Morita conjectures

Transcription

PETER: Welcome to Authors at Google and welcome to Alex Bellos. I don't know if any of you saw his earlier talk a couple of years ago around his first book, "Adventures in Numberland," which was a huge hit. And on a personal business, it was a huge hit for me because my 10-year-old son at the time I couldn't get him to read any books at all. And in desperation, we were on holiday, and I kind of handed him "Adventures in Numberland" and said have a look at this. And he quickly became extremely hooked on and spent the entire holiday going yam, tan, tethera which I'm sure you'll know is the ancient English way of counting sheep. And when you get to 15, it's bumfit I think I'm right in saying. And he spent the entire holiday talking about bumfit. So it sort of both hooked him in an interest in maths and gave him a license to use bad language so it was kind of perfect. So the new book is "Alex Through the Looking-glass." Alex is going to talk for half an hour or so and then take some questions? ALEX BELLOS: A little bit more than that but, because I want to be very informal, so if there's anything you don't understand or you want more, just shout out, put your hand up. So let's kind of have the kind of Q and A as we go I think because we're a little bit delayed. I'm going to have to-- PETER: And I'll hand if over [INAUDIBLE]. Keep going. Cheers. ALEX BELLOS: Usually, I'm used to giving talks to book festivals where the kind of mathematical numeracy is going to be a lot lower than you guys are here so I might whip through it but you might have slightly more high-brow questions. So thank you very much, Peter. I'm Alex Bellos. This is me. Oh, my god. I just pressed something. That's me with a weird stain on my red jumper. I don't know why I was given that. And this is my new book, "Alex Through the Looking-glass," which sounds very much like it's a sequel to "Alex's Adventures in Numberland" but it isn't really. Both can be read separately. You don't need to have read the first one to read this. If there is a difference, the first one was just an exploration about mathematical abstraction and what we get when we use it. This is a little bit more kind of applied. It's about using abstraction to learn about the world, mathematics being the language that we use to talk about the world, and also how life reflects numbers and numbers reflect life. I also wanted to get into kind of psychology of numbers and interesting psychological responses that we have to numbers which are quite surprising I think and definitely under-researched. Now, how do you write a book on maths and numbers and make it interesting. It's quite difficult because the nature of the subject is dry and sometimes it's conceptually challenging. So I've got a degree in maths and philosophy and then I became a journalist. So my approach is always to tell stories. And I think that by telling stories, if you can interest someone in the story, then you don't need to say, oh, by the way I'm going to teach you some maths. You just learn the maths on the way. And where do you look for stories? Well, the kind of classic place to look for a story is in oneself. And so, actually, my story as a popularizer of mathematics is really where this book begins. And I have to say I don't think it happened at my talk at Google last time, but pretty much every talk that I did when the first book came out, I would give what I thought was a fantastically fascinating talk about maths and numbers. And someone would put their hand at the end of it and they would say, yes, sir. And they would just say, so what's your favorite number? And I can just remember thinking, oh, have you learned nothing? Who has a favorite number? I'm grown-up. You're a grown-up. And I would just think they were kind of teasing me. It was way of belittling the subject on myself. Until I kept on being asked this question. I would give talks at universities, secondary schools, books festivals. People were always asking it. So once I just said, oh, I didn't know. What's your favorite number? But rather than saying, ah, they went, oh, 12, gave me a considered response. And I just was like, what? And then the person next to them said, oh, no, my favorite number is 13. I said, well, who here's got favorite numbers? And at least half the audience of grown-ups put their hands up. And this happened a few more times. So I thought, well, this is interesting. From being something that I thought was completely idiotic, I thought, well, if grown-up people who are literate and numerate have serious [INAUDIBLE] emotional reactions numbers, this is interesting and why don't I try and research it? And so what I did is I thought I would undergo a survey to try and find the world's favorite number. But essentially it was to try and quantify emotional responses to numbers. So I set up this website here, favouritenumber.net. I realized it was going to have to be international, so I also bought this one, favoritenumber.net, just so no one was excluded. And within a few weeks-- at that time, four years ago, I maybe had only a few thousand followers on Twitter. I just put it on Twitter. What's your favorite number? Join my survey. Within a few weeks, more than 30,000 people from around the world had replied. And so this was already vindicating this idea. This is something that serious people feel passionately about it. Now, what were the results? The results were this. 7, 3, 8, 4, 5, 13, 9, 6, 2, 11. Which "Metro" said was the most confusing top 10 list ever written. Which it is. Now, if you go back to favoritenumber.net, I have all the results up there. There's a big Excel file of all. There's more than 1,000 individual numbers and all their frequencies. And you can get them there if you are interested to do an analysis on them, if you have a statistical mind. But what was interesting was that when I launched the results, it was an actual news story. Again, showing that people are interested in this stuff. It was a news story in the "Times." It was a news story in the "Daily Mail" of all places. But it also started going around the world. This is "Glamour" magazine in Paris. And I don't think I found one country that didn't report it in some way. Just going to mention a few of them. Hungary. Hungary has a huge brilliant mathematical tradition. And per capita, I think that more Hungarians entered the survey than any other nation, so they were really fascinated by it. Obviously, it was [INAUDIBLE] Greek, Greece being the beginnings of Western thought. By the time, it got to India, I was now a Brazilian mathematician. But India-- there are certain countries where mathematics and the idea of doing arithmetic is really an important part of national identity. And so it was all over the regional press in India. Vietnam, and obviously I can't read Vietnamese. And it just became quite interesting what happens in the different photo archives countries when you put in seven. This is what you get when you're in Vietnam. And when you're in China, this is what you get. So it was a new story. But what is the moment that you really know that it has touched the public consciousness, it's become a real news story? It's when you're a question-- volume, please? Three days afterwards, it was on "Have I Got News for You." [VIDEO PLAYBACK] -Alex Bellos asked 44,000 people to submit their favorite number-- ALEX BELLOS: So yeah there were 44,000 people in the end. But I only did the cleaning of the data for 33,000. The Nigel [INAUDIBLE] numbers. ---top ten world's favorite numbers. -Yes. -Yeah. ALEX BELLOS: The reason why this is funny is because we actually do have feeling-- things are only fun if there's an element of something serious about it. -Number ten, the tenth most popular is, in fact, 11. ALEX BELLOS: [INAUDIBLE] mathematical joke. -14. -No. -21? -No. At number nine is the number two. The most popular number in eighth position is six. And seven, the seventh most popular number is nine. And the sixth most popular number is 13. OK. Here's the top five. -This is going to be on Channel Four for a whole evening. There'll be some talking heads in a minute. What did you of six? Eh, not it. ALEX BELLOS: This is actually my idea-- -I've always liked five because it's a working class number. That's what I like about five. But I think that three [INAUDIBLE] and I thought three was a magnificent num-- oh, I'm sorry. I'll stop auditioning. Sorry. Where am I? -Let's firstly complete the top five. The number five is five. And number four is four. And number three is eight. And number two is three, the magic number. And, of course, seven is the top one, number one. [END VIDEO PLAYBACK] ALEX BELLOS: OK. It's interesting that everyone laughed at five being five and four being four because, essentially, we like small numbers. And the bigger the numbers get, we like them least. This is what the survey says. So a number should, if there are no other considerations punch more or less exactly their numerical value. So those are almost the ones that are the least funny or the least interesting. And what happens and what you see is the odd numbers and especially prime numbers are way up the rankings and round numbers really, really way back down the rankings. We do not really like round numbers, but we really like prime numbers. And that is something which, even though the quantitative research you can dismiss and say, oh, this is silly because it's 44,000 people but it was 44,000 people who already feel passionately about it who selected themselves to join the survey. Even amongst them, there are very coherent patterns of these emotional responses. So on the survey, I said, what's your favorite number? But I also said, why? I just left it blank. There's a little space. You could write it in there. And to me, it was the qualitative results of the survey that were actually a bit more interesting and a bit more surprising and that felt a bit like proper new research. So I'm going to just take you through some of the entries with some of the reasons. And we start to understand what it is about numbers that make people excited by them. So this number was selected, 10 over root two. Now, in this audience, does anyone know what 10 over root two is in decimals? It's 10 times 1 over root two. Which if you work out is actually square root of 50. 7.07 which in America is called the Boeing number because of 707. And I had originally thought that Boeing was called 707 just because this looks a little bit like a plane, two wings and a fuselage. Apparently that's not the reason why. But you will find that certain numbers work in brands really well based on just how they look and how they sound. Because even though numbers are supposedly these abstract entities just representing quantity and order. How do we get to them? They exist within culture, within language. They have to look like something as a symbol. And all these things actually influence how we understand even the numerical properties of numbers. I'm going to read you the reason why the person chose 707. It wasn't because they were an aviation enthusiast. People wrote down their nationalities, their genders, their level of mathematical ability, and their age. And this was from a woman in Canada, aged 34, with university-level maths. She said, this number is 10 root two. I'd been doing a lot of trig homework for a calculus class I was taking in undergrad and this number appears a lot. At the time, I was sort of weirded out by the fact that I kept waking up at 7:07 in the morning instead of 7:30 when my alarm was set for. Anyway, one Saturday, I went to my local art supply store and bought some paintbrushes. To my surprise, the total came to $7.07. And I sort of blurted out, oh, that's 10 over root two all over again. So the very cute cashier, for whom I had a rather pathetic crush and who I was constantly embarrassing in front of, after explaining myself, he was duly impressed and began embarrassing himself in front of me whenever I came to the store. And from that point on, I realized there's a brand of arty guys that like nerdy girls and this still makes me happy some 15 years later. OK. This is written in a tiny little white space on a website. Just the story, the passion. There's so much stuff that you could analyze. Also the fact that when you start to think emotionally about a number you just see it absolutely everywhere even though numbers don't really predominant one over the other in such a strong way. 1,000,000,007. Why was this chosen? This was chosen by 21-year-old male Russian because it's the largest prime number I can remember. So the fact that people take to remembering large prime number. A prime number I'm sure you now. A prime number is a number that only divides by itself and one. So they start with two. Two is the only even prime number, then it goes 3, 5, 7, 11, 13, 17, 19, et cetera. Now, 1,000,000,007 is interesting because 1,000,000,009 is also a prime number. In fact, you get this things called twin primes which are prime numbers that are two apart. And it's conjectured that there are an infinite number of them. But the largest one that's know of the form one, loads of zeroes, then seven, one, loads of zeroes, then nine is 1,000,000,007 and 1,000,000,009. So you can kind of think you're really clever to remember [INAUDIBLE] prime number. It would be clever if it was the second highest prime number I can remember. 219. Now, this was chosen by an 18-year-old Irishman because it's the lowest whole number that does not have its own Wikipedia entry. So it's too boring. It's the lowest boring number, basically. Every number from one to 218 has it's own Wikipedia entry. And, actually, this was a couple years ago now. So I've looked. There's an inflation in lowest boring numbers. And now the lowest boring number is 224. OK. But you might not think that Wikipedia is a very good judge of what makes a number interesting or not. And so I thought I would look in the online encyclopedia of integer sequences, which is a kind of Bible of mathematics and really of all science. It contains about 250,000 separate sequences. And a sequence is just a list of numbers. They can be infinite. They can be finite. So the prime numbers, that's a sequence. The square numbers, at one, four, nine, sixteen, twenty-five. That's a sequence. What it be interesting to find out what is the lowest number that has never appeared in the online encyclopedia of digit sequences, which you could probably say is the only number-- well, probably the lowest number never to have to appeared in any mathematical piece of research in the last 3,000 years since, really, there's been mathematics. And that number is 14,228, which I thought was quite low personally. But it's also interesting. It's very, very divisible, isn't it? Divided by two obviously. This idea that numbers that are really interesting are the prime numbers, the odd numbers. Finally, this number 11, which is also a cheap plug because I also used to live in Brazil and wrote a book on Brazilian football, which has also been updated for the World Cup. So if you're interested in finding out about Brazil and the World Cup buy my book about it. It has [INAUDIBLE], who is the-- well, he's not the captain, actually, at the moment, but he's the talisman of the Brazilian national team. And the reason why 11 was chosen was from an American, 36 years old, and I like to think that he was from the deep South because if he wasn't, it doesn't really make much sense because he said, I like 11 because it sounds like lovin'. So, why? What are the reasons why seven was the world's favorite number? I made a little animation, hoping the sound is OK. [VIDEO PLAYBACK] -We've been obsessed by the number seven for as long as we know. Go back to the earliest writings there is, on Babylonian clay tablets, and they're just full of sevens. Then you have seven dwarves, seven sins, seven seas, seven sisters. The list just goes on and on. So why is seven so special? One argument is that there are seven planets in the sky visible to the naked eye. To me, that's just coincidence. There's a much more compelling reason. Seven is the only number among those that we can count on our hands-- that's those from one to 10-- that cannot be divided or multiplied within the group. So one, two, three, four, and five, you can double them. six, eight and 10, you can half them. And nine you can divide by three. Seven is the only one that remains. It's unique. It's a loner, the outsider. And humans interpret this arithmetical property in cultural ways. By associating seven with a group of things, you kind of makes them special too. The point here is that we're always sensitive to arithmetical patterns and this influences our behavior even if we're not conscious of it and irrespective of our ability at maths. [END VIDEO PLAYBACK] ALEX BELLOS: There's another thing which kind of ties into this. If you were to ask someone-- and this is a psychology experiment that's been repeated many times-- just think of a number off the top of your head between one and 10, most people say seven. And why they say seven, essentially-- again, you're doing mathematics without realizing. You think, well, I'm not going to say one because that's not off the top of my head. That's just too obvious. It's not sort of arbitrary enough. I'm not going to say 10. I'm not going to say five. Well, you've just done the two times table. And then you sort of say, well, two, that's also. I wouldn't choose two. Then you eliminate two, four, six, eight. You do exactly what I was doing in there and you're left with seven. Seven-- it's the most difficult. It's the one that's [INAUDIBLE] to stretch where it feels the most random. Now, if we go back to the qualitative results of the favorite number survey, I was trying to work out, is there any way that I could grasp or explain the kind of collective views about numbers? And I was thinking, well, how can I do this. And I was reading all this more than 30,000 reasons why people like numbers. And I noticed that the language used to describe each number was different. So I'm going to pluck the adjectives that we use to describe different numbers. Number one was described-- this is each word in a different reason-- as independent, strong, honest, brave, straightforward, pioneering, competitive, dramatic, egoistic, hot-headed, and lonely. So imagine we were a creative writing class. And I said this is like the main protagonist of our story. Those are really quite coherent personality traits there. Two. These are words used to describe the number two. In fact, sorry. That was number one. Two. So look at number two and see if you visualize these traits. Simple, elegant, cautious, wise, observant, pretty, fragile, open, sympathetic, comfortable, flexible, subtle, inconspicuous. This is from lots of different people and they've got lots of different reasons but together, again, there is a kind of coherence there. And what struck me most of all was that if these were a creative writing class and we were having to imagine characters for the numbers one and two, one is really male characteristics or traditionally male characteristics. And two, traditionally female characteristics. So the cartoonist of the book did exactly that. This is how she interpreted it. We could do a fantastic kind of rom-com or romantic story with one as the man and two as the woman. Now, I saw this and I thought, well, this is really quite striking. But the more that I read-- and I read quite a lot of stuff on the history of maths and the growth of intellectual ideas and also about psychology-- it's really quite common, [INAUDIBLE]. So numbers were invented about 8.000 years ago, say, in Sumeria, which is present-day Iraq. The very first words that we know that we used by humans for the number one and the number two-- the word for the number one is the same words as the word for phallus and the word for two was the same word for woman. So right from the beginning, there-- for whatever reasons. We don't know. You can imagine, I suppose. One is male and two is female. Actually, a friend of mine who's a quite well known novelist, she said well, Alex, I was reading your stuff about masculine and feminine. And I just thought that everyone thought that. I've always thought-- and she's one of the smartest people that I know. And she was saying, yeah, well because it's obvious that one and odd numbers are masculine and two and even numbers are feminine. They obviously are because two you can kind of split apart. They can become two new things. It's like they kind of give birth. And obviously, I had never thought that at all. But the fact that someone-- there are people who are smart who think these. We do relate to numbers in emotional ways. Pythagoras who's the beginning of Greek mathematics had this Pythagorean brotherhood. And they believed that odd numbers are masculine and even numbers are feminine. And I discovered some research that had been done quite recently in which respondents or participants were asked to look at babies who are six weeks old of indeterminate gender and all you have to do is just say, is this a boy or is this a girl. Always, they had numbers at the bottom of them. But the people were told. Don't look at the numbers. There's going to be a number. But just don't look at it, just look at the child. And what the experimenters were doing is that sometimes it was all three even numbers, sometimes all three all numbers, all kind of randomized, proper science and that. And it turns out that, with odd numbers, you're 10% more likely to think that a baby of indeterminate gender is male and with even numbers, 10% more likely that the baby is female. So there is still this kind of subconscious association with odd numbers with masculinity and even numbers with femininity. Also, when I [INAUDIBLE] started to get interested in this idea that numbers have personalities. And even though if I had been talking to myself a few years ago I would have thought that absolutely ridiculous. When I started to think about it, I though, yeah, well I use numbers all the time. Yeah, certain numbers do kind of feel a bit different to others. And then when you start looking at even Shakespeare noticed this. This is from "Merry Wives of Windsor," Falstaff, They say there is divinity in odd numbers, either in nativity, chance or death. And it's true. All our most mystical religious symbolic numbers are odd numbers in the Western tradition. So three, Christianity, seven, 13, five a little bit. They're all low primes and this is interesting. And there is psychological research, proper psychological research, not just my online survey, that starts to give us a bit of an inkling about why this might be the case. So one of the experiments was you're just shown two digits. Either they're both odd or they're both even or one is even and one is odd. All you need to do is just press a button when they're both the same. So either when they're both even or when they're both odd. Turns out you're much slower pressing the button when they're both odd and you make more mistakes. In other words, it kind of takes us longer just to process it, just to think about them, not to do any math with them, just to think about odd numbers than it does with even numbers. And this sort of gives this idea that somehow there's sort of more room, there's more space, to have a kind of emotional interaction with them. The final experiment I want to talk about in terms of these psychological and emotional responses to numbers is all about branding. And I'm sure we're going to be seeing a lot more of this because this is quite recent work. The scientists, the academics, wanted to know if we are more likely to be attracted to want to buy and pay more money for brands if they have different types of numbers on them. So WD-40 is a very popular brand. That's an even number. Oxy10. That's an even number. There's something sensible, safe about even numbers. And it turns out that, for household products, for things that you want to work and be reliable, people are much more likely, will pay more money, if they have an even number on them. So the experiment here was two packs of contact lens, one is the brand called Solus36, one is Solus37. They're absolutely exactly the same. You do all these tests about how to evaluate the brand. Solus36 is much more popular. People pay more money for it. They want it. They like it better. OK. So this is the first of the experiment. Then they add in this tagline, 6 colors. 6 fits. Just to see if this affected how much you wanted the brand how much you liked it, how much money you were prepared to pay for it. And what happened is that Solus36 becomes even more liked and Solus37, which wasn't liked very much in the first place, becomes even less liked. And the argument put forward, the hypothesis, by the scientists is that the reason why we like Solus36 is that we-- how do we learn numbers? We learn numbers by learning times tables by rote when we're children. So we are much more familiar with numbers that are in times tables. So 36. Whenever we do the six times tables, we get it. We do the 12 times tables, we get to it. And what happens? We never get to 37 because it's a prime number. It's not on any times table. It will be the answer to very few-- basically, as a child, you will never utter or think about the word 37 but you will use 36 all the time. So this idea that it's what you're familiar with is that fluency of processing. And you misattribute this fluency of processing for liking of the product. So it kind of feels good. Oh, I like that. Oh, it's like a friend, a friendly product. And this is what's exacerbated by the tagline because subconsciously we're seeing six, six, 36, and it's like, oh, yeah. It's like a nursery rhyme from school. I remember that. I like that. I really like those contact lenses. And with six, six, 37, there's a kind of mathematical cacophony. We don't like it. So that brand is really unattractive. And they did a lot more experiments. And it's interesting that when you have an ad for something and if you subtly put numbers in it that relate arithmetically together, you can increase quite drastically the liking for that brand. So in the future I'm sure there'll be lots of numbers subtly positioned within ads. Which you tend to think, oh, god, I hated learning maths by rote. I hated learning my times tables, but actually that either can be kind of leveraged to make you feel good about a product is ironic to say the least. That is essentially what is the beginning of my book, the story about me and the favorite numbers and emotional connections to numbers which is how life reflects numbers. The rest of the book, so the majority of it, is kind of proper more serious maths. And I'm going to just talk to you about one part of it now that begins with-- we're talking about prime number. So let's just repeat prime numbers. These are the prime numbers. The numbers that only divide by themselves and one. Those are the first seven-- I think, yeah. --of them two, three, five, seven, eleven, thirteen, seventeen. We live in the age of algorithms as we're told every day. What was probably the very first algorithm in mathematics and, therefore, in civilization? It was probably what's called the sieve of Eratosthenes which was the algorithm to try and find all the prime numbers under a certain number. So this is a grid of the numbers from one to 100. I've done them in six rows. We're going to use the sieve of Eratosthenes to try and find all the prime numbers. OK. What do you-- you start at the bottom, you get to a number, if the number has not been eliminated, it's a prime number. And then you eliminate all multiples of it. So let's start with one. One's not a prime number because we start at two. Get to two. Then we eliminate all multiples of it which is basically all even numbers. And it's nice when you do it in rows of six. So next one. We get to three. Three's a prime number. Then we eliminate all multiples of three which is just one more line because the bottom row is all multiples of three but we've already eliminated that. Four's been eliminated. We get to five. That's a prime number. Eliminate all multiples of that. Six has been eliminated. You get to seven, eliminate all multiples of that. Nice little crisscross pattern. Then up to 11. And you only need to-- when doing the sieve, go to the square root of the number that you're seiving which is root of 100 which is 10. So we've got to 10 already so therefore we can guarantee that all of those numbers in their are primes. And there are few interesting things to say. First is that all prime numbers are either one or one less or one more than a multiple of six because that can only be in line one or in line five. If this went on to infinity, it would be the same. But also-- which is why people are kind of fascinated by prime numbers-- is that before you get to them, there's no way of knowing if a number is going to be prim or not. They appear to be kind of randomly sprinkled around the number line. And that is one of the reasons why they are still one of the first things to be studied in numbers and there's still-- lots of things about them are just not known. Lots of books about prime numbers. So in 1963, Stanislav Ulam, who was one of the great mathematicians of the 20th century, was doodling, got bored in a math lecture and started to doodle. And he did a great but he put the one in the middle and then started to spiral and then started to cross off the prime numbers. And he noticed something that no one had noticed before-- that prime numbers tend to sit on lines together. And so this is something after several thousand years no one had ever bothered to do this. And this is up to 100, which is the numbers that we saw on the grid before. But if we do it up to 20,000, you see it a bit more. There are these lines that prime numbers seem to sit on. And it's called an Ulam spiral. And really, in maths, you kind of have total order and that's a bit boring. You have total chaos. That's a bit complicated. When you have this sort of in the bit between order and chaos which you have here is where things get really interesting. So this is a really kind of simple yet incredibly deep image of where are all these primes and what relations do they have to each other? This is Ulam right here on the right. And he's sitting. This is just after the Second World War near Los Alamos with John Von Neumann on the left, who I hope that Google has a shrine to John Von Neumann somewhere because basically he invented the computer with Alan Turing of course. But a lot of the kind of conceptual framework of computing was basically invented by John Von Neumann. In the middle is Richard Feynmann, the famous American physicist. This is an amazing picture because you could quite easily argue that these three men sort of changed the 20th sort of more than anyone else because they were instrumental in creating nuclear weapons, the computer, Von Neumann-- possibly the greatest mathematical genius of the 20th century. Whatever he tried to do he invented like a new field. And it's one of the fields that he and Ulam invented I'm just going to finish off talking about now which hopefully is right up your street because it talks about essentially the first computer craze that there ever was. So Von Neumann was inventing so much stuff. And it was in the 1940s, early '50s, started to worry about what he was creating. Computers, robots, what's going to happen to robots? What would it take, he asked, for a robot or a machine to replicate itself? And this is actually a kind of psychologic problems. It's like a mathematical problem. It's not really a biological problem. It's a kind of conceptual idea because just say you've got the machine and you've got, say, the instructions. If you put the instructions in the machine, can the machine make itself from the instructions? Because the instructions, obviously, are part of the machines. So you cannot replicate. And basically it can't in a finite world. Just say you've got a machine with a set of instructions. What you want to do is create exactly that. A machine and a set of instructions. Just say the machine reads the instructions and it's got instructions how to rebuild a machine. Does that set of instructions have instructions on how to build a new set of instructions because if it does, then that set of instructions within the instructions. It's got to have some other instructions. And then you get this infinite regress. It's a finite system. It can't happen. So he realized that if you have a machine with a set of instructions to self-replicate, you need to treat the instructions in two ways. You need to read it to build the machine and then you need to duplicate it and then present it to the new machine. So there are two ways-- basically, you need these two elements. And when, a few years ago, Watson and Crick discovered DNA, they realized that essentially the same thing happens in humans more in general terms. The DNA is the instructions. It has instructions for the cell. But DNA does not contain instructions for DNA because DNA replicates. It's a double helix and one of the helixes splits off and then you create another one. So this is another fantastic example of something which the mathematics was conceptually discovered years before it was shown to be that was the way-- it's like the Higgs boson. It was kind of mathematically proved and then, decades later, it was discovered. What John Von Neumann wanted to do-- he's also like a fantastic entertaining character. He loved parties and sometimes would hold parties just because he liked working to the sound of parties. So he would be there kind of mixing together and then go back to his room and do some amazing maths. He wanted to think about building a self-replicating robot, something which can use instructions, then duplicate them. But it was just too complicated actually working out how to build it in the real world so Ulam who, as we've seen, loves grids and things like that and patterns, said, let's invent a new kind of mathematical abstraction. Which they did, called the cellular automaton. And a cellular automaton essentially is a grid of cells. The cells can have different states but the behavior of the cells depends only on neighboring cells. I'm going to explain now a very simple cellular automaton called the Game of Life because Von Neumann invented it to try and find something that would self-replicate and he managed to do that conceptually. And then, a decade or so later, at the end of the '60s, a British mathematician called John Conway, who's still alive-- he's at Princeton at the moment-- invented what is one of the most famous examples of what's called recreational maths but it's also quite serious called the Game of Life. It's not a game. It's a cellular automaton. Life because it emulates evolution of life. So this is the grid. The game of Life-- I'm going to explain how it works. All of it takes place on a two-dimensional grid. The grid can be infinite, and it has these orthogonal cells. The cells can have two states, either alive or dead. This is a live cell. All around it are dead. And it's immediate neighbors-- there are eight of them-- are these ones here all around it. So its behavior depends only on those eight cells around it. And these are the rules that it obeys. There are the Game of Life genetic laws. One, if you have a live, if it's surrounded by zero or one live neighbors, it dies by loneliness. If it has two or three live neighbors, it survives. Four or more live neighbors, it dies by overcrowding. Really, really simple. Basically, it a live cell is surrounded by two or three, it survives. If it's not, it dies. A dead cell, if it has exactly three live neighbors, it becomes alive. And in all other cases, it stays dead. So Conway-- this was the late '60s-- he had one of those grids. It was basically a Go board. There were no computers then. And he used Go counters just to see what would happen. So let's see what happens. This is a pattern with three shapes. What the Game of Life does is what we will always do-- we will start with a pattern. And the idea is to put an interesting pattern on the board and then you don't, in fact, touch it at all. It then just evolves and changes. So what we want to see is how it evolves. And Conway chose his rules to make the most unpredictable and interesting way that things evolve. So what we're going to do here-- let's look at this one here, surrounded by one live neighbor, it will die. Surrounded by two live neighbors, it will survive. Surrounded by one live neighbor, it will die. A dead cell surrounded by three, it will survive. So what happens is that you work out how each one is going to behave and then you apply the rules on everything and then you get a new generation. So what would happen here? The new generation would be that. OK. And then it dies because there are only two left. This one here. That becomes that. And then it dies. This one here. We're going with shapes with three live cells. Becomes that. And then it just stays there. It's called a stable form. So at the beginning, they were like zoologists. They've got this new world. They were just trying to work out what was there, finding the patterns and what would happen. This one here, another three cell pattern. Becomes that, then that, and then that. That's known as the blinker. OK. Not that interesting yet. When you start to add more, most things end up dying, but sometimes you get amazing bursts of life. So this four-cell pattern becomes this then this then this then this. It's four blinkers, which is called traffic lights. Let's go and look at a five cell pattern. So they had done four-cell, see what happened to them. They were cataloging them. And then they noticed something amazing. And this is when basically the study of life really took off because look at this five-star pattern. It repeats. It's called a glider. Every four generations, it's moved one space down and one cell along on the grid. So essentially it moves. It looks like it's moving. And this was the-- Conway called these spaceships in the kind of etymology of the kind of animals or the living things that you get. And he wondered and in Scientific American he said, does anyone know if there's any pattern that can grow and have more and more live cells. Because this one it all stays at five cells all the way down. Something that can grow and get infinitely big. And he put a reward in Scientific American. And the Martin Gardener article that wrote about it was the most read article he ever wrote. And this is a guy who is like the guru of all kind of popularization of maths and science. And this was Conway in Cambridge writing about it. Martin Gardener in America writing about it. But the real development started to happen at MIT. The Game of Life came along at a time where computing was just beginning. But you would only have computers if you were in a big kind of organization or if you were at a university like MIT. And I don't know if you know where the word "hacking" comes from. The word "hacking" originally comes from the MIT Model Railway Club, and a hack is when you kind of customize something just for the hell of it, just to make it fun. And so the hackers were the people who worked at the MIT Model Railway Club. But when computers came along, they became not so interested in model railways and worked on computers. And the thing with Game of Life, as you saw-- incredibly simple rules but incredibly complex behavior. It became basically the first computer craze that the hackers at MIT-- and this is essentially the people whose-- the forefathers or forebrothers, say, of Steve Jobs, Bill Gates, et cetera, were the hackers at MIT working on the Game of Life, which was basically the biggest deal in computing in the early '70s. And Bill Gosper, who was the kind of king of the hackers, discovered and won the prize for the first pattern that grows without bound, which is this. And it looks like a kind of pair of lungs. And what it's doing is called a glider gun. And basically it just, every period, spits out a glider. And that will carry on, getting bigger. And the total of live cells just gets bigger and bigger. And the more that people looked at this, they started to find other patterns that did other interesting things. So this thing here is called an eater. And what an eater does is an eater will eat anything that you send at it. So this gave the first indication that this might actually have some use because it's like how to build self-repairing object. But also it meant that you could start to just have these amazing kind of engineering patterns where you could build massive patterns and all the debris you could have eater just clean them up. So let's have a look at the eater, what happens. So this is a glider coming in it to an eater, gets eaten. This is another spaceship, just gets eaten. And let's now look at-- this is the sort of thing that people started to build. So this is a glider moving. And I'm going to be going out of it. What's happening? So it's going to this other massive pattern. What's going to happen to the glider? Bing. It bounces. So that's basically a glider reflector. It will bounce a glider at 90 degrees. So as we keep on going out we see we've got these gliders in kind of outer space. And what's interesting is that now you look at it, you're thinking it's like these little ants walking around. The grid, every cell, is only responding to the eight cells around it. So there is no kind of overall guiding thing, but yet there is. So Game of Life is such a great way to understand how things at different scales take on different appearances. So we keep on going out of this pattern. All these are gliders in streams. And here we are. In fact, what this is-- it's a gun. It goes back in in a second, magnifies. These are the two kind of barrels. And everything is working in such synchronicity that-- going in. This is a spaceship, a special slightly larger spaceship than a glider. That will be shot out. So this thing will keep on going forever, just shooting out those spaceships. Now, what kind of things can you build using the Game of Life? And I wanted to show you this pattern. So this is a pattern, 20,000 cells. It's the sieve of Eratosthenes. What it's doing-- it's a gun that's shooting out spaceships only in the prime number positions, OK? This is two, three. Four's not prime so it's not there. That's Five. Six is not there. That's seven. That's nine. Nine's not going to come back because a glider gets shot at it. OK. So what is actually happening here is that, as that goes along there, this thing here shoots a glider every three. In other words, anything that's divisible by three it's going to kill. This shoots something every five, this every seven, this every nine. And if I go to the next, this one here. Hang on. Yeah. So again. So this shooting every three. That's seven. This is nine. Because we know all even numbers aren't prime, we only shoot at the odds. That is nine. That's 11. That's going to get through. So it's the sieve of Eratosthenes turned into this kind of intergalactic shootout between gliders and spaceships. That's 11. The next one is 15 so 15 is divisible by that's the three coming in. That's the five coming in. That's 15. That's going to get shot. So we could let this pattern go forever and it would just shoot out gliders. I've one slide to go left, which is this one here. So what is the extent of the Game of Life? What can the Game of Life do? It can find prime numbers. Actually, speeding up so we have enough time, the Game of Life can do absolutely anything that any computer can do. Right? Because what is a computer at its most basic level? It's a bunch of wires with a pulse going through it. It's binary. You can emulate a wire with a pulse going through it by a stream of gliders like here. So when there's a space it's a zero. When there's a glider, there's a one. What else do you need? You need logic gates. You can make the three logic gates. There are patterns for the three logic gates. And you need wires to be able to cross each other. You can do that just by thinning out the wires so they don't hit. And you need a memory register. You can do that with a block, which is the square. And a block at a certain distance, keeping the memory. So the mathematical term for that would be that Life is universal because it's capable of universal computation. So even though it would be a very inefficient way to do it, you could actually use the Game of Life, which incredibly simple rules can do anything that the most complicated, sophisticated computer could ever do. And I guess on that note, which is kind of quite a wow. We come to the end of the talk. That's the name of the book, which has more information both about emotional responses to numbers, loads of things in between, culminating in cellular automata. And the whole idea about cellular automata is that it's really good for emulating replication. And people have now started to get patterns that do self-replicate in the sort of Von Neumann sense of self-replicating. And mathematicians there's only a matter of time that you will be able to create self-replicating creatures that live on a kind of big infinite grid of the Game of Life. So the kind of game is turning into life. And there are philosophers who say that, actually, the best explanation for the world and the universe is a bit like the Game of Life-- that there's some initial configuration of kind of discrete elements of some way and that we are just squillions of generations beyond that beginning grid. So, again, it's interesting to think philosophically about that. That's the first book, "Numberland." That's the football book that I did. I blog, "Alex's Adventures in Numberland," at the "Guardian" every couple of weeks. And I'm on social media if you want to follow any of the maths popularization work that I do. Thank you very much. Sorry for talking for a bit longer than I probably expected to. Thank you. PETER: Thank you so much. That was crazy stuff. Brilliant. And people have to run, I'm afraid. It's not rudeness. I think people have meeting at 2:00. But I think we've probably run out of time for questions. Except you neglected to say what your favorite number was. ALEX BELLOS: Well, that's the last thing. I don't have a favorite number which is why I got interested in it in the first place. I don't have one. PETER: He'll be around for a few minutes. ALEX BELLOS: I will totally be around. PETER: If you want to ask questions one-to-one, feel free to do so. Thank you very much indeed. [APPLAUSE]