The notation was the argument

Before Winning Ways the facts in this essay were known — several of them for decades — and were stated as tables. A table saying who wins this position; another table for that one; and nothing joining them.

The notation that replaced those tables — ↑ for up, ∗ for star, and $\{L \mid R\}$ for a position written from its options — is routinely described as convenient shorthand. It is not shorthand. It is the claim that these things add, and the claim is false for the objects the tables described.

The demonstration, in one line #

Take two positions, both first-player wins. What is their sum?

It depends, and not on anything the phrase “first-player win” records.

∗ plus ∗ is zero — a second-player win. The two positions cancel exactly: whatever the first player does in one, the second player copies in the other.

∗ plus ∗2 is ∗3 — a first-player win. No cancellation at all.

Same two outcomes going in. Different outcomes coming out, and no property of the inputs available to a table of outcomes distinguishes the two cases. So the outcome of a sum is not a function of the outcomes of the parts, and a theory built on outcomes cannot get started.

What the notation is doing instead #

A value is a finer object than an outcome, and the notation names values.

∗ and ∗2 are different values with the same outcome, and the notation distinguishes them, so the sum can be computed. That is the whole trick and it is the reason the symbols were worth inventing: not brevity, but that the symbol is what adds.

Once values are named, the arithmetic follows the ordinary rules a reader expects. ↑ + ↑ is 2·↑ and is written that way. ↑ + ∗ is ↑∗, and the concatenation is deliberate — it is a sum, spelled as though it were a compound word, because that is what it is.

A value is positioned by comparison rather than by size: ↑ is above 0 and below every positive number there is, and ∗ is none of greater, smaller or equal. The notation encodes exactly the comparisons that hold, which is what makes expressions in it computable, and what the infinitesimals actually are is argued elsewhere on a ladder drawn for the purpose.

What the tables looked like #

It is worth being concrete about what the notation replaced, because “case analysis” is vague.

Before values, a result about a class of positions was stated as a rule with conditions: a position of this shape, with this many components of that kind, is a win for the player to move unless the following holds. Each such statement was about one family, proved separately, and combined with nothing.

Two of them could not be added because there was no operation to add them with. A reader wanting the answer for a position built from two families had to go back to first principles, and the two results contributed nothing.

The whole vocabulary available before values was four labels — Left wins, Right wins, the first player wins, the second player wins — exhaustively describing a single position and supporting no operation. Two positions in that scheme carry four bits of information between them and nothing to do with it.

A notation with an operation replaces $n$ separate results with one. That is the practical form of the argument, and it is why the change was not cosmetic: results stopped being one-off and started composing.

Three notations, and what each can express #

It is worth being explicit about the ladder, because the steps are not obvious.

Outcome classes. Four labels: Left wins, Right wins, first player wins, second player wins. This is Zermelo’s information and it is complete for a single position considered alone. It supports nothing.

Grundy values. One number per position, for impartial games only, and the numbers add by exclusive-or. This is Sprague and Grundy, and it is a genuine notation in the sense this essay means: a symbol that supports an operation.

Game values. A value for every position, partizan or not, and the values form a group under the disjunctive sum with a partial order on top. This is Conway’s, and the symbols of Winning Ways are how it is written down.

Each step is a strict increase in what can be written and therefore in what can be computed, and each is bought by the same move: name the thing finely enough that the operation is well defined on the names.

Each step is bought by the same move made twice, and the last column is uniform across kinds: a number, a nimber and an infinitesimal are written in one notation and added by one rule, so the arithmetic does not need to know which sort of thing each part is.

Why the symbols look like that #

The specific choices are not arbitrary either, and each encodes a claim.

∗ for star, an asterisk, because the position is marked rather than measured — it has no size, it is confused with zero, and a numeral would have implied a magnitude it does not have.

↑ and ↓ for up and down, arrows rather than letters, because they are directional and tiny: positive but below every positive number. An arrow suggests a direction with no length, which is exactly right.

Multiples written as $2 \cdot {\uparrow}$ rather than as a new symbol, because they genuinely are sums of copies and behave like it — $2 \cdot {\uparrow}$ is bigger than ↑ in the ordering, which a fresh symbol would have hidden.

And concatenation for sums: ↑∗ is up plus star. A notation that used a plus sign there would have been clearer and longer, and the compound reads as a single object because it usually behaves as one.

The surprise: the notation forced a discovery #

The best evidence that a notation is doing work is that it makes a question askable that nobody would have asked.

Once positions have values and values add, it is natural to write $G - H$ and ask whether it is zero. That is comparison, and it turns out to have four answers rather than three: $G > H$, $G < H$, $G = H$, or $G$ confused with $H$, written $G \parallel H$.

The fourth answer has no analogue in ordinary arithmetic and nobody looking at tables of outcomes would have found it, because it is a statement about a difference and tables of outcomes cannot be subtracted. It exists because the notation permitted the subtraction.

Two positions are compared by playing their difference out, and the fourth relation is what a partial order looks like when it is genuinely partial: ↑ is above zero, ∗ is confused with zero, ↑ is above ∗, and a half is above a quarter — four pairs, four relations, and only three of them have a name in ordinary arithmetic. Comparing two positions is where that is drawn; what matters here is that the question needed a subtraction to be askable at all.

What the picture cannot show #

Every figure here writes values as strings — ↑, ∗2, ↑∗ — and a string suggests the value is a name the position was given. It is not. It is a canonical form, computed by reduction, and the computation can be expensive.

Two positions written completely differently may have the same value, and finding out is a search rather than a comparison of strings. The notation makes the answer expressible; it does not make it cheap, and no figure showing tidy symbols conveys the cost of getting them.

The brace form is the one that matters #

Of everything Winning Ways introduced, the piece that does the most work is the least decorative: writing a position as $\{L \mid R\}$, its Left options before the bar and its Right options after.

That is not a symbol for a particular value. It is a general form, and it is what makes every other symbol definable rather than stipulated. $0 = \{\ \mid\ \}$, $* = \{0 \mid 0\}$, $1 = \{0 \mid\ \}$, $\uparrow = \{0 \mid *\}$ — each is a position written out, not a name assigned.

So the special symbols are abbreviations and the brace form is not. That inverts the usual reading, in which the arrows and stars are the notation and the braces are bookkeeping.

It also carries the site’s own figure rule into difficulty, which is worth admitting. A brace expression is a compression of a position and nothing else, and an essay of nested braces with no positions in it has stopped being figure-first — which is why every figure here that uses the form also shows what it describes.

How much of the theory the symbols actually name #

The inversion above — the braces are the notation and the arrows are abbreviations — can be checked rather than argued, by counting how many values get a symbol.

Among the 22 values born by day two, 15 have a short name: the integers and halves, $\ast$, $\ast 2$, $\uparrow$, $\downarrow$, $\uparrow\!\ast$, $\downarrow\!\ast$, $1\ast$ and $-1\ast$. The other seven are written out from their options — $\{1 \mid 0\}$, $\{1 \mid \ast\}$, $\{0, \ast \mid -1\}$ and their relatives. So on day two the special symbols cover about two thirds of everything there is.

One day later they cover 42 of 1,474 — under three per cent.

That is the inversion made quantitative, and the size of it is the point. The symbols do not fall behind gradually; they cover most of the second day of the construction and almost none of the third, and there is no reason to expect the fourth to be kinder. A notation whose special symbols name three per cent of its objects is a notation whose real content is the general form, and the arrows and stars are a vocabulary for the handful of values that keep turning up.

The same collapse has been measured on this site from another direction: the numbers, which are the totally ordered part of the universe, go from about a third of day two to one per cent of day three. Two different distinguished families, both shrinking to nothing as a fraction, and for the same underlying reason — a fixed cast of named objects sitting inside a universe that squares.

Which is why the abbreviations are the ones they are #

Read that way, the choice of which values got symbols stops looking like taste and starts looking like a survey.

Nobody invented a symbol for $\{1 \mid \ast\}$, and it is born on day two alongside $\uparrow$. What separates them is not age or complexity; it is that $\uparrow$ turns up over and over in real positions — in all-small games, in Clobber rows, in pawn endings, wherever a player has a tempo and nothing else — while $\{1 \mid \ast\}$ turns up in the enumeration and almost nowhere a person is looking.

So the named values are the ones with recurrence, not the ones with priority. That is the ordinary way a vocabulary forms in any subject, and it explains a feature of the notation that otherwise looks inconsistent: $\uparrow$ and $\ast$ get glyphs, $2\!\cdot\!\uparrow$ gets an arithmetic expression, and $\{1 \mid \ast\}$ gets nothing but its own definition. Three levels of abbreviation, assigned by how often a reader will need to say the thing.

And it sharpens the essay’s central claim rather than weakening it. The argument is that the notation is a claim that these objects add — and the objects that add are all 1,474 of them, not the 42. What the brace form supplies is exactly that: a way of writing every value, uniformly, such that the addition is defined on the writing. The arrows are what makes the resulting arithmetic readable, and readability is a real contribution, but it is a second contribution and a smaller one.

A reader who takes the arrows to be the notation will conclude that the theory is about a small menagerie of exotic values with names. The count says otherwise: the menagerie is three per cent of one day, and the theory is about the other ninety-seven.

Where the model stops #

The notation is for normal play. Under misère play the values do not compose, so there is nothing for a symbol to denote that would support an operation — and the misère literature is correspondingly written in terms of much larger objects with no compact notation at all.

And “canonical” needs a convention. Two forms of one value reduce to the same canonical form, and the reduction removes dominated and reversible options in an order that turns out not to matter. That it does not matter is a theorem, not a definition, and the notation quietly relies on it every time a value is written as though it were unique.

What a notation cannot do #

For balance, because the argument above is enthusiastic.

It does not make anything computable that was not. Every value expressible in this notation was determined by the position before anybody wrote it down; the notation records answers rather than producing them.

It does not make anything cheap. Reducing a position to canonical form is a search, and writing the answer compactly afterwards does not reduce the cost of finding it.

And it does not extend itself. Loopy games needed new notation and got it awkwardly; misère play needed some and never got any, because there was nothing well behaved to denote. A notation is a record of a structure, so where the structure fails the notation has nothing to record.

That last point is the one that ties this essay to the rest of the field. The reason misère play has no compact notation is not that nobody has tried; it is that a symbol supporting an operation requires an operation to support, and there is not one.

Who found it, and when #

The notation did not arrive all at once, and the two books split the work: On Numbers and Games establishes that values exist and how they behave, and Winning Ways is where the symbols and the games are put together for a reader who wants to play something.

Winning Ways was published in 1982 by Elwyn Berlekamp, John Conway and Richard Guy, and Conway’s On Numbers and Games in 1976 laid the foundation.

The notation was not adopted immediately or universally, and some of it — ⧾ and ⧿ for tiny and miny, the loopy game names — remains specialised. What did take is the core: braces for a position, ∗ for the nimbers, ↑ and ↓, and the convention that a value is written as its canonical form.

A test of the claim #

The essay asserts that the notation is a result rather than a convenience, and there is a way to test an assertion of that shape: remove the notation and see whether the results survive.

Take the three facts on this page. ∗ + ∗ = 0. ∗ + ∗2 = ∗3. ↑ + ∗ = ↑∗, which is a first-player win.

Stated without values, the first becomes: a position whose only move for either player is to a position with no moves, placed beside another such position, is a second-player win. That is sayable, awkwardly, and it is a fact about one pair of positions.

The second becomes a similar sentence about a different pair. The two sentences have nothing in common that anybody could act on, and there are infinitely many such pairs.

With values, the three facts are three lines of arithmetic in one system, and the system settles every other pair too. That is the difference, and it survives the test: the results do not survive the removal, because “the results” were never about the specific positions.

The same test applied to a single value says it in one line. Up is above zero, below every positive number there is, and confused with star is three relations about one object, and there is no way to express any of them in a vocabulary of four outcome labels — the labels have nowhere to put a comparison, because a comparison is a statement about a difference and a label cannot be subtracted from a label.

The cost of adoption, which was real #

Notations are not free and this one was resisted, for reasons worth recording rather than dismissing.

It requires learning a vocabulary before any result is available. A reader meeting ↑, ∗, ⧾ and the brace form has to absorb all of it before the first theorem makes sense, and the payoff arrives late.

It looks like whimsy. The names — up, star, tiny, loopy, on, off — were chosen for memorability and read as unserious to anybody expecting the register of a mathematics paper. That has cost the subject readers.

And it makes the material look easier than it is. ↑ + ∗ = ↑∗ is three symbols and hides a computation; a reader who follows the arithmetic can be a long way from being able to evaluate a position.

None of that argues against the notation. It argues that a notation is an investment, and the case for this one is the case any investment makes: what it buys is worth what it costs, and the buying is the arithmetic in this essay.

A summary in one comparison #

The whole argument fits in a comparison between two ways of recording the same knowledge.

A table of outcomes answers who wins this position for each position it lists, and nothing else. Its size grows with the number of positions and it never composes.

A value answers who wins, and who wins every sum, and how the position compares with every other, and what it is worth in many copies — from a single expression, computed once.

The second is not a compressed version of the first. It records a different thing, and it happens to imply the first as a special case.

That is what a good notation does, and it is the reason the symbols in Winning Ways are a result rather than a convenience: they are the first way anybody found of writing down the second sort of object.

Where the ladder goes next #

This anchor’s first rung is what a notation buys. The direction from here is toward the notations that were tried and abandoned, and toward the places where the current one strains — loopy games, where a value may be an infinite expression, and misère play, where there is nothing to write.