From the Vault: Fantastic Numbers, with Antonio Padilla

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Stuff to Blow Your Mind. My name is Robert.

If we could back up a whole lot, I guess and just ponder the basics here, what exactly is a number?

five that describes the loaves of bread. And then you really sort of when you sort of make that, you disconnect the two and you start to build the idea of like what I call an emancipated number and number that's independent of the thing that it's describing. Then you're really sort of making quite quite an intellectual leap. So that for me is what is what a number is? It's kind of emancipated from the thing that it's describing. Whether such a thing really exists in a philosophical sense

is a whole new debate that you can have. But yeah, that for me is the key mathematical lead that I think was made, you know, a long time ago. And yeah, it's really important.

of space time as like an abstract concept. It's not something that can affect the things in space time. It can't affect the material objects that we have around us. You also have the nominalists, who says basically that numbers and maths only exist to of understand stuff. So in some sense, we talked about the five, you know, five jars of oil, the five five loaves of bread. That's the only reason that the number five exists to describe the jars of oil, to describe the loaves of bread.

And then of course you've got the third sort of you know school, which is perhaps in some sense the most extreme, which just says the numers don't exist at all. They're just a useful tool that we used to describe the universe surrounds us. And I guess the analogy people use here is it's like saying, well, you could be an atheist, but you could still believe with some of the sort of moral messages that you read in the

Bible or the Koran. It doesn't mean that you know, you can't be inspired by them, but you just don't have to believe in every element of it. I guess as a physicist, for me, it's kind of hard to sort of go with that fictionist idea. And yet see universe that is so amazingly described by mathematics. Now, is that something that's embedded in the universe or not? I guess it's really difficult to know. We certainly not seen any evidence that it is.

me that does that. So when you know you can have these wonderful mathematical concepts ideas like Graham's number three three, these truly bizarre and wonderful numbers, to have a wonderful place in mathematics, but then you really bring them to life when you try to sort of squeeze them into our physical world. So that, for me, is what makes

a number fantastic. It's almost like what makes a number fantastic is the fantastic physics that it can lead you towards and lead you to imagine and whatever that might be.

most of us. But in principle you might say that that that that would be the limit. Of course, if you then go beyond that and start to say, well, what if I could somehow get my head to find a way to actually store information store concepts more efficiently than just the usual idea of neurons firing on it off. Let's suppose that it could do that somehow. Then they numbers get get much bigger, and you start to the things that limit it are literally preventing your head collapsing

to form a black hole. Because black holes what they do is that they're they're the best thing at storing information. So if you want to get something the size of a head of a human head, and you want to say, what's the best thing the size for human head that can store information, it's a black hole the size for human head. That's nothing can do it better. And so so that that places a new limit, and you can ask, well, again,

what is that limit would be? Well, he's certainly way below grains now, but you're not going to get anywhere near the magnificence of Graham's number, you could probably get a digit that that's about ten to the seventy, but it's about ten to the seventy digits long, so less than a Google digits long. Having said that, you could imagine a number like a Google plex, a google plex

has a Google digits. Now I've just said that you can't imagine at Google digits, not possible, but a google plex you could because what you know about a google plex is that it's a one followed by a Google zero. So you know that all the numbers that come later on as zeros, So there's not much information in that, so it doesn't cost as much as many bits. You don't have to put as many bits in your head

to imagine that. So what we're really talking about now are really a random assortment of digits, a completely random assortment digits the kind that would appear in Graham's number. And I don't think you can get passed around ten to the seventy, which is a one with with seventy zeros. You couldn't get past that many digits completely randomly sort of allocated. At that point, your head's going to collapse. Into a black hole.

like a one followed by one hundred zeros. And he said, well, okay, that's really small compared to infinity, right, even though something really big is actually really small compared to infinity. So he came up with this one with one hundred zeros. He wanted a name for this number, so at the time he asked his nephew who was nine years old at the time. He was called Milton Soota. He said, can you qu up with a name for this? And Milton said, well, at Google, which is an absolute stroke

of genius, right, it's such a great name. And then so they wanted to then develop things further. So then they wanted an even bigger number, again building on this idea that it's nothing compared to infinity. And so he said, well, okay, I'm going to quote with the idea of a google Plex. It's going to be an even bigger number. Well how big? So Kasner then goes to Milton. He says, well, how big should it be? And Milton's like, well, it should be a one, not followed by one hundred zeros, but

zeros until you get tired. But Kasner is like, you know, a sort of a you know, esteemed academic at Columbia and all that. That's just not precise enough for him. So he went with a which a much more sort of well defined idea, which is a google plex should be a one followed by a Google zeros. So a Google's already massive, that's a one followed by one hundred zeros. A google plex is a one followed by a Google zero. So it's a whole new level of big compared to what we're normally used to.

And then a Google quadruplex is a one followed by a Google triplex zeros. And you can see each time you're growing the number just by so much, by such an unimaginably large and that's what You're not just adding a zero every time, you're kind of really ballooning the number of zeros on the end of this number in Gargangian proportions, and that's what and it's this power of mathematical recursion that allows you to do that.

managed to slow down time. And when he was he was running in the World Championships and I think Berlin and he set the world record. And this is due to the effects of relativity, so that when when somebody actually moves quickly, they actually slowed. Time actually slows down for them. And this is the amount by which Usain Bolt was actually able to slow down time due to the effects of Einstein's theory. And it's compared to the people in the stadium, for example. This is this was

the difference that he experienced. So it's one of the one of the weird consequences of it is that you can actually it's not that you say Bolt, actually even though he slowed down time, it's not that he that he actually ran the race any quicker. He still runs the race at roughly ten meters per second. It's actually an even more strange consequence. He actually the track also shrinks for him a little bit, so so he actually

runs it in less time, but in the same speed. Therefore, the track shrinks because relative to him, the track's moving. And this is another effect of relativity, one of the remarkable things. And and yes, you could perhaps argue that he didn't actually finish the race because the tracks rank so he didn't run quite one hundred meters.

It's not a lot, but it's still pretty amazing when you think about it.

but you also have a lot of helpful illustrations. So I was curious, since you are a regular communicator of this topic, is it is it more challenging or in some cases almost too challenging to describe some of these numbers without the visual aids or the actual numerals to like visually present somebody with.

symbolism to sort of actually even describe the number. So you've got to introduce that. There's just no getting away from it. But I guess what you can do is describe the physics associated with it, and that you can certainly do, you know, just just just with words. And you know, in the case of a number like Graham's number, you can talk about how you just can't picture it in your head because your head will will collapse to

form black hole. And that's already going to make people think, wow, that numbers there's something big of something big and crazy about that number or a google plex, you know, and you can talk about a universe that's that's a google plex meters across, and then you can ask, well, if the universe is that big, if the universe is ritually that large, then it's likely that you would find multiple copies of yourself, like literally exact doppelgangers elsewhere in this ginormous universe.

not exist and it's really remarkable. So what we're saying is that there are two ways in which you can describe the physics that we see around us. On the one hand, we can imagine three dimensional world with a gravitational force and the force of gravity doing its thing with planets around the Sun and so on and so forth. On the other hand, there's a completely equivalent description of the same phenomena which just uses two dimensions and no gravity.

So think of it a bit like, you know, on the one hand, somebody's you know, in English, we say if we see a plate of meatballs, we call them meatballs, but a Spaniard might call them album the gas. They're both describing the same things, they're just using a different language.

And that's kind of what the holographic truth says. It says that you can have a theory like a three dimensional world with gravity, and you can use that to describe all the physical phenomena you see, or use this different language which has no gravity and only requires two dimensions of space. So is it true of our world? We don't know. It's a conjecture. It's a conjecture that has sort of evidence coming from the physics of black holes.

There are actually concrete examples that we know of of sort of toy universes, so not our universe, but but space times that maybe the higher dimensional they may be warped in weird and wonderful ways. And you can think about gravity in these in these simple toy universes, and you can show that there's an equivalent description in one dimension less like a holographic description, and it's called a hologram because that's essentially what holograms do. Right, If you

think of a HOLOGRAMD, what have you got? You've got an image on a that's stored on a holographic plate. You know, it's just some light and dark bands on a holographic plate, a two dimensional plate. It stores a bunch of information that way, but that's just one way of looking at the information. You can decode it in a different way by shining monochromatic light through it and creating a three D image. You're not creating any new information.

It's the same information, just stored either in two dimensions or three. And it seems to be that that seems to be a fundamental property of gravitation, of gravitational worlds that you can think of them as as like, as I said, three, the world with gravity, or you just forget about gravity and consider a world with one dimension less and you can describe exactly the same physical phenomena.

is something that's just just beyond our finite realm. That that that's you know, if you keep on counting forever, you know, it's kind of the at the end of that, or sort of almost beyond the end of that. Now, that's in some sense thinking of infinity as not a number, as a limit of say, you know, the whole numbers. But what cant or George cantor the you know, the great German mathematician from the late Victorian times, what he did was actually taught us how to count beyond infinity.

So literally, using really smart ideas associated with something called set theory, he was able to show that actually you can have all the sort of finite numbers, and beyond that you can have infinity. But that's just one layer of infinity. You can have the infinity, which is all the whole numbers, but you can also have a different layer of infinity, which is all the numbers between zero and one. So think of the continuum of the numbers

between zero and one. That's you think there's an infinite number of numbers between zero and one, but that's actually a different infinity to all the whole numbers. So you've got you know, these discrete infinities continuum infinities, and they have different sizes, and they you have many layers of what can be an infinite number. And this is what Cantor really really began to explore and develop, and he met a lot of resistance when he was doing it.

He actually people thought he was crazy. He sort of fell into a lot of depression. You know, he was in battles with with someone called Chronicer, who was kind of, you know, the big guy in Berlin at the time, the elite university in Germany. He thought that Cantor was just delving into sort of witchcraft and he was a shot. He called him a charlatan, a corruptor of youth. And this really bothered Cantor and actually is quite a sad story. I mean, Cantor actually sort of really fell into into

quite bad depression. Whether it's because of this or whether he was he was predisposed anyway, it's not clear. But he actually ended his days sort of very sort of emaciated in it in a sanity Toorium, essentially starving because of the effects of the First World War at the time and not having enough foods. So it's quite a tragic tale in the end, but he was certainly a tremendous mathematician, and now all his ideas are really you know, I think people acknowledge him for the genius that he was.

But then you can have these high infinities, which are the you know, things like the set of the continuum, essentially the continuum between zero and one. So not just all the fractions and rational numbers, but also the irrational numbers, numbers like one over the square root of two, that kind of thing. And this is a new letter. He actually proved that they're actually that's a bigger infinity, and it's not easily obvious, but he did show it, and

it's remarkable. And there's so many sort of things about infinity. There's so many paradoxes associated with them. For example, one thing you can say is you think about the number of are there more square numbers or whole numbers? And you think, well, you think naively, obviously there are more whole numbers than square numbers, because one is a square, but two isn't a square, and three isn't a square. Okay, four is So it seems that there's obviously more whole

numbers than square numbers. But actually it's not true. And the reason you know that's not true because you just take a square number and you can map it to its square roots and you get the whole numbers. So the number of whole numbers is actually exactly the same as the number of square numbers. It's completely crazy. I mean,

these these parrot it's the same. There are same number of even numbers as there are even in odd numbers, and there's all these one there's the same number of numbers between zero and one as there are between zero and two. There's all these paradoxes that emerge the minute you start to think about infinity. And that's why most mathematicians for a long time just stayed away from it. But Canto was brave enough to climb into this infinite heaven and explore it.

a tree that has appeared before. So if I draw like, you know, one particular tree, then later on you can't draw a bigger tree that's got my tree stuck in it somehow, it's just not allowed. That would end the game. So there's a bunch of rules in how you draw these trees and build up this particular game, which I call the game of trees. Now, how long the game lasts depends on how many different types of seeds you have.

So you could have, for example, just black seeds, okay, or maybe you could have black seeds and you're also got white seeds, or maybe you've got black seeds, white seeds, and yellow seeds. You know, there's a whole bunch of possibilities. How many seeds you play with sort of sort of changes how long the game can last. For now, if you've just got one seed, the game can only last one move. You can just write down one seed and

that's it. You can't write down anything else because anything else that follows is going to contain the tree that went before. Okay, you've got two seeds, like a black and a white seed, the game can last up to you can draw up to three trees, and the game will automatically end after just three moves. It can't go beyond three moves. So you've got this this sort of sequence. So you've got one seed, you can play only one move. If you've got two seeds, you can play three moves.

And so then you go to three seeds. And you might think, well, I started off with one and it went to three, and now I've got three seeds, maybe maybe I can play ten moves or something called fifteen moves. It's not gonna be some it shouldn't be anything crazy. Well it is. So this sequence just goes bang. It just goes from one. So just for one seed you get one move, two seeds you get three moves, and then three seeds you get three. Three moves is where the game will last too. And this is a number

which just blows everything else. So we talked about Google and a Google plays, well, that's just nothing compared to three three. Talk about Graham's number, which will collapse your head's form black hole. That's nothing compared to three three three three is just it. It's impossible. I mean, I actually think it is impossible to imagine how ridiculously big this number is. And it's just so mundane. Where it comes from? Is this game so if he starts off you.

So so you're playing this game with two seeds. This game keeps ending after three moves, and if somebody comes along and adds a different colored seed, and you're like, okay, how is how long can the game last? Now? And somebody says tree three and this is three three. It's just a number that's actually too big for the universe. It just whow where did that leap come from? Leaps should not be that big. But that's so that's in a essence what tree three is, and it is too

big for the universe. So one of the things I worked out was suppose you're playing this game involving these trees. So you're writing drawing these trees, right, so you play one, go draw a tree. Play next, go draw a tree, and so on. You've got three seed, three different colors of seeds. So we know the limit of the game is three three moves treating three three different trees in the forest. How long are you going? Could you finish

the game? And one other thing I imagine is you know you're playing this game at high speed, so you're playing it as fast as space time will allow. So you literally if you play any faster space time will break due to quantum effects. So you play it super super fast, and so you play again, you play again, You'll play it through a lifetime. You'll get nowhere near tree three. After you die, and maybe you replace yourself

with some artificial intelligence. You've got two art Ai machines playing against each other, you know, powered by the light of the sun. They'll keep playing the game at this crazy pace, and they keep going, and they keep going. The sun gets bigger, you know, it goes to a red giants. All these things happen. Eventually it falls back

forms a white dwarf. Over many billions of years, and still this these two ais are still playing the game because they have got nowhere near tree three, and they're playing at breakneck speed as well, and so eventually they lose power. They can't get any power because the sun dies, right, so they need to somehow develop some new technology which gets energy from I don't know, the cosmic microwave background radiation. And the game goes on, and the game goes on,

and the game goes on. In fact, the game will go on way beyond the sort of heat death of the universe, and still you will not get to the end of tree three. And actually there's a phenomena called Puancore recurrence, which says that in any system, in any finite system, you'll eventually get back to where you started. And that applies to our universe too. So you can

imagine a pack of cards. You know, if you shuffle a pack of cards enough times, you'll a lot of times, but enough times you'll eventually get back to the point where all the cards are in order. It'll take a long time, but it will happen eventually. It's the same with our universe. You shuffle the universe enough times, you allow it to evolve for long enough, eventually you'll get

back to where it started. It will reset. And that reset time for our universe is actually shorter than the time it would take to play this game of trees all the way up to three three moves, playing as fast as you possibly can. And so even even if you could do it, even if you could live past all these you know, Gargangian timescales, the universe is just going to go, nah mat, game over. We're resetting. You ain't going to get to the game which you ain't

going to end this game. So three three is actually a number that's that's actually too big for the universe. That's how big it is.

on the edge of the black hole? And these are ideas which which which leads you to to to the to the holographic truth, to the idea that actually, maybe the information in our world isn't stored inside the world. Maybe it's stored on the boundary of the world, at the edge, on the walls that surround it. And in

that sense, that's why it's it's holographic. All the ideas, all the limits that we're talking about, you know, counting how much information you can store in a head, you know, and when it's going to turn into a black hole, you know, counting how long it takes for our universe to reset it up. All these ideas come back to the question of how our universe stores its information. Does it store it inside, and if so, how does it

store it? Well, actually, no, it turns out it's seem like it stores it on the edge of space, and that allows you to count how much information there is in that space and how many different ways you can combine things. But it all comes back to that holographic truth.

I'm not as versed in mathematics. There's a lot of people out there, and one of the things that I kept thinking about reading the book was just a one quick joke from the season one episode of the British comedy look Around You, in which the narrator the episode is about math, and the narrator tells us that the largest known number is around forty five million, but the larger numb umbers might exist, and they like speculate forty five million in one could be an number, and you know,

of course that's absurd, and that's absurd ast humor. But there's something about that that seems to sort of ring true with a lot of these these concepts, and I was wondering what you thought about the role of absurdity in contemplating big numbers.

in our universe. Now, you might imagine other universes which could accommodate it, and in a you know, a sort of multiverse scenario, like maybe you get from something like string theory, could you get universes that can contain tree three. Well maybe we don't know, right, we don't know enough about about the multiverse of string theory, but it's not inconceivable potentially so, but certainly in our world you can't.

It's interesting. One of the things I did a video quite quite recently actually about the biggest number that nobody will ever think of. And I did these sort of quite a bunch of estimates based on a bunch of dubious sort of you know, sort of assumptions, which I acknowledge with quite jubious, But I think I came up with an estimate that if you think of a random seventy three digit number, also something of that order, then probably nobody's going to ever ever think of it. Other

than you. I mean, you know, so, I'm not saying I just think of a one followed by seventy two zero. It's clearly not something like that, but just completely random, random seventy two to seventy three digit numbers something like that. Chances are nobody in the history of humanity, either before or to come, we'll ever think of that number. And it's kind of that's kind of mind blowing. I think it's kind of yours. Just think of it, and that's

yours forever. So just everybody should just write down a seventy three digit numb and name after themselves.

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