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Speaker 1: a single episode. Thanks for listening. Hey, you welcome to
Speaker 1: Stuff to Blow Your Mind. My name is Robert.
Speaker 2: Lamb and I'm Joe McCormick and it's Saturday, so we
Speaker 2: are reaching into the vault to pull out an older
Speaker 2: episode of the show for you. This one originally published
Speaker 2: July twenty eighth, twenty twenty two, and Rob this was
Speaker 2: your interview with Antonio Padilla, author of the book Fantastic
Speaker 2: Numbers and Where to Find Them.
Speaker 1: Yeah, this is a fun chat, I think ultimately a
Speaker 1: very accessible chat, but one that will also probably break
Speaker 1: your brain a little bit. So have a good time
Speaker 1: with this one.
Speaker 3: Welcome to Stuff to Blow Your Mind, production of iHeartRadio.
Speaker 1: Hey, welcome to Stuff to Blow Your Mind. My name
Speaker 1: is Robert Lamb. My co host Joe is away from
Speaker 1: work today, so I am conducting an interview here with
Speaker 1: Professor Antonio Padella, author of the new book Fantastic Numbers
Speaker 1: and Where to Find Them, a fascinating read about big numbers,
Speaker 1: fantastic numbers, black holes, and more. This is a really
Speaker 1: fun chat. I think you're all going to enjoy it,
Speaker 1: So go ahead and jump right in with me right now. Hi, Tony,
Speaker 1: Welcome to the show.
Speaker 4: Hi, Hi Rop How you doing. Oh?
Speaker 1: Pretty good? Pretty good. I'm really excited to talk about
Speaker 1: the new book Fantastic Numbers and Where to Find Them.
Speaker 1: A wonderful read, and it's a book that gets into
Speaker 1: some pretty wonderful, mind rending cosmological territory as we'll no
Speaker 1: doubt I'll discuss here. But first I wanted to start
Speaker 1: with just a really basic sort of grounding question. I
Speaker 1: guess we encounter numbers every day, and you discuss some
Speaker 1: numbers that most of us don't encounter really every day.
Speaker 1: If we could back up a whole lot, I guess
Speaker 1: and just ponder the basics here, what exactly is a number?
Speaker 4: Well, I mean, this is this is an idea I
Speaker 4: sort of, you know, delve into in my book because
Speaker 4: of course, when you go really back into history, back
Speaker 4: to sort of the ancient Sumerians or something like that,
Speaker 4: you know, obviously they really began to use numbers to
Speaker 4: talk about, well, I've got five jars of oil, I've
Speaker 4: got five loaves of bread. But then it sort of
Speaker 4: begs the question, is that five the same five is
Speaker 4: the five that describes the jars of oil, the same
Speaker 4: five that describes the loaves of bread. And then you
Speaker 4: really sort of when you sort of make that, you
Speaker 4: disconnect the two and you start to build the idea
Speaker 4: of like what I call an emancipated number and number
Speaker 4: that's independent of the thing that it's describing. Then you're
Speaker 4: really sort of making quite quite an intellectual leap. So
Speaker 4: that for me is what is what a number is?
Speaker 4: It's kind of emancipated from the thing that it's describing.
Speaker 4: Whether such a thing really exists in a philosophical sense
Speaker 4: is a whole new debate that you can have. But yeah,
Speaker 4: that for me is the key mathematical lead that I
Speaker 4: think was made, you know, a long time ago. And yeah,
Speaker 4: it's really important.
Speaker 1: Now getting back into that sort of philosophical territory. This
Speaker 1: is one that I know that you tackle a lot.
Speaker 1: It's pretty standards for philosophical math question. But is mathematics
Speaker 1: more of a human discovery or more of a human invention?
Speaker 4: Yeah, I mean I don't think this is straightforward answer
Speaker 4: to this. Of course, this sort of you know, boils
Speaker 4: down to like a sort of alluded to whether numbers exist,
Speaker 4: whether maths exists, and as kind of I mean, I'm
Speaker 4: not a philosopher, but I know that philosophers talk about
Speaker 4: this in sort of this kind of three different angles
Speaker 4: that you can take on it. So on the one hand,
Speaker 4: you've got the Platonists who will say that numbers and
Speaker 4: mathematics is true and it exists, but it exists outside
Speaker 4: of space time as like an abstract concept. It's not
Speaker 4: something that can affect the things in space time. It
Speaker 4: can't affect the material objects that we have around us.
Speaker 4: You also have the nominalists, who says basically that numbers
Speaker 4: and maths only exist to of understand stuff. So in
Speaker 4: some sense, we talked about the five, you know, five
Speaker 4: jars of oil, the five five loaves of bread. That's
Speaker 4: the only reason that the number five exists to describe
Speaker 4: the jars of oil, to describe the loaves of bread.
Speaker 4: And then of course you've got the third sort of
Speaker 4: you know school, which is perhaps in some sense the
Speaker 4: most extreme, which just says the numers don't exist at all.
Speaker 4: They're just a useful tool that we used to describe
Speaker 4: the universe surrounds us. And I guess the analogy people
Speaker 4: use here is it's like saying, well, you could be
Speaker 4: an atheist, but you could still believe with some of
Speaker 4: the sort of moral messages that you read in the
Speaker 4: Bible or the Koran. It doesn't mean that you know,
Speaker 4: you can't be inspired by them, but you just don't
Speaker 4: have to believe in every element of it. I guess
Speaker 4: as a physicist, for me, it's kind of hard to
Speaker 4: sort of go with that fictionist idea. And yet see
Speaker 4: universe that is so amazingly described by mathematics. Now, is
Speaker 4: that something that's embedded in the universe or not? I
Speaker 4: guess it's really difficult to know. We certainly not seen
Speaker 4: any evidence that it is.
Speaker 1: And yet now your book deals with, as the title indicates,
Speaker 1: fantastic numbers. What defines for you a fantastic number? And
Speaker 1: are there categories of categorizations of numbers other than that
Speaker 1: that we need to have in our heads before we
Speaker 1: can get to the idea of what is truly fantastic.
Speaker 4: Yes. So for me and my own relationship with numbers,
Speaker 4: kind of it comes from on the one hand, you
Speaker 4: have a number, whatever that number might be, and for me,
Speaker 4: I always want to bring that sort of personality alive.
Speaker 4: There's sort of real spirit of the number, sort of
Speaker 4: to the four, and so it's always been physics for
Speaker 4: me that does that. So when you know you can
Speaker 4: have these wonderful mathematical concepts ideas like Graham's number three three,
Speaker 4: these truly bizarre and wonderful numbers, to have a wonderful
Speaker 4: place in mathematics, but then you really bring them to
Speaker 4: life when you try to sort of squeeze them into
Speaker 4: our physical world. So that, for me, is what makes
Speaker 4: a number fantastic. It's almost like what makes a number
Speaker 4: fantastic is the fantastic physics that it can lead you
Speaker 4: towards and lead you to imagine and whatever that might be.
Speaker 1: You also talk about I believe you specifically, you're talking
Speaker 1: about Graham's number pretty early on in the book, and
Speaker 1: you point out that if you if you try and
Speaker 1: actually picture it in your head, your head collapses into
Speaker 1: a black hole. And this this made me wonder, like,
Speaker 1: what are the largest numbers roughly speaking then an average
Speaker 1: person can fit into their head by one definition or another, Like,
Speaker 1: at what point does it just become this this other
Speaker 1: enterprise entirely? Yees.
Speaker 4: So it's a good question. So, I mean, it kind
Speaker 4: of depends on how how you sort of defined the question.
Speaker 4: In some sense, if you're just thinking about neurons, how
Speaker 4: many neurons have you have? You got in your in
Speaker 4: your brain as about one hundred billion neurons, and so
Speaker 4: you might say that you can use them if you
Speaker 4: manage to clear your mind of every of the thoughts
Speaker 4: to imagine one hundred billion digit number. Okay, that might
Speaker 4: not be particularly practical, it might be quite challenging for
Speaker 4: most of us. But in principle you might say that
Speaker 4: that that that would be the limit. Of course, if
Speaker 4: you then go beyond that and start to say, well,
Speaker 4: what if I could somehow get my head to find
Speaker 4: a way to actually store information store concepts more efficiently
Speaker 4: than just the usual idea of neurons firing on it off.
Speaker 4: Let's suppose that it could do that somehow. Then they
Speaker 4: numbers get get much bigger, and you start to the
Speaker 4: things that limit it are literally preventing your head collapsing
Speaker 4: to form a black hole. Because black holes what they
Speaker 4: do is that they're they're the best thing at storing information.
Speaker 4: So if you want to get something the size of
Speaker 4: a head of a human head, and you want to say,
Speaker 4: what's the best thing the size for human head that
Speaker 4: can store information, it's a black hole the size for
Speaker 4: human head. That's nothing can do it better. And so
Speaker 4: so that that places a new limit, and you can ask, well, again,
Speaker 4: what is that limit would be? Well, he's certainly way
Speaker 4: below grains now, but you're not going to get anywhere
Speaker 4: near the magnificence of Graham's number, you could probably get
Speaker 4: a digit that that's about ten to the seventy, but
Speaker 4: it's about ten to the seventy digits long, so less
Speaker 4: than a Google digits long. Having said that, you could
Speaker 4: imagine a number like a Google plex, a google plex
Speaker 4: has a Google digits. Now I've just said that you
Speaker 4: can't imagine at Google digits, not possible, but a google
Speaker 4: plex you could because what you know about a google
Speaker 4: plex is that it's a one followed by a Google zero.
Speaker 4: So you know that all the numbers that come later
Speaker 4: on as zeros, So there's not much information in that,
Speaker 4: so it doesn't cost as much as many bits. You
Speaker 4: don't have to put as many bits in your head
Speaker 4: to imagine that. So what we're really talking about now
Speaker 4: are really a random assortment of digits, a completely random
Speaker 4: assortment digits the kind that would appear in Graham's number.
Speaker 4: And I don't think you can get passed around ten
Speaker 4: to the seventy, which is a one with with seventy zeros.
Speaker 4: You couldn't get past that many digits completely randomly sort
Speaker 4: of allocated. At that point, your head's going to collapse.
Speaker 4: Into a black hole.
Speaker 1: Now now backing up to the Google and the google Plex,
Speaker 1: can't can you? Can you walk us briefly through the
Speaker 1: difference between a Google, a google Plex, and and and
Speaker 1: maybe realms beyond that? This is about the only area
Speaker 1: of fantastic numbers that I'd really heard anything about prior
Speaker 1: to reading your book.
Speaker 4: Yeah, so at Google is is it's a number, which
Speaker 4: is which is a one followed by one hundred zeros.
Speaker 4: So I think everybody would agree that sounds like quite
Speaker 4: a big number. It goes back to a physicist called
Speaker 4: Edward Kasner who was Columbia, and he was writing a
Speaker 4: popular science book, and he was trying to sort of,
Speaker 4: you know, convey that he really wanted to show how
Speaker 4: big infinity he was, and so he wanted to cou
Speaker 4: up with numbers that we all think are really big,
Speaker 4: like a one followed by one hundred zeros. And he said, well, okay,
Speaker 4: that's really small compared to infinity, right, even though something
Speaker 4: really big is actually really small compared to infinity. So
Speaker 4: he came up with this one with one hundred zeros.
Speaker 4: He wanted a name for this number, so at the
Speaker 4: time he asked his nephew who was nine years old
Speaker 4: at the time. He was called Milton Soota. He said,
Speaker 4: can you qu up with a name for this? And
Speaker 4: Milton said, well, at Google, which is an absolute stroke
Speaker 4: of genius, right, it's such a great name. And then
Speaker 4: so they wanted to then develop things further. So then
Speaker 4: they wanted an even bigger number, again building on this
Speaker 4: idea that it's nothing compared to infinity. And so he said, well, okay,
Speaker 4: I'm going to quote with the idea of a google Plex.
Speaker 4: It's going to be an even bigger number. Well how big?
Speaker 4: So Kasner then goes to Milton. He says, well, how
Speaker 4: big should it be? And Milton's like, well, it should
Speaker 4: be a one, not followed by one hundred zeros, but
Speaker 4: zeros until you get tired. But Kasner is like, you know,
Speaker 4: a sort of a you know, esteemed academic at Columbia
Speaker 4: and all that. That's just not precise enough for him.
Speaker 4: So he went with a which a much more sort
Speaker 4: of well defined idea, which is a google plex should
Speaker 4: be a one followed by a Google zeros. So a
Speaker 4: Google's already massive, that's a one followed by one hundred zeros.
Speaker 4: A google plex is a one followed by a Google zero.
Speaker 4: So it's a whole new level of big compared to
Speaker 4: what we're normally used to.
Speaker 1: And then it just it keeps building on that. Right,
Speaker 1: there's there's even like what a Google plexian is that
Speaker 1: the next level?
Speaker 4: So so yeah, I mean this is this is a
Speaker 4: really nice, nice idea. You can really now start to
Speaker 4: really build very big numbers, very very quickly using this
Speaker 4: this mathematical technique called recursion. So for example, you can
Speaker 4: develop the idea of a Google duplex or what's a
Speaker 4: Google duplex, Well, it's a one followed by a Google
Speaker 4: plex zeros, and then you could go to a Google triplex.
Speaker 4: Well you can probably guess what it's going to be.
Speaker 4: It's gonna be a one followed by a Google duplex zeros.
Speaker 4: And then a Google quadruplex is a one followed by
Speaker 4: a Google triplex zeros. And you can see each time
Speaker 4: you're growing the number just by so much, by such
Speaker 4: an unimaginably large and that's what You're not just adding
Speaker 4: a zero every time, you're kind of really ballooning the
Speaker 4: number of zeros on the end of this number in
Speaker 4: Gargangian proportions, and that's what and it's this power of
Speaker 4: mathematical recursion that allows you to do that.
Speaker 1: Now, you also talk about fantastic numbers that are I
Speaker 1: guess you would say smaller. The main example that comes
Speaker 1: to mind, you refer to this several times in the
Speaker 1: book is a number associated with Olympic sprinter Hussein Bolt.
Speaker 1: Would you tell us a little bit about this number?
Speaker 4: Yeah, yeah, so well, actually this is one of my
Speaker 4: big numbers. Actually, even though it doesn't seem that thing,
Speaker 4: it's actually it's one of my big numbers. So I
Speaker 4: can read out the number. What it is, Okay, one point.
Speaker 4: I think it's fifteen zeros eight five eight, So it's
Speaker 4: just a number just slightly north of one. So it's
Speaker 4: it doesn't seem like a big number, but in my book,
Speaker 4: I say it is a big number. And the reason
Speaker 4: is it's it measures the amount by which Usain Bolt
Speaker 4: managed to slow down time. And when he was he
Speaker 4: was running in the World Championships and I think Berlin
Speaker 4: and he set the world record. And this is due
Speaker 4: to the effects of relativity, so that when when somebody
Speaker 4: actually moves quickly, they actually slowed. Time actually slows down
Speaker 4: for them. And this is the amount by which Usain
Speaker 4: Bolt was actually able to slow down time due to
Speaker 4: the effects of Einstein's theory. And it's compared to the
Speaker 4: people in the stadium, for example. This is this was
Speaker 4: the difference that he experienced. So it's one of the
Speaker 4: one of the weird consequences of it is that you
Speaker 4: can actually it's not that you say Bolt, actually even
Speaker 4: though he slowed down time, it's not that he that
Speaker 4: he actually ran the race any quicker. He still runs
Speaker 4: the race at roughly ten meters per second. It's actually
Speaker 4: an even more strange consequence. He actually the track also
Speaker 4: shrinks for him a little bit, so so he actually
Speaker 4: runs it in less time, but in the same speed. Therefore,
Speaker 4: the track shrinks because relative to him, the track's moving.
Speaker 4: And this is another effect of relativity, one of the
Speaker 4: remarkable things. And and yes, you could perhaps argue that
Speaker 4: he didn't actually finish the race because the tracks rank
Speaker 4: so he didn't run quite one hundred meters.
Speaker 1: Wow. I was really blown away with this, because you know,
Speaker 1: you often hear the standard analogies concerning airplanes and pyramids
Speaker 1: and so forth when it comes to time dilation and
Speaker 1: so forth. But I hadn't I hadn't heard this particular
Speaker 1: example before.
Speaker 4: This is great. Yeah, I mean it's true if like
Speaker 4: taxi drivers, if you imagine a taxi driver that's driving
Speaker 4: around I don't know, any city in New York wherever,
Speaker 4: you know, sort of for forty to fifty years of
Speaker 4: their life because of that extra extra speed that they're
Speaker 4: picking up, that's going to accumulate over time, and actually
Speaker 4: they can probably leap forward in time by probably I
Speaker 4: think about a microsecond over the course of their career.
Speaker 4: It's not a lot, but it's still pretty amazing when
Speaker 4: you think about it.
Speaker 1: So they've got the knowledge, and then they have that
Speaker 1: as well, right.
Speaker 4: Oh yeah, of course, yeah exactly, not just the knowledge, Yeah,
Speaker 4: they actually get it. They actually get a little bit younger.
Speaker 1: So your book makes use of written numbers, and of
Speaker 1: course you have this wonderful YouTube series number file, and
Speaker 1: in that you benefit not only from some fantastic descriptions
Speaker 1: and pop culture tie ins as you do in the book,
Speaker 1: but you also have a lot of helpful illustrations. So
Speaker 1: I was curious, since you are a regular communicator of
Speaker 1: this topic, is it is it more challenging or in
Speaker 1: some cases almost too challenging to describe some of these
Speaker 1: numbers without the visual aids or the actual numerals to
Speaker 1: like visually present somebody with.
Speaker 4: Yeah, I think so this is where the physics comes
Speaker 4: in in some respects. Right. So, on the one hand,
Speaker 4: if you really want to describe the number, like I say,
Speaker 4: a number like Grames number, you do need those visual
Speaker 4: aids because it's not a number that you're going to
Speaker 4: sort of stumble across in any kind of normal environments. Right,
Speaker 4: It's not a number you're going to see on a
Speaker 4: price tag, at least you'd hope not. And you know,
Speaker 4: so these are you need new notation, new sort of
Speaker 4: symbolism to sort of actually even describe the number. So
Speaker 4: you've got to introduce that. There's just no getting away
Speaker 4: from it. But I guess what you can do is
Speaker 4: describe the physics associated with it, and that you can
Speaker 4: certainly do, you know, just just just with words. And
Speaker 4: you know, in the case of a number like Graham's number,
Speaker 4: you can talk about how you just can't picture it
Speaker 4: in your head because your head will will collapse to
Speaker 4: form black hole. And that's already going to make people think, wow,
Speaker 4: that numbers there's something big of something big and crazy
Speaker 4: about that number or a google plex, you know, and
Speaker 4: you can talk about a universe that's that's a google
Speaker 4: plex meters across, and then you can ask, well, if
Speaker 4: the universe is that big, if the universe is ritually
Speaker 4: that large, then it's likely that you would find multiple
Speaker 4: copies of yourself, like literally exact doppelgangers elsewhere in this
Speaker 4: ginormous universe.
Speaker 1: Yeah. I wasn't. I was I wasn't prepared for doppelgangers
Speaker 1: to enter into the scenario. So another great part about
Speaker 1: the book for me, and another thing 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 idea
Speaker 1: of the holographic truth.
Speaker 4: Yeah, so the holographic truth is. I mean, it's an idea.
Speaker 4: It's probably the most important idea I would say that's
Speaker 4: emerged from theoretical physics in the last thirty years, and
Speaker 4: it's actually mind blowing when you really think about what
Speaker 4: it pertains to it. It's this following statement that essentially
Speaker 4: one of the dimensions of space that we experience around us.
Speaker 4: So we normally talk about say three dimensions of space, well,
Speaker 4: one of them could well be an illusion. It might
Speaker 4: not exist and it's really remarkable. So what we're saying
Speaker 4: is that there are two ways in which you can
Speaker 4: describe the physics that we see around us. On the
Speaker 4: one hand, we can imagine three dimensional world with a
Speaker 4: gravitational force and the force of gravity doing its thing
Speaker 4: with planets around the Sun and so on and so forth.
Speaker 4: On the other hand, there's a completely equivalent description of
Speaker 4: the same phenomena which just uses two dimensions and no gravity.
Speaker 4: So think of it a bit like, you know, on
Speaker 4: the one hand, somebody's you know, in English, we say
Speaker 4: if we see a plate of meatballs, we call them meatballs,
Speaker 4: but a Spaniard might call them album the gas. They're
Speaker 4: both describing the same things, they're just using a different language.
Speaker 4: And that's kind of what the holographic truth says. It
Speaker 4: says that you can have a theory like a three
Speaker 4: dimensional world with gravity, and you can use that to
Speaker 4: describe all the physical phenomena you see, or use this
Speaker 4: different language which has no gravity and only requires two
Speaker 4: dimensions of space. So is it true of our world?
Speaker 4: We don't know. It's a conjecture. It's a conjecture that
Speaker 4: has sort of evidence coming from the physics of black holes.
Speaker 4: There are actually concrete examples that we know of of
Speaker 4: sort of toy universes, so not our universe, but but
Speaker 4: space times that maybe the higher dimensional they may be
Speaker 4: warped in weird and wonderful ways. And you can think
Speaker 4: about gravity in these in these simple toy universes, and
Speaker 4: you can show that there's an equivalent description in one
Speaker 4: dimension less like a holographic description, and it's called a
Speaker 4: hologram because that's essentially what holograms do. Right, If you
Speaker 4: think of a HOLOGRAMD, what have you got? You've got
Speaker 4: an image on a that's stored on a holographic plate.
Speaker 4: You know, it's just some light and dark bands on
Speaker 4: a holographic plate, a two dimensional plate. It stores a
Speaker 4: bunch of information that way, but that's just one way
Speaker 4: of looking at the information. You can decode it in
Speaker 4: a different way by shining monochromatic light through it and
Speaker 4: creating a three D image. You're not creating any new information.
Speaker 4: It's the same information, just stored either in two dimensions
Speaker 4: or three. And it seems to be that that seems
Speaker 4: to be a fundamental property of gravitation, of gravitational worlds
Speaker 4: that you can think of them as as like, as
Speaker 4: I said, three, the world with gravity, or you just
Speaker 4: forget about gravity and consider a world with one dimension
Speaker 4: less and you can describe exactly the same physical phenomena.
Speaker 1: Wow. Now here's another question that came up reading the
Speaker 1: book that I don't know if of all of our
Speaker 1: listeners necessarily would have thought of this question. I don't
Speaker 1: think some of them would have. And that comes to infinity. Infinity,
Speaker 1: Like sometimes it's easy to think of like, Okay, infinity
Speaker 1: is the If we think of it as a number,
Speaker 1: we think it's the aid 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 as? How do
Speaker 1: we classify infinity?
Speaker 4: So I love this question because the answer is that
Speaker 4: it's both not a number and lots of numbers. This
Speaker 4: is a wonderful thing about infinity. So it depends how
Speaker 4: you want to think about infinity. And I think most
Speaker 4: of us when we intuitively think about infinity, we kind
Speaker 4: of think of like I don't know, the infinite distance,
Speaker 4: you know, or infinite time, And what we're really thinking
Speaker 4: there is we're thinking of it is like a limit
Speaker 4: is something that's just just beyond our finite realm. That
Speaker 4: that that's you know, if you keep on counting forever,
Speaker 4: you know, it's kind of the at the end of that,
Speaker 4: or sort of almost beyond the end of that. Now,
Speaker 4: that's in some sense thinking of infinity as not a number,
Speaker 4: as a limit of say, you know, the whole numbers.
Speaker 4: But what cant or George cantor the you know, the
Speaker 4: great German mathematician from the late Victorian times, what he
Speaker 4: did was actually taught us how to count beyond infinity.
Speaker 4: So literally, using really smart ideas associated with something called
Speaker 4: set theory, he was able to show that actually you
Speaker 4: can have all the sort of finite numbers, and beyond
Speaker 4: that you can have infinity. But that's just one layer
Speaker 4: of infinity. You can have the infinity, which is all
Speaker 4: the whole numbers, but you can also have a different
Speaker 4: layer of infinity, which is all the numbers between zero
Speaker 4: and one. So think of the continuum of the numbers
Speaker 4: between zero and one. That's you think there's an infinite
Speaker 4: number of numbers between zero and one, but that's actually
Speaker 4: a different infinity to all the whole numbers. So you've
Speaker 4: got you know, these discrete infinities continuum infinities, and they
Speaker 4: have different sizes, and they you have many layers of
Speaker 4: what can be an infinite number. And this is what
Speaker 4: Cantor really really began to explore and develop, and he
Speaker 4: met a lot of resistance when he was doing it.
Speaker 4: He actually people thought he was crazy. He sort of
Speaker 4: fell into a lot of depression. You know, he was
Speaker 4: in battles with with someone called Chronicer, who was kind of,
Speaker 4: you know, the big guy in Berlin at the time,
Speaker 4: the elite university in Germany. He thought that Cantor was
Speaker 4: just delving into sort of witchcraft and he was a shot.
Speaker 4: He called him a charlatan, a corruptor of youth. And
Speaker 4: this really bothered Cantor and actually is quite a sad story.
Speaker 4: I mean, Cantor actually sort of really fell into into
Speaker 4: quite bad depression. Whether it's because of this or whether
Speaker 4: he was he was predisposed anyway, it's not clear. But
Speaker 4: he actually ended his days sort of very sort of
Speaker 4: emaciated in it in a sanity Toorium, essentially starving because
Speaker 4: of the effects of the First World War at the
Speaker 4: time and not having enough foods. So it's quite a
Speaker 4: tragic tale in the end, but he was certainly a
Speaker 4: tremendous mathematician, and now all his ideas are really you know,
Speaker 4: I think people acknowledge him for the genius that he was.
Speaker 1: Yeah, it of course brings to mind those like the
Speaker 1: infinity hotel discret scenarios that are used to describe infinity.
Speaker 1: I've always found those to be super interesting and and
Speaker 1: mind blowing.
Speaker 4: Yeah, I mean, that's so, that's what I mean. So, so,
Speaker 4: as I said, cancer sort of had these different layers.
Speaker 4: So you can sort of imagine the first infinity, which
Speaker 4: he called alif zero, which is he defined as the
Speaker 4: set of all of all the whole numbers, essentially all
Speaker 4: the natural numbers you know, one, two, three, four, all
Speaker 4: the way up to well infinity, all of them basically,
Speaker 4: so that that's what he called the sort of first infinity.
Speaker 4: But then you can have these high infinities, which are
Speaker 4: the you know, things like the set of the continuum,
Speaker 4: essentially the continuum between zero and one. So not just
Speaker 4: all the fractions and rational numbers, but also the irrational numbers,
Speaker 4: numbers like one over the square root of two, that
Speaker 4: kind of thing. And this is a new letter. He
Speaker 4: actually proved that they're actually that's a bigger infinity, and
Speaker 4: it's not easily obvious, but he did show it, and
Speaker 4: it's remarkable. And there's so many sort of things about infinity.
Speaker 4: There's so many paradoxes associated with them. For example, one
Speaker 4: thing you can say is you think about the number
Speaker 4: of are there more square numbers or whole numbers? And
Speaker 4: you think, well, you think naively, obviously there are more
Speaker 4: whole numbers than square numbers, because one is a square,
Speaker 4: but two isn't a square, and three isn't a square. Okay,
Speaker 4: four is So it seems that there's obviously more whole
Speaker 4: numbers than square numbers. But actually it's not true. And
Speaker 4: the reason you know that's not true because you just
Speaker 4: take a square number and you can map it to
Speaker 4: its square roots and you get the whole numbers. So
Speaker 4: the number of whole numbers is actually exactly the same
Speaker 4: as the number of square numbers. It's completely crazy. I mean,
Speaker 4: these these parrot it's the same. There are same number
Speaker 4: of even numbers as there are even in odd numbers,
Speaker 4: and there's all these one there's the same number of
Speaker 4: numbers between zero and one as there are between zero
Speaker 4: and two. There's all these paradoxes that emerge the minute
Speaker 4: you start to think about infinity. And that's why most
Speaker 4: mathematicians for a long time just stayed away from it.
Speaker 4: But Canto was brave enough to climb into this infinite
Speaker 4: heaven and explore it.
Speaker 1: Now one of the numbers that comes up a lot
Speaker 1: in your in your book, and I know you've done
Speaker 1: videos on this as well. I'm also afraid to ask
Speaker 1: about it because it just seems kind of I get
Speaker 1: confused anytime I read anything about it. And that's this
Speaker 1: idea of I'm not even sure if I'm saying it correctly?
Speaker 4: Is it?
Speaker 1: Do we say tree? Three?
Speaker 4: Yeah, that's right? Yeah, yeah, yeah, three three?
Speaker 1: So yeah, what is this?
Speaker 4: What is three? Three? So? So there's a particular game
Speaker 4: that was that was developed involving some trees. So details
Speaker 4: aren't too important, it's just but basically, you draw these
Speaker 4: little stick trees and you have some seeds, you have
Speaker 4: some lines which are kind of like the branches, and
Speaker 4: you build these trees. Right, So one of the rules
Speaker 4: of the game is is that you know, for example,
Speaker 4: you can't have a tree that's got a bit of
Speaker 4: a tree that has appeared before. So if I draw like,
Speaker 4: you know, one particular tree, then later on you can't
Speaker 4: draw a bigger tree that's got my tree stuck in
Speaker 4: it somehow, it's just not allowed. That would end the game.
Speaker 4: So there's a bunch of rules in how you draw
Speaker 4: these trees and build up this particular game, which I
Speaker 4: call the game of trees. Now, how long the game
Speaker 4: lasts depends on how many different types of seeds you have.
Speaker 4: So you could have, for example, just black seeds, okay,
Speaker 4: or maybe you could have black seeds and you're also
Speaker 4: got white seeds, or maybe you've got black seeds, white seeds,
Speaker 4: and yellow seeds. You know, there's a whole bunch of possibilities.
Speaker 4: How many seeds you play with sort of sort of
Speaker 4: changes how long the game can last. For now, if
Speaker 4: you've just got one seed, the game can only last
Speaker 4: one move. You can just write down one seed and
Speaker 4: that's it. You can't write down anything else because anything
Speaker 4: else that follows is going to contain the tree that
Speaker 4: went before. Okay, you've got two seeds, like a black
Speaker 4: and a white seed, the game can last up to
Speaker 4: you can draw up to three trees, and the game
Speaker 4: will automatically end after just three moves. It can't go
Speaker 4: beyond three moves. So you've got this this sort of sequence.
Speaker 4: So you've got one seed, you can play only one move.
Speaker 4: If you've got two seeds, you can play three moves.
Speaker 4: And so then you go to three seeds. And you
Speaker 4: might think, well, I started off with one and it
Speaker 4: went to three, and now I've got three seeds, maybe
Speaker 4: maybe I can play ten moves or something called fifteen moves.
Speaker 4: It's not gonna be some it shouldn't be anything crazy.
Speaker 4: Well it is. So this sequence just goes bang. It
Speaker 4: just goes from one. So just for one seed you
Speaker 4: get one move, two seeds you get three moves, and
Speaker 4: then three seeds you get three. Three moves is where
Speaker 4: the game will last too. And this is a number
Speaker 4: which just blows everything else. So we talked about Google
Speaker 4: and a Google plays, well, that's just nothing compared to
Speaker 4: three three. Talk about Graham's number, which will collapse your
Speaker 4: head's form black hole. That's nothing compared to three three
Speaker 4: three three is just it. It's impossible. I mean, I
Speaker 4: actually think it is impossible to imagine how ridiculously big
Speaker 4: this number is. And it's just so mundane. Where it
Speaker 4: comes from? Is this game so if he starts off you.
Speaker 4: So so you're playing this game with two seeds. This
Speaker 4: game keeps ending after three moves, and if somebody comes
Speaker 4: along and adds a different colored seed, and you're like, okay,
Speaker 4: how is how long can the game last? Now? And
Speaker 4: somebody says tree three and this is three three. It's
Speaker 4: just a number that's actually too big for the universe.
Speaker 4: It just whow where did that leap come from? Leaps
Speaker 4: should not be that big. But that's so that's in
Speaker 4: a essence what tree three is, and it is too
Speaker 4: big for the universe. So one of the things I
Speaker 4: worked out was suppose you're playing this game involving these trees.
Speaker 4: So you're writing drawing these trees, right, so you play one,
Speaker 4: go draw a tree. Play next, go draw a tree,
Speaker 4: and so on. You've got three seed, three different colors
Speaker 4: of seeds. So we know the limit of the game
Speaker 4: is three three moves treating three three different trees in
Speaker 4: the forest. How long are you going? Could you finish
Speaker 4: the game? And one other thing I imagine is you
Speaker 4: know you're playing this game at high speed, so you're
Speaker 4: playing it as fast as space time will allow. So
Speaker 4: you literally if you play any faster space time will
Speaker 4: break due to quantum effects. So you play it super
Speaker 4: super fast, and so you play again, you play again,
Speaker 4: You'll play it through a lifetime. You'll get nowhere near
Speaker 4: tree three. After you die, and maybe you replace yourself
Speaker 4: with some artificial intelligence. You've got two art Ai machines
Speaker 4: playing against each other, you know, powered by the light
Speaker 4: of the sun. They'll keep playing the game at this
Speaker 4: crazy pace, and they keep going, and they keep going.
Speaker 4: The sun gets bigger, you know, it goes to a
Speaker 4: red giants. All these things happen. Eventually it falls back
Speaker 4: forms a white dwarf. Over many billions of years, and
Speaker 4: still this these two ais are still playing the game
Speaker 4: because they have got nowhere near tree three, and they're
Speaker 4: playing at breakneck speed as well, and so eventually they
Speaker 4: lose power. They can't get any power because the sun dies, right,
Speaker 4: so they need to somehow develop some new technology which
Speaker 4: gets energy from I don't know, the cosmic microwave background radiation.
Speaker 4: And the game goes on, and the game goes on,
Speaker 4: and the game goes on. In fact, the game will
Speaker 4: go on way beyond the sort of heat death of
Speaker 4: the universe, and still you will not get to the
Speaker 4: end of tree three. And actually there's a phenomena called
Speaker 4: Puancore recurrence, which says that in any system, in any
Speaker 4: finite system, you'll eventually get back to where you started.
Speaker 4: And that applies to our universe too. So you can
Speaker 4: imagine a pack of cards. You know, if you shuffle
Speaker 4: a pack of cards enough times, you'll a lot of times,
Speaker 4: but enough times you'll eventually get back to the point
Speaker 4: where all the cards are in order. It'll take a
Speaker 4: long time, but it will happen eventually. It's the same
Speaker 4: with our universe. You shuffle the universe enough times, you
Speaker 4: allow it to evolve for long enough, eventually you'll get
Speaker 4: back to where it started. It will reset. And that
Speaker 4: reset time for our universe is actually shorter than the
Speaker 4: time it would take to play this game of trees
Speaker 4: all the way up to three three moves, playing as
Speaker 4: fast as you possibly can. And so even even if
Speaker 4: you could do it, even if you could live past
Speaker 4: all these you know, Gargangian timescales, the universe is just
Speaker 4: going to go, nah mat, game over. We're resetting. You
Speaker 4: ain't going to get to the game which you ain't
Speaker 4: going to end this game. So three three is actually
Speaker 4: a number that's that's actually too big for the universe.
Speaker 4: That's how big it is.
Speaker 1: It's just so astounding that 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 notmes, like when you're
Speaker 1: talking about the Googles and the Google Plexus, it's easy
Speaker 1: to think, well, those those big numbers live out there
Speaker 1: like they're like in the deep water. But then this
Speaker 1: seems to illustrate that now the deep water is is
Speaker 1: far closer than you think.
Speaker 4: And it's not I wouldn't even call it deep water.
Speaker 4: It's it's water. That's you know, you're sort of like, yeah,
Speaker 4: you're just sort of tiptoeing across the you know, through
Speaker 4: the shallows and there, and then bang it just gives
Speaker 4: away underneath you. And there's just it's it's bottomless as
Speaker 4: far as you're concerned. You know, I wouldn't even got
Speaker 4: a d watter is beyond deep. It's too it's too
Speaker 4: deep for the universe.
Speaker 1: Wow. So ultimately, what do fantastic numbers reveal about the cosmos?
Speaker 1: Like what is the I guess what is the lesson
Speaker 1: of big numbers? Fantastic numbers et cetera.
Speaker 4: So for me, I think all the ideas that I
Speaker 4: talk about in the context of the big numbers in
Speaker 4: the book, they all come back to the same thing
Speaker 4: which we've talked about, which is the holographic truth, the
Speaker 4: idea that a lot of the ideas associated with black
Speaker 4: holes and and how much information you can fit inside
Speaker 4: a black hole. Where that information stored, for example, is
Speaker 4: it stored inside the black hole or is it stored
Speaker 4: on the edge of the black hole? And these are
Speaker 4: ideas which which which leads you to to to the
Speaker 4: to the holographic truth, to the idea that actually, maybe
Speaker 4: the information in our world isn't stored inside the world.
Speaker 4: Maybe it's stored on the boundary of the world, at
Speaker 4: the edge, on the walls that surround it. And in
Speaker 4: that sense, that's why it's it's holographic. All the ideas,
Speaker 4: all the limits that we're talking about, you know, counting
Speaker 4: how much information you can store in a head, you know,
Speaker 4: and when it's going to turn into a black hole,
Speaker 4: you know, counting how long it takes for our universe
Speaker 4: to reset it up. All these ideas come back to
Speaker 4: the question of how our universe stores its information. Does
Speaker 4: it store it inside, and if so, how does it
Speaker 4: store it? Well, actually, no, it turns out it's seem
Speaker 4: like it stores it on the edge of space, and
Speaker 4: that allows you to count how much information there is
Speaker 4: in that space and how many different ways you can
Speaker 4: combine things. But it all comes back to that holographic truth.
Speaker 1: I have to ask about this because I, again most
Speaker 1: I'm not as versed in mathematics. There's a lot of
Speaker 1: people out there, and one of the things that I
Speaker 1: kept thinking about reading the book was just a one
Speaker 1: quick joke from the season one episode of the British
Speaker 1: comedy look Around You, in which the narrator the episode
Speaker 1: is about math, and the narrator tells us that the
Speaker 1: largest known number is around forty five million, but the
Speaker 1: larger numb umbers might exist, and they like speculate forty
Speaker 1: five million in one could be an number, and you know,
Speaker 1: of course that's absurd, and that's absurd ast humor. But
Speaker 1: there's something about that that seems to sort of ring
Speaker 1: true with a lot of these these concepts, and I
Speaker 1: was wondering what you thought about the role of absurdity
Speaker 1: in contemplating big numbers.
Speaker 4: Yeah, absolutely, no, I really do think so. When you
Speaker 4: think of something like three three, at least within our universe,
Speaker 4: you can't fit it in. It cannot fit in. There's
Speaker 4: nothing that could, you know, you could describe, because it's
Speaker 4: just too big for anything that we can talk about
Speaker 4: in our universe. Now, you might imagine other universes which
Speaker 4: could accommodate it, and in a you know, a sort
Speaker 4: of multiverse scenario, like maybe you get from something like
Speaker 4: string theory, could you get universes that can contain tree three.
Speaker 4: Well maybe we don't know, right, we don't know enough
Speaker 4: about about the multiverse of string theory, but it's not
Speaker 4: inconceivable potentially so, but certainly in our world you can't.
Speaker 4: It's interesting. One of the things I did a video
Speaker 4: quite quite recently actually about the biggest number that nobody
Speaker 4: will ever think of. And I did these sort of
Speaker 4: quite a bunch of estimates based on a bunch of
Speaker 4: dubious sort of you know, sort of assumptions, which I
Speaker 4: acknowledge with quite jubious, But I think I came up
Speaker 4: with an estimate that if you think of a random
Speaker 4: seventy three digit number, also something of that order, then
Speaker 4: probably nobody's going to ever ever think of it. Other
Speaker 4: than you. I mean, you know, so, I'm not saying
Speaker 4: I just think of a one followed by seventy two zero.
Speaker 4: It's clearly not something like that, but just completely random,
Speaker 4: random seventy two to seventy three digit numbers something like that.
Speaker 4: Chances are nobody in the history of humanity, either before
Speaker 4: or to come, we'll ever think of that number. And
Speaker 4: it's kind of that's kind of mind blowing. I think
Speaker 4: it's kind of yours. Just think of it, and that's
Speaker 4: yours forever. So just everybody should just write down a
Speaker 4: seventy three digit numb and name after themselves.
Speaker 1: Well that's wonderful. Well, Tony, thanks for taking time out
Speaker 1: of your day to chat with us. I want to
Speaker 1: make sure we're hitting all the plugs here. The book
Speaker 1: Which Which is? Which? Is out? I believe it's out now? Correct?
Speaker 4: Yeah? Yeah, it's actually released today in the US. I
Speaker 4: probably should say today, should I?
Speaker 1: I guess it'll be. It will have been released two
Speaker 1: days ago when we published this, So yeah, it's it's out.
Speaker 1: It's fantastic numbers and where to find them? And then
Speaker 1: the YouTube series is number file, correct?
Speaker 4: Yeah? So I appear on number File. There's another channel.
Speaker 4: I appear on which is more physics based called sixty Symbols.
Speaker 4: So they're both made by by Brady Harran. And yeah,
Speaker 4: so I appear regularly on both of those. So it's
Speaker 4: a lot of fun. But yeah, it's I hope people
Speaker 4: enjoy enjoy the book. It's and I just don't think
Speaker 4: too recklessly about Grahams number because you going to have.
Speaker 1: Yeah, we don't want anybody's heads to collapse into black holes.
Speaker 4: Absolutely.
Speaker 1: All right, Well, thanks for coming on the show. I
Speaker 1: hope you have a great day, Thanks Chub. All right, well,
Speaker 1: thanks once again to Tony for taking time out of
Speaker 1: his day to chat with me here. The book again
Speaker 1: is Fantastic Numbers and Where to Find Them. Highly recommend
Speaker 1: it for anyone who is at all intrigued by what
Speaker 1: we were talking about here today. As always, if you
Speaker 1: want to reach out to us and ask any questions,
Speaker 1: share your relationship with Fantastic Numbers, well, you can find
Speaker 1: us in a number of ways. Let's see if you
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Speaker 1: There's also the Stuff to Blow Your Mind discussion module
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 content Act. It's Stuff
Speaker 1: to Blow your Mind dot com.
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