Timeline for What's wrong with the surreals?
Current License: CC BY-SA 4.0
21 events
| when toggle format | what | by | license | comment | |
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| Nov 11, 2021 at 17:52 | comment | added | Martin Sleziak | Some links seem to be dead, but at least for the abstracts, there is a version saved in the Wayback Machine. | |
| Nov 2, 2021 at 12:37 | history | edited | Timothy Chow | CC BY-SA 4.0 |
Updated dead link
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| Nov 9, 2017 at 20:17 | comment | added | Christopher King | @JoelDavidHamkins yeah. For example invented cellular automaton, called it "a zero player game". The look and say sequence has a pretty funny definition as well. I think it actually helps to get the public interested in serious math (cellular automaton, for example), but I could see the disadvantages as well. | |
| Nov 9, 2017 at 13:38 | comment | added | Joel David Hamkins | @PyRulez That is very interesting! It aligns closely with my own attitude... | |
| Nov 9, 2017 at 6:10 | comment | added | Christopher King | It should be noted that after his big paper in group theory, Conway decided to approach all mathematics in a "just-for-fun" way (and it has worked out pretty well for him). | |
| Feb 23, 2016 at 10:53 | comment | added | მამუკა ჯიბლაძე | @LeeMosher Many thanks for the link! | |
| Feb 21, 2016 at 17:25 | comment | added | Lee Mosher | Perhaps this is the final version of the article on the Absolute Arithmetic Continuum in Joel's answer? ohio.edu/people/ehrlich/Unification.pdf | |
| Jun 27, 2010 at 0:30 | comment | added | Mike Shulman | The link to the draft article doesn't work for me. | |
| Jun 25, 2010 at 3:52 | comment | added | KConrad | A Boyarsky turning to Dwork for an example. Am I the only one amused by this? | |
| Jun 25, 2010 at 1:21 | comment | added | Boyarsky | For the first 50 years after Hensel, $p$-adic analysis was championed by a few but viewed as esoteric by most; intrinsic beauty wasn't enough. Then Dwork used it to prove a Weil Conjecture and Tate invented rigid-analytic spaces to make analytic continuation possible over totally disconnected fields (with applications in number theory), and now it's a huge industry. Surreals are in need of a Dwork and Tate. (Although the set theory they require is elementary to set theorists, I conjecture it is a real obstacle for many without such expertise. Look at the history of non-standard analysis.) | |
| Jun 25, 2010 at 0:04 | comment | added | Terry Tao | It seems the main thing that would be needed to broaden the interest of surreals would be to provide some tools from transferring results back from the surreals back to more classical number systems (analogous to the transfer principle in nonstandard analysis). The situation here reminds me of that of generalised functions in analysis. There are many, many ways to generalise the concept of a function, but only distributions have really been successful, because there are ways to get from distributions back to classical functions, e.g. by convolving with a test function. | |
| Jun 24, 2010 at 21:50 | vote | accept | user2498 | ||
| Jun 24, 2010 at 18:19 | comment | added | Joel David Hamkins | Boyarsky, I am sorry that you seem to resist temptation by the beautiful surreals! The level of set theory required seems elementary to me, and involvement of several subjects would seem to be part of any unifying theory. | |
| Jun 24, 2010 at 18:09 | comment | added | Boyarsky | Oh, if they're not a set then that answers "what's wrong with the surreals" from the viewpoint of those whose main areas of interest aren't set theory and logic (since everyone -- or at least almost everyone? -- learns algebra, analysis/PDE, and geometry within the framework of set theory). So experts in set theory and/or logic who have solid background in differential geometry and/or PDE will probably have to be the ones who look further to see if the talk of promising developments in more analytic/geometric directions can be realized. I still wonder what the "promise" is meant to achieve. | |
| Jun 24, 2010 at 15:04 | comment | added | Joel David Hamkins | They are not a set---the form a definable proper class, but have a form of set-completeness, in the sense that any set has a first-born least upper bound (which on the next birthday is suberted as least by a newly added cut). Ehrlich mentioned theorems that characterize exactly the initial segments (in terms of birth order) that are groups, rings, fields, complete orders, and so on, in terms of the closure poperties of the ordinal. As for manifolds, DEs and geometry, I leave it for the experts, but many people have pointed to the surreals as promising for such developments. | |
| Jun 24, 2010 at 13:55 | comment | added | Boyarsky | @Joel: there are many real-closed fields; is there something which distinguishes the surreal numbers (if they are in fact an ordered field, and in particular a set)? Can one define notions of manifold, differential equations, and analytic geometry over the surreal numbers? I'm having a hard time seeing what it is that one is meant to do with them beyond set-theoretic investigations (via ordinals, etc.). I'm not seeing what kinds of insights we should hope to obtain in analysis or number theory by study of the surreal numbers. Note that $p$-adic analysis was a backwater until Dwork and Tate. | |
| Jun 24, 2010 at 12:58 | comment | added | Joel David Hamkins | Boyarsky, the surreals unify the transfinite ordinals with a robust theory of infinitesimals, in a single ordered field having an extremely rich structure theory. Thus, it unifies disparate number concepts arising in set theory, analysis and number theory. In the surreals, one has numbers such as $\sqrt{\frac{1}{\omega}+e^{\pi/2-\omega^2}}. How can a mathematician not be attracted to the possibilities of this? The tree was there all along, of course, so in principle there is nothing new there, but Philip uses it to provide canonical representations that he says are simplifying. | |
| Jun 24, 2010 at 8:29 | comment | added | Neel Krishnaswami | Apparently the Continental philosopher Alain Badiou regards the surreals as being of substantial philosophical importance, since they supply a way of talking about the infinite which still satisfy the ordered field axioms (in contrast with, say ordinal arithmetic). (I can't honestly claim to understand Badiou, but I do enjoy reading him: it's like reading about the foundations of mathematics from some parallel universe where Russell became a disciple of Husserl instead of Frege..!) | |
| Jun 24, 2010 at 4:45 | comment | added | Boyarsky | Unifying nature in what sense? That is, what are examples of mathematical (as opposed to philosophical) concepts from outside of pure set theory were once seen as unconnected but now are seen as unified by means of this work? And what does the tree allow one to do which couldn't be done before? Or is the interest internal to logic and set theory? A skim of the draft copy did not clarify this for me. | |
| Jun 24, 2010 at 4:32 | history | edited | Joel David Hamkins | CC BY-SA 2.5 |
added 239 characters in body
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| Jun 24, 2010 at 3:40 | history | answered | Joel David Hamkins | CC BY-SA 2.5 |