TITLE: Why are the surreals considered "recreational" mathematics?
QUESTION [9 upvotes]: 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)?

REPLY [4 votes]: 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.