not. Anyhow, it seems unlikely to happen and my initial comment was not entirely serious anyway. | |
| Dec 1, 2014 at 22:33 | comment | added | Emil Jeřábek | @AndreasThom: It’s possible exactly as much as it is possible that the reals $0$ and $1$ are equal, or that there are only finitely many primes, or that there exists a nonconstant polynomial over $\mathbb C$ without a root or ... (you get the idea). Are any of these (equivalent) statements “conjectures/problems whose resolution would significantly advance mathematics”? Matter of opinion, but IMHO, no. For one thing, they already have been mathematically resolved. | |
| Dec 1, 2014 at 22:22 | review | Reopen votes | |||
| Dec 2, 2014 at 0:37 | |||||
| Dec 1, 2014 at 22:04 | history | edited | Kim Morrison | CC BY-SA 3.0 |
added 180 characters in body
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| Dec 1, 2014 at 22:03 | comment | added | Kim Morrison | Please, everyone, vote up the comment above about the meta post so that people see it. | |
| Dec 1, 2014 at 21:43 | comment | added | Colin Reid | In many cases, it will depend on how the conjecture is resolved (not just the answer, but the method of proof). Fermat's last theorem and the 3-dimensional Poincaré conjecture are both examples where the result proved was more important than the most famous conjecture. In the case of FLT, if a direct proof had been found before the development of modern number theory, that might have actually slowed down our understanding of mathematics. | |
| Dec 1, 2014 at 21:36 | history | closed |
Eric Wofsey Gerald Edgar Andreas Thom Stefan Kohl♦ Felipe Voloch |
Needs more focus | |
| Dec 1, 2014 at 21:08 | comment | added | Andreas Thom | @EmilJeřábek: I did not claim consistency cannot be proved. My point was just that the existence of a concrete contradiction is still possible (and I thought "are" was to dispute this.) | |
| Dec 1, 2014 at 21:08 | comment | added | Emil Jeřábek | Actually, @AndreasThom: please read Timothy Chow’s answer to the question linked by quid, it explains it quite well. If you want a quote: "In the case of Con(PA), the aforementioned "normal conditions" for removing its "open problem" status have been met, and in fact exceeded." | |
| Dec 1, 2014 at 20:57 | comment | added | Emil Jeřábek | @AndreasThom: I’m using the word in a perfectly sound technical mathematical way, I’m not talking about Gentzen’s proof, and even if I did, there is a huge difference between “securing classical number theory”, and just proving something as a theorem. Kleene’s issue is with philosophical implications of Gentzen’s proof for the purposes of Hilbert’s program, not with the mathematics involved in the proof. I’m not going to comment on the infamous Voevodsky’s talk, which is a distillation of popular misconceptions about Gödel’s theorems. | |
| Dec 1, 2014 at 20:47 | comment | added | Andreas Thom | @EmilJeřábek: You don't use the verb "are" in a mathematically sound way. Kleene on Gentzen's work "To what extent the Gentzen proof can be accepted as securing classical number theory [...] is in the present state of affairs a matter for individual judgement [...]." | |
| Dec 1, 2014 at 20:42 | comment | added | user9072 | @EmilJeřábek Does Voevodsky agree meanwhile? :-) | |
| Dec 1, 2014 at 20:27 | comment | added | Emil Jeřábek | @AndreasThom: Peano arithmetic and first order logic are consistent. That’s a (fairly easy) theorem, not a conjecture, notwithstanding a host of popular misconceptions about Gödel’s theorems. | |
| Dec 1, 2014 at 20:14 | history | edited | Ricardo Andrade |
added relevant tags
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| Dec 1, 2014 at 19:28 | answer | added | Charles Matthews | timeline score: 4 | |
| Dec 1, 2014 at 18:36 | review | Close votes | |||
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| Dec 1, 2014 at 17:55 | comment | added | Hauke Reddmann | What about specializing to applications? My life wouldn't change if the Riemann hypothesis is proven/disproven, and knowing P=/!=NP also not necessarily would have THAT impact with computers. | |
| Dec 1, 2014 at 17:45 | comment | added | Andreas Thom | What about a contradiction in Peano Arithmetics or First Order Logic? However, it's hard to argue about consequences (at all) - any way out would certainly be a true advance. | |
| Dec 1, 2014 at 17:23 | history | reopened |
David E Speyer Joel David Hamkins Karl Schwede Andrey Rekalo Neil Strickland |
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| Dec 1, 2014 at 5:43 | comment | added | Theo Johnson-Freyd | @user74230 Good to know! As I hope was clear from my comment, I am not an expert. | |
| Dec 1, 2014 at 0:54 | review | Reopen votes | |||
| Dec 1, 2014 at 10:33 | |||||
| Nov 30, 2014 at 23:28 | comment | added | Emil Jeřábek | Meta post: meta.mathoverflow.net/questions/2000/should-we-reopen . | |
| Nov 30, 2014 at 7:46 | comment | added | user74230 | @TheoJohnson-Freyd: Deligne's later Weil II provides tremendously powerful results on the cohomology of $\ell$-adic sheaves going way beyond anything that could be extracted from the standard conjectures. There's so much more to the role of $\ell$-adic cohomology in mathematics than for motives, and I've always wondered if Grothendieck ever re-evaluated his opinion after Weil II was produced (though he had left the scene by then). | |
| Nov 30, 2014 at 5:55 | history | closed |
Andy Putman Alexandre Eremenko Henry Cohn Stopple Steven Landsburg |
Opinion-based | |
| Nov 30, 2014 at 3:20 | comment | added | Theo Johnson-Freyd | My memory is that in algebraic geometry, Grothendieck hoped for a program to develop quite a lot of machinery in order to prove some outstanding (Weil?) conjectures. When Deligne proved them without all the machinery Grothendieck hoped would be developed, Grothendieck, according to the story, was quite upset: the point wasn't the result, but the rest of the theory. I bring this up to support the idea that "open conjectures that would advance mathematics" might make such an advance because they seem to need dramatically new machinery or ideas, which would then be available for other uses. | |
| Nov 30, 2014 at 2:40 | comment | added | Henry Cohn | One somewhat objective way to answer this question is by asking about the known consequences. In that case it's probably hard to beat the Riemann hypothesis and P vs. NP, but deducing consequences directly from a single statement is a pretty limited notion of advancing mathematics. The deeper question is what the proof techniques could tell us if we had them, but that's really not easy to predict. (For example, before the elementary proof of the prime number theory, people imagined it would lead to a revolution in number theory, but that turned out not to happen.) | |
| Nov 30, 2014 at 2:18 | comment | added | Joseph O'Rourke | An MO reference re @Venkataramana: "Has there been any progress on the standard conjectures on algebraic cycles?." | |
| Nov 30, 2014 at 1:36 | answer | added | Gerald Edgar | timeline score: 1 | |
| Nov 30, 2014 at 1:26 | comment | added | Venkataramana | Grothendieck's standard conjectures? | |
| Nov 30, 2014 at 0:38 | answer | added | Paul Siegel | timeline score: 13 | |
| Nov 30, 2014 at 0:30 | history | made wiki | Post Made Community Wiki by Todd Trimble | ||
| Nov 29, 2014 at 23:55 | review | Close votes | |||
| Nov 30, 2014 at 5:57 | |||||
| Nov 29, 2014 at 23:41 | comment | added | Johannas | BSD wins for me, but I am biased because it is something that I am working on. However, my biases aside, the problem is currently producing lots of advances in mathematics (e.g. progress in p-adic L-functions and Iwasawa theory by Urban et al, and all of the work by Bhargava surrounding the solution to BSD "on average"). | |
| Nov 29, 2014 at 23:38 | comment | added | Joseph O'Rourke | @AndyPutman: It is indeed both broad and opinon-based. Which does not mean we cannot learn from responses. | |
| Nov 29, 2014 at 23:36 | comment | added | Andy Putman | I feel that this question is too broad and unfocused, as well as being too "opinion-based". | |
| Nov 29, 2014 at 23:25 | comment | added | user74230 | Fermat's Last Theorem was not blocking progress, since it was an isolated (albeit historically important) puzzle whose statement was disconnected from the main developments of algebraic number theory. However, its reduction to modularity of Galois representations (a problem that, expressed in multiple ways, was a key blockage) by Ribet did provide inspiration for Wiles to work on the problem and the techniques invented to thereby prove modularity theorems indeed opened the floodgates to a tremendous number of subsequent advances. | |
| Nov 29, 2014 at 23:02 | history | asked | Joseph O'Rourke | CC BY-SA 3.0 |