Timeline for Nonstandard analysis in probability theory
Current License: CC BY-SA 3.0
22 events
| when toggle format | what | by | license | comment | |
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| Feb 2, 2023 at 14:24 | comment | added | Willem Fouche | It should be mentioned though that NSA is quite central in Ramsey theory and combinatorial number theory. See for example Mauro Di Nasso • Isaac Goldbring • Martino Lupini Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory springer.com/series/304 | |
| Jul 6, 2015 at 17:40 | answer | added | Bartek Bozic | timeline score: 3 | |
| Apr 12, 2013 at 8:12 | answer | added | Mikhail Katz | timeline score: 8 | |
| Sep 27, 2012 at 13:39 | comment | added | Alexander Pruss | I've used NSA heuristically when doing probability theory, but then moved to standard analysis for the final proof. There may be more people doing that. | |
| Sep 26, 2012 at 13:29 | history | edited | an12 | CC BY-SA 3.0 |
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| Sep 26, 2012 at 0:00 | vote | accept | an12 | ||
| Sep 25, 2012 at 23:44 | comment | added | François G. Dorais | I am clearly biased (and not ashamed of it), but I find it nonstandard that so many people have sharp opinions about nonstandard analysis. | |
| Sep 25, 2012 at 7:32 | history | edited | an12 | CC BY-SA 3.0 |
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| Sep 25, 2012 at 2:59 | comment | added | user21349 | There are generic reasons why NSA has not had more impact, and some or all of them may apply here. (1) There were overblown claims that NSA would lead to new results in the rest of math that could never have been found without it. Not true. At most it made arguments more transparent, or allowed an argument to be made with a quantifier depth that was lower by 1. (2) Physicists and engineers, who have never stopped using infinitesimals, don't know about NSA and don't need its blessing. (3) After ca. 1890, mathematicians got out of the habit of using infinitesimals, and now they don't need them. | |
| Sep 25, 2012 at 1:44 | answer | added | Kevin O'Bryant | timeline score: 16 | |
| Sep 25, 2012 at 1:03 | history | edited | an12 | CC BY-SA 3.0 |
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| Sep 24, 2012 at 22:35 | comment | added | user21349 | @Qiaochu Yuan: But NSA is not so much a new idea as a more rigorous way of applying old ideas about infinitesimals. Fourier was already essentially doing Dirac delta functions ca. 1800, Dirac popularized them in 1930, and physicists and engineers generally still think of a Dirac delta function as a spike of infinitesimal width $\epsilon$ and height $1/\epsilon$. Distribution theory came ca. 1935. So if there is a new idea that needs to be proved worthwhile in transition costs, it's distribution theory -- which has never been widely adopted by physicists and engineers. | |
| Sep 24, 2012 at 15:58 | comment | added | kjetil b halvorsen | here is another usefull book: Nonstandard Methods in Stochastic Analysis and Mathematical Physics (Dover Books on Mathematics) by Sergio Albeverio, Jens Erik Fenstad, Raphael Hoegh-Krohn, Tom Lindstrom | |
| Sep 24, 2012 at 14:02 | comment | added | Gerald Edgar | I observed (at least at one time) that those working on nonstandard methods in probability would attend conferences and give talks NOT at probability meetings, but at logic meetings. So my impression was: they were not probabilists trying to find new methods to solve their problems; instead they were nonstandardists trying to find places to apply their method. | |
| Sep 24, 2012 at 10:21 | answer | added | Michael Greinecker | timeline score: 33 | |
| Sep 24, 2012 at 6:15 | answer | added | Robert Israel | timeline score: 24 | |
| Sep 24, 2012 at 4:48 | comment | added | an12 | @Qiaochu Yuan, I totally agree with you. I hope that my question will not be misunderstood as a criticism to probability theory. I will edit it to clarify. | |
| Sep 24, 2012 at 4:48 | history | edited | an12 | CC BY-SA 3.0 |
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| Sep 24, 2012 at 3:54 | history | edited | an12 |
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| Sep 24, 2012 at 3:13 | comment | added | R Hahn | I speculate from Geyer's intro that this approach mainly appeals to a relatively smallish group: statisticians with a desire for rigor and simultaneously very specific applications (stat inference) in mind. Learning this new approach may pay off in licensing the heuristic of thinking about all integrals as sums and all conditioning as division. I do wonder how this approach squares with recent results to the effect that there exist computable joint distributions which yield non-computable conditionals. | |
| Sep 24, 2012 at 2:00 | comment | added | Qiaochu Yuan | The burden of proof is not on probabilists to object to this approach but on this approach to show that it is so much more useful than the standard approach that adopting it will be worth the transition costs. This is a fairly general principle about new ideas, I think. | |
| Sep 24, 2012 at 1:28 | history | asked | an12 | CC BY-SA 3.0 |