Kent Palmer's interest in the Surreal Numbers got me to thinking about them. Robert Matthews, science correspondent of The Sunday Telegraph, described Surreal Numbers in New Scientist (147 (2 Sep 1995) 36) as presented by Martin Kruskal of Rutgers at Cambridge in 1995: Surreal Numbers, invented by John Conway, include all the natural counting numbers, together with negative numbers, fractions, and irrational numbers, and numbers bigger than infinity and smaller than the smallest fraction. It was this huge scope of Conway's invention, that prompted Knuth to christen them "surreal" numbers, from the French "sur" meaning "above" the reals.Despite their astonishingly broad membership, surreal numbers are simply sequences of that most fundamental of notions, a binary choice: yes/no, off/on. Surreals also seem able to help theorists tackle the perennial headache of divergent series and integrals. When attacking questions about, say, the fundamental forces of the Universe, theorists usually start from a simple situation and then add complexity as a "perturbation". The result is an infinite series of terms made up of the simple case followed by an endless stream of subsequent terms that bring in progressive levels of complexity. What theorists would dearly like is for just the first few terms to really matter, so that all the others could be discarded as insignificant. All too often, however, the series is divergent, with the extra terms actually being far more important that the first few.According to Kruskal, these problems could disappear if theorists use infinitesimals, numbers smaller than any imaginable positive real numbers. A series that involves such numbers can be prevented from diverging essentially because infinitesimals are so small that they "mop up" any tendency a series might have to zoom off to infinity. "The surreals give us a way of working with infinitesimals, and thus perhaps of working with divergent series," says Kruskal. Divergent integrals, another common bugbear in theoretical physics, may also bow to the surreal approach. Although they permit most usual mathematical operations like addition, raising to powers and taking logs, it's proving hard to integrate with surreals - crudely, to add up infinite amounts of infinitely small quantities and get sensible results. Without integration, surreals lose much of their interest in physics, which relies heavily on this mathematical operation.Ironically, the trouble stems from the sheer size of the surreal kingdom. Conventional integration takes place over the familiar pastures of real numbers, which contain no nasty surprises. But the far greater scope of the surreals also spans enormous numbers of "holes", which riddle the realm of the surreals. One, for example, pops up at the frontier between the finite and infinite numbers, while another divides the finite from the infinitesimally small numbers. Unless mathematicians can find some way of leaping across these gaps, integration is always going to be a problem for surreal numbers. Surreal Numbers are just sequences of binary choices, and constructing them is something of a game. It begins with the simplest surreal number, an empty sequence made up of nothing at all: this is written as 0, and is the starting place of what mathematician Martin Kruskal calls the Binary Number Tree. To the upper right of 0, one then puts a single upward-pointing arrow. This represents the simplest surreal number greater than 0. The rules of surreal arithmetic then show that, naturally enough, this single up arrow is the surreal representation of the ordinary number 1. To the lower right of 0, one has a downward-pointing arrow, representing the simplest negative surreal, equal to -1. From then on, this "branching" process continues, giving the "binary tree" its name. To the upper right of the single upward pointing arrow, a surreal consisting of two upward-pointing arrows is drawn: this represents the number 2. To the lower right, however, a surreal made up of one up and one down arrow appears. What is this? Finding out becomes the first exercise in decoding the arrow notation. The basic rule is that each successive arrow in a surreal number says a little bit more about it. For example, if it starts with an upward-pointing arrow, that means that the surreal is positive; an initial downward-pointing arrow means that it's negative. After that, the arrows alter the description of the overall surreal number in the "simplest" way possible, with upward meaning greater and downward meaning less. Thus two up arrows define the simplest positive surreal greater than 1, and is thus equivalent to the ordinary number 2. Similarly, two down arrows represent the simplest negative surreal less than -1, in other words -2. Clearly, all surreals made up of a finite number of entirely upward-pointing arrows represent the positive integers, while those made up of a finite number of only down-arrow strings are the negative integers. So what does up-down mean? The leading upward arrow means the surreal is positive, and the downward arrow that follows means that it's the simplest surreal that is positive but less than 1. Given that we've already got zero, the most natural choice is exactly half way between 1 and zero, that is 1/2, and this in fact follows from the rules of surreal arithmetic. Similarly, down-up is equivalent to -1/2. It turns out that, in fact, all the rational numbers have an arrow representation of this kind.The irrationals such as the square root of 2, the transcendentals such as pi and even infinite numbers are brought into the fold through arrow sequences that go on forever. For example up-up-up-..., which is written as up^hat, represents the infinitely large number w, while up-down^hat represents the infinitesimally small number iota. All these latter arrow sequences lie along a vertical line connecting w to -w and mark a kind of frontier. The "familiar" reals all lie somewhere on or to the left of this demarcation line. But the realm of the surreals does not end there. To the right of the line are surreals vastly larger than infinity.
Another way to represent infinitesimals as an extension of the real numbers is by Robinson's Nonstandard Analysis. Robinson's Nonstandard numbers are a subfield of the Surreals. Kruskal is quoted (by James Lawry on usenet sci.math.research) as saying in his Cambridge lectures that it was unknown whether any of the infinite Surreal integers are prime, but that it is known that there exist prime infinite Nonstandard integers.
To begin, choose a lattice spacetime of a Feynman Checkerboard, and call it the lattice spacetime of "our" universe in the ManyWorlds. For simplicity, look at one of the space-time dimensions. Represent its vertices by the red dots - the integers of the Surreal Numbers. Then consider the "set" of all "other" lattice spacetimes of the other universes in the ManyWorlds. Let the blue dots represent one of the "nearest neighbor" lattice spacetimes, and let the green dots represent the other "nearest neighbor" lattice spacetime. Then go one step further, and let the purple and gold dots represent the two "next nearest neighbor" lattice spacetimes that are "accessible through" the blue spacetime, and also (not shown on figure) go to the two "next nearest neighbor" lattice spacetimes that are "accessible through" the green spacetime. Continuing to fill out the Surreal Number Binary Tree, you have a representation of all the universes of the ManyWorlds by the "set" of all Surreal Rationals with finite expansions.
As Onar Aam has noted, the Surreals have natural mirrorhouse structure. The two "nearest neighbors" of the origin in this 1-dimensional Surreal "spacetime", Sur^1, correspond to the set {+1, -1} (represented in the diagram by {blue, green}) which are reflected into each other through the origin (represented by red). The reflection group is the Weyl group of the rank-1 Lie group Spin(3) = SU(2) = Sp(1) = S3. If we look at k-dimensional Surreal spacetimes Sur^k, we see that the "nearest neighbors" of the origin correspond to the root vectors of the Weyl reflection groups for the largest rank-k Lie group that contains as a subgroup the rank-k Lie group Spin(2k). Here I am using the term "nearest neighbors" to include both nearest and next-nearest neighbors in the case of Weyl groups whose root vectors are not all of the same length, as, for example, G2, F4, and Spin(2k+1) for k greater than 1. For instance: The origin of Sur^2 has 4+8 = 12 "nearest neighbors", corresponding to (12+2)-dim G2 containing (4+2)-dim Spin(4). The "nearest neighbors" are in a Star-of-David pattern, with 30-degree angle between adjacent root vectors. Since 2x30 = 60, 3x30 = 90, and 4x30 =120, the G2 lattice of all "neighbors" combines both the square Gaussian lattice and the triangular Eisenstein lattice. Sur^2 has complex structure. The origin of Sur^4 has 24+24 = 48 "nearest neighbors", corresponding to (48+4)-dim F4 containing (24+4)-dim Spin(8). The "nearest neighbors" are in a double 24-cell pattern, and all "neighbors" form a double D4 lattice. Sur^4 has quaternionic structure. The origin of Sur^8 has 112+128 = 240 "nearest neighbors", corresponding to (240+8)-dim E8 containing (112+8)-dim Spin(16). The "nearest neighbors" are in a Witting polytope pattern, and all "neighbors" form an E8 lattice. If the other 6 of the 7 E8 lattices are included, then there are 480 "nearest neighbors". Sur^8 has octonionic structure, and can therefore represent the D4-D5-E6 physics model. The "neighbors" of the origin of Sur^16 form a /\16 Barnes-Wall lattice. The "neighbors" of the origin of Sur^24 form a /\24 Leech lattice. The line of "first-order" infinite, transcendental, irrational, infinitely-expanded, and infinitesimal Surreals represents the "set" of "ALL ManyWorld universes" "near" our universe in the sense that they are within a finite number of steps of "our universe".
The "higher-order" infinite-infinitesimal Surreals, shown in the image at the top of this page, represent more and more "distant" sets of ManyWorlds universes.
Note that the Surreal Rationals with finite expansions are more nearly in 1-1 correspondence with the universes of the ManyWorlds than with the rational numbers, and that the holes of the Surreals are not problematic in representing the universes of the ManyWorlds, as they are when they are taken to represent mere numbers.
Note also that the infinitesimal structure of the Surreals means that they may be able to represent TOPOLOGICALLY the exotic structure of R^4 and other exotic manifolds, especially the possibly exotic structure of S1 x S3.
The Surreal Number binary decision tree is similar to the binary decision tree that produces Markov Processes and the binary decision tree that produces Bernoulli Numbers:
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