It's a big universe we live in, so it comes as no surprise that big numbers are needed to describe it. There are roughly 1022 stars in the observable universe, and about 1088 particles altogether. But these numbers are nothing compared to some of the truly ginormous quantities that mathematicians have found to talk about, with inscrutable names like Graham's Number and TREE(3). Could such immense numbers have any meaningful relationship with the physical world? In his recent book Fantastic Numbers and Where to Find Them, theoretical physicist Antonio Padilla explores both our actual universe and the abstract world of immense numbers, and finds surprising connections between them.
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Antonio (Tony) Padilla received his Ph.D. in physics from the University of Durham. He is currently a Royal Society Research Fellow in the School of Physics and Astronomy at the University of Nottingham. He is a frequent contributor to the YouTube series Sixty Symbols and Numberphile.
Click to Show Episode Transcript
0:00:00.1 Sean Carroll: Hello, everyone. Welcome to the Mindscape Podcast. I'm your host, Sean Carroll. Sometimes when talking about cosmology, I will remind people that our universe, which is about 14 billion years old, or roughly order of magnitude 10 to the 10 years old, right, which sounds pretty old, is nevertheless pretty young, in some sense. Our universe is just a baby compared to how old it will be.
0:00:23.5 SC: Of course, we don't know exactly how old the universe will get, but according to the leading cosmological models that we have right now, the universe will get infinitely old. There's no reason for it to ever end. And anyway, it will use up its fuel in some finite amount of time. The sun's shining, the stars are shining, galaxies are shining, but this shining won't go on forever. Stars are going to burn up their fuel in about 10 to the 15 years, so that's a hundred thousand times the current age of the universe. And then the last black hole will evaporate away, we think, roughly speaking, about 10 to the 100 years from now. In other words, about a googol years from now, in the original notion of the word googol, before the search engine took it over.
0:01:11.7 SC: The idea of a number 10 to the 100 was actually invented, sort of almost as a joke, just as to stand in for a really big number you would never actually think to use. Today's guest, Antonio Padilla, Tony Padilla, as he's known, thinks otherwise. And he's written a wonderful new book called Fantastic Numbers and Where to Find Them: A Cosmic Quest from Zero to Infinity, where he talks about these big numbers, googol and of course its cousin the googolplex, which is 10 to the power of a googol. But then even way bigger numbers than that, number theorists, mathematicians, theoretical computer scientists have devised clever ways to represent these ginormously big numbers, much bigger than you can really wrap your head around in a very real sense, as Tony will point out as we talk.
0:02:00.0 SC: But what I really like about it is that Tony's day job is as a cosmologist. He's a theoretical cosmologist, writes similar papers to papers that I've written. He thinks about the cosmological constant and the cosmic microwave background and dark matter and dark energy and things like that. And so he finds ways to relate these ginormously big numbers to the physical world. You might think that some of these numbers are just so big, they have no possible physical relevance, and then as soon as you say that, someone is going to find out a way to bring them down into physical relevance.
0:02:31.4 SC: And so, which raises, by the way, some interesting philosophical questions. So are these numbers even real? If you could find the number that is so big that it is impossible to contemplate any physical relevance of it, then is that even a potentially real thing? Is that, is that something we should include in our set of numbers? That gets too philosophical. We're not going to talk about that stuff that much. But the point is, the very existence of big numbers is not only useful to physicists, but stretches our brains in thinking about them in fun ways. And so this is going to be a fun conversation about exactly that. Let's go.
[music]
0:03:25.4 SC: Tony Padilla, welcome to the Mindscape Podcast.
0:03:26.9 Antonio Padilla: Hi, Sean. How are you doing?
0:03:29.0 SC: I'm doing alright. Now, you've written a wonderful book and we'll be talking about stuff in the book, and it's about big numbers, but you're a physicist, you're not a mathematician. You and I are very similar in the papers we've written over the years, right? There's a lot of overlap there. We've cited each other. But you use the big numbers of math as a way of talking about the universe, which I think is great. And so to get there, let's start with some numbers that the person on the street might think are big but the professional mathematician or cosmologist would scoff at, like 10 to the 10, right? That's a pretty big number. What does that mean? What do we, how do we think about exponentials and other ways of making bigger numbers?
0:04:13.9 AP: Yeah, so I mean, when you think of something like 10 to the 10, obviously that's 10 multiplied by itself 10 times. So it's a 1 followed by 10 zeros. So you see that, and it clearly looks like a huge number. But that's kind of quite a, this idea of sort of repeated multiplication giving you exponentiation, it's just the first in a whole chain of ideas that you can build in mathematics. And yeah, it's just one of... It's really just the seed for something that can grow much larger and therefore can grow much bigger numbers.
0:04:44.7 SC: Well, the famous one of course is the googol and then the googolplex. And there's fun stories about how that came to be.
0:04:51.1 AP: Yeah, so of course a Google, which obviously most people associate with the search engine, it's spelt differently when we talk about a number, it's G-O-O-G-O-L, right, of course. But what so what is a googol? It goes back to Edward Kasner, who I think those of us who work in cosmology will know from, he has a spacetime named after him.
0:05:10.6 SC: Yeah, pretty good.
0:05:11.2 AP: Which people have used to think about the early universe and that sort of stuff. But what Kasner was actually doing was Kasner was thinking about, he was writing a popular science book, which we've both done. And he was trying to really convey the ideas about infinity and really how any finite number, no matter how big it might seem to us in our sort of day-to-day lives, is actually negligible, essentially zero compared to the infinite. So he just sort of, he came up with a big number. Well, what's a big number? A 1 followed by, never mind 10 zeros that you talked about, but a hundred zeros, right? So one followed by a hundred zeros. Now, no one's going to deny that's a big number, right, by any earthly measure that we use.
0:05:50.9 AP: And yet, and he wanted a name for this number. So he consulted his nephew, who was nine years old at the time, his name was Milton Sirotta. And he said, Well, can you come up with a name for this big number? And Milton said, Well, a googol. So it came from his nephew. But then the story didn't stop there 'cause of course Kasner wanted to go bigger and still talk about how even bigger numbers were sort of small compared to the infinite. And so he came up with a new number, which was a googolplex, which was supposed to be loads bigger than a googol, right? And so again, he asked his nephew, Milton, you know, what would be a good a good definition for this? So Milton said, Well, it should be a 1 followed by zeros until you get tired.
[laughter]
0:06:39.5 AP: Which I think for somebody like Kasner and any mathematician, that's a little imprecise. So he didn't go with that definition.
0:06:47.0 SC: Not very rigorous. Yeah.
0:06:47.6 AP: Yeah, yeah. So he went with a more rigorous one, so whereas a googol is a 1 followed by 100 zeros, a googolplex is a 1 followed by a googol zero. So it's a whole new level of size that you're talking about. And this sort of develops an idea of recursion in maths, which is really powerful. So from a googolplex, you can go to, well, what some people call a googolduplex, which is a 1 followed by a googolplex zero, and then you can go to a googoltriplex, which is a 1 followed by a googolduplex, so and you can see every time you do this, you're going bigger and bigger and bigger and bigger, and that's the power of maths.
0:07:25.9 SC: Yeah, no. I mean, that recursion is basically the secret, right? You make a big number by some process, and then you do the process again, and the number becomes way, way bigger than you started.
0:07:37.1 AP: Yeah, recursion, repetition, it's... That's what really gives you the calculational power to grow numbers really, really quickly and yeah, it's the beauty of it.
0:07:47.8 SC: And maybe it's useful to ground us in the real world a little bit. We live in a world that is pretty big. There's 8 billion people on Earth almost and... But the professional cosmologist deals with even bigger numbers, the number of galaxies or stars in the universe, but none of them are that big compared to a googol. Why don't you give us some of like the size and scale of the universe and compare it to these big numbers?
0:08:15.3 AP: Yeah. I mean, especially if you think about the universe. I mean, various things, numbers you can pluck out. It's things like how big is the universe? What's its size to the sort of the cosmological horizon just as far as we can see? Well, that's about 10 to the 26 meters. So that's 1 followed by 26 zeros. So that's the distance in meters to the cosmological horizon, which is what we normally think of as our universe. Other numbers, big numbers that you can think about in the context of the universe... Well, it probably doesn't really get any bigger than the number of particles that exist in the universe, right? And that's 10 to the 80, which is 1 followed by 80 zeroes. And that's already way smaller than a googol, right?
0:09:00.1 AP: So a googol is 1 followed by 100 zeros, and so 10 to the 80, it's 20 orders of magnitude smaller. So one of the things I talk about in the book is what could you do if you were a googol heir, if you had a googol pounds, right? You can literally buy every particle in the universe at a really inflated rate and still have plenty of money left over.
0:09:19.4 SC: Now, I might actually be misremembering this, but I thought that was 10 to the 88th particles if we're counting photons and neutrinos and so forth.
0:09:26.9 AP: Yeah. Okay. That's the... That's... Right. I mean, the typical estimate if you're just going on baryonic particles.
0:09:32.1 SC: Bary... Yeah. So 10 to the 80 protons and neutrons and stuff like that. Perfectly fair, but I know I have listeners. They're going to write in. So this is good.
0:09:43.1 AP: Don't worry. I know the feeling.
0:09:43.2 SC: And so what that makes you think... Oh, let's also get on the table Avogadro's number or some number dealing with biology somehow just to make sure that the biologists don't have bigger numbers than the cosmologists do.
0:09:54.5 AP: Oh, now you're testing me, Sean, because I stopped doing biology when I was about 13, so I'm not going to know. So Avogadro's, is that the one that's 10 to the 23? Is that the one that...
0:10:04.9 SC: It is. You got it right. And believe me, I live in fear of people asking me that, too, but I had to recently look it up. So yeah, 6 times 10 to the 23.
0:10:12.7 AP: Right. Yeah, exactly. And that's the number... What's that? The number of particles in a mole of gas or something like that? Is it?
0:10:17.7 SC: Yeah, yeah, that's right.
0:10:19.0 AP: Oh, see, this is really me digging back my...
0:10:21.8 SC: It's good.
0:10:22.1 AP: My old high school education.
0:10:23.4 SC: British education. Fair enough.
0:10:26.1 AP: Yeah. It's not something I've been working with in my day job, to be honest, but yeah, of course, the number of particles in a mole of gas, which is a... It's a huge number, right? But what I think that number really shows, demonstrates to us is there is actually huge numbers lurking right beneath our nose, and it's basically because of the hierarchy of scales that we see in nature, right? So it's on the one hand, we've got the scale that we... Of us, right? Roughly, a typical human who's around between a meter and two meters tall, and that's a typical scale of our size, but then you compare it to the sort of subatomic world and all that and even all molecules and particles and all that, that's such a much smaller scale. And that's why you can get big numbers like Avogadro's constant just coming out of quite mundane things. It's really the difference between particles and the world that we live in.
0:11:20.3 SC: And if the universe around us only has 10 to the 88th particles in it, and so typical macroscopic human-scale things have Avogadro's number of particles in them, isn't there... The naive guess would be, you never need a number as big as a googol. And 10 to the 100 is bigger than all of those numbers.
0:11:42.1 AP: Well, yeah, I guess that's an interesting way of looking at it. But like I said, it depends what you define to be our universe, right? On the one... And this is the one thing I always wanted to do with these numbers. So these numbers, they are in some sense beyond our universe, right? And so what I try to do in the book is drag them into our universe to try to get some sort of physical personality from them, which is... It's not easily done, because you're absolutely right, they don't have a natural place in there. But when you try to bring them in and you can really reveal some quite remarkable physics and remarkable ideas of physics, that's the idea.
0:12:18.8 AP: It's always been the case for me that numbers... I started studying maths in university, and for me, there is a beauty in numbers and there's an elegance in numbers, but I've always needed a little bit more, and what I've needed is that little bit of personality. And that personality, I think, comes from physics and bringing the physics into the game. So you're absolutely right. How do you drag a number like a googol or even a googolplex into the physical world? Well, you have to go a little bit beyond our universe and what we normally talk about our universe, and you have to start to think about maybe the universe beyond the cosmological horizon, and imagine a universe that maybe is much larger, and then you can really start to play games with these truly gargantuan numbers.
0:13:00.3 SC: The difference between mathematics and physics as occupations or past-times is a fascinating one, 'cause I do think that people who are neither one of those might think that they blur together, right? I mean, you're pushing around equations and you're trying to solve them, but the motivations are really quite different, right? The things that get a mathematician really, really excited are very often not the things that get the physicist excited.
0:13:24.7 AP: I think that's entirely true. I think sometimes we can get sort of fed up with the pedantry of our mathematician colleagues, right? It's like, "This is this." Like, "Why are you worrying about this detail? It doesn't matter. It's not going to change the physics, right," 'cause, you know, we have this idea of decoupling in physics that we don't worry about little tiny, microscopic details. They're not going to affect what's happening on macroscopic scales. And so we don't need to worry about your little pedantry. It's not important in the broader game and scheme of things.
0:13:52.0 AP: And I think that's probably where the mathematicians and physicists differ in their approach. And it was always, I'll give you an example of what it really was that triggered my change in my attitude to this was, I was doing, it was shortly after I first went to uni. As I said, I did a master's degree, and I had this proof to do and I wrote out the proof and I had all the arguments correct, and there was no issue with it. But I got zero, I got zero for this proof.
0:14:20.2 AP: And it was really frustrating. I was like, "Why have you given me zero?" And I thought, "What's going on here?" And it, and he, the don at the time acknowledged that the proof is correct, but he didn't like where I'd put the implication signs in the layouts of the proof, right. They were all on one side of the paper. And he goes, "Well, there's nothing on the other side of the paper." And I was like, "Well, this is ridiculous, right? This is a level of pedantry, which I, which it's too much for me. And I need more from my numbers." And that's where I started to look towards physics.
0:14:49.7 SC: I could tell from reading about it in the book that this incident really scarred you for life. You remember?
0:14:53.9 AP: It really did. It really did. I mean, I kind of got it. Well, looking back I kind of got it 'cause you know when you mark exam scripts and if things aren't laid out well it can be really frustrating, really. But wow, this I thought was a pedantry too far. But yeah.
0:15:09.6 SC: The way that I think about it is if you have a billion examples of some mathematical construction and there's some property that is true for all but one of them, the mathematicians really really, care about the one where it's not true and the physicists only care about the others where it really is true.
0:15:25.0 AP: Yeah, that's true. Although, at the same time we do focus on the things about our universe. When we think about the properties of our universe, it's the things that don't make sense that we care about.
0:15:33.4 SC: Also, that's fair.
0:15:33.8 AP: It's the things that don't add up that we care about. All the wonderful things that are just, yeah, that's how it should be. We're not going to worry about those details, but we're drawn towards the puzzles. As is everybody, I guess.
0:15:42.8 SC: So one of the things that the mathematicians do with this attitude of being interested in the math for its own sake rather than to describe physics, is they find ways, either because that's what they want to do or because they stumble upon them in the process of their regular math, find ways to generate way bigger numbers than a googol or a googolplex or a googol duplex, etcetera. So why don't you... Why don't you tell us just some of your favorite gigantic numbers? 'Cause there's a lot of sort of philosophical aspects of what these numbers mean when you get right down to it.
0:16:16.0 AP: Yeah. So do you want me to tell you a little bit about how you generate these really big numbers?
0:16:19.6 SC: Yeah.
0:16:20.2 AP: And... Okay, cool. So there's a really... So when you really talk about big, big numbers, like the kind of number like Graham's number, which was the largest number ever to appear in a mathematical proof for a long time. When you're talking about numbers like that are, you can't just use our conventional mathematical language that people learn in school to describe them. They're just too big. They're just too big. So you need to develop new ideas, right?
0:16:43.8 SC: Yeah.
0:16:44.2 AP: So, one of the things you can do is you can develop this sort of, there's, Donald Knuth who I believe is in Stanford, is a computer scientist over there. He developed a new notation to sort of build very, very big numbers. And it kind of uses, again, this idea of recursion and repetition. So very simply, so if you think about, really go back to sort of fundamental level and you think about, well, what is what is multiplication? Well, when I say multiplication, I'm really doing repeated addition. So 3 times 4 is 3 added to itself 4 times, right. That's what it is.
0:17:21.3 AP: And so, okay, so you've gone from sort of addition to multiplication using this idea of repetition and recursion. Well, you can go, now you can go one step up again, you can say, well, what is repeated multiplication? Well, we're still in the high school realm now, we're talking about exponentiation. So when you say what is 3 to the 4? Well, that's 3 to the power of 4. That's 3 times, times by itself 4 times, alright? So exponentiation is repeated multiplication. Now, at high school level, you stop at that point, right?
[laughter]
0:17:52.6 SC: That's enough. Yeah.
0:17:54.7 AP: But mathematicians don't, right. They don't, so they carry on. And so, so instead of saying that 3 to the power of 4 is, instead of saying it like that, they say 3 arrow 4. Okay. That's the notation they would use. And then they go a step further and they say, well, what's next...
0:18:09.8 SC: Sorry. Just to be... Because we can't see in the audio podcast, this is like a little upward going arrow in this notation, right?
0:18:16.1 AP: Yeah, yeah, exactly. So you have a little upward arrow. So instead of saying 3 to the power of 4, you say 3 upward arrow 4.
0:18:20.7 SC: Good.
0:18:21.2 AP: Okay. And so that just means the same thing.
0:18:23.8 SC: Yeah.
0:18:24.3 AP: Now you can go a step further. You've got repeated addition gives multiplication, repeated multiplication gives exponentiation, well, what's repeated exponentiation? Okay, well that's called tetration, right? And you can denote that with two arrows. So you have now 3 two arrows 4. That would mean 3 exponentiated 4 times. So 3 to the 3 to the 3 to the 3 to the 3, like that. And you can build like this. And of course, once you've got double arrows, you can have triple arrows, which are repetitions of double arrows.
0:18:57.0 AP: And you can have quadruple arrows, which are repetitions of triple arrows. And you very quickly get very, very big numbers. So for example, let me give you an example. So 3 to the power of 3 is 27, so 3 arrow 3 is 27. What's 3 double arrow 3? Well, that's 3 to the power of 27, which I think is about 7.6 billion. So already you get very big.
0:19:18.6 SC: Pretty big.
0:19:18.9 AP: Now you do more and more of these repetitions and you get really, really huge numbers. And this technique allows you to grow to very large numbers, as large as Graham's number.
0:19:30.0 SC: And the rules of the game, I guess, are that we have a finite number of symbols to work with, right? So we're trying to figure out ways to denote big numbers without just listing all the digits or whatever. That's impractical. There are not enough atoms in the universe, but we can compactly represent some of these giant numbers.
0:19:50.0 AP: Exactly.
0:19:51.1 SC: And then we can ask how big we can get.
0:19:53.4 AP: Exactly. So in the example we talked about, the... Each new arrow in this notation allows you to grow things much more quickly. But what you can then do is you can then start to sort of almost label the arrows themselves, the number of arrows themselves, and grow the number of arrows in this hugely powerful way. And then you can keep doing that and keep doing these tricks. And again, it's like you say, it's compactifying the notation that can allow you to sort of grow very quickly. And then you always put a little label on it, and then that label, then, there's some idea of what you do at the next step, what do you do when you do the repetition. And then you can grow the labels, you can grow everything. Mathematicians love having games with symbols and labels. And in some sense getting the right notation is really what gives you the mathematical power.
0:20:40.2 SC: And so what is Graham's number?
0:20:42.6 AP: So Graham's number is this extremely large number that was... It came about from the mathematician Ron Graham. He used it in a mathematical proof, he was trying to solve a problem in a branch of mathematics known as as Ramsey theory. Now, Ramsey theory, very loosely, it's the idea of finding order from chaos. So you have some random assembly of things and you're trying to find some structure and cliques and some order within that chaos. So for example: If you think about the Houses of Parliament in the UK, and you've got a whole sort of gossiping, bunches, shouting, and it seems very chaotic and everybody seems to be disagreeing and it's just... It's chaos. But you might ask, is there some order within that chaos? And you say, well, maybe there is, maybe there's a bunch of trade unionists. That there's a clique of trade unionists there. Within that seeming chaos there is agreement and order. And intuitively, Ramsey theory is the study of that intuitively.
0:21:46.5 AP: Now, what Ron Graham was doing, he was looking at a problem involving hypercubes, which are generalizations of cubes that are higher dimensions, you know, 5, 6, 7, 8, 9, whatever number dimensions you might be interested in. And he was looking for a particular property of these cubes and he wanted to know how many dimensions did you have to have to guarantee that this particular mathematical property would exist, would always be guaranteed to hold. And what he showed was that it's... The number had to be I think bigger than six dimensions, but it had... That the upper limit was this Graham's number, which is this truly gargantuan number that you can't write in any conventional way, you need this fancy mathematical tricks and notation to write it.
0:22:34.4 AP: Now, for me as a physicist, I did my usual thing where I like to think about, "Well, okay, this is all fine. And it's all nice thinking about hypercubes and thinking about this number. And there's some wonderful maths in there, and it's great and it's lovely, but can I bring it into the physical world?" Well, most people when they think about a number, they don't think of Knuth's arrows or some fancy mathematical recursion tricks. They don't think about that. They just think of it written out, right, using the standard way that we write out numbers in our decimal expansions that we use, right? That's what people do.
0:23:08.2 AP: So I said, "Well, suppose somebody thinks about Graham's number, what would happen then if they tried to picture it in their head?" And I quickly realized that if you do that, then you have a bit of a problem, because Graham's number is so large. And what you must remember is that each digit in Graham's number contains a little bit of information.
0:23:27.7 SC: Sure.
0:23:28.0 AP: Okay. There's information stored in each one of those digits. And so if you're going to picture Graham's number in your head, you're storing a hell of a lot of information in your head, and information weighs. And if information weighs, then every time you picture one of Graham's number's digits, you are going to store some information in your head, you're going to add some mass to your head. Might only be a very microscopic amount, but it will be a little bit.
0:23:53.7 SC: Yeah.
0:23:54.3 AP: And when you've got the whole of Graham's number in there, well, you're basically trying to cram too much into your head and the inevitable outcome will be that your head would collapse into a block. [0:24:03.1] ____.
[laughter]
0:24:04.9 SC: Has this ever happened?
0:24:07.1 AP: I'm not aware of it ever happening. Maybe it's one of those things we have to worry about, like they've created a black hole at the LHC, [laughter] somebody thinking about Graham's number.
0:24:17.2 SC: So just to get this right, if someone just asked you, "How many digits in Graham's number?" There's no known answer to that, right? It's not... Just to say the number of digits would be some incredibly large number.
0:24:30.2 AP: Yeah, it'd be a huge number. So yeah. No, I don't think you could... Is there a way to calculate it? It's probably of order Graham's number, this is ironically the answer, or maybe the log of it. It's probably the log of Graham's number.
0:24:45.4 SC: It doesn't help very much, it's still enormously large.
0:24:48.4 AP: Yeah. It'd be the logarithm of Graham's number. So but already... What you can calculate, the thing that you can calculate is how much information you can contain inside a space the size of a human head, that's something you can calculate. And the reason you know what the maximum amounts of information you can contain in the space the size of a human head, it's 'cause you know the thing that contains information the best. And the thing that stores information the best in the universe is a black hole. Nothing stores information better than black holes, they're brilliant information stores. And so you just ask yourself the question, "How much information can I store in a black hole the size of it, of a human head?" And it's way less than Graham's number.
0:25:32.2 SC: Yeah. So I have some deep philosophical questions about the existence of these numbers, but I just want to get one more number on the table first, which is TREE [3]. I love this number.
0:25:43.6 AP: Yeah.
0:25:43.8 SC: You talk a little bit about it, but I don't think I'd ever heard... I'd heard of Graham's number before, but not of TREE [3].
0:25:49.3 AP: Yeah. So TREE [3] is... So the way I talk about it in the book is I link it to this idea of the game of trees. So this is... So very intuitively, you have a game where you have seeds and you have branches and you build trees, right? And there are various rules to the games, like you start off with a single seed and then you can start adding and then somebody writes down a picture of a tree with a single seed and then the next person writes down another tree with maybe branches and different seeds and so on, and each time you start drawing these trees. Now there's... There's various rules, but one of the rules is that if somebody writes down a tree which contains a bit of a tree that's gone before, then the game's over.
0:26:38.1 AP: So what that mean... What do I mean by that? So let's suppose we're playing, Sean, and you draw one particular tree and then five goes later, I draw a different tree and sort of set within my tree is a bit of... Is something that looks a lot like your tree.
0:26:52.5 SC: Yeah. Okay.
0:26:52.9 AP: Then I would lose the game.
0:26:53.9 SC: You lose. Okay.
0:26:55.2 AP: Okay? So the question you can ask is, how long can this game last? Right? You might think...
0:27:00.1 SC: So, how long could you keep going drawing trees without any previous tree being in it?
0:27:06.0 AP: Yeah, yeah. Is the game guaranteed to end, for example?
0:27:08.9 SC: Right.
0:27:09.2 AP: Is it guaranteed to end? And if so, how long can it go on for? These are the kind of questions you can ask. Now, it turns out that the game is always guaranteed to end no matter how many different types of seed you use. So one of the things you can ask is, Well, how many... Suppose I have one type of seed, so let's say, a black seed, right? Then that's all I've got, then I play the game and see how far it can go. Now, it turns out that game can only last one move.
[laughter]
0:27:37.2 AP: Okay? 'Cause as soon as I draw the black seed, you've got to draw another tree and it's got to contain my black seed, and game over.
0:27:42.7 SC: There you go. Yeah. And so that means that tree of 1 equals 1?
0:27:48.6 AP: Yeah, exactly. Exactly. So now you play the game with two seeds...
0:27:52.6 SC: I was going to ask. Good.
0:27:53.7 AP: Different types of seeds, so maybe a black and a white seed.
0:27:57.8 SC: Right.
0:27:57.9 AP: Right? And... Okay, now you can play the same game again, and you start writing down the things that you're allowed to do, and it turns out the game cannot last beyond three moves. Okay? That's the maximum. There's no way the game is lasting beyond three moves. Okay? So tree of 2 is 3.
0:28:16.4 SC: Equals 3. Okay. Our sequence so far is 1,3, tree of 3?
0:28:22.0 AP: Yeah. Exactly. So now you play the game with three different types of seeds. So let's say you've got a black seed, a white seed and a red seed, and you can make trees that are combinations of these seeds that have got branches and all sorts of fun and games. And again, there are various rules about what you can lay down initially, but you'd agree to those rules. So we're playing this game now, Sean, and again, you can ask how long can it go on for. And you might think, "Well, okay. Well, for one seed it only lasted one go, for two seeds it lasted three goes. So maybe it's... I don't know, maybe it's going to be 10 goes for three seeds or 15 goes or something like that." And it's not, it goes bang. It just...
[laughter]
0:29:01.6 AP: It literally goes off on one. It's the most crazy jump in a sequence you could ever imagine. It goes to this number Tree [3] which... We just talked about Graham's number, which is too big to fit in your head. Well, I think Tree [3] is too big to fit in the universe, it's just a number which is... It's beyond anything. It dwarfs these already gargantuan numbers. It's an absolute monster. And yeah, it comes from this thinking about this game, and there's lots of... One of the things it's used for actually, one of the things it's useful for is proof theory and understanding what you can prove, what you can't prove within mathematics. And these kind of tree games are a good vehicle for that.
0:29:47.7 SC: But it's interesting because Tree [3], you can state what it is. It is the largest number of games or... What is exactly the definition? The largest number of moves that you could do?
0:30:00.1 AP: It's the kind of maximum length of the game of trees with three different seeds.
0:30:03.9 SC: Gotcha. And I can say those words out loud, but I can't, in a down to earth sense, calculate the number. It doesn't fit into the universe, right?
0:30:12.7 AP: No. It really can't fit into the universe. And this is the kind of thing that... This is again, where you tried to bring physics into the game, right? And start to say, "Well, okay, let's think about actually playing this game." So me and you, Sean, we start playing this game, right? And we're going to play with three seeds. And it's 'cause... Who's going to... We're really good at this game, so we make sure we don't get knocked out, right? And so we play this game, we play this game. We play as fast as we can. Well, how fast can we play it? We certainly can't play a go, a single go, faster than a Planck time, right?
0:30:44.7 SC: Yeah.
0:30:44.9 AP: Because if we... Which is 10 to minus 43 seconds. If we do it faster than the Planck time, we break spacetime. So we're not going to do that, so that's as fast as we go. We're going as fast as spacetime will allow us.
[laughter]
0:30:55.4 AP: And so we're doing it, we're playing this frantic game and we're drawing trees, drawing trees, drawing trees, drawing trees, drawing trees. Well, we'd be old men long before this game was over, okay? So maybe... Okay, so we pass away and we replace ourselves with some fancy AI device which carries on playing the game beyond our mortal lives. This game's playing. So there's AI Sean, and there's AI Tony playing this game of trees and playing and playing and playing. And they would carry on playing beyond the age of the solar system, the sun would die. So who knows what's powering these AI devices? But suppose they carried on, maybe they're getting their energy off the cosmic microwave background, I don't know. So they carry on playing this game, they carry on playing, and they would go far beyond this, the heat death of the universe.
0:31:50.9 AP: And they would go even beyond that. And eventually, the game would go on and on and on and on and on. And then you'd say, "Well, but surely it's going to end at some point, right?" And the problem is it never can, because one of the things we know about the universe is that we think it's a finite system. And in any finite system you get something called a Poincare recurrence, which is where everything comes back to where it began. So for example, if you think about a pack of cards, right? So if you think about pack of cards, when you open a pack of cards you normally... They're all laid out in the order of suits and it's really nice. And then somebody starts shuffling it and they get messed up and so on. And you play lots of games of cards and they get regularly messed up. Well, if you keep randomly shuffling the cards, eventually, and it will take a very long time, but eventually you'll get back to all the cards lined up as they were when you bought them. It will happen, okay?
0:32:42.2 SC: Mm-hmm.
0:32:42.7 AP: And it happens after this Poincare recurrence time. And this is something that's true of any finite system. And so if our universe is a finite system, which we believe it is, then that means that the universe will undergo a Poincare recurrence. And the time it would take for that Poincare recurrence is not enough time for you to finish the game of trees...
0:33:01.5 SC: Yep.
[laughter]
0:33:02.2 AP: So the universe with the tree would just reset itself.
0:33:05.4 SC: I'll put on the table that I'm not sure that the universe is a finite system. I think it's an excellent question that we don't quite know the answer to, but it at least is absolutely plausible that it is. I'll go that far.
0:33:16.1 AP: Yeah, I guess I'm basing that on the idea that we're dominated by dark energy, which in fact is going to dominate the universe for a long period and then... Yes. And I guess I'm basing it on the ideas that things like de Sitter, thinking about the de Sitter universe and the amount of degrees of freedom associated in...
0:33:35.9 SC: And I do want to talk about that. I just want to point out that we don't know, which is perfectly a safe thing to say.
0:33:41.7 AP: Fair enough.
0:33:43.2 SC: But I guess the question about these numbers like Graham's number and Tree [3] is we can define them, but do we really know what they are if we can't calculate them? What does it mean to know what a number is?
0:34:00.7 AP: What does it mean... Oh, right, yeah, I mean, what does it mean for it to be meaningful in our... I guess it seems to be what it is to be meaningful and what it is for us to know whether it exists or not. I guess this comes back to the idea of whether any number exists, in some sense.
0:34:11.1 SC: Well, there's that.
0:34:12.7 AP: Which is a whole new philosophical debate. And these ideas that numbers only exist, that they kind of exist outside the physical world and... Or they're only there to describe the things that are in it. There's no such thing in some sense as an emancipated number, there's only the number that can describe the number of cups of tea I've got, or the number of magic beans I have, or whatever, right? But there's all these different ways about thinking about whether a number exists, and I don't think there's any consensus within philosophers about that. And I think that probably also applies to big numbers as well. One of the things you can ask is... And I did a video about this quite recently, it's a bit of a controversial video. But it was about what's the biggest number anybody will ever think of?
0:35:02.2 AP: You can ask yourself a question, if I think... Sorry, I didn't say that right. You could say, what is the number... How big a number do I have to get to find a number that nobody will ever think of? So how big do I have to go? And the estimate I came up with is that if you think of a random assortment of a 73 digit number or so, random assortment of digits, then chances are nobody in the history of humanity will ever think of that number. And it's just some kind of silly trick with numerology a little bit, and trying to think about how many numbers that an individual person would think of. And you think about Benford's law and the distribution of numbers and that sort of thing. But I think it's true that certain numbers that are... Maybe not Graham's number, because we've thought about Graham's number, it's been conceived of. But if you think of the numbers that are just in and around Graham's number, that are maybe just... I don't know. They're just somewhere around it, it doesn't matter, anyway.
0:36:00.2 AP: No one's ever going to think of that number. No one's ever going to have anything ever to do do with that number. Not even any person, but probably no alien or anybody. The universe will definitely... Something terrible is going to happen to the universe before that number is ever conceived of, I think.
0:36:15.3 SC: And that's what, it's a little bit humbling to think about this. If you invent the rules of the game being that you have a finite alphabet of symbols to write things down with and a finite number of such symbols that you can put in your piece of paper, book or whatever, and these symbols are supposed to represent numbers, as you get out there to bigger and bigger numbers of smaller and smaller fraction of all the numbers are representable, even in principle. It's literally impossible to even, if you just stick to the integers, it's literally impossible to denote all of these different numbers, which is kind of wild.
0:36:53.0 AP: And I'd say the flip side, though, is, there is a flip side to this as well, is the unreasonable effectiveness of mathematics in our universe. You're talking about mathematics which seemingly isn't in our universe. And one of the things I've tried to do in the book is bring into our universe and you get all this extreme physics as a result, but at the same time, there's the question of we have our universe, and yet maths has this uncanny ability to describe it. And there's no reason why that should be true. Maths is a man made thing. Maths is, maybe that's something we could debate, but it feels it's a man-made creation. And yet it's doing this amazing job of describing the universe. I think that Vigna, the great physicist Vigna, has this lovely idea where he thinks about... So you think of distribution of bread amongst people and how this will contain the number Pi when you think... In the distribution of bread in a random community, it'll contain the number Pi.
0:37:57.0 AP: And at the same time, somebody can say, "Well, yeah, what's that number Pi?" Oh, yeah, that's something to do with the radius of a circle. That's the difference between the circumference or the radius of a circle. And you're like, "Well, what's that got to do with the distribution... A random distribution of bread?" It's not got anything to do with it. And yet the same number appears, and I think this is what's... And it's amazing that the universe is so mathematical and one can ask, would it always be? Is it guaranteed to always be mathematical or is it mathematical up to a point? Maths has done an amazing job of describing our universe, but is that something we can rely on forever? I don't know.
0:38:32.4 SC: And one of the fun things you do in the book is you... Just to connect our previous discussion of cosmic numbers that are pretty darn big to these wild math numbers that are just hilariously big, let's make that the technical term, hilariously big numbers. It turns out that even though there's only 10 to the 88th particles in the universe, you can still formulate sensible-sounding physics questions that require us to think about numbers much bigger than that. And the question that you dwell on a little bit is the doppelganger question. Why don't you set that up for the audience?
0:39:11.7 AP: Yeah, so one of the things I wanted to think about was... Well, I was really trying to do this thing where I tried to make a big number, like in this case a googolplex. Tried to think about it in a physical setting. And so we talked about how the universe reaches to 10 to the 26 meters to the cosmological horizon. Now, it's not that you get to that cosmological horizon and there's a big wall and you can't pass it. That's not how it goes. Who knows what's beyond the wall? I mean, there may be wildlings beyond the wall, who knows? But so the universe could in principle be much, much larger. It's entirely possible.
0:39:51.2 AP: And so I wanted to imagine, well, what if it was a googolplex across, in meters, say. It doesn't really matter whether you use meters, inches, furlongs, whatever, it's not going to make much difference. So you imagine a really big universe which is this big, a googolplex across, in, say, meters. And I thought what would be the consequences of such a universe? And the kind of remarkable thing is that you realize that doppelgangers would be kind of an inevitability in such a universe. And why is that? So very crudely, you can think about it as sort a human being. So let's take you, Sean, and let's think about the volume of space that you occupy.
0:40:28.1 AP: And you can think about the number of ways in which you can arrange sort of the... If we speak quite crudely about it, we can imagine atoms and arranging the atoms in that volume. But if we were more sophisticated about it we'd say the number of quantum states that describe that volume of space. How many are there? And we expect that this is finite. This is one of the things that we expect. And so if that's finite, there's only finite number of ways of arranging that volume of space, so one of them would correspond to you. One would correspond to a cow, one would correspond to a Donald Trump, I'm sorry to say. [laughter] There's all these different possibilities that you could imagine. Empty space would be another one.
0:41:11.3 AP: But there's finite number. So assuming the laws of physics don't change if you go across the universe, which there's no reason to believe that they do, then you can imagine sampling this, the volume of space next to you. And we say, right, do we have another Sean here? No, we don't. Okay, let's go to the next volume of space. Another Sean. No, we don't find another Sean there. And we keep going, and we traverse across this magnificently large universe which is a googolplex meters across. And then we find out because the number of possibilities that could have described that original Sean-sized volume of space is finite and less than a googolplex, in fact a lot less than a googolplex, you realize that if you go a googolplex across, in meters across in the universe, then eventually you have to start seeing repetitions. Now, you might say that someone like you, Sean, is quite rare.
0:42:06.4 SC: Very special, yeah.
0:42:07.2 AP: You're very special, Sean. You're very special. [laughter] Obviously you're quite an unusual state. Obviously, I would imagine empty space is by far the most common thing that you're going to find. So the probability of finding a Sean is very low. But actually, the difference between the number of possibilities and a googolplex is so large that your chances of overwhelming those probabilities become... It almost becomes implausible to think that you couldn't encounter the doppelganger.
0:42:33.4 SC: Yeah. And this is maybe even a little bit more specifically we look around our observable universe and in the observable universe conditions are pretty similar from place to place. And so if that just extends infinitely far out or even super duper far out, then there's only a finite number of things that can happen. Everything's going to happen over and over again. That's basically what it comes down to.
0:42:57.3 AP: I think that's basically it. Yeah. And so what I was doing was putting out a number on that, really. And so you estimate the typical distance to your doppelganger and it's... I think the number I have is 10 to the 10 to the 68 meters is roughly the distance you might estimate.
0:43:12.2 SC: Would it be weird for me to say that doesn't seem that far [laughter]? I would've guessed it was larger.
0:43:17.7 AP: I think it's... Yeah, I mean, it's pretty far, Sean, I think you couldn't... I don't think you're going to be going there on your holidays.
0:43:24.1 SC: But the implication is also that there's also every copy of me with every possible small variation, right?
0:43:34.1 AP: Yes. Yes. Exactly. So this is one of the things I talk about. One can ask the question, "Well, what is a copy of you?" Am I talking about something that's just something that looks like you? Or am I being much more precise than that? Can I talk about something that looks like you and has the same thoughts simultaneously, and all the atoms are arranged the same way, all the neurons are firing in exactly the same way in your brain. And I can really start to go right down to the quantum state and I can demand that that it has exactly the same quantum state.
0:44:02.4 AP: Now, of course, to actually show that I find... To actually measure your quantum state exactly. Well, that wouldn't be very good for you, Sean. That would be not good at all. That would destroy you. You're going to turn to plasma if I do that. 'Cause I'd have to measure the state of every single one of your atoms, and that's not going to be... That's not going to end well. But at least in principle you can ask the question. You have some particular quantum state which I don't know precisely and I would have to kill you to know it precisely, but it's presumably there and there are only a finite number of possibilities till I find another one.
0:44:41.5 SC: Well, I think... I mean, this is actually something that is worth getting into the weeds a little bit about, because the subtleties matter here. I mean, if it weren't for quantum mechanics, if it were classical mechanics, and I thought that I was made up of the set of electrons and protons and neutrons described by the laws of classical mechanics, then this kind of statement would be harder to make, because each particle is located somewhere in space. And for just one particle in principle I need an infinite precision to locate where it is. But quantum mechanics says something a little bit different, right?
0:45:21.9 AP: I think it allows you to just discretize in some sense. It's kind of... Well, what quantum mechanics gives you, it gives you a fundamental unit which is measured in units of Planck's constant, of course, in some sense. And I think that's what quantum mechanics does give you, that you can really break up phase [0:45:36.3] ____ space into discrete building blocks in a way that you can't do with classical physics yet, I think.
0:45:43.7 SC: And just to be cosmologically correct, we don't actually know the universe is as big as 10 to the 10 to the 68th meters across. It may be. It might be, we don't know.
0:45:55.1 AP: No, we don't, but you can ask by what process might it get so big, and I guess one way it would do that would be this idea of eternal inflation, for example, in which you can have a universe... So we have this... One of the things we think about our own universe is that in very early times it grew very big very quickly. Well, there's no dispute that our universe has got quite big, right?
0:46:16.1 SC: Yeah.
0:46:16.5 AP: And so how did it do that? And there's this process called inflation where there's this field which pushes the universe apart very fast, and it's known as the inflaton field. Now, one of the questions that we have is how did that process start? And that's a bit of a puzzle. So one of the solutions to it is that you have this inflaton field and what it does is it jumps about quantum mechanically from value to value, and then at some point it hits this sweet spot where it has just the right value that it causes the universe to go crazy and it pushes this around... Pushes the universe apart very quickly.
0:46:51.4 AP: So if that's the set up that you have, then there's absolutely no reason that in distant parts of the universe this inflaton field isn't doing its little quantum jumps and hopping around and then occasionally hitting a sweet spot and making that little pocket in the universe it found itself in suddenly gargantuan, and this can keep happening. You just get this inflaton doing its little dance and then creating gargantuan universes every so often. And if that keeps going and keeps going and keeps going, it's almost like a recursion within the universe. Because you creates a big universe and then you have jumps within that universe which create new big universes and then jumps within those universes to create those new big universes, and this is kind of the universe is doing its own recursions in a sense.
0:47:33.0 AP: And so that's how it can get to quite, very big numbers. So that would be the scenario on which you can imagine a universe this large, I think, but yeah, we don't know that the universe is that big, but we don't know that it isn't either.
0:47:45.2 SC: Then just because there's lots of different variations on this theme, and I want to be clear about what the possibilities are. We don't, to make these statements about my or your doppelgangers appearing very far away, none of this involves the multiverse as we usually think about it or the string theory landscape or anything like that. It's just saying the universe is big, and you just sketched a reason why that might be true, but even without that reason it could just be the universe is big.
0:48:12.5 AP: I think it's two things. I think it's that the universe is big and it's the thing you alluded to earlier, it's quantum mechanics at the end of the day, I think. And that then makes those counting... How we count stuff finite in some sense, it discretizes the world in which we live.
0:48:25.4 SC: But there is the third thing, which is actually one of my favorite things to think about, so I'll highlight it, which is the existence of gravity and black holes. Because if it weren't for gravity, then even in a finite region of space we could do an infinite number of different things. In quantum field theory we can do an infinite number of different things. But gravity comes along and changes the game, and I think this is going to matter for your calculations.
0:48:49.3 AP: Yeah, so certainly gravity provides this cut-off on a number of things that you can put in any finite space, because eventually it comes back to if I want to... One of the things you can do is if you think about a volume of space, you can count how many different bits of information can I hide in there. How many different possibilities are there. And then again, we talked again about how information weighs, and you realize that if you put too much information and you allow for too many sort of hidden possibilities, then the only option is to form a black hole, and black holes are the best hider of all the hidden possibilities that there is.
0:49:22.6 AP: And gravity just kicks in and makes sure that that happens, and there's nothing you can do to avoid it. And that's the remarkable truth. And indeed, if you don't have gravity, is the gravitational coupling goes to zero, these limits that I'm putting on how much you can fit into one volume of space, they go infinite. It is absolutely true that the limit on the... That gravity plays a hugely important role in this.
0:49:54.3 SC: And gravity also pushes us in the direction that you really put a lot of emphasis on in the book, which is the holographic idea. The idea that it's the way that we get these countings of how many things can happen in a region is not by separately adding up what can happen at each location. There is some global constraints there that make the world in some sense have less possibilities than you might have guessed.
0:50:21.9 AP: It's really amazing when you think about it. So they're very naive. Even with gravity you might naively say "Huh, how do I count the number of ways I can build a Sean-sized volume of space? How many different possibilities are there?" And you'd say, "Well, maybe I break up that space into the smallest volumes that I can imagine." Which are in the case of gravitational physics they are Planck volumes, so 10 to the minus 35 meters across. And I could say, "Well, maybe each of those different volumes has a certain number of possibilities it can... States it can be in." And then I just add them all up. But if you did that you would get a massive over-counting, and that's because what gravity actually does is it forbids loads of possibilities, loads of possible combinations.
0:51:07.7 AP: It just does. And actually, that would be the wrong calculation. It doesn't store its information in the interior of the space, it seems, in the way that I've just described there, where you break the space up into lots of tiny little bits. It seems to store it on the edge of the space. That seems to be where the counting is done. And so when you look at a Sean-sized volume of space you should really... You want to make out how many possibilities there are. You need to break up the surface that surround you into their tiny building blocks and calculate how many possibilities that are there.
0:51:37.8 AP: But this is the... I think the holographic principle is... It started to develop around the time I was doing my PhD, and I think it's mind-blowing and fascinating and so profound in so many ways... Yes, indeed. Indeed. It's using the holographic ideas that give you the better estimates for how many different possibilities there are.
0:51:58.4 SC: I think that lurking in the back of my mind and your mind here is the idea of entropy, but maybe we haven't connected that explicitly, if you want to do that.
0:52:06.8 AP: Yes. Yes. So yes, I am talking about entropy. I was saying information, but really in some sense what I really mean here is entropy. Well, entropy and information are kind of one and the same thing. So yeah, so when I say that black holes are the best storers of information, what I'm really saying there is that they're the best storers of entropy. So what is entropy? Entropy is in some sense the number of different ways you can have a... The number of different ways you can get the same macroscopic object... Macroscopic observables from different possibilities. So I take an egg. So I take a typical egg and I look at an egg. It looks like an egg, and it's got a given temperature. It's got a given size. It's got a given pressure, it's the egg that I see. Now, what I don't know is where all its precise atoms and molecules are located.
0:53:01.4 AP: It's not information I need. It's not information I need to know. Temperature, it's not information I need to know. It's an egg. It's fine. So all those little bits of information, which are hidden in some sense, they all contribute to the entropy, 'cause different possibilities could give the same egg, and the number of possibilities you count, in other words, the number of bits of hidden information, are what give you the entropy. And when we talk about black holes as being the best storers of information, what we're really saying is they're the objects which have the most entropy for their size. And that's the thing that's really... We're really using here.
0:53:39.2 SC: And it goes as the area of the boundary, and that was the beginning of holography.
0:53:45.7 AP: Absolutely, yes. Absolutely. So one of the most remarkable things about black holes, perhaps the most surprising thing from back in the day, is that when you calculate the entropy of the black hole, indeed it goes like the surface area of the event horizon. So that's how it grows. If I want to double the amount of information stored in a black hole, if I want to double its entropy, I don't double the volume of the black hole, I have to double its surface area. And that's quite different to how you normally think of how you might grow information. If I were to take a dinosaur, which is not a very gravitational object.
[laughter]
0:54:25.3 AP: Okay. If I take a dinosaur. If I want to double how much information's stored inside there, I would think about doubling the volume. That's probably what you would do. But a dinosaur is not a very gravitational object. It's not a deep [0:54:39.9] ____, a black hole is. And so with black holes the game is different. It behaves differently. And indeed, you have to double the surface area to double the amount of information it can contain. And that insight is what really led to this idea of holography. That actually in some sense the gravity... When gravity is... Sometimes when you think about the gravity in, say, three dimensions of space, it's in some sense connected to a theory in two dimensions, which maybe lives on the boundary of that space.
0:55:14.6 SC: But we... I like how vague you were in that enunciation right there, because we don't actually know how to do that. We don't know how to describe the real world in terms of a complete theory one dimension lower. But that might be something we're aspiring to.
0:55:30.4 AP: I think so. So what we do have is we have... So maybe we should describe what a holographic principle really is, right?
0:55:34.6 SC: Yeah.
0:55:35.6 AP: So these ideas about black holes and how they store information. They've led to this conjecture that on the one hand you can have a theory of gravity and let's say, for example, in three dimensions of space. But you don't have to restrict yourself to three dimensions of space, but just to be concrete. Let's say we have a theory of gravity in three dimensions of space. And that's one description of the physics. But what the holographic conjecture would say is that it's an entirely equivalent description of the same physics which has no gravity, zero gravitational sort of... You're not doing Einstein's equations or anything like that. You're just describing a theory without gravity but in one dimension less.
0:56:23.7 AP: And... But you describe exactly the same physics. Exactly the same physical phenomena can be described in both languages. So in some sense it's like when I just talk about a plate of meatballs. I talk about a plate of meatballs, I say, if I'm English, I'll say "It's a plate of meatballs." If I'm Spanish I'll say it's albóndigas. But still, we're both describing the same thing, just using different language. So I think this is what the holographic principle really is. It's just saying you've got physical phenomena which you're going to use mathematics to describe. Well, which mathematics are you going to use?
0:57:01.3 AP: Okay, I can use gravity with three dimensions of space or I can use... Just not bother with gravity and I just use this quantum field of theory in one dimension less. And I can find a way to describe the physics using either language and I can get the same results and as long as I can have a dictionary that tells me how to go between the two, then I can do this. Now, you're absolutely right, Sean, in saying that does this apply to our universe. Well, we don't know. What we do know is that there are examples of sort of, if you like, toy universes, where this conjecture, the evidence for it is absolutely compelling.
0:57:32.2 AP: So in particular, this goes back to [0:57:35.8] ____. There are examples of universes that are kind of five-dimensional, they're warped through de Sitter spaces. And there you can think about gravity or maybe more precisely string theory in these universes. And you can describe physical phenomena that might occur in those settings, and then you can also show that there's an equivalent theory which describes the same physical phenomenon, which is given by what's called N equals 4 Super Yang-Mills that lives in one dimension less on the boundary of that space.
0:58:08.3 AP: And so we have these examples of where we can really show that you don't always need gravity. You can always just do away with it and live on one dimension less. And so the conjecture is, is that true of our universe. And it seems to... This idea that gravity is, somehow wants to push you towards one less dimension, I think it's really something deep that may well be true of quantum gravity that, you know, when we really understand it better in the long run.
0:58:36.3 SC: Right. But having said all that, and the very nice version of the sales pitch for the holographic principle, like you said that's not the universe that we live in, the one where Maldacena, in anti-de Sitter space, as we say, imagined a world with a negative cosmological constant, negative energy density in empty space. But our world, as you know as well as anyone, has a positive energy density in empty space, a positive cosmological constant. And that's both a fact that we need to face up to and also a little bit of a puzzle or maybe a clue, depending on how you look at it.
0:59:12.3 AP: Yeah, I mean, people have tried to think about these holographic ideas even in such a setting. I absolutely agree. The ideas are not as robust and as clean as they are in anti-de Sitter space. And anti-de Sitter space seems to lend itself very naturally to a holographic prescription, in a way that de sitter doesn't. But work is ongoing, I think it's fair to say. And then people... I mean, there's... I don't want to go into too much technical detail, but I know Eva Silverstein, for example, a few years back was looking at these, you know, sort of lower dimensional examples, and trying to push them from AdS into de Sitter with a certain deformation of the theory and coming up with some quite... Quite, this is this TT bar stuff that they were playing with. So there's some nice stuff there that's being done. But I agree, it's absolutely... It's not something I'm in any way expert on. And it's... But I know there's people thinking about it.
1:00:05.9 SC: But I mean, maybe we can talk about what you are expert on, which is in the fact that our energy density in our universe does seem like a small positive number. And this is confusing to us, and we would like to try to figure this out. It gets into the small numbers rather than the large numbers that we were talking about before.
1:00:23.6 AP: Yeah. So what I talk about in the book is that small numbers... So there's a whole section on small numbers. And the idea there is that small numbers kind of portray the unexpected. They tell you... They signify a puzzle. When a small number appears it tells you that something's probably not making sense. And the big... Big, small, whatever way you want to look at it.
[laughter]
1:00:42.3 AP: The big example of this is the cosmological constant. So one of the things we see our universe, right? So we see that it's very, very large scales. It's accelerating. The universe seems to be pushing galaxies apart at an ever-increasing rate. And one thing you can ask is, well, what can cause this acceleration? I mean, gravity you normally think of, it's an attractive force, but so why would it cause the universe to accelerate? It would surely slow down the expansion of the universe. But actually, something can push the universe apart at an ever-increasing rate, and that's the energy of empty space itself, the vacuum energy.
1:01:26.1 AP: And you might say, well, how does the universe... How does the vacuum have an energy? It's just empty space. How could it possibly have energy? Well, this is where quantum mechanics comes in, and quantum mechanics tells you that that, actually, the vacuum's a really exciting place. [chuckle] It's not as boring as you think. You could... You know, it really is this bubbling broth of virtual particles that the... Not real particles, but virtual particles that sort of pop in and out of existence. And they can essentially endow the vacuum with an energy.
1:01:53.7 AP: Now, the problem arises when you try to calculate how much energy they give the universe, and then you compare that to how much energy you think there needs to be to get the acceleration we see. And what you find is, is that the observed vacuum energy is a factor of 10 to the minus 120 of the theoretical expectation, the amount that you calculate. So it's a huge mismatch between what your calculation's saying you should get and what we actually see. The value that we see is far, far smaller, more than a googol times smaller, than what our theoretical estimates are. And the truth is, if the universe really did have the huge vacuum energy that our theoretical calculations based on quantum field theory tell us it should have, well, then it would've been crushed out of existence within a moment of creation. It would've been...
[chuckle]
1:02:48.5 AP: The amount of vacuum energy would've just torn and twisted the universe into oblivion.
1:02:53.2 SC: Do you know about Shannon's number? Have you ever heard of that one?
1:02:57.3 AP: That's not one I know about. Go on, Shannon's number.
1:03:00.3 SC: Yeah. Well, I actually discovered it while sort of puttering around doing research for this podcast. Claude Shannon of the Shannon information theory fame, asked, you know, how many different games of chess there would be, possibly?
1:03:12.1 AP: Okay.
1:03:13.0 SC: Plausibly. And I forget whether it was like total games or total strategies or something like that. But he was not able to calculate it, of course. It's a very hard number to calculate, but he gave a lower limit, and it turns out to be 10 to the 120.
1:03:26.3 AP: Okay. Wow.
1:03:28.1 SC: Which is coincidentally the cosmological constant in Planck units. So I think that can't be a coincidence, right. Maybe chess is revealing the secrets of the universe somehow.
1:03:36.7 AP: Yeah. Now, I'm starting to pitch the universe as a chessboard and then to think [laughter] what does all that mean, right? Yeah. That's cool. Each one's a little Planck... Each little board's a little Planck size across or something like that, right? So I don't know.
1:03:48.8 SC: So, anyway...
1:03:49.9 AP: No, I mean, it's a puzzle, right?
1:03:52.9 SC: Yeah, and that is probably... Has nothing to do with chess, just so everyone knows that we're just kidding around here. But it's... So the puzzle is that we have a guess as to how big the energy of the vacuum could be, then we go observe it. It's much, much, much smaller. Do you personally have a favorite explanation for that?
1:04:13.6 AP: Yeah, I mean, so it's something that I've thought about a lot over the years. I mean, of course, I'm not going to say it's my favorite explanation, but probably the standard explanation is the anthropic on, which says that the universe is... If the vacuum energy hadn't been tiny, then, you know, we never would've had galaxies forming. Now, the universe would've expanded too quickly early on for galaxies to form or it would've crunched into oblivion early on before galaxies form. And if galaxies don't form and planets don't form, then complex life doesn't evolve to sort of ask questions and record podcasts, asking about how big the vacuum energy of the universe actually is. So that's the standard anthropic story.
1:04:53.7 AP: Whilst that's fine, I think it still behooves us to think about other ideas to try to understand why the cosmological constant is so small. I think for me, and I know it's something you've worked on as well, Sean, is this idea that there's something very special about the cosmological constant which makes a difference to every other sort of source of energy and momentum in the universe. And that's, that it's constant.
[laughter]
1:05:25.4 AP: It does what it says on the tin. Everything outside that, sort of you think about planet or a human being, it's a localized source of energy and momentum. It drops off, maybe both in time and space, potentially. And that's not true of vacuum energy. At least it's a constant, right? It's sort of... At least a baseline guy. It's constant. And so that's what sets it apart, I think. And so when we think about what we might do to address this strange question about why we don't see this huge vacuum energy, maybe we need to look at gravity on the scale that it's reacting to these constant sources, okay. So we need to look... So if we think about this, what this means is, is that we need to look at gravity, not on the local scale, but on a global scale, where we're really most sensitive to that constant source.
1:06:18.6 AP: And so one of the things that I've worked on, and I know you have too, is to try to think about global modifications of Einstein's theory. So what that means is, is that you're only changing the theory on the scale of the whole universe. So it's only the... Sorry, it's the longest wavelength modes of the theory that, when the theory reaches out to the very, very far distances that you start to see changes. The really, literally the furthest distances you can imagine. Any local physics, like what happens in the solar system, well, that's got nothing to do with vacuum energy anyway. That's just the physics of the solar system. That obeys the rules of... The idea would be that obeys the rules of general relativity and does all the wonderful things general relativity does.
1:06:58.5 AP: But if you... But what you could think about is modifying the theory on the global scale. So really just for... So it reacts to constant sources slightly differently. And then what you can say is, well, it's fine. Let's say the vacuum energy is large, it's large, it just is large. But my theory of gravity reacts to constant sources slightly differently to how it does in general relativity. And so I can somehow screen away this large, large vacuum energy and just get the amount of acceleration that I see. Now, it's a tricky game. You've got to come up with a consistent theory that does this, that's consistent with quantum mechanics and consistent with relativity and all of that. But that's something that we've been trying to develop in recent years through this sequestering proposal.
1:07:41.5 SC: Yeah. And I think this, I mean, it's a great little insight for the people out there who don't do science for a living, about how theoretical physics gets done. You know, you have some puzzle you're trying to understand, then, so you in words or pictures conjecture an idea, but then the hard work is turning that into equations, right? And I guess just so people know, correct me if I'm wrong here, about how you think about this, ordinary general relativity or other theories of gravity we would write down, what they care about is the amount of energy at each point, and that's what matters. But you're imagining changing it so that what matters is not just how much energy there is at each point, but how it's distributed, like if it's constant versus if it's lumpy. And then that's a kind of a radical imagination of how to change gravity.
1:08:28.5 AP: Yeah, yeah, kinda. In a way, that's one way of thinking about it. I think what I'm saying is, is that in some sense you've got like a filter. So you've got this sources of energy and momentum, and it's a filter, but how that filter works is sensitive to the properties of the source. So if the source is a planet, the filter doesn't filter it at all, it just lets it through.
1:08:47.1 SC: It's a high-pass filter.
1:08:47.9 AP: Okay. Yeah, exactly. Like a high-pass filter. Exactly. But if the source is constant, the filter shuts it down. That's the idea. So, there's something in the theory which sort of, is acting like this cut-off. Like this filter that's saying, well, you may pass, you may not pass. But doing that in a consistent way that's consistent with the sort of underlying principles of general relativity and it's consistent with quantum mechanic is not easy. You know, one of the things that I found about... A career thinking about gravity and trying to mess around with Einstein's theory, is that you mess with Einstein's theory at your peril.
1:09:22.1 SC: You do. [laughter]
1:09:22.4 AP: It's really difficult to mess with Einstein's theory and not unleash all manner of beasts and horrible creatures and cause all sorts of problems. Instabilities, new particles appearing that we don't see in nature, there's all sorts of problems that can arise. But one of the things you can do by focusing on those changes only on the global scale, so only on spacetime as a whole, then you can actually sort of evade a lot of these problems.
1:09:49.3 SC: Well, you know, since we've reached the hour-long moment at the podcast, we're allowed to let our hair down a little bit and get a little even crazier than we've been getting. So I wanted to take it back to this finite versus infinite question, because secretly there's a relationship to the cosmological constant. You talked about holography and the information in a black hole, and we talked about there's a cosmological constant that is accelerating the universe apart. But that's why implicitly you said at the beginning of the podcast that there's reason to believe the universe has a finite set of information in it, right? Because there's a horizon around us with a finite entropy. And maybe you could explain what I just said in more easily digestible terms.
1:10:35.5 AP: Yeah. So, of course, we have this, if we... So let's say somehow some... There's some theory somewhere which solves the puzzle of why the vacuum energy isn't large. So it's just some small number. So the vacuum, the energy density of the universe is very low, but it's positive and it's causing the universe to accelerate very gently. Now, what that means is, is that we live in what's called a de Sitter universe, and we're surrounded by this, this horizon, this cosmological horizon, this de Sitter horizon in that case, it would be. And when you start trying to think about how many different ways there are to arrange sort of that universe, the quantum states of that universe, you think about the entropy of that universe. It's again, this weird property of gravity that always projects everything onto the surface area that surrounds it.
1:11:26.6 SC: Right.
1:11:26.8 AP: And in this case, it's this de Sitter horizon, and that de Sitter horizon that surround... This shroud that surrounds our universe, that surrounds each and every one of us individually, has a finite area, if the vacuum energy of the universe is small and positive. And yeah, so that's where I came back to these ideas about the universe having a Poincare recurrence. Such a universe would have a Poincare recurrence, because you would think of it as a finite system.
1:11:58.7 SC: And I think one of the things... This is not even a question, I'm just going to set that out there, 'cause I think people are not thinking about it enough, is that a universe that has a finite number of states, a finite number of degrees of freedom, and it's in this theoretical description, that's a universe that is not described by quantum field theory, right, which is what we always use to describe things. Quantum field theory has an infinite number of things going on, and I don't think we've adapted... I don't think we've digested that shift quite as much as we should have.
1:12:30.6 AP: I think that's because you have to bring an interplay between... You have to bring gravity into the game, and of course, then we're talking about quantum gravity, and we don't really know what quantum gravity is yet...
1:12:39.3 SC: Fair enough.
1:12:39.5 AP: I think that's the fundamental obstacle. Of course, as you know, as you remove gravity, as you decouple gravity, all these entropies that we're counting... So if I count what the entropy of de Sitter space would be, as I decouple gravity, it goes infinite. If I calculate the entropy of a black hole as I decouple gravity, it goes... All these numbers go infinite, because basically the entropy is the area divided by Newton's constant. And as Newton's constant goes to zero, all these entropies diverge.
1:13:07.3 SC: So I'll give you a chance to... I'll ask a very simple question to end the podcast, but you can decide how... At what length you want to answer it. Is the universe, is reality infinite or finite?
1:13:23.0 AP: I think it's finite. Okay.
1:13:26.3 SC: Wow. Alright.
1:13:28.4 AP: I just... And it's complete prejudice. I have no experiential evidence for knowing this. I just... Part of me feels that there's no room for infinity in nature. I don't know. That's a complete... There's probably loads of examples... Everyone's going to get really upset with me saying this now. But in truth, I just feel like the idea of a universe which is compact and spatially compact, it seems much more elegant to me than having to meddle with the infinite world. That's not to say that the infinite world isn't mathematically in itself wonderfully interesting and something that you can think about and... But do we really need Cantor's sort of infinite heavens to describe our universe? I don't know, maybe we do.
1:14:17.9 SC: Yeah.
1:14:20.2 AP: But my pure prejudice is that we probably don't.
1:14:22.3 SC: I asked, 'cause I myself don't really know. I don't even have a prejudice there. On the one hand, I feel the worries about infinity... The mathematics of infinity gets tricky, and maybe it's just because there isn't any real infinity in the physical description of reality. But on the other hand, infinity is a nice number [chuckle] in a way that 10 to the 120 isn't, right? If the universe is finite, then there's some specific size that it has, and who ordered that? I'm not really sure.
1:14:55.5 AP: Well, no, I think... But then you come straight back at you, with which infinity is it, Sean? That's the next question you have to ask, right?
1:15:03.1 SC: I know. It is.
1:15:04.9 AP: Is it aleph 0? Is it aleph 1? Is it... Whoever it is... The story doesn't end with infinity. You've still got those same questions. So if we keep ourselves in the finite realm, then things feel a little bit... I just feel a bit cosier.
1:15:16.5 SC: Alright. Let's feel cosy. I think that's a good way to wrap up, feeling like we're cosy in a finite little universe that we live in. So Tony Padilla, thanks so much for being on the Mindscape Podcast.
1:15:25.7 AP: Thanks, Sean.
[music]
Only one number: 1, or +1.
All other numbers can be explained between 0 and 1, moved abut in economies of scale. Defining large numbers is a good proof against the reality of numbers, proof it is a language.
Fascinating topic and discussion.
It seems reasonable to assume ‘The Universe’ has always existed. Not our so called ‘observable universe’ which supposedly came into existence about 13.8 billion years ago according to the ‘Big Bang Theory’. By definition ‘The Universe’ is everything that ever existed, including space and time. It seems almost beyond comprehension (at least to me) that nothing, not even space nor time, existed before the Big Bang. Assuming, for the sake of argument, that ‘The Universe’ has always existed another important question to consider is, how big is ‘The Universe’? Is it finite or infinite in extent? If it is truly infinite extent, then then there is NO number which can be used to define how large it is. On the other hand, if ‘The Universe’ is finite in extent, then there MUST be some finite number which can be used to define its surface area in square meters, and another finite number could be used to define its volume in cubic meters.
If I had to guess, I would guess that ‘The Universe’ is infinite in extent, mainly because of the difficulty in coming up with a finite number to define either the surface area or the volume of ‘The Universe’.
This is so fascinating to think about! Great episode! To take things further, suppose numbers do meaningfully exist “out there,” outside of the situation where a number has to be a number of something. In that case, there would be *many* more unthinkably large numbers in existence than numbers that can be comprehended. TREE [3] would be closer to a random representative of the thing called a number than ordinary 3. If you start with the basic premise that people can use numbers to talk about reality & stop there for the moment, I feel like it should then be surprising and strange that we’re situated so solidly and definitively in the shallow end of the pool.
I want to say that the reason our daily lives contain all of these small numbers rather than more ‘typical’ representatives seems to be tied up with the difficulty of having a thing that’s like another thing. If I have a rock and a stick, I have one rock and one stick because the rock is not like the stick. If I have two rocks, I’m only allowed to say I have two because I was somehow able to find a second thing that’s enough like a rock to count as another rock. The thing that’s rare and difficult and limiting isn’t the incomprehensibly huge number that could be said to exist in theory, the thing that’s rare and difficult and limiting is the countable entity. There must be something about existing, even as a tiny particle, that’s so expensive and rare that our reality made up of things that exist is only playing with a corner of the tablecloth of what numbers-as-numbers might theoretically have to offer.
Probably the most puzzling question in all of physics is why there are two sets of laws, one set for the microscopic world of elementary particles and atoms, where the probabilistic theories of quantum mechanics reign supreme and another set for the macroscopic world of stars, planets, and people, where the deterministic theories of special and general relativity reign supreme.
The basic entities of the microscope world seem to be made up of discrete size chunks, while the basic entities of the macroscopic world seem to be continuous. Most attempts at unification involve making changes to the theory of relativity so that fundamental entities like space, time (or spacetime) and even gravity are no longer considered continuous, but also come in discrete chunks, just like the fundamental entities of quantum mechanics.
Supposedly if this unification is successful, it would imply that at the deepest level all the laws of nature are probabilistic, like Heisenberg and Bohr suggested and Einstein, who like most classical physicists believed the laws of nature are deterministic, was wrong.
As of yet all attempts at unifying relativity and quantum mechanics have been unsuccessful, so it can’t be said for sure who was right and who was wrong. In the long run it may take an entirely new theory, one not involving quantum mechanics nor relativity, to explain why nature acts differently at the microscopic level then it does at the macroscopic level and to know if the laws of nature are really deterministic or probabilistic at all levels- or the question may never be resolved!
I must say, the book is far more engaging than the talk, and the talk was engaging. A hit! A great Sunday morning respite and read.