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?

wouldsee surreal numbers everywhere. $\endgroup$ – Ryan Budney Apr 28 '11 at 18:34