# Quanta Writers and Editors Discuss Trends in Science and Math

## Introduction

Why doesn’t our universe make sense? What is time? What is life? On Friday, more than 200 readers joined writers and editors from *Quanta Magazine* at the Simons Foundation for a wide-ranging panel discussion that examined the newest ideas in fundamental physics, biology and mathematics research, including the questions of whether our universe is “natural,” the nature of time, the origin and evolution of life, whether mathematics is invented or discovered and what role it plays in science and society. These are just some of the topics presented in *Quanta*’s two new books published by The MIT Press: *Alice and Bob Meet the Wall of Fire* and *The Prime Number Conspiracy*.

If you missed the event, we are making a podcast of the conversation available here (click “play” to the right) and on iTunes. (November 29 update: A video of the event is now available below.)

**PANELISTS**

• Robbert Dijkgraaf, director of the Institute for Advanced Study

• Kevin Hartnett, senior writer at *Quanta Magazine*

• John Rennie, deputy editor at *Quanta Magazine*

• Natalie Wolchover, senior writer and editor at *Quanta Magazine*

**MODERATOR**

• Thomas Lin, editor in chief at *Quanta Magazine*

**PURCHASE BOOKS ON AMAZON
**•

*The Prime Number Conspiracy*

•

*Alice and Bob Meet the Wall of Fire*

## Transcript

**Thomas Lin:** Thank you for coming tonight. Welcome to the *Simons Foundation Presents* panel discussion featuring writers and editors from *Quanta Magazine*. I’m Thomas Lin; I’m the editor of the magazine, and tonight we have a lot of interesting things to talk about. We will be discussing some of the new ideas and emerging themes in physics, biology and mathematics, some of the same topics covered in the magazine and that are also presented in our two new books: *Alice and Bob Meet the Wall of Fire* and *The Prime Number Conspiracy*. One quick note about the video you just saw — we will be saving audience questions for the end and will be bringing microphones to the aisles, so you don’t need to use the speakers in front of you on the chairs. I want to thank the Museum of Mathematics for being here tonight to help make the books available to all of you at this event, and also to MIT Press for working with us to publish the books.

Tonight, as those of you who read *Quanta* are probably familiar with, we will be talking about some fairly profound ideas and somewhat highly technical — somewhat technical ideas — but we will, again like the magazine, try to make it as accessible and as jargon free as possible.

It is a great pleasure to introduce our four panelists tonight. To my left is Natalie Wolchover; she is the senior physics writer and editor at *Quanta*. She has been winning lots of awards — which are all well deserved. In 2016 she won both the Excellence in Statistical Reporting Award and the Everett Clark/Seth Payne Award for Young Science Journalists, and just last year she won the prestigious American Institute of Physics Science Communication Award. Next to her is John Rennie, our deputy editor, who leads our biology coverage. He is the former editor in chief of *Scientific American*; he was also the host of a TV series on the Weather Channel called Hacking the Planet. Next to him is Kevin Hartnett, our senior math writer, and he has had several articles republished in the annual Best Writing on Mathematics anthology. Last but not least we have Robbert Dijkgraaf. He’s a mathematical physicist and a *Quanta *columnist. He is the director of the Institute for Advanced Study in Princeton, New Jersey, and he’s also a painter and extraordinary science communicator. So I’m glad to have all of you here. Thank you for coming and thank you all for being here tonight, both the live audience and those watching on the live feed online. Now we don’t have a lot of time here, we only have an hour, and we have a whole universe of ideas to talk about. And so we’re going to just jump right in and start with the first question, which comes from our science collection, *Alice and Bob Meet the Wall of Fire*.

The first question in the book asks: Why doesn’t our universe make sense? Many of you are probably aware — and in 2012 there was a major discovery in physics: The Higgs boson was detected at the Large Hadron Collider in Switzerland. But physicists were hoping to also detect some other particles as well that would help them make sense of our universe. Those particles have not been detected, which leads to questions of whether our universe is natural — which I’ll ask Natalie about in a second — and also to this fear that physicists have had for a number of years that if they don’t detect these particles, that they will essentially face what they call a nightmare scenario. So with that I want Natalie — if you could start off by just setting this up for us: What do physicists mean by the universe being natural or unnatural, and what is it that is behind this nightmare scenario?

**Natalie Wolchover:** Yeah, so I think a lot of people, when they hear the idea that the universe is unnatural, that just inherently seems like an oxymoron. And so over the past five years I’ve kind of — hopefully — gotten better at trying to explain what this means. And so what naturalness means in a scientific sense, is basically that the variables in the equation can take different values. And the values themselves of different constants and things in the equations should all kind of work together in a way that makes sense. And there’s a more precise way of thinking about it, but with the Higgs boson mass, the problem is that this particle — the Higgs boson — has a mass which is many, many, many orders of magnitude different from what’s seen as the fundamental mass in nature. And so this enormous hierarchy between the two values basically seems to physicists like something that needs to be explained. And they had this brilliant explanation, which was supersymmetry, that has been sort of the paradigm for a long time, even before supersymmetry was even searched for. People just kind of believed that it was true, and that you would just find the particles because it was such a beautiful way to make the constants all kind of work out together.

But instead they did not find these supersymmetric particles at the Large Hadron Collider. And that’s really created a fascinating time in physics, where this kind of paradigm and principle of naturalness that was leading people to build the Large Hadron Collider to look for these particles — that all basically failed, and now physicists don’t know how to make the equations natural. And they also don’t even know if this naturalness idea is even — was a principle they should have been holding in the first place.

**Lin:** Another way to think about it, I think, is that much of what we have now in terms of some of our bedrock theories in physics came from the idea that the universe would be natural, right? And yet now, if in fact these variables or constants that you’re talking about are in essence fine-tuned, then it almost leads you to believe that our universe is unnatural, and maybe just one of many universes out there. And so talk a little bit about what that means — if there’s actually a multiverse, and do physicists really like that idea?

**Wolchover: **Yeah, so, exactly, if — if the equation, constants in the equations, kind of all make perfect sense with each other then you can imagine that our universe is this kind of inevitable structure where everything is kind of cohesive, and there’s some underlying kind of theory that — that predicts everything based on, you know, pi and two and, you know, maybe a factor of four somewhere, but very simple kind of math that underlies everything. But if you have these really weirdly, you know — and when I say the constants are weird I mean like one is something like a billion billion billion billion times bigger than another constant in the equations, and so there’s this huge hierarchy. The only — a way to get that is if everything is just kind of random in our universe and the constants — there was some sort of miraculous — almost really, really unlikely fine-tuning, where, you know, the Higgs mass was really tiny even though the fundamental mass is really huge, and that difference allowed our universe to give rise to life, and so that’s why —

**Lin**: Other universes would have a different Higgs mass —

**Wolchover: **— and in order to have that kind of very unlikely event, then you have to have a huge variety of other universes in order for ours to be realized. So that’s where the multiverse idea kind of comes in, but it’s — people hate that idea because it’s untestable —

**Lin**: Let’s take a poll — who hates the idea?

**Wolchover:** — I mean it’s kind of like people who have already acclimated to the idea of the multiverse, so, like, this makes perfect sense. There’s many, many universes, why should our universe be the only one? Why should — why should there just be one universe that happens to have life? And not a — people who are against the multiverse think that it’s untestable, and that this is kind of a cop-out, saying, well, we can’t explain this big hierarchy so let’s just say that it’s all random and there’s no explanation.

**Lin:** So I want to also ask Robbert about this since he’s the director of the Institute for Advanced Study. Some of the researchers at the institute are studying this exact problem and — and I wonder if, you know — I want to get your perspective on whether there’s a sense or — that the universe is in fact natural and we just need to find the right theory — or if things are trending toward sort of acceptance that it’s unnatural, and sort of buying into in thinking about how the multiverse idea might help us.

**Robbert Dijkgraaf:** First of all, all my opinions are, here — are just my own, [LAUGHTER] I’m not speaking on behalf of others, because it’s actually something very much in debate. And I think actually this is kind of wonderful expression where Feynman explains how particle physics developed in his lifetime. He says, no, we have this great beauty — everything fits, and then there are little experiments and they do not quite fit. And you try to hold things together — at some point you have to give up. It looks very ugly, and then after a very confusing period it starts to hang together again, and it’s beautiful, but it’s beautiful in a more subtle, more deeper sense. So we know, we go through these transitions. So I feel particle physics is very much at that point, because there was an obvious scenario with the Higgs particle where indeed, you know, certain other particles would be found and everything would fit together. They do not. But you know, it’s not clear that we have to switch to the other frame and say, well, it’s all chaos, it’s all random etc. So I think we are really in this kind of mixed state.

But I think an important question we should realize even about the Standard Model, as it is: There are lots of numbers that appear there and just — you know — even in the masses of the particles that we all know, there are huge differences, you know. They differ as much as a whale compared to a fly. And none of us have any idea whether these are real constants of nature or are they determined by some other set of equations. Are they the solutions of equations, or are they constants of nature that somehow nature picked out for once and for all? We simply don’t know. But if you are — if you are theoretical physicists and you write down any equation that should describe the universe — and it could be a kind of cartoon version of the equations we have — it very easily happens that you create more than one universe, right? Because in some sense it’s — if you think about it we almost are saying that, you know, if there is a unique universe we are searching for laws of nature to describe physics or science in terms of you have an input, then something happens and we calculate the output. But we even are trying to determine there’s only one solution, and that solution is us. So that’s — that’s rare. I don’t know of any equation in science that has only one solution. You could have a little potential or something** —** so think of it as a mountain landscape, but very simple — but only one valley. It’s very easy — there’s a little bump, and there’s a second valley, and then what happens to that solution?

**Lin:** Right, right.

**Dijkgraaf:** So that’s something that, you know, I think cosmology confronts physics with. And I think, you know, it’s a different set of questions because we are not describing what’s often said — you know, science is about answering the question “What’s next?” But it’s really, you know, why this whole thing? And I think that’s actually where we’re struggling with it. I think it’s very exciting to see that although all of us hate, I think, the multiverse — in the sense, you know, it’s free for all — we do know that we had a whole sequence over many centuries where our sphere of what we think of as the world, the universe, has been growing — from our planets to our solar system to our galaxy, so there are these increasing — and we feel entirely comfortable talking about, perhaps, galaxies that we will never see, that are outside the Hubble radius. That’s simply, you know, because of the expanding universe we will not be able to ever witness.

**Lin:** Right. So in some sense it’s almost as though —

**Dijkgraaf: **So we feel comfortable with that. I think, you know, there are things I feel quite comfortable — something happening, you know, a few blocks away. I can’t observe it but probably it’s out there —

**Lin:** So in some sense it’s — in thinking only of our own universe —

**Dijkgraaf: **We might be provincial. We might be trying to do something — and so I think we should always be aware of this feeling that Feynman expressed, that, you know, you feel, well, it’s going in the wrong direction. But what you’re doing is something — also like tunneling through a barrier — and you might end up in a new valley that’s much greener than the one that we are in right now.

**Lin:** So this leads very well to the next question I have which, you know — when you think about the multiverse idea, you think about things like string theory. These are all ideas that are powerful tools and ways for thinking about our universe, but there’s really no empirical evidence for these theories. And so I want to talk a little bit about what we mean by evidence in science.

Natalie, in 2015 you covered a debate in Munich just about this question of, you know, given that it’s getting harder and harder and may not even be possible to find empirical evidence for some of the — these most sort of exciting theories that are happening at the forefront of physics right now, what exactly does it mean to — what counts as evidence in that case — you know, is something that’s mathematical enough? Do we want to follow sort of the Karl Popper idea that something has to be falsifiable in this way, which a lot of these ideas that we’ve been talking about today are not. Talk to us a little bit about what were some of the key arguments that you covered during that debate.

**Wolchover: **It was a really amazing opportunity to go to this conference in Munich and kind of have this really big theme of physics right now come to a climactic point — where you had people from both sides coming together and basically arguing about whether string theory and the multiverse and these ideas that are not directly testable are scientific, or whether science is kind of going off course by focusing on these kinds of theories. And one big — I think one big theme that emerged is there — there’s so much anger and passion on both sides — and that is usually a mark of confusion and a confusing time where things are subject to change rapidly. And I think, what was kind of — one thing that was clear, is that — it’s not really about physicists kind of going down the wrong path and becoming more interested in speculative theories when they should be more grounded. It’s really that we as a humanity, we’ve kind of gotten to the ends of the points where we can really push further. Like we — we can no longer build instruments that can probe to a much smaller scale than we have; we can see out to the cosmic horizon but we’ll never see further than that. So we’re kind of coming to the end of where we can actually test our ideas. But theorists have, of course, because of just logical kind of thinking about how things would extend, have gone much further in both directions. So they’re thinking about the multiverse on large scales, and they’re thinking about strings and string theory on the small scales, even though we’ll never see that far down.

So a big part of the debate is whether there are other kinds of evidence besides direct empirical evidence. Some people think it’s just blasphemy to even suggest that there might be ways of kind of gaining confidence in a theory that is not having to do with experiments and testing the theory. But other people think that just by looking at the logic of an idea you can gain confidence in it. And so the big question is, with string theory — are physicists just kind of fooling themselves or dazzled by the beauty of it, or are they seeing real kind of qualities in string theory that seem compelling enough to actually believe it.

**Lin:** We’re going to come back to this question in a bit. Thanks for setting up that debate for us. You know, thinking about evidence, in the realm of physics there are clear examples of very testable ideas and experiments and — and empirical evidence for what we know in physics. But there are, as Natalie was saying, many newer theories that rely more on mathematics to sort of prove that at least it’s something that’s logically consistent. Kevin, in the world of pure mathematics, which does not rely on the physical world and deals with, you know, abstractions, where evidence is not inherently empirical, what is evidence in mathematics, what counts as evidence?

**Kevin Hartnett:** Sure, I mean what counts as knowledge is a proof, and short of that, you know, kind of what matters — we ran an article in *Quanta* a couple weeks ago on this concept of the rank of elliptic curves and whether or not the rank is bounded: Is there a cap on how big it can be? There has been this — kind of this — these sea changes over the last century. A hundred years ago people thought for sure it had to be a cap, and in the ’60s and ’70s they thought, no way, it must be unbounded. And now they’re kind of — opinion has shifted back toward there being a cap. But kind of, why? Right? No one has proven it one way or the other, or even really come that close.

And so I think they’re kind of — there are different types of evidence that matter and that move opinion. There’s computational evidence, and as computers get better you can do more computations. And if every time you find a solution to the Riemann zeta-function, it lies on the critical strip that makes you think every solution lies on this on — every zero of the function lies on this strip. So there’s computational evidence. There are heuristics — if you kind of compare something — if you’re interested in understanding a property of Object A and you can’t prove that property, you know, but you know that Object A is kind of like Object B, and Object B has this analogous property, and we can prove that the property holds for B — that’s again heuristic. That lets us think that the property also holds for A. And you see that happening sometimes in mathematics, and physics often provides a kind of a source of evidence for mathematical things being true.

I know in the collection Robbert has a column that talks about mirror symmetry, and mirror symmetry is something we’ve covered in *Quanta* as well — so physicists kind of proposed that there are these two models of string theory that seemed equivalent, and when you think about those models in mathematics, it tells you that these two types of geometric spaces are equivalent — these symplectic spaces and these complex spaces. So physics kind of proposes that their physical theories work if these two types of spaces are equivalent. And then mathematically you might then go out and actually try to prove that these two spaces are equivalent. So that’s kind of a source of evidence in math, and I guess I would say, what is the upshot here? Because, you know, evidence is not a proof. And evidence can in some ways** — **often doesn’t lead you at all closer to a proof than you were before. But I think the main thing it does is it kind of emboldens mathematicians to pursue lines of inquiry. I talked to a mathematician once who told me that he’d been working on this problem for a long time and then he heard that some other mathematician proved — you know, proved the conjecture was true, and he didn’t know what they did to prove it was true. But just knowing that it had been proven true, like, you know, the next day he figured out his own proof [LAUGHTER], because he just suddenly had this new faith, that it probably worked. So I think that is kind of a major — that’s something that evidence can do in mathematics.

**Lin:** That’s great. Thank you, Kevin.

**Lin:** You know, we live in a time right now where I think evidence — at least arguably evidence — and facts are almost existential issues for our democratic nation. And, but you know, I wanted to ask Robbert — as a mathematical physicist, because you work sort of in both worlds — to put nonempirical evidence into context for us: When is it useful, how does it help us move science and mathematics forward when we cannot rely on evidence from the physical world?

**Dijkgraaf: **Well, by the way, your — your anecdotes suggest that you should have fake news outs [LAUGHTER] that the famous conjecture was proven, because then somebody might feel, oh, I can do this, too.

**Lin:** Like if we didn’t — if we don’t have empirical evidence for — for, say, a physics theory, can we just then say: I like this theory better?

**Dijkgraaf: **Even in pure math, there is kind of almost the art form of the conjecture, right? So what makes a good conjecture? It’s by definition something that hasn’t been proven. You have some suggested evidence, perhaps some cases, etc. — so I think both the math and physics — kind of the dotted line where we want to go — is as important as, actually, you know, taking the steps and proving the points. In fact, I think, you know, it’s these more visionary elements both in math and physics that are motivating us. Why are you working on a certain problem? Because it’s not just one step further, but it’s a step along a route that will take you to a completely different world.

So in some sense you have to make sense of a few facts. If you’re a number theorist, you know, perhaps you can work on concrete cases. If you are a theoretical physicist, now nature has given us a few hints that the world is not yet finished. So if you think of the standard model, there are things like neutrino masses, there’s the Higgs particle that doesn’t really quite work, you know, there is the difference between particles and antiparticles — so you somehow know this is, this has to — nature has found a solution to it, and I think it’s that kind of soft thinking — which is incredibly important because it actually guides our thoughts. And you know there’s a long tradition of, you know, theoretical physicists getting things wrong, but there’s also a long tradition of somehow being — that their theories are smarter than the person who proposed it, that they’re holding back.

Now Einstein is famous for many of the consequences of general relativity — he felt uncomfortable with, you know, Dirac basically on paper predicting antiparticles, but nobody had seen these particles, so you held back. So I think it’s very important that we also, you know, encourage guided imagination: guided with previous examples, guided with previous experiments, etc., push theorists**, **because you can also make the mistake —somebody said in one of the discussions, who needs the thought police? And I think indeed — in thinking we don’t need policing, we just need smart thinking. And it should be based on sometimes what we call intuition. But I think intuition is our way which by experience we learn to deal with kind of lack of data, error bars in our thinking, and trying to find whether there is something that, you know — can you kind of fit in some sense a curve through this, and that’s — so part of theory should also be that you’re out in front of the data and guide — and it’s actually quite literally what we need. Because in modern experiments people don’t realize that when they create billions of collisions a second, but the machine can only keep, so to say, a few. So you — even the experimenter, the one thing they want to know is what should we look for, because they can’t look for everything at the same time.

**Lin:** So in a way you’re speaking to this dynamic between theory and experiments. And sometimes the experiments give rise to new theories, so as you need theory to know what to look for with your experiment —

**Dijkgraaf: **You need both. So in that sense, I think if you have a diverse community where people are searching in many different directions, that’s good. If we are all, like, on one little cluster thinking the same way, and we all say yes to this or no to that, I think that would be very dangerous.

**Lin:** Thank you. Now I want to turn to the idea of biological evidence. Because, you know, in physics — at least in theoretical particle physics —we’re sort of — it seems like we’re kind of hitting a wall in some ways. But with biology it’s, you know, biology — biologists, I think in general, insist on empirical evidence, right? And yet biology is a very complex subject, also. The systems are extremely complex. So it’s not necessarily easy to make sense of the data that you have, and make sense of the evidence that you’re collecting. And also, if you’re thinking about things like evolution, and studying how evolution works, obviously — well, not obviously to everyone, but I think to people who care about evidence, obviously — there’s an overwhelming amount of evidence for evolution itself. There’s no question that evolution itself occurred, but there are questions about specific mechanisms by which it works. And thinking back — obviously we can’t go back in time to see exactly what happened — so there’s all sorts of challenges to understanding biological evidence. John, I’d love for you to talk a little bit about what new kinds of data and evidence we’re gaining in biology to help us better understand things like evolution and what life is.

**John Rennie: **Sure, so obviously all kinds of information available to us can find a way of working a way into different sorts of biology, and biology is many different things in many different situations, just as life itself is. I feel like biology in some ways is faced with a similar sort of problem to the ones you were talking about with physics, in — in that we in — in the biology world, there’s also a certain sort of level of tension because our understanding of life as being a result of evolution is that there’s obviously a huge element of randomness that is the germ of the — the source of all the variation on which life begins. But then other things that are applying much more regularity to then lead us to certain sorts of ends. So contingency is obviously very important in our understanding of biology, in the history of life on Earth and anyplace else. But what can be intellectually unsatisfying about that is that you don’t want us to just think that life is just a result of some random thing that just happened. We would like to think that we understand certain sorts of principles that made certain kinds of things more likely than others. So there’s a certain kind of balance that we’re trying to go to with that.

And I think where that’s surfaced a lot in — in a couple of — the two ways that I think strike me as really particularly fascinating these days, and would drive a lot of the biology: One is that our traditional view of life is one that’s based on sort of being something kind of like what we are. You know, we would talk about — there is no universally agreed-upon definition for life. We talk about it as being sort of like a thermodynamically open system; we talk about its being made of cells; we talk about it having like a property of homeostasis, and that these life reproduces and so forth, but what’s interesting is that when we’ve been learning more about life and the diversity of life, what we’re seeing is that a lot of the things that we thought of as sort of the mainstream — the stuff like us — but we are weirdly more of an edge case than we thought. And that a lot of the things that we thought were the edge cases are much more mainstream.

So — viruses. Everybody — whoever has that sort of sense of, oh life is made of — you know, life is independent and it perpetuates itself and it’s made of cells — you would look at viruses and you would think well, they’re a weird sort of parasitic thing. Are they alive? Eh — let’s try to move past that. Well, what we’re discovering is that viruses are in fact a gigantically central part of all of life, that there are more viruses and more types of viruses and an abundance of — abundance of viruses in nature. That’s hugely more important than we ever thought. And in fact, that there are viruses that are bigger than some types of bacteria. So all the different sorts of things that we were using to sort of marginalize some types of biological phenomena because they weren’t like us turn out to be much more important than we thought.

So one thing is that in that search for the right sort of evidence — we have to pay more attention to those things that we didn’t think were important. The other thing is that we’ve traditionally thought of — certainly Post-Darwin — that we have tended to think that we needed to really look to evolution as being the source of all of the order that we would find in life. But what we’re increasingly finding, and what is a fascinating area of research that I’m lucky that we actually get to write about so much at *Quanta*, is the way in which life can be treated as a mathematical system or a physical system, and that a lot of the sources of order that we see in the living world are already basically there to be taken advantage of within physical systems anyway, and that life didn’t have to evolve certain sorts of systems. So the fact that cells, the sort of default shape of a cell is being round, it’s not round because it was evolved to be round. It’s round because a sphere is the natural minimal shape of being a way to contain something inside a membrane. And we’re finding more and more that some kinds of physical phenomena with increasing levels of sophistication can be sources of vitally important structure and functions in biology, and that’s a huge frontier for us.

**Lin:** Right. It’s certainly been a recent rise in the power of biophysics and trying to merge things from these, you know, sort of — kind of from, you know, the small from physics in terms, you know, at a more basic particle level to the sort of larger, more emergent state that — that biology tends to — to examine. You know, thinking about evolution and life and even our own experiences — everything that we experience and think about in terms of these things involves a directionality to time, right? That’s how we experience things. I’m looking up at the clock, you know, every now and then, to see, you know, where we are in terms of our time here [LAUGHTER], but in physics the constant of time isn’t really well defined. The fundamental laws of physics — things can kind of move forward or backward in time. What are physicists doing about that and, you know is — as I understand it based on your reporting and the story in our science collection, is it connected in some way to quantum entanglement? Is that helping us understand what time is from a physics perspective?

**Wolchover:** Yes, sort of. So, I think there’s many different mysteries about time that are all kind of related, or maybe they’re the same mystery. So there’s —there’s like time as kind of this background, this dimension that we’re moving in. And then there’s the arrow of time, so the fact that there’s a directionality of time, but that arrow might kind of be on top of the background of existing time. And then we also have, like, this perception of time that changes, but then that also — our perception might also be more related to actual time than we think. So I feel like there’s — when people talk about time, it’s even hard to understand what the question is sometimes. But for sure, it’s one of the big mysteries of physics. And I think it’s sort of — I mean, I would say it’s maybe the big mystery of physics.

I don’t know how you feel about that, but I just —I think it’s sort of very, very tied with all the other things that are most mysterious. So, like quantum gravity and the nature of the Big Bang and all of these things. I think if we knew — if we understood time, we would also understand all these other things. But there is — but, yeah, there is progress in understanding the arrow of time, so why time can move forward but you can’t kind of — you know even — we can move in three dimensions of space, we can’t move backwards in time, and I think that is becoming more and more understood as being associated with — with the growth of quantum entanglement. So, the growth of things, kind of — all particles, all interacting in the universe, and getting more and more, basically, where their own state can’t be defined individually, their state is very dependent on the states of other things. And so information is kind of shared between many, many different particles. So this gradual growth of entanglement of particles and stuff in the universe is more and more being associated with the flow of time, because it’s very hard to unentangle — things don’t become unentangled over time. But even that, I think, is — a lot of people would view that as a separate question from, just, why is there this sense — why is everything not happening at one instant, right now? Like how do you get this — how is everything unpacked so that you’re allowed to have, like, particles becoming entangled, more and more entangled. I think that’s completely unanswered.

**Lin:** Sure, Sure. So I do want to get Robbert’s take on this as well. I mean —

**Dijkgraaf:** Quite right. It’s one of the great paradoxes. So I think on one hand we can be quite confident that space and time are one and the same thing, so Einstein told us, and everything — all the evidence points out that if you understand space and time, you can basically have any combination of it. And then on top of that, I think we have much more understanding about the kind of, oh let’s say, like the granular nature of space. So we know that if you want to study space at the smallest distances, we would have to concentrate energy so tightly that a little black hole would form and it would basically eat a little hole in the space. So you could never see it at the smallest distances, and there are great theoretical models how space can appear, emerge out of quantum fluctuations, quantum information, etc. So it’s, like, not fundamental, something that comes out. So if you take these two arguments, you said time should also emerge, it should also — we should have a kind of a granular nature of time. It would be terrific because people always ask you, what happened at the Big Bang, or before the Big Bang? But if time is something like a river, that you go upstream, at some point you will just have little drops of water — the whole concept of a river would disappear, and you don’t have a paradox. Rivers have beginnings. That’s perfectly fine.

But I think obviously nobody has any idea how to make this work. And what we find in particular, I think, that quantum mechanics, which often is seen now in this battle between general relativity, gravity and quantum mechanics, that like — quantum mechanics has the last word. Because all the theories still that we have, like string theory, etc., they use kind of ordinary quantum mechanics. Quantum mechanics makes a huge distinction between time and the other directions, because again it describes how things evolve in time, whether it’s a wave function or entanglement, etc. But this can’t be the last word, so we kind of have to give up. And so I think there’s a lot of physicists who think in this battle between gravity and quantum, where somehow quantum seems to have won the first round in our ideas [LAUGHTER], but in the second round we’ll have to give up. Because the concept of time is totally conventional in quantum mechanics. And as Natalie’s saying, that has to somehow disappear. So who knows?

That’s actually extremely exciting, and it’s actually so fascinating. But, of course, because you’ve all, you know — physicists are pushing, you know, and they can’t build a microscope that’s small enough or telescope that’s big enough to explore — actually time is happening right now. It’s something that’s happening here, we’re experiencing it. So physics to a large extent is also about the things that just surround us. And you can ask very deep questions — we can be totally confused about everyday concepts, which is actually humbling, because people have been thinking about time for millennia, but it shows that we do not necessarily have to build huge experiments. We can just think about it, and — and wonder about it. And the biggest puzzles of physics — perhaps time is it — it’s actually something that we are experiencing right, now, this moment, whatever that sentence means.

**Lin:** Right. Do you want to add something about time?

**Rennie:** Just a small thing. I mean I love to read the articles about — about time and the rest purely as a kind of, like, you know, a tourist going through this area [LAUGHTER] of things, because I’m fascinated just to see what comes up in it. But what also really strikes me about that all the time **—** sorry, inevitable. [LAUGHTER] — is the fact that our understanding, our appreciation of time this way comes in the fact that we evolved to have some kind of experience of time. It was something, for whatever reasons, out of necessity, or again contingency, or whatever — we evolved to have an appreciation of time that works the way that it does, and it’s deeply ingrained inside the nature of life itself. I mean if you — if all things down to the cells, down to the most primitive level, they all make — they all make use of time, they all keep track of time to organize their activities in different ways, and I mean it’s really fascinating to understand why that is the case, and whether there is an option for it to be any different. About, like, whether is — is there something about — that we appreciate time because this is the only way, that we would have to perceive it in that way? Or, you know, could something be like the — the aliens in, I think, Vonnegut’s book *Slaughterhouse-Five,* where it’s like the aliens, the Tralfamadorians — there it is — who appreciated, they — they experienced all times simultaneously— could something do that? Or is — or is there something built into the nature of the physics that then in biology has to come out of — that makes that impossible?

**Lin:** The big question, right: Is time fundamental? Is it emergent? Is it just a figment of our imagination? Is it an illusion? This is something that physicists and biologists at a different level are very keen to understand, and — and philosophers, too.

So, getting back to the question of math and evidence, you know I wanted us to think a little bit about the question of math as the universal language of science. Because certainly a lot of the science that we cover at *Quanta* tends to involve some very rigorous math, and, you know, there’s a pretty well-known essay from 1960 by the Nobel Prize-winning physicist Eugene Wigner in which he talked about the unreasonable effectiveness of mathematics. So we all are familiar with math that can come out of physical things. We can think of a circle, we think about, you know, pi and the relationship between the circumference and diameter. There’s a look at motions of — orbits of planets, things like that, but there’s been a lot of mathematics developed separate from what we can observe, and somehow that mathematics — some of it, at least — has found its way back into physics and back into descriptions of the real world. And so that’s, I think, what he was referring to, and it also sort of speaks to the — sort of the deep question of whether mathematics is, you know, invented or discovered. And some of the mathematicians I’ve spoken to tend to say it’s both, right? That both things are happening. But I wanted to ask Robbert about this — well, actually, I’m sorry, I’m going to start with Kevin first, because Kevin wrote about a strange link between physics and math in one of your stories in the science book. Talk a little about that.

**Hartnett**: Right —

**Lin:** — about particle physics and pure mathematics.

**Hartnett:** Yeah, this was a story about a mathematician at Oxford named Francis Brown who was just kind of exploring this correspondence between these important numbers in math and these important numbers in particle physics. So when particle physicists run — you know — collide two particles together, and they want to kind of follow that chain reaction forward and try and determine what happens, they use this thing called the Feynman diagram. And each Feynman diagram you associate by, say, a probability that the — the collision played out kind of the way your diagram said it played out. And so you have this number describing these collisions called the amplitude, and then there are all these other numbers in pure math that are called motives or periods of motives. But the point being, they’re just like a characteristic number of algebraic equations, and these — these numbers, these probabilities, for these Feynman diagrams seem to kind of match these — these periods of motives in this field of algebraic geometry. And so Francis Brown, in coordination with physicists — really it’s a quite a dynamic back and forth they’re having — has been trying to explore, like, well, why would these — what is — what is the structure of these numbers that we’re seeing coming out of these physics experiments? Because it does seem like they have a structure, they have this thing called a group structure, that mathematicians would be interested in understanding. And at the same time, if there is this structure, what does that mean physically? Because if these — these periods have, or these amplitudes have, this group structure, it’s suggesting something symmetrical is happening in the way these particle collisions are having, and that symmetry should have an actual physical meaning. So it’s kind of this dynamic back and forth between the two, and I think it’s one instance I would just say — I’ve just — I think one of the most exciting things going on in math today is this — I feel, like a growing collaboration between number theory and physics. Geometry and topology and physics have been kind of intertwined for several decades, but number theory and physics are really starting to communicate a lot more. This article in the — in the book, “Strange Numbers Found in Particle Collisions” — is one instance of that.

**Lin:** And that connection is — was a complete surprise, right? I think in the story you said something like it’s — it’s like, you know, counting grains of rice and seeing prime numbers, or something like that.

**Hartnett:** Right, right. That’s no reason to believe, you know, that they should be there.

**Lin:** So now I want to turn to Robbert because you have a different — you have a column in the book where you refer to something that sounds like what Eugene Wigner was saying, but it’s a little bit different. You said there appears to be an unreasonable effectiveness of quantum theory in mathematics, that somehow quantum theory is leading to new mathematics.

**Dijkgraaf:** Well, you encourage interaction between, I would say, mathematics and the sciences, and but — it’s indeed a little bit of a — kind of an arms race of these two fields. So, for instance, we would say, well, math is wonderful to describe the motions of planets, etc., but Newton had to invent calculus to do this, right? The math wasn’t there yet, and it was — now we see this as a standard part of mathematics, but very much inspired by — by mechanics. And so I would say, you know, it’s — math is a very forceful tool for scientists, but also nature is a great mathematics teacher and teaches us mathematics in a very deep way. And I think these lessons have only just started. Now we know that, you know, nature is quantum mechanical — so that’s one thing we know and it’s nothing an approximation — really — it’s — if you — what’s the fundamental way to think about nature is quantum mechanical. And yet I think most of our geometrical concepts and — or just our all our mathematical concepts — are kind of classical nature — we think in points and lines and curves and velocity, and so all of these would make perfect sense in classical mechanics. So our math, I think, is still almost like 19th-century a little bit and we haven’t fully embraced quantum mechanics. Now it turns out that certain ideas in quantum physics which are very unnatural to us human beings because we are not elementary particles — so we never are at two places at the same time, we’re never entangled with somebody else — it’s — although we use these words but we are not [LAUGHTER] — technically we are not —

**Lin:** Right.

**Dijkgraaf:** — and so we have very little intuition. You might say our brains have been shaped by basically classical mathematics. So another question is, can the quantum world be a good math teacher? And it turns out that, you know, there’s certain things in quantum mechanics which are like perfect for mathematics, for modern mathematics. And modern mathematics doesn’t want to look at one object — the modern style of mathematics is you look at everything, you know, every member of the family. And you don’t look at one object, but at their — the mutual relations — relations among relations, etc. You build this very kind of communal view, and actually it’s exactly what quantum mechanics does. It’s — it’s, you know, it’s the famous sum over histories. If you go home today, you know, if you feel you take one path, but quantum mechanically you take all possible paths, each with a certain probability — very unlikely, only elementary particles have to deal with this — but actually, it turns out that also it’s wonderful for mathematicians to deal with that, to kind of try to think like a quantum particle. And it — and again here the proof or the — the evidence for the usefulness of this unreasonable effectiveness of these ideas is that it actually leads to concrete proof of mathematical theorem. So there is a whole series of results proven in the last few decades — which I would say been proven or understood using quantum logic. And the remarkable thing I feel is that we are able to push our intuition, even our mathematical intuition, in an area — the world of small particles — where none of us have any concrete experience.

**Lin: **That’s fascinating. I’m taking mental notes here, by the way, for future Quanta stories. [LAUGHTER]

**Dijkgraaf:** I mean that’s such a great thing, you know. Think, for instance, of the life sciences. John, what you were saying — what would happen — literally happen — if we move math into that area which is as ill understood, so to say, and we have as little intuition as we have elementary particles. And so science can really inform new math in very deep ways.

**Hartnett:** I would say the — we did a profile in the last year of a mathematician named Minhyong Kim, who is very much using some quantum ideas to try and understand the rational solutions to equations. It’s hard to find the rational solutions to equations, but Kim has this idea that if you think about these spaces of spaces that Robbert was kind of just talking about, you can kind of — the rational solutions to these equations, so what you want to find, are like the classical paths traveled, and the — kind of the — all the other solutions are like the path not taken in the classical world. And an interesting thing about that article, though, was Kim has kind of harbored this idea — that these ideas from physics should inform these ideas from math — for more than a decade. And he, like, didn’t want to tell anyone [LAUGHTER]. Because he was really afraid number theorists would, you know, would think it was just quackery —

**Dijkgraaf:** Yes, yes.

**Hartnett:** — so he kind of worked up the courage over the course of a decade, and now was trying to, you know, get — gained some adherents. But it takes a while, I think, to kind of break out of these ways of thinking about these math — mathematical — questions and kind of admit a new perspective.

**Dijkgraaf:** And none of us would ever think of this ourselves — you can meditate for a millennium but you will never find the rules of quantum mechanics — experiment, nature forces us to think like these crazy ways.

**Lin:** Yeah, wow. So I’m going to ask another question, but while we’re talking our volunteers here will be bringing microphones up to the aisles here in case anybody from the audience has any questions. So please start making your way to the aisle if you have a question. And we’re going to line up behind two microphones that they’re setting up right now, as we speak, and we’ll just continue on. But in a few minutes we’re going to start taking questions.

So, so far tonight we’ve talked about the nightmare scenario in fundamental particle physics, we’ve talked about the difficulty that we have with evidence in science and what that means, we’ve talked about how we don’t know what time is — it’s been kind of a little bit negative. [LAUGHTER] And I apologize for that — that’s my fault for leading the conversation this way.

But, yeah, I want to end sort of on a positive note, and thinking and asking every one of our panelists to talk about what is the most exciting thing that you’re seeing in the area that you cover and write about, or research, and, you know, what is it likely to lead to, and how is it going to bear fruit in the future? So in physics, for example, what are you seeing out there that’s most promising?

**Wolchover:** Well, so there is so much happening in physics and I didn’t — definitely didn’t mean to convey that there isn’t. I think we’ve been talking about fundamental physics, so cosmology and particle physics, but of course there’s so much happening in condensed matter physics, so — better understanding materials, and it’s actually fascinating stuff happening there. And also, just, like, turbulence. So there are all kinds of different areas that aren’t kind of getting at, like, the heart of what’s going on with the universe.** **

**Lin:** But even at — even at the larger sort of, you know, sort of larger scale. LIGO, for example, has been hugely transformative in terms of our ability to see what’s out there, right?

**Wolchover:** Yeah. And so we’re now detecting gravitational waves from colliding black holes, and that’s helping us kind of map out and understand the universe, and especially the role of black holes in the universe. But I would say, actually, what excites me most is that in this period of — kind of — turmoil and crisis, basically, in fundamental physics, there is — there are a lot of new ideas kind of bubbling up. And mostly what it is, is people are questioning the assumptions that they’ve held for a long time. Or they just kind of felt like we would be playing out this plan that we’d laid out of, you know, looking for supersymmetry, and finding it, and then understanding kind of how all the particles in nature fit together. And then kind of, you know — and maybe getting evidence for string theory somehow, and then we’re done. And that obviously has not happened. So people are going back and really questioning the assumptions like naturalness — they’re trying to better understand the patterns in the equation. So kind of mysterious things about the equations that we have already, to try to understand why they are the way they are. And I think there is just a lot of fascinating stuff coming out.

**Lin:** Absolutely, yeah. We talked about, you know — there’s this sort of, you know, moniker of the nightmare scenario out there, but really talking to, especially, some of the younger physicists out there, they actually see a big, wide-open space. They see a lot of opportunity. They no longer have to follow their mentors into any field, they can just create their own programs now. They can think about, you know, anything that they can imagine to — to try to create new physics. What about in — in your realm, in biology?

**Rennie:** So I’m going to slightly cheat because I sort of alluded to one of these before, which is I think that, really, in terms of a big concept, the idea that embedded within nature just — just the sheer nature of the physical world is such that it can drive a lot of the form in what we find in nature. So there — there are scientists like Jeremy England at M.I.T., for example, who’ve been developing arguments that suggest that there’s a kind of inevitability to life because life is a natural manifestation of a tendency in the material world to keep finding better and better ways to dissipate entropy. So just — so I think that that is kind of a fascinating idea, that way. That basically that life, that — there’s an inevitability to life that way, and just that the complexity that we see arising in life over time, that there might actually be an innate directionality toward more complexity that comes to that. But even beyond that —I mean, just the thing that I think is just endlessly fascinating is all the stuff that we — you would think we would understand about life by now, how often we really don’t. You know, Craig Venter in his group a few years ago, they said let’s take all that we understand about genomics and let’s try to create the simplest living cell we possibly could. And so they took all the kind of things and they tried to figure out what would be the minimum number of cells — of genes they could put into an artificial cell. And they tried and tried and tried, and it was a dismal failure because they couldn’t keep those things alive. So they tried a different way, where they started off with a really small cell and then just started randomly knocking things out to see what would be left. And what they were left in the end was this artificial organism that had only, like, 473 genes, I think, but what — which you know really, really small, it’s like a 10th the size of what you’d find in something like an E. coli. But here’s the kicker: Of those 473, like 150 of them — we have no idea what those genes do. So even when we tried to get it down to the minimum number of stuff, we found that we still don’t even understand why those things were alive. And so I — there’s so much still to just learn at that basic level of what our genome is doing.

**Lin:** A lot to learn, yes. Kevin?

**Hartnett:** Yeah. I think there’s, I mean, a lot of exciting work going on in math. I mentioned the search for rational solutions to polynomial equations. There’s — there’s like several kind of exciting initiatives in that respect. One mathematician recently — I know — I think that mathematicians are optimistic — that we will be kind of identifying rational solutions in a more systematic way. I think kind of a work that, kind of, puts analysis and algebra in conversation with each other and allows mathematicians to kind of port results back and forth, there’s like a lot of exciting work there. And that opens up kind of solutions on both sides. Yeah, those are definitely true.

**Lin:** Certainly a lot that’s going on in mathematics. And things are — maybe it’s harder to talk about them all as one thing, but that’s — those are great examples. Thank you, Kevin. And what about you, Robbert?

**Dijkgraaf:** I would highlight the, kind of the, collision of the small and the very large, so where cosmology and quantum theory are colliding, you know. I think we all are excited about what we do not understand, so it’s, I think most of you know, 95 percent of the universe is ill-understood. It’s dark matter, dark energy. We know there are black holes, we know there’s something like the Big Bang, we know that nature found a way to resolve the logical problems, so these are, I think, often said — these are the paradoxes of the 21st century, as the atom was the paradox for the 20th century. So though they’re out there somehow, nature found a way around, and what I find extremely exciting for theoretical physics and mathematics, actually, I feel it’s like a jigsaw puzzle. We started on the very edge, the easy parts, perhaps the corners, and now we’re going into the middle part where you actually see ideas for many, many fields coming together. So when you study gravity or quantum physics or quantum information or very pure math, perhaps even number theory, somehow it all comes together. So if you’re a young person right now, you can kind of have this great pleasure of moving in all these worlds and still thinking about the same problem. So, I think that’s actually extremely exciting, and I think that actually makes — I think like in science you can say every year is the most exciting year ever [LAUGHTER] — I think you have been right for every year.** **

**Lin:** So I took you sort of down, you know, a little bit at first, but we ended on a positive note. I want to take some questions now from the audience. I think you were first, sir, on this side.

**Audience member:** There are two outstanding mysteries in quantum mechanics that you haven’t touched on tonight, so I’d like to get your opinion. The one I’m not really asking your opinion on is: What does quantum mechanics really tell us about the nature of the universe? I fear that Einstein was wrong and God does throw dice, but that’s not my question. Another outstanding problem is the problem of measurement that you all know about. And John Bell, probably close to his death, wrote an essay called “On Measurement” in which he challenged quantum physicists to incorporate measurement into quantum theory, and I’d like your comments by any of you on how much success, how — how things are going on that course.

**Wolchover:** So yeah, I’ve written a lot about interpretive issues with quantum mechanics, basically the question is what — what does it mean, what is reality if things are so probabilistic on the quantum scale — nothing’s real until you measure it, you know — it brings up all these questions. I tend to think that that — we just, that — in the next revolution in physics, when we have a better, deeper understanding, that it will be clear that our current thoughts on this issue are naïve. And that we will no longer be, we won’t — it won’t be that we turned — it turned out that a single interpretation was right and the other ones weren’t. It’ll be just unimaginably different.

**Lin: **Thank you, and [UNINTELLIGIBLE] I think we should probably also allow others to ask [UNINTELLIGIBLE] — and I just want to say real quick, that was a great question, and please do make sure your questions are questions, and keep them brief so that we can get a few more questions in today. So let’s start on this side.

**Audience member:** This is not exactly a specific question, but if you could comment on the statement: Is it meaningful and what does it mean for an axiom to be true? That’s somewhat of —

**Lin:** Mathematics —

**Audience member:** Yeah. It’s mathematical and philosophical, but I think you guys could have some good answers or comments.

**Lin:** Thanks for that deep question. Kevin?

**Hartnett: **You’re not looking at me, are you?

**Lin**: You are our math panelist —

**Hartnett:** What does it mean for an axiom to be true? I don’t know. I’m afraid I don’t know. [LAUGHTER]

**Wolchover:** I mean, those are things that are often based in physics, right? Like we — we start with axioms in math that seem true, based on our understanding.

**Dijkgraaf:** This relates directly to this point of math being discovered or invented, so we think often, you know, if it’s natural mathematical structures they are discovered. So somebody finds like symmetry groups, you know, oh, OK, that’s a natural thing. But you can have very contrived actions, and then you have a creative world. Well, clearly you can discover things in that world, but a whole world seems to be kind of made-up, right? So the question is, what are in some sense the natural — I mean that’s how I read — what are the natural actions? And here you might think that, no, perhaps indeed our science/physics/reality experiences are guiding us because we do not want to have them completely — now you can have very sterile or very contrived mathematical worlds that feel made-up.

**Lin:** There’s even the incompleteness theorem — there are things that can sort of undermine your attempts to find out, you know, sort of the fundamental axioms. This side, please.

**Audience member:** In my inbox the *Quanta Magazine* has a soft little sibling called *Nautilus,* and recently there was an article in *Nautilus* about quantum — quantum entanglement and consciousness. Penrose apparently has been dabbling in this and looking at microelements of the brain — specifically, I believe, it was something like microtubules. So I was wondering whether that’s a question that you all have encountered. It sounded like a fair amount of the world dismisses it out of hand, and then there are a couple of people who think that he — well, maybe there is something there. So I was curious to hear whether anyone had any thoughts on this. And then in particular relation to another, I think growing, tradition of thought around a much more decentralized architecture of the human neural system. So it’s not just the brain, because in this case you had quantum entanglement with elementary or microelements of the brain, but if you have a larger system that includes the gut and the toes and so on and so forth. So any thoughts?

**Rennie:** I’ll jump in at least sort of briefly on a couple of those. Obviously, one dismisses Roger Penrose casually at one’s peril. That said, I think you’re right in saying that I think that the view of most neuroscientists is that an idea of trying to pin notions of consciousness, and where that comes down to some sort of quantum effects associated specifically with microtubules, seems questionable for a variety of different reasons that I am certainly not capable of being able to say at the moment. But I know that is the general state of it, I think. As with all these sorts of issues, the right experiments coming along could easily change opinions about that. Because it’s not as though we’re — have all sorts of other great ideas right now about where consciousness is coming from, but I don’t think that’s a very favored explanation right now. I think that there — I think your other comment about the sort of, the decentralization of a lot of functions that we’ve tended to associate with the brain itself, I think that is a very active area of research. Although I think my own reading of the literature on that is that most of that has less to do with the notion — with actual quantum effects — and more of a growing sense that in fact — that a lot of things that we thought that had to be just purely localized inside the brain can in fact be distributed more outside to other parts of the nervous system because that information can be communicated and shared in other ways. That’s a sort of quick gloss on that, anyway.

**Lin:** Thank you John, and because we started a little bit late we’re going — I’ll take two more questions, and then afterwards we’re going to have a little bit of a reception. There’ll be some refreshments outside, we’ll do a book signing for those of you who have books, and so you’ll have an opportunity to talk with panelists individually and ask your questions there. So we’ll take just two more questions.

**Audience member:** Humankind has needed to find beauty in reality, and we see this in mathematics, in physics, and any geometrical explanation, interpretation of nature leads to certain beauty. Just think of symmetry, Fibonacci series, etc. etc. There is a long tradition. Can you speak of an aesthetics of cosmology — is there such a thing?

**Lin:** That’s an interesting question. Want to give that a shot, Robbert?

**Dijkgraaf:** Well, I often remark that beauty is becoming something very contentious in the arts, but in some sense it’s now migrating to science. [LAUGHTER] So we are the ultimate refuge of beauty. [LAUGHTER] And — and if — I think, you know, somehow our definition of so — it’s quite remarkable that these equations and objects and that we have a universal sense of beauty, you know. Mathematicians from all different backgrounds agree, this is a beautiful equation. I don’t think there’s a real good definition of it, but has something to do, I would almost say, of the power per symbol of the equation. So the fact that you, with a very brief set of concepts, can explain a lot. And so I would say, if you look at the great beauty I see in physics now — and I’ve written about it in one column in the physics collection — you find it at the two ends of physics— at the very small and very large, where things start to simplify, basically because you either zoom in so closely that you don’t see any of the messy details, or you zoom out so largely you only see the big picture. And in these two areas, I think the equations, say Einstein’s equations or the equations of the standard model, are incredibly beautiful because they are so powerful. So just the fact that with a very simple set of equations for describing 14 billion years of evolution, again, the surprise — again — has been that people think of all kind of bells and whistles to add to it, and nature proves them wrong. So there is something — I find it very comforting that, you know — I think we’re trying to read the character of nature, and think what she likes and I think it’s the kind of equations that we like, too.

**Lin:** On the subject of mathematical beauty, I mean, for anybody who hasn’t heard of this, a number of years ago the mathematician Michael Atiyah and some neuroscientists did a paper where they looked at fMRIs or something of mathematicians’ brains as they were looking at different equations to see which patterns matched the patterns of other people looking at beautiful art and sort of the aesthetics of that. And they found that the one equation that most mathematicians found to be the most beautiful was Euler’s equation, Euler’s identity** — ***e ^{iπ }*

*+ 1 = 0*— and it does embody all the things that Robbert was just talking about.

**Lin:** One last question.

**Audience member:** Hi. So Robbert’s comments on quantum mechanics being a mathematical teacher made me think about, you know, the recent computing systems involving the blockchain and decentralized computing, how that’s pushed sort of the frontiers of cryptography and a lot of other stuff. And so I wonder, first of all, when do you think the first hobby quantum computer will be available? [LAUGHTER]

**Hartnett:** We’re giving one away tonight actually — [LAUGHTER]

**Audience member:** And what questions and — and topics, you know, do you think, you know, could be addressed in terms of quantum mechanics and mathematics that this potential computer would sort of spur?

**Wolchover:** They always say 20 years —

**Lin:** — and then in 20 years it would be another 20 years, of course.

**Rennie:** I can tell you exactly when it will be available, but not where. [LAUGHTER]

**Lin:** I don’t know if we can give you a better answer than that to be honest. [LAUGHTER]

**Lin:** Thank you everyone, and again we will have some refreshments outside, a little bit of a reception, and if you have books you’d like us to sign, the panelists — except for Robbert, I think you have to leave? Unfortunately, we won’t have Robbert — but we thank you for being here and for your amazing insights that you shared with us, and all the panelists today. Thank you.

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