Speaker 1: Welcome to stot to Blow Your Mind, production of My
Speaker 1: Heart Radio. Hey, welcome to Stuff to Blow Your Mind.
Speaker 1: My name is Robert Lamb. My co host Joe is
Speaker 1: away from work today, so I am conducting an interview
Speaker 1: here with Professor Antonio Padella, author of the new book
Speaker 1: Fantastic Numbers and Where to Find Them, a fascinating read
Speaker 1: about big numbers, fantastic numbers, black holes, and more. This
Speaker 1: is a really fun chat. I think you're all going
Speaker 1: to enjoy it, So go ahead and jump right in
Speaker 1: with me right now. Hi, Tony, Welcome to the show. Hi,
Speaker 1: Hi up, how you doing? Oh? Pretty good? Pretty good? Um?
Speaker 1: Really excited to talk about the new book Fantastic Numbers
Speaker 1: and Where to Find Them? A wonderful read, and it's
Speaker 1: a book that get into some pretty wonderful mind rending
Speaker 1: cosmological territory as well. Note I'll discuss here, but first
Speaker 1: I wanted to start with just a really basic sort
Speaker 1: of grounding question. I guess we encounter numbers every day,
Speaker 1: and you discuss some numbers that most of us don't
Speaker 1: encounter really every day. If we could back up a
Speaker 1: whole lot, I guess, and just ponder the basics here,
Speaker 1: what exactly is a number? Well, I mean, this is
Speaker 1: this is an idea I sort of, you know, delve
Speaker 1: into him in my book, because of course, when you
Speaker 1: go really back in into history, back to sort of
Speaker 1: the ancient Sunarians or something like that, you know, obviously
Speaker 1: they really began to use numbers to talk about And
Speaker 1: while I've got five jars of oil, I've got five
Speaker 1: loaves of bread. But then it sort of begs the question,
Speaker 1: is that five the same five is the five that
Speaker 1: describes the jars of oil, the same five that describes
Speaker 1: the loaves of bread. And then you really sort of
Speaker 1: when you sort of make that disconnected too, and you
Speaker 1: start to build the idea of like what I call
Speaker 1: an emancipated number, but it's independent and of the thing
Speaker 1: that it's describing, then you're really sort of making quite
Speaker 1: quite an intellectual leap. So that, for me is what
Speaker 1: is what a number is. It's kind of emancipated from
Speaker 1: from the thing that it's describing. Whether such a thing
Speaker 1: really exists in a philosophical sense, is a whole new
Speaker 1: debate that you can have. But yeah, that for me
Speaker 1: is is the key mathematical leap that I think was
Speaker 1: made you know, a long time ago, and and yeah,
Speaker 1: it's really important now getting back into that sort of
Speaker 1: philosophical territory. This is one that I know that you
Speaker 1: you tackle a lot. Uh, it's pretty standard sort of
Speaker 1: philosophical math question. But is mathematics more of a human
Speaker 1: discovery or more of a human invention? Yeah, I mean
Speaker 1: I don't think there's a straightforward answer answer to this.
Speaker 1: Of course. This this sort of you know, boils down
Speaker 1: to like a sort of alluded to whether numbers exist,
Speaker 1: whether whether maths exist, and it's kind of I mean,
Speaker 1: I'm not a philosopher, but but but I know that
Speaker 1: philosophers talk about this, and sort of this kind of
Speaker 1: three different angles that you can take on it. So
Speaker 1: so on the one hand, you've got the platonists who
Speaker 1: will say that numbers and mathematics is true and it exists,
Speaker 1: but it exists outside of space time. Is as like
Speaker 1: an abstract concept. It's not something that can affect the
Speaker 1: things in space time. It can't affect the material objects
Speaker 1: that we have arounders. You also have the nominalists, who
Speaker 1: says basically that numbers and maths only exist to sort
Speaker 1: of understand stuff. So so in some sense, we talked
Speaker 1: about the five, you know, five jars of oil, the
Speaker 1: five five loaves of bread. That's the only reason that
Speaker 1: the number five exists to describe the jars of oil,
Speaker 1: to describe the loaves of bread. And then, of course
Speaker 1: you've got the third sort of you know school, which
Speaker 1: is perhaps in some sense the most extreme, which just
Speaker 1: says the numbers that exist at all, that they're just
Speaker 1: a useful tool, that that that we used to describe
Speaker 1: the universe arounders. And I guess the analogy people use
Speaker 1: here is it's like saying, well, you could be an atheist,
Speaker 1: but you can still believe with some of the sort
Speaker 1: of moral messages that you read in the Bible or
Speaker 1: the Koran. It doesn't mean that, you know, you can't
Speaker 1: be inspired by them, but you just don't have to
Speaker 1: believe in every element of it. And I guess as
Speaker 1: a physicist, for me, it's kind of hard to sort
Speaker 1: of go with that fictionist idea and yet see a
Speaker 1: universe that is so amazingly described by mathematics. Now, is
Speaker 1: that something that's embedded in the universe or not. I
Speaker 1: guess it's really difficult to know. We've certainly not seen
Speaker 1: any evidence that it is. And yet now your book
Speaker 1: deals with, as the title indicates, fantastic numbers. Uh, what
Speaker 1: what defines for you a fantastic number? And are there
Speaker 1: categories of categorizations of numbers other than that that we
Speaker 1: need to have in our heads before we can get
Speaker 1: to the idea of what it's truly fantastic. Yes, it's
Speaker 1: so for me and my own relationship with numbers kind
Speaker 1: of and it comes from, on the one hand, you
Speaker 1: have a number, whatever that number might be, and if me,
Speaker 1: I always want to bring that sort of personality alive,
Speaker 1: the sort of real spirit of the number, sort of
Speaker 1: to the four. And so it's always been physics for
Speaker 1: me that that does that. So when you know you
Speaker 1: can have these wonderful mathematical concepts ideas like Graham's number
Speaker 1: three three, these truly bizarre and wonderful numbers to have
Speaker 1: a wonderful place in mathematics, but then you really bring
Speaker 1: them to life when you try to sort of squeeze
Speaker 1: them into our physical world. So that for me is
Speaker 1: what what makes a number fantastic. It's almost like, what
Speaker 1: makes a number fantastic is the fantastic physics that it
Speaker 1: can lead you towards and lead you to imagine and
Speaker 1: whatever that might be. You also talk about I believe
Speaker 1: specifically you're talking about Graham's number pretty early on in
Speaker 1: the book, and you point out that if you if
Speaker 1: you try and actually picture it in your head, your
Speaker 1: head collapses into a black hole. And this this made
Speaker 1: me wonder, like, what what are the largest numbers roughly
Speaker 1: speaking then an average person can fit into their head
Speaker 1: by one definition or another, Like, at what point does
Speaker 1: it just become this this other enterprise entirely? Yes, So
Speaker 1: it's a good question. So, I mean, it kind of
Speaker 1: depends on how how you sort of define the question.
Speaker 1: In some sense, we're just thinking about your rs. How
Speaker 1: many neurons have you have you got in your in
Speaker 1: your brain as about a hundred billion neurons, and so
Speaker 1: you might say that you can use them if you
Speaker 1: managed to clear your mind of every other thought to
Speaker 1: imagine a hundred billion digit number. Okay, that might not
Speaker 1: be particularly practical, it might be quite challenging for most
Speaker 1: of us. But but in principle you might say that
Speaker 1: that that that would be the limit. And of course,
Speaker 1: if you then go beyond that and start to say, well,
Speaker 1: what if I could somehow get my head to find
Speaker 1: a way to actually store information store concepts more efficiently
Speaker 1: than just the usual idea of neurons firing on and off.
Speaker 1: Let's suppose that it could do that somehow. Then then
Speaker 1: the numbers get get much bigger, and you start to
Speaker 1: the things that limited are literally preventing your head collapsing
Speaker 1: to form a black hole. Because black holes what they
Speaker 1: do is that they're they're the best thing at storing information.
Speaker 1: So if you want to get something the size of
Speaker 1: a head of a human head, and you want to say,
Speaker 1: what's the best thing the size of human head that
Speaker 1: can store information, it's a black hole the size for
Speaker 1: human head that that's the nothing can do it better,
Speaker 1: and so so so that that places a new limit,
Speaker 1: and you can ask, well, again, what what is that
Speaker 1: limit would be? But he's certainly way below grains. No,
Speaker 1: but you're not gonna get anywhere near the magnificence of
Speaker 1: Graham's number. And you could probably get a digit that's
Speaker 1: that's about tens of the seventy and number that's about
Speaker 1: ten to the seventy digits long, and so less than
Speaker 1: a Google digits long. Having said that, you could imagine
Speaker 1: a number like a google plex. A google plex has
Speaker 1: a Google digits. Now I've just said that you can't
Speaker 1: imagine a Google Digit's not possible, but a google plex
Speaker 1: you could, because what you know about a google plex
Speaker 1: is that it's a one followed by a Google zero.
Speaker 1: So you know that all the numbers that come later
Speaker 1: on as zeros, and so there's not much information in that.
Speaker 1: So it doesn't cost as much as many bits. You
Speaker 1: don't have to put as many bits in your head
Speaker 1: to imagine that. So what we're really talking about now
Speaker 1: are really a random assortment of digits. Are completely random
Speaker 1: assortment digits the kind that would appear in Graham's number.
Speaker 1: And I don't think you can get passed around tens
Speaker 1: of the seventy, which is a one with seventies zero.
Speaker 1: You couldn't get past that many digits completely randomly sort
Speaker 1: of allocated. At that point, your head is going to
Speaker 1: collapse into a black hole. Now, now backing up to
Speaker 1: the Google and the google plex. Can't can you? Can
Speaker 1: you walk as briefly through the difference between a Google
Speaker 1: and google plex, and and and maybe realms beyond that?
Speaker 1: This is this is about the only area of fantastic
Speaker 1: numbers that i'd I'd really heard anything about prior to
Speaker 1: reading your book. Yes, So, so Google is is um
Speaker 1: it's a number which is which is a one followed
Speaker 1: by a hundred zero. So I think everybody would agree
Speaker 1: that sounds like quite a big number. And it goes
Speaker 1: back to to a physicist called Edward Kasner who is Columbia,
Speaker 1: and he was writing a popular science book and he
Speaker 1: was trying to sort of know convey He really wanted
Speaker 1: to show how big infinity he was, and so he
Speaker 1: wanted to quote with numbers that we all think are
Speaker 1: really big, like a one followed by a hundred zeros.
Speaker 1: And he said, well, okay, that's really small compared to infinity, right,
Speaker 1: even though something really big is actually really small compared
Speaker 1: to infinity. So he came up with this one with
Speaker 1: a hundred zeros. He wanted a name for this number,
Speaker 1: so at the time he asked his nephew who was
Speaker 1: nine years old at the time. He was called Milton Serata.
Speaker 1: He said, can you come up with a name for this?
Speaker 1: And and Milton said, well, a Google, which is an
Speaker 1: absolute stroke of genius, right, It's such a great name.
Speaker 1: And and then so they wanted to then develop things further,
Speaker 1: so then they wanted an even bigger number, again building
Speaker 1: on this idea that it's nothing compared to infinity and
Speaker 1: and so so he said, well, okay, I'm gonna quote
Speaker 1: with the idea of a Google plex. It's going to
Speaker 1: be an even bigger number. Well how big? So Kasna
Speaker 1: then goes to to Milton. He says, well, how big
Speaker 1: should it be? And Milton's like, well, it should be
Speaker 1: a one, not followed by a hundred zeros, but zeros
Speaker 1: until you get tired. But Kasner is like, you know,
Speaker 1: a sort of you know, steamed academic at Columbia and
Speaker 1: all that. That's just not precise enough for him. So
Speaker 1: he went with it which are much more sort of
Speaker 1: well defined idea, which is a google plex should be
Speaker 1: a one followed by a Google zeros. So a Google
Speaker 1: is already massive. That's a one followed by a hundred zeros.
Speaker 1: A google plex is a one followed by a Google zero.
Speaker 1: So it's a whole new level of big compared to
Speaker 1: what we normally used to. And then it just it
Speaker 1: keeps building on that. Right, there's there's even like what
Speaker 1: a google plexian is that the next level. So so yeah,
Speaker 1: I mean this is this is a really nice, nice idea.
Speaker 1: You can really now start to to really build very
Speaker 1: big numbers, very very quickly using this this mathematical technique
Speaker 1: called recursion. So for example, you can develop the idea
Speaker 1: of a Google duplex. What a Google duplex, Well, it's
Speaker 1: a one followed by a Google plex zeros. And then
Speaker 1: you can go to a Google triplex. Well, you can
Speaker 1: probably guess what it's gonna be. It's gonna be a
Speaker 1: one followed by a Google duplex zeros. And then a
Speaker 1: Google quadruplex is a one followed by a Google triplex zeros.
Speaker 1: And you can see each time you're growing the number
Speaker 1: just by so much, by such an unimaginably large amounts,
Speaker 1: And that's what. You're not just adding zero every time,
Speaker 1: You're kind of really ballooning the number of zeros on
Speaker 1: the end of this number in Gargangian proportions. And that's
Speaker 1: what and it's this power of mathematical riccasion that allows
Speaker 1: you to do that. Now you also talk about fantastic
Speaker 1: numbers that are I guess you would say smaller than
Speaker 1: the main example that comes to mind. You refer to
Speaker 1: this several times in the book is a number associated
Speaker 1: with Olympic sprinter Hussain Bolt. Would you tell us a
Speaker 1: little bit about this number? Yeah, yeah, so so well,
Speaker 1: actually this is one of my big numbers. Actually, even
Speaker 1: though it doesn't seem that it's it's actually it's one
Speaker 1: of my big numbers. So I can read out the number.
Speaker 1: What it is one point, I think it's fifteen zero
Speaker 1: eight five eight, So it's just a number just slightly
Speaker 1: north of one. So it's it doesn't seem like a
Speaker 1: big number. But but in my book, I say it
Speaker 1: is a big number. And the reason is it's it
Speaker 1: measures the amount by which Usain Bolt managed to slow
Speaker 1: down time. And when he was he was running in
Speaker 1: the World Championships and I think Berlin Um and he
Speaker 1: set this the world record. And this is due to
Speaker 1: the effects of relativity, so that when when somebody actually
Speaker 1: moves quickly, they actually slow time actually slows down for them.
Speaker 1: And actually this is the amount by which Usain Bolt
Speaker 1: was actually able to slow down time due to the
Speaker 1: effects of Einstein's theory. And it's compared to the people
Speaker 1: in the stadium, for example, this is this was the
Speaker 1: difference that that he experienced. So it's one of the
Speaker 1: one of the weird consequences of it is that you
Speaker 1: can actually it's not that you say, but actually, even
Speaker 1: though he slowed down time, it's not that he that
Speaker 1: he actually ran the race any quicker. He still runs
Speaker 1: the race at roughly ten ms per second. It's actually
Speaker 1: an even more strange consequence. He actually the track also
Speaker 1: shrinks for him a little bit, so so he actually
Speaker 1: runs it in less time, but in the same speed. Therefore,
Speaker 1: the track shrinks because the relative to him, the tracks moving,
Speaker 1: And this is another effect of relativity, one of the
Speaker 1: remarkable things. And and yes, you could perhaps argue that
Speaker 1: he didn't actually finish the race because the tracks rank
Speaker 1: so he didn't run quite a hundred Wow. I was
Speaker 1: really blown away with this, because you know, you often
Speaker 1: hear the standard analogies concerning airplanes and pyramids and so
Speaker 1: forth when it comes to time dilation and so forth.
Speaker 1: But but I hadn't. I hadn't heard this particular example
Speaker 1: for this is great. Yeah, I mean, it's true. It's true.
Speaker 1: If like taxi drivers, if you imagine a taxi driver
Speaker 1: that's driving around I don't know any city in New
Speaker 1: York wherever, you know, sort of forty fifty years of
Speaker 1: their life because of that extra extra speed that they're
Speaker 1: picking up. That's going to accumulate over time. And actually
Speaker 1: they can probably leap forward in time by probably I
Speaker 1: think about a micro second over the course of their career.
Speaker 1: It's not a lot, but it's still fairly amazing when
Speaker 1: you think about it. So they've got the knowledge, and
Speaker 1: then they have that as well, right, Oh yeah, of course,
Speaker 1: yeah exactly, not just the knowledge, Yeah, they actually got it.
Speaker 1: They actually get a little bit younger. So your book
Speaker 1: makes makes use of written numbers, um. And and then
Speaker 1: of course you have this wonderful YouTube series number file,
Speaker 1: and in that you benefit not only from some fantastic
Speaker 1: descriptions and pop culture tie ends as you do in
Speaker 1: the book, but you also have a lot of helpful illustrations.
Speaker 1: Uh So I was I was curious since you are
Speaker 1: a regular communicator of of this this topic. Um is
Speaker 1: it is? It? Is? It? Is? It more challenging or
Speaker 1: in some cases almost too challenging to describe some of
Speaker 1: these numbers without the visual aids or the actual numerals
Speaker 1: to like visually present somebody with. Yeah, I think you
Speaker 1: so this is where the physics comes in in some respects. Right.
Speaker 1: So on the one hand, if you really want to
Speaker 1: describe the number, like I said, a number like Grams number,
Speaker 1: you do need those visual aids because it's not a
Speaker 1: number that you're going to sort of stumble across in
Speaker 1: any kind of normal environments. Right, It's not a number
Speaker 1: you're going to see on on a price tag, at
Speaker 1: least you'd hope not. And you know, so these are
Speaker 1: you need new notation, new sort of symbolism to to
Speaker 1: sort of actually even describe the number. So you've got
Speaker 1: to introduce that. There's just no getting away from it.
Speaker 1: But I guess what you can do is described the
Speaker 1: physics associated with it, and and that you can certainly do,
Speaker 1: you know, just just just with words. And you know,
Speaker 1: in the case of a number like Grahams number, you
Speaker 1: can talk about how you just can't picture in your
Speaker 1: head because your head will will collapse to for a
Speaker 1: black hole. And that's already going to make people think, wow,
Speaker 1: that number. There's something big of something big and crazy
Speaker 1: about that number. Or a google plex, you know, when
Speaker 1: you can talk about a universe that's that's a google
Speaker 1: plex meters across. And then you can ask, well, if
Speaker 1: the universe is that big, if the universe is literally
Speaker 1: that large, then it's likely that you would find multiple
Speaker 1: copies of yourself, like literally exact apple Gangers elsewhere in
Speaker 1: this ginormous universe. Yeah, I wasn't. I was. I wasn't
Speaker 1: prepared for dopple gangers to enter into the scenario. So
Speaker 1: there was there was another great part about the book
Speaker 1: for me um and another thing that that comes up
Speaker 1: in the book that I was very intrigued by. I
Speaker 1: was wondering if you might talk about is the the
Speaker 1: idea of the of the holographic truth? Yes, so the
Speaker 1: holographic truth is I mean, it's an idea. It's probably
Speaker 1: the most important idea I would say that's emerged from
Speaker 1: theoretical physics in in the last thirty years. And it's
Speaker 1: it's actually mind blowing when you really think about what
Speaker 1: it pertains to. It's it's it's this following statement that
Speaker 1: essentially one of the dimensions of space that we experience
Speaker 1: around it. So we normally talk about say three dimensions
Speaker 1: of space, Well one of them could well be an illusion.
Speaker 1: It might not exist, and it's really remarkable. So what
Speaker 1: we're saying is that there are two ways in which
Speaker 1: you can describe the physics that we see around as
Speaker 1: On the one hand, we can imagine three dimensional world
Speaker 1: with a gravitational force and the force of gravity doing
Speaker 1: its thing, with planets around the Sun and so on
Speaker 1: and so forth. On the other hand, there's a completely
Speaker 1: equivalent description of the same phenomena which just uses two
Speaker 1: dimensions and no gravity. So think of it a bit like,
Speaker 1: you know, on the one hand, somebody's you know, in English,
Speaker 1: we say if we see a place of meat balls,
Speaker 1: we call them meat balls, but a Spaniard might call
Speaker 1: them album the gas. They're both describing the same things,
Speaker 1: they're just using a different language. And that's kind of
Speaker 1: what what the holographic truth says. It says that you
Speaker 1: can have a theory like a three dimensional world with gravity,
Speaker 1: and you can use that to describe all the physical
Speaker 1: phenomena you see, or use this different language which has
Speaker 1: no gravity and only requires two dimensions of space. So
Speaker 1: is it true of our world? We don't know. It's
Speaker 1: a conjecture. It's a conjecture that has sort of evidence
Speaker 1: coming from from the physics of black holes. There are
Speaker 1: actually concrete examples that we know of of sort of
Speaker 1: toy universe is so not our universe, but but but
Speaker 1: space times that may be that the higher dimensional there
Speaker 1: may be warped in weird and wonderful ways. And you
Speaker 1: can think about gravity in these in these simple toy universes,
Speaker 1: and you can show that there's an equivalent description in
Speaker 1: one dimension less like a holographic description, and it's called
Speaker 1: a hologram because that's essentially what what holograms do? Right?
Speaker 1: If you think of a hologram, what have you got?
Speaker 1: You've got an image on a that's stored on a
Speaker 1: holographic plate. You know, it's just some light and dark
Speaker 1: bands on a holographic plate, a two dimensional plate. It's
Speaker 1: stores a bunch of information that way, But that's just
Speaker 1: one way of looking at the information. You can decode
Speaker 1: it in a different way by shining monochromatic light through
Speaker 1: it and creating a three damage. You're not creating any
Speaker 1: new information. It's the same information, just stored either in
Speaker 1: two dimensions or three. And it's that seems to be
Speaker 1: that that seems to be a fundamental property of gravitation,
Speaker 1: of gravitational worlds that you can think of them as
Speaker 1: as like as I said, a three D world with gravity,
Speaker 1: or you just forget about gravity and consider a world
Speaker 1: with one dimension or less and you can describe exactly
Speaker 1: the same physical phenomena. Now here's another question that that
Speaker 1: that came came up reading the book that that I
Speaker 1: don't know if of all our listeners are necessarily would
Speaker 1: have thought of this question. I think some of them
Speaker 1: would have. And that comes to infinity um infinity, Like
Speaker 1: sometimes it's easy to think of like, okay, infinity is
Speaker 1: the it's it's if we think of it as a number.
Speaker 1: We think it's the eight on its side representing infinity.
Speaker 1: Is infinity a number? And if it's not a number,
Speaker 1: like what do we think of it at? How do
Speaker 1: we classif high infinity? So I love this question because
Speaker 1: the answer is that it's both not a number and
Speaker 1: lots of numbers. This is the wonderful thing about infinity.
Speaker 1: So it depends how you want to think about infinity.
Speaker 1: And I think most of us when we intuitively think
Speaker 1: about infinity, we kind of think of like I don't
Speaker 1: know the infinite distance, you know, or infinite time, And
Speaker 1: what we're really thinking there is we're thinking of it
Speaker 1: is like a limit is something that's just just beyond
Speaker 1: our finite realm. That that that's you know, if you
Speaker 1: keep on counting forever, you know, it's kind of the
Speaker 1: at the end of that, you're sort of almost beyond
Speaker 1: the end of that. Now, that's in some sense thinking
Speaker 1: of infinity as not a number, as a limit of saying,
Speaker 1: you know, the whole numbers. But what Cancelor Judge cancel that,
Speaker 1: you know, the great German mathematician from the late Victorian times,
Speaker 1: what what what he did was actually taught us how
Speaker 1: to count beyond infinity. So literally, using really smart ideas
Speaker 1: associated with something called set theory, he was able to
Speaker 1: show that actually you can have all the sort of
Speaker 1: finite numbers, and beyond that you can have infinity. But
Speaker 1: there's that's just one layer of infinity. You can have
Speaker 1: the infinity, which is all the whole numbers, but you
Speaker 1: can also have a different layer of infinity, which is
Speaker 1: all the numbers between zero and one. So think of
Speaker 1: the continuum of the numbers between zero and one. That's
Speaker 1: you think there's an infinite number of numbers between zero
Speaker 1: and one, but that's actually a different infinity to all
Speaker 1: the whole numbers. So you've got, you know, these discreet infinities,
Speaker 1: continuum infinities, and they they have different sizes, and they
Speaker 1: have you have many layers of what can be an
Speaker 1: infinite number. And this is what Cancel really really began
Speaker 1: to explore and and and develop. And he met a
Speaker 1: lot of resistance when he was doing it. He actually
Speaker 1: people thought he was crazy. He sort of fell into
Speaker 1: a lot of depression. Um, you know, he was in
Speaker 1: battles with with someone called Chronicker, who was kind of,
Speaker 1: you know, the big guy in Berlin at the time,
Speaker 1: the elite university, and Jereman. He he thought that Cancer
Speaker 1: was just delving into sort of witchcraft and he was
Speaker 1: a shot. He called him as Charlatan, a corruptor of youth.
Speaker 1: And this really bothered Cancer and actually quite a sad story.
Speaker 1: I mean, Cancer actually sort of really fell into into
Speaker 1: quite a bad depression. Whether it's because of this or
Speaker 1: whether he was he was predisposed anyway, it's not clear.
Speaker 1: But he actually ended his day sort of very sort
Speaker 1: of emaciated in a in a sanatorium, essentially starving because
Speaker 1: of the effects of the First World War at the time.
Speaker 1: And I'm not having enough food, so it's quite a
Speaker 1: tragic tale in the end, but he was certainly a
Speaker 1: tremendous mathematician, and now all his ideas are really you know,
Speaker 1: I think people acknowledging for the genius that he was. Yeah,
Speaker 1: of course brings to mind those um like the infinity
Speaker 1: hotel discrete scenarios that are used to describe infinity. I've
Speaker 1: always found those to be super interesting and and men
Speaker 1: mind blowing. Yeah, I mean that's so, that's that's what
Speaker 1: I mean. So, so, as I said, cancer sort of
Speaker 1: had these these different layers. So you can sort of
Speaker 1: imagine the first infinity, which he called alve zero, which
Speaker 1: is he defined as the set of all of all
Speaker 1: the whole numbers, essentially all the natural numbers you know,
Speaker 1: one to three for all the way up to well infinity,
Speaker 1: all of them basically, so that that's what he called
Speaker 1: the sort of first infinity. But then you can have
Speaker 1: these higher infinities, which are the you know, things like
Speaker 1: the the the set of the continuum, essentially the continuum
Speaker 1: between zero and one, so not just all the all
Speaker 1: the fractions and irrational numbers, but also the irrational numbers
Speaker 1: numbers like one over the square out of tow that
Speaker 1: kind of thing um and and this is a new
Speaker 1: letter that he actually proved that they're actually got a
Speaker 1: bigger infinity. And it's not immedially obvious, but but he
Speaker 1: did show it, and it's it's it's remarkable, and and
Speaker 1: there's so many sort of things about infinity. There's so
Speaker 1: many paradoxes associated with them. For example, one one thing
Speaker 1: you can say is you think about the number of
Speaker 1: the more square numbers or whole numbers, and you think, well,
Speaker 1: you think naively, obviously there are more whole numbers and
Speaker 1: square numbers because one is a square, but but two
Speaker 1: isn't a square, and three isn't a square, Okay, four
Speaker 1: is So it seems that there's obviously more whole numbers
Speaker 1: than square numbers. But actually it's not true. And the
Speaker 1: reason you know that's not true because you just take
Speaker 1: a square number and you can map it to its
Speaker 1: square roots and you get the whole numbers. So so
Speaker 1: the number of whole numbers is actually exactly the same
Speaker 1: as the number of square numbers. It's completely crazy. And
Speaker 1: these these parents and it's the same. There are the
Speaker 1: same number of even numbers, there are even at odd numbers,
Speaker 1: and there's all these one there's the same number of
Speaker 1: numbers between zero and one as there are between zero
Speaker 1: and two. There's all these paradoxes that are merged just
Speaker 1: the minute you start to think about infinity. And that's
Speaker 1: why most mathematicians for a long time just stayed away
Speaker 1: from it. But Cancer was brave enough to climb into
Speaker 1: this infinite heaven and explore it. Now. One of the
Speaker 1: numbers that that comes up a lot in your in
Speaker 1: your book, and you've done your videos on this as well. Um,
Speaker 1: I'm also afraid to ask about it because it just
Speaker 1: seems kind of I get confused anytime I read anything
Speaker 1: about it. And that's this idea of and I'm not
Speaker 1: even sure what I'm saying it correctly? Is it? Do
Speaker 1: we say tree three? Yeah? That's right, Yeah, yeah, three three?
Speaker 1: So yeah, what is this? What is tree three? So?
Speaker 1: So there's a particular game that was that was developed
Speaker 1: involving some trees, right, So, so the details aren't too important,
Speaker 1: it's just but basically you draw these little stick trees
Speaker 1: and you have some seeds, you have some lines which
Speaker 1: are kind of like the branches, and you and you
Speaker 1: build these trees. Right, So, so what are the rules
Speaker 1: of the game is is that you know, for example,
Speaker 1: you can't have a tree that's got a bit of
Speaker 1: a tree that that's has appeared before. So so if
Speaker 1: I draw, like, you know, one particular tree, then later
Speaker 1: on you can't draw a bigger tree that's got my
Speaker 1: tree stuck in it somehow. It's it's just not allowed.
Speaker 1: That would end the game. And so there's a bunch
Speaker 1: of rules in how you draw these trees and build
Speaker 1: up this this particular game, which I call the game
Speaker 1: of trees. Now, how long the game depends on how
Speaker 1: many different types of seeds you have. So you could have,
Speaker 1: for example, just black seeds, okay, or maybe you can
Speaker 1: have black seeds and you're also got white seeds, or
Speaker 1: maybe you've got black seats, white seeds and yellow seeds.
Speaker 1: You know, there's a there's whole bunch of possibilities. How
Speaker 1: many seeds you play with sort of of changes how
Speaker 1: long the game can last. For now, if you've just
Speaker 1: got one seed, the game can only last one move.
Speaker 1: You can just write down one seed and that's it.
Speaker 1: You can't write down anything else because anything else that
Speaker 1: follow is going to contain a tree that went before. Okay,
Speaker 1: you've got two seeds, like it's like a black and
Speaker 1: a white seed. The game can last up to you
Speaker 1: can draw up to three trees and the game will
Speaker 1: automatically and after just three moves, it can't go beyond
Speaker 1: three moves. So you've got this this sort of sequence.
Speaker 1: So we've got one seed, you can play only one move.
Speaker 1: You've got two seeds, you can play three moves. And
Speaker 1: so then you go to three seeds, and you might think, well,
Speaker 1: I can I start for one and it went to
Speaker 1: three and I've got three seeds, Maybe maybe I can
Speaker 1: play ten moves or something called fifteen moves something. It's
Speaker 1: not gonna be some it shouldn't be anything crazy. Well
Speaker 1: it is. So this sequence just goes bang. It just
Speaker 1: goes from one. So just from one seed you get
Speaker 1: one move, two seeds, you get three moves, and then
Speaker 1: three seeds you get tree. Three moves is where the
Speaker 1: game will last too. And this is a number which
Speaker 1: just blows everything else. So we talked about Google and
Speaker 1: a Google place, Well that's just nothing compared to three three.
Speaker 1: Talk about Graham's number which will collapse, your heads fall
Speaker 1: and black hole, that's nothing compared to three three Tree
Speaker 1: three is just it's it's impossible. I mean, I actually
Speaker 1: think it is impossible to imagine how ridiculously big this
Speaker 1: number is. And it's just so mundane. Where it comes
Speaker 1: from is this game. He starts off you. So you're
Speaker 1: playing this game with two seeds, as game keeps ending
Speaker 1: after three moves, and then somebody comes along and adds
Speaker 1: a different color of seed, and you're like, okay, how
Speaker 1: long is the how long can the game last? Now?
Speaker 1: And somebody says tree three, and this is tree three,
Speaker 1: just and not but it's actually too big for the universe.
Speaker 1: Just whoa where did that leap come from? The leap
Speaker 1: should not be that big, but that that's that's so
Speaker 1: that's in essence what what tree three is. And it
Speaker 1: is too big for the universe. So one of the
Speaker 1: things I worked out was suppose you're playing this game
Speaker 1: involving these trees. So you're writing drawing these trees, right,
Speaker 1: So you play one, go draw a tree. Play next,
Speaker 1: go draw a tree. And so you've got three seeds,
Speaker 1: three different colors of seeds. So we know the limit
Speaker 1: of the game is tree three moves treating three three
Speaker 1: different trees in the forest. How long could you finish
Speaker 1: the game. And one of the thing I imagine is,
Speaker 1: you know, you're playing this game at high speed, so
Speaker 1: you're playing it as fast as spacetime will allow. So
Speaker 1: you literally, if you play any fastest, spacetime will break
Speaker 1: due to quantum effects. So you play it's super super fast.
Speaker 1: And so you played again, You played again, You play
Speaker 1: it through a lifetime. You'll get nowhere near tree three.
Speaker 1: After you die, and maybe you replace yourself with some
Speaker 1: artificial intelligence. You've got two AI machines playing to each other,
Speaker 1: you know, powered by the light of the sun. They'll
Speaker 1: keep playing the game at this crazy pace, and they
Speaker 1: keep going and keep going. The sun gets bigger, you know,
Speaker 1: he goes to a red giant. All these things happen.
Speaker 1: Eventually it falls back forms a white dwarf. Over many
Speaker 1: billions of years, and still this these two ais are
Speaker 1: still playing the game because they have got nowhere near
Speaker 1: tree three, and they're playing at breakneck speed as well,
Speaker 1: and so eventually they lose power. They can't get any
Speaker 1: power because the sun dies, right, so they need to
Speaker 1: somehow develop some new technology which gets energy from I
Speaker 1: don't know, the cosmic by background radiation and the game
Speaker 1: goes on, and the game goes on, and the game
Speaker 1: goes on. In fact, the game will go on way
Speaker 1: beyond the sort of heat death of the universe, and
Speaker 1: still you will not get to the end of tree three.
Speaker 1: And actually there's a phenomenal called Puan career recurrence which
Speaker 1: says that in any system, in any finite system, you'll
Speaker 1: eventually get back to where you started. And that applies
Speaker 1: to our universe too. So you can imagine a pack
Speaker 1: of cards. You know, if you shuffle a pack of cards,
Speaker 1: he off times. You're a lot of times, but enough
Speaker 1: times you'll eventually get back to the point where all
Speaker 1: the cards are in order. It'll take a long time,
Speaker 1: but it will happen eventually. It's the same with our universe.
Speaker 1: You shuffle the universe enough times, you allow it to
Speaker 1: evolve for long enough, eventually you'll get back to where
Speaker 1: it started. It will reset, And that reset time for
Speaker 1: our universe is actually shorter than the time it would
Speaker 1: take to play this game of trees all the way
Speaker 1: up to three three moves, playing as fast as you
Speaker 1: possibly can. And so even even if you could do it,
Speaker 1: even if you could live past all these you know,
Speaker 1: gargantia and time scales. The universe is just going to go. Now, mate,
Speaker 1: game over, we're resetting. You ain't going to get to
Speaker 1: the game which you ain't gonna end this game. So
Speaker 1: three three is actually a number that's that's actually too
Speaker 1: big for the universe. That's how big it is. It's
Speaker 1: just so astounding that it as you describe it, it's
Speaker 1: just it's such a short step to reach that point.
Speaker 1: Because because a lot of these names, like when you're
Speaker 1: talking about the Googles and the google Plexes, it's easy
Speaker 1: to think, well, those those big numbers live out there
Speaker 1: like they're like in the d deep water. But then
Speaker 1: this seems to illustrate that the deep water is is
Speaker 1: far closer than you think, and it's not I wouldn't
Speaker 1: even call it deep water. It's it's water. That's you know,
Speaker 1: you're sort of like, yeah, you're just sort of tiptoeing
Speaker 1: across the you know, through the shallows and they're doing
Speaker 1: and then bang, it just gives away underneath you. And
Speaker 1: and there's just it's a it's bottomless as far as
Speaker 1: you're concerned. You know, I wouldn't even got a d
Speaker 1: what it's beyond deep it's too it's it's too deep
Speaker 1: for the universe. So ultimately, what do fantastic numbers reveal
Speaker 1: about the cosmos? Like what is the what I guess,
Speaker 1: what is the lesson of big numbers, fantastic numbers, etcetera.
Speaker 1: So so for me, I think all the ideas that
Speaker 1: I talk about in the context of the big numbers
Speaker 1: in the book, they all come back to the same
Speaker 1: thing which we've talked about, which is the holographic truth,
Speaker 1: the idea that that a lot of the ideas associated
Speaker 1: with black holes and and how much information you can
Speaker 1: fit inside a black hole. Where that information stored for example,
Speaker 1: is it stored inside the black call or is it
Speaker 1: stored on the edges of the black hole? And these
Speaker 1: are ideas which which which leads you to to to
Speaker 1: the to the holographic truth, to the idea that actually,
Speaker 1: maybe the information in our world isn't stored inside the world.
Speaker 1: Maybe it's stored on the boundary of the world, at
Speaker 1: the edge, on the walls that surround it. And in
Speaker 1: that sense, that's why it's it's holographic. All the ideas,
Speaker 1: all the limits that we're talking about, you know, counting
Speaker 1: how much information you can store in ahead, you know,
Speaker 1: and when it's going to turn into a black hole,
Speaker 1: you know, counting how long it takes for our universe
Speaker 1: to reset itself. All these ideas come back to the
Speaker 1: question of of how our universe stores its information, does
Speaker 1: its story inside and if so, how does it story? Well,
Speaker 1: actually no, it turns out it's seemed like it stores
Speaker 1: it on the edge of space, and that allows you
Speaker 1: to count how much information there is in that space
Speaker 1: and how many different ways you can combine things. But
Speaker 1: it all comes back to that holographic truth. Um. I
Speaker 1: have to ask about this because I again am ad
Speaker 1: is um his versed in mathematics. Uh, there's a lot
Speaker 1: of people out there. And one of the things that
Speaker 1: I kept thinking about reading the book was just a
Speaker 1: one quick joke from the season one episode of the
Speaker 1: British comedy look Around You, in which the narrator that
Speaker 1: the episode is about math, and the narrator tells us
Speaker 1: that the largest known number is around forty five million,
Speaker 1: but that larger numbers might exist and they like speculated
Speaker 1: forty five million in one could be another number and
Speaker 1: you know, of course that's absurd and that's absurdist humor.
Speaker 1: But Um, there's something about that that seems to sort
Speaker 1: of ring true with with a lot of these uh,
Speaker 1: these these these concepts. And I was wondering what you
Speaker 1: thought about the role of absurdity in contemplating big numbers. Yeah, absolutely, no,
Speaker 1: I really do think so. When you think of something
Speaker 1: like three three, at least within our universe, you can't
Speaker 1: fit it in. It cannot fit in. There's nothing that could,
Speaker 1: you know, you could describe because it's that's too big
Speaker 1: for anything that we can talk about in our universe. Now,
Speaker 1: you might imagine other universes which could accommodate it. And
Speaker 1: in a you know, a sort of multiverse scenario, like
Speaker 1: maybe you get from something like string theory, could you
Speaker 1: get universes that can contain three three? Well maybe we
Speaker 1: don't know, right, we don't know enough about about the
Speaker 1: multiverse of string theory. But but it's not inconceivable potentially so,
Speaker 1: but certainly in our world you can't. It's interesting. One
Speaker 1: of the things I did a video quite quite recently
Speaker 1: actually about the biggest number that nobody will ever think of.
Speaker 1: And I did these sort of quite a bunch of
Speaker 1: estimates based on a bunch of dubious sort of you know,
Speaker 1: sort of assumptions, which I acknowledge with quite dubious. But
Speaker 1: but I think I came up with an estimate that
Speaker 1: if you think of a random seventy three digit number,
Speaker 1: um also something of that order, then probably nobody's going
Speaker 1: to ever ever think of it other than you. I mean,
Speaker 1: you know, so I'm not saying, like, just think of
Speaker 1: a one followed by SEO is clearly not something like that,
Speaker 1: but just completely random, random seventy seventy three digit number
Speaker 1: something like that. Chances are nobody in the history of humanity,
Speaker 1: either before or to come, we'll ever think of that number.
Speaker 1: And it's kind of that kind of mind blowing. I
Speaker 1: think it's kind of yours. Just think of it, and
Speaker 1: that's yours forever. So just everybody should just write down
Speaker 1: a seventy three digit number and name after themselves. Well
Speaker 1: that's wonderful. Well, Tony, thanks for taking time out of
Speaker 1: your day to chat with us. I want to make
Speaker 1: sure we're we're hitting all the plugs here. The book
Speaker 1: which which is? Which? Is out? I believe it's out now, correct? Yeah, yeah,
Speaker 1: it's actually released today in the US. I probably should
Speaker 1: say today, should I guess it'll be it'll all will
Speaker 1: have been released two days ago when we published this,
Speaker 1: so yeah, it's it's out. It's Fantastic Numbers and Where
Speaker 1: to Find Them? Um. And then the YouTube series is
Speaker 1: number File, correct, Yes, so I appear on Number File.
Speaker 1: There's another channel I appear on which is physics Base
Speaker 1: called sixty Symbols. Um. So they're both made by by
Speaker 1: Brady Harron and yeah, so so I pay regularly on
Speaker 1: both of those so it's a lot of fun. But yeah,
Speaker 1: it's um. I hope people enjoy enjoy the book. It's
Speaker 1: and just don't think too recklessly about Grave's number, because
Speaker 1: what's gonna have. Yeah, we don't want anybody's heads to
Speaker 1: collapse into black holes absolutely. All right, Well, well thanks
Speaker 1: for coming on the show. Have I hope you have
Speaker 1: a great day. Thanks all right, Well, thanks once again
Speaker 1: to Tony for taking time out of his day to
Speaker 1: chat with me here. The book again is Fantastic Numbers
Speaker 1: and Where to Find Them? Highly recommended for anyone who
Speaker 1: was at all intrigued by what we were talking about
Speaker 1: here today. As always, if you want to reach out
Speaker 1: to us and ask any any questions, share your relationship
Speaker 1: with fantastic numbers. Well, you can find us in a
Speaker 1: number of ways. Let's see if you email us and
Speaker 1: I'll give you that email. On a second, you can
Speaker 1: have access to the discord where you can discuss show
Speaker 1: matters with with with other Stuff to Blow your Mind listeners.
Speaker 1: There's also the Stuff to Blow your Mind discussion Mondule
Speaker 1: that is on Facebook. You can find that and seek
Speaker 1: access to that as well. And of course thanks as
Speaker 1: always to Seth Nichols Johnson for producing the show here
Speaker 1: and yeah, if you want to get in touch with us,
Speaker 1: you can simply email us at contact at stuff to
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