How a Letter Between Two Mathematicians in 1654 Changed the Way We View the Future

MAUREEN CAVANAUGH (Host): We may not be able to predict the future, but we can develop a pretty good idea of what's likely to happen. We do this by examining the possibilities and comparing them to what's happened in the past. This idea of calculating likely scenarios may seem obvious, but that's only because we live in the modern world. It took a handful of brilliant mathematicians in the 17th century to devise a method of figuring out likely events. Two Frenchmen in particular, were key to the creation of Probability Theory. Their work is the subject of a book by NPR's "Math Guy" and Stanford mathematician, Keith Devlin.

Tom Fudge interviewed Keith Devlin about his book last fall. Here's that interview.

TOM FUDGE (Interviewer): I'm Tom Fudge. You're listening to These Days. What's going to happen next week? Will it rain? Will you get in a car accident? God knows, and we don't. But while we may not be able to predict the future, we can develop a pretty good idea of what's likely to happen. We do this by examining the possibilities and comparing them to what's happened in the past. Therefore, we may be able to say that there's a 1-in-20 chance that we'll get some rain in San Diego next week. If you're going to the casino tonight, you may have a 1-in-5 chance of actually winning any money. If this kind of thinking seems very ordinary, that's only because we live in the modern world. There was a time when Probability Theory was not part of the human mindset. That changed when a handful of brilliant mathematicians in the 17th century figured it out. Two Frenchmen, in particular, Blaise Pascal and Pierre de Fermat were key to the creation of Probability Theory. Their work and their correspondence is the subject of a new book by Stanford mathematician Keith Devlin. Keith Devlin is known to Public Radio listeners as 'the Math Guy' on NPR. He's also co-founder and executive director of Stanford University's H-STAR Institute. The book that he's just written is called "The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern." And, Keith, thank you very much for coming in.

KEITH DEVLIN (Author): Tom, thanks for having me back again.

FUDGE: Well, what is Probability Theory? What is that?

DEVLIN: It's assigning numbers to the way the future might turn out. The standard description we give our students of science is you assign numbers to the world we live in, like numbers temperature, forces, mass and weight and so forth. And then you do science by studying the relationships between those numbers. In Probability Theory, we assign numbers to things that might happen in the future and then we try to understand the future by looking at the numerical patterns and relationships between those probabilities.

FUDGE: The interesting thing about the story you tell in this book is that these guys, Pascal and Fermat and a lot of their friends, seem to enjoy gambling.

DEVLIN: Yeah, indeed. The Probability Theory, the stuff that is behind most of modern life, from predicting the weather to actually having mortgages in the first place, and a lot of the present trouble in the mortgage crisis is that people started to ignore those mathematical models. But, yeah, the – having those models—and we've only had them for three or four hundred years now—is really the basis for an awful lot of the way we live our lives today. We just take these things for granted.

FUDGE: You mentioned the financial bailout and the financial risk that comes from the mortgages, how would you say that Probability Theory applies to what we're talking about in Washington right now?

DEVLIN: Well, you know, to me, as someone who knows about mathematics and Probability Theory, and it is a very interesting story how that came about and let's come back to that one, but to me the interesting thing is that mortgages exist in the first place. I mean, you think about it. People of ordinary means accept the fact that we can buy houses and it's going to take us 25, 30 years to pay it back. Why would anyone in a competitive financial market lend you money that's going to take you 30 years to pay back. The answer is, they can use Probability Theory to put likelihoods on what may or may not happen in the future and to lay off the risks that they may be exposing themselves to. And, therefore, to the people who lend money, if they follow that mathematics of Probability Theory, it's a reliable way of making a living. And we're so used to doing that now that we forget that it's really rather remarkable. And prior to the middle of the 17th century, such a thing not only was not possible but was not even thought to be possible.

FUDGE: It's interesting that we're – well, we were talking about gambling and…

DEVLIN: Yeah, right.

FUDGE: …the fact that gambling was very popular among these mathematicians that you write about. But it's only been the creation of Probability Theory that has allowed casinos to say, okay, we're going to make money.

DEVLIN: Yeah, yeah.

FUDGE: And we know exactly…

DEVLIN: Absolutely, yeah.

FUDGE: …how much money we're going to make because we can put a – use mathematical calculations to figure it out.

DEVLIN: Right, yeah, and I jumped into the mortgage thing a minute ago because that was your previous guest and I was listening to the show…

FUDGE: Yeah.

DEVLIN: …myself. But, yeah, to get back to the fact that it was about gambling. It was Pascal, in particular, had several friends who were keen gamblers and it was because of the desire to win money in the gaming rooms of Europe that they got mathematicians to work on these problems and, ultimately, to develop these methods. And it's just as in the present day, the financial markets—and that's another form of gambling—insurers are basically gambling on your lives and your automobile skills and so forth.

FUDGE: Yeah.

DEVLIN: They also employ mathematicians to work on the mathematics so that they can make a reliable living selling insurance. It's still gambling, it's just in one case it's in the gaming rooms with dice and roulette wheels, and in the other case it's an insurance company with your life, my life, and our automobiles.

FUDGE: Prior to the days of Pascal and Fermat, what did people think of the future? Did they just think that it could not be predicted in any sort of way?

DEVLIN: Absolutely. Not only did they not – did they think it couldn't be but they wrote. There were many quotations I've got, and some of them are in my book, from mathematicians who said you simply could not do this. It goes back to the ancient Greeks. And for 2,000 years, people thought that you had no handle on the future. You just had to accept what was going to be: What will be, will be. There's no way of offsetting risks or for making decisions to try and minimize risks because you simply had no handle on the future. It was really a watershed event when Pascal wrote this letter to Fermat, and we know the exact date. It was Monday, August the 24th, 1654. This is one of the few times in the history of science or, indeed, life in general, when we can pinpoint the moment when the switch changed from thinking that the future was in the hands of the gods, nothing you could do about it, to actually having a way – a means of precisely saying what the relative likelihoods are in numerical terms of what may or may not happen. That was a huge watershed moment, and that's why the subtitle of the book is 'it made the world modern.' The modern world depends on that little bit of mathematics that was in this 3,000 word letter.

FUDGE: Keith Devlin is co-founder, executive director of Stanford University's H-STAR Institute. He's a mathematician and author of a new book called "The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern." If you have any questions about Probability Theory and how we use it, how to fix our lives, you can call us at 1-888-895-KPBS. By the way, Keith is going to be signing copies of his book this afternoon, at noon, actually at noon, at the UCSD Bookstore. For more information, go to the These Days page of KPBS.org. What kind of a time was it when Pascal and Fermat were living? It sounds like it was a very exciting time in terms of the whole development of mathematics.

DEVLIN: Yeah, it was exactly the same time as calculus was developed. And, in fact, Pascal – well, Fermat, in particular, was involved to a great extent in that. And, in fact, what happened was the people who were working on calculus, which, again, is middle of the 17th century, beginning with Newton and Leibniz, many of them also gravitated very quickly into this new area we've been able to calculate, probabilities. And so it was almost like being in Silicon Valley in the sixties. There was so many bright people around that one idea was quickly by – gave advance to many more. In that very brief period of 20 or 30 years in the middle of the 17th century, you have calculus developing, all of modern science and technology on its heels, you have probability developing, all of modern financial risk management techniques developing. And, in fact, as I make clear in the book, you've only got to go ten years beyond the writing of this letter and you've got modern society. You have insurance, annuities, the ability to lend money precisely, to predict the future. It was in a ten-year span modern statistics began, modern actuarial science. This all came in a huge rush on the – immediately after this letter, within 10 to 20 years. In fact, in 50 years of this letter, modern society was in place.

FUDGE: And this was a time when some of the associates of these people are – were people like Rene Descartes, Leonardo da Vinci…

DEVLIN: Oh, yeah. Yeah.

FUDGE: …this was the Renaissance.

DEVLIN: This was the huge time when everything happened and it was a huge explosion. It's sort of remarkable that you've got these two explosions at the same time, the one that gave us modern science and technology, which is calculus, exactly the same time as Probability Theory, which gives us modern financial risk management. There was no real reason that – the mathematics is different. There was no real reason why they had to have happened at the same time. They just did.

FUDGE: Tell us the story of this correspondence and this conversation that Pascal and Fermat were having. It sounds like they were talking about playing a game of dice.

DEVLIN: Yeah, it was a game. There was a problem that went back several hundred years called 'the unfinished game.' (That's the title of the book.) And that was where two players are gambling with each other, they're throwing – it doesn't matter what the game is, the game they looked at and the one I discuss in the book, is throwing a pair of dice. And it's a game of best of five throws. And so you've played the game and one player is ahead two-to-one and something happens, they have to call off the game before it's finished. Now they've all put money in the pot, they've partly played the game, the question is, how do you divide the pot fairly based on the positions they have in the game? So it's an unfinished game, no way of finishing it, divide the pot fairly. That had been around several hundred years. Many of the best mathematicians around had tried to solve it and failed. Pascal got ahold of it and tried to solve it. He got a sort of solution; he wasn't sure he was right. And then he tried another method, it seemed to fail. He then corresponded with Fermat and Fermat, who was one of the best mathematicians the world has ever seen, we don't know how long it took him because this was a – there was a correspondence went over several months between the two of them. Some of the letters have been lost. The letter I concentrated my book is the one that really nails the solution. The letter is written by Pascal but the solution he describes is actually one that Fermat produced. Today, we can teach that solution to high school students after a couple of hours of talking. It's really simple. But you read that letter, two of the best mathematicians in the world, are struggling to understand this. There's a little subtitle of the book, it's called 'a tale of how mathematics is really done.' And for – to my mind, one of the most interesting things is that this story – you know, it's said that if you like sausages, you shouldn't see them being made. If you like law, living in a legal society, you shouldn't see laws being made. We tend to think of mathematics as being this pristine stuff that sort of is – it comes down on tablets from Heaven.

FUDGE: Uh-huh.

DEVLIN: No, it's developed in a very messy, stumbling, fumbling way by mathematicians. And one of the things that fascinated me about this story when I looked into the details, was how these two great mathematicians make mistake after mistake after mistake. They stumble, they go back. It takes them a lot of effort to produce what today a high school student can understand in a couple of hours.

FUDGE: Well, let me go back to the solution that you describe.

DEVLIN: Yeah.

FUDGE: What was the solution? Was it a matter of figuring out how many different possibilities could happen in the future?

DEVLIN: Yeah, it was literally a way of modeling the future. You're saying, how many – let's enumerate all the different ways the game could have come out and then we'll count the ones that Pascal, that one player would have won and then we'll count the ones that the other one would have won and that's how we'll proportion the pot. It's simply who wins more games than others of the unfinished – of the games that have yet to be played, the rounds that have yet to be played. Now that seems to us today such a natural thing. We think, well, how come these guys didn't see that? Well, it's only natural because they showed us to look at the world that way. We now look at the world in terms of looking at the ways it might turn out and counting the good ones and the bad ones. You know, you look at tomorrow, nine out of ten days that are like today, it will rain tomorrow. One out of ten, it won't. So you don't organize a barbecue because there's a 90% chance of rain.

FUDGE: Uh-huh.

DEVLIN: That's the way we live our lives. Prior to 1654, people didn't live that way. In fact, people thought it was impossible to do that. The real stumbling point that prevented Pascal and Fermat and all of the mathematicians before them from getting the right answer was they thought it was an impossibility, that you simply cannot count things in the future. You cannot have that access to the future.

FUDGE: I think you've made mention of one of the great Greek intellectuals. Was it Aristotle who said there are three…

DEVLIN: Right.

FUDGE: …kinds of events, events that have occurred, events that are predictable, and events that are unpredictable?

DEVLIN: That's right. Yeah, there was certain knowledge and then there was reliable knowledge—there was necessary knowledge—and then there was reliable knowledge and then there was this unpredictable stuff. And the unpredictable stuff really was unpredictable. You're wasting your time, they would say, trying to develop a mathematics of the future because it's just not doable.

FUDGE: But it never occurred – Prior to the Renaissance, it never occurred to anybody to say, well, we could look at the past, assume that the future is going to behave like the past, and…

DEVLIN: Oh…

FUDGE: …make some calculations about what's going to happen?

DEVLIN: In fact, the only thing you could do was assume that the future's going to be like the past. Now the trouble, of course, in a gambling context is we know that someone's going to roll the dice but what the past tells us is we don't know what's going to happen in the future. They didn't seem to have realized that there are actually regular patterns when you repeat these things. They had some sense of that but the idea of looking into the future really did – I mean, it was just literally regarded as something impossible. So it was one of these cases where the big breakthrough was hindered because people thought it was impossible. Once they got beyond that, they realized, oh, not only is it possible but it's really fairly straightforward. And so today we do teach elements of Probability Theory to high school students.

FUDGE: Let's go to Ian in Solana Beach. Ian, go ahead, you're on with Keith Devlin.

IAN (Caller, Solana Beach): Good morning, Tom, and good morning, Keith.

DEVLIN: Hi, Ian.

IAN: My question relates to the development now of applied stochastics and Markov's theory which is now possible with very, very high speed computers that we have today.

DEVLIN: Umm-hmm.

IAN: For listeners who may – who are not familiar with Markov's theory, essentially you can calculate the area of a circle by throwing darts at a dartboard and count the number of times it hits the circle and how many times it goes outside, and that way you can solve a problem by throwing the dart millions and millions of times to solve the problem.

FUDGE: So if you need an answer, you can write a bunch of answers on a board, throw a dart and maybe it will right…

DEVLIN: Yeah.

FUDGE: …land on the right one. So, Ian, what's your question?

IAN: My question is, your guest's comment on the growth of applied stochastics similar to game theory but a different kind of branch of mathematics to solve problems that were heretofore unsolvable because of the advent of modern computers.

DEVLIN: Yeah, yeah. I think it went in many directions. One of the – Several of the people who did the early development work in Probability Theory, it was a family of Swiss mathematicians called the Bernoullis and at least three or four of them actually got involved. They were involved in developing calculus. They got involved in developing Probability Theory. And one of them particularly, combined the work in Probability Theory with the work in calculus looking at the motion of fluids and, in fact, a lot of the modeling that's done now, the stochastic modeling that's done now, assumes a sort of – a model that's based upon fluid flow where you've got sort of random flows and you're assigning numbers to the way things go, a lot of the modeling in the financial markets assumes that money flows like rivers, like water down rivers. And so what Ian was talking about is one of the many ways of applying the patterns of Probability Theory to the world that came tumbling onto us immediately after this letter was sent.

FUDGE: Aside from Pascal and Fermat, you talk about a lot of very interesting, very colorful characters in…

DEVLIN: Yes, right.

FUDGE: …this book, and one of them was an Italian who became known as the greatest doctor in the world, or at least the greatest doctor in Milan. And who was that and what did he contribute?

DEVLIN: That was Girolamo Cardano who was – he was actually the first person that developed some mathematics of gambling. He was the person that showed if you roll two dice you have to multiply the probabilities together, and so he was the one that developed an earlier – this was just prior to the Pascal-Fermat correspondence. He was the one that developed some of the beginnings of what's – what we now call Probability Theory, although that term didn't come until long after Pascal and Fermat, in fact. He actually did try to solve this problem of the points and failed, the unfinished game problem. He was one of the people who failed to do that but he was an extremely interesting character. In fact, interesting enough, after I'd finished writing this book and I was talking about it to people, I was chatting about it to someone I know who's a screenwriter in Hollywood and he says, this sounds like the grist for a movie. And I actually sent him an advance copy of the manuscript and he got back to me within days and said, boy, this is an exciting story but the best story might be this guy Cardano because not only did he do some great work but he was a multi – he was a poly-math, he did medicine and various things. He was a very, very interesting character.

FUDGE: Well, good luck with your movie. Keith Devlin is author of a book called "The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern." He's going to be signing copies of "The Unfinished Game" this afternoon at noon at the UCSD Bookstore. Keith, thanks for coming in.

DEVLIN: Thanks for having me on. It's been a pleasure.

FUDGE: I'm Tom Fudge. You've been listening to These Days on KPBS.