Timeline for Proofs that require fundamentally new ways of thinking
Current License: CC BY-SA 4.0
14 events
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| Jan 1, 2022 at 0:47 | history | edited | LSpice | CC BY-SA 4.0 |
Typos, while this is on the front page
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| Oct 5, 2021 at 5:34 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
fixed arxiv front-end links
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| Dec 13, 2010 at 16:56 | comment | added | Gil Kalai | [Cont:] There is another difficulty which is not so much about "new" but about "fundamentally". a') Putting a metric on ideas and telling what is really FN seems also difficult. I like to think about mathematics as a fractal-like beast and in this thinking you can have an idea which is FN in one scale which is almost identical to another idea looked at another scale. | |
| Dec 11, 2010 at 19:40 | comment | added | Gil Kalai | a) it is very difficult; understanding the flow of ideas and influences is extremely hard; b) I do not see how it can be useful. Let's say that I understand why Dvir's and Guth's results are FN while Guth and Katz is not FN. This seems as useful for achieving something similar to their achievements as a detailed understanding of the choices of last week's lottery winner. c) (when we consider recent discoveries) It is also personal, and I am not sure we have good tools to discuss such matters avoiding it being loaded. | |
| Dec 11, 2010 at 19:38 | comment | added | Gil Kalai | That's very interesting account, Terry. (The account still agrees with 1 & 2 & 3 being reasonable reactions to Dvir's discovery and 1 & 2 perhaps correct-in-hindsight.) Regarding the general issue, I think that talking about what computers can or cannot do in the context of this question is vastly premature. Forgetting about computers, we can still discuss what is fundamentally new (FN) and what is not but this I also find rather discomfortable for various reasons[cont]: | |
| Dec 11, 2010 at 17:12 | comment | added | Terry Tao | Perhaps what this example shows is that a computer trying to generate mathematical progress has to look at more than just the 1-skeleton of mathematics (B is solved by C; A is close to B; hence A might be solved by C) but also at the 2-skeleton (B is solved by C; D is to A as C is to B; hence A might be solved by D) or possibly even higher order skeletons. It seems unlikely though that these possibilities can be searched through systematically in polynomial time, without the speedups afforded by human insight... | |
| Dec 11, 2010 at 16:48 | comment | added | Terry Tao | In other words, his contribution was that D:A=C:B (algebraic topology is to continuous incidence geometry as algebraic geometry is to discrete incidence geometry), which was definitely a very different way of thinking about these four concepts that was totally absent in previous work. (After Guth's work, it is now "obvious" in retrospect, of course.) | |
| Dec 11, 2010 at 16:47 | comment | added | Terry Tao | Gil, many people (including myself) tried options (1) and (3) (with C equal to algebraic geometry, and more precisely the polynomial method), but for continuous problems (such as incidences between balls and tubes, which is basically what Kakeya is) it failed dramatically. Guth's breakthrough was to observe that A should be attacked instead using method D (algebraic topology, and more precisely the polynomial Ham sandwich theorem). [Cont] | |
| Dec 11, 2010 at 12:03 | comment | added | Gil Kalai | I am not sure Guth's work "broke through" any conventional wisdom. Suppose that you have a famous problem A and a new related problem B and you believe that 1) Problem A is very hard, 2) Progress for problems A and B is very related. Now, Problem B is easily settled using method C. This is in some tension with your earlier beliefs so you need to update them. So the conventional wisdom (or Bayesian thinking) will lead you to think that: 1) Method C may be useful for problem A; 2) Maybe problem A and B are not as closely related as we believed, 3) Maybe problem A is not as hard as we believed. | |
| Dec 10, 2010 at 13:30 | comment | added | Gil Kalai | But, Terry, are the adjectives "radical" or "fundamentally new" really justified in the description of any of these examples? and of our business as a whole? | |
| Dec 9, 2010 at 18:07 | history | edited | Terry Tao | CC BY-SA 2.5 |
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| Dec 9, 2010 at 18:01 | history | edited | Terry Tao | CC BY-SA 2.5 |
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| Dec 9, 2010 at 17:53 | history | edited | Terry Tao | CC BY-SA 2.5 |
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| Dec 9, 2010 at 17:47 | history | answered | Terry Tao | CC BY-SA 2.5 |