**Krista Tippett, host:**Keith Devlin sees mathematical equations like sonnets. They are reflections, he says, of the inner worlds of our minds. But what most of learn in school doesn’t begin to convey this. And that’s not just because some of us — like me — seem congenitally bad at math. Our brains are not primarily logical, but analogical. And what we’ve asked everyday math to do up to now can be better done by computers.

Keith Devlin is also a leader on the frontier of MOOCs, massively open online courses. And he believes that on-line teaching is helping us finally learn how we learn. To be with this mathematician is to glimpse the beauty of “mathematical thinking” - illuminating what it means to be human in unexpected ways.

Mr. Keith Devlin:Just as a trained musician who can read music can look at a musical score and in their head — in their mind — they can hear that music playing. For a mathematician, the same thing is true. Providing it's in a part of mathematics you're familiar with, you can look at those symbols and in your mind, this mathematical world is created. And you can see the flow and ebb of ideas; you can see it going on. It comes to life in your mind.

**Ms. Tippett:**I’m Krista Tippett, and this is

*On Being*.

*[Announcements]*

**Ms. Tippett:**Keith Devlin co-founded and directs H-STAR at Stanford University, the Human-Sciences and Technologies Advanced Research Institute. He’s also a senior researcher at Stanford’s Center for the Study of Language and Information. He grew up in England, in Kingston upon Hull.

Ms. Tippett:Were you born, bred, raised in Yorkshire? Is that right?

Mr. Devlin:Yeah. I was born in Hull, in what apparently was the worst snowstorm of the decade, in 1947. So I had a very exciting, melodramatic entry into the world.

Ms. Tippett:Which you don't recall.

Mr. Devlin:I don't recall, no. But my mother, who is long passed away now, she told me of the difficulty getting a midwife in and the doctor and anything. The lady next door actually delivered me because nobody could get out. There were snow drifts outside the house of several feet deep, apparently.

Ms. Tippett:And were you drawn to math, mathematics, pretty early in your life?

Mr. Devlin:Uh, no. Not at all. I used to like playing soccer and rugby and rough-and-tumble things. It was sort of a classic, working-class childhood. The mathematics came much later. The only role model I had in the family of anyone who sort of had a degree and a professional career was an uncle by marriage, and he was a chemist. His house was full of books, he would travel around, he had an extensive collection of jazz records, and he lived the life of an intellectual.So my first introduction to anything to do with science or technology or learning in general was to chemistry. But that was pushed sideways a little bit. Just when I was about to or just when I started at high school, Sputnik went up. So I sort of began to sort of think in terms of physics. Once I decided I was going to go into physics, then of course I realized that I needed to be good at mathematics because that was the key to physics.

Ms. Tippett:I want to ask you this: There's such a contrast between the way, not just the way you speak about mathematics, but the way you, define it. And there's such a contrast between that and the — well, I'd say, the understanding and even the experience of mathematics that so many people have right at that age of school, and then maybe never have another experience.

Mr. Devlin:Yeah, that's right. If the last experience you have of mathematics is what you learned certainly up to the middle level of high school, and to a large extent even right through the end of high school the way it's taught, you've basically never seen mathematics. And so most people in the world haven't a clue what mathematics is.

Ms. Tippett:Right. So let's talk about some of the ways that you — just reading you, your imagination is — starting with one definition is the science of patterns.

Mr. Devlin:Yeah. I was actually one of the people who really pushed that in the 1980s. I mean, psychology is the science of patterns of the mind; sociology is the science of patterns of social interaction. So you could call anything the science of patterns. What makes it mathematics is the kind of patterns you study and the methodology you use.But the reason I pushed it was to very much get away from this idea that it was about calculation and computation. It's nonsensical, but it needed to be done; it needed to establish itself as different. More recently, I've actually started using the phrase "mathematical thinking." I'm giving one of these Massive Online Courses from Stanford — I'm about to give it the third time.

Ms. Tippett:Yeah, I want to talk about that in a little while.

Mr. Devlin:We'll come back to that, but the thing to note is, I don't call it the Introduction to Mathematics; I call it the Introduction to Mathematical Thinking, to say this that isn't about computation and calculation and solving equations. It's about thinking about the world in a certain way that we have learned over the centuries is extremely powerful. Not the only way, not the best way, a valuable way.

Ms. Tippett:Hmm. Yeah, I mean, here's another definition that you use variations of a lot: that mathematics is a "vast and beautiful mental landscape that generations of human beings have created over 3,000 years." A mental landscape.

Mr. Devlin:And that's what stumbled — that's when I became a mathematician; that's what I stumbled on at age 15 or 16 when here I was learning all this mathematics because I needed it. I had a utilitarian view of mathematics. I was learning it because I needed to solve the equations because I was going to be solving them in physics. And then, at the age of about 16 or 17, it all fit because it all came together in my mind. It was no longer this disjointed collection of techniques you could use to solve problems. It all fell into the place, into this wonderful landscape. It was as if I'd been stumbling around in a forest, and suddenly I've climbed to the top of a tree and looked out and thought, this is the most beautiful place in the world. You can't tell it when you're down in the trees, which I had been, but the moment you reach an elevation where it all falls into place and you can see the whole topographic display in front of you, then the beauty is incredible. And the moment I discovered it, I said, um, I want to study mathematics. And I've been studying it ever since.

Ms. Tippett:Right. This idea that also what you saw, what you work in, is not just about the physical world or about abstract equations but the inner worlds of our mind? And that, in fact, that numbers — and this is such an interesting point you make — that numbers are abstractions, that they only exist in our mind than what we do with them.

Mr. Devlin:Yeah. There's a strong feeling that you have when you do mathematics that it's actually a sort of an objective reality, that the numbers — it's known as the platonic realm. It sort of goes back to Plato's writings and ideas — that you're discovering things that are in this sort of ethereal world out there. And in a sense, that's the case. But when you start to dig deeper, you realize that that ethereal world itself has been created by generations of human beings.And so mathematics exists as a psychological and a social construct. Which means that when you're doing mathematics, you're actually — even if you're doing mathematics about quarks or electrons or whatever it is, superstrings — if you're doing mathematics about physics or something in the world, in some sense, yes, what you're doing is learning more about the world. But in a deeper sense, when you're doing mathematics, what you're really learning about is how the human mind encounters and makes sense of the world. It's really a mirror. So mathematics in some sense is a lens through which you look at the world. But in a deeper sense, it's a mirror through which you look at yourself in a very abstract and penetrating way.

Ms. Tippett:And I know this is frustrating to you, because when you describe it that way, it feels so essential to who we are as a species, right? That somehow, each of us should have some sense of that, even if we're not professional mathematicians. Or, you know, this line of Galileo that you also evoke often, that language is — sorry, that mathematics is the language in which nature was written, which the universe was written. And then there is the great question about why we don't really — most of us don't walk around knowing this.

Mr. Devlin:Because no one's told us. And it's hard to see it unless someone opens the door for you. I was lucky. I discovered mathematics in spite of my education, as most of us, I think do. Occasionally, you come up against a really great math teacher that can open your eyes. In my case, I didn't have bad math teachers, but their view of mathematics was the same as everybody else's. It was a utilitarian subject that you need to know. But they prepared the groundwork, helped prepare the groundwork for me to make that discovery.

*[music]*

Ms. Tippett:You know, I have no — I'm one of these people who gave up on myself mathematically a long time ago. But I interview a lot of scientists, a lot of physicists and astronomers. I have to say, I'm not sure if I've interviewed many pure mathematicians, but I've interviewed people who use mathematics.And so what I've come to, and I think also my listeners come to, is an appreciation, you know, I think it may be too late for many of us to actually grapple directly with mathematics, but to start to understand the importance of this as part of who we are and as part of the universe. I mean, I think one of the ways you talk about this and have worked with it that does help to bring it down to earth a little bit is some of the connection you've made between language and mathematics. Could you talk a little bit about that?

Mr. Devlin:Yeah. You know, people get turned off mathematics in various ways. If you teach it as sort of just stuff you need to know to balance your checkbook — which is nonsense because none of us balance our checkbooks; computers do that for us. On the other hand, because language is so important to us as living creatures, everyone is interested in language one way or another, be we language mavens or just interested in listening to the radio or reading or novels. You know, language is a fundamental part of what we ask.In fact, in a book I wrote in 2000, called

The Math Gene, I actually made a case based on sort of rational reconstruction of human evolutionary development, I made the case that actually mathematics and language are actually two sides of the same coin in terms of evolutionary development. Human beings, when we developed the capacity for language — and nobody knows when that was; it might be as recent as 50,000 years ago — but when our ancestors developed language capacity, at that moment they developed the capacity for mathematics. It's the same capacity. It just plays out in different ways.

Ms. Tippett:Hmm-mm. Hmm-mm. I'm just thinking, in my own family, I had a grandfather who had a second-grade education, but he had this prodigious capacity to do things with numbers in his head. You know, it was completely mysterious.

Mr. Devlin:Yeah.

Ms. Tippett:Completely mysterious. Not taught. I don't think he had any clue why he could do it. And then, I have a son, now, I'll say, who's 14 — he's doing well in everything but math — and he said, "My brain just doesn't work that way." You know? And I think he's not the only kid who says that. I think I probably said that at some point.So then, in the context of that, I'm reading you and this sense that there's something innate in us. And then why does that not make itself manifest?

Mr. Devlin:It's true. Everyone is on a spectrum, with physical abilities, mental abilities. We're all on a spectrum. But a lot of the problem in mathematics is that an awful lot of what goes on in the school system is basically trying to train the mind to do what a $10 calculator can do: follow rules and algorithms and procedures. And one thing that we do know is, that the human brain does not find that natural. The human brain is analogical, not logical. And so, when we try to force it to be procedural and exact, the brain simply doesn't like it.It was important for many thousands of years to be able to do computation and calculation because that was the basis of commerce and trade and buying and selling. And you had to do it in your head or with an abacus board or something. So for hundreds of years, it was actually important to train the mind to follow rules to do computations and get the right answer. Well, now we've automated that. And we carry around devices in our pockets that can do that. Which means that we can spend more time letting the brain do things that the brain is really well suited for that computers can't do very well: making value judgments, making analogical leaps.

The trouble is, the education system is at least 50 years behind the changes in technology in society.

Ms. Tippett:Right. That takes me back to you saying in the beginning that you don't talk about mathematics but about mathematical thinking. And what we learn in school is that old-fashioned, narrow idea of mathematics. But you're saying — that that's not natural to learn, as you say, to come up with the right answers. But how interesting that could be if, even at a young age, we were offered the invitation to do mathematical thinking.

Mr. Devlin:Yeah. It's as if you went along and you said, I want to be an architect. And you go to architecture school and the first five years they teach you how to lay bricks accurately, in nice straight lines. You know, taking advantage of being an architect does depend on the fact that people can lay bricks.

Ms. Tippett:Right.

Mr. Devlin:Bricklaying and architecture are not the same thing. And mathematics is the architecture, it's the creative stuff you can do with these bricks. And it's no longer necessary for people to be able to lay bricks themselves because we now have machines that handle the bricklaying for us.

Ms. Tippett:Right. Right. Right.

Mr. Devlin:Actually, so do builders.

*[music]*

**Ms. Tippett:**I'm Krista Tippett and this is

*On Being*. Today with mathematician Keith Devlin.

Ms. Tippett:I want to talk to you about something else, another connection that you shine a light on an interesting way that's always been very intriguing to me, which is the connection between mathematics and music, right? Like we all know that that connection is there, but I've never really been able to comprehend it. I'm not sure I can comprehend it now, but, again, I feel like you open this up in an interesting way. One place you said, the most beautiful equations are like sonnets.

Mr. Devlin:Yeah. Yeah.

Ms. Tippett:And you said that mathematics and music are both captured in dry symbols, but they both come alive when interpreted by the human mind.

Mr. Devlin:Yeah. Certainly in the days when the only way to store and distribute music — before recording devices — was to write it down in symbolic notation. Then, of course, if you couldn't master that notation, you couldn't get to the music. Mathematics is still largely still like that. And maybe always will still be like that. It is still a hurdle that you have to master mathematical notation to get to the mathematics that it encodes. But just as a trained musician who can read music can look at a musical score and in their head — in their mind — they can hear that music playing.

Ms. Tippett:Yeah.

Mr. Devlin:For a mathematician, the same thing is true. Providing it's in a part of mathematics you're familiar with, you can look at those symbols and in your mind, this mathematical world is created. And you can see the flow of the ebb of ideas; you can see it going on. It comes to life in your mind.

Ms. Tippett:And take in the beauty of it, which is a word that you and a lot of mathematicians use constantly.

Mr. Devlin:Yeah. And so you see through the symbols to what they're about. Just — the interesting thing is, I don't read music. So to me, music is just [inaudible]. I can really resonate with people who can't get mathematics from looking at a book, because I can't look at a musical score and hear the music.

Ms. Tippett:Yeah.

Mr. Devlin:Someone has to interpret it for me, to play it.

Ms. Tippett:So, you know, you did this interesting project a couple of years ago, called "Harmonious Equations," which I had so much fun discovering.

Mr. Devlin:That was a hoot, yeah.

*[music: "Euler’s Identity”*

Ms. Tippett:Yeah. And I think we'd like to layer some of that music into this production.

Mr. Devlin:Oh, please do. Yeah.

Ms. Tippett:So let's just talk about a few of those. So there's this "Euler's" — is that how they say it? — "Euler's Identity." You know, again, it's very intriguing language at the supreme level of abstraction, "seemingly different concepts sometimes turn out to have surprisingly intimate connections."And then, as you narrate, the numbers that are part of that equation, you know, what becomes clear to me — even though I don't understand this — is there is kind of a world within each of these numbers.

Mr. Devlin:Yeah.

Ms. Tippett:And then you talk about them becoming a perfect mathematical composition.

Mr. Devlin:Right. Right. Absolutely.

Ms. Tippett:Say a little bit about what "Euler's Identity" is for someone who doesn't know.

Mr. Devlin:Yeah, because I do work for the media and things, I always get asked the question: what is your favorite equation? What is your most beautiful equation? So even though, in a sense, the scientist in me says I don't have a favorite, I love them all — that's like saying you love all your children, that kind of thing.

Ms. Tippett:That’s right.

Mr. Devlin:I actually always give an answer and my answer is always "Euler's Identity." And that's certainly a justifiable one, from my point of view.

*[music: “Euler’s Identity"]*

Mr. Devlin:There were very few basic constants in mathematics. So you’ve got these five most fundamental constants: zero, which has to do with addition; one, which has to do with multiplication; π, to do with geometry of circles; e, to do with natural logarithms; and the square root of negative one, which is to do with algebra. And given that they're coming from different parts of mathematics, or so it seems, there should be no reason why these things connect together. But it turns out, there's the most beautiful, elegant identity that connects them together, which says that e raised to the power i times π, plus one, equals zero.

*[music: “Euler’s Identity"]*

Mr. Devlin:You've got an equation — each one is mentioned once, all it involves is a plus sign, an equal sign, and an exponentiation sign and multiplication sign for the i and π, and you've got this identity. There's no reason on the face of it why that should be the case.I remember when I first met that, as a teenager, I was completely blown away. I mean, if anything could have come close to convincing me that there was this thing called God out there, that was surely it.

*[music: “Euler’s Identity"]*

Mr. Devlin:"Euler's Identity" is an instantiation of the fact that what we thought previously were separate things actually always were, within the realm of mathematics, part of the same whole. As with good art, not only does it reveal — not only does the beauty, the intrinsic beauty of the equation itself — in this case, not the beauty of the symbols but of the logical connections between them, the elegance of the connection between them. So it's beautiful on that sort of technical level, but it's also beautiful because it tells us — it's a great insight into the world. You know, just like the Mona Lisa's smile captures an awful lot about being a human. And you can — I mean, people discuss endlessly what was going on in her mind, assuming it was a she that was the model, what was going on in that person's mind when that painting was done and what did the painter capture.Well, it's the same with "Euler's Identity." It tells you an awful lot about the mathematical world that we're exploring.

Ms. Tippett:Mmm.

Mr. Devlin:It's an absolutely beautiful piece of art as well as a beautiful piece of mathematics.

Ms. Tippett:Mmm. That's wonderful. Thank you. And then in the "Pythagoras' Theorem" — this is another piece that you commissioned with these musicians.I just want to ask you about this statement that you made. Pythagoras said that "number is the ruler of forms and ideas and the cause of gods and demons."

*[music: "Pythagoras' Theorem"]*

Ms. Tippett:And you took that phrase as the theme for the song. So what did he mean by that? What do you understand in that statement by Pythagoras?

Mr. Devlin:[laughs]Uh, well we've got to go back and put ourselves in the mind of Pythagoras. It was sort of mystical; it was a secret society. There was also sort of a cult element to it. One of these things about mathematics — this comes up when people talk about things like the golden ratio and all this kind of stuff — is that even people who don't have professional mathematical ability, there's a sort of fascination with number.

Ms. Tippett:Yeah.

Mr. Devlin:I've written many times about the golden ratio, because people say that the golden ratio has also all sorts of wonderful properties, most of which are completely false and bogus and have nothing to do with the path. It's not the dimensions of the most perfect rectangle — it's all complete nonsense.So you ask yourself, why do people get so passionately involved with number? And they do. And the only answer to that I can come up with is that deep in our psyches, we know that the world we now live in, the lives we lead, depend in a fundamental way on the discovery and the invention of numbers, just as they do on language. Most people don't publicly acknowledge that number is part of what we are. But I think deep down in their psyche, they know we were.

Ms. Tippett:But there's some mystery to it. The fact that we don't understand it makes it mysterious — makes that intuition mysterious.

Mr. Devlin:That adds to it. Absolutely that adds to it. But we all sort of know that they are important. And I think that goes back to what Pythagoras was saying. Even back then in the days when it was sort of more cultish, folded in with religious beliefs and all kinds of things, even back then, people realized that numbers were a big, big deal. And I think that's what Pythagoras' quote was referring to.

*[music: "Pythagoras' Theorem"]*

**Ms. Tippett:**You can listen again or share this show with Keith Devlin at onbeing.org. There we also have links to more of his favorite equations set to music with the choral group Zambra. There are musical odes to Leibniz's series for pi and Einstein’s energy equation.

*[music: "Pythagoras' Theorem"]*

**Ms. Tippett:**Coming up…how teaching thousands of students online can be more intimate than a classroom.

I’m Krista Tippett. *On Being* continues in a moment.

*[Announcements]*

**Ms. Tippett:**I'm Krista Tippett and this is

*On Being*. Today: with mathematician Keith Devlin. We’ve been exploring his view that what most of us learned as math in school doesn’t convey the beauty and purpose of this endeavor. We’re catching some of his delight in mathematical thinking - as something that reflects the inner world of the human mind and can help us navigate the outer world in unexpected ways. Keith Devlin leads Stanford University’s H-STAR: the Human-Sciences and Technologies Advanced Research Institute.

Ms. Tippett:I'd like to talk a little bit about this, the H-STAR project at Stanford that you're involved in. It's interesting the different kinds of things that you've gotten involved with, even at Stanford alone: the study of languages — the Center for the Study of Language and Information, now this H-STAR project, which focuses on, as I understand it, the human applications and ramifications of technology. And I'm finding that this is a subject everyone wants to talk about now, kind of the enormity of the role of technology in our lives is settling in. The reality of that and our understanding of that. And then people are saying, you know, what do we do with this? I mean, how do we make sense of it and how do we make it purposeful and livable?I mean one of the things that you — really surprised me that you mentioned this, in terms of mathematics, is that computers have only really affected mathematics around the edges. That, in fact, mathematics still needs free-form scribbling to survive, which is so interesting.

Mr. Devlin:I mean, mathematics takes place in the human mind, or increasingly, when human minds interact together. We still have this nice romantic image of the lone mathematician working in the attic late at night. And those people still exist. But if you sort of look at mathematics as a whole enterprise, it's a huge enterprise, then an awful lot of mathematics these days, like everything else, is done collaboratively in teams that communicate both by physically co-locating themselves at conferences and seminars and workshops, and collaborating over the Internet.But it's still about people talking to and with people. And the technology — other than the fact that we use email and things like that, video conferencing — the technology is still not really affecting the product itself. It's just around the edges.

Ms. Tippett:And I really do think that you have your finger on a pulse of this, this disconnect — let me just say it this way: that, you know, on the one hand, as you and I have been discussing, many of us, I think most of us are illiterate in the language of mathematics. And then there is the increasing reality that the indispensable tools of our everyday lives, you know, starting with our mobile phones, as you say, rely on masses of abstract math.

Mr. Devlin:Yeah.

Ms. Tippett:So we're living with these things, we need these things, we can't begin to grasp what's behind them.

Mr. Devlin:Yeah. The scale of that has gotten huge, but actually, that began with the very beginnings of mathematics because arguably mathematics began — and I would say this is when it began — with the invention of numbers. And we think that took place in Sumeria about maybe 8,000 years ago. And numbers were introduced essentially to provide as a form of money. Numbers were things that mediated trading goods. The moment that we invented numbers, we did it to introduce into interpersonal reactions into society, to introduce abstractions that could mediate. Because numbers are pure abstraction, money is an abstraction. I mean, a $10 bill is just a physical manifestation of the number 10. I mean, that's all it is. You've just got a physical token of 10, and you've got a mental token of 10. And nowadays, we have bits on discs in banks. I mean, we don't even need the $10 bill. It's just a configuration of bits on a computer.So we are mediated in terms of the kind of life we lead, the length of our lives — almost everything about our lives is mediated by money. And money is just numbers and numbers are abstractions. And all that has happened recently is the scale has gotten much bigger and the rate of change has been faster. But our lives became irrevocably changed and bound up with numbers the moment we invented numbers and money, about 8,000 years ago.

Ms. Tippett:Hmm.

Mr. Devlin:And we've gotten used to that. And people actually don't think of money as abstract.

Ms. Tippett:No they don't. They don’t.

Mr. Devlin:I'm old enough to remember when credit cards came along, and I do remember that initially there was a lot of unwillingness to go to a credit card because people wanted dollar bills. In my case, it was pounds and coins …

Ms. Tippett:It wasn't real money.

Mr. Devlin:… and so, but now we've gotten used to that. And then, of course, when you start having mobile phone transactions and things, you don't even need the credit card anymore. You know, you just go in and you wave your mobile phone in front of a scanner. So, and the rate of change has gone up dramatically, and the extent has gone up, but it's actually not a new phenomenon. We are more aware of it because we are living through the latest transition.

Ms. Tippett:You know, this is — this is kind of tangential to that, but it feels a little bit related. One of the interesting ways you talk about how mathematics has evolved is that across history, there's abstract math going on, which has no conceivable application at the time in which it's being done. But, for example, you talked about how encryption systems that now run the Internet followed on work with prime numbers a couple of centuries ago, at which time it was outlandish to imagine that these things would ever have practical applications. And now, fundamental aspects of our reality depend on them.

Mr. Devlin:Yeah. In fact, one of the leading people, G.H. Hardy in Cambridge, actually went on record in a book and said — he was quite — because someone had challenged him about the fact that you can use these things for practical uses in mathematics. And he said — he went on record as saying, nothing he had done in his professional career could ever find practical application. And by golly, within a hundred years, this is the basis for the Internet in modern society and security.

Ms. Tippett:[laughs]

Mr. Devlin:So, if there's one thing history tells us, it's never, ever look at something and say this will never be used because in the case of mathematics, time and time again things come around and get used. I'm actually of two minds about this. I'm always surprised and I think, ‘Wow, this is what Eugene Wigner described as "the unreasonable effectiveness of mathematics.’"We develop mathematics to be useful in the first place, and so at first it is obviously going to be of use. But then we sort of follow our intuitions, we go off on these tangents, we pursue these abstract ideas, and we end up producing something that just looks like a pure art form, just having no connection with reality, to whatever. We've let our imaginations run free, subject to the constraints of mathematical logic. And we produce these new theories and their beautiful and their entirely coherent and they don't seem to have any application. And then, a hundred years later, maybe longer, someone comes along and finds a really important application.

So part of me says, that is absolutely surprising. On the other hand, if you go back to the point I made earlier, that mathematics is just really looking at the way the human mind encounters its environment, then if that's what we're doing, you should expect that things we develop — even though we can't understand how it's going to be relevant now — will eventually be relevant. Because what it really tells us is, at the time we've done that mathematics, we haven't developed as a society or individuals to the point where we can appreciate what we've just seen.

Ms. Tippett:Mmm. We have to catch up with our own …

Mr. Devlin:We have to catch up with it.

Ms. Tippett:Interesting.

Mr. Devlin:And so, in a sense, it's not a mystery; it's just part of what mathematics is and how it works.

**Ms. Tippett:**I'm Krista Tippett with

*On Being*. Today with Stanford mathematician Keith Devlin. Stanford has been a pioneer of MOOCS, Massively Open free Online Courses. And Keith Devlin has been one of the first professors to join in, offering “Introduction to Mathematical Thinking” to thousands of students at a time, all over the world.

Ms. Tippett:So tell me about these Massively Open Online Courses you're doing. I mean, what is it like to teach 27,000 people.[laughs]

Mr. Devlin:It's like going on the radio and talking to however many million people we're talking to now, actually. I've always been driven for various reasons towards trying to spread the word — there's a sort of evangelical streak in me here — just to spread the word …

Ms. Tippett:Absolutely. You are totally a mathematics evangelist. I know that.

Mr. Devlin:But whenever there's a new communication medium, I sort of look at it and I can't resist seeing if I can use that — or how I can use it, I know I can use it — how can I use that to communicate mathematics? You do it in different ways. You know, the way — in a movie, you do it one way; on TV you do it a different way ...

Ms. Tippett:Hmm-mm.

Mr. Devlin:And in a text book, you do it one way; And so I gave my first MOOC a year ago. And I changed it a bit and gave it again in the spring of this year. And in September this year, I'm going to give it a third time in a third incarnation.And I think education, to me — I approached the MOOC the same way. It's not about me talking to 100,000 students, it's about me talking to one student. In fact, in many ways, I find MOOCs more intimate than giving a lecture to a class of 30 students. And most of my time, I give lectures to classes of university students of maybe between 15, 25, 30 students sometimes. And I give big lectures to sort of a few hundreds of students. But those are very impersonal. They are theatrical displays.

But part of doing a MOOC is you're — it's you and one student, because by its very nature, it's one person, sitting at a computer in front of a screen, interacting with you. Now, the interaction is mostly one-way, but part of the trick of putting these courses together is to compensate for the fact that the student can't directly address you back. Now in a big class of 100 students the students can't do that either but they have TAs and they can interact with each other in the classroom and so there is a lot of interaction going on.

With a MOOC, the challenges are more acute, when you think of individual students in remote places who aren't even in the same classroom as other students and don't have access to anyone to ask questions for. That makes it a very interesting thing to design. And most of the mathematics is done with a cheap little camcorder mounted on a tripod above my desk.

Ms. Tippett:Mmm.

Mr. Devlin:But I did go into the campus TV studio here and spend an exhausting six-hour period, actually, recording all of the intercam punctuations, which actually, if you look at them — I'm not talking about mathematics. I'm establishing human contact, by looking into that camera, eye to eye, looking at that student. Because it was all about establishing human connection and getting them to feel comfortable with me as their friend and instructor.

Ms. Tippett:Hmmm. I'd kind of like to take you back to the book you wrote in the late '90s,Goodbye, Descartes: The End of Logic and the Search for a New Cosmology of the Mind.

Mr. Devlin:Ah, yeah.

Ms. Tippett:I mean, I found that really wonderful book, and you — you know, one of the things you were pointing out is that just in about the last 300 years, we'd glorified rational thought and logic in a way that had not been done before in history. Even in the history of science, kind of treating our minds as calculators. But that we're now getting this much more sophisticated understanding of our minds and brains and learning and intelligence.And then the reason I thought that I would like to talk to you about this now is, you know, that was I think '96 or '97. So here we are these decades later and these decades of rapid transition and rapid new fields of knowledge about, in particular, our brains and our minds and our bodies. So how would you look now at this notion of, you know, what the world looks like beyond logic, beyond Descartes?

Mr. Devlin:Yeah. Well, if anything, what I've said, what has happened since then, is simply reinforce the feelings I had then. It was interesting the transformation that led me to write Goodbye, Descartes. It was very expressly goodbye to the sort of Cartesian logic based approach to life. In a sense, I was finally renouncing my former self. Because I went into mathematics — I mean, I don't think I was ever a sort of total Platonist. But I went into mathematics seduced by the fact that the power of the logic of the reasoning. And in the 19th century, mathematics ruled supreme as the basis of physics, of engineering, calculus, all the more, economics. Mathematics literally built the modern world. To the degree where people in mathematics, including myself, began to think, well, we've got this powerful mechanism. We can take it into the realms of psychology, sociology, linguistics, and start to make sense of these soft sciences, you know.

Ms. Tippett:And kind of solve every problem with this logic …

Mr. Devlin:We can certainly try to solve every problem. That had already been tried a bit a few decades early by the people who try to build artificial intelligence systems and expert systems. And in both cases, they had very limited success but not great success.But you know, undaunted by that, I thought, well, maybe we can do it in natural language, understanding language impulses. So when Stanford founded its Center for the Study of Language and Information in 1983, I got very interested and came out as a visit in '85.

Now, what I didn't expect was that when I got to CSLI, I would be brought face to face with all these psychologists, these linguists, these psycholinguists, these social linguists, sociologists, the etymologists. I was brought in with all of these people in what I used to call the soft sciences …

Ms. Tippett:Right. Right.

Mr. Devlin:… as opposed to the hard sciences, and thinking — you know, I wasn't so naive I would be able to say, here's some mathematics; it will make your discipline a little more robust. Although, some of my colleagues seem to have that belief.I was prepared to sort of be taught new things. What I wasn't prepared for, but what happened to me, was I ended up realizing the huge limitations on mathematics in those domains. And this happened to Chomsky when he tried to sort of do the same to linguistics in the '50s.

Ms. Tippett:Right.

Mr. Devlin:And it became clear there were — it was good stuff and lots of stuff has come out of it, but it wasn't coming close. It was like trying to reach the moon by building taller and taller ladders. That was not the way. Didn't mean to say that you wouldn't get to the moon, but you weren't going to get there by building a ladder. And building the ladder that got us to the moon certainly involved a lot of mathematics. And building the ladder that would involve us to understanding — sort of really understanding people and society, mathematics can certainly play a role. But it's not the rule of the king and laws that it was in physics. It's the rule of a trusted worker who can provide stuff as and when it's needed.We needed to stop trying to build mathematical ladders that would help us reach this moon of understanding of human beings and society, and simply view mathematics as a tool that could play a useful role in a more broad and holistic interdisciplinary study that would help us to understand it.

Along with that, came, in my case, the final acknowledgement that mathematics isn't the discovery of an external platonic realm, that it's actually the building and study of a realm within our own minds.

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Ms. Tippett:I mean, I don't know if you would think of this in this context at all, but it seems to me that in this century, you know, and as late as 2008, with the economic downturn, the crisis, we've kind of lived more deeply into this recognition that — that we, that we had called things rational that weren't rational, right? That …

Mr. Devlin:Yeah.

Ms. Tippett:… this principle of logic and pretending that when numbers are involved it is logical. Somehow, it was in precisely those systems — our financial systems, that work with numbers — that turned out to be these monuments to irrationality. And then, suddenly, there's this recognition also in the field of economics, which I think had had that Cartesian view that somehow at bottom, this is all rational. That, in fact, this is what you're pointing out as well: we're getting a more complex sophisticated understanding of ourselves from a lot of different directions.

Mr. Devlin:Yeah. Yeah. And ultimately, mathematics is black and white, and the world that we live in is an infinite variety of shades of gray. I mean, mathematics can work in localized ways and be useful. And it can also provide interesting metaphors to help us make decisions. There's all sorts of great uses mathematics has in sort of making sense of today's world and building the world of tomorrow from it.But as I said a moment ago, the days of the 19th century, when mathematics ruled how you built your machines and how the machines worked, those are gone. And if anything, the financial crisis was a great warning that says, if you build systems based on mathematics, you are heading for disaster. This circles back to this business about NSA and security. Those systems are built on mathematics. And they, if not checked, they will lead to disaster every bit if not worse than the financial crisis — for the very same reasons. Because mathematics works when you're talking about a clockwork mechanical universe; it doesn't work when you're talking about a human society.

Ms. Tippett:[chuckles]Unfortunately, that's not the world we live in. It's so important. And I also do want to circle back to your love of mathematics. And, in fact, you know, it's very interesting to me the spiritually and theologically evocative way, the aesthetic way, that physicists and mathematicians speak of mathematics.There's this quote of the English mathematician and philosopher Bertrand Russell that I’ve seen you use a few times. I'm just going to read this because it's very beautiful and I hadn't seen it before: "Mathematics, rightly viewed, possesses not only truth, but supreme beauty. A beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show."

Mr. Devlin:Yeah. Isn't that wonderful?

Ms. Tippett:Yeah.

Mr. Devlin:Actually, as you were saying that, I remembered your very first question was, you know, when I was a child, do I have — am I spiritual. And I actually tangented off immediately to talk about not having religious beliefs in a sort of white-bearded god in the sky, and that kind of thing. But I mean, the reality is, of course, that I'm deeply spiritual. It's just that my spirituality is in mathematics and through mathematics understanding ourselves and the place we have in the environment we're in.So in the sense — if spirituality means revering existence, revering my fellow creatures on the planet, reflecting in them, thinking about them— in my case — through the eyes and the mind of a mathematician, then I'm as spiritual as it gets. And you know, if that manifests itself with poetic language and overblown use of language, then that's just the price you pay for that spirituality.

Ms. Tippett:How do you think this work you do, this life you lead, this mathematical thinking with which you move through the world — how do you think it forms your sense of what it means to be human or affects the way you live your life?

Mr. Devlin:I see the world differently from anybody who's not a mathematician. In fact I see the world differently from many mathematicians. So, in some sense, I've seen things that other people are not seeing. But I think it's almost — actually, I guess I have to use the words "shades of gray," although that's been co-opted for a different purpose these days.

Ms. Tippett:[laughs]

Mr. Devlin:But anyway, I think we all see the world through shades of gray. There's a period — when most of us learn mathematics, there's a period in our early career when we're in our teens or 20s, when we think that mathematics provides the black-and-white view of the world — the view of the world. I remember distinctly having that view because it seemed to be so logically correct. Now it is logically correct more so than any other discipline, but the question is, at what cost for connecting to the rest of the world, especially the human social world.So there's that period. But then you go beyond that — or you should go beyond that to realize that mathematics is just another shade of gray that gives you a different perspective on the world. But it's still the same world. The grayer world to one person is not that much different from the grayer world to another [person].

Ms. Tippett:So this really is my last question. Is there any frontier, right now, that you're aware of, where new things are being learned or new processes are being experimented with that you are watching …

Mr. Devlin:Wow! Where do we begin?

Ms. Tippett:Well, just tell us about one thing that's unfolding that you're really watching with great interest.

Mr. Devlin:Oh! It's the one that I'm right in the middle of, which is online learning. Not because of the fact that we're reaching thousands of people. We're finally beginning to understand how people learn. Here's an analogy: If you went back to the 18th century and you got sick, good luck, because you might find somebody who was instinctively a good doctor or who was lucky, but medicine was just hit and miss by then and depended on talent and whatever it was, on luck. Because it took the development of, first, chemistry, then biology, evolutionary biology, and the developments of modern science on which we can ground medical training.So, when you go to a doctor today, the chances are that person is going to make you well, providing you at least have a recoverable disease, because we've learned the science on which medicine is built. Science in 200 years, science went from hit-and-miss hunches to a scientifically grounded discipline where the chances were very high that the doctor would be able to help you and make you well.

Right now, in education, we are in the 18th century. We are actually, I think, within a couple decades — of being able to actually improve mathematics education to the point where people actually do begin to get it and get over the hurdles. Because we're beginning to understand it better..

Education is now starting to make that transition, to the fact that, yes, there'll be good teachers and bad teachers and better teachers and worse teachers, but it will all be grounded in understanding. And in the process of understanding how we learn, we are, of course, doing what mathematics has been doing since the very beginning. And as I've been saying, we've been learning about ourselves. Because the more we learn about how we learn, the more we learn about what it is to be human.

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**Ms. Tippett:**Keith Devlin is executive director of H-STAR, the Human-Sciences and Technologies Advanced Research Institute at Stanford University. His books include

*Introduction to Mathematical Thinking*and

*Life by the Numbers*.

To listen again, share this show, or subscribe to our podcast go to our website at onbeing.org. On Facebook, we’re at facebook.com/onbeing. On Twitter, you can follow our show @beingtweets. I share my thoughts @Krista Tippett.

*On Being* is produced on-air and online by Chris Heagle, Lily Percy, and Stefni Bell.

Trent Gilliss is our senior editor.