Robbert Dijkgraaf, a mathematical physicist and director of the Institute for Advanced Study in Princeton, New Jersey, describes how art school in the Netherlands taught him how to do science. Then he and host Steven Strogatz discuss the matrix model revolution in string theory and why space and time might not be the most fundamental things in the universe. This episode was produced by Dana Bialek. Read more at QuantaMagazine.org. Production and original music by Story Mechanics.
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Steven Strogatz: Can you describe, were you curious about science from a young age, or —
Robbert Dijkgraaf: Well, I often joked that I, intellectually, I kind of peaked when I was eight years old. Because I remembered the time I had a friend and we were always together and we were in school but that was only our side job. And after school we would go, actually to the attic where we’d build a huge laboratory doing science experiments, and, but also, you know, making animation movies. Which is quite remarkable at such a young age. So for me, that period was like the renaissance and I was like embracing everything in a very, a very bottom-up fashion.
Strogatz (narration): From Quanta Magazine, this is The Joy of x. I’m Steve Strogatz. In this episode, Robbert Dijkgraaf.
Strogatz: Mm, well, can you give us a feeling for what kinds of experiments? Are we talking chemistry sets or electrical things, or?
Dijkgraaf: Yes. Well, what’s the usual thing? I think chemistry sets with all the kind of explosions that happened, you know, when parents were not there.
Dijkgraaf: There were, I remember, you know, playing a lot, constructing and deconstructing telescopes and microscopes and, you know, playing with spectra and lenses.
Strogatz: Robbert Dijkgraaf is a theoretical physicist who works in string theory and quantum gravity. He’s originally from the Netherlands. Honestly, I never had a chance to meet him before this interview. Though I had seen him give a performance once. I call it a performance because even though it was a lecture it felt more like a theatrical show of some kind. He had a little boy running around on stage who was playing the young Einstein.
And this, this way of making the material so engaging made me feel like talking to him was going to be a super dynamic experience.
Strogatz: It’s interesting that it was done in collaboration with this friend of yours, right?
Strogatz: I mean so much of science is collaborative. Although there is this mythology about the lone genius, you know, toiling away. But I — I’m struck by your experience that it was a partnership from very early age.
Dijkgraaf: It was a complete partnership and I noticed with my friend that we had complementary strength. So I think, you know, my strength was actually finishing projects and really thinking things through and trying to push for, you know, better quality. And his strength was to kind of completely surprise me with new ideas. So — so we were halfway into a project and then he would come and say, “Well, now for something completely different.” I must say that, you know, in my whole career I’ve always been collaborating. I mean that hardly any papers, scientific papers, I’ve written by myself.
And just the pleasure of, you know, doing yourself so much better with the help of your friends and your colleagues.
Dijkgraaf: I think that stayed with me my whole life.
Well, remember I had a — a math teacher who I still very much remember him giving me, uh, it was an — an East German encyclopedia of mathematics. And, you know, it was perhaps the most boring book ever. But for me, this was like a catalog of everything I didn’t know I didn’t know. And so there were all these different concepts. So, for me, that triggered a tremendous amount of my imagination. I had a chemistry teacher who made clear to me that chemistry was the most advanced topic I could do in high school. Because it was truly quantum mechanical.
Dijkgraaf: So I loved that. He was really pushing me in quantum mechanics. You know, these — these two teachers I think were crucial for me. And, you know, if you’re, at that stage, so, you know, my background — I mean my parents didn’t go to university or college.
Dijkgraaf: And, they were very curiously, they were very interested. My father was very interested in history. My mother had some of a background in — in the arts, in the visual arts. But none of them had any interest in science or math. And so, you know, you — you just need kind of a first contact, you know, you have to be made aware there’s this new continent that you can go. And so for me, the real amazing experience was when I was a high school student. And not only could I kind of set foot on that continent, but, you know, you could travel, there were no barriers.
I — I mean somehow the language of mathematics, which is the language spoken in that far away land was very natural to me. So for me, it was a big surprise that there were no barriers.
Now, I was struggling every day with Latin and Greek and French and German and, you know, that’s an uphill struggle. You need all the grammars, but for math there, was no such barrier. The — the grammar was very crisp and clear and then it just could start, you know, learning more about all the wonderful things that inhabit that world.
Strogatz: Hm, it’s interesting to hear that your parents were not only not scientists, but you say did not — neither one went to college? Did I — did I understand you correctly?
Dijkgraaf: Yes, yes.
Strogatz: That’s interesting to me because my — that was true for my parents also. Possibly from the same generation.
Strogatz: I mean my parents grew up in the Depression, and didn’t have a chance to go to college and they always were — were supportive but didn’t quite understand what I was doing or what… You know, “where are you ever going to use mathematics?” they would sometimes ask. But not — not in an aggressive way, they just genuinely didn’t know.
Dijkgraaf: Well, for me that was true, too. So, uh, I think I was in this very prestigious high school. Almost all other students had families, you know, with an academic background.
Dijkgraaf: And in fact, there was a lot of pressure. Basically, there were two career choices. You either became a medical doctor if you liked the sciences or you became a lawyer if you’re more interested in humanities. And the fact that I actually chose to study physics was seen as a crazy career choice.
Dijkgraaf: And often, I think, you know, I only made the choice because, um, you know, I had parents that in, uh — They were very supportive, but they had basically no clue, what university was and so they just said, “Well, if it feels good to you, please go ahead.” And so that was kind of liberating, I felt.
I went as many of my, you know, contemporaries, went to study in Utrecht because Gerard ’t Hooft, who would become a Nobel Prize winner, was like the leading light in particle physics, so I wanted to work with him. But I never saw him, you know, we just saw people teaching you calculus and classical mechanics and electrodynamics and I just found it extremely boring.
Dijkgraaf: I must say, you know, one thing that really attracted me from the very beginning of science is the fact that it’s this, you know, this limitless world, you know, you can do anything you want. You, you know, you — the things that you meet in science are even crazier than your imagination can — can think of. And somehow I was disappointed so that, or that was not the way, physics was taught. It was about taking exams and actually I was very good in taking exams. I always had perfect scores.
Strogatz: Okay, uh. [LAUGHS]
Dijkgraaf: But I’m, you know, I — I had perfect score but I had the, like zero interest. And I think that was not a good combination.
Strogatz: [LAUGHS] 100 on results, zero on interest.
Dijkgraaf: After a few years, I remember my — my wife, who was then my girlfriend, said, “Robbert, you know, um, um, what’s happening? You know, you — I don’t see you doing any calculations. But your whole room is full of paintings?” And, you know [LAUGHS] somehow, slowly, without me actually realizing it, my attention had shifted away from physics.
So I applied to the — the most prestigious art school in the Netherlands and, which was wonderful to get in because I think now only one in 20 got in.
Dijkgraaf: So I — I wasn’t more proud of myself as the day I was admitted. One thing I did very consistently is keep from them that I have any background in science because my — my great fear would be to be recognized as the best painter among physicists. Or the best physicist among painters.
Strogatz: [LAUGHS] Uh-huh.
Dijkgraaf: And none of them sounded very attractive [LAUGHS]. So I wanted to be seen for an artist and that’s it. And I must say I — I spent two years there and it was just absolutely wonderful. Because, you know, the remarkable thing in art school is it’s not at all about taking exams, it’s not at all about whether you have a perfect score on a calculation.
It’s about how adventurous are you. Are you willing to go into other fields and — particularly in the beginning. They pushed us to do photography or graphics or sculpture or architecture, all of it. So for me, that felt like a wonderland, you know, you could just explore, explore, explore.
Dijkgraaf: And then I still remember the day, you know, I was walking into the university bookstore and I said, “You know what?” I saw the physics shelves and I think, “You know, that’s wonderful. I can go and look at the physics books and I’m not burdened in any way that this is my career.” And I remember picking up a few books of physics, well, that’s so wonderful, [LAUGHS], and now I can read them and really enjoy them. And I think then in some sense, the — the fire that I thought it was ex — extinguished, started to slowly burn again. And then I was reading these books and thought, oh, that’s terrific, and it was so exciting [LAUGHS].
I remember reading books about particle physics, something about — this was the early ’80s about the standard model and, and then I thought, wow, you know, I’m missing my old life.
Dijkgraaf: And one thing which was I think quite — I still remember very well that then I had to break the news to my art school that I would go back into physics.
Dijkgraaf: And this was kind of interesting conversation because that had — that message had two independent bits of information. Because they didn’t know my background in physics.
Strogatz: [LAUGHS] So you kept it secret the whole time?
Dijkgraaf: I kept it secret.
Dijkgraaf: People often tell me, “Wow. You know, you took this big detour going through art school.” But I like to point out it actually was a shortcut.
Dijkgraaf: Because for me actually art school, honestly, I learned what it means to do research. Um, after a week of hard work, the only thing that mattered in art school was whether you — how high was the pile of drawings or projects? How much did you explore that week?
And so when I went back to physics, I felt let’s keep that. You know, at the end of the week, I want to see a pile of calculations, of sketches. It doesn’t matter whether these are original ideas or not. They are original because, you know, you made them.
Dijkgraaf: And it should be an active subject. It’s, uh, physics is something you do, math is something you do. It’s not something you absorb. And so for me, that two years in art school were like a tremendous boost for what it means to be a research scientist. So, yeah, I really credit that experience as something well, to not only I — regenerating my love for the field but also giving some very serious advice. And basically, the advice is that, you know, research is much more about the process, about the path than the goal or the result.
Strogatz: So often we hear people talking about can we reconcile art and science. Or the two halves of our brain or whatever and you’re — you’re telling something much — you’re — you’re phrasing it so much more strongly. That — that art was actually enabling for you in — in guiding you to the right way to do the science you’d always sort of wanted to do?
Dijkgraaf: Well, I mean one thing as a scientist that you’re always struggling with is how to deal with the unknown.
Dijkgraaf: By definition, science is going into an area where nobody has gone before. And it could be the smallest, tiniest calculation, or could be the bigger theory, the most amazing experiment or result. But how do you deal with the unknown? And art always deals with this. The moment you put a pencil on paper and you start drawing. You’re already make something that nobody has ever made before. Or the possible shapes of objects of art are often infinitude, it’s much larger than, you know, what we can explore in science.
So artists have found ways to, you know, get comfortable enough and get confident enough to venture out into this big open space. And I think that’s something you can learn. For one thing, you can learn from your colleagues. One thing I really learned in art school, how important it is to jump into the deep together. Because, you know, at — at best you can hold onto each other.
Dijkgraaf: And when I went back to physics, that’s also something that, for me was a huge difference from my undergraduate experience. That very early on in graduate school I got to know some wonderful colleagues and, you know, we — we hang together. And we felt, you know, we had a lot of confidence I would say, overconfidence in what we could achieve.
Dijkgraaf: We were afraid of nobody or nothing.
Dijkgraaf: I must say, my graduate school time was a golden era. Because it was, you know, we just were the group of students, we were hanging together, we were, you know, we had — in principle, we could have the most amazing supervisor. Gerard ’t Hooft, you know, he’s one of the great, you know, all-time great physicists of the 20th century, I feel. You know, we were basically given this freedom to do whatever we want. And that was tremendous.
And so I remember, you know, we would, you know, just read all the pre-prints, at that time string theory was taking off. So we felt, well, that’s wonderful, you know, it’s like, a big reset of physics. And, you know, we all start with zero points and, you know, as a young graduate student you could jump in and be part of that. And it was a little bit like, the time when I was with my elementary school friend in the attic, you know we were just — we’re just going around reading papers. I remember, looking at some very complicated paper and then just throwing it in the corner of our offices saying, “Well, I’m sorry, we’re not interested in this.”
So kind of cocky behavior that now I would strongly object to as a professor. But, actually as graduate students, we had a ball.
Strogatz: Even as somebody who’s been thinking about physics for close to 50 years now, Robbert really stretched me in the next part of our conversation. His — he took me to the outer reaches of what I know in — in math and in physics. He and his colleagues are working at the extreme cutting edge of modern physics. It’s a field that we call string theory. And for them, the stakes are very high. They — they would say, and I think it’s not an exaggeration, that physics is in, uh, a time of crisis. They have two magnificent theories that both seem right, but they can’t both be right because they disagree with each other. So one of them comes out of Einstein’s work and it’s his theory of gravity and curved space and time that we call general relativity. And then there’s another theory, quantum theory, quantum mechanics, that describes the smallest particles, uh, subatomic particles.
It’s just our deepest and most accurate theory of the way nature works. And — and yet, when they try to put these two theories together as physicist like to do, they love to unify.
There’s been a great history of unification in physics. People unified the theory of electricity with magnetism and now we have electro-magnetism as a single object. Or Einstein showed that mass and energy were equivalent through his famous E=mc2. This idea that nature is sort of all of one piece almost — it’s almost a religious thing for them. Like when you hear about monotheism, you know, that there is one God as opposed to the older idea that there are many, many gods. This monotheist impulse in physics is very strong. And — and so when we hear, nowadays, that quantum theory, the theory of the very small, atoms and subatomic particles, doesn’t really play nicely with the theory of the very large, that cosmological theories of, um, general relativity and gravity, the difficulty of reconciling those two theories is considered one of the greatest, if not the greatest outstanding unsolved problem in physics.
Robbert and people like him work in this area of string theory, which is the — the great hope for bringing these two subjects together.
Dijkgraaf: But I think the real breakthroughs, came through, for me at least personally, through what are called matrix models.
Dijkgraaf: And, matrix models are like a car — cartoon versions of the gauge theories that describe, you know, all the forces in nature. All the other forces except gravity, so the strong and the weak interactions, standard model. And the concepts, the crucial concept in matrix models is symmetry. And so you can have, for instance, n ✕ n matrices, which means it’s like symmetry in the end dimensional space.
And there’s a long line of thinking that, um, out of these matrices somehow space can emerge. And actually, my own research I’ve had several incarnations of that idea. The very first one I remember well because it’s — it happened basically on my very first day, uh, as a fully grown-up physicist. That is my very first day when I was a postdoc here in Princeton. And, you know, I — the — the plane landed, I arrived, it was in 1989 and my friends came to me, “Did you hear it, did you hear it? There’s a revolution that just started.” And, uh, and literally on that day —
Strogatz: Wait a minute, so while you’re in the air something happened?
Dijkgraaf: Something happened, yes, literally that — that day I landed three papers had appeared that, or, what now are known as the kind of matrix model revolution. They showed that out of these matrices you could build strings, you could build gravity, you could build spacetime essentially.
And, I still remember well that afternoon walking into the office of Edward Witten, who explained to us how all of this is related to his grand vision of mathematics. It’s related to the geometry of Reimann surfaces and all of that. And that was a very exciting period because it — it showed that this is one of these surprises that are totally from left field. Now, everybody was trying to quantize space and time. Somehow bending gravity and space and time according to the traditional ways in which we would quantize other fields. And — and then this was like something completely different because space and time were no longer there. What you basically got — you got a theory of gravity together with space and time, all of it emerging out of something more fundamental.
Strogatz: So man, I’ve got to look at Bert, Bert’s looking at me. What are they saying in Star Trek? To — no, not just to go boldly go, but something about space, that’s it.
[AUDIO CLIP: The final frontier, these are the journeys of the star — ]
Space, the final frontier. Space is the final frontier, not — and I’m not just talking here about outer space, I’m talking about the space that we all move in, left and right, up and down, forward and backward. The three dimensions of space. I mean we think of space as a very fundamental, basic thing. We don’t even really think about it, it’s okay, it’s got these three different dimensions, the three axes, but there it is. And then we also talk about this other thing that’s even more mysterious, time.
We move forward in time from the — the present to the future. We wish we could sometimes go backward in time, we haven’t figured out how or whether that’s possible. But see, we got space and time, and these seem like the bedrock for everything that when you describe things happening in the world as a physicist it’s — it’s forces and particles and it’s all taking place in this arena of space and time.
Dijkgraaf: Basically, a long part of my research career is that, you know, that we might have things upside down, that we always thought of space and time as being, you know, on the bottom floor of the building. It’s the most fundamental thing there is. And in fact, Einstein wanted to construct everything out of space and time, even particles. But nowadays, we think, well, in some sense there — it might be one of the top floors.
Dijkgraaf: Or — or at least there is a basement, you know, you can go below space and time. Where you have mathematical structures that, you know, are very rich and interesting but one thing they do not have is space and time. Because space and time will emerge as an indirect property out of them.
Strogatz: So in Robbert’s metaphor, space and time are the ground floor. They are the bottom and everything is built on top of that and — and the history of physics has been figuring out the rules of the forces, gravity and electricity and magnetism.
And the types of matter, you know, electrons and protons and everything that moves around in this arena of space and time. And so those other things are seen as higher floors built on top of this bedrock of space and time. But the — the mind-blowing thing that Robbert brings up is that as physicists understand it today, and this is still very speculative and it may not turn out to be right, but the best current guess is that at the deepest level, space and time are not actually the ground floor.
There’s something below them in the basement, there is some more fundamental nature of reality that is almost like math itself in the form of matrices or something like that. That gives rise to what we used to think of as the most fundamental thing of all, space and time.
So Robbert was part of — he calls it a matrix model revolution. In the late 1980s there was a discovery and a series of papers that this point of view that matrices were fundamental. That by thinking about matrix models, mathematical descriptions of these matrices and their properties, other things popped out of them. Strings that people had been studying in string theory, gravity itself, and space and time. They all could be derived from this — this deeper level of reality of matrix models.
Dijkgraaf: No, we would like to see the great conundrum of quantizing gravity, of kind of marrying quantum mechanics and general relativity, gravity and say gauge theories or symmetries. That we would do it in a two-step process, now. First we have space and time. And then on top of the spacetime there are these gravity fields and they have to be quantized.
Strogatz: When you try to make Einstein’s theory of gravity match up with quantum theory to create a — a theory that people call quantum gravity, it just doesn’t seem to work very well. Certain numbers that you’d like to calculate come out to be infinite. And that’s crazy because we know that we can measure these numbers, properties of like particles. We want to measure the mass of the electron or the charge of the electron or things like that. They should just come out to be some interesting number but the theory in it’s naïve form keeps giving infinity if you just do the simplest way of quantizing gravity.
So this is a killer for the physicists, and they have not solved it yet. But the best hope for solving it seems to be to say that points that we used to think about as fundamental — Like if you picture an electron as just a point with no dimensions at all, just the tiniest conceivable speck. If you somehow imagine that a speck is not the smallest thing in nature anymore, but that it has to be a tiny loop, a little loop of what they call string, kind of tongue in cheek. That — that nature is not made of point-like particles, it’s actually made of tiny loops of some ineffable string. It turns out if you do that, that all these problems about the infinities go away.
This was an astonishing discovery that — that led to string theory becoming sort of the It Girl or the It Boy of theoretical physics.
Dijkgraaf: The matrix models that I started about were low-dimensional examples of this.
Dijkgraaf: It was gravity in two-dimensions.
Dijkgraaf: But now we have four and five and other dimensions. But, you know, it’s — it’s — it’s case by case.
Strogatz: What he’s saying is we’re trying to practice, we’re doing a warm-up problem for what we hope to do someday with string theory and these matrix models give us a warm-up problem. It’s like a sandbox for a little kid to play in. We’re going to play with these matrix models, we’re going to grow. up and get good at them and maybe someday, maybe a century from now, we’ll finally figure out how string theory really works and then we’ll have it, we’ll have the real theory of the universe.
When we get back, Robbert and I are going to get all soft and gooey about the most emotional dimension, the dimension of time.
Strogatz: Now we’ll go back to Robbert’s metaphor of the universe as a building. Let’s start on the first floor.
Dijkgraaf: If on the first floor if we have all these possible shapes, think of it as a large sculpture gallery where we have all these possibly beautifully curved spaces, of cosmologies or black holes or, you know, combinations of black holes.
And now in the basement, we have all these quantum mechanical machines. But we have a similar zoo there, yeah. So if you ask me or anybody what is the general pattern and, you know, how do I know that something is describing approximately a — a spacetime? I think these are fundamental questions that at this point, uh, we cannot answer. We are doing more — we’re more like botanists at this point than, you know, we have a kind of, uh, grand theory as, Darwin had of life.
Strogatz: Did — did this, uh, zoo of theories always predict that there’s one time dimension? As opposed to many possible times?
Dijkgraaf: That’s a terrific question, you know, whether the role of time and I must say, the — up to now, most success has been in theories with, building kind of these space — these emergent space directions.
Dijkgraaf: I would say emergent time is still very, very fuzzy. There are just a handful ideas, they’re much less rigorous. And, you know, this points out to perhaps a great defect of quantum mechanics. Because quantum mechanics is a completely revolutionary theory, it’s in, puts all our preconceived ideas away about what it means to predict the role of, you know, what does it mean to — to be something. You know, you have this wonderful property of being in many places at the same time. But somehow quantum mechanics treats time in a very conventional way.
So, I think, you know, the, probably the big breakthroughs in all this field, this whole field will come when we have a better understanding of the role of time, of emergent time.
Dijkgraaf: And then it’s a fascinating question whether you can have more than one time direction. I think this is often seen as a little bit of a crackpot subject.
Dijkgraaf: But I’m very proud that I’ve written a paper about multiple times.
Strogatz: Oh, good.
Strogatz: [LAUGHS] How many do you have?
Dijkgraaf: I think we —we actually had, you know, various kind of signatures. So if I were in that paper, we — we explored basically arbitrary number of times. So we tried to explore that, it’s not inconceivable that it’s allowed within string theory. And, you know, it’s — it’s one of these fascinating questions, you know. It’s a good example of a question that’s often asked by people who, you know, have very little understanding of physics. And it’s a wonderful question. You know, and it’s, you know, it’s often the so-called stupid questions that we don’t have answers to.
Strogatz: Well, time especially — I mean, getting emotional for a moment, feels to me if it even makes sense to say this, the most emotional of the dimension. That there’s something poignant about time.
Dijkgraaf: It is.
Strogatz: We all know that, right? Memory and loss and future and space doesn’t compare to time as far as emotional significance.
Dijkgraaf: I agree, and you know, I have my own spacetime theory about the arts. So, you know, you can—
Strogatz: Let’s hear it.
Dijkgraaf: So, so you — you can think of all — if you explore space you have typically, you know, the visual arts, you know, the sculpture, painting, et cetera. If you go in time you have music. And, you know, I think I’m fair to say that on average the most emotional form of the arts is music. That, you know, people can really start crying if they hear a certain song or to even a certain movement. It’s, it’s a different way and I think it’s because, you know, we experience space in a very different way than time. You know, we experience space very freely, we can move, you can — we wave our hand and, you know, we go left and right, it’s no problem. Time you only go forward, so I feel we are really — we’re strapped into a little car on the rollercoaster, which is time.
Dijkgraaf: And — and we push forward and that’s why I think indeed time is much more emotional because we have to experience it second by second. You know, there’s no way in which we at once can see the whole plot of a movie or play or we, you know, you — you can look at the score of a symphony. But you’re not as emotionally touched as if you hear it note by note.
Dijkgraaf: So there’s something in which we experience time I think as human beings, completely different from space. But then in Einstein comes and say, “Well, they’re interchangeable.” No, you, the moment you start moving, you’re somehow confusing a little bit space and time. And this is, again, one of the great conundrums, um, the way in which we formulate physics and certainly the way we experience reality. We definitely do not experience space and time on the same footing.
Dijkgraaf: Even our physical theories make a great distinction. You know, often it’s joked that physics science is all about one question, you know, “what happens next?” So it tries to predict the future knowing the present.
And that’s a very, very different approach than we have to space direction, so yeah. Time is still, kind of sticking out in many ways.
Strogatz: [LAUGHS] Well, so when we talk about the emotional aspects of these different dimension, it — it raises in my mind this broader question of — of feelings in general. When we do science — since feelings, of course, are — well, traditionally important in art. I don’t know if in modern art we still talk about feelings. But — but what about these questions of aesthetic feelings and their role in scientific or mathematical creativity?
Dijkgraaf: Yeah. Well, for me it always been, a remarkable experience that first of all, there’s certain areas in math and physics that you strongly emotionally react to. You know, we use the word beautiful or elegant without any hint of irony. You know, I think actually truly feel this is a strong ascetical, emotional response.
And then for me, the great surprise has been that you talk to your colleagues and they actually have the same feelings, you know? They are — they’re equally enthusiastic and they — the — and they — they might even point you to something that’s even more beautiful, you know, and they’re right. And then you enjoy it. So there is this great kind of consensus in physics and math, I would say, largely, about what we find beautiful. And sometimes I joke, you know, because the words “beauty” and “beautiful” are — these words are very little used in arts these days. So, you know, because you — a piece of art, an artwork is, uh, interesting, it’s provocative, it’s stimulating, you know?
Dijkgraaf: You have all these other, descriptors. And I feel, you know, beauty is being chased out of arts and it felt, found refuge in science.
So the remarkable thing it’s like almost an upside-down world, you know, where the sciences which are often were accused of being, you know, in the — the romantic periods of being, you know, heartless and rational. And only about results and about mathematics. You know, the — the famous saying that, uh, Newton unweaved the rainbow.
Dijkgraaf: And that nowadays, I think, t he only place I can freely talk about beauty in some senses in the world of — of science where, at least it has, kind of, uh, good understanding, you know. We have some consensus about it.
Dijkgraaf: And it’s — it’s I think it’s kind of remarkable the role it has. It’s, it’s — I think it’s an emotional response but often I also think of it as, it’s part of our intuition about — it’s a compass almost guiding us into the unknown.
Dijkgraaf: So I think, you know, what — when with talk about, uh, the beauty in math sometimes I joke, you know, it’s the impact per symbol in a mathematical formula. You know, it has to do with the conciseness about the depth of the ideas. It’s basically how we, uh, intuitively value it’s future use. You know, how far it will push us ahead.
And it’s a remarkable fact that, some of us and I think we the truly great scientists, mathematicians, have this and even much greater amount of this kind of intuition. They are able to pick up certain elements in math and say wait a moment, you know, this is — this is a little diamond. This is something that, you know, will — has great value, this will help us and this will guide me in my research. And sometimes I think that beauty is an acquired taste. So it might be the first time you read about it and you hear about it, you know, it looks very convoluted. It’s very, almost hidden.
But then you spend more time reading about it, learning about it, and then sometimes this kind of beauty really appears to you. And you feel it.
Dijkgraaf: And it’s, it’s not only very useful in a sense, it guides your future research, but let’s — also be fair, you know, it’s one of the great joys of being a scientist.
Dijkgraaf: To experience that beauty, it’s like being, you know, a mountaineer and standing on a summit and seeing this vast landscape. It’s, you know, it’s just a sheer joy of being at that moment and — and feeling the depth of these ideas. It’s astonishing.
Strogatz: Well, uh, this, so [LAUGHS]. There’s a little bit of tension though in — in science between beauty and truth. Not tension may not be quite the right word, but, there’s a subtle relationship between them that was brought home to me in a remark I heard you make in — in one of your talks that I — I happened to listen to on the internet where you used the analogy of Odysseus being — asking to be tied to the mast of the ship because of the — you know he wants to hear the beautiful Siren song. But he knows how dangerous it is, it’ll carry him into the rocks if [LAUGHS], you know, if he guides the ship that way.
Dijkgraaf: Yeah, exactly.
Strogatz: So there are, of course, possibilities that beauty could be the Siren song for us in physics.
Dijkgraaf: Yeah, beauty can be very — mathematical beauty can be very seducing. And of course, you know, if you put on your hats as a physicist you want to describe the structures in reality, in the world out there.
Dijkgraaf: Or I would say, you know, the possible world, you know, it — we not only, of course, describing physics but really out there we — we’ve described possibilities, you know, you’re also looking at, you know, other possible solutions to the laws of nature. So — and one thing, of course, we have learned that, you know, math can be extremely relevant in describing the world.
And mathematical beauty can coincide with the truth of describing reality. But, you know, sometimes you can declare victory too early.
Dijkgraaf: So I think there are many examples in the past where, you know, physicists were seduced by mathematical beauty and they thought, you know, the idea is so beautiful it has to be true. And this can be very painful. Essentially, I think if you look back on all these cases I would say that they were wrong, not because they were valuing, overvaluing, overestimating the beauty of the — this particular branch of mathematics. But they were simply not aware there were even more beautiful branches of mathematics.
Dijkgraaf: Honestly, uh, and —
Strogatz: Okay, you’re all in. You’re all in for beauty.
Dijkgraaf: No, and I think there — but this is, I think there is a wonderful phrase, I think Feynman describes this. He says, you know, you’re typically — the life of a theoretical physicist, you know, you have this wonderful theory. And you’re all happy and it’s all beautiful and fine and elegant. And then reality, uh, intervenes. And then your experiment is old. There might be another particle, there might be a number, and it doesn’t fit. And you feel, “What’s this, you know? I’m — I’m somebody is, destroying my — my, uh, work of art, you know, there’s like scratches on it and it’s got — got uglier and uglier and the —” That is a very depressing thought.
But often, what happens, and we have — we have went through this transition a few times. That, when at some point, when the dust settles there are many more pieces and they fit together in a new scheme that’s more beautiful because it’s more powerful.
Dijkgraaf: It’s — it’s something you have to learn and sometimes — let’s be fair too, that, you know, physicists are not always the most sophisticated mathematicians.
And it’s almost like an art, you know, people might love, you know, a realistic art and when they for the first time see an impressionist painting or they see a Van Gogh or they see, you know, abstract expressionist painting they might think, “What’s this? You know, we’re going in the wrong direction.” But after sometimes — you may say, “Wait a moment,” you know, or “This piece of art expresses much more, it’s much more powerful.”
Dijkgraaf: And so it think that has happened again and again. So in some sense, I think the art here is to cherish these pieces of mathematics. Keep them in your toolbox. But not try to use them, so to say, you know, to force — force this upon nature. And what’s the saying, you know, if you have a hammer everything looks like a nail? And there have been examples of this, you know, of certain branches of mathematics who try to hit everything with that particular mathematical concept. If fact, new mathematical concepts can emerge and if — if I’m try to make a bet, say in 50 years, if we see what the kind of mathematics is that’s being seen as beautiful and powerful and elegant and relevant to nature, so — truthful too — it’ll probably be a branch of mathematics that isn’t with us right now.
Strogatz: Mm. One of the most fascinating developments in both physics and mathematics these days is — is that the flow of ideas has started recently to go from physics into mathematics especially because quantum mechanics is such a bountiful source of radical ideas that — that math itself is being reshaped.
Dijkgraaf: Yeah, I think that’s one of the most wonderful developments. That the hard-won intuitions about quantum theory in physics which were really, you know, pressed upon us by experiments — you know, I’d like to say, you know, you could meditate for centuries, you would never come up with the rules of quantum mechanics, you know, they’re really, really bizarre.
But actually, it’s describing reality, 100 percent of reality. And so the ideas from quantum mechanics are also useful in understanding abstract mathematics. In fact, are, could be the kernel of developing completely new mathematical concepts. It’s fascinating, there’s great need of this new kind of mathematics, I like to call it quantum mathematics in physics. But I think it’s also very important movement just within pure math itself.
Strogatz: When he speaks of quantum mathematics, he means infusing ideas that come from physics, from the study of the real world into the pure ideal world of mathematics. That there’s normally a feeling that we use math to study nature. And that the flow goes in that direction from math into physics. But quantum mathematics is this reversal of the current that has been happening in the past few decades with some stunning successes. Where ideas from physics are now being imported back into math and revolutionizing math itself, separate from its connection to reality.
Dijkgraaf: And if you think about it, there actually is a lot of resonance between the ideas in quantum physics and modern math. For one thing, quantum physics does — you know, you have this sum of our histories. Basically, everything happens at, you know, at the same time in quantum mechanics. It’s just with different kind of probabilities. So if you’re a quantum physicist, you’re forced to look at the whole collection of natural phenomena and somehow organize them. And I would say this is also very much the modern mathematical point of view. Modern mathematics is not so much about studying one particular member of the family of spaces, of numbers, of symmetry groups. But looking at the whole family and their mutual relationships and perhaps even the relationships, the modern relationships so it’s a much more comprehensive point of view.
The great thing of quantum theory is that it, you know, allows you to calculate with this abstract principle. So the wonderful thing is that, you know, although in quantum theory, as I said, you know, everything is happening at the same time and we have this bizarre concept of the sum of all histories … in the end, you get a number out, you know, a probability amplitude, a complex number. So quantum theory is also a calculational scheme to deal with this infinitude of possibilities. And I think that’s exactly what the doctor ordered to mathematics.
Dijkgraaf: Because it’s — it’s a way to study, you know, many different spaces at the same time. To get various topologies or count objects or there’s so many kind of almost combinatorial problems in math that are like perfectly set up for a quantum treatment.
Dijkgraaf: So I think mathematicians are slowly embracing this point of view. I must say, there’s still a lot of work to be done.
And, you know, and this perhaps shared the — the biggest frustration in this is that, you know, the concepts in physics are very fuzzy. So I, you know, there are wonderful ideas about space and time and quantum theory. But we’re really struggling with these concepts and that makes the communication with mathematicians sometimes difficult. Because if I talk about a number or a space or a group or any other well-defined object that I could find in my mathematical encyclopedia, you know, I’m — I’m — I can talk to my mathematical colleagues.
Unfortunately, the concepts that we’re struggling with right now — what is quantum gravity, what are emergent space times — these are all really fuzzy, uh, unclear objects. And I can’t give an axiomatic definition so sometimes mathematicians, you know, I noticed they only start listening to you when you have a very well-defined conjecture or result.
Strogatz: [LAUGHS] Well, that, of course, that depends on the nature of the mathematician. There are some who, you know, in the more applied areas like that fuzziness because we know that there’s — I say we because I think of myself in that space in the applied math world. That — that there — there’s so much fertility in the fuzziness.
Dijkgraaf: I think that’s right, and — and so, um, I wouldn’t be surprised that as often happens, you know, with ideas that come out of physics, you think about, you know, the birth of geometry, you know, the ancient world or the birth of calculus, as you know very well, that described wonderfully coming out of studying of mechanics. And the impact that, you know, calculus or analysis or geometry in all its inclinations have had on math, you know, they’re really the big, huge areas in math. That the ideas of quantum theory should be equally productive.
Dijkgraaf: And if you’re honest, you know, you would say, wait a moment, quantum theory is the chosen language of reality. Nature actually picked quantum theory, the language of quantum mechanics. And — and preferred it, you know, compared to classical mechanics and, and geometry and all these concepts that we love. You know, differential functions and lines and, and, and planes and points. So nature probably has an even more powerful set of ideas. So I think, you know, it’s — I’d be really surprised if in the long run, the impact of quantum theory, which has taken over almost all of physics, will take over a large chunk of mathematics.
Strogatz: Mm. Well, let’s leave it there, Robbert. That is a very fascinating forecast for the future. It reminds me a little bit of something a colleague of mine once said. I — I was at MIT and my colleague, Dan Freedman, who — who you may know from supergravity.
Dijkgraaf: Yes, I know him very well.
Strogatz: Yeah. So Dan said to me that I needed to learn quantum mechanics. He said if — if you just knew some quantum mechanics you could be dangerous.
Strogatz: And it think that may go for all of us in mathematics.
Dijkgraaf: I think that’s a wonderful line. [LAUGHS]
Strogatz: All right.
Dijkgraaf: It is very dangerous, it is.
Strogatz: Next time on The Joy of x, Corina Tarnita will reveal the magic of teeny, tiny termites. The Joy of x is a podcast project of Quantum Magazine, we’re produced by Story Mechanics. Our producers are Dana Bialik and Camille Peterson. Our music is composed by Yuri Weber and Charles Michelet. Ellen Horne is our executive producer. From Quanta Magazine, our editorial advisors are Thomas Lin and John Rennie. Our sound engineers are Charles Michelet, and at the Cornell University Broadcast Studio, Glen Palmer and Bertrand Odom-Reed, though I know him as Bert. I’m Steve Strogatz, thanks for listening.
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