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I had this hotel room for a week. On the sixth day I finished it.
On the seventh day I rested; and on the seventh day
I tried to write a letter to my secretary
to tell her how I wanted this to be typed up
and I couldn't finish a sentence!
It was gone; the muse was gone. I would start writing and all of a sudden
I'd say "Now what was I going to say?"
It was flowing for six days and then it stopped
and so I think it's an inspiration that I can't just turn on and turn off, either
Well, I understand you've been considering
all kinds of numbers under the sun
and probably the greatest kind of numbers,
because it includes everything,
is the surreal numbers.
The name actually came to me in the middle of the night; I have no idea...
I have a vague recollection that I saw, at one point,
a French paper that talked about 'nombres sur ordinaux'
I don't pronounce it well, but sort of sur- ordinal numbers
I've never been able to track down that paper and I don't know if I heard about it
before or after I thought of the name "surreal" but you'll see that it's
a great name after you understand what surreal numbers are.
John Conway invented the concept, and his first description of it was
"All numbers great and small"
Not only big and small, but *inside* and everything.
So people know what the real numbers are, and I call them surreal because
there are infinitely many surreal numbers between any two real numbers.
[off-camera] I thought the numbers between all the other numbers already existed, so...?
-Yes, that's what almost everybody thought except Conway
So I learned about it...
I happened to be visiting Calgary in Canada at the same day that
John Conway was up there, and we had lunch together and he had recently discovered
this concept, which we're going to describe. During our lunch
he wrote out on a napkin all the rules of...
a definition of surreal numbers. I was enchanted by this idea. Of course I
had lots of other stuff to do, but I kept the napkin--I took it home with me.
The basic idea: you have a new way to describe all numbers: you build up all
numbers starting from nothing, and actually nothing is kind of the
representation of 0.
But then you can use 0 to describe another number which turns out to be 1.
And you can use 0 and 1 to make two other numbers, one of which is 2 and
one of which is 1/2. And you can continue creating more and more numbers.
It's something like the creation of a universe. Here we're creating a
universe of numbers. But the numbers that you get are always... the denominator is
always 2 or 4 or 8 or 16 or something...
You don't get everything... you'll never get 1/3 throughout the first
part of this process, but you keep on creating numbers until you go to
infinity. Then a big bang occurs and all of a sudden you get all the real numbers.
The next day after that then you create infinity plus 1 and infinity minus 1
You also have 1 over infinity which is a number that's bigger than 0
but it's less than any positive real number. That's why I started calling them
surreal, because there's a number like pi plus one over infinity
This is between pi and every real number that's greater than pi
And also we keep on going and keep on creating, so at some point we get
1/2 of infinity, and we get square root of infinity, and we get infinity to the infinity.
All these numbers that we get we can add, subtract, multiply, and divide them
So it goes way beyond any kind of numbers that people ever conceived of before
[off-camera] These sound like tiny slivers that we're dropping into gaps that we didn't even know were there.
-We got the tiny slivers and we also have the reciprocal, 1 over a tiny sliver,
which is a gigantic thing, larger than we've ever...
people ever thought of before. So that's why Conway called them "All numbers
great and small" One night in the middle of the night I said, "Hey, I bet
why don't I call them surreal numbers?" But the the main inspiration that I had was,
I said "Wait a minute,
this is so cool, I think maybe I ought to write a book for high school students
that would teach them what surreal numbers are." There was a need for a book
that shows how mathematical research is done, and how you take some simple
laws and create gigantic universes out of simple laws.
And the process of discovering is something that's usually not
taught in schools.
You just are taught the facts, but you don't get to see the thrill of
developing these facts. I woke my wife up.
We were on sabbatical year in Norway and I...
Well maybe I didn't wake her up till the morning, but anyway I had the idea in
the middle of night and I couldn't go to sleep and I said "Wait a minute,
a book like this ought to be written." I was already years behind on books
that I'd promise to write, and I'm still not done writing... that was 50 years ago
But I told my wife, you know
Jill, I decided I have to write another book, but it's only gonna take me a week.
She said, well you know that's actually kind of a nice idea...
...time in your life when you can do this, here we are in sabbatical and so we
worked it out that a few weeks later I would rent a hotel room and
downtown Oslo and spend a week writing this book. She would come visit
me a couple times that week, because we always wondered what it would be like to
have an affair in a hotel room! So that was the plan, and it actually worked out
so marvelously, it was probably the greatest week of my life.
I started out every morning having a big Norwegian breakfast.
My hotel was very near where Ibsen used to live so I could get some of his
vibes at breakfast.
There were a group of 50 American students and I could listen to what they
were saying, because I had decided to write this book in dialogue
between two characters:
Alice and Bill. Might as well show you the book. Here's the book and you can see Alice
and Bill here. My wife did the illustrations.
[reading] How two ex-students turned on to pure mathematics
and found total happiness
If you look on the web you find out that that there's a bimodal distribution of
reader comments:
there are those who are looking for beautiful mathematics and those people
are rating it at five, and then there are people who are looking for a novel and
and some cool sex scenes and things like this, and they're reading in a zero and
they're saying it's the stupidest thing I ever did
and "Why would anybody waste his time?" So it's a kind of a litmus test
as to what you like
Oh by the way, A is Alice and B is Bill, but there's one line here which
is said by Conway and I went and visited John Conway a few months later to
try to make sure that I could quote him. So in this particular part, he's...
Conway says "Rubbish. Wait until you get to infinite sets."
Alice says "What was that? Did you hear something? It sounded like thunder." Bill says "I'm
afraid we'll be getting into the monsoon season pretty soon." [off-camera] Is that the only line
from Conway? or is he here multiple times? - I believe it's the only line that Conway
has in here. In the book, he's the creator and he's not only J.H. Conway,
but he's J. H. W. H. Conway, which is an allusion to the tetragrammaton, the
Hebrew name for God. And the characters in the book discover two axioms on a
stone tablet and also some markings on the stone tablet: a surreal number
consists of two parts and you indicate it by brackets and a colon. You have a bracket
on the left. To the left of the colon your put some number
and on the right, er...
some set of numbers and on the right you put some set of numbers, and then you
create more numbers. So you start out with this number which is 0
and if you put 0 on the left,
you get a number which called 1 and if you put 0 on the right you
get a number which is called minus 1. And this stone that Alice and Bill find has
these markings on it, but they have to decode what it means for
these markings to do. Even though I have hardly any plot in this book
the fact that I do refer to nature and have a little story going on, meant
that I was watching everything in the world much more intensely.
Not only was I listening to what these students said at breakfast but I'm also
seeing the colors..., the leaves on the trees... everything outside
There was more, too, because every morning I was actually trying to create
the theory myself.
Conway had told me the thing
and I saved the napkin on which he wrote it down, but I lost it,
somehow, in the next month, so I had to try to remember it and
then I had to try to remember how did he prove all these marvelous things that he
had told me about during the lunch? And I purposely didn't plan that in advance,
but every day I would work on it and then the mistakes that I would make,
the characters in my story would make those mistakes. But then
eventually, you know, I got further and further and was able to
develop the theory.
[off-camera] Had Conway not published it himself and he just...?
-No, he had the report: "All numbers great and small" was his report.
Then he was writing this book
on numbers and games that came out a few years later. But at that point it was
brand new
[off-camera] If you are so much pleasure from this form of writing, why don't you do it more often?
-Oh no, it's too scary.
It's not sustainable! [laughing]
I think you... These are moments that come once in a lifetime. But I
think also there was... it wasn't just me, there was something... the ideas
were coming to me as if they were being dictated, by some muse. The last thing
at night, I'd turn out the light
the next several sentences would flow into my head. I couldn't sleep, so I'd turn
on the light,
I would write down those sentences, but they were coming so fast, I only
had time to write the first letter of every word, so I jotted that down,
turn out the light, go to sleep. The next morning I can figure out what I was going to say
and continue. But if I had been doing that all my life, I think I'd be dead
long ago! So the characters discovered this rock and it's covered with
Hebrew writing
that explains two rules, and everything flows from two rules:
The first rule is: every number corresponds to two sets of previously
created numbers
and furthermore, there's a left set and a right set
Nothing on the left is greater than or equal to anything on the right
That's the first rule. This makes a new number. The second rule
describes what does it mean to be greater than or equal?
So this is the number that we call 0, there's nothing on the left and nothing
on the right.
Here we put 0 on the left and nothing on the right, and this is a number that we now
call 1, but on the stone that Alice and Bill discover, it was indicated by a
vertical line. And if we put 0 on the right and nothing on the left
then we get something that on the stone was indicated a by horizontal line. Now this
sort of Day Zero is when we got the number 0. The next day we got the number
1, we'll call it Day One. On Day Two,
now we can have a set of numbers, so we can put 0 and 1 on the left and nothing
on the right, for example. And this is actually... turns out to be 2.
Or we could put 0 on the left and 1 on the right
this is the number that turns out... is going to be 1/2. But now,
there was a classical definition of real numbers by Dedekind called
Dedekind cuts: you have a set of rational numbers on the left and a set
of rational numbers on the right. That defines a cut between the two and that
gives real numbers. So Conway's genius was that...
keep on going and allowing, besides rational numbers, to allow any set of
surreal numbers. After i get to a point where
0 and 1... (let me put a comma here) And a 1 and 2, let's call it,
and you get all of these here,
but nothing on the right, this is infinity. Then we find out what's
infinity plus 1? And that turns out... actually got an infinity here, ...
These rules might have been invented before Dedekind's rules and that everybody
for 100 years had learned about this in school, and we'd considered that
this is the way numbers are. Then physicists would have said that this
is actually part of the real universe, that these numbers aren't strange but that
that these are basic. And people probably would have discovered the real
numbers as a special subset of the surreal numbers. There's no reason for
us to think that the universe obeys the laws of real numbers.
People used to think that the real world had Euclidean geometry, but now
people know that space bends. The patterns that are revealed here are as
mind-boggling as anything in mathematics and are far from being explored. Every year
more and more converts come along and and we find people winning prizes
because they've advanced the theory of surreal numbers.
...which is a subtle point...
So, I'd lost a napkin
but it turned out that the rule that I gave in my first draft during those
six days was not his rule actually, but it was subtly different.
Here, for example, is Chinese translation. [Conway] ...don't like thinking of my impending
death, [pauses] and you know, I haven't got all that many years left...
I don't quite know how many