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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

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A2 初級

シュールな数字(最初の本を書くこと) - Numberphile (Surreal Numbers (writing the first book) - Numberphile)

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    林宜悉 に公開 2021 年 01 月 14 日
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