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One of my college math professors once remarked to me that it was interesting that John Conway's two "biggest" contributions to math were both recreational: the Game of Life and the Surreals. No one, he said, was really working in the Surreals. I understand why Life is considered recreational, but why aren't the Surreals considered interesting by working algebraists (or was my professor just wrong)?

Eric Wofsey
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kuzzooroo
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  • Surreal numbers are associated with combinatorial game theory. Although it's game theory, I wouldn't essentially call it recreational. I highly doubt that there are no people working in combinatorial game theory. – EuYu Feb 12 '14 at 21:42
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    I think it is rather dismissive to say that Conway's biggest contributions are the Game of Life and the surreals; maybe "most well known", but certainly not "biggest". – Zhen Lin Feb 12 '14 at 21:43
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    Conway has made major contributions to group theory, knot theory and the exposition and analysis of the theory of quadratic forms amongst other things, and to the inspiration of young mathematicians over a generation. "The Symmetries of Things" is a brilliant book. – Mark Bennet Feb 12 '14 at 21:55

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There are a few different things worth addressing in your question, and I'll try to tease them apart.

1. Is anyone really working in the Surreals?

Well, few people are still working on the Surreals. Euyu commented that Surreals are associated with combinatorial game theory (CGT), but the bottom line is that everything about the Surreals that has any real bearing on CGT has been known for decades. However, outside of CGT proper people are still working on things. For example, a paper was posted on arXiv about Surreal analysis back in July.

2. Are the Surreals considered interesting by mathematicians?

Absolutely! Alling even wrote a whole serious book about them (and significant chunks of them). More recently, "our own" JDH used them in a set theory paper. There are lots of interesting things to say about them that I won't elaborate on here, because that's not what you asked for.

3. Are the Surreals recreational mathematics?

Probably. The basic definitions aren't very complicated (in some sense they were first introduced in a work of fiction!). They can be fun to think about, but all that mathematicians who aren't trying to do things like search for integration or appreciate the beauty of their structure need to know about them can be summarized into a few sentences, or perhaps less. ("All the ordered fields glued together; it works out."?)

Community
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Mark S.
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Surreal numbers originated in the analysis of combinatorial games, very much including some games actually played by humans, such as Go endgames. At the same time, many of the examples in Winning Ways came from specifically "recreational mathematics" sources such as games that appeared in publications of Henry Dudeney and Sam Loyd. So there is an indisputable connection to recreation and to "recreational mathematics", and many of the deeper constructs related to surreals come from combinatorial game theory (CGT).

In addition, no field other than CGT has strong ties to surreal numbers. Although Conway has stressed the analogy with Dedekind cuts, with set theory (now with two types of membership relation), with the ordinals, and the potential for use as infinitesimals in analysis --- the surreal numbers remain a necessity in CGT but a curiosity everywhere else, with non-CGT research on surreal numbers being purely exploratory and not with any particular connection to other mathematics.

No one, he said, was really working in the Surreals.

Other than the authors of Winning Ways, and the many combinatorial game theorist who use but do not research the surreal numbers, there is Jacob Lurie who seems to have maintained his early interest in surreal numbers and taught a class on them (or CGT) at Harvard a few years ago. According to the online lecture notes taken by a student, Lurie proposed that there is additional mysterious structure in the surreals related to a sort of circular or projective compactification that he predicted should exist. I do not know if he or others have developed those ideas since the lectures, but the message seemed to be that there is a big piece of un-researched deep structure in the surreals that deserves to be unearthed.

zyx
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    I agree with most of what you said, but I think most of the structure/theorems about the surreals are not even necessary for CGT. Does, say, the multiplicative structure of the surreals have any bearing on Games? – Mark S. Feb 15 '14 at 14:22
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    That was approximately what I had in mind when writing the answer. For example, the Go endgame analysis did involve some new theory, but it was developed by one of the "authors of Winning Ways". That exception and the "CG theorists who use but do not research the surreals" was a way of expressing the idea in your first sentence. Multiplication does not seem to be used in analysis of games. – zyx Feb 15 '14 at 15:39
  • The online lecture notes don't seem to be working for me. It says something like "Missing 404. The file you tried to access is missing or protected." Is there anywhere else I can find Lurie's lecture notes as a PDF? –  Aug 28 '14 at 15:35
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The surreals are currently considered as "recreational" or "marginal" because the existing models of ordinal numbers and set theory have proved adequate. Although in "On Numbers and Games" aka "ONAG" Conway suggests that the arithmetic of "surreals" is more easily theorised and more simply understood than the conventional treatments of the "reals" (avoiding multiple cases* when dealing with negatives), there are two other issues.

First that the surreals "can't naturally stop" before you get to infinity and beyond.

Second, he suggests that a set theory with two types of membership (left and right) may be required.

*The cases can be enumerated, even if this is rarely done. Careful formulations reduce the number of cases.

Mark Bennet
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  • I see that "can't naturally stop" is already in scare quotes but I'm afraid I don't follow. What do you mean? – kuzzooroo Feb 12 '14 at 22:01
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    So the Surreals arguably constitute an improvement over the reals in some ways (so says Conway), but they introduce new problems large enough that the cure is worse than the disease. My interest in math is admittedly recreational, but to me the Surreals seem interesting enough to study in their own right, not just as a fix for known annoyances in $\mathbb{R}$. – kuzzooroo Feb 12 '14 at 22:05
  • @kuzzooroo It isn't scare quotes. Note that (in the terminology of ONAG) you need to go to the $\omega^{th}$ birthday to get the reals, but you get $\omega$ along with them, unless you are careful - and there is no reason a priori to stop at $\omega$. My second printing of the original edition (with corrections) has comments on this on pp25-27 and pp64-67. I suggest you read the original, which I have tried to reflect fairly in a brief comment. – Mark Bennet Feb 12 '14 at 22:20
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    If you don't want infinite numbers, take numbers bounded by finite integers (or bounded by "something with finite ordinal birthday"). If you want to define the reals, they're "the finite surreals $r$ simplest less than all $r+q$ and greater than all $r-q$ for positive rationals $q$" (You could replace rationals with dyadic rationals, the surreals with finite birthdays, if you prefer.) It is absolutely not the case that a new set theory is required to define the surreals (there is no logical problem). However, the definition might be a little nicer-looking in such a "two-sided set theory". – Mark S. Feb 15 '14 at 03:16