On Numbers and Games by John H. Conway

John H. Conway recently died from COVID-19.

When Conway inevitably challenges the Devil to a game,

I would not envy the Devil.

When the Devil asks him what game he chooses,

Conway will simply answer:

"All of them. At once."

A Divine Game

Imagine you are God.

Literally.

Perfect in every way.

You create the universe,

naturally,

because this is what God is supposed to do.

You set the initial conditions perfectly,

because you are perfect in every way,

and let the rules of physics unfold from this initial condition

into a beautiful,

blossoming world.

This can take a while.

Since you created things perfectly,

there is nothing to do,

no need to intervene.

Things could get boring.

You need some way to entertain yourself.

So you let the Left arm play a game against the Right arm.

A Go game,

rich in majesty,

unimaginably big in scope.

Both the Left arm of God and the Right arm of God are,

of course,

great players.

They make no simple mistakes.

So after a while,

the middle of the board fills up.

The Divine Go game is now made up of many sub-regions

that do not interact.

When it is the Left arm's turn,

it picks a region,

and puts a stone.

When it is the Right arm's turn,

it does the same --

but not necessarily the *same* region.

A few billion years in,

God Himself looks at the Divine Go board.

"I wonder,"

He asks Himself,

"who is winning".

God wonders if it is possible to find some

"arithmetic of games",

so that if you analyzed each sub-region separately,

you would do some computation and figure out the state of the board.

Human Games

Since echoes of the Divine world are found in our world,

the same thing happened.

John H. Conway invented the Game of Life:

a 0-player game where initial conditions unfold into beautiful

guns shooting spacheships and,

potentially,

doing any computation imaginable.

Just like God,

when Conway was done creating a perfect world,

he decided to amuse himself with games between

Left and Right.

Just like God,

Conway looked for an arithmetic of games.

The first thing Conway did was seperate the games that favor Left

from the games that favor Right.

He called the first kind "positive"

and the second kind "negative".

The games which favored neither he called "zero"

if the second player could win,

and "fuzzy" if the first player could win.

The reason that favoring the second player gets the nicer name

is because an empty game,

one with no legal moves,

favors the second player:

the first to play finds themselves with no legal moves,

and must concede defeat immediately.

Conway defined the "negative" of a game as one where the roles of

Left and Right are switched.

All of these,

any mortal human could do.

It was the next definition that secured Conway his Divine Right:

addition of games.

In order to add two games,

you imagine that at each point,

at each turn,

the player chooses one of those games,

and chooses a legal move.

Imagine adding chess and checkers,

and letting

Left begin.

Left likes checkers,

so they play a move on the checkers board.

Right prefers chess,

so they play a move on the chess board.

On Left's turn,

they could play again on the checkers board.

If you only looked at the checkers board,

you would see white move

twice:

clearly not a usual move in checkers.

If we know what addition is,

and what the negative is,

we can define subtraction: A-B=A+(-B).

Now we can define an order on the games: A is less than or equal to B if A-B favors right or the second player.

Numbers

Now Conway does something that every mathematician,

and even God Himself,

would envy,

He defines a number.

A number is a game where each Left option

(the game after Left took one turn)

is less than each Right option

(the game after Right took one turn),

and each option is itself a number.

So the empty game is a number

(conveniently called "zero"),

since there are no options for either player.

The game {0|} where Left can move to the empty game

but Right has no legal moves is clearly biased in favor of L:

it is positive,

and we call it 1.

Its negative, -1, is {|0}.

The game H={0|1}, where Left has the single option of moving to the empty game,

and Right has an option to move to 1,

which is a game they lose,

is clearly still biased in favor of Left.

But how biased?

We have defined addition and subtraction,

so with a little legwork, we can see that

H+H-1 favors the second player:

in other words,

it is zero, and

H is exactly 1/2.

It is still biased in favor of Left,

but in a strict sense,

less biased than 1.

It is possible to define multiplication on games,

but this definition is more subtle.

Conway himself struggled with the definition for many weeks.

But,

for the moment,

take it as a given that we can define multiplication,

and the expected results hold: (1+1) times 1/2 gives 1,

for example.

How many of the numbers we know are games?

"All of them" is a Huge understatement.

All real numbers are games.

So are all infinities.

You can embed all multi-variate real-coefficient polynomials in it.

They all can be added, subtracted, multiplied, divided

(except by zero, for this is a rule not even God or Conway can violate)

and compared.

Worlds

Conway built a machine to analyze the game of Go,

which splits naturally into separate sub-games where each player

chooses a sub-game and makes a move.

The "addition" operation is exactly the operation needed to analyze

such a game.

But what he got was a machine that

spits out a structure that takes a few weeks in an upper-level math class

to build,

all in the space of a few paragraphs.

Describing it as a math book does it injustice:

it is a physics book of a fictional universe...

or several fictional universes.

Together with No,

the field of numbers defined above,

Conway introduces us to the magical land of Oz,

and the mirror universe of On.

For much like God,

Conway wanted to create many worlds to maximize total happiness.

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