Where do surreal numbers come from and what do they mean?

Question

I know about Conway's original discovery of the surreal numbers by way of games, as well as Kruskal's way of viewing surreal numbers in terms of asymptotic behavior of real-valued functions, leading to connections between surreal analysis and the theory of o-minimal structures (if Kruskal isn't the right attribution here, please feel free to correct me and educate everyone else). But I feel that, even with those two viewpoints available, surprisingly few connections between surreal numbers and the rest of mathematics have emerged over the past four decades. I say "surprising" because one would expect something so beautiful and natural to have all kinds of links with other things!

I think one reason the surreal numbers have found so few points of contact with the rest of contemporary mathematics is that the simplicity relation $a$-is-simpler-than-$b$ does not have any translational or dilational symmetries. (The simplest number between $-2$ and $+2$ is $0$, but the simplest number between $-2+1=-1$ and $+2+1=3$ is not $0+1=1$ but $0$. Likewise, the simplest number between $1$ and $3$ is $2$, but the simplest number between $2 \times 1=2$ and $2 \times 3=6$ is not $2 \times 2=4$ but $3$.) In the wake of Bourbaki, mathematicians have favored structures that have lots of morphisms to and from other already-favored structures, and/or lots of isomorphisms to themselves (aka symmetries), and the surreal numbers don't fit in with this esthetic.

Are there any new insights into how the surreal numbers fit in with the rest of math (or why they don't)?

See also my companion post What are some examples of "chimeras" in mathematics? .

It occurred to me after I posted my question that there is a weak $p$-adic analogue of the 2-adic surreal-numbers set-up, in which one relaxes the constraint that every interval contains a unique simplest number (that's a lot to give up, I admit!). If one defines $p$-adic simplicity in ${\bf Z}[1/p]$ in the obvious way (changing "2" to "$p$" in Conway's definition, so that integers are small if they are near 0 in the usual sense and elements of ${\bf Z}[1/p]$ are small if they have small denominator), then the following is true for $a_L,a_R,b_L,b_R$ in ${\bf Z}[1/p]$: if there is a unique simplest $a$ in ${\bf Z}[1/p]$ that is greater than $a_L$ and less than $a_R$, and there is a unique simplest $b$ in ${\bf Z}[1/p]$ that is greater than $b_L$ and less than $b_R$, then there is a simplest $c$ in ${\bf Z}[1/p]$ that is greater than $a+b_L$ and $a_L+b$ and less than $a+b_R$ and $a_R+b$, and it satisfies $c=a+b$. (Conway's multiplication formula works in this setting as well.) Is this mentioned in the surreal numbers literature, and more importantly, does the observation lead anywhere?

Answer 1

"...at least for now, I'm finished. All the questions that have yet to be answered are too hard." -- Jacob Lurie on surreal numbers, 1996

Answer 2

Not numbers exactly, but certainly Conway's games (and related game-like structures) have aroused plenty of interest in certain category theorists and logicians, as they can be used to give categorical (non-posetal) semantics for substructural logics. In other words, under the Curry-Howard paradigm of "propositions as types", where propositions are promoted to objects and proofs of propositions $p \Rightarrow q$ are promoted to morphisms $p \to q$, the structures of formal deductions are embedded in strategies for games.

It was originally observed by André Joyal that Conway games are the objects of a compact closed category, where morphisms between games $G \to H$ are second-player winning strategies for $-G + H$. Closer to logical concerns, in

• Andreas Blass, A game semantics for linear logic, Ann. Pure Appl. Logic 56 (1992), 183-220

Andreas gave a game semantics for Girard's recently introduced linear logic (which on the "type" or categorical side correspond to $\ast$-autonomous categories); you can read some of his thoughts here for example, and elsewhere on his web page. This began a sort of cottage industry, where various refinements of games were developed to give soundness and completeness theorems for various forms of linear logic (see for example the Hyland-Ong reference in the online article).

(In an act of shameless self-promotion, I'll mention a little project with James Dolan to use certain types of games to model (free) cartesian closed categories, which was partly written up here.)

Numbers per se are used to measure strengths of positions in games, and sometimes this has been put to cunning use (as for example in analyzing some fairly specific but difficult positions in Go), but I don't know whether they have been exploited in game semantics along the lines above. Maybe Andreas can weigh in?

Answer 3

An algebraic perspective on the surreal numbers is that they are ``the maximal totally ordered field.'' Of course, since the surreal numbers are a class and not a set they cannot really be a field. Nevertheless, I mentally file the surreal numbers in the same folder as various universal constructions: the algebraic closure of a field, the absolute Galois group of the rationals, the fraction field of a ring, etc...

Conway's construction is really beautiful, but it doesn't seem to lend itself to algebraic manipulations like the axioms of a totally ordered field. For there to be interesting algebraic applications of the surreal numbers, one has to weigh the benefits of having ``one object'' against the set-theoretic headaches of dealing with a proper class.

Answer 4

There is a book by Norman Alling, entitled Foundations of Analysis over Surreal Number Fields, published by North-Holland in 1987. I am not aware that it had much influence, but it looks like quite an interesting read.

Answer 5

See this paper in regard to the p-adic analogue: http://arxiv.org/abs/1108.0962