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Antonio Padilla, "Fantastic Numbers and Where to Find Them: A Cosmic Quest from Zero to Infinity" (FSG,2022)
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- [00:01]
Gregory McNiff
Welcome to the New Books Network. Welcome to the New Books Network. I'm your host Gregory McNiff and I'm excited to be joined by Antonio Padilla, the author of Fantastic Numbers and Where to Find the Cosmic Quest from Zero to Infinity, published by Brar, Strauss and Garou in July of 2022. I chose fantastic Numbers because it uniquely weaves modern physics with the deep history and philosophy of mathematics, making difficult ideas both intellectually serious and genuinely beautiful. While Tony does include formulas in the book, they're relatively straightforward. He walks you through them and it is definitely worth your time to invest in understanding the formulas. Tony is a leading theoretical physicist and cosmologist at the University of Nottingham. He is Associate Director of the New Nottingham center of Gravity and has served as the Chair of UK Cosmology for over a decade. In 2016, he and his collaborators shared the Book Author Cosmology Prize for their work on the cosmological constant and we will talk about that later. He is also a star of the number field YouTube network where his most popular videos include a discussion of Ramanand Yujan Sum of all positive integers, which has been viewed more than 7 million times. That in itself is a pretty impressive feat and I highly recommend tuning into his YouTube channel. Tony, thank you so much for joining me today to discuss your book.
- [01:28]
Antonio Padilla
Thanks Greg.
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Gregory McNiff
Pleasure to be here, Tony. I always start with the obvious question, why did you write Fantastic Numbers and who is the target audience?
- [01:37]
Antonio Padilla
Why I wrote it is. It's quite a sad story in a way. So it goes back a very close friend of mine that I noticed since childhood took ill with cancer and we wanted to raise money for possible treatments abroad and stuff that you couldn't get in the uk so we, we all went on a major fundraising sort of drive and, and myself concluded and I started giving lectures where I would try to raise money at the lectures. And the topic of the lectures was, was those lectures were about hinged on a certain number of videos that I, that I'd made that put brought sort of extreme physics and extreme maths together. So I was giving these lectures and trying to raise money for my friend anyway and I realized as I was giving them that actually the content here would actually could sort of amalgamate into a book. They were very well received, the lectures. They were down really well and I thought this could make a book. And then I was approached by publisher about writing one and it was around this time and so it seemed natural to me that okay, well let's do that then. So I guess you Asked about the target audience, and I would say basically anybody with a love of maths and physics, right, that sees. Looks at the world, wonders why it is what it is, wonders why. I guess maths does this wonderful job of describing it, but also loves maths for. For its own reasons. I think. For me, the book is about many things, but it's about that interface between maths and physics. It's how physics can make maths wonderful and maths can make physics wonderful, and the two really go hand in hand. And one without the other is just incomplete, at least in my world.
- [03:31]
Gregory McNiff
No, absolutely. I think hopefully in my intro I referenced that. But yeah, I think you did a great job of tying physics and mathematics together. A lot of times you see pop science books that ignore one to emphasize the other, and they're obviously, they go hand in hand. Tony, you start the book off with a bang, or I should say a bolt, and namely Dane Bolt. And you use him as an example to talk about Einstein's theory of relativity and time dilation and length contractions. Could you maybe touch on that? And specifically this chapter, I hope I get this right, is titled 1, dot. I believe it's 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 16 zeros, and then 8, 5, 8, which is a very small number, but one that has a very specific and important significance.
- [04:27]
Antonio Padilla
Yeah, so that, that, I mean, it's a little bit tongue in cheek, but that. That is the sort of the amount by which I claimed Usain Bolt was able to slow down time when he. When he ran his. His. So I use this as a. And I say it's therefore a world record achievement in itself, as you mentioned. I use this as a sort of platform to talk about relativity. So you go back to when Usain Bolt was sort of running incredible races, breaking world records left, right and center. And it was obviously running really fast. We could all agree. And then we sort of looked at how fast he was running and you sort of think to yourself, you know, he says, I think he had this quip that, you know, how did you run? Everyone's asking, how did you run so fast? And he was like, well, I just eat chicken nuggets. Right. So ridiculous like that. So you sort of see this and you think to yourself, well, is there any limit to how fast he could go if he ate enough chicken nuggets? Right. But then, you know, the answer, of course, is there is a limit in it. And the limit is, of course, the speed of light, which is the Universal speed limit in nature. So I thought it was a fun. A fun sort of, you know, platform in which to talk about relativity. So the story there is Bolt runs through this race really fast and you can actually think about the. You know, when you have sort of one person, in this case Usain Bolt running relative to. Moving relative to, say, his parents watching into the stadium, then it actually turns out that their clocks, their individual sort of watches, are ticking at slightly different rates. And you can calculate how big you'd expect this effect to be. It all just comes from Einstein's theories. And you can figure it out. And it turns out that the bog's clock would have been ticking slightly slower than his parents clock watching in the stadium. And so you start to wonder, well, does that mean that he. So he ran 100 meters, but he recorded a shorter time, so does that mean he actually ran a bit more quickly? It doesn't mean that. It actually means. Something even more remarkable is that. So when you think about life from Usain Bolt's perspective, he's there. His watch records a small, smaller time because of relativity and because of the way these time effects work. But also when you think of it from his perspective, the track is moving relative to him, so he's moving roughly 10 meters a second relative to the track, and the track's moving roughly 10 meters a second relative to him, but he records a shorter time. So what that actually means is, from his perspective, if you've got the same time, but, sorry, a shorter time but the same speed, that means the distances must also change. So it turns out from Bolt's perspective, the track also shrinks. So there's all these wonderful effects going on. And when you think of them in the framework of Usain Bolt running the rage, you realise his clock slows down. You realize the track actually shrank from his perspective, so he didn't actually run 100 meters. So there's all these sorts of things. Maybe he should take his gold medal away or something, I don't know, but this is what I thought it was. Of course, as I said, it's a little bit tongue and tricky. You can then go on to sort of more powerful, more realistic scenarios where these effects are much, much larger. With satellites going around the Earth, for example, at very high speeds, you have to include these relativistic effects and how fast clocks are ticking, sort of at different speeds and at different points in altitude. You have to factor all this in for your GPS to work, for example. So this is real physics, right? And, yeah, I thought it was fun to talk about it in the context of Usain Bolts.
- [07:59]
Gregory McNiff
Great chapter. And you do a nice job of explaining both special relativity and general relativity towards the latter, which involves acceleration. You write the problem with trying to accelerate up to the speed of light is that inertia blows up to infinity. Could you briefly explain that if possible?
- [08:17]
Antonio Padilla
Yeah. Yes. So people often talk about light being this, the speed of light being this sort of speed that you can't go past. Right. So it's like, well, why? Why is that? What stops you? Just couldn't I just, you know, if we go back to Usain, Bob, if I just eat a few more chicken nuggets, can I not. If I keep accelerating, eat more chicken nuggets, keep accelerating, can I not eventually pass through that barrier and go even faster? Right. And the answer is no. And the reason is it comes back to, like I said, this business of inertia. So what is inertia? Inertia is your resistance to motion. Okay? So we know Newton's second law. We know force equals mass times acceleration. So for us, our mass is our inertia. It's our resistance to motion, essentially. So when I say that, so what happens? This is the physical thing that stops you from hurdling past the speed of light. It actually turns out that the closer and closer you get to the speed of light, your inertia, your mass, your resistance to motion actually increases. And actually, at the point that you hit the speed light, it goes infinite. So it doesn't matter how many chicken nuggets you eat, how much more force you put into the system, you've got so much inertia, so much resistance to motion at that point that you can't go any faster. And that's the point, really, I think, and that's really why the speed light is this barrier in nature that you can't get past, because to get past it, you would have to start, your mass would become infinite. And there's no way you're moving once your masses hit infinity.
- [09:52]
Gregory McNiff
Yeah, no, it's so counterintuitive. In addition to a great book on mathematics and physics, you've got some great biographies in here. And I really, I want to circle back at an individual at the end when you talk about infinity. But one individual you introduced in the beginning of this discussion is Minkowski. And you write, by combining his spacetime coordinates with a measure of space time distance, Minkowski was starting to build a remarkably elegant picture of physics in terms of four dimensional geometry. When Maxwell's equations are written in this new language, they take on an incredibly simple form. Could you just talk about how Minkowski's mathematical equations made Maxwell's electromagnetism equations more simple, more accessible.
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Antonio Padilla
I mean, it's actually ridiculous how much simpler they become, right? It goes. So Maxwell's equations, which describe the laws of electromagnetism, they very much, you've got sort of electricity and magnetism on one hand and they're sort of put in different parts, but the equations show how they're related. And they're very complicated, messy, dirty equations, right? They're correct, but they're complicated. And part of the reason that they're complicated is because they are sort of. It's because you're trying to strong arm them into sort of three dimensional space, right? What with Minkowski, when you sort of realize that you're actually part of this relativistic world where you really should be bringing space and time together. And we see with the Usain Bolt example why you should definitely do that. I talked about, when I was talking about Usain Bolt, I was saying, right, his clock slowed down, so his time slows down and the track shrinks. So you've got these effects which are clearly affecting both time and space in equal measure in some sense, right? So it's clear that time and space are really a combined dynamical thing that you shouldn't really be splitting. So Minkowski really ran with that. So when you write, and then when you apply that to Maxwell's equations, instead of having this sort of this long, ugly set of equations, what you have is really one beautiful equation that's really short and it's just so elegant and literally you can write it in a centimeter. If you write it in normal size, you can Write it as 2cm across. Whereas I'm writing Maxwell's equations as they were originally written. Know I need author page.
- [12:17]
Gregory McNiff
Tony, if, if there's one sentence in the book that made me go, wow, did I read that properly? It's the following one. Gravity is fake. And I really might be some tongue in cheek. I'll let you answer that, but could you clarify that?
- [12:31]
Antonio Padilla
Yeah. So I'm actually teaching, teaching my advanced gravity class at the. At the moment. It's ongoing at the moment and I often open the class with that, with that line just to, just to wind them up a little bit, really. But I think in many ways it is a fake force. I stand by. Why is it fake? Well, one reason it's fake is that you can always get rid of it. If you can get rid of something, then it's not really there. So what do I mean by that? So if you were to. And I talk about this example in the book, if you were to sort of climb up at the top of the Burj Khalifa and you would sort of put yourself in a black box, you can't see out the window, right? Burj Khalifa, it's the tallest building in the world. And you throw that box off the edge, okay. And then if I neglect their resistance, then I'm going to accelerate down, you know, with the acceleration due to gravity. Okay, but you're sat inside the box, you can't see outside. Just your entire world is inside that box. What will actually happen to you in that box is you'll actually become weightless. You, just because you're accelerating with gravity, you don't feel that force of gravity, you actually become weightless. So in that very simple example, you're already seeing that you're doing away with gravity, you're losing it. It's gone. All he did was throw yourself off a building. Okay, that's quite a dramatic thing to do. But nevertheless, that's all he did. And gravity's gone. And actually in UC Davis, which I visited a lot in the past, is that there's a lift in the physics department and there's a chair on it, which, as you move in the lift, you can sit on it and it's sort of. It's on.
- [14:09]
Gregory McNiff
Like.
- [14:11]
Antonio Padilla
It moves. It can move up and down. So you can feel yourself get more weightless and downwards, your weight increase. So it's basically playing with exactly this idea that gravity is fake. And of course gravity is fake. I mean, it's not a real force in the way the other forces. The force is. What is it? It's the curvature of space and time. That's what it is. It's just the shape of the geometry in which we live. Whereas it's not a force like if I push, kick a football or something like that. That's. Clearly, I'm doing something there, I'm kicking up. But gravity is just the shape of space and time in which I find myself.
- [14:45]
Gregory McNiff
Yeah, no, that was a really interesting thought experiment. Moving to the next chapter, entitled A Gogol. You talk about entropy. And again, another fascinating biography here, Stadi Carnot and his contribution to our understanding of energy conservation. Like a lot of individuals, he wasn't appreciated in his own time and unfortunately seemed to have past relatively young. Could you briefly talk about his contribution as. Is it the father of thermodynamics?
- [15:15]
Antonio Padilla
I think he was very much the father of thermodynamics. Yeah, I wouldn't necessarily call him the father of energy conservation. I would go back to more Julius von Meyer, who I also talk about in that chapter. But, yeah, so I think the ideas of entropy and understanding what that is, which of course is key to where later on I'm going to talk about what entropy really is. And so maybe we could talk about. Talk about that a bit later on. But it's counting the number of ways in which you can make something, a number of different ways you can make something and make it look the same. It's essentially what entropy is. But, yeah, Carnot, I mean, it's a very tragic story as well. Of course, because he died of cholera, and because he died of cholera, a lot of his manuscripts that he'd written, they had to be burnt, so they were never seen. So we don't know what he actually had in there. And it's sort of been lost forever from history, actually, just because of this. You know, there's this medical need to sort of burn everything that he'd been in contact with, and that included some of his unpublished work. So we'll never know what was actually in there. But, yeah, very much. I would say he was one of the founding fathers of thermodynamics. Absolutely. No doubt. Yeah.
- [16:29]
Gregory McNiff
To show you how much a nerd I am, I actually have a baseball card of Saddy Carnot published by an American firm, that. Baseball cards. Still trying to track down his autograph, but reading figure, I do want to move into entropy right now. You have another great line, and I suspect there's a double meaning here. Entropy is what counts. Could you talk about that?
- [16:50]
Antonio Padilla
Yeah, yeah, yeah. So, yes, of course, the dawn reading there. So it kind of alludes to what I was saying earlier. So the way to think about entropy. So people get very confused about what entropy is. People talk, oh, it's a measure of disorder and all that sort of stuff, which is. That's not really what it is. It's a useful way to think about it, maybe in the right context, but it's not what entropy is. Entropy is the following. If I look at something. So let's take an example. Let's take my daughter's bedroom as an example. My daughter's bedroom's usually a mess. Okay. So if I were to, you know, okay, so there's two states that I can imagine. My daughter's bedroom. One is a mess, one is it's tidy.
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Gregory McNiff
Right.
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Antonio Padilla
The rare example is when it's tidy. If it's tidy. If I stick my head through the door and look at it, right, I'll see, okay, there's a book there, there's a pen there. You know, I know where they are. There's sort of one real sort of, you know, with that quick glance that I make, that quick measurements that I make in my head of what the room looks like, I can build a pretty accurate representation of it. And that's pretty much what it is. Okay? That is what it is. There's that book there, there's that pen there. Right? Okay. On the more common occasions where I stick my head into her room and have a quick look around and ask, what does it look? What? You know, take a snapshot, make a measurement effectively, and it's a mess. Well, what then? Okay, everything's scattered around. There could be a book there, but there might not be. There could be a pen, there might not be. It's all a mess. There's. In my head, there's a whole load of different scenarios that actually amount to the same thing. Okay? So that corresponds to situation with higher entropy. So why entropy is counting the number of you make a measurement. Now, in thermodynamics, we might talk about temperature, things like that. You make a measurement of a macroscopic object, and it's the number of different. So those macroscopic measurements, it's the number of different microstates, a number of different microscopic sort of arrangements that you could have for that macroscopic object that would yield the same measurements. So in the example I've just talked about, the macroscopic object is my daughter's room. The different microstates are all the different intricate arrangements of the books, of the teddies, of the pens, Everything all around, all spread around the room. There's lots of different ways in which you could do that, lots of different microstates for that room. But as far as I'm concerned, when I stick my head in and have a quick look, they all look the same. It's just a mess, right? So that's what entropy is. It's counting those different ways we should get the same macroscopic outcome as far as your measurements are concerned. Okay, so that's all entropy is. It's counting those microstates.
- [19:35]
Gregory McNiff
No, that's an example. That really brings it home. I can hear parents nodding in that. Tony, I want to move to the next chapter of Googleplex. And again, another fascinating thought experiment. I'll just hit you with it. If the universe is big enough, your doppelganger is out there. Is my doppelganger out there, Tony?
- [19:53]
Antonio Padilla
Depends whether the universe is big enough. Exactly. So what I try to do here is what this really is. Vehicles to found out. This is trying to. So we take you, Greg, we take a Q, and you're a macroscopic object. And we can ask the question, how many microstates are there that make you you, right? And then we can ask that, right? And there's different ways, the different levels at which you can ask that question. But there is ultimately the most fundamental level as well. When you really get down to the quantum DNA, really talking about quantum gravity as well, it's important that this is also gravitational. And we can say that for any object of your size because of the way that we know that there's. Just by looking at it, we can tell for any object of your size, which is roughly like, let's say, a cubic meter of space, okay? For any object of that size, there's a maximum amount of entropy in which you can store in that space. So that means there's a maximum amount of different arrangements that you can have for a cubic meter of space. Okay? You occupy a cubic meter of space. There's a maximum entry you can fit into that. Why is there a maximum entry you can fit into a cubic meter of space? Because if you go too high, you start creating black holes, okay? Gravity puts this barrier in, okay? So there's a maximum of entropy associated with a cubic meter of space. And therefore, there's a maximum of arrangements that you could have for a cubic meter of space. And so that really puts it a cap on what you can have there. There's really no. There's only so many arrangements that could make up U. Right. Or any cubic meter of space. So. So let's take a cubic meter of space. There's one arrangement that is you. There's one arrangement that is me. There's one arrangement that there's a cow. There's one arrangement that's just a vacuum. There's one arrangement that's an alien. There's one arrangement that's Donald Trump. There's a whole bunch of different things that you could have that are a cubic meter of space right? Now, when you actually calculate, based on black hole physics and our understanding of gravity and quantum gravity, quite how many that is. It's a very big number. It is a very big number, but it's nowhere near as big as a Googleplex. Googleplex is far, far, far bigger. That's how ridiculously big a Googleplex is. It's way bigger than this already big number, that is the number of arrangements that you could possibly have in a cubic meter of space. So now we ask the question, okay, I take a cubic meter of space. It's you, Greg. Okay, what's the state that this cubic meter of space is in? It's in a Greg state. Brilliant. Let's go to the state, let's go to the cubic meter of space alongside him. Now I don't know what's alongside you Greg, but whatever's there is another state. And I keep going across the universe like this, sampling cubic meter after cubic meter after cubic meter and eventually I'm going to start seeing repetitions. It's absolutely inevitable. Now most of those repetitions are going to be pretty boring repetitions. There'll be repetitions of say a vacuum state or something like that, but they're not all going to be. And you can start to calculate. Yes, Greg is quite an unlikely state. He is. There is Greg, you're very special. You're an unlikely state, but you're not so unlikely that a Googleplex can't start to overwhelm the probabilities. And so it starts to become implausible in a Google plissan universe, a universe with the Googleplex across that there is not another Greg there. The probabilities, you basically sample so many, so many states and you start to overwhelm the small probability that it is a Greg. Okay, then you have to ask the question, well, is the universe that big? Well, the observable universe certainly is not that big. That's certainly true. But could the actual universe be that big? And it's possible, right? One of the theories of the universe that is very popular is this idea of what we call internal inflation. And basically what that means is that you, you find yourself with baby pocket universes growing up and expanding very quickly. So here, over here I have some. A universe sort of will pop up and grow very quickly. Another one here pop up, grow very quickly and these expand really fast and very quickly. You end up with an enormous gargantuan universe. And in that kind of setup I think it's entirely plausible that it could be a Googleplex across, in which case there are doppelgangers.
- [24:13]
Gregory McNiff
That is fascinating. And you teed up. What I should have introduced at the beginning, namely your book is divided into three parts, as I'm sure our readers or listeners have already figured out. The first part is large numbers. And on that note I want to move to a chapter titled Graham's number and you introduce a very cool phrase there And I'll ask him this question. What is Graham's number, and why can it lead to black hole head?
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Antonio Padilla
So Graham's number is so named after Ron Graham, who was a famous mathematician. He passed away fairly recently, actually. And it sort of came to sort of notoriety, this number, because it held the world record for the largest number to have ever appeared in a mathematical proof. That's actually not actually true. The number that was in the mathematical proof is not the number people call Graham's number. The number of people called Graham's number is a very closely related number that Ron Graham cooked up for a Popular Science article to describe the work he'd been doing. But it's not actually quite the same number. But we won't worry about that kind of detail. It's an enormous number. I mean, it really is an enormous number. We talked about Google, which is a one followed by 100 zeros. There's a Googleplex, which is a one followed by Google zeros. Graham's number dwarfs all of them by loads. I mean, it's not even remotely close how much this dwarfs them. You can't write it out in any kind of meaningful sense that using traditional decimalized numbers or anything like that, you need to introduce a new notation to try to describe it. But, okay, you do that. But one of the things I asked about was, okay, let's imagine you did write out its decimal expansion, right? You literally write it out as an ordinary number, like you do 10 or 15 or whatever, right? What then? Okay, so you're learning how to write it out on a piece of paper. Could you even picture it in your head? Could you get it in your head? Now, the point is that every number carries a little bit of information. Every digit in that expansion carries a little bit of information. Okay? I think it's about four bits of information, if I remember rightly, each digit. So there's information in every digit. So as you imagine Graham's number sort of coming into your head, like digit by digit by digit, I think it ends in a seven, if memory serves me correctly. So you see the seven come in, okay, you've taken in a little bit of information. You next see the next digit, the next digit, each time more information coming into your head. Now, it turns out that information and entropy are very closely related things. They are in some ways the same thing, I would say. So what that means is that you're putting. If that information's coming into your head, then entropy is coming into your head, okay? And so that entropy is coming into your head. And so what's going to happen, your entropy in your head is going to build. It's going to build, it's going to build, it's going to build. In all likelihood, what would inevitably happen if you really kept putting this information is if you could keep doing it without having a seizure, right? Then in all likelihood your temperature of your head would start to increase and you'd start to. Your brain had boiled. So if you could avoid that, right, so you could avoid the seizure, you avoid your head boiling, but you keep putting these digits in one bit by bit by bit by bit into your head, all these digits of Graham's number. Well, eventually there is this point in which no matter what you do to avoid a seizure or avoid your brain boiling, there is a point which you can never pass. And that's the point which is at which your head has no choice but to collapse and form a black hole. Because I think I sort of touched on it earlier. Black holes are really, are the most entropic objects in the universe. Nothing can hold more entropy than a black hole. It's just not possible, right? So you can ask, what is the most entropy you could fit into a space the size of your head? It is in the black hole the size of your head. So if you shove so many digits of Graham number into your head, great. And then ask, and then eventually you've put so much entropy into such a small space, then there's no choice but for that thing to be. That was your head to become a black hole. And you suffer from black hole head death. And that's it. And you don't even close the graves number by the time you get there.
- [28:37]
Gregory McNiff
Excellent explanation. And you anticipate my next question, which is, what is the information paradox?
- [28:42]
Antonio Padilla
Ah, yes. So the information paradox is. So that's where we start to think about where this information goes inside black holes. So what I think about is, so imagine you had, you woke up one morning and you see a black hole at the bottom of your garden, right? And you're like, okay. And black holes don't give up much information about themselves. They the only things you can measure. There's a famous statement, black holes have no hair. What that, what that says is, is that you can only really, when you look at a black hole, measure its mass, its electric charge and how fast it's spinning. Those are the only things you can, you can figure out just by looking at it. So you see this black hole in your garden and then, and then you measure those things. Great. And then the Next day you look out again and it's still there. And you measure those things and let's say those things have changed so that the mass of it, for example, maybe has changed by the mass of an elephant. And you say, well, what happened? Did it swallow an elephant maybe? Or did it swallow a car which also has the mass of an elephant? There's actually no way for you ever to know. It's just not there. It's like the information has just disappeared. Now then you can say, okay, well, doesn't it give off information through Hawking radiation? Stephen Hawking applied quantum mechanics to a black hole and found that actually, you know, black holes can actually radiate. They'll eventually irradiate and evaporate. Okay. When you apply quantum mechanics, but, but the spectrum of that radiation, Hawking show, doesn't contain that information that you're talking that you would need to tell if it swallowed an elephant or if it swallowed a car. It's just the information is not there right in the radiation. So that's the information loss problem. It's like, where did the information go? The rules of quantum mechanics tell us that information shouldn't be lost. What goes in should eventually come out. But black holes seem to be violating that. I think most physicists don't think that they will, obviously, with these famous bets that Hawking took and with Kip Thorne and people like that, the guy behind interstellar. So you know, about whether information really was lost or whether it wasn't lost. I think most people now would say it's not lost. But the question of how it's not lost is still an open one and it's not understood. There's been a lot of very interesting recent work on this, but I think it's still an open question.
- [31:04]
Gregory McNiff
Yeah, as an aside, and I want to ask you about string theory and a theory of everything. It seems like black holes might be the laboratory to figure out, you know, these deep questions. I mean, the physics of black holes are just amazing. Related to galaxy formation and the information paradox, de sitter horizon, all of this, it's just fascinating. But I want to move on to the next chapter. I believe it's pronounced tree three, but I'll let you correct me there. And I want to read you a brief passage here. The Poinacare recurrence of our universe is out there, but no one will catch it with experiment. In a way, it's Goodell's incompleteness, but with physics rather than math and unprovable truth of the physical realm, we could say the same of Tree three and the game of trees. It exists in principle, but it is so big, the laws of our universe will never let it happen. What do you mean by that?
- [32:05]
Antonio Padilla
Okay, so we have to ask, what is Poincare recurrence? So if you've got any kind of finite, essentially a finite system, any system that's finite, then you can start somewhere in that system, so some particular state, right? So I don't know. I start off with. So let's take the arrangement of my desk. It's a finite system. Let's imagine it's a finite system. I start off with everything here, everything laid out as it is, right? And then over time, you know, I move a pan across here, move the iPad across here. Everything's getting moved around. I'm going to, to new states. I'm exploring the space of possible states that are available to me, and I rearrange my desk, okay? Now what it says is that eventually, eventually I'll get back to where I started from. I'll get arbitrarily close to it. This is what the statement of Poincare occurrences. Think of a pack of cards, right? It's basically the same as saying if I keep shuffling a pack of cards, eventually I'll get back to the start of the original deck. And it's to do with finite systems. And it's a very long time. It takes a very, very, very large number of steps to return back to sort of some initial point. It's extremely large number. Now you might say, well, what's that got to do with the universe? Well, it turns out that actually, because we actually live in a universe which is. We talk about the expansion of the universe, how the distance between galaxies is growing at an ever increasing rate, and that expansion, rather than slowing down, is actually speeding up. It's quite unexpected. It's counterintuitive. You wouldn't expect that with gravity. It's an attractive force, but it is. This is what's happening. And it turns out that when you have these universes where the expansion is speeding up, they're a bit like universes in a box in a weird way. There's this point beyond, because the universe is speeding up, you get regions which just speed away from you. And it's. If you live in a little local box, which is just your. You're the bit of the universe that you can see, and you ain't going to go beyond it, and you can never go beyond it, there's going to be parts of the universe which are just always disconnected. From you, that boundary between what we can see and what we can't see in this context is called the de sitter horizon. But that's just a word. The point is we find ourselves in this kind of like cosmic box. Because we're in this cosmic box because of quantum mechanics and because of gravity, it turns out that were in a finite system, just like my desk or just like a pack of cards. And so because it's a finite system, you can ask yourself, well then eventually I'm going to have a career recurrence. So I have a state of the universe at one point, let's say now look out the window state of the universe now. And eventually, just like any find light system, just like my desk, just like a pack of cards, it's going to find itself back where it started from. After it'll move from one step to another, to another, to another to another and explore all these finite possibilities and eventually come back to where it started. It's an extremely long time, A really, really long time. But it's not as big as tree three, okay? The number of steps, there's no idea to think it's tree three. So what that means is this tree three is this number associated with a particular mathematical game. So if you try to play this mathematical game, then the number of steps you would need would just take you way beyond the Poincare recurrence time of the universe and therefore you can never complete the game.
- [35:33]
Gregory McNiff
That's another mind blowing concept. Moving to the next mind blowing concept, Tony, you're right. The holographic principle is the most important idea to have emerged in physics in the last 30 years. What is the holographic principle and why is it so important?
- [35:49]
Antonio Padilla
Yeah, so the holographic principle is the idea that in some ways our world is sort of three or four dimensional world, if you count time is kind of an illusion and that actually we actually live on the boundary of that world in a space which has one dimension less. And so in that sense it's like a hologram. So you think of a hologram, there's a sort of lower dimensional representation of something higher dimensional. Right. So two dimensional representation of a three dimensional object. In some ways what we perceive as the world around us is this sort of representation of something which is fundamentally lower dimensional. You know, there's a dimension here which is in some sense an illusion. Why do we believe that? It seems like a weird thing to believe. Well, it's to do black holes. Actually black holes were the first real sign that that might be True. The engine tree of a black hole, normally stuff like an egg or a dinosaur or whatever, its entropy will grow with its volume. So you double the amount of egg, you double the number of arrangements, you know, that you could, you could, you know, double the size of an egg, you double the number of arrangements that egg could have, and therefore you double the entropy. That's normally how things work. Same with a dinosaur, same with me and you, Greg. We're not grab, we're not deeply gravitational objects. So that's, that's how those things work. But for deeply gravitational objects like black hole, when you really start to probe the structure of quantum gravity, things behave differently. And it turns out the entry of a black hole doesn't scale like its volume. You don't double the volume of a black hole and double the entropy. To double the entropy of a black hole, you have to double its surface area. So entropy is counting these states. It's also a measure of counting information. And so that means in a sense, it's a bit like the, the information in a black hole is kind of stored on its surface area, on its event horizon. That's a lower dimensional surface. Okay. And that's the first idea that holography might be, might be real. And of course, Juan Maldacena, who's this brilliant physicist, then actually found a concrete example of holography in action where on the one hand you have this higher dimensional theory with gravity and all those kinds of forces, all very wonderful. And then there's an equivalent description of exactly the same theory that exists in one dimension or less. Okay. It's a toy model. It's not something, it's not our world. But it's a construction where you can talk about gravity on one side of the equation and talk about this holographic theory on another, and the two together describe the same physics. And it's remarkable. Well, sorry, I should. And so why is it the most important idea? Well, it's fundamentally changing how we thinking about gravity. It's intuitively remarkable. It really gives us a new approach to think about gravitational physics and to check aspects of our ideas about quantum gravity in a way we would never have imagined, I think, and that's why it's so important.
- [39:03]
Gregory McNiff
And Tony, just to clarify that, Lisa, gravity. You do talk about, Shannon, in this concept as well. Is this the idea that there's information embedded within this concept of gravity or this entropy? What makes this idea of gravity so unique or new?
- [39:19]
Antonio Padilla
I guess I think it's not so much that gravitational objects are entropic because or carry information because basically every object does, right? Every macroscopic object is entropic and carries information. What's funny about black holes is the way they carry it. As I said, it's as if the information a black hole, you don't double the amount of information in a black hole by doubling its volume. You double it by doubling its surface area. So that intuitively implies and this was really the thing that triggered the holographic ideas that maybe the information on a black hole is not stored inside the inside it, but on its surface area. And that's why black holes because they're deeply gravitational objects in a way that an egg or a dinosaur isn't.
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- [41:39]
Gregory McNiff
Yeah, no, like I said, black holes are just so fascinating. I want to move to part two of the book, which is appropriate, titled Little Numbers and the first chapter is called Zero. And you do a really nice job of explaining, I guess, the historical development of the concept of zero. I just want to hit you with two quotes. You write we can glimpse at the connection between symmetry and zero by imagining the accounts of a large organization. And you later write that the connection between zero and symmetry is more than just mathematics and philosophy could you briefly talk about that connection between zero and symmetry?
- [42:14]
Antonio Padilla
Yeah. So the example of the accounts. So let's suppose you had a large organization where you had a million pounds going out like it cost a million pound cost, and then there's. Do you sell something for, again, order of a million pounds, but not exactly a million pounds, maybe two million pounds. And you've got these ingoings and outgoings and da, da, da, da, da. And there's a whole bunch of them, right? And then at the end of the year, you come in and you add them all up and you find you get exactly zero. The ins have completely balanced the outs. Well, if that were to happen, it feels like there has to be an explanation. Something's gone on. Right. Why have the ins completely perfectly balanced the outs? Okay. When I'm dealing with general transactions that are a million pounds or a million dollars, and yet at the end of the day, it all balances to zero. It's just very unexpected. Right. And this is why. So it suggests when things sort of miraculously cancel like that, it sort of points towards a missing explanation. Okay. And it's certainly our experience of physics is that when things balance out perfectly to zero, there is an explanation for it. And when we don't understand why things are balancing, then we start panicking and we start looking for an explanation. And that's what often drives the directions for new physics in research.
- [43:43]
Gregory McNiff
Okay, I want to move to the next chapter which talks about the Higgs boson article, and it's appropriately titled zero dot decimal point followed by 15 zeros and then one. And in this chapter you reference this notion of spontaneous symmetry breaking, which you call a very profound idea. Could you explain that?
- [44:05]
Antonio Padilla
Spontaneous symmetry breaking. Yeah, yeah. So. So. So actually, without it, we wouldn't exist. So spontaneous symmetry breaking is what actually. So. So when we. So things like you're built, made up of sort of, you know, the quarks and protons and, you know, the build up protons and neutrons inside your body, inside the nuc, you've also got electrons floating around those nuclei. Where do those objects, those quarks, those electrons, where do they get their mass from? Because they get it from spontaneous symmetry breaking. So it turns out there's this mysterious field called the Higgs field, which sort of lives around, and it starts off in the universe in one particular state, a highly symmetric state. And in that situation, it turns out that, well, in all situations, the mass of the electrons and the quarks and all the things that make you up, those guys depend on where the Higgs is sat. If the Higgs is sat in its original symmetric state, then the masses of those quarks and those electrons is going to vanish. In fact, they won't even exist. Right? So if you want for them to get a mass, what the Higgs has to do, it has to break the symmetry. It has to find a new position in its possible field space. It needs to find some new place to go where it breaks the symmetry. It reaches some state where it's not zero. We've talked just now about how zero is quite like. Is quite like you might associate it with symmetry, right? With an explanation, with symmetries. Those sorts of things are balanced, right? But in order to get mass for quarks and electrons and all the stuff that make up human beings and planets and stuff like that, you need the Higgs to break that symmetry so that it goes to a new place where it's not zero and where it will. So that those things can start to build up mass. At that point, you can start to build up objects like me and you.
- [46:17]
Gregory McNiff
Regarding the Higgs, you say, quote, finding a Higgs like ours is like finding a snowman in the fires of hell.
- [46:24]
Antonio Padilla
Yeah. So there's a mystery in it that the Higgs has a funny thing about it. Right? So it, too, has a mass. Right. It also has a mass like the. And it gets its. You know, we can understand where its mass comes from. What we don't understand is why it takes the value it takes. Because when we add quantum corrections, when we ask about the quantum effects on the mass of the Higgs, they turn out to be huge. So what you should think of is you think of the mass of the Higgs is you can think of it in a classical world. It just has a mass. But actually, it doesn't really live in a classical world. It lives in a quantum world. It's surrounded by an ambient quantum world where. Where quantum mechanics is happening, that is where the Higgs lives, is where you and I live, right? And so because of that, it will feel that world. It will feel that quantum world. And one of the effects that it can have, that, you know, that ambient quantum world around it, it will change the mass of the Higgs. It can change what it is. It's a bit like, say, if I take a cannonball and I have a cannonball in air, and then. So that's a bit like the classical world, and then I plunge the cannonball into water, right? I will change its effective inertia in the water because it's surrounded by all this material that's like, plunging from the air, which is like the classical world, into the water, which is like the ambient quantum world. So just like the cannonball, it starts to sort of feel its surroundings. That's also true of the Higgs. And so its mass changes due to the ambient quantum world. And we can calculate by how much. And it's a huge amount. Okay. The corrections are enormous. So what that meant was, is that there had to have a certain value outside, and there's this huge balancing act, which is just so unlikely and so unexpected to get the observed value. The corrections are so big that it means he must have started out with something which is balancing those corrections to get the relatively small value of the mass of the Higgs that we see. So it's a really unlikely scenario. And we. Well, there are some explanations for why it might be true, why this is happening. What I will say is that none of them have been experimentally proven yet.
- [48:38]
Gregory McNiff
Yeah, that's a nice segue about experimentally proven. To my next question, you write, we can think of super symmetry as the most complete symmetry of space and time, as the beauty beyond all beauty. Only there is a snag. No one has ever seen such beauty. How do we know it's so beautiful if we haven't seen it?
- [48:58]
Antonio Padilla
Well, because this comes back to maybe one of the overarching themes of the book, because we can do the mathematics right, we can push the mathematics to its most beautiful extreme, and then we're naturally very much led to supersymmetry. And supersymmetry has this nice property that when you ask about those ambient quantum corrections to the mass of the Higgs, how the Higgs responds to quantum environments, if supersymmetry exists, it gives you the beautiful balancing act that you need to not get huge corrections. So that's why we like it. The problem is it's never been observed in nature. We're looking for it. The Large Hadron Collider. Those experiments are absolutely looking for evidence of supersymmetry. It could be that supersymmetry is just beyond the experimental reach of those experiments, and it's just around the corner. But that begs the question, how long do you keep looking for? It's pretty expensive experiments, no?
- [49:54]
Gregory McNiff
Absolutely. And definitely in the billions. I want to move to another fascinating chapter 10, to the minus 120th, in which you discuss Einstein and the cosmological constant. Why do you call the cosmological constant embarrassingly small?
- [50:11]
Antonio Padilla
Yeah, so we talked about the Higgs, the Higgs feels the ambient quantum environment and the corrections to its mass are much smaller than the corrections to its mass should be large. But there's some weird balancing act going on when it comes to the cosmological constant. What is the cosmological constant? It's the energy density of literally empty space. So you take empty space, you might think it's got no energy in it, but it's not true. It does. Quantum mechanics tells us that virtual particles are sort of popping in and out of existence. And one of the effects of those virtual particles popping in and out of existence is that they give the empty space an energy density. They give it some energy, and you can calculate that. And it's huge. It's really, really big. And then you can ask, well, okay, if empty space has this energy, then shouldn't that energy gravitate? Right. Shouldn't it be part of how the universe evolves under gravity? And it should be, because that's how general relativity works. So you can ask, okay, I have this huge energy of empty space. I throw it into my equations that Einstein taught me, and I ask what happens? And what happens is the universe would have crunched itself out of existence within less than a heartbeat of being formed. So that's obviously not our universe. And so it turns out that the energy of empty space in our universe is absolutely minuscule. It's minuscule to be consistent with what we see in observations, to have such an old and large universe. So something's going wrong in our estimate of what the energy of empty space should be based on quantum mechanics. So we've got quantum mechanics telling us it should be huge, and we've got the observations and general relativity telling us it's tiny. So it's a huge mismatch. It's probably the worst prediction in theoretical physics. Huge on one side, tiny actual value that you see. So you expect it to be huge. It turns out to be tiny. It's a terrible prediction. We don't know why it is, what's going wrong.
- [52:21]
Gregory McNiff
Yeah, another great chapter. I want to move to the final third of the book, titled Infinity, and I'd argue the subtitle could be There is a Fine Line between Genius and Madness. And with that as a opening, tell me about Georg Cantor and his contribution to infinity. And candidly, just what a fascinating and maybe slightly tragic figure he was.
- [52:47]
Antonio Padilla
Yeah. So what did Cantor try to do? So he said he tried to count beyond infinity. This is essentially what he tried to do. So we have our finite numbers, and we keep adding one. We go up and up and up and up and up, and we climb the ladder of numbers and infinity lives there sort of in the far distance, inevitably reached.
- [53:06]
Gregory McNiff
Right.
- [53:06]
Antonio Padilla
That's the sensible way to think of it. Well, Cantor wasn't happy with that. He wanted to go beyond infinity. And so that's what he did. He found a way to start categorizing infinities, different types of infinity, and how to count beyond infinity and how to do this in a kind of rigorous way. Well, yes, a rigorous way, I would say. But I guess playing in that landscape of it's in danger of sort of, it's made me so beyond normal life that maybe it took him, drove him slightly mad. I think there was another issue with Cantor was that he was met with a lot of opposition within the, you know, the mathematical community in his time. And there were some very powerful. He had some very powerful enemies who tried to suppress his career. So I think that was probably more than anything else the thing that led to his sort of his poor mental state towards the end as much as the kind of, you know, really intense mathematics that he was trying to do and, you know, the transfinite world in which he found himself. But I think history has actually looked upon him quite favorably, as it turned out, even if his contemporaries didn't so much.
- [54:22]
Gregory McNiff
No. That's nice to hear. I know you point out Cantor believe Shakespeare is a fake, and then two pages later you have beware the tides of gravity, which I assume was a reference to a paraphrase, the eyes of March. So nice job on that. Which means I'm going to ask you a pretty heavy question now. How likely are we to find a theory of everything? And is string theory our best approach to doing so?
- [54:49]
Antonio Padilla
So I would say yes, string theory is our best approach. It's the, for me, the best candidate. And it's. I didn't get time to chance to talk about as much why in the book so much. But there are reasons why I think there's, there's, there's often a sort of a lazy narrative that string theory should came up with this idea that, that, oh, what if the universe was made up of tiny strings rather than particles? Couldn't I then just. Let's just see what happens with that. Oh, it seems to give me quantum gravity. It's not quite how it worked. It was far more methodical than that. And actually it involved a problem. They weren't even trying to find a quantum theory of gravity or a theory of everything at all. They were trying to do something very Different. They were trying to, trying to solve a very simple mathematical problem that had to do with what happens. So you scatter particles together and then you find a breakdown in your understanding at high energies and you try to mathematically fix it. This is essentially what they were doing. And they found a way to mathematically fix this problem. And they tried to then interpret what the mathematical fix was. Nobody had mentioned the strings at this point. But then somebody realized actually the thing theory that was underpinning this mathematical fix and it was purely mathematical fix was strings. And this is often how it goes, right, is that you do. This is the tool that maths gives you. It gives you the power to sort of attack a problem from a purely well defined mathematical perspective to come up with an answer which is within the, you know, the rules of the game which you've set is potentially even unique. And then you can come along later and try to interpret physically what that mathematical answer is giving you. And that's what happened with string theory. So it was again to solve this mathematical problem involving going to very high energy scattering of particles and then try to reinterpret what it actually meant. And it pointed to this theory of strings. But the theory of strings then gave so much more than we could have hoped for. It started to reveal itself as a potential quantum theory of gravity. It contained there was low energy limits where it could look a lot like our universe, which doing that perfectly is tricky. But certain features which are very similar to our universe which you can get things to look a lot like. Newton's law for example, and other theories of quantum gravity simply can't do that. So I think it's a very powerful theory. Is it perfect? Absolutely not all the low hanging fruit was sort of picked. And to fully understand string theory as a truly fundamental theory of physics, we're still some way short. It's got mathematically very, very difficult. We need a real revolution in how we're thinking about it. A lot of people criticize string theory because they say, oh, it can't be experimentally tested and stuff like that. To some extent that's true, but what I would say is, is that that's true of any fundamental theory that includes gravity. Because testing gravity at the scale at which it becomes quantum is not going to be accessible to us in any experiment, any time of that lifetime. So this is true of any quantum theory of gravity. Not just string theory, but string theory has a kind of mathematical uniqueness and a mathematical elegance that none of the others I would say do. It can be connected to the low energy world in a way that the others can't. But it's not perfect and we do need a revolution in our understanding of it. I would say.
- [58:19]
Gregory McNiff
Yeah, wow, that's a great answer. And I want to circle back to the relationship between mathematics and our understanding of the universe. But before I ask that, as the last question, you have a really nice, I guess, coda or ending. And could you briefly talk about the relationship between K2 the mountain and K3 surfaces?
- [58:36]
Antonio Padilla
K2 the mountain and K3. What did I say around that? Okay, now I talk about K3 reset. Yeah, yeah, yeah, yeah. What did we say Now? So the K3 service is related to sort of. So what is K3 service? So K3 service is. So in string theory, one of the things about string theory is that it does live in four dimensional space time. It actually lives as a consistent theory in higher dimensional space time. 10 dimensions, in fact. So to get from 10 dimensions down to four, you need to wrap your. You need to wrap six of these missing extra dimensions up on a very. To wrap them up tiny, tiny, tiny, so you can't see them. Now there's a certain way in which you should wrap them up, right, in order to get sensible looking worlds in four, you know, leftover four dimensional bits. And one of those involves wrapping things on something called a K3 surface. Now, Ramanujan, who's this great mathematician, he did some physics, which seems to mysteriously. And he was looking at, I think, sums of cubes and sort of, you know, these. I'm trying to remember exactly what it was now, you know, things that sort of nearly. He was trying to solve Fermat's last theorem, I think. So he was finding, looking for solutions, you know, counterexamples to Fermat's last theorem. So he's taking very large cubes, added them together and seen if they were giving another cube, for example, to try to sort of find a counter example to Fermat's last theorem. And he had lots of very near misses. Never quite. Of course, Fermat's theorem is true, right? So he was never going to get that, but he got lots of very near misses. And the technology he was using to do this, weirdly, is very closely related to the technology of these K3 surfaces, which are the kinds of surface that you need in string theory to get down from 10 dimensions down to a more realistic lower dimensional world.
- [60:41]
Gregory McNiff
Great answer, and I should have read you your quote. The climber George Bell had once described K2 as a savage mountain that tries to kill you. K3 surfaces can be just as hostile, at least in the eyes of those mathematicians brave enough to study them. But you prefer.
- [60:56]
Antonio Padilla
Stop that stated.
- [60:58]
Gregory McNiff
I thought that was very clever. Last question, a little more serious. And this has sort of been a recurrent theme throughout the book in our interview. But is mathematics a universal language? I think Galileo thought the universe was written in mathematics. Others have said comparable, had comparable insights or claims. Or do you view it as a tool we use to describe the universe? How do you think about mathematics as we continue to probe the deepest mysteries of physics and the universe?
- [61:28]
Antonio Padilla
I think it's a great question. I don't have a definite answer. I have maybe a prejudice would probably be the only way I could describe it. So I think we live in a world where we have this. We have this tool, this amazing tool which is mathematics, which we as humans have invented. And this tool seems to do this incredible job of describing the physical world around us. It's incredible, right? Even things like imaginary numbers, which you think would have no right to have anything to do with the physical world. Well, yes, they do. They appear in the equations of quantum mechanics. So there's something to do with it. And quantum mechanics is part of the real world. So even wild mathematics like that seems to have a place in the real world now. But you can ask the question, so we have this tool, it does an incredible job. You can ask the question, is everything in our universe able to be described by this amazing tool that we've got, this mathematics? And I don't think the answer to that is obvious. I don't think the answer to that is answered, actually. Of course, it could be that there are just some things which are beyond mathematics. You could flip that question around and let's say everything in our universe was mathematical, could be described by mathematics. Then you could say, well, is our universe all of mathematics then? Or is there some mathematics which just has nothing to do with our universe? In which case, why not? So this question, I think just you can really get lost in a rabbit hole. And what I will say is that we have an amazing tool and so far it's done brilliantly well and hasn't come unstuck yet. But if you really push it to extreme, I don't think we know that's.
- [63:09]
Gregory McNiff
A great answer and a great place to end this interview. Again, the book is fantastic. Numbers and where to Find A Cosmic Quest from Zero to Infinity by Antonio Padilla. Tony, thank you so much for taking the time to chat with us today. And thanks for writing such a great book. Like I said, it really is mind exploding, mind bending without necessarily causing black hole head depth.
- [63:33]
Antonio Padilla
Good stuff. Foreign.
- [63:46]
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