Timeline for Proofs that require fundamentally new ways of thinking
Current License: CC BY-SA 3.0
102 events
| when toggle format | what | by | license | comment | |
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| Jan 1, 2022 at 0:24 | answer | added | Cloudscape | timeline score: 3 | |
| Mar 1, 2020 at 20:25 | answer | added | the L | timeline score: 6 | |
| Mar 1, 2020 at 17:00 | answer | added | K Hughes | timeline score: 1 | |
| Nov 12, 2019 at 19:53 | answer | added | EGME | timeline score: 0 | |
| Mar 26, 2019 at 0:00 | review | Close votes | |||
| Mar 26, 2019 at 2:31 | |||||
| Aug 21, 2018 at 14:24 | history | edited | Kimball |
removed proofs tag
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| Dec 11, 2017 at 10:03 | history | reopened |
Stefan Kohl♦ Henry.L Yemon Choi Lucia David Handelman |
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| Dec 10, 2017 at 14:46 | review | Reopen votes | |||
| Dec 11, 2017 at 10:03 | |||||
| Jun 29, 2017 at 1:00 | review | Reopen votes | |||
| Jun 29, 2017 at 2:23 | |||||
| Oct 31, 2015 at 2:06 | review | Reopen votes | |||
| Oct 31, 2015 at 20:59 | |||||
| Jul 24, 2015 at 23:27 | comment | added | j0equ1nn | Wow. So you can get 147 votes and still be closed as off-topic. Doesn't the fact that 147 math researchers liked it, alone, attest to its relevance? ...anyway surprised nobody mentioned Cantor's cardinality proofs. Before that, proofs by contradiction were not considered valid. | |
| May 29, 2015 at 11:46 | review | Suggested edits | |||
| May 29, 2015 at 12:45 | |||||
| Feb 15, 2014 at 14:48 | review | Reopen votes | |||
| Feb 16, 2014 at 14:23 | |||||
| Oct 13, 2013 at 23:05 | history | closed |
David White Suvrit Chris Godsil Vidit Nanda Fernando Muro |
Not suitable for this site | |
| Oct 13, 2013 at 19:08 | review | Close votes | |||
| Oct 13, 2013 at 22:13 | |||||
| Oct 13, 2013 at 18:52 | comment | added | David White | It seems to me that this question has been around a long time and is unlikely garner new answers of high quality. It also seems unlikely most would even read new answers. Furthermore, nowadays I imagine a question like this would be closed as too broad, and if we close this then we'll discourage questions like it in the future. So I'm voting to close. | |
| Oct 13, 2013 at 4:37 | answer | added | Koushik | timeline score: 1 | |
| Aug 15, 2013 at 9:59 | answer | added | kjetil b halvorsen | timeline score: 3 | |
| Aug 14, 2013 at 22:51 | history | edited | Ricardo Andrade | CC BY-SA 3.0 |
added hopefully relevant tags (since question was on the front page)
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| Aug 14, 2013 at 17:48 | answer | added | Liviu Nicolaescu | timeline score: 6 | |
| Jul 3, 2013 at 5:14 | answer | added | Włodzimierz Holsztyński | timeline score: 5 | |
| Jun 27, 2013 at 16:47 | answer | added | Ronnie Brown | timeline score: 2 | |
| May 9, 2013 at 23:09 | answer | added | Dan Sălăjan | timeline score: 2 | |
| Jul 26, 2012 at 7:26 | answer | added | Jesko Hüttenhain | timeline score: 12 | |
| Jul 26, 2012 at 5:53 | answer | added | none | timeline score: 6 | |
| May 25, 2012 at 7:25 | answer | added | Daniel Moskovich | timeline score: 7 | |
| May 22, 2012 at 16:27 | answer | added | Turbo | timeline score: 2 | |
| Apr 16, 2012 at 1:00 | answer | added | FelipeG | timeline score: 0 | |
| Mar 3, 2012 at 18:55 | answer | added | Grant Rotskoff | timeline score: 10 | |
| Jan 2, 2012 at 20:33 | answer | added | Nate Ackerman | timeline score: 2 | |
| Aug 30, 2011 at 16:40 | answer | added | Buschi Sergio | timeline score: 5 | |
| Aug 30, 2011 at 16:18 | answer | added | Phil Isett | timeline score: 6 | |
| Aug 29, 2011 at 14:45 | answer | added | Pace Nielsen | timeline score: 14 | |
| Aug 29, 2011 at 9:18 | answer | added | Manya | timeline score: 10 | |
| Jan 19, 2011 at 11:37 | answer | added | Alan Bundy | timeline score: 3 | |
| Jan 18, 2011 at 9:50 | answer | added | jzadeh | timeline score: 1 | |
| Dec 17, 2010 at 13:21 | answer | added | Gil Kalai | timeline score: 3 | |
| Dec 17, 2010 at 12:02 | answer | added | Kevin H. Lin | timeline score: 13 | |
| Dec 17, 2010 at 10:37 | answer | added | Zhen Lin | timeline score: 5 | |
| Dec 14, 2010 at 11:32 | comment | added | Gil Kalai | It is surprising how (successful) fundamentally new ways of thinking are clustered. Cantor idea is FN and yet very closely related to the ancient liar paradox, in this cluster also Russell's proof that his set is not a set and Goedel's theorem. The FN idea of non constructive methods, and in particular, probabilistic proofs. Homology is FN method in classifying topological spaces and fixed point theorems, and then (Emerton's answer) through fixed point theorems in number theory, and also there is the FN mysterious method to classify representations based on actions on homology. | |
| Dec 13, 2010 at 19:24 | comment | added | Gil Kalai | One more (rather obvious) remark is that sometimes "fundamental new way of thinking" in general, and such new ways that lead to proofs, emerges gradually from a large body of work by many people. | |
| Dec 13, 2010 at 19:16 | comment | added | Gil Kalai | A related MO question: mathoverflow.net/questions/21562/… | |
| Dec 13, 2010 at 19:08 | answer | added | Dirk | timeline score: 93 | |
| Dec 13, 2010 at 10:23 | answer | added | Unknown | timeline score: 2 | |
| Dec 12, 2010 at 17:59 | answer | added | Victor Miller | timeline score: 5 | |
| Dec 12, 2010 at 16:53 | answer | added | Unknown | timeline score: 0 | |
| Dec 12, 2010 at 9:09 | answer | added | Unknown | timeline score: 0 | |
| Dec 11, 2010 at 20:57 | answer | added | Ben Krause | timeline score: 6 | |
| Dec 11, 2010 at 19:30 | answer | added | Vamsi | timeline score: 5 | |
| Dec 11, 2010 at 18:44 | answer | added | Deane Yang | timeline score: 97 | |
| Dec 11, 2010 at 18:26 | answer | added | Deane Yang | timeline score: 27 | |
| Dec 11, 2010 at 11:09 | answer | added | Andreas Thom | timeline score: 22 | |
| Dec 11, 2010 at 11:06 | answer | added | Andreas Thom | timeline score: 18 | |
| Dec 11, 2010 at 8:04 | comment | added | Gil Kalai | A comparison of the two scenarios is relevant for trying to understand what computers can do. Even if we agree that "fundamentally new (=FN)" arguments is the hardest element to automatize, it still seems harder (for a computer and perhaps also for a human) to find an FN argument at an unknown place down the proof than right at the beginning. | |
| Dec 11, 2010 at 7:17 | comment | added | gowers | @Gil: I agree that the two scenarios exist. I'm not sure I see the need to compare them. | |
| Dec 11, 2010 at 7:16 | comment | added | gowers | @Ryan, as I hope my remarks make clear, I completely agree. In other words, I hope that by asking for fundamental newness I have set an impossible challenge. Maybe I could refine the question further: I am looking for proofs that appear to be so different from what went before that they require some special and characteristically human "genius" to be discovered. | |
| Dec 11, 2010 at 5:17 | comment | added | Ryan Budney | My feeling is that when someone says "X is fundamentally new" (for various values of X) in reference to some mathematics, IMO this usually demands as a prerequisite that one has a pretty narrow perspective on the kinds of thinking that came beforehand in order to believe the statement. This doesn't take anything away from novel mathematics, it's just that fundamentally new is almost always too hyperbolic expression for the mathematics it describes. I imagine the main reason mathematicians use such hyperbolic terminology is that hype draws people's attention, and that helps ideas propagate. | |
| Dec 10, 2010 at 15:13 | comment | added | Gil Kalai | I see another conceptual difficulty with the spirit behind the question: Suppose we have to compare two proofs for two a priori equally important theorems. The first proof is based on a fundamentally new way of thinking (whatever it means, but let's assume that it is meaningful). In the second proof the proof of Lemma 12.7 is based on a fundamentally new way of thinking. How do we compare these two scenarios? | |
| Dec 10, 2010 at 14:22 | answer | added | John Sidles | timeline score: 15 | |
| Dec 10, 2010 at 12:42 | comment | added | Gil Kalai | While this is a common belief even for strong proponents of computerized mathematics, it is not clear if these types of ideas/proofs would be harder for computer systems (fully automatic or interactive). For example, the "probabilistic method" had major impact and led to surprising proofs/concepts in different areas in different times. So the idea: "Use a probabilistic argument to prove the existence of the required objects" or "Add probabilistic ingredient to this notion" could have been offered (and still can be offered) rater automatically. | |
| Dec 10, 2010 at 10:01 | answer | added | Dylan Wilson | timeline score: 48 | |
| Dec 10, 2010 at 8:39 | answer | added | Sebastian | timeline score: 25 | |
| Dec 10, 2010 at 4:05 | answer | added | paul Monsky | timeline score: 15 | |
| Dec 10, 2010 at 4:05 | answer | added | P C | timeline score: 6 | |
| Dec 10, 2010 at 3:32 | answer | added | Emerton | timeline score: 27 | |
| Dec 10, 2010 at 2:52 | answer | added | Timothy Chow | timeline score: 45 | |
| Dec 10, 2010 at 1:21 | answer | added | Timothy Wagner | timeline score: 4 | |
| Dec 9, 2010 at 23:52 | answer | added | Qiaochu Yuan | timeline score: 22 | |
| Dec 9, 2010 at 23:14 | answer | added | Michael Nielsen | timeline score: 18 | |
| Dec 9, 2010 at 22:15 | answer | added | Gerry Myerson | timeline score: 12 | |
| Dec 9, 2010 at 21:20 | answer | added | Akhil Mathew | timeline score: 42 | |
| Dec 9, 2010 at 19:45 | answer | added | J.C. Ottem | timeline score: 12 | |
| Dec 9, 2010 at 19:42 | answer | added | Johannes Ebert | timeline score: 95 | |
| Dec 9, 2010 at 19:09 | comment | added | gowers | @Luke: My motivation is that I believe that computers ought to be able to do mathematics. To explore that view, it is very helpful to look at problems of this type, since either one will be able to explain how certain ideas that seem to come from nowhere can in fact be generated in an explicable way, or one will end up with a more precise understanding of the difficulties involved. Of course, I'm hoping for the former. | |
| Dec 9, 2010 at 19:07 | answer | added | M.G. | timeline score: 2 | |
| Dec 9, 2010 at 19:04 | answer | added | Fedor Petrov | timeline score: 22 | |
| Dec 9, 2010 at 19:02 | comment | added | Jon Bannon | What about Hilbert's approach to the "fundamental problem of invariant theory"? I.e. the one that supposedly provoked the remark "This is not mathematics, but theology". | |
| Dec 9, 2010 at 18:56 | answer | added | Orbicular | timeline score: 25 | |
| Dec 9, 2010 at 18:52 | answer | added | Harry Gindi | timeline score: 19 | |
| Dec 9, 2010 at 18:46 | answer | added | Tim Dokchitser | timeline score: 143 | |
| Dec 9, 2010 at 18:39 | answer | added | P C | timeline score: 9 | |
| Dec 9, 2010 at 18:11 | comment | added | Luke Grecki | I'm still a bit confused about your motivation. Are you trying to understand why you think certain proofs are hard to be generated by computers? Or are you interested in this list for its own sake? Or something else? | |
| Dec 9, 2010 at 18:04 | comment | added | gowers | I agree that they are difficult, but in a sense what I am looking for is problems that isolate as well as possible whatever it is that humans are supposedly better at than computers. Those big problems are too large and multifaceted to serve that purpose. You could say that I am looking for "first non-trivial examples" rather than just massively hard examples. | |
| Dec 9, 2010 at 17:50 | answer | added | John Stillwell | timeline score: 40 | |
| Dec 9, 2010 at 17:47 | answer | added | Terry Tao | timeline score: 34 | |
| Dec 9, 2010 at 17:34 | comment | added | Minhyong Kim | In response to edit: On the other hand, I think those big theorems are still reasonable instances of proofs that are difficult to imagine for a computer! Incidentally, regarding your example 2, it seems to me Dirichlet's theorem on primes in arithmetic progressions might be a better example in the same vein. | |
| Dec 9, 2010 at 17:31 | answer | added | Kimball | timeline score: 58 | |
| Dec 9, 2010 at 17:25 | answer | added | Michael Hardy | timeline score: 116 | |
| Dec 9, 2010 at 16:58 | answer | added | Gerhard Paseman | timeline score: 110 | |
| Dec 9, 2010 at 16:52 | answer | added | Andrés E. Caicedo | timeline score: 91 | |
| Dec 9, 2010 at 16:47 | answer | added | Gerhard Paseman | timeline score: 5 | |
| Dec 9, 2010 at 16:42 | comment | added | D. Savitt | Never mind the application of Fourier analysis to number theory -- how about the invention of Fourier analysis itself, to study the heat equation! More recently, if you count the application of complex analysis to prove the prime number theorem, then you might also count the application of model theory to prove results in arithmetic geometry (e.g. Hrushovski's proof of Mordell-Lang for function fields). | |
| Dec 9, 2010 at 16:40 | history | edited | gowers | CC BY-SA 2.5 |
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| Dec 9, 2010 at 16:21 | comment | added | roy smith | Of course, there was apparently a surprising and simple insight involved in the proof of FLT, namely Frey's idea that a solution triple would give rise to a rather exotic elliptic curve. It seems to have been this insight that brought a previously eccentric seeming problem at least potentially within the reach of the powerful and elaborate tradition referred to. So perhaps that was a new way of thinking at least about what ideas were involved in FLT. | |
| Dec 9, 2010 at 16:11 | answer | added | Roy Maclean | timeline score: 4 | |
| Dec 9, 2010 at 16:07 | answer | added | Steven Heston | timeline score: 6 | |
| Dec 9, 2010 at 16:06 | answer | added | anonymous | timeline score: 16 | |
| Dec 9, 2010 at 15:50 | history | edited | gowers | CC BY-SA 2.5 |
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| Dec 9, 2010 at 15:43 | history | edited | gowers | CC BY-SA 2.5 |
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| Dec 9, 2010 at 15:30 | comment | added | Kevin Buzzard | @Minhyong: right! All of these proofs involved fundamental new insights, but probably the proof of an arbitrary statement that was known to be hard (in the sense that "the usual methods don't seem to work") and was then proved anyway ("because a new method was discovered") seem to fit the bill... | |
| Dec 9, 2010 at 15:18 | comment | added | Minhyong Kim | Perhaps you could make the requirements a bit more precise. The most obvious examples that come to mind from number theory are proofs that are ingenious but also very involved, arising from a rather elaborate tradition, like Wiles' proof of Fermat's last theorem, Faltings' proof of the Mordell conjecture, or Ngo's proof of the fundamental lemma. But somehow, I'm guessing that such complicated replies are not what you have in mind. | |
| Dec 9, 2010 at 15:08 | history | asked | gowers | CC BY-SA 2.5 |