Why are the surreals considered "recreational" mathematics?

Question

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)?

Answer 1

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."?)

Answer 2

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.

Answer 3

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.