The numbers came out of the game

The surreal numbers are usually introduced like this: here is a construction, it produces a vast ordered field, and — pleasantly — the games of combinatorial game theory turn out to be a generalisation of it.

The discovery went the other way, and the order matters, because it explains every feature of the definition that otherwise looks arbitrary.

Conway was looking at Go endgames. A Go endgame breaks into independent regions; each region is worth something; the whole is the sum. Working out what a region is worth produced positions behaving exactly like fractions — a region worth half a move, a region worth a quarter — and the fractions came with a rule for which one, which is the simplicity rule. The number system was read off the games.

Why the definition looks strange #

A surreal number is a pair of sets of previously constructed surreal numbers, $\{L \mid R\}$, with every member of $L$ strictly less than every member of $R$.

Read cold, two things about that are odd.

It quantifies over what has already been built. A Dedekind cut quantifies over all rationals at once — a real number is a partition of $\mathbb{Q}$ — and there is no notion of one real being available before another. The surreal definition is explicitly staged: day $n$’s numbers are built from day $n-1$’s.

And it is two-sided in a way nothing forces. A cut is one set with a complement. Here there are two sets, and the requirement is an inequality between them rather than exhaustiveness.

Both features are inherited directly from games. $L$ and $R$ are Left’s options and Right’s options. They are two sets because there are two players. They are drawn from what already exists because a game’s options are positions reachable from it, and a position is built before the position that moves to it. Nothing about numbers demanded either feature; the games did.

Drawn as a tree of birthdays, the same construction says the same thing from the other side: each number’s day is the number of moves it takes to reach the end from the position representing it. That is a fact about a game showing through as a fact about a number, and it is the shape of every claim this essay makes.

The staging is not a formality #

Here is where the two constructions genuinely come apart, and it is sharper than “different presentations of the same thing”.

Run the surreal construction for any finite number of days and every number produced is a dyadic rational — a fraction with a power of two underneath. Days 0 to 4 produce 31 of them, and the check that they are all dyadic is run by the figure rather than assumed.

One third is not among them, and never will be. Not on day 5, not on day 500. A Dedekind cut gives $\tfrac13$ immediately, because it quantifies over arbitrary sets of rationals and $\{q : q < \tfrac13\}$ is such a set.

So the two constructions produce different collections at every finite stage, and the difference is not an artefact of presentation. The staged construction reaches $\tfrac13$ only on day $\omega$ — past every finite day — as $\{0, \tfrac14, \tfrac5{16}, \ldots \mid 1, \tfrac12, \tfrac38, \ldots\}$, the two sequences of binary approximations closing in from each side.

The rule deciding which number goes into each gap is the simplicity rule: the earliest-born number strictly between the two options, which is not the midpoint and is not any average. That distinction is what makes the construction produce dyadics in order of denominator rather than producing arbitrary reals, and it is another clause that reads as a choice from the numbers side and as a theorem from the games side.

Why the games only produce dyadics #

This is the fact that makes the connection tight rather than decorative, and it has a one-line reason.

A Hackenbush string is worth a number, and the number is read off the string in binary. A finite string is a finite binary expansion. A finite binary expansion is a dyadic rational. There is nothing else it could be.

More generally: a position with a finite game tree has a finite birthday, and the simplicity rule at each step picks a number of the simplest available denominator — which doubles at worst per day. Finite play, finite denominator, power of two.

The dyadics are not a restriction that was imposed. They are what finite games are worth.

That argument is short enough to be suspicious, so it is worth watching the two halves of the numeral appear separately. The leading run of one colour is the integer part, one unit per edge, and nothing about it is a fraction at all.

The fractional part is the tail, and it is where the powers of two come from. Put one blue edge on the ground and pile red edges on it: each red edge can only be cut by Right, and each one halves what Left is left holding.

Those two figures are the whole of the argument. An integer part with a finite number of edges, a fractional part with a finite number of halvings, and the sum of them is a finite binary expansion — which is a dyadic rational and cannot be anything else.

Why the count doubles, and what that says #

Thirty-one is not an arbitrary number and the way it arises is the clearest statement of what the staging does.

Count the numbers born on each day: 1, 2, 4, 8, 16. One on day nought, and a doubling thereafter, so the total born by day $n$ is $2^{n+1} - 1$.

The doubling has a one-line reason. Suppose $k$ numbers exist. They cut the line into $k + 1$ gaps — $k - 1$ between consecutive numbers, plus the two unbounded ends. The next day puts exactly one number in each gap: the simplest number strictly between its neighbours, which is what the simplicity rule delivers and which is unique because “simplest” is a well-ordering. So the day produces $k + 1$ new numbers, the total goes from $k$ to $2k + 1$, and starting from one the counts are $1, 3, 7, 15, 31$.

Two things follow that are worth having.

The construction is binary all the way down. Each day adds one bit of resolution and nothing else — a gap either gets its midpoint-in-simplicity or it does not, and every gap gets exactly one. That is the same binary structure the Hackenbush reading exhibits from the other side, and it is why the denominators are powers of two rather than merely happening to be.

And nothing can be produced out of turn. A number’s day is fixed by the construction, not chosen: $\tfrac14$ cannot appear before $\tfrac12$, because $\tfrac12$ has to exist to make the gap $\tfrac14$ sits in. The staging is not a bookkeeping convention that could be relaxed — it is a strict order on the numbers, and it is exactly the order in which the corresponding positions can be built.

Which is why one third has nowhere to be born #

The doubling also gives the sharpest form of the essay’s central contrast.

A gap is filled by the simplest number strictly inside it, and at every finite stage every gap is between two dyadics. The simplest number strictly between two dyadics is another dyadic — the construction cannot reach outside the dyadics, not because it is forbidden to but because there is never a step whose answer is anything else.

So $\tfrac13$ is not skipped over. It is never anybody’s gap-filler, at any finite day, because every gap containing it is bounded by dyadics and has a dyadic strictly inside it that is simpler. The construction keeps producing better approximations — $\tfrac14$, $\tfrac5{16}$, $\tfrac{11}{32}$ — and each of them fills a gap that $\tfrac13$ was also inside, so the thing being approximated is passed over an unbounded number of times without ever being produced.

The same thing can be watched happening in a position, which is worth doing because “approached and never reached” sounds like a statement about limits and is here a statement about strings. Alternate the two colours and read off the value each time an edge is added.

Two thirds is not one third, and it is the same situation: a fraction whose denominator is not a power of two, closed in on from both sides by strings that are always dyadic. A reader who wants the limit has to let the string be infinite, which is exactly what letting the day be infinite means on the other side of the correspondence.

A Dedekind cut has no such mechanism, which is the whole difference. A cut is not built from anything and does not fill a gap; it is a set of rationals, presented whole, and $\{q : q < \tfrac13\}$ is as available on the first day as any other set.

That is the staging showing its teeth. The two constructions are usually described as arriving at the same place by different routes, and at every finite stage they are not even in the same country: one has thirty-one numbers and the other has all the reals. What they share is a destination reached at day $\omega$, which is past every day this site’s evaluator or any finite game can reach.

Reading the definition as a game again #

It is worth taking the surreal definition and translating every clause back, because each one comes from somewhere.

$\{L \mid R\}$ — a position with Left’s options and Right’s options.

Every member of $L$ is strictly less than every member of $R$ — neither player wants to move. A position where Left has an option at least as good as one of Right’s is a position where somebody is eager, and eagerness is exactly what makes a position not a number: it is a switch, something both players are fighting over.

Built from previously constructed numbers — the game is finite and terminates, so the options were reached first.

The value is the simplest number strictly between — the simplicity rule, which is not an extra axiom but a theorem about what the position is worth under normal play.

Every clause of a definition that reads as a technical choice turns out to be a fact about play. That is what it looks like when a construction is discovered rather than designed, and it is the best evidence for the historical claim this essay makes.

One theorem makes the point better than the definition does. In a sum containing a number and something that is not a number, a player should never move in the number. Read as arithmetic that statement is not false but meaningless, because numbers do not have moves; read as a game it is among the most useful rules in the subject, and it is quoted more often than anything else the theory produces.

The surprise: the games are the bigger object #

The standard framing has games as a generalisation of numbers, and that is true and understates it.

Most games are not numbers. A position is a number only when every Left option is strictly less than every Right option — when the two players’ claims do not overlap. The moment they overlap, the position is something else: star, which is confused with zero; up, which is positive and smaller than every positive number; a switch, which both players want to move in.

So the numbers are a thin slice of the games, singled out by an inequality. And the historical order makes sense of that: somebody studying games would have no reason to expect the numbers to be interesting, and would notice that some of the positions behave like arithmetic. Somebody studying numbers would never have found the rest.

The thinness of the slice is visible in a Hackenbush string too, and it takes exactly one edge to leave the numbers behind. A green edge is one either player may cut, so it belongs to nobody, and a string carrying one is not a number however the rest of it is coloured.

That is the whole distance between the numbers and the games, measured in edges. Three of those four stalks are outside the number system, and nothing was done to them except giving one edge to both players at once.

The evidence for the claim, such as it is #

A claim about the order in which somebody thought of things is not the kind of claim this site normally makes, and it is worth being explicit about what supports it.

The published record. On Numbers and Games opens with the numbers and says so; Conway’s accounts of the discovery describe the Go endgame route. That is testimony rather than proof.

The shape of the definition, which is the argument this essay actually makes. Every feature of the construction that reads as arbitrary from the numbers side is forced from the games side, and there are four of them — staging, two-sidedness, the inequality between the sets, and the simplicity rule. A definition arrived at numbers-first would have no reason for any of them.

And the dyadics. A construction designed to produce the reals would not produce only dyadic rationals at every finite stage; that is a defect from the numbers side and a theorem from the games side. It is the strongest of the three, because it is checkable: the birthday figure verifies it over every number it draws.

Where the model stops #

This site’s evaluator handles finite option sets only. game() interns a position from a finite list of options, so ω, its reciprocal, and $\tfrac13$ are all outside what any figure here can draw. The hero figure names them and prints the reason rather than drawing them, which is invariant 7 applied to the one place it bites hardest.

And “day” is being used loosely for games. For numbers the birthday is unambiguous. For games in general, the birthday is the day the canonical form is born, which may be earlier than the day the written form was born — a position written with redundant options is born later than its value. Every birthday claimed here is a birthday of a canonical form.

What the picture cannot show #

The birthday figures draw a line with numbers on it, and a line is exactly the wrong shape for what is being constructed.

Surreal numbers form a proper class, not a set — there are too many to be a set, in the technical sense — and they include infinite and infinitesimal quantities that no position on a drawn line represents. The line in these figures is the finite dyadic part, which is the part that fits.

Worse, the line suggests density: between any two marks there is room, and the construction seems to be filling it in. That is right for the dyadics and wrong about the destination, because the finished object has ω and 1/ω and ω − 1 in it, and none of those is between two marks on any drawn line.

What was actually being computed #

It is worth being concrete about the Go endgames, because “Conway was looking at Go” is the kind of sentence that carries no information.

A Go endgame late in the game consists of independent regions, each of which is worth something to whoever moves there. Some regions are worth a whole point to whoever takes them; some are worth a point but only if the opponent does not answer; some are worth a fraction of a point in a sense that is precise and was not, before this, precisely stated.

The last kind is what produced the numbers. A region where Left moving gains a point and Right moving gains nothing behaves like $\tfrac12$ — and the reason is not analogy: it is that two copies of the region together are worth exactly one move, which is a statement that can be checked.

Accounted for that way, an endgame is a list of independent regions with a value against each and a total at the bottom, and the fractions in the list are not approximations to anything. They are exact values of positions, obtained by the same recursion that produces the number system — which is why the accounting is arithmetic rather than an estimate.

That check — two halves make a whole — is what makes the identification with arithmetic real rather than suggestive, and it is available only because the regions add. The disjunctive sum is what turns a collection of positions into arithmetic, and without it the fractions would be a naming convention.

What the reverse presentation costs a reader #

There is a practical consequence to the numbers-first order, and anybody who has tried to learn this material has met it.

Presented numbers-first, the reader spends a long time on a construction whose motivation is withheld. Why two sets? Why previously constructed? Why does the simplicity rule pick that number? Each question has an answer and the answer is a game, and the game has not been introduced yet.

Presented games-first, every one of those is answered before it is asked, at the cost of the reader having to accept for a while that positions have values — which is easier, because a position is a concrete thing and its value can be checked by playing.

This site takes the games-first order for that reason, and the order is visible in the ladder: Hackenbush comes before the number tree, and the tree comes before this essay.

Who found it, and when #

Conway worked this out around 1970 and published On Numbers and Games in 1976. Donald Knuth wrote Surreal Numbers in 1974 — a novella, of all things, in which two characters reconstruct the theory from an inscription — and it is Knuth’s book that gave them the name.

The publication order is itself the inversion this essay is about: the number book came out before the game book, so the world met the numbers first and the games second, which is the reverse of how they were found and is why the standard presentation runs that way to this day.

Why the inversion is worth recording #

It would be easy to treat “which came first” as trivia. It is not, and the reason is that the standard presentation makes two of the theory’s features look like design decisions when they are consequences.

Presented numbers-first, the restriction to dyadic rationals at finite stages looks like a limitation of a slightly awkward construction — a construction that gets to the reals eventually, but takes ω days to reach a third. Somebody meeting it that way reasonably asks why one would use it.

Presented games-first, the dyadics are the answer to a question: what is a finite position worth? And the answer being dyadic is a result, checkable on any Hackenbush string, rather than a shortfall.

The same inversion applies to the two-sided definition, to the staging, and to the simplicity rule. Every one of them reads as a choice from the numbers side and as a theorem from the games side.

A word about “surreal” #

The name is Knuth’s, and Conway did not choose it. In On Numbers and Games they are simply the numbers, which is the right name from inside the theory: they contain the reals, the ordinals and a great deal else, so calling them a variety of number with a modifier understates what has happened.

The modifier has nonetheless been useful, because it separates the number system from the games. On this site the distinction matters constantly — a value may be a number or may not be, and most are not — so having a word for the numerical part is worth the slight inaccuracy.

Where the ladder goes next #

The rungs below this one build the number system and its rules. This one is why the rules take the shape they do. The rung above is what happens when the construction is run without a stopping condition at all — the recursion this site cannot run — where the objects stop being computable and the figures have to say so.