Speaker 2: So I think I was introduced, so I don't, I'm Robert Krulwich from national public radio and from radio lab
Speaker 2: I should warn you in advance that what I know about mathematics higher mathematics is very, very little, which is the excitement to me of trying to do this. This is for me, it would be as trapeze work. So, let me introduce the guests. first known as the math guy on national public radio, he is the author of 30 books and over 80 published research articles, Keith Devlin is a recognized mathematician. He recognized mathematician, which I guess means that he's really a plumber and looks like a mathematician and a researcher at Stanford university. And here is Keith
Speaker 2: And next Jonathan Barwon is a mathematics professor at the university of new castle in Australia. He's the director of the center for computer assisted research mathematics and it's applications. You kind of know when they do that, got to be some acronym it's called karma. So, he's a noted expert on the number PI and a leading thinker in the field of experimental mathematics. So here is Jonathan. next Marcus DeSotoy, a professor of mathematics at the university of Oxford, a mathematician, a researcher, occasionally on Radiolab. He studies prime numbers and symmetry. and finally a science journalist producer with a PhD in particle physics, Simon Singh documentary about Fermat's last theorem was nominated for an Emmy, his publication on the same subject. Fairmont's enigma was the first book about mathematics to become the number one bestseller in the UK. Here he is silent. let's see how to start this. First of all, this is the question posed is what is mathematics there's I just, I don't know way of describing nature, art form language, brains waves, seeing the world exploration, competitive sport God's mind revealed. let's start with a pattern sense. When you were a kid, did you take an unusually long gazes at pineapple surfaces or sunflower
Speaker 3: No, it took me a long while actually to get into math. So I didn't, or math as you call it. yeah, I don't know why it's, I think it's a very plural subjects, but mathematics, maybe mathematics is a better way to, but it was around, you know, cause I think the real distinction here is that, early on your new career, you're doing dealing with numeracy and arithmetic and that's not what mathematics is about. And I think for many people, that's the misconception. They think it's, they say the bad at math, but actually they may be bad at numeracy. Most mathematicians are bad at arithmetic. and actually math is about pattern searching it's and it's what we've been evolutionary program to do. You know, we've survived in this world because we're very good at spotting patterns in the jungle. If you see something with a bit of symmetry, it's likely to be, an animal, which either you could eat or it might eat you. So it's good to recognize symmetry. Yeah.
Speaker 2: So when you were seven, like did you go to the, out into the garden and say- one ant, two ant...
Speaker 3: No, no, no. I was actually much more interested in languages. I wanted to become a spy and to be in the foreign office. and it was only when I started to realize that mathematics is actually actually I hated languages because there are all these irregular verbs and strange spellings. And it was when I suddenly realized mathematics is a fantastic language. It has no exceptions, but lots of surprises, but it's gotten a regular verbs and it's a fantastic language for describing the world around you. And I think it was about 12, 13
Speaker 3: That I started to get a little bit nerdier and start to spot spirals in pineapples. But that's how that happened if I did. And I now can't look at a pineapple without going Oh, eight, 13. And you just, did you happen to be a pattern sensitive Not at all. No.
Speaker 4: The year before I went to high school back in England was the year the wishlist put Sputnik in. I wanted to go into the space race. I mean, it was just space exploration. I wanted to do physics. And so I had started learning mathematics in order to become a good physicist. And I didn't like mathematics until I was 16 and I met calculus. Anyone who's read the blog. I put up on the, on the world science festival blog. I put a blog up about why I became a mathematician, the very moment where all my classmates got turned off, calculus that turned me on. First of all, it was obviously something was wrong and it was obviously important for getting rockets into space and getting them back again. But it was also very powerful and I couldn't understand it. It's the first thing you meet at school that you actually can't understand if you put your mind to it, it baffled me. It was clear that there was a mystery. I mean, they were talking to them
Speaker 4: Was such a mystery that you felt personally, you wanted to solve it.
Speaker 5: Oh, I couldn't stand that. I was the kind of kid that used to take the ultimate a apart. I wanted to see how things worked. And who was this Calculus was invented by a guy who was age 22 because the college was closed. You know, he was going to go to Isaac Newton was going to Cambridge. It was the plague.
Speaker 4: Well, are you a Newton fan or what Okay. Yeah.
Speaker 5: What would you do if the college was closed? You'd probably go to have a good time and go to the beach. Isaac Newton invented calculus. So you've got this 22 year old invents calculus. It's powerful. I could use it. I could do all the problems and get them right. I had no idea why it worked and I wanted to tick it out.
Speaker 4: Oh, that's that's maybe when you played sports, did you think, okay. If I stand here, the arc of the will go to the declining, I think, or were you odd or were you what I would call ordinary
Speaker 6: I was never gonna have a sportsman to do anything except say, I wish it goes somewhere else. Please let somebody else catch it.
Speaker 4: My, my father is quite a distinguished mathematician
Speaker 6: he has been president of Canadian mathematical society. And I think he's now the sixth oldest member of London, mathematical society.
Speaker 4: Well, they keep a list. It's a little bit because it's the mathematics society. Indeed, indeed
Speaker 6: Win a prize. But only once for being the oldest person at an American math society banquet, really, you can't keep doing it for obvious reasons.
Speaker 4: Well, wait, wait, wait, you said you're doing your annual banquets and you're like to get back to your central. Why, if you're 93, can't you make it to your 94th because they want to give somebody else younger a chance. Oh, so you're not invited to the bank. You can come, but you can't win the prize.
Speaker 5: And since the oldest,
Speaker 6: Viennese or, Austrian mathematician some years ago was via tourists who was also the Austrian and died about age 108. You can see that you wouldn't want to give us pies out only once every 20 years. But getting back to your central question, I also discussed
Speaker 2: Such a strange envy. So these people in Britain are going to this dinner. Envying. The Australian is two years older than me. I don't know people do lie about their age in the opposite direction after all right. Yeah.
Speaker 6: They're older. I like, since my dad was a mathematician, he taught me a few pieces of math to prove, to win bets with faculty in St. Andrew's. So he taught me to solve two by two simultaneous equations when I was six.
Speaker 2: Well, when would you like that When you're saying, because he wanted 5,000
Speaker 6: Pounds of cheese and a five pound note, which is quite a lot in the fifties.
Speaker 2: No he did. But if you're, if you're a six year old using whatever you just said, independent situation with six year old, I don't see how that would be an occasion that would ever come up. Well, I suppose I must have liked the pattern because I had no idea what I was doing.
Speaker 6: But after that, I went to university to study history and I go up to this far in the days of punch cards for registration or being about to drop my cards in second year into a history box. And I thought if I do this in 10 years time, I won't be able to integrate anything or work out the equations of motion. But if I go back and find the math cards in five years time, I can still read about the treaty of Vienna I did. And it's turned out to be the truest decision I ever made. So I can tell you the day I became a mathematician, but I came to it a bit, like, it sounds like you out of the arts, not out of the sciences directly.
Speaker 2: Well, so, so then what is it exactly Is it, is it an instinct that humans are born with I mean, there are, there are people in the Amazon who have one and two and three, but they don't have four or five or six or seven. And they do perfectly well. I mean, they seem to know the difference between more and less. I don't know what they do if I had three fish and you had two apples and I wanted to raise the price of fish, I don't know how that conversation would go. But do you think it's an instinct in us I don't know you, I,
Speaker 7: I don't think it's an instinct in, in the way that we think about math too. When we think about numbers and precision, you can do experiments. I think this experiment with the Raven Raven, was, on the top of a tower and a Hunter would approach the tower and the Raven would disappear and the Raven would, would disappear until the Hunter left the tower and gone away again. And then the, the, the, just the war. How clever is this Raven So then two hunters were going, the tower, Raven would disappear. One Hunter would leave, but the Crow, the Raven wouldn't return until the second hundred left. So, so the Raven could somehow count, but if five hunters went in and four came out, then the Raven couldn't differentiate between five and four, it would come back and get shot by the remaining Hunter.
Speaker 7: So, so it's as though it's an approximate. It is a Hunter hunters where like, tablespoons of sugar. And I can differentiate between two spoonful's of sugar and one spoonful of sugar in a pile. But I can't really differentiate between five and four. So unless we can begin to put symbols or words to numbers, that it's only, then I think that we can begin to really manipulate them. I should just say, I have the people here on the panel. I like everybody here. I didn't really like Matt as a child. And I didn't realize it as an adult either. Really No. I mean, I became like light like Keith. I was interested in science and he's a very obvious desire to want and understand the universe, transcend where the universe came from. What stuff's made of where life came from. This is really obvious stuff to be interesting. I think nearly all children are interested in that. And then gradually we maybe lose that curiosity, but to be interested in math is a bit odd. And I think that's why people are sometimes scared of math or don't understand why math is a pattern. I only became interested in math because then as a scientist, I became a science journalist as a science journalist. I ended up writing about math and I'd always seen math as something you have in order to do science,
Speaker 3: But I think that's how it, yeah, that's how it started. Yeah, it was, you know, in order to navigate the world, to measure land areas, to tax them, to build new buildings, you find all to answer these big questions of science. You find that math is the best language, and then it starts to take on a life of its own. And you start to get interested it for in its fruit, for its own sake and the properties of numbers and the patterns there. So the measuring thing,
Speaker 6: If, if, if a girls were going to jump rope, so it just basically won't, you know, and they go and they had those patterns. I didn't put duper, duper, duper, duper another girl. So I don't know, but, I don't know why I say things like
Speaker 7: That, but
Speaker 6: It isn't, isn't singing, dancing and playing games sort of playing with segments of time a lot. And so isn't that math sort of,
Speaker 7: So mathematics really changed the, the, the, the elementary parts of mathematics, the stuff that everyone would have had in school that does come out of the world. It's sort of formalizing ways of thinking that we use to survive and have done for the history of humankind. But then the moment when I got into it at calculus, it sort of flips. It becomes larger something we invent and create ourselves becomes very abstract. It's still actually useful, but it doesn't. I mean, in calculus, you're talking about infinite decimals. There are the infinite festivals in the world. We invent them in part to understand the world. But those of us that have been mathematics play with them because they're fun to play with. And they really trust me.
Speaker 6: Well, the audience probably knows it. Certainly what the encyclopedia it's thought that a while French was a perfectly good language for science only German was really set for emotion and love. These are things that people think about other languages, but the reason I'm bringing it up is that what has kept me attracted to mathematics It's two things. It's all of what Keith said, this wonderful set of tools, unreasonably efficacious tools, for making more progress often than seems reasonable. But at the same time, it's a language in which you can express things. You didn't know, you wanted to express until you could speak some of the language. So the language is called numbers. I guess the language is called mathematics numbers are some of the words I do. I remember reading somewhere that a baby born on the first day, if you stood in front of them on the first day of its life, and you went peep, peep, peep, peep, peep, peep, peep, peep for a very long time.
Speaker 6: And then you changed, Oh, it's only up to three. Yeah. And if you change the number, the BB startles or looks more closely. So they're just some sense of, of pattern anyway, but now the N the number, that was invented or where that comes from, how old it is. Yes. It's interesting. You can even do some fairly road bust kind of cladistics. And you'll discover that the number two is older than the number one in terms of where it's shared and the tree of language. Really. Yeah. So, you know, it's all conjectural, that's the lovely thing about archeology and paleo, whatever. You're never going to be proved wrong, but it seems pretty plausible that we, well, that's a good thing, isn't it
Speaker 3: And he's very ancient bones. They like the Shango boat, which is tens of thousands of years old with notches, you know, people keeping track of things and wanting to know how many, I mean, it's not quite clear what it's keeping track of, whether it's, perhaps a calendar or something, but, you know, I, I think, you know, navigate it, seeing patterns in the stars is probably where, you know, you start to skip the mathematical mind working the fact that things repeat themselves. And if they repeat themselves, then you can make predictions about the future. And that this language gives you incredible power. And, and, you know, you've seen that
Speaker 6: Predict a flood or eclipse. Yeah.
Speaker 3: The, the flooding of the denial, they spotted patterns to this. And they're that, that, that matters.
Speaker 7: Well, once you've had the flooding of the Nile, not only, you want a bit, not only do you want to be able to predict the flooding of the Nile, you then want to be able to rebuild your fields afterwards, you need to have measurement and geometry be able to do so. It's a purely functional,
Speaker 6: Useful tool
Speaker 7: That, that society is. If we're going to trade, if we're going to send ships, we're going to plot them. If we're going to be interested in astrology, we need mathematics. But the really interesting transition, and I think what was interesting about the little video clip we had at the beginning was a lot of that was applied mathematics and it's verging on the science. And we kind of all understand why science is interesting, useful, and important, but the really weird transition. I'm not sure when this happens. This is maybe going back two and a half thousand years is people who begin to study numbers purely for the hell of it, for the fun of it, for the surprisingness. The fact that there's this thing called a perfect number six, a one plus two plus three is six and one and two and three are the only numbers that go into six and 28 is one, four, seven, and 14. Am I missing one And to add up to 28 and divided the 20th, but then I don't know where the next one is up in the hundreds. Isn't it think that.
Speaker 2: Do you think these people sort of, did you have to get, cause I'm actually wondering whether you get notches first or maybe circles, you think circles or lines proceed, numbers or circles lines or trying not to miss them.
Speaker 5: Truck numbers are incredibly recent as about eight, 10,000 years ago most. And it was essentially money in Sumeria. Sumerian society reached a surge of complexity where the trading was obviously much better if it was mediated by something that basically, if you want to know where numbers do, it's like following political careers, find where the money comes from, that's where the numbers came from, but nothing new in the world, especially in Ninja
Speaker 3: Elementary as well. And I mean, you see, in the Ryan papayas, you see a formula for the volume of a pyramid and beautifully proved as well, using an early form of the calculus. But of course that was a practical need to know how many stones are we going to have in this pyramid. But, but then you see a sort of abstract love of the mathematics and the game starting to come in and it takes on a sort of power of its own. And I think that's right. It's perfect. Numbers are perfectly,
Speaker 5: It's perfectly pointless. Perfect. Useful. Yeah. But you kind of think, well, that's kind of cool. These are,
Speaker 2: Are you imagining that this is cause you're sorta like th like, is this like a, a prayer that you're telling us here, or is this history that you could, I mean, we're not going to meet
Speaker 6: This is history. And I think it's interesting to ask how much of it is, contingent on how much it was necessary. So for example, it, usual accounting would be that the Greeks, the class, the classic Greeks brought us the sense of trying to build a deductive system, being interested in the properties of numbers for their own right or mixed with ideas of harmony and religion. But imagine, and I've just been doing some work with my colleague, David Bailey at Lawrence Berkeley labs, and flabbergasted to describe a 13 decimal accuracy, square roots in very old Indian documents, pre Christian. so sophisticated that you wonder what would have happened if those techniques had made it to the West through Spain in 719 1,408. And for some reason, all of the manuscripts that we learned about Aristotle or Pythagoras had actually got burned. And so what we had inherited was a very much more functional set of how to do things. Is it necessary that we would have later said, we need theory. I'm not so sure
Speaker 2: Keith is positing that, that the first guy was just a businessman on the fast track. And he just got himself at first detach a case or something. Right. The other way to think of it would be that the first mathematician was sort of an artist like you would think maybe like he just wanted to play,
Speaker 5: Right Match goes back so far in time that we had to feed ourselves. We had to keep warm. We had to fight off bears and wolves and Matt evolve in that environment where you desperately needed to survive. And it's the same with science.
Speaker 2: Oh, you were the artist guy. I thought you were trying to imagine this a fellow who was playing,
Speaker 5: But I think math starts off
Speaker 7: As being functional. Science starts off science. Doesn't start off, you have technology. And then people start realizing if they want to build technology. It's actually quite useful to understand some of the basic science. So, so the pure side emerges only much later.
Speaker 6: The great Canadian geometer center was a friend of Asher's, the Dutch artists who most people come to these sort of things I think probably likes, and late in his life cook, cooks or lived a very long life live to age 97 and gave lectures to about age 96. good lectures. He, wrote a paper, maybe one of his very last papers and he's explaining one of Essure's, continuing joint constructions. And he said, Asher did this as art I had to use trigonometry. Huh So I, I,
Speaker 2: You associate, there are things like elegant proofs and ugly proofs where it's like that tweet. If it feels to me a little bit, is that, is that art is just freer. It, it th the real art means if you paint a painting, you're not constrained.
Speaker 3: No I'm going to in most artists will disagree with you and say that their best creative art art is done under huge constraints. I mean, Stravinsky used to say that he was only creative when he put constraints on himself. So I think if you have too much freedom, very often, you become uncreative. And in mathematics is an incredibly creative subject. We're making a lot of choices in the sort of things we want to celebrate as theorems. I'll talk about in a seminar right up in a, an a paper. I mean, I studied the world of symmetry. I can get a computer just to generate arbitrary new symmetrical objects. What's interesting is to choose the ones which have something special about them. And it's just the same. You can get a computer to churn out music, but you make a choice as a composer. And I think there's an incredibly creative science to, well, why is this exciting There's an exciting connection with another bit of mathematics, which is totally unexpected. And this is why I want to tell you this story. So I think there is a lot of choice in mathematics.
Speaker 6: Nice description of, of the aesthetic response is the perception of order in what seems first to be chaotic. And again, that calls on whether it's a few or a structured piece of poetry to have that underlying structure and that to some degree, the aesthetic response, the aha equivalent, or an aha mathematically, as we say, Oh, okay. It does make sense. It's not just Jangles.
Speaker 2: Here's the difference. If, if we all agree that this will be the number three, yes, we all agree. And we all agree that this symbol will mean we're going to add something once we've agreed on those terms, three means a triplet triplet of some sort. And plus me, then if you, then, if I say three plus three equals it. At that point, it has to be six. That's not the case for Shakespeare. Hamlet could have killed his father any time, bill Shakespeare,
Speaker 3: But you're, you're comparing two things which are not alike. It's like talking about English grammar. This, this word is spelled like this and the humans only, but mathematics is not about spelling and grammar. It's about the big stories, the stories, these are numbers of cemetery. These are the things that we stories we choose
Speaker 6: To celebrate. And there, they have a truth about them. You can't change them there. So it feels like there's more constraints on mathematics, but still the stories we choose to tell are as exciting and have the drama. And we choose, we choose to tell those stories because they have the same effect on us as a piece of Shakespeare. You're just comparing to wrong things. Yeah, exactly. Because,
Speaker 7: I feel
Speaker 2: I have like the Chinese menu of all time. I know that as soon as I got three plus three, I don't, I, I'm not allowed
Speaker 7: To turn it into Cinnabon.
Speaker 6: Exactly. But if you've got hamlets, you will say, well, I'm going to spell it. H a M L E T. And then if you have it later on in the play and you'd misspell it, it's sort of, that's uninteresting.
Speaker 7: It's not what she thinks mathematics. Isn't about. We're all gonna be very, very aggressive. with very elementary math, it's dry, it's ugly. It's tedious. Just like very ugly prosaic English. The, the jump you have with the spoken word is from the spoken functional word to poetry. And sudden you say, wow, that's, that's beautiful. That that's something I want, I want to embrace it. Something I want to play with. I want to create a prizes. I want to make people laugh. And, and with Matt, again, you have these very mundane prosaic numbers, and you, aren't very basic question like multiplication square roots. And, and you say, well, what happens if I take away three from six Well, that's three. That's pretty obvious. What about if I take away six from three, Hey, that's mind blowing the first person to ask that question suddenly has to invent the idea of negative numbers and this, this I can't hold negative three pebbles. This is that's mindless. And then you start saying, well, okay, square root of nine is three, but what about the squirted Negative three. What does that mean And that, because curious and odd, and then you say, well, okay, let's allow that to be a number. And we'll call that route three R. Is that right Good. Thank you.
Speaker 2: I'm just thinking one thing is that I don't want to push this too hard. I think you guys are of the, you'll say one, an odd number, plus one, another odd number makes an even number. And that like shocks you. And it's like storytelling in a way.
Speaker 7: It doesn't, it doesn't
Speaker 6: Come in. And you're at the Harbor. You find it beautiful. The harness, something really complicated. But I could say all of computer science really originated when George Gould said no one plus one doesn't equal two. It equals zero.
Speaker 2: Who said that
Speaker 6: We're back to the basis of the idea that you might be adding in. What's called clock notation or binary. And the effect of two ones is a zero. And that gives you the whole idea as a modern, modular arithmetic article, finite fields and on. And so where the creativity in math comes is to say, not that it was wrong to say one plus one equals two, but there may be interesting ways of abstracting this, where it's better to think of it as being something else. And so, again, we don't try and see if you've got a dictionary you can Shakespeare,
Speaker 2: Right We do try to say, if you've learned the language, you probably appreciate some of the Shakespeare. Well, let me switch the question a little bit. Let me, why are some people better at this than others Is this, is this a talent Is it genetic Is it just a language with different degrees of difficulty
Speaker 5: Or interest It's just interest just in few interested, you become good at it and you do it. I mean, the two of these guys didn't even begin in mathematics. I didn't begin. I began in physics and not mathematics and one isn't but we all enjoy it. We appreciate it. And it's just a matter of interest.
Speaker 3: I think there is a, there's a, there is a reason why mathematics people do seem to be sometimes bad at it, which is mathematics is a bit like building a pyramid. And each year you add more of the permit. If you have one bad year in your mathematical education, it's impossible to build anything on it. You're lost from there onwards. And that doesn't happen so much in other subjects. You know, if you, if you've got a bad history teacher, teaching the Victorians, it doesn't kind of mess you up for the history of the 20th century. You can get back into it a little bit, like you said, with your experience of when I can reread the history later on, but with a massive I'll miss out. So I think that what happens is that you need to have a very consistent, education because one bad layer and it really messes you for the rest of the day.
Speaker 2: Well, it's the key thing. When you say it's interest, is it, is it that you're like the problem with math as opposed to some other subjects, is that there is a right answer. So you, you, if you're learning, you know, it's hard to fit often. Yeah. So, so it's hard to you just going to fail or succeed. It's sort of binary. And that may be, I mean, that's one of its chunks, I think. Okay.
Speaker 5: Yeah, there are sort of right answers, but that's what mathematics is really about. What's right. It's why is something right or wrong It's the why it's of interest. Not whether it's right or wrong. That's a fallout from the way we often teach it that it's all about right and wrong. There's a right and wrong list. There just like, there's a right and wrong way to spell things. But Shakespeare, isn't sort of to go back to the example, isn't about how you spell the words. Is it a good story And why do people behave the way they do in mathematics It's not what's right. And what's wrong, but why is this right in this circumstance
Speaker 2: Well, that may be there. The difference is some people make this personal. They say, this is my question. I want to know the answer to put it as you did. I'm interested. Oh yeah. Others of us think I'll do what the teacher says and you never somehow get across. It never becomes useful.
Speaker 5: You know, that's how I passed English literature at school. I've got these cliff notes and I'd read them. And I answered the exams and it took me 10 years to like Shakespeare. After I graduated from high school, Pete people don't have to like math people, people have perfect right to hate math. And I have many, I hate languages. I'm terrible at languages, but there is a world out there that people may be unaware of. And just came back to the point. Keith made, there's a lovely problem called the four color map problems. If you've got a map and people may remember this from school, how many colors do you need to color a map Any, any conceivable
Speaker 7: Map so that no two regions have the same color. And for a hundred years, people knew that you didn't need more than five, but they weren't sure whether you could get away with four. And that's kind of a weird, interesting, odd question. And it's just something that gets you think, what do you need Four, do you need five Why is it four Why is it five How, how would I go about trying to prove it So it's an odd, quirky, interesting bit of playful maths. And it turned out, I think it was about in the seventies, a couple of American guys, Apple and hearken proved it. And they proved it by saying that every map can basically be built from several thousand other constituent maps. And if you can prove that the other thousand or so constituent maps only need four colors, then all maps only need four colors. Now we now know you only need four colors, but it's not a very satisfying proof. It doesn't really give you any insight. It doesn't make you smile. It doesn't, we're not that surprised because we already kind of knew it was four. So that's an area of math, which is, has been curious or in quirky. But w I think mathematicians are still waiting for that. Aha moment is it doesn't have that beauty and elegance that something like Euclid's, pro-nuclear had a proof about the number of privates
Speaker 6: That comes to my present passion, which is I I'm much influenced by people like Greg chase in the complexity theorist. I don't believe that just because something has let's use the word. Beautiful fact. We don't know if the infinitely many perfect numbers, those numbers like one and six and 28. We do know it relates to how many primes there are of the form, two to the N minus one, but, and we don't know any odd, perfect numbers. And we have known that for 2,500 years, that's a problem that classic Greek sausage, is there a number like six, which is the sum of its proper divisors and is odd. Nobody knows. And nobody has the slightest idea how to prove it. Well, we grew up in a 2000 years of post Greek philosophizing about mathematics, which essentially identified truth and provability. And we've known for a hundred years. That's not true. And it's perfectly reasonable in my now, the way I organize the world to think that might be true because it's not false. It wouldn't make it unbeautiful, but there's a lot of stuff out there that we should try and work out what we think about it and not everything that beautiful fact we'll have a beautiful proof, not everything that's provable will have a nice fruit. And that changes your view of Matthew,
Speaker 3: Because do you hear what you're saying The way you're already in love with the question What happens in a lot of schools are particularly in junior high schools and high schools in America is that people want the answers. The teachers want the answers, but they never asked you to love the question, right Well, I think it's so comfortable because this idea, actually, that very often, when we're teaching math, we just teach the grammar and the vocabulary. Although it's like learning a musical instrument where you're only allowed to play scales and arpeggios, and nobody ever plays your real piece of music. The reason why I fell in love with mathematics is because my math teacher at my school, when I was 13 and pulled me out of the masks gloves at the end of the lesson, I thought it was in trouble. It took me around the back of the mask book. I
Speaker 6: Thought it was really in trouble now. and
Speaker 3: Then he said, you know, I think you should find out what maths is really about. And he started telling me about the Fibonacci numbers and about primes and recommended a few books to me suddenly I saw what this subject was really about. And I think we, we cheat our kids by, by just saying, Oh, it's about scales and arpeggios, but you get to listen to a piece of bark or some, some blues. Then you say, that's what I want to do. I don't know how to play that yet. I don't know how to compose it, but that's where I'm heading. And that's what I was
Speaker 4: Teachers are there like that. Not enough when I decided, when I loved calculus and what to do, that's exactly what my teachers did. They took me out to the class and said, here's my college textbooks, teach yourself. I'll try and remember what it was like to do those and help you out. And it was this hundred, this incredible world opened up a real mathematics that I didn't know existed. And Simon and I were looking at, I mean, Marcus and I were lucky Simon wasn't
Speaker 7: Yeah, I was in a class of two people got taken out. I was wondering, well, I'm cleaning up are things called non-trans Tifft item. Non-traffic yeah, I've got a dice and it doesn't have one to six on it. I've got some old numbers on it and I give you a dice and I give Keith a dice. And typically, if you and I roll this, these are our own two dices. You typically beat my dice. Whatever these numbers are, your dice generally are going to show a higher number than my dice. And when you play the dice game with Keith Keith's dice typically beats your dice. And that's the way it goes. Your dice is better than my dice. Keith dice is better than yours, but when I played dice against Keith, my diet beat Keith, that's a really weird thing. It's like saying that one is less than two. Two is less than three, but three is less than well, it's a mind blowing concept. And then the really weird thing is if we double our dice, the order reverses. So if you show that to a 13, 14 year old kid, who's, who's played with normal arithmetic and multiplication square roots, that's again, that's a mind blowing world.
Speaker 4: The other thing that means that you want to make it, you want to attach it to the world. You want to say, if you know this, then you can do that. But that might be
Speaker 6: You like playing Mario brothers. It doesn't have to some, a B like working out the velocity of a bicycle. It could just be letting you do something you want you to do.
Speaker 7: Then you need to be a scientist. I mean, there's, there's something like
Speaker 4: Math classes. I don't know. There's probably a good answer to this, but it seems to me that there's the two problems are that the teacher doesn't sort of make it, let you make it personal. Doesn't let you fail without punishing you for it. And third doesn't seem to make it of the world where you live.
Speaker 6: And that's something because the skills needed to do that. When it gets cold in educational language, unpacking concepts are a great deal, more sophisticated than the skills that are needed to do fractions. If you've just learned enough math and we don't pay, and we don't have enough teachers ready to come, the trenches who are secure about their mathematical knowledge enough that they've actually studied why, and might have a good idea about me saying it wrong because they just try to stay one step ahead. And this is massive difference between being able to do the arithmetic or the manipulations and being able to step back and say, Oh, I see how that fits in. Here's the contrast. When we, in most parts of the world, we put teachers in the classroom to teach math who have very little math in their training.
Speaker 2: Well, let me ask another question different. It does. Does math describe reality or is math itself out there and, and real, the answer is yes. It just felt well,
Speaker 7: It does. It's a genome. So it looks in both directions on it, legitimate. It looks at both directions and that's some of it.
Speaker 2: Thanks. What does that mean That, that math does tell you about the world, but any more perfect way and that we live in some sort of a shadow of perfection,
Speaker 7: And I'm not even sure that physics tells us about the world. Physics tells us about how the human mind conceives of the world, the mathematics is part of that. It tells us as much about the human mind as it does about the environment, about if you want to regard those two things as separate entities, it's a, it's like a mirror on the mind. It, what it really does is tell you how the human mind perceives the environment that it's in. So is it about the world that we're in or is it about the mind The answer is it's both, but it's even better when it's turned in on itself. So you have this game, you play a game of chess. We can have some rules and the rules of chess allow you to have a very beautiful, intricate game, which for a thousand years still has, has entertained and effectuated achievements. The game of maths, which has these basic rules, like one plus one is two. And so again, it's an infinitely rich game. And if you just play it around the world of maths, you end up discovering truths that are true forever because they're logically consistent. And it's, it's a, it's a perfect game. What's the greatest, let, let me just, so the key thing here is that when you do something in maths and it's purely done in a mathematical realm, if you prove that it's true, it's true for
Speaker 6: My brother who was also a mathematician. So I think some of it's got to do with innate abilities, not to training because at a very different things, the three of us have all ended up as mathematicians. He said, you know, we may do it better than Newton, but we don't do it righter. Yeah.
Speaker 2: When you say always, right, can you, is there some, some thing that a mathematician discovered that is just so beautiful and so gigantic that it's like the super-duper best story ever.
Speaker 3: I think one of the stories that I was told when I was a kid, which is the proof that there are infinitely many primes, I mean, primes, they're the atoms of mathematics. They, what we build all numbers from the indivisible numbers, like five, seven, 17. So all numbers 105 is built by multiplying three times, five times. So they're the most basic things. So, but how, how many are there all these numbers we've got two, three, five, seven, 11, but maybe they're just a finite number. Like the periodic table has, I don't know what it is now, 118 atoms. Maybe there are fine nightly, many primes, which can build all numbers from well, Euclid proved with just a beautiful little arguments 2000 years ago, that there'll always be more and more Pines. However far you go up to the universe of numbers and he did it like this.
Speaker 3: Suppose you have a finite number of primes. Suppose there are a final number of primes. He said, okay, well take, this is probably one of you says, Oh, I don't believe you. I don't think so. Here's a list of all the primes. So usually it says, multiply all those primes together. And then here was his act of genius. He added one to that number. And then he says, okay, so this new number that I've just built you, what's that built out of it must be built out of some of your primes. You say you've got all the primes. Well, if you're trying to divide by any of the primes on your list, you get remainder one because of the way this number was built. So there must be some primes missing from your list. You haven't got them all. So you say, okay, well, I'll just add those. Then nuclear says, aren't play the same trick. I multiply all those together. Add one, if you still miss them. And here's the beauty, you know, infinity does it exist in the physical world I don't know. Probably not, but in the mathematical world, but this Sinai little argument that Euclid made he's proved that you'll never ever run out of these prime numbers. For me, that was a magical moment. And it's a proof which is lost forever. It's it's, it's terminal,
Speaker 7: I'm at physics, but what's it what's its essence. have you heard that proof before Yes. Okay. And when you heard it for the first time, did that not didn't they not change Yes. It, it didn't rock your world. Well, my rock is a smaller rock than you're coming back to the idea that that GH Hardy said that, you know what the Greek said about medicine. We laugh at what the Greek said about astronomy. We think it's a joke, but what the Greek said about math, we still teach it today because that mathematical truth remains forever. And he may have a, because I asked you this way, I guess you're saying who rave for the mind of a Greek that could think that up aren't we clever, but the no, no, no, no. I think we tapped into a unit. My question was, did the numbers care
Speaker 7: If, if you got number 3 trillion, 6,950, the, the, the, the some prime way out there, that's what I'm saying. I knew that it's not, it's not about how clever these people are. It's not about how clever Euclid was. It's about high, beautiful in four lines of argument, you can grip grapple with infinity for it. It wouldn't matter whether it was Euclid wouldn't matter where was any, but it wouldn't matter whether it was a pet cat, whoever did it. It's the idea that that's so precious might be called Garfield serum. Yeah. But nothing it's meant to
Speaker 3: From the sciences. I mean, we're at a science festival and I think it's important to distinguish that mathematics builds on the shoulders of giants. I mean, w what that proof is true, and it'll be true forever. And in science, there's a much more evolutionary process. You propose a model of the world and you find it doesn't quite fit. And a new one comes over and knocks that one out that doesn't happen in mathematics. Proofs gives us this certainty. And we, it allows us to really stand on the shoulders of giants and build something new. So that's the charm of it. I know that the mathematical theorems that I proved will last forever. It's a bit of immortality, and that's why I love this subject.
Speaker 2: But, but if you were to, if you were to, I don't know how I asked this. If you were to become a number, are you there Whether or not we're here I mean, are you just there
Speaker 3: is about to all, we play tennis at heart, and I think most mathematicians, I mean, I'm interested to see, but I think most mathematicians feel, you know, a 317 is a prime number. And not because the mind thinks it is, it's just a property of that number. And I certainly feel there's a platonic reality out there called mathematics, which is the, where I spend my time exploring this world. and sometimes it says things about the real world that I live in, but frankly, I'm not interested in that. I'm more interested in just exploring this extraordinary world, which I think has a reality of its own. And when you,
Speaker 2: When you say that, is there, is there, is there a castle Is there like, is this, is this a metaphysic which got like legs I mean, is there like when you, like,
Speaker 3: It has structure and pattern and, and it's, it's a bit like more like, I think, I mean, more like the musical world that it's got the same set of texts,
Speaker 2: Like maybe you are actually more religious than you, maybe you would like to admit, maybe you think that they succeeded
Speaker 3: Richard Dawkins is in Oxford. So
Speaker 7: Hopefully you'll get, say, Oh God, no, but why not
Speaker 2: Well, if you believe that numbers are out there, whether or not we're here, if you believe there is that logic that exists independent of all of us and all of us that ever will be,
Speaker 6: I believe numbers are out there, but I don't believe that's true. All mathematics. I feel I'm inventing most of the time.
Speaker 2: I believe the vending or discovery, I believe
Speaker 6: Any choice about discovering the numbers. And they're modern cognitive scientists making some neurological arguments about that, but that's above my pay scale, but I don't think that to all mathematics, it gets proved, had to be invented. It could just have never been.
Speaker 7: So there's a link there with religion in terms of invention of ideas. And so off I go that, I mean, it's, it's it, you know, these, these are concrete hyper. I'm not the mathematician. So I, when I look at math, I look at with a purpose, when you're looking at maths, you're looking at math because if it's this game that you'll play, it it's, I think it's more than a game.
Speaker 3: I think that know, I think that always, I always resent it being compared to chess because it sort of makes it, you know, I think it's more important than that.
Speaker 2: Well, can I, let me ask different, let me ask it a different
Speaker 4: Way. it, it, it doesn't matter if we know it's true or not, like, is there some math that's just simply undo. Is there a proposition you can think of that is permanently and totally undecidable and therefore, whether we may not ever know it, but nevertheless, it may have its existence independent, as soon as you formalize what you mean by decide to build the answer is yes, there were undefinable things
Speaker 6: And arguably true, and that cannot be proved really. And there are things where this is a safe
Speaker 3: Discovery made in the 1930s by Kurt. Good.
Speaker 4: Oh, when I've heard of this devastating food, a lot of people, very nervous, frustrating for philosophers, but
Speaker 6: Largely ignored for 52
Speaker 4: Ways. You can get around it in different ways. Mean every math problem does not have an answer.
Speaker 6: No, that's why I mentioned the idea of odd, perfect numbers. We've grown up in a culture, which says, if you can ask the question and it's a well formed question, we kind of expect you to have an answer and probably a well-formed answer. And I believe there's no reason why I'll tell you. There are no odd, perfect numbers, and nobody will ever know why. Say that again. There are no odd, perfect numbers. There's no odd number, which is the sum of its proper divisors and nobody will ever know why. So then how do you know it's true. I, I have an article I'm telling you, it's true.
Speaker 4: He's the expert. He's the world expert. So that's religious, isn't it I mean, no, I think, you know,
Speaker 3: Good. We'll prove that within first order logic, which is the logic that we use for doing things like number theory, there'll be statements which are true, which will not be able to be proved within that system now. But actually we don't use first order logic. Mostly when we do mathematics, we use second order logic, third orders.
Speaker 4: But I think it's oversold this disorder. You'll say that's an undecidable conjecture, I'm conjecturing. But certainly what we've seen in the
Speaker 6: Last few years are more and more natural language sounding versions of what are called girdle statements. So the original girdle statements are kind of snakes, swelling, their own tails. imagine, imagine a theorem that says, if you have a proof of this theorem, you have a shorter proof of its negation. Yes. Well, you start working through it. Could I have a proof of it I don't know.
Speaker 4: Well, if you did, you'd have a shorter version
Speaker 6: Investigation and at least according to standard Aristotelian logic, you would have proved something and not something, and all rules would be gone. You could prove anything. So if mathematics is consistent, which we tend to hope it is there. If I could show you that, that statement, if I could prove that statement and what I can't prove that statement, but I could show it's true. This is what the Goodwill and people off trim did. But now we've even got so-called Goodwill statements, which seem to be saying fairly natural things about measureables of physical systems are unprovable are undecidable undecided. And we know one at a very famous things that Paul Kuhn, one, a fields medal for many, many years, 50 years ago, it was a what's called the continuum hypothesis. And it doesn't matter precisely what it is, but it asks whether there is a set essentially, is there a set bigger than the natural numbers and smaller than the real numbers And it's now been shown to be independent of the normal rules that mathematicians are willing to accept is God three in one or just one, but the different, well, that doesn't bother me. It bothered Augustine. That's the difference. And the difference
Speaker 3: You want the differences. And this is really important is you haven't defined your terms. And in mathematics, we're very good at defining our terms, our axioms, and w how are you going to use one thing from another, but you've just introduced something gold, which, you know, you haven't defined, so we can't stop
Speaker 4: Ruben. I can not pull it. Isn't that kind of a gurgle light kind of went through his death
Speaker 6: Among other things in his notes is an attempt to logically prove the existence of God. Goodwill didn't really believe what he'd done. He felt that meant mathematics wasn't properly fitted together.
Speaker 4: Well, let's talk about you guys. What do you do when you go to work Do you like sit in the chair or like, you're not like other dads, you don't go to the office. I wouldn't think you do, but maybe you don't like, do work like behaviors. So let me ask it a different way. Do you do this alone, your work, or we've been recorded I'm not, I'm putting into the end of the hall. Now. It is at work is my new subject. How do you do it A very famous mathematician once actually characterize mathematics as something that you lay down, close your eyes and work like hell. And if you could say that and mean it, then you're a mathematician and this of course means you're doing it by yourself. Actually more fun. A lot of mathematics is so historically it was done by people on their own. And very often it is, but these days it's increasingly becoming a team.
Speaker 7: Okay. There's a lovely website. Just started in the last couple of years called polymath where the field's met, Tim, Tim gal is in Cambridge. So he throws out this problem on a blog. And he just says to everybody in the math community or any other community, if you can contribute to trying to solve this problem, add a comment and they've solved at least one quite significant problem by just people throwing in ideas and bouncing off each other. And that's almost the opposite extreme of this idea of the lone genius.
Speaker 3: You don't think, you know, your, your book about Andrew Weil's proving Fermat's last theorem. That was such a prize that he, he didn't talk to anybody else because he wanted to be one of the ones to prove it. And yeah,
Speaker 7: Andrew was is, is an extreme case of somebody who who's a problem, 350 years old and as well, much less theory. That's right. So he 350 or so, he said, the more that people fail to prove it, the more it became desirable. And when Andrew Weil's were based in Princeton, by that time, once he realized the technique that might get him there, he close shop. He largely worked at home. Didn't go to the department very much.
Speaker 2: He was scared that if he whispered his process to someone else, they'd say,
Speaker 7: I think, I think it was partly, it was fear of, of being beaten to the prize, right To the fear of being thought to be crazy, because if no one else has done it for 350 years, why does he think he is so smart And also just the intense focus that's required to take on such a major problem. And, and for seven years he did nothing but work on this problem. But on the other, roughly the same time you have somebody like Paul air dish, who is an extraordinary as a wonderful book called the man who loved only numbers very, very much.
Speaker 2: Was this the fellow who had ring your doorbell and saying, I'm open your mind. I'm here for Monday.
Speaker 7: I collaborated with 500 mathematicians during his career published over 1500 papers. And
Speaker 2: That was the Willy Loman of math that he would just show up as the salesman go, ding dong. And he then spent two weeks with you and you have to give him, yeah,
Speaker 7: He was here once with Ron Graham on the East coast and they'd got up early and he only had two suitcases that was the entire world possessions. And he was with Ron Graham and he said, Oh, we're stuck. You know, we, we don't know what to do. Ron Graham said, it's okay. I know a guy on the West coast who can solve this problem and just said, well, let's bring him there. And Ron Graham said, look, it's 8:00 AM in the morning in California, it'll be 5:00 AM. And then I just said, well, that's great. That means he'll be in it for him. It was purely mathematics. That's where the whole world revolved.
Speaker 2: And, and, and which is the more common way to do it by yourself or, on the phone or online, or, ringing the doorbell or I think more and more, it's uncommon to be entirely alone. Yeah. Uncommon, uncommon.
Speaker 3: It's a combination. I mean, it's, I find what I do best is to work on my own, then go and share ideas, bounce, and you get much further and you come back and you need to spend, and I think there's some other conferences, right We go to conferences, we go and visit each other. We, I find maths quite difficult to do online because, by email, because actually, I think there's a lot of unspoken, sort of, trying to verbalize an idea in your head. You haven't quite got the language. There's a lot of with my collaborators and I can't do that by email and somehow my collaborative. Yeah. I think I see what you're seeing and then we can start to formulate,
Speaker 2: But I, I would, we do quite a lot of that with Skype or, or Skype plus. Absolutely. And it's really a matter of how good the broadband is. But if you have a problem, if you have a problem, that's, you know, 13 pages in the figuring and you have this fragile sort of thought in your head and it's got all these attenuating numbers. So you go, Oh my God, man, you're sitting at a bar with someone and he's got his problem, which is 30 pages long. I don't quite understand how that conversation because
Speaker 4: I there's a tension here. I don't want to hear about your confusions. I've got my own.
Speaker 6: I have a good friend category theorist, which is not my brand of mathematics. And in fact, there's a Columbia is going to make his mathematics in many ways is a single subject. But in other ways, I often have much more in common with certain parts of physics or engineering than I have with other parts of mathematics because of the way I approach my subject. And if I don't happen to speak to given the sub subject, just haven't been taught enough. It's just as alien as if somebody's speaking Russian and I'm trying to speak Italian. But I had a friend who was a category theorist and his first wife left after 10 years of hearing all the category theory, anyone could hear in 10 years, it didn't matter that she didn't have any, he still wanted the audience. And a lot of mathematicians have that experience that in the, in the need to articulate it, it really doesn't matter often if you're getting answers, because you've had to say
Speaker 4: So much more clearly, do you guys have pencils or talk about, maybe we don't have, just give me the pencils and shop you work with a pen. You write something down.
Speaker 3: Yeah, yeah, yeah. You're trying out sort of, do you all have computers No. Computers are not something that I was like unkosher no, no, no, no. It depends on your sort of mathematics
Speaker 6: That might be at the other end. I'm a real passion for using the computer as a, as a colleague. So I'd be firing thousands of questions. And I, and the next thing I'm going to is a conference on symbolic numeric computing, mainly to try and make the point that I think we've largely restricted what we use the computer for because of the way we were trained and the way computers used to be. But any piece of math that I'm showing, I can usually make more progress at a computer without it. And part of it's because I can Google because
Speaker 4: I can Google. I mean, there's an enormous amount. Google what Well Googled the math literature, pull out a paper from 1940. If I know how to do it.
Speaker 3: No, but I think there's this kind of misconception that the computer has kind of put the mathematician out of business because of course it can do arithmetic much faster than I can. But, but actually if you're trying to explore, you know, these infinitely many prime numbers, well, a computer isn't much help because there's a finite beast and it can't have the same sort of, thought processes yet as I do. So I don't find it a very, you know, it can find another big prime number, but that's not very helpful in trying to understand the pattern that underlies all of these prime numbers,
Speaker 7: All your questions about do people work alone with these computers. It, mathematicians are no different than anybody else at this world science festival, except in the group we have here in their extreme purity, in the absolute abstractness and pointlessness of what is done and yet, and yet every not just every so often relentlessly something pops up that has a major application back in the real world. Whether you know, encryption is an example where prime numbers are fundamental. And yet you pry, you find that you study prime numbers for the, of it,
Speaker 5: Fair miles, little theorem, again, applicable to, to, to cryptography, but invented by a pure mathematician 350 years ago, the tools, the community, everything is the same, except the utterly attic. If you think string theory is abstract, you should think about the mathematics that underpins it.
Speaker 2: One last computer question. Do you know what the computer, if, if you ask the computer, so how much of double double the blood and it goes thing and give you an answer. Do you know how it reached that answer
Speaker 6: Sometimes the answer is yes. In principle. Sometimes the answer is no, but ideally what you have is a certificate. You can have an answer that comes out and it says, if you don't believe me, check this, that's an ideal sort of answer where you don't know how it found it, but there are many cases in which checking an answer. Once you're told it is much easier than finding it really Yes. And those are the kinds of cases where, where the, where you feel like you've won because not only did you get an answer, you couldn't find, but you get an easy way to check.
Speaker 2: Okay. Well, let me ask about Gregoire Perlman, Keith. this is a very weird man, but he, his problem is something called the plank, Herve, conjecture. What is that Who is he Who is Gregoire
Speaker 5: So he's a very unusual case. We're going to be up. We're going to be talking about a very atypical case, all a very famous one. But the point career conjecture goes back to the beginning of the 20th century, Henri Poincare. And one way to think about it was think of the following question. You go back to ancient Greece. The Greeks lived on what seemed like a flat earth, but by using mathematics, there were able to figure out what the world was. In fact, spherical, and even calculated the diameter with good accuracy. So using mathematics, you can step out of the world, you live in and see what it must look like. We didn't look at the, we didn't actually see that the image of the world. We didn't know with our eyes, that it was vertical till NASA sent spacecraft. We took photographs, but the Greeks had figured it out by using mathematics.
Speaker 5: Could we as creatures now, living in a three-dimensional world, could we understand the shape of the universe we live in by using mathematics So as it were step outside the universe, for example, the universe may be like the inside of a sphere and that you could move around freely, or maybe it's like the inside of a big inner tube that you can keep going round around or something more complicated, like a pretzel. So there were lots of ships. The universe might be, could we figure it out using mathematics Planko air came up with a method that can be understood in those terms. And here's what plank reading. We've actually got a little video. We can show about the conjecture because it's on their Pelham and actually was the one that moved this 2002. So if we can roll up to it, let's imagine we want to understand the shape of the universe is in.
Speaker 5: So we got to get in a spacecraft and we're going to go out and we're going to start splaying out a rope behind us. We're going to go all around the universe on this tour. And we're eventually going to come back again and create a big loop that will track the path that we'd followed. So here we go round, and it's a big world. It's going to take us a while to get round, but we're going to come back. And when we come back, we're going to be able to find the tail that we left behind the end of that rope. And then we're going to start to pull the rope tight. And when we start to pull it tight, one of two things can happen. You start chugging or it stopped. Maybe you're in the inside of an inner tube. When you can't pull it any tighter or the following might happen, you pull it and pull it and pull it pulls down.
Speaker 5: So those are the two possible ways it could come about. If you can't pull it together, you know that the universe is not like the inside of a sphere. It be, it's more like an inner tube or something like that. If you could always, if you did this infinite there, when it's abs, if you could always pull it together, the plan career conjecture says, then you will be able to conclude that you are indeed on the inside of a sphere. That's three sphere. If you can always do that, this is not practical in the way that the Greeks were. But it does say that with mathematics, there is no limit to the ability of what we can find out with mathematics. In principle, we can find out the shape of the universe we live in without stepping outside of that universe, we can look through the eyes of God if you like it, our universe using mathematics. And Pellman proved that in 2002, although it took seven or eight years before the rest of their mathematical career, which one did he proved That was the, that you can in fact find the shape of the universe by going up one things. If that way, if you can always,
Speaker 7: Well, it's again a strange guy though, because he loves a prize.
Speaker 5: It didn't want the million dollar prize. It didn't want, it didn't want the fields metal. He's a very unusual guy. I've never met him. Very few mathematicians have cos he's very reclusive. that made it a great new story. On the other hand, the mathematical discovery itself was the front-page story on science magazine, the mathematical discovery of the years, the science discovery, the science discovery of the year. Yeah,
Speaker 7: But I, I'm not so sure it's so atypical. No, I don't think so. this is one of the seven great problems in math in the year 2000, the clay foundation said these are the seven grade problems in the new millennium that mathematicians should think about. And the Poincare conjecture was one of them and it's been the first one to be improved by Perelman. There's a million dollars for it are. They rang up parallel and they said, Hey, do you want the million dollars He said, no, I don't do the math for the money. So then they offered him the field's metal, which is the math equivalent of a Nobel prize. Even better than a Nobel prize because you only get it every four years. And they invited into Madrid to collect it at the international Congress of mathematicians. And he didn't turn up and they rang him up and they said, look, why didn't you come and get the fields, man He said, I live in Russia. We're taking a data flight to Madrid, data, collect the prize, dates, do the interviews, data fly back to Russia. That's four days wasted when I could have been doing maths and air dish to get air is exactly the same. I think that's a very description
Speaker 6: Of a paramount.
Speaker 4: Yeah. By all accounts, she's a little more Australian than that. Can I ask it This is a delicate question. And then I'm going to do something. This, this conference is this festival. Doesn't usually let questions come from.
Speaker 7: I'm really curious about this because very few people have met him and he is an ultra busy. I would say that he, that this absolute purity and dedication like air dish, air dish won the Wolf prize. 50,000 pounds with 50,000 pounds was worth a lot. And he didn't buy a hat. She never had never had a house in his entire life. Instead he offered the money as prizes for other people to solve other matters.
Speaker 6: Most of you don't know Perlman, but I knew Erdos quite well. I first met him when I was one. And
Speaker 4: When I was one that you remember it clearly I'll tell you the strange thing about Perlman though, is that he stopped doing mathematics. That's a strange thing. That is, you know, that is a total lie. Can I suggest that there may be just a hint of Mr. Asperger in the rug Oh, there's a lovely
Speaker 6: Recent book. I think it's called perfect rigor.
Speaker 4: Perfect rigor. Yeah. It is
Speaker 6: Essential description of a high achieving ultra Asperger and a very, very interesting description of the sociology of how math kids going to the Olympiad were being trained in a position in a time in which there was tremendous antisemitism. So two places in the elite university system for Jews a third, if you got a gold medal on the Olympiad team. Yeah.
Speaker 4: The reason I said he was atypical was I know thousands of mathematicians and hardly any of them are like that. They're regular people. Yeah.
Speaker 7: Hate to think that as a journalist who meets lots of mathematicians, lovely people, lots of
Speaker 4: But what I'm saying,
Speaker 7: What I'm trying to say is that two of the greatest discoveries to the grace mathematicians of hour of your time, your generation have utterly, utterly bizarre attitudes to the way they live their lives. And in science, in physics and cosmology, there are people that are dedicated and obsessed, but not to that extreme. So we're looking at the very long end of the tale, but it's a tale that doesn't exist. Even in other abstract
Speaker 4: Newton you in the first room, Mr. Newton fan. But he wasn't one of these guys who would just kind of walk around and get so lost in thought he would draw things on the path. And no one, no one seemed to understand anything he was saying. He was an odd duck also. Right Yeah.
Speaker 7: Robin wins told a story about Norbert Wiener the other day. I don't know if it's true, but the Wiener founder had just moved house. And so he was in his new house and he went off to the office and they said, look Norbert, remember where we live Remember you've moved house. So went off to work, came back and went back to his old house. I was sat on the doorstep, really depressed. Didn't know what He didn't know where to go. And it's little girl came up to him. He said, Hey, little girl, do you know where the Wiener family live And she said, yes, daddy is just over here. So on that note,
Speaker 4: You guys want to ask questions of the panelists to see, Oh, well we're only at five minutes. Well, what the hell the night is young. Let's just do the, I know, I don't know where they're going to turn up the lights. Cause this is illegal. We're not supposed to let you ask questions, but what is, so yes, you'll have to shout them, but you're in the front row. Go ahead.
Speaker 8: And I'll repeat. I I've heard that. Just like, you reach a peak at age 27.
Speaker 4: The question is, is athletes sometimes reach their peak very young, say 26, depends on the sport. What about, what about peak math years now That's false.
Speaker 6: It was a great article in the New York, weekend section a few years ago called a young geniuses and old masters, not about mathematics, but about art and saying don't try and compare Picasso to teaching. So I think it's absolutely true that that, radical ideas tend to be the work of younger mathematicians. And maybe if you've not done anything earth shaking by the age of 35, it's unlikely you're going to do something fantastic later, but a recent book by, Ruben Hirsch called loving and hating mathematics. And one of the, the cliche is that he tries to break down is just that. So I think like in any other subject, you don't do the same things at 60 that you do to 20, or if you do, it's probably been wasted. I don't know, very many really creative mathematicians who say, Oh, I'm 35. It's time to stop.
Speaker 4: Did air dish when he rang the bell and he was like 60 and he was ringing the bell of a 24 year old. Did the 24 year old, the old guy, or did the guy go, Oh, I'm so glad to see him.
Speaker 6: One night, my mother called me when you were two.
Speaker 7: No, I was
Speaker 6: I was 29 and I was actually naked in the dark in bed. And how Vikas and I thought only one person would call it this time of the night. I picked up phones on my mother and she said, somebody would like to speak to you. And a little voice got on the phone and said,
Speaker 7: Hello, this is Paul. Do you know the question with 29 points in the plane
Speaker 6: And it was just continuing a discussion. We'd had several years earlier. It actually had,
Speaker 7: He'd had it with
Speaker 6: My brother. I realized after a while, but I thought, I didn't need to just disabuse him about that. But no, he was completely a Galatarian his mind was open. He wasn't a mathematician of the same, status as the greatest mathematicians of the 20th century. But he was maybe one of the most influential because he went everywhere and I was getting a little grouchy about comparing him to Perelman. he was an odd man out here to share dosh. Yeah. But he was odd in idiosyncratic gentle ways. And, he didn't have many needs and that's largely because people like Ron Graham look,
Speaker 7: But, but I would say, not a tool or in the heat find the thing he loved, which was doing math. And
Speaker 5: If you, if you love that math, maths, that much, then any other behavior would have been odd. So that's the way I kind of do this in swimming. He was in the pool all the time, because he just loved being in the pool. You read about the ones that produce the pinnacle results by definition, they are unusual. And that often manifests by the way. I think that was the other thing with us. They ask one more time. that waving person. Yeah.
Speaker 8: I was wondering, is there any hope for those of us who fell off or were knocked off or from the prosaic of the sublime the appropriate time found an interest in later
Speaker 5: What if I was good at the beginning and then I wasn't good enough. And now I'm still really interested. Do I still have a prayer Right
Speaker 5: Except that I wasn't really that good. Yeah. I think there's always a chance to restart the game because it's such a logical subject. You just have to take yourself back to the point where, you know, you understand and start again, going down the path that the mathematicians have laid out for you. So I think if you've got determination and this is why, you know, people carry on doing mathematics, however old they are, because they've got a passion for wanting to know the answer provided. You've got that passion. It doesn't matter when or what age you were when you started. If you want to know what the next step is, you'll be able to get there.
Speaker 6: One of the things that seems to be determined about genius in general forms is you see it with Newton and others too. The capacity to actually not leave your desk for 24 hours independent of Saudi. I was going to say that I think the answer is a little, like I wish I'd learned another language when I was six. It's a lot easier when you're six than when you're 36. So the chances that you will learn it with the fluency that you would have learned, if you hadn't had the interruption a zero, but, but the chance that you can learn it to do it well enough for it to be a source of great pleasure and accomplishment. That's a matter of persevere.
Speaker 5: A good starting point is Martin Gardner's books and recreational math. And that's a good way just to get, get into the swing of things and just to learn some of the curious math that's around, it's getting easier because of the internet and the accessibility. There's the Wikipedia resource it's actually Wikipedia in mathematical areas is really pretty darn good. The higher you will pick. It's very reliable, the materials and I every week or so, because I've got an MBA. I was a math guy. I get emails from people with that very same question. You can reach people and we're not going to give an hours time in a response, but we'll say, try this book or try to talk to this person at your nearest university. It's actually relatively easy. Now, you know, if you're a mathematician, you actually, your heart lifts. If someone says, I'd like to get back into mathematics, we usually spend a few minutes trying to help them out. We have so few friends, we need one. We can get two more and then we'll yes. Over there. Yes.
Speaker 9: I think, I dunno if I experienced that when I look at like a science textbook or like physics or chemistry or something, that's more like a story I can sort of read through and say, okay, get this, get that. And then I look at fat and the first step starts out. let's assume this is a product you get to the end and say, well, it's not prime. So this thing is false. Wait, did we assume that it was a prime number Like I can't remember what the assumptions that would be. So how do I read a math textbook compared to something like
Speaker 2: The question is, if you read a science textbook, it seems like storytelling and you can get involved and you get interested and you want to know how it's going to turn out. You read a math textbook and you go into it and they make assumptions. And you can't remember whether you made those same assumptions and suddenly you are brought to the area. There's the answer.
Speaker 6: The comparison there would be like reading a cookery book of an ultimate, big of a pen by Bunuel. All four of us have written books about mathematics, which tell the story. That's why we have all the stage of costs. But I think
Speaker 3: The proofs that you find in books, you know, for them as proof that, every prime, which has remained a one, when you divide it by four can be written as two squares. It's a fantastic story. And how you get from, you know, something like 41 to 16 plus 25 and whatever the prime is. If it's got remainder one on division my fall, you could always write it as two squares. It's an extraordinary story. And I want, I don't want to just know the answer. I want to know how did this, this hero gets from here to here.
Speaker 2: It's my suspicion that the answer to your question is some people like I once went to breakfast with my boss, Larry Tisch at CBS, and he brought H Ross Perot was then just, tycoon. And they, they sat around and they opened up the wall street journal. Was it a breakfast table at the Drake hotel And they, they looked at these just sheets of numbers and they would point and they giggle and they gave little, I like, I guess we're going to make him this guy. So, no, I, I, but I thought to myself, wow, these people read numbers the way I would read ASAP. You know, it's just, they just are different.
Speaker 9: There'll be,
Speaker 2: She, she can't make the step.
Speaker 6: I think you're identifying a real problem that Marcus mentioned early on, which is that mathematical is a massive mathematics is a very large scaffold and, and hierarchical. And it's hard to learn something before another thing. And so we've tended in writing textbooks to look for very efficient, very focused proofs and kind of trust that the reader knows why they want to be there. And that's not always a particularly good solution. The alternative and a mathematician called Moore in Texas, tried to do the opposite with great students, which is, do motivate every step, prove nothing. Get the students to prove everything from cells. And in theory, it was a fantastic idea in practice. It meant that by the end of a four-year degree, you were in semester two last
Speaker 2: Trade off the evening. see, I hate to do the physical in the front. So Hey, is there somebody in the back way in the back that's naked arm up
Speaker 1: There. Yeah. Good now. Yeah.
Speaker 5: And hear that. The question is, is it because of the good ELLs, this uncertainty that he introduces, Oh boy, am I going to have trouble Did you say non axiomatic, maths I mean, it's almost one of these things. What is mathematics Because you can define mathematics. And for many years it was the stuff that follows on the axioms, but most of what mathematicians or the living today do is actually not axiomatic. It's just exploring things in an analytic way, thinking rationally, it's not based on those actions of this sort of pure axiomatic mathematics that sort of has a history more comparison. But if you just actually say of all the people who earn money by doing mathematics, hardly any of them do ask them. I think, mathematics, they deal with problems that arise in the real world or within mathematics, but not in an axiomatic framework that was for axiomatic mathematics, or was he this innocence girdle prove that it was a pointless trying to do that because he wasn't going to capture everything how we saw. Have we answered your question
Speaker 1: You're welcome. Thanks everybody for coming.
Speaker 2: I should warn you in advance that what I know about mathematics higher mathematics is very, very little, which is the excitement to me of trying to do this. This is for me, it would be as trapeze work. So, let me introduce the guests. first known as the math guy on national public radio, he is the author of 30 books and over 80 published research articles, Keith Devlin is a recognized mathematician. He recognized mathematician, which I guess means that he's really a plumber and looks like a mathematician and a researcher at Stanford university. And here is Keith
Speaker 2: And next Jonathan Barwon is a mathematics professor at the university of new castle in Australia. He's the director of the center for computer assisted research mathematics and it's applications. You kind of know when they do that, got to be some acronym it's called karma. So, he's a noted expert on the number PI and a leading thinker in the field of experimental mathematics. So here is Jonathan. next Marcus DeSotoy, a professor of mathematics at the university of Oxford, a mathematician, a researcher, occasionally on Radiolab. He studies prime numbers and symmetry. and finally a science journalist producer with a PhD in particle physics, Simon Singh documentary about Fermat's last theorem was nominated for an Emmy, his publication on the same subject. Fairmont's enigma was the first book about mathematics to become the number one bestseller in the UK. Here he is silent. let's see how to start this. First of all, this is the question posed is what is mathematics there's I just, I don't know way of describing nature, art form language, brains waves, seeing the world exploration, competitive sport God's mind revealed. let's start with a pattern sense. When you were a kid, did you take an unusually long gazes at pineapple surfaces or sunflower
Speaker 3: No, it took me a long while actually to get into math. So I didn't, or math as you call it. yeah, I don't know why it's, I think it's a very plural subjects, but mathematics, maybe mathematics is a better way to, but it was around, you know, cause I think the real distinction here is that, early on your new career, you're doing dealing with numeracy and arithmetic and that's not what mathematics is about. And I think for many people, that's the misconception. They think it's, they say the bad at math, but actually they may be bad at numeracy. Most mathematicians are bad at arithmetic. and actually math is about pattern searching it's and it's what we've been evolutionary program to do. You know, we've survived in this world because we're very good at spotting patterns in the jungle. If you see something with a bit of symmetry, it's likely to be, an animal, which either you could eat or it might eat you. So it's good to recognize symmetry. Yeah.
Speaker 2: So when you were seven, like did you go to the, out into the garden and say- one ant, two ant...
Speaker 3: No, no, no. I was actually much more interested in languages. I wanted to become a spy and to be in the foreign office. and it was only when I started to realize that mathematics is actually actually I hated languages because there are all these irregular verbs and strange spellings. And it was when I suddenly realized mathematics is a fantastic language. It has no exceptions, but lots of surprises, but it's gotten a regular verbs and it's a fantastic language for describing the world around you. And I think it was about 12, 13
Speaker 3: That I started to get a little bit nerdier and start to spot spirals in pineapples. But that's how that happened if I did. And I now can't look at a pineapple without going Oh, eight, 13. And you just, did you happen to be a pattern sensitive Not at all. No.
Speaker 4: The year before I went to high school back in England was the year the wishlist put Sputnik in. I wanted to go into the space race. I mean, it was just space exploration. I wanted to do physics. And so I had started learning mathematics in order to become a good physicist. And I didn't like mathematics until I was 16 and I met calculus. Anyone who's read the blog. I put up on the, on the world science festival blog. I put a blog up about why I became a mathematician, the very moment where all my classmates got turned off, calculus that turned me on. First of all, it was obviously something was wrong and it was obviously important for getting rockets into space and getting them back again. But it was also very powerful and I couldn't understand it. It's the first thing you meet at school that you actually can't understand if you put your mind to it, it baffled me. It was clear that there was a mystery. I mean, they were talking to them
Speaker 4: Was such a mystery that you felt personally, you wanted to solve it.
Speaker 5: Oh, I couldn't stand that. I was the kind of kid that used to take the ultimate a apart. I wanted to see how things worked. And who was this Calculus was invented by a guy who was age 22 because the college was closed. You know, he was going to go to Isaac Newton was going to Cambridge. It was the plague.
Speaker 4: Well, are you a Newton fan or what Okay. Yeah.
Speaker 5: What would you do if the college was closed? You'd probably go to have a good time and go to the beach. Isaac Newton invented calculus. So you've got this 22 year old invents calculus. It's powerful. I could use it. I could do all the problems and get them right. I had no idea why it worked and I wanted to tick it out.
Speaker 4: Oh, that's that's maybe when you played sports, did you think, okay. If I stand here, the arc of the will go to the declining, I think, or were you odd or were you what I would call ordinary
Speaker 6: I was never gonna have a sportsman to do anything except say, I wish it goes somewhere else. Please let somebody else catch it.
Speaker 4: My, my father is quite a distinguished mathematician
Speaker 6: he has been president of Canadian mathematical society. And I think he's now the sixth oldest member of London, mathematical society.
Speaker 4: Well, they keep a list. It's a little bit because it's the mathematics society. Indeed, indeed
Speaker 6: Win a prize. But only once for being the oldest person at an American math society banquet, really, you can't keep doing it for obvious reasons.
Speaker 4: Well, wait, wait, wait, you said you're doing your annual banquets and you're like to get back to your central. Why, if you're 93, can't you make it to your 94th because they want to give somebody else younger a chance. Oh, so you're not invited to the bank. You can come, but you can't win the prize.
Speaker 5: And since the oldest,
Speaker 6: Viennese or, Austrian mathematician some years ago was via tourists who was also the Austrian and died about age 108. You can see that you wouldn't want to give us pies out only once every 20 years. But getting back to your central question, I also discussed
Speaker 2: Such a strange envy. So these people in Britain are going to this dinner. Envying. The Australian is two years older than me. I don't know people do lie about their age in the opposite direction after all right. Yeah.
Speaker 6: They're older. I like, since my dad was a mathematician, he taught me a few pieces of math to prove, to win bets with faculty in St. Andrew's. So he taught me to solve two by two simultaneous equations when I was six.
Speaker 2: Well, when would you like that When you're saying, because he wanted 5,000
Speaker 6: Pounds of cheese and a five pound note, which is quite a lot in the fifties.
Speaker 2: No he did. But if you're, if you're a six year old using whatever you just said, independent situation with six year old, I don't see how that would be an occasion that would ever come up. Well, I suppose I must have liked the pattern because I had no idea what I was doing.
Speaker 6: But after that, I went to university to study history and I go up to this far in the days of punch cards for registration or being about to drop my cards in second year into a history box. And I thought if I do this in 10 years time, I won't be able to integrate anything or work out the equations of motion. But if I go back and find the math cards in five years time, I can still read about the treaty of Vienna I did. And it's turned out to be the truest decision I ever made. So I can tell you the day I became a mathematician, but I came to it a bit, like, it sounds like you out of the arts, not out of the sciences directly.
Speaker 2: Well, so, so then what is it exactly Is it, is it an instinct that humans are born with I mean, there are, there are people in the Amazon who have one and two and three, but they don't have four or five or six or seven. And they do perfectly well. I mean, they seem to know the difference between more and less. I don't know what they do if I had three fish and you had two apples and I wanted to raise the price of fish, I don't know how that conversation would go. But do you think it's an instinct in us I don't know you, I,
Speaker 7: I don't think it's an instinct in, in the way that we think about math too. When we think about numbers and precision, you can do experiments. I think this experiment with the Raven Raven, was, on the top of a tower and a Hunter would approach the tower and the Raven would disappear and the Raven would, would disappear until the Hunter left the tower and gone away again. And then the, the, the, just the war. How clever is this Raven So then two hunters were going, the tower, Raven would disappear. One Hunter would leave, but the Crow, the Raven wouldn't return until the second hundred left. So, so the Raven could somehow count, but if five hunters went in and four came out, then the Raven couldn't differentiate between five and four, it would come back and get shot by the remaining Hunter.
Speaker 7: So, so it's as though it's an approximate. It is a Hunter hunters where like, tablespoons of sugar. And I can differentiate between two spoonful's of sugar and one spoonful of sugar in a pile. But I can't really differentiate between five and four. So unless we can begin to put symbols or words to numbers, that it's only, then I think that we can begin to really manipulate them. I should just say, I have the people here on the panel. I like everybody here. I didn't really like Matt as a child. And I didn't realize it as an adult either. Really No. I mean, I became like light like Keith. I was interested in science and he's a very obvious desire to want and understand the universe, transcend where the universe came from. What stuff's made of where life came from. This is really obvious stuff to be interesting. I think nearly all children are interested in that. And then gradually we maybe lose that curiosity, but to be interested in math is a bit odd. And I think that's why people are sometimes scared of math or don't understand why math is a pattern. I only became interested in math because then as a scientist, I became a science journalist as a science journalist. I ended up writing about math and I'd always seen math as something you have in order to do science,
Speaker 3: But I think that's how it, yeah, that's how it started. Yeah, it was, you know, in order to navigate the world, to measure land areas, to tax them, to build new buildings, you find all to answer these big questions of science. You find that math is the best language, and then it starts to take on a life of its own. And you start to get interested it for in its fruit, for its own sake and the properties of numbers and the patterns there. So the measuring thing,
Speaker 6: If, if, if a girls were going to jump rope, so it just basically won't, you know, and they go and they had those patterns. I didn't put duper, duper, duper, duper another girl. So I don't know, but, I don't know why I say things like
Speaker 7: That, but
Speaker 6: It isn't, isn't singing, dancing and playing games sort of playing with segments of time a lot. And so isn't that math sort of,
Speaker 7: So mathematics really changed the, the, the, the elementary parts of mathematics, the stuff that everyone would have had in school that does come out of the world. It's sort of formalizing ways of thinking that we use to survive and have done for the history of humankind. But then the moment when I got into it at calculus, it sort of flips. It becomes larger something we invent and create ourselves becomes very abstract. It's still actually useful, but it doesn't. I mean, in calculus, you're talking about infinite decimals. There are the infinite festivals in the world. We invent them in part to understand the world. But those of us that have been mathematics play with them because they're fun to play with. And they really trust me.
Speaker 6: Well, the audience probably knows it. Certainly what the encyclopedia it's thought that a while French was a perfectly good language for science only German was really set for emotion and love. These are things that people think about other languages, but the reason I'm bringing it up is that what has kept me attracted to mathematics It's two things. It's all of what Keith said, this wonderful set of tools, unreasonably efficacious tools, for making more progress often than seems reasonable. But at the same time, it's a language in which you can express things. You didn't know, you wanted to express until you could speak some of the language. So the language is called numbers. I guess the language is called mathematics numbers are some of the words I do. I remember reading somewhere that a baby born on the first day, if you stood in front of them on the first day of its life, and you went peep, peep, peep, peep, peep, peep, peep, peep for a very long time.
Speaker 6: And then you changed, Oh, it's only up to three. Yeah. And if you change the number, the BB startles or looks more closely. So they're just some sense of, of pattern anyway, but now the N the number, that was invented or where that comes from, how old it is. Yes. It's interesting. You can even do some fairly road bust kind of cladistics. And you'll discover that the number two is older than the number one in terms of where it's shared and the tree of language. Really. Yeah. So, you know, it's all conjectural, that's the lovely thing about archeology and paleo, whatever. You're never going to be proved wrong, but it seems pretty plausible that we, well, that's a good thing, isn't it
Speaker 3: And he's very ancient bones. They like the Shango boat, which is tens of thousands of years old with notches, you know, people keeping track of things and wanting to know how many, I mean, it's not quite clear what it's keeping track of, whether it's, perhaps a calendar or something, but, you know, I, I think, you know, navigate it, seeing patterns in the stars is probably where, you know, you start to skip the mathematical mind working the fact that things repeat themselves. And if they repeat themselves, then you can make predictions about the future. And that this language gives you incredible power. And, and, you know, you've seen that
Speaker 6: Predict a flood or eclipse. Yeah.
Speaker 3: The, the flooding of the denial, they spotted patterns to this. And they're that, that, that matters.
Speaker 7: Well, once you've had the flooding of the Nile, not only, you want a bit, not only do you want to be able to predict the flooding of the Nile, you then want to be able to rebuild your fields afterwards, you need to have measurement and geometry be able to do so. It's a purely functional,
Speaker 6: Useful tool
Speaker 7: That, that society is. If we're going to trade, if we're going to send ships, we're going to plot them. If we're going to be interested in astrology, we need mathematics. But the really interesting transition, and I think what was interesting about the little video clip we had at the beginning was a lot of that was applied mathematics and it's verging on the science. And we kind of all understand why science is interesting, useful, and important, but the really weird transition. I'm not sure when this happens. This is maybe going back two and a half thousand years is people who begin to study numbers purely for the hell of it, for the fun of it, for the surprisingness. The fact that there's this thing called a perfect number six, a one plus two plus three is six and one and two and three are the only numbers that go into six and 28 is one, four, seven, and 14. Am I missing one And to add up to 28 and divided the 20th, but then I don't know where the next one is up in the hundreds. Isn't it think that.
Speaker 2: Do you think these people sort of, did you have to get, cause I'm actually wondering whether you get notches first or maybe circles, you think circles or lines proceed, numbers or circles lines or trying not to miss them.
Speaker 5: Truck numbers are incredibly recent as about eight, 10,000 years ago most. And it was essentially money in Sumeria. Sumerian society reached a surge of complexity where the trading was obviously much better if it was mediated by something that basically, if you want to know where numbers do, it's like following political careers, find where the money comes from, that's where the numbers came from, but nothing new in the world, especially in Ninja
Speaker 3: Elementary as well. And I mean, you see, in the Ryan papayas, you see a formula for the volume of a pyramid and beautifully proved as well, using an early form of the calculus. But of course that was a practical need to know how many stones are we going to have in this pyramid. But, but then you see a sort of abstract love of the mathematics and the game starting to come in and it takes on a sort of power of its own. And I think that's right. It's perfect. Numbers are perfectly,
Speaker 5: It's perfectly pointless. Perfect. Useful. Yeah. But you kind of think, well, that's kind of cool. These are,
Speaker 2: Are you imagining that this is cause you're sorta like th like, is this like a, a prayer that you're telling us here, or is this history that you could, I mean, we're not going to meet
Speaker 6: This is history. And I think it's interesting to ask how much of it is, contingent on how much it was necessary. So for example, it, usual accounting would be that the Greeks, the class, the classic Greeks brought us the sense of trying to build a deductive system, being interested in the properties of numbers for their own right or mixed with ideas of harmony and religion. But imagine, and I've just been doing some work with my colleague, David Bailey at Lawrence Berkeley labs, and flabbergasted to describe a 13 decimal accuracy, square roots in very old Indian documents, pre Christian. so sophisticated that you wonder what would have happened if those techniques had made it to the West through Spain in 719 1,408. And for some reason, all of the manuscripts that we learned about Aristotle or Pythagoras had actually got burned. And so what we had inherited was a very much more functional set of how to do things. Is it necessary that we would have later said, we need theory. I'm not so sure
Speaker 2: Keith is positing that, that the first guy was just a businessman on the fast track. And he just got himself at first detach a case or something. Right. The other way to think of it would be that the first mathematician was sort of an artist like you would think maybe like he just wanted to play,
Speaker 5: Right Match goes back so far in time that we had to feed ourselves. We had to keep warm. We had to fight off bears and wolves and Matt evolve in that environment where you desperately needed to survive. And it's the same with science.
Speaker 2: Oh, you were the artist guy. I thought you were trying to imagine this a fellow who was playing,
Speaker 5: But I think math starts off
Speaker 7: As being functional. Science starts off science. Doesn't start off, you have technology. And then people start realizing if they want to build technology. It's actually quite useful to understand some of the basic science. So, so the pure side emerges only much later.
Speaker 6: The great Canadian geometer center was a friend of Asher's, the Dutch artists who most people come to these sort of things I think probably likes, and late in his life cook, cooks or lived a very long life live to age 97 and gave lectures to about age 96. good lectures. He, wrote a paper, maybe one of his very last papers and he's explaining one of Essure's, continuing joint constructions. And he said, Asher did this as art I had to use trigonometry. Huh So I, I,
Speaker 2: You associate, there are things like elegant proofs and ugly proofs where it's like that tweet. If it feels to me a little bit, is that, is that art is just freer. It, it th the real art means if you paint a painting, you're not constrained.
Speaker 3: No I'm going to in most artists will disagree with you and say that their best creative art art is done under huge constraints. I mean, Stravinsky used to say that he was only creative when he put constraints on himself. So I think if you have too much freedom, very often, you become uncreative. And in mathematics is an incredibly creative subject. We're making a lot of choices in the sort of things we want to celebrate as theorems. I'll talk about in a seminar right up in a, an a paper. I mean, I studied the world of symmetry. I can get a computer just to generate arbitrary new symmetrical objects. What's interesting is to choose the ones which have something special about them. And it's just the same. You can get a computer to churn out music, but you make a choice as a composer. And I think there's an incredibly creative science to, well, why is this exciting There's an exciting connection with another bit of mathematics, which is totally unexpected. And this is why I want to tell you this story. So I think there is a lot of choice in mathematics.
Speaker 6: Nice description of, of the aesthetic response is the perception of order in what seems first to be chaotic. And again, that calls on whether it's a few or a structured piece of poetry to have that underlying structure and that to some degree, the aesthetic response, the aha equivalent, or an aha mathematically, as we say, Oh, okay. It does make sense. It's not just Jangles.
Speaker 2: Here's the difference. If, if we all agree that this will be the number three, yes, we all agree. And we all agree that this symbol will mean we're going to add something once we've agreed on those terms, three means a triplet triplet of some sort. And plus me, then if you, then, if I say three plus three equals it. At that point, it has to be six. That's not the case for Shakespeare. Hamlet could have killed his father any time, bill Shakespeare,
Speaker 3: But you're, you're comparing two things which are not alike. It's like talking about English grammar. This, this word is spelled like this and the humans only, but mathematics is not about spelling and grammar. It's about the big stories, the stories, these are numbers of cemetery. These are the things that we stories we choose
Speaker 6: To celebrate. And there, they have a truth about them. You can't change them there. So it feels like there's more constraints on mathematics, but still the stories we choose to tell are as exciting and have the drama. And we choose, we choose to tell those stories because they have the same effect on us as a piece of Shakespeare. You're just comparing to wrong things. Yeah, exactly. Because,
Speaker 7: I feel
Speaker 2: I have like the Chinese menu of all time. I know that as soon as I got three plus three, I don't, I, I'm not allowed
Speaker 7: To turn it into Cinnabon.
Speaker 6: Exactly. But if you've got hamlets, you will say, well, I'm going to spell it. H a M L E T. And then if you have it later on in the play and you'd misspell it, it's sort of, that's uninteresting.
Speaker 7: It's not what she thinks mathematics. Isn't about. We're all gonna be very, very aggressive. with very elementary math, it's dry, it's ugly. It's tedious. Just like very ugly prosaic English. The, the jump you have with the spoken word is from the spoken functional word to poetry. And sudden you say, wow, that's, that's beautiful. That that's something I want, I want to embrace it. Something I want to play with. I want to create a prizes. I want to make people laugh. And, and with Matt, again, you have these very mundane prosaic numbers, and you, aren't very basic question like multiplication square roots. And, and you say, well, what happens if I take away three from six Well, that's three. That's pretty obvious. What about if I take away six from three, Hey, that's mind blowing the first person to ask that question suddenly has to invent the idea of negative numbers and this, this I can't hold negative three pebbles. This is that's mindless. And then you start saying, well, okay, square root of nine is three, but what about the squirted Negative three. What does that mean And that, because curious and odd, and then you say, well, okay, let's allow that to be a number. And we'll call that route three R. Is that right Good. Thank you.
Speaker 2: I'm just thinking one thing is that I don't want to push this too hard. I think you guys are of the, you'll say one, an odd number, plus one, another odd number makes an even number. And that like shocks you. And it's like storytelling in a way.
Speaker 7: It doesn't, it doesn't
Speaker 6: Come in. And you're at the Harbor. You find it beautiful. The harness, something really complicated. But I could say all of computer science really originated when George Gould said no one plus one doesn't equal two. It equals zero.
Speaker 2: Who said that
Speaker 6: We're back to the basis of the idea that you might be adding in. What's called clock notation or binary. And the effect of two ones is a zero. And that gives you the whole idea as a modern, modular arithmetic article, finite fields and on. And so where the creativity in math comes is to say, not that it was wrong to say one plus one equals two, but there may be interesting ways of abstracting this, where it's better to think of it as being something else. And so, again, we don't try and see if you've got a dictionary you can Shakespeare,
Speaker 2: Right We do try to say, if you've learned the language, you probably appreciate some of the Shakespeare. Well, let me switch the question a little bit. Let me, why are some people better at this than others Is this, is this a talent Is it genetic Is it just a language with different degrees of difficulty
Speaker 5: Or interest It's just interest just in few interested, you become good at it and you do it. I mean, the two of these guys didn't even begin in mathematics. I didn't begin. I began in physics and not mathematics and one isn't but we all enjoy it. We appreciate it. And it's just a matter of interest.
Speaker 3: I think there is a, there's a, there is a reason why mathematics people do seem to be sometimes bad at it, which is mathematics is a bit like building a pyramid. And each year you add more of the permit. If you have one bad year in your mathematical education, it's impossible to build anything on it. You're lost from there onwards. And that doesn't happen so much in other subjects. You know, if you, if you've got a bad history teacher, teaching the Victorians, it doesn't kind of mess you up for the history of the 20th century. You can get back into it a little bit, like you said, with your experience of when I can reread the history later on, but with a massive I'll miss out. So I think that what happens is that you need to have a very consistent, education because one bad layer and it really messes you for the rest of the day.
Speaker 2: Well, it's the key thing. When you say it's interest, is it, is it that you're like the problem with math as opposed to some other subjects, is that there is a right answer. So you, you, if you're learning, you know, it's hard to fit often. Yeah. So, so it's hard to you just going to fail or succeed. It's sort of binary. And that may be, I mean, that's one of its chunks, I think. Okay.
Speaker 5: Yeah, there are sort of right answers, but that's what mathematics is really about. What's right. It's why is something right or wrong It's the why it's of interest. Not whether it's right or wrong. That's a fallout from the way we often teach it that it's all about right and wrong. There's a right and wrong list. There just like, there's a right and wrong way to spell things. But Shakespeare, isn't sort of to go back to the example, isn't about how you spell the words. Is it a good story And why do people behave the way they do in mathematics It's not what's right. And what's wrong, but why is this right in this circumstance
Speaker 2: Well, that may be there. The difference is some people make this personal. They say, this is my question. I want to know the answer to put it as you did. I'm interested. Oh yeah. Others of us think I'll do what the teacher says and you never somehow get across. It never becomes useful.
Speaker 5: You know, that's how I passed English literature at school. I've got these cliff notes and I'd read them. And I answered the exams and it took me 10 years to like Shakespeare. After I graduated from high school, Pete people don't have to like math people, people have perfect right to hate math. And I have many, I hate languages. I'm terrible at languages, but there is a world out there that people may be unaware of. And just came back to the point. Keith made, there's a lovely problem called the four color map problems. If you've got a map and people may remember this from school, how many colors do you need to color a map Any, any conceivable
Speaker 7: Map so that no two regions have the same color. And for a hundred years, people knew that you didn't need more than five, but they weren't sure whether you could get away with four. And that's kind of a weird, interesting, odd question. And it's just something that gets you think, what do you need Four, do you need five Why is it four Why is it five How, how would I go about trying to prove it So it's an odd, quirky, interesting bit of playful maths. And it turned out, I think it was about in the seventies, a couple of American guys, Apple and hearken proved it. And they proved it by saying that every map can basically be built from several thousand other constituent maps. And if you can prove that the other thousand or so constituent maps only need four colors, then all maps only need four colors. Now we now know you only need four colors, but it's not a very satisfying proof. It doesn't really give you any insight. It doesn't make you smile. It doesn't, we're not that surprised because we already kind of knew it was four. So that's an area of math, which is, has been curious or in quirky. But w I think mathematicians are still waiting for that. Aha moment is it doesn't have that beauty and elegance that something like Euclid's, pro-nuclear had a proof about the number of privates
Speaker 6: That comes to my present passion, which is I I'm much influenced by people like Greg chase in the complexity theorist. I don't believe that just because something has let's use the word. Beautiful fact. We don't know if the infinitely many perfect numbers, those numbers like one and six and 28. We do know it relates to how many primes there are of the form, two to the N minus one, but, and we don't know any odd, perfect numbers. And we have known that for 2,500 years, that's a problem that classic Greek sausage, is there a number like six, which is the sum of its proper divisors and is odd. Nobody knows. And nobody has the slightest idea how to prove it. Well, we grew up in a 2000 years of post Greek philosophizing about mathematics, which essentially identified truth and provability. And we've known for a hundred years. That's not true. And it's perfectly reasonable in my now, the way I organize the world to think that might be true because it's not false. It wouldn't make it unbeautiful, but there's a lot of stuff out there that we should try and work out what we think about it and not everything that beautiful fact we'll have a beautiful proof, not everything that's provable will have a nice fruit. And that changes your view of Matthew,
Speaker 3: Because do you hear what you're saying The way you're already in love with the question What happens in a lot of schools are particularly in junior high schools and high schools in America is that people want the answers. The teachers want the answers, but they never asked you to love the question, right Well, I think it's so comfortable because this idea, actually, that very often, when we're teaching math, we just teach the grammar and the vocabulary. Although it's like learning a musical instrument where you're only allowed to play scales and arpeggios, and nobody ever plays your real piece of music. The reason why I fell in love with mathematics is because my math teacher at my school, when I was 13 and pulled me out of the masks gloves at the end of the lesson, I thought it was in trouble. It took me around the back of the mask book. I
Speaker 6: Thought it was really in trouble now. and
Speaker 3: Then he said, you know, I think you should find out what maths is really about. And he started telling me about the Fibonacci numbers and about primes and recommended a few books to me suddenly I saw what this subject was really about. And I think we, we cheat our kids by, by just saying, Oh, it's about scales and arpeggios, but you get to listen to a piece of bark or some, some blues. Then you say, that's what I want to do. I don't know how to play that yet. I don't know how to compose it, but that's where I'm heading. And that's what I was
Speaker 4: Teachers are there like that. Not enough when I decided, when I loved calculus and what to do, that's exactly what my teachers did. They took me out to the class and said, here's my college textbooks, teach yourself. I'll try and remember what it was like to do those and help you out. And it was this hundred, this incredible world opened up a real mathematics that I didn't know existed. And Simon and I were looking at, I mean, Marcus and I were lucky Simon wasn't
Speaker 7: Yeah, I was in a class of two people got taken out. I was wondering, well, I'm cleaning up are things called non-trans Tifft item. Non-traffic yeah, I've got a dice and it doesn't have one to six on it. I've got some old numbers on it and I give you a dice and I give Keith a dice. And typically, if you and I roll this, these are our own two dices. You typically beat my dice. Whatever these numbers are, your dice generally are going to show a higher number than my dice. And when you play the dice game with Keith Keith's dice typically beats your dice. And that's the way it goes. Your dice is better than my dice. Keith dice is better than yours, but when I played dice against Keith, my diet beat Keith, that's a really weird thing. It's like saying that one is less than two. Two is less than three, but three is less than well, it's a mind blowing concept. And then the really weird thing is if we double our dice, the order reverses. So if you show that to a 13, 14 year old kid, who's, who's played with normal arithmetic and multiplication square roots, that's again, that's a mind blowing world.
Speaker 4: The other thing that means that you want to make it, you want to attach it to the world. You want to say, if you know this, then you can do that. But that might be
Speaker 6: You like playing Mario brothers. It doesn't have to some, a B like working out the velocity of a bicycle. It could just be letting you do something you want you to do.
Speaker 7: Then you need to be a scientist. I mean, there's, there's something like
Speaker 4: Math classes. I don't know. There's probably a good answer to this, but it seems to me that there's the two problems are that the teacher doesn't sort of make it, let you make it personal. Doesn't let you fail without punishing you for it. And third doesn't seem to make it of the world where you live.
Speaker 6: And that's something because the skills needed to do that. When it gets cold in educational language, unpacking concepts are a great deal, more sophisticated than the skills that are needed to do fractions. If you've just learned enough math and we don't pay, and we don't have enough teachers ready to come, the trenches who are secure about their mathematical knowledge enough that they've actually studied why, and might have a good idea about me saying it wrong because they just try to stay one step ahead. And this is massive difference between being able to do the arithmetic or the manipulations and being able to step back and say, Oh, I see how that fits in. Here's the contrast. When we, in most parts of the world, we put teachers in the classroom to teach math who have very little math in their training.
Speaker 2: Well, let me ask another question different. It does. Does math describe reality or is math itself out there and, and real, the answer is yes. It just felt well,
Speaker 7: It does. It's a genome. So it looks in both directions on it, legitimate. It looks at both directions and that's some of it.
Speaker 2: Thanks. What does that mean That, that math does tell you about the world, but any more perfect way and that we live in some sort of a shadow of perfection,
Speaker 7: And I'm not even sure that physics tells us about the world. Physics tells us about how the human mind conceives of the world, the mathematics is part of that. It tells us as much about the human mind as it does about the environment, about if you want to regard those two things as separate entities, it's a, it's like a mirror on the mind. It, what it really does is tell you how the human mind perceives the environment that it's in. So is it about the world that we're in or is it about the mind The answer is it's both, but it's even better when it's turned in on itself. So you have this game, you play a game of chess. We can have some rules and the rules of chess allow you to have a very beautiful, intricate game, which for a thousand years still has, has entertained and effectuated achievements. The game of maths, which has these basic rules, like one plus one is two. And so again, it's an infinitely rich game. And if you just play it around the world of maths, you end up discovering truths that are true forever because they're logically consistent. And it's, it's a, it's a perfect game. What's the greatest, let, let me just, so the key thing here is that when you do something in maths and it's purely done in a mathematical realm, if you prove that it's true, it's true for
Speaker 6: My brother who was also a mathematician. So I think some of it's got to do with innate abilities, not to training because at a very different things, the three of us have all ended up as mathematicians. He said, you know, we may do it better than Newton, but we don't do it righter. Yeah.
Speaker 2: When you say always, right, can you, is there some, some thing that a mathematician discovered that is just so beautiful and so gigantic that it's like the super-duper best story ever.
Speaker 3: I think one of the stories that I was told when I was a kid, which is the proof that there are infinitely many primes, I mean, primes, they're the atoms of mathematics. They, what we build all numbers from the indivisible numbers, like five, seven, 17. So all numbers 105 is built by multiplying three times, five times. So they're the most basic things. So, but how, how many are there all these numbers we've got two, three, five, seven, 11, but maybe they're just a finite number. Like the periodic table has, I don't know what it is now, 118 atoms. Maybe there are fine nightly, many primes, which can build all numbers from well, Euclid proved with just a beautiful little arguments 2000 years ago, that there'll always be more and more Pines. However far you go up to the universe of numbers and he did it like this.
Speaker 3: Suppose you have a finite number of primes. Suppose there are a final number of primes. He said, okay, well take, this is probably one of you says, Oh, I don't believe you. I don't think so. Here's a list of all the primes. So usually it says, multiply all those primes together. And then here was his act of genius. He added one to that number. And then he says, okay, so this new number that I've just built you, what's that built out of it must be built out of some of your primes. You say you've got all the primes. Well, if you're trying to divide by any of the primes on your list, you get remainder one because of the way this number was built. So there must be some primes missing from your list. You haven't got them all. So you say, okay, well, I'll just add those. Then nuclear says, aren't play the same trick. I multiply all those together. Add one, if you still miss them. And here's the beauty, you know, infinity does it exist in the physical world I don't know. Probably not, but in the mathematical world, but this Sinai little argument that Euclid made he's proved that you'll never ever run out of these prime numbers. For me, that was a magical moment. And it's a proof which is lost forever. It's it's, it's terminal,
Speaker 7: I'm at physics, but what's it what's its essence. have you heard that proof before Yes. Okay. And when you heard it for the first time, did that not didn't they not change Yes. It, it didn't rock your world. Well, my rock is a smaller rock than you're coming back to the idea that that GH Hardy said that, you know what the Greek said about medicine. We laugh at what the Greek said about astronomy. We think it's a joke, but what the Greek said about math, we still teach it today because that mathematical truth remains forever. And he may have a, because I asked you this way, I guess you're saying who rave for the mind of a Greek that could think that up aren't we clever, but the no, no, no, no. I think we tapped into a unit. My question was, did the numbers care
Speaker 7: If, if you got number 3 trillion, 6,950, the, the, the, the some prime way out there, that's what I'm saying. I knew that it's not, it's not about how clever these people are. It's not about how clever Euclid was. It's about high, beautiful in four lines of argument, you can grip grapple with infinity for it. It wouldn't matter whether it was Euclid wouldn't matter where was any, but it wouldn't matter whether it was a pet cat, whoever did it. It's the idea that that's so precious might be called Garfield serum. Yeah. But nothing it's meant to
Speaker 3: From the sciences. I mean, we're at a science festival and I think it's important to distinguish that mathematics builds on the shoulders of giants. I mean, w what that proof is true, and it'll be true forever. And in science, there's a much more evolutionary process. You propose a model of the world and you find it doesn't quite fit. And a new one comes over and knocks that one out that doesn't happen in mathematics. Proofs gives us this certainty. And we, it allows us to really stand on the shoulders of giants and build something new. So that's the charm of it. I know that the mathematical theorems that I proved will last forever. It's a bit of immortality, and that's why I love this subject.
Speaker 2: But, but if you were to, if you were to, I don't know how I asked this. If you were to become a number, are you there Whether or not we're here I mean, are you just there
Speaker 3: is about to all, we play tennis at heart, and I think most mathematicians, I mean, I'm interested to see, but I think most mathematicians feel, you know, a 317 is a prime number. And not because the mind thinks it is, it's just a property of that number. And I certainly feel there's a platonic reality out there called mathematics, which is the, where I spend my time exploring this world. and sometimes it says things about the real world that I live in, but frankly, I'm not interested in that. I'm more interested in just exploring this extraordinary world, which I think has a reality of its own. And when you,
Speaker 2: When you say that, is there, is there, is there a castle Is there like, is this, is this a metaphysic which got like legs I mean, is there like when you, like,
Speaker 3: It has structure and pattern and, and it's, it's a bit like more like, I think, I mean, more like the musical world that it's got the same set of texts,
Speaker 2: Like maybe you are actually more religious than you, maybe you would like to admit, maybe you think that they succeeded
Speaker 3: Richard Dawkins is in Oxford. So
Speaker 7: Hopefully you'll get, say, Oh God, no, but why not
Speaker 2: Well, if you believe that numbers are out there, whether or not we're here, if you believe there is that logic that exists independent of all of us and all of us that ever will be,
Speaker 6: I believe numbers are out there, but I don't believe that's true. All mathematics. I feel I'm inventing most of the time.
Speaker 2: I believe the vending or discovery, I believe
Speaker 6: Any choice about discovering the numbers. And they're modern cognitive scientists making some neurological arguments about that, but that's above my pay scale, but I don't think that to all mathematics, it gets proved, had to be invented. It could just have never been.
Speaker 7: So there's a link there with religion in terms of invention of ideas. And so off I go that, I mean, it's, it's it, you know, these, these are concrete hyper. I'm not the mathematician. So I, when I look at math, I look at with a purpose, when you're looking at maths, you're looking at math because if it's this game that you'll play, it it's, I think it's more than a game.
Speaker 3: I think that know, I think that always, I always resent it being compared to chess because it sort of makes it, you know, I think it's more important than that.
Speaker 2: Well, can I, let me ask different, let me ask it a different
Speaker 4: Way. it, it, it doesn't matter if we know it's true or not, like, is there some math that's just simply undo. Is there a proposition you can think of that is permanently and totally undecidable and therefore, whether we may not ever know it, but nevertheless, it may have its existence independent, as soon as you formalize what you mean by decide to build the answer is yes, there were undefinable things
Speaker 6: And arguably true, and that cannot be proved really. And there are things where this is a safe
Speaker 3: Discovery made in the 1930s by Kurt. Good.
Speaker 4: Oh, when I've heard of this devastating food, a lot of people, very nervous, frustrating for philosophers, but
Speaker 6: Largely ignored for 52
Speaker 4: Ways. You can get around it in different ways. Mean every math problem does not have an answer.
Speaker 6: No, that's why I mentioned the idea of odd, perfect numbers. We've grown up in a culture, which says, if you can ask the question and it's a well formed question, we kind of expect you to have an answer and probably a well-formed answer. And I believe there's no reason why I'll tell you. There are no odd, perfect numbers, and nobody will ever know why. Say that again. There are no odd, perfect numbers. There's no odd number, which is the sum of its proper divisors and nobody will ever know why. So then how do you know it's true. I, I have an article I'm telling you, it's true.
Speaker 4: He's the expert. He's the world expert. So that's religious, isn't it I mean, no, I think, you know,
Speaker 3: Good. We'll prove that within first order logic, which is the logic that we use for doing things like number theory, there'll be statements which are true, which will not be able to be proved within that system now. But actually we don't use first order logic. Mostly when we do mathematics, we use second order logic, third orders.
Speaker 4: But I think it's oversold this disorder. You'll say that's an undecidable conjecture, I'm conjecturing. But certainly what we've seen in the
Speaker 6: Last few years are more and more natural language sounding versions of what are called girdle statements. So the original girdle statements are kind of snakes, swelling, their own tails. imagine, imagine a theorem that says, if you have a proof of this theorem, you have a shorter proof of its negation. Yes. Well, you start working through it. Could I have a proof of it I don't know.
Speaker 4: Well, if you did, you'd have a shorter version
Speaker 6: Investigation and at least according to standard Aristotelian logic, you would have proved something and not something, and all rules would be gone. You could prove anything. So if mathematics is consistent, which we tend to hope it is there. If I could show you that, that statement, if I could prove that statement and what I can't prove that statement, but I could show it's true. This is what the Goodwill and people off trim did. But now we've even got so-called Goodwill statements, which seem to be saying fairly natural things about measureables of physical systems are unprovable are undecidable undecided. And we know one at a very famous things that Paul Kuhn, one, a fields medal for many, many years, 50 years ago, it was a what's called the continuum hypothesis. And it doesn't matter precisely what it is, but it asks whether there is a set essentially, is there a set bigger than the natural numbers and smaller than the real numbers And it's now been shown to be independent of the normal rules that mathematicians are willing to accept is God three in one or just one, but the different, well, that doesn't bother me. It bothered Augustine. That's the difference. And the difference
Speaker 3: You want the differences. And this is really important is you haven't defined your terms. And in mathematics, we're very good at defining our terms, our axioms, and w how are you going to use one thing from another, but you've just introduced something gold, which, you know, you haven't defined, so we can't stop
Speaker 4: Ruben. I can not pull it. Isn't that kind of a gurgle light kind of went through his death
Speaker 6: Among other things in his notes is an attempt to logically prove the existence of God. Goodwill didn't really believe what he'd done. He felt that meant mathematics wasn't properly fitted together.
Speaker 4: Well, let's talk about you guys. What do you do when you go to work Do you like sit in the chair or like, you're not like other dads, you don't go to the office. I wouldn't think you do, but maybe you don't like, do work like behaviors. So let me ask it a different way. Do you do this alone, your work, or we've been recorded I'm not, I'm putting into the end of the hall. Now. It is at work is my new subject. How do you do it A very famous mathematician once actually characterize mathematics as something that you lay down, close your eyes and work like hell. And if you could say that and mean it, then you're a mathematician and this of course means you're doing it by yourself. Actually more fun. A lot of mathematics is so historically it was done by people on their own. And very often it is, but these days it's increasingly becoming a team.
Speaker 7: Okay. There's a lovely website. Just started in the last couple of years called polymath where the field's met, Tim, Tim gal is in Cambridge. So he throws out this problem on a blog. And he just says to everybody in the math community or any other community, if you can contribute to trying to solve this problem, add a comment and they've solved at least one quite significant problem by just people throwing in ideas and bouncing off each other. And that's almost the opposite extreme of this idea of the lone genius.
Speaker 3: You don't think, you know, your, your book about Andrew Weil's proving Fermat's last theorem. That was such a prize that he, he didn't talk to anybody else because he wanted to be one of the ones to prove it. And yeah,
Speaker 7: Andrew was is, is an extreme case of somebody who who's a problem, 350 years old and as well, much less theory. That's right. So he 350 or so, he said, the more that people fail to prove it, the more it became desirable. And when Andrew Weil's were based in Princeton, by that time, once he realized the technique that might get him there, he close shop. He largely worked at home. Didn't go to the department very much.
Speaker 2: He was scared that if he whispered his process to someone else, they'd say,
Speaker 7: I think, I think it was partly, it was fear of, of being beaten to the prize, right To the fear of being thought to be crazy, because if no one else has done it for 350 years, why does he think he is so smart And also just the intense focus that's required to take on such a major problem. And, and for seven years he did nothing but work on this problem. But on the other, roughly the same time you have somebody like Paul air dish, who is an extraordinary as a wonderful book called the man who loved only numbers very, very much.
Speaker 2: Was this the fellow who had ring your doorbell and saying, I'm open your mind. I'm here for Monday.
Speaker 7: I collaborated with 500 mathematicians during his career published over 1500 papers. And
Speaker 2: That was the Willy Loman of math that he would just show up as the salesman go, ding dong. And he then spent two weeks with you and you have to give him, yeah,
Speaker 7: He was here once with Ron Graham on the East coast and they'd got up early and he only had two suitcases that was the entire world possessions. And he was with Ron Graham and he said, Oh, we're stuck. You know, we, we don't know what to do. Ron Graham said, it's okay. I know a guy on the West coast who can solve this problem and just said, well, let's bring him there. And Ron Graham said, look, it's 8:00 AM in the morning in California, it'll be 5:00 AM. And then I just said, well, that's great. That means he'll be in it for him. It was purely mathematics. That's where the whole world revolved.
Speaker 2: And, and, and which is the more common way to do it by yourself or, on the phone or online, or, ringing the doorbell or I think more and more, it's uncommon to be entirely alone. Yeah. Uncommon, uncommon.
Speaker 3: It's a combination. I mean, it's, I find what I do best is to work on my own, then go and share ideas, bounce, and you get much further and you come back and you need to spend, and I think there's some other conferences, right We go to conferences, we go and visit each other. We, I find maths quite difficult to do online because, by email, because actually, I think there's a lot of unspoken, sort of, trying to verbalize an idea in your head. You haven't quite got the language. There's a lot of with my collaborators and I can't do that by email and somehow my collaborative. Yeah. I think I see what you're seeing and then we can start to formulate,
Speaker 2: But I, I would, we do quite a lot of that with Skype or, or Skype plus. Absolutely. And it's really a matter of how good the broadband is. But if you have a problem, if you have a problem, that's, you know, 13 pages in the figuring and you have this fragile sort of thought in your head and it's got all these attenuating numbers. So you go, Oh my God, man, you're sitting at a bar with someone and he's got his problem, which is 30 pages long. I don't quite understand how that conversation because
Speaker 4: I there's a tension here. I don't want to hear about your confusions. I've got my own.
Speaker 6: I have a good friend category theorist, which is not my brand of mathematics. And in fact, there's a Columbia is going to make his mathematics in many ways is a single subject. But in other ways, I often have much more in common with certain parts of physics or engineering than I have with other parts of mathematics because of the way I approach my subject. And if I don't happen to speak to given the sub subject, just haven't been taught enough. It's just as alien as if somebody's speaking Russian and I'm trying to speak Italian. But I had a friend who was a category theorist and his first wife left after 10 years of hearing all the category theory, anyone could hear in 10 years, it didn't matter that she didn't have any, he still wanted the audience. And a lot of mathematicians have that experience that in the, in the need to articulate it, it really doesn't matter often if you're getting answers, because you've had to say
Speaker 4: So much more clearly, do you guys have pencils or talk about, maybe we don't have, just give me the pencils and shop you work with a pen. You write something down.
Speaker 3: Yeah, yeah, yeah. You're trying out sort of, do you all have computers No. Computers are not something that I was like unkosher no, no, no, no. It depends on your sort of mathematics
Speaker 6: That might be at the other end. I'm a real passion for using the computer as a, as a colleague. So I'd be firing thousands of questions. And I, and the next thing I'm going to is a conference on symbolic numeric computing, mainly to try and make the point that I think we've largely restricted what we use the computer for because of the way we were trained and the way computers used to be. But any piece of math that I'm showing, I can usually make more progress at a computer without it. And part of it's because I can Google because
Speaker 4: I can Google. I mean, there's an enormous amount. Google what Well Googled the math literature, pull out a paper from 1940. If I know how to do it.
Speaker 3: No, but I think there's this kind of misconception that the computer has kind of put the mathematician out of business because of course it can do arithmetic much faster than I can. But, but actually if you're trying to explore, you know, these infinitely many prime numbers, well, a computer isn't much help because there's a finite beast and it can't have the same sort of, thought processes yet as I do. So I don't find it a very, you know, it can find another big prime number, but that's not very helpful in trying to understand the pattern that underlies all of these prime numbers,
Speaker 7: All your questions about do people work alone with these computers. It, mathematicians are no different than anybody else at this world science festival, except in the group we have here in their extreme purity, in the absolute abstractness and pointlessness of what is done and yet, and yet every not just every so often relentlessly something pops up that has a major application back in the real world. Whether you know, encryption is an example where prime numbers are fundamental. And yet you pry, you find that you study prime numbers for the, of it,
Speaker 5: Fair miles, little theorem, again, applicable to, to, to cryptography, but invented by a pure mathematician 350 years ago, the tools, the community, everything is the same, except the utterly attic. If you think string theory is abstract, you should think about the mathematics that underpins it.
Speaker 2: One last computer question. Do you know what the computer, if, if you ask the computer, so how much of double double the blood and it goes thing and give you an answer. Do you know how it reached that answer
Speaker 6: Sometimes the answer is yes. In principle. Sometimes the answer is no, but ideally what you have is a certificate. You can have an answer that comes out and it says, if you don't believe me, check this, that's an ideal sort of answer where you don't know how it found it, but there are many cases in which checking an answer. Once you're told it is much easier than finding it really Yes. And those are the kinds of cases where, where the, where you feel like you've won because not only did you get an answer, you couldn't find, but you get an easy way to check.
Speaker 2: Okay. Well, let me ask about Gregoire Perlman, Keith. this is a very weird man, but he, his problem is something called the plank, Herve, conjecture. What is that Who is he Who is Gregoire
Speaker 5: So he's a very unusual case. We're going to be up. We're going to be talking about a very atypical case, all a very famous one. But the point career conjecture goes back to the beginning of the 20th century, Henri Poincare. And one way to think about it was think of the following question. You go back to ancient Greece. The Greeks lived on what seemed like a flat earth, but by using mathematics, there were able to figure out what the world was. In fact, spherical, and even calculated the diameter with good accuracy. So using mathematics, you can step out of the world, you live in and see what it must look like. We didn't look at the, we didn't actually see that the image of the world. We didn't know with our eyes, that it was vertical till NASA sent spacecraft. We took photographs, but the Greeks had figured it out by using mathematics.
Speaker 5: Could we as creatures now, living in a three-dimensional world, could we understand the shape of the universe we live in by using mathematics So as it were step outside the universe, for example, the universe may be like the inside of a sphere and that you could move around freely, or maybe it's like the inside of a big inner tube that you can keep going round around or something more complicated, like a pretzel. So there were lots of ships. The universe might be, could we figure it out using mathematics Planko air came up with a method that can be understood in those terms. And here's what plank reading. We've actually got a little video. We can show about the conjecture because it's on their Pelham and actually was the one that moved this 2002. So if we can roll up to it, let's imagine we want to understand the shape of the universe is in.
Speaker 5: So we got to get in a spacecraft and we're going to go out and we're going to start splaying out a rope behind us. We're going to go all around the universe on this tour. And we're eventually going to come back again and create a big loop that will track the path that we'd followed. So here we go round, and it's a big world. It's going to take us a while to get round, but we're going to come back. And when we come back, we're going to be able to find the tail that we left behind the end of that rope. And then we're going to start to pull the rope tight. And when we start to pull it tight, one of two things can happen. You start chugging or it stopped. Maybe you're in the inside of an inner tube. When you can't pull it any tighter or the following might happen, you pull it and pull it and pull it pulls down.
Speaker 5: So those are the two possible ways it could come about. If you can't pull it together, you know that the universe is not like the inside of a sphere. It be, it's more like an inner tube or something like that. If you could always, if you did this infinite there, when it's abs, if you could always pull it together, the plan career conjecture says, then you will be able to conclude that you are indeed on the inside of a sphere. That's three sphere. If you can always do that, this is not practical in the way that the Greeks were. But it does say that with mathematics, there is no limit to the ability of what we can find out with mathematics. In principle, we can find out the shape of the universe we live in without stepping outside of that universe, we can look through the eyes of God if you like it, our universe using mathematics. And Pellman proved that in 2002, although it took seven or eight years before the rest of their mathematical career, which one did he proved That was the, that you can in fact find the shape of the universe by going up one things. If that way, if you can always,
Speaker 7: Well, it's again a strange guy though, because he loves a prize.
Speaker 5: It didn't want the million dollar prize. It didn't want, it didn't want the fields metal. He's a very unusual guy. I've never met him. Very few mathematicians have cos he's very reclusive. that made it a great new story. On the other hand, the mathematical discovery itself was the front-page story on science magazine, the mathematical discovery of the years, the science discovery, the science discovery of the year. Yeah,
Speaker 7: But I, I'm not so sure it's so atypical. No, I don't think so. this is one of the seven great problems in math in the year 2000, the clay foundation said these are the seven grade problems in the new millennium that mathematicians should think about. And the Poincare conjecture was one of them and it's been the first one to be improved by Perelman. There's a million dollars for it are. They rang up parallel and they said, Hey, do you want the million dollars He said, no, I don't do the math for the money. So then they offered him the field's metal, which is the math equivalent of a Nobel prize. Even better than a Nobel prize because you only get it every four years. And they invited into Madrid to collect it at the international Congress of mathematicians. And he didn't turn up and they rang him up and they said, look, why didn't you come and get the fields, man He said, I live in Russia. We're taking a data flight to Madrid, data, collect the prize, dates, do the interviews, data fly back to Russia. That's four days wasted when I could have been doing maths and air dish to get air is exactly the same. I think that's a very description
Speaker 6: Of a paramount.
Speaker 4: Yeah. By all accounts, she's a little more Australian than that. Can I ask it This is a delicate question. And then I'm going to do something. This, this conference is this festival. Doesn't usually let questions come from.
Speaker 7: I'm really curious about this because very few people have met him and he is an ultra busy. I would say that he, that this absolute purity and dedication like air dish, air dish won the Wolf prize. 50,000 pounds with 50,000 pounds was worth a lot. And he didn't buy a hat. She never had never had a house in his entire life. Instead he offered the money as prizes for other people to solve other matters.
Speaker 6: Most of you don't know Perlman, but I knew Erdos quite well. I first met him when I was one. And
Speaker 4: When I was one that you remember it clearly I'll tell you the strange thing about Perlman though, is that he stopped doing mathematics. That's a strange thing. That is, you know, that is a total lie. Can I suggest that there may be just a hint of Mr. Asperger in the rug Oh, there's a lovely
Speaker 6: Recent book. I think it's called perfect rigor.
Speaker 4: Perfect rigor. Yeah. It is
Speaker 6: Essential description of a high achieving ultra Asperger and a very, very interesting description of the sociology of how math kids going to the Olympiad were being trained in a position in a time in which there was tremendous antisemitism. So two places in the elite university system for Jews a third, if you got a gold medal on the Olympiad team. Yeah.
Speaker 4: The reason I said he was atypical was I know thousands of mathematicians and hardly any of them are like that. They're regular people. Yeah.
Speaker 7: Hate to think that as a journalist who meets lots of mathematicians, lovely people, lots of
Speaker 4: But what I'm saying,
Speaker 7: What I'm trying to say is that two of the greatest discoveries to the grace mathematicians of hour of your time, your generation have utterly, utterly bizarre attitudes to the way they live their lives. And in science, in physics and cosmology, there are people that are dedicated and obsessed, but not to that extreme. So we're looking at the very long end of the tale, but it's a tale that doesn't exist. Even in other abstract
Speaker 4: Newton you in the first room, Mr. Newton fan. But he wasn't one of these guys who would just kind of walk around and get so lost in thought he would draw things on the path. And no one, no one seemed to understand anything he was saying. He was an odd duck also. Right Yeah.
Speaker 7: Robin wins told a story about Norbert Wiener the other day. I don't know if it's true, but the Wiener founder had just moved house. And so he was in his new house and he went off to the office and they said, look Norbert, remember where we live Remember you've moved house. So went off to work, came back and went back to his old house. I was sat on the doorstep, really depressed. Didn't know what He didn't know where to go. And it's little girl came up to him. He said, Hey, little girl, do you know where the Wiener family live And she said, yes, daddy is just over here. So on that note,
Speaker 4: You guys want to ask questions of the panelists to see, Oh, well we're only at five minutes. Well, what the hell the night is young. Let's just do the, I know, I don't know where they're going to turn up the lights. Cause this is illegal. We're not supposed to let you ask questions, but what is, so yes, you'll have to shout them, but you're in the front row. Go ahead.
Speaker 8: And I'll repeat. I I've heard that. Just like, you reach a peak at age 27.
Speaker 4: The question is, is athletes sometimes reach their peak very young, say 26, depends on the sport. What about, what about peak math years now That's false.
Speaker 6: It was a great article in the New York, weekend section a few years ago called a young geniuses and old masters, not about mathematics, but about art and saying don't try and compare Picasso to teaching. So I think it's absolutely true that that, radical ideas tend to be the work of younger mathematicians. And maybe if you've not done anything earth shaking by the age of 35, it's unlikely you're going to do something fantastic later, but a recent book by, Ruben Hirsch called loving and hating mathematics. And one of the, the cliche is that he tries to break down is just that. So I think like in any other subject, you don't do the same things at 60 that you do to 20, or if you do, it's probably been wasted. I don't know, very many really creative mathematicians who say, Oh, I'm 35. It's time to stop.
Speaker 4: Did air dish when he rang the bell and he was like 60 and he was ringing the bell of a 24 year old. Did the 24 year old, the old guy, or did the guy go, Oh, I'm so glad to see him.
Speaker 6: One night, my mother called me when you were two.
Speaker 7: No, I was
Speaker 6: I was 29 and I was actually naked in the dark in bed. And how Vikas and I thought only one person would call it this time of the night. I picked up phones on my mother and she said, somebody would like to speak to you. And a little voice got on the phone and said,
Speaker 7: Hello, this is Paul. Do you know the question with 29 points in the plane
Speaker 6: And it was just continuing a discussion. We'd had several years earlier. It actually had,
Speaker 7: He'd had it with
Speaker 6: My brother. I realized after a while, but I thought, I didn't need to just disabuse him about that. But no, he was completely a Galatarian his mind was open. He wasn't a mathematician of the same, status as the greatest mathematicians of the 20th century. But he was maybe one of the most influential because he went everywhere and I was getting a little grouchy about comparing him to Perelman. he was an odd man out here to share dosh. Yeah. But he was odd in idiosyncratic gentle ways. And, he didn't have many needs and that's largely because people like Ron Graham look,
Speaker 7: But, but I would say, not a tool or in the heat find the thing he loved, which was doing math. And
Speaker 5: If you, if you love that math, maths, that much, then any other behavior would have been odd. So that's the way I kind of do this in swimming. He was in the pool all the time, because he just loved being in the pool. You read about the ones that produce the pinnacle results by definition, they are unusual. And that often manifests by the way. I think that was the other thing with us. They ask one more time. that waving person. Yeah.
Speaker 8: I was wondering, is there any hope for those of us who fell off or were knocked off or from the prosaic of the sublime the appropriate time found an interest in later
Speaker 5: What if I was good at the beginning and then I wasn't good enough. And now I'm still really interested. Do I still have a prayer Right
Speaker 5: Except that I wasn't really that good. Yeah. I think there's always a chance to restart the game because it's such a logical subject. You just have to take yourself back to the point where, you know, you understand and start again, going down the path that the mathematicians have laid out for you. So I think if you've got determination and this is why, you know, people carry on doing mathematics, however old they are, because they've got a passion for wanting to know the answer provided. You've got that passion. It doesn't matter when or what age you were when you started. If you want to know what the next step is, you'll be able to get there.
Speaker 6: One of the things that seems to be determined about genius in general forms is you see it with Newton and others too. The capacity to actually not leave your desk for 24 hours independent of Saudi. I was going to say that I think the answer is a little, like I wish I'd learned another language when I was six. It's a lot easier when you're six than when you're 36. So the chances that you will learn it with the fluency that you would have learned, if you hadn't had the interruption a zero, but, but the chance that you can learn it to do it well enough for it to be a source of great pleasure and accomplishment. That's a matter of persevere.
Speaker 5: A good starting point is Martin Gardner's books and recreational math. And that's a good way just to get, get into the swing of things and just to learn some of the curious math that's around, it's getting easier because of the internet and the accessibility. There's the Wikipedia resource it's actually Wikipedia in mathematical areas is really pretty darn good. The higher you will pick. It's very reliable, the materials and I every week or so, because I've got an MBA. I was a math guy. I get emails from people with that very same question. You can reach people and we're not going to give an hours time in a response, but we'll say, try this book or try to talk to this person at your nearest university. It's actually relatively easy. Now, you know, if you're a mathematician, you actually, your heart lifts. If someone says, I'd like to get back into mathematics, we usually spend a few minutes trying to help them out. We have so few friends, we need one. We can get two more and then we'll yes. Over there. Yes.
Speaker 9: I think, I dunno if I experienced that when I look at like a science textbook or like physics or chemistry or something, that's more like a story I can sort of read through and say, okay, get this, get that. And then I look at fat and the first step starts out. let's assume this is a product you get to the end and say, well, it's not prime. So this thing is false. Wait, did we assume that it was a prime number Like I can't remember what the assumptions that would be. So how do I read a math textbook compared to something like
Speaker 2: The question is, if you read a science textbook, it seems like storytelling and you can get involved and you get interested and you want to know how it's going to turn out. You read a math textbook and you go into it and they make assumptions. And you can't remember whether you made those same assumptions and suddenly you are brought to the area. There's the answer.
Speaker 6: The comparison there would be like reading a cookery book of an ultimate, big of a pen by Bunuel. All four of us have written books about mathematics, which tell the story. That's why we have all the stage of costs. But I think
Speaker 3: The proofs that you find in books, you know, for them as proof that, every prime, which has remained a one, when you divide it by four can be written as two squares. It's a fantastic story. And how you get from, you know, something like 41 to 16 plus 25 and whatever the prime is. If it's got remainder one on division my fall, you could always write it as two squares. It's an extraordinary story. And I want, I don't want to just know the answer. I want to know how did this, this hero gets from here to here.
Speaker 2: It's my suspicion that the answer to your question is some people like I once went to breakfast with my boss, Larry Tisch at CBS, and he brought H Ross Perot was then just, tycoon. And they, they sat around and they opened up the wall street journal. Was it a breakfast table at the Drake hotel And they, they looked at these just sheets of numbers and they would point and they giggle and they gave little, I like, I guess we're going to make him this guy. So, no, I, I, but I thought to myself, wow, these people read numbers the way I would read ASAP. You know, it's just, they just are different.
Speaker 9: There'll be,
Speaker 2: She, she can't make the step.
Speaker 6: I think you're identifying a real problem that Marcus mentioned early on, which is that mathematical is a massive mathematics is a very large scaffold and, and hierarchical. And it's hard to learn something before another thing. And so we've tended in writing textbooks to look for very efficient, very focused proofs and kind of trust that the reader knows why they want to be there. And that's not always a particularly good solution. The alternative and a mathematician called Moore in Texas, tried to do the opposite with great students, which is, do motivate every step, prove nothing. Get the students to prove everything from cells. And in theory, it was a fantastic idea in practice. It meant that by the end of a four-year degree, you were in semester two last
Speaker 2: Trade off the evening. see, I hate to do the physical in the front. So Hey, is there somebody in the back way in the back that's naked arm up
Speaker 1: There. Yeah. Good now. Yeah.
Speaker 5: And hear that. The question is, is it because of the good ELLs, this uncertainty that he introduces, Oh boy, am I going to have trouble Did you say non axiomatic, maths I mean, it's almost one of these things. What is mathematics Because you can define mathematics. And for many years it was the stuff that follows on the axioms, but most of what mathematicians or the living today do is actually not axiomatic. It's just exploring things in an analytic way, thinking rationally, it's not based on those actions of this sort of pure axiomatic mathematics that sort of has a history more comparison. But if you just actually say of all the people who earn money by doing mathematics, hardly any of them do ask them. I think, mathematics, they deal with problems that arise in the real world or within mathematics, but not in an axiomatic framework that was for axiomatic mathematics, or was he this innocence girdle prove that it was a pointless trying to do that because he wasn't going to capture everything how we saw. Have we answered your question
Speaker 1: You're welcome. Thanks everybody for coming.