July 28, 2022 • 38 mins
In this episode of Stuff to Blow Your Mind, Robert chats with theoretical physicist and cosmologist Professor Antonio Padilla, author of the new book "Fantastic Numbers and Where to Find Them." Strap in for big numbers, fantastic numbers, black holes and more.
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Speaker 1 (00:03):
Welcome to stot to Blow Your Mind, production of My
Heart Radio. Hey, welcome to Stuff to Blow Your Mind.
My name is Robert Lamb. My co host Joe is
away from work today, so I am conducting an interview
here with Professor Antonio Padella, author of the new book
(00:28):
Fantastic Numbers and Where to Find Them, a fascinating read
about big numbers, fantastic numbers, black holes, and more. This
is a really fun chat. I think you're all going
to enjoy it, So go ahead and jump right in
with me right now. Hi, Tony, Welcome to the show. Hi,
(00:49):
Hi up, how you doing? Oh? Pretty good? Pretty good? Um?
Really excited to talk about the new book Fantastic Numbers
and Where to Find Them? A wonderful read, and it's
a book that get into some pretty wonderful mind rending
cosmological territory as well. Note I'll discuss here, but first
I wanted to start with just a really basic sort
(01:10):
of grounding question. I guess we encounter numbers every day,
and you discuss some numbers that most of us don't
encounter really every day. If we could back up a
whole lot, I guess, and just ponder the basics here,
what exactly is a number? Well, I mean, this is
this is an idea I sort of, you know, delve
into him in my book, because of course, when you
go really back in into history, back to sort of
(01:31):
the ancient Sunarians or something like that, you know, obviously
they really began to use numbers to talk about And
while I've got five jars of oil, I've got five
loaves of bread. But then it sort of begs the question,
is that five the same five is the five that
describes the jars of oil, the same five that describes
the loaves of bread. And then you really sort of
(01:52):
when you sort of make that disconnected too, and you
start to build the idea of like what I call
an emancipated number, but it's independent and of the thing
that it's describing, then you're really sort of making quite
quite an intellectual leap. So that, for me is what
is what a number is. It's kind of emancipated from
from the thing that it's describing. Whether such a thing
really exists in a philosophical sense, is a whole new
(02:15):
debate that you can have. But yeah, that for me
is is the key mathematical leap that I think was
made you know, a long time ago, and and yeah,
it's really important now getting back into that sort of
philosophical territory. This is one that I know that you
you tackle a lot. Uh, it's pretty standard sort of
philosophical math question. But is mathematics more of a human
(02:37):
discovery or more of a human invention? Yeah, I mean
I don't think there's a straightforward answer answer to this.
Of course. This this sort of you know, boils down
to like a sort of alluded to whether numbers exist,
whether whether maths exist, and it's kind of I mean,
I'm not a philosopher, but but but I know that
philosophers talk about this, and sort of this kind of
three different angles that you can take on it. So
(02:58):
so on the one hand, you've got the platonists who
will say that numbers and mathematics is true and it exists,
but it exists outside of space time. Is as like
an abstract concept. It's not something that can affect the
things in space time. It can't affect the material objects
that we have arounders. You also have the nominalists, who
(03:19):
says basically that numbers and maths only exist to sort
of understand stuff. So so in some sense, we talked
about the five, you know, five jars of oil, the
five five loaves of bread. That's the only reason that
the number five exists to describe the jars of oil,
to describe the loaves of bread. And then, of course
you've got the third sort of you know school, which
(03:41):
is perhaps in some sense the most extreme, which just
says the numbers that exist at all, that they're just
a useful tool, that that that we used to describe
the universe arounders. And I guess the analogy people use
here is it's like saying, well, you could be an atheist,
but you can still believe with some of the sort
of moral messages that you read in the Bible or
the Koran. It doesn't mean that, you know, you can't
(04:02):
be inspired by them, but you just don't have to
believe in every element of it. And I guess as
a physicist, for me, it's kind of hard to sort
of go with that fictionist idea and yet see a
universe that is so amazingly described by mathematics. Now, is
that something that's embedded in the universe or not. I
guess it's really difficult to know. We've certainly not seen
(04:24):
any evidence that it is. And yet now your book
deals with, as the title indicates, fantastic numbers. Uh, what
what defines for you a fantastic number? And are there
categories of categorizations of numbers other than that that we
need to have in our heads before we can get
to the idea of what it's truly fantastic. Yes, it's
(04:44):
so for me and my own relationship with numbers kind
of and it comes from, on the one hand, you
have a number, whatever that number might be, and if me,
I always want to bring that sort of personality alive,
the sort of real spirit of the number, sort of
to the four. And so it's always been physics for
me that that does that. So when you know you
can have these wonderful mathematical concepts ideas like Graham's number
(05:07):
three three, these truly bizarre and wonderful numbers to have
a wonderful place in mathematics, but then you really bring
them to life when you try to sort of squeeze
them into our physical world. So that for me is
what what makes a number fantastic. It's almost like, what
makes a number fantastic is the fantastic physics that it
can lead you towards and lead you to imagine and
whatever that might be. You also talk about I believe
(05:29):
specifically you're talking about Graham's number pretty early on in
the book, and you point out that if you if
you try and actually picture it in your head, your
head collapses into a black hole. And this this made
me wonder, like, what what are the largest numbers roughly
speaking then an average person can fit into their head
by one definition or another, Like, at what point does
it just become this this other enterprise entirely? Yes, So
(05:52):
it's a good question. So, I mean, it kind of
depends on how how you sort of define the question.
In some sense, we're just thinking about your rs. How
many neurons have you have you got in your in
your brain as about a hundred billion neurons, and so
you might say that you can use them if you
managed to clear your mind of every other thought to
imagine a hundred billion digit number. Okay, that might not
(06:16):
be particularly practical, it might be quite challenging for most
of us. But but in principle you might say that
that that that would be the limit. And of course,
if you then go beyond that and start to say, well,
what if I could somehow get my head to find
a way to actually store information store concepts more efficiently
than just the usual idea of neurons firing on and off.
(06:37):
Let's suppose that it could do that somehow. Then then
the numbers get get much bigger, and you start to
the things that limited are literally preventing your head collapsing
to form a black hole. Because black holes what they
do is that they're they're the best thing at storing information.
So if you want to get something the size of
a head of a human head, and you want to say,
(06:58):
what's the best thing the size of human head that
can store information, it's a black hole the size for
human head that that's the nothing can do it better,
and so so so that that places a new limit,
and you can ask, well, again, what what is that
limit would be? But he's certainly way below grains. No,
but you're not gonna get anywhere near the magnificence of
Graham's number. And you could probably get a digit that's
(07:21):
that's about tens of the seventy and number that's about
ten to the seventy digits long, and so less than
a Google digits long. Having said that, you could imagine
a number like a google plex. A google plex has
a Google digits. Now I've just said that you can't
imagine a Google Digit's not possible, but a google plex
(07:41):
you could, because what you know about a google plex
is that it's a one followed by a Google zero.
So you know that all the numbers that come later
on as zeros, and so there's not much information in that.
So it doesn't cost as much as many bits. You
don't have to put as many bits in your head
to imagine that. So what we're really talking about now
(08:02):
are really a random assortment of digits. Are completely random
assortment digits the kind that would appear in Graham's number.
And I don't think you can get passed around tens
of the seventy, which is a one with seventies zero.
You couldn't get past that many digits completely randomly sort
of allocated. At that point, your head is going to
collapse into a black hole. Now, now backing up to
(08:23):
the Google and the google plex. Can't can you? Can
you walk as briefly through the difference between a Google
and google plex, and and and maybe realms beyond that?
This is this is about the only area of fantastic
numbers that i'd I'd really heard anything about prior to
reading your book. Yes, So, so Google is is um
it's a number which is which is a one followed
(08:45):
by a hundred zero. So I think everybody would agree
that sounds like quite a big number. And it goes
back to to a physicist called Edward Kasner who is Columbia,
and he was writing a popular science book and he
was trying to sort of know convey He really wanted
to show how big infinity he was, and so he
wanted to quote with numbers that we all think are
(09:05):
really big, like a one followed by a hundred zeros.
And he said, well, okay, that's really small compared to infinity, right,
even though something really big is actually really small compared
to infinity. So he came up with this one with
a hundred zeros. He wanted a name for this number,
so at the time he asked his nephew who was
(09:26):
nine years old at the time. He was called Milton Serata.
He said, can you come up with a name for this?
And and Milton said, well, a Google, which is an
absolute stroke of genius, right, It's such a great name.
And and then so they wanted to then develop things further,
so then they wanted an even bigger number, again building
on this idea that it's nothing compared to infinity and
(09:48):
and so so he said, well, okay, I'm gonna quote
with the idea of a Google plex. It's going to
be an even bigger number. Well how big? So Kasna
then goes to to Milton. He says, well, how big
should it be? And Milton's like, well, it should be
a one, not followed by a hundred zeros, but zeros
until you get tired. But Kasner is like, you know,
(10:09):
a sort of you know, steamed academic at Columbia and
all that. That's just not precise enough for him. So
he went with it which are much more sort of
well defined idea, which is a google plex should be
a one followed by a Google zeros. So a Google
is already massive. That's a one followed by a hundred zeros.
A google plex is a one followed by a Google zero.
(10:32):
So it's a whole new level of big compared to
what we normally used to. And then it just it
keeps building on that. Right, there's there's even like what
a google plexian is that the next level. So so yeah,
I mean this is this is a really nice, nice idea.
You can really now start to to really build very
big numbers, very very quickly using this this mathematical technique
called recursion. So for example, you can develop the idea
(10:55):
of a Google duplex. What a Google duplex, Well, it's
a one followed by a Google plex zeros. And then
you can go to a Google triplex. Well, you can
probably guess what it's gonna be. It's gonna be a
one followed by a Google duplex zeros. And then a
Google quadruplex is a one followed by a Google triplex zeros.
(11:15):
And you can see each time you're growing the number
just by so much, by such an unimaginably large amounts,
And that's what. You're not just adding zero every time,
You're kind of really ballooning the number of zeros on
the end of this number in Gargangian proportions. And that's
what and it's this power of mathematical riccasion that allows
(11:37):
you to do that. Now you also talk about fantastic
numbers that are I guess you would say smaller than
the main example that comes to mind. You refer to
this several times in the book is a number associated
with Olympic sprinter Hussain Bolt. Would you tell us a
(12:00):
little bit about this number? Yeah, yeah, so so well,
actually this is one of my big numbers. Actually, even
though it doesn't seem that it's it's actually it's one
of my big numbers. So I can read out the number.
What it is one point, I think it's fifteen zero
eight five eight, So it's just a number just slightly
(12:22):
north of one. So it's it doesn't seem like a
big number. But but in my book, I say it
is a big number. And the reason is it's it
measures the amount by which Usain Bolt managed to slow
down time. And when he was he was running in
the World Championships and I think Berlin Um and he
set this the world record. And this is due to
(12:43):
the effects of relativity, so that when when somebody actually
moves quickly, they actually slow time actually slows down for them.
And actually this is the amount by which Usain Bolt
was actually able to slow down time due to the
effects of Einstein's theory. And it's compared to the people
in the stadium, for example, this is this was the
(13:05):
difference that that he experienced. So it's one of the
one of the weird consequences of it is that you
can actually it's not that you say, but actually, even
though he slowed down time, it's not that he that
he actually ran the race any quicker. He still runs
the race at roughly ten ms per second. It's actually
an even more strange consequence. He actually the track also
(13:25):
shrinks for him a little bit, so so he actually
runs it in less time, but in the same speed. Therefore,
the track shrinks because the relative to him, the tracks moving,
And this is another effect of relativity, one of the
remarkable things. And and yes, you could perhaps argue that
he didn't actually finish the race because the tracks rank
so he didn't run quite a hundred Wow. I was
(13:47):
really blown away with this, because you know, you often
hear the standard analogies concerning airplanes and pyramids and so
forth when it comes to time dilation and so forth.
But but I hadn't. I hadn't heard this particular example
for this is great. Yeah, I mean, it's true. It's true.
If like taxi drivers, if you imagine a taxi driver
that's driving around I don't know any city in New
York wherever, you know, sort of forty fifty years of
(14:10):
their life because of that extra extra speed that they're
picking up. That's going to accumulate over time. And actually
they can probably leap forward in time by probably I
think about a micro second over the course of their career.
It's not a lot, but it's still fairly amazing when
you think about it. So they've got the knowledge, and
then they have that as well, right, Oh yeah, of course,
yeah exactly, not just the knowledge, Yeah, they actually got it.
(14:30):
They actually get a little bit younger. So your book
makes makes use of written numbers, um. And and then
of course you have this wonderful YouTube series number file,
and in that you benefit not only from some fantastic
descriptions and pop culture tie ends as you do in
the book, but you also have a lot of helpful illustrations.
Uh So I was I was curious since you are
(14:51):
a regular communicator of of this this topic. Um is
it is? It? Is? It? Is? It more challenging or
in some cases almost too challenging to describe some of
these numbers without the visual aids or the actual numerals
to like visually present somebody with. Yeah, I think you
so this is where the physics comes in in some respects. Right.
So on the one hand, if you really want to
(15:13):
describe the number, like I said, a number like Grams number,
you do need those visual aids because it's not a
number that you're going to sort of stumble across in
any kind of normal environments. Right, It's not a number
you're going to see on on a price tag, at
least you'd hope not. And you know, so these are
you need new notation, new sort of symbolism to to
(15:33):
sort of actually even describe the number. So you've got
to introduce that. There's just no getting away from it.
But I guess what you can do is described the
physics associated with it, and and that you can certainly do,
you know, just just just with words. And you know,
in the case of a number like Grahams number, you
can talk about how you just can't picture in your
(15:53):
head because your head will will collapse to for a
black hole. And that's already going to make people think, wow,
that number. There's something big of something big and crazy
about that number. Or a google plex, you know, when
you can talk about a universe that's that's a google
plex meters across. And then you can ask, well, if
the universe is that big, if the universe is literally
that large, then it's likely that you would find multiple
(16:16):
copies of yourself, like literally exact apple Gangers elsewhere in
this ginormous universe. Yeah, I wasn't. I was. I wasn't
prepared for dopple gangers to enter into the scenario. So
there was there was another great part about the book
for me um and another thing that that comes up
in the book that I was very intrigued by. I
was wondering if you might talk about is the the
(16:37):
idea of the of the holographic truth? Yes, so the
holographic truth is I mean, it's an idea. It's probably
the most important idea I would say that's emerged from
theoretical physics in in the last thirty years. And it's
it's actually mind blowing when you really think about what
it pertains to. It's it's it's this following statement that
essentially one of the dimensions of space that we experience
(17:00):
around it. So we normally talk about say three dimensions
of space, Well one of them could well be an illusion.
It might not exist, and it's really remarkable. So what
we're saying is that there are two ways in which
you can describe the physics that we see around as
On the one hand, we can imagine three dimensional world
(17:20):
with a gravitational force and the force of gravity doing
its thing, with planets around the Sun and so on
and so forth. On the other hand, there's a completely
equivalent description of the same phenomena which just uses two
dimensions and no gravity. So think of it a bit like,
you know, on the one hand, somebody's you know, in English,
(17:41):
we say if we see a place of meat balls,
we call them meat balls, but a Spaniard might call
them album the gas. They're both describing the same things,
they're just using a different language. And that's kind of
what what the holographic truth says. It says that you
can have a theory like a three dimensional world with gravity,
and you can use that to describe all the physical
phenomena you see, or use this different language which has
(18:05):
no gravity and only requires two dimensions of space. So
is it true of our world? We don't know. It's
a conjecture. It's a conjecture that has sort of evidence
coming from from the physics of black holes. There are
actually concrete examples that we know of of sort of
toy universe is so not our universe, but but but
(18:26):
space times that may be that the higher dimensional there
may be warped in weird and wonderful ways. And you
can think about gravity in these in these simple toy universes,
and you can show that there's an equivalent description in
one dimension less like a holographic description, and it's called
a hologram because that's essentially what what holograms do? Right?
If you think of a hologram, what have you got?
(18:46):
You've got an image on a that's stored on a
holographic plate. You know, it's just some light and dark
bands on a holographic plate, a two dimensional plate. It's
stores a bunch of information that way, But that's just
one way of looking at the information. You can decode
it in a different way by shining monochromatic light through
it and creating a three damage. You're not creating any
(19:08):
new information. It's the same information, just stored either in
two dimensions or three. And it's that seems to be
that that seems to be a fundamental property of gravitation,
of gravitational worlds that you can think of them as
as like as I said, a three D world with gravity,
or you just forget about gravity and consider a world
with one dimension or less and you can describe exactly
(19:31):
the same physical phenomena. Now here's another question that that
that came came up reading the book that that I
don't know if of all our listeners are necessarily would
have thought of this question. I think some of them
would have. And that comes to infinity um infinity, Like
sometimes it's easy to think of like, okay, infinity is
the it's it's if we think of it as a number.
(19:52):
We think it's the eight on its side representing infinity.
Is infinity a number? And if it's not a number,
like what do we think of it at? How do
we classif high infinity? So I love this question because
the answer is that it's both not a number and
lots of numbers. This is the wonderful thing about infinity.
So it depends how you want to think about infinity.
And I think most of us when we intuitively think
(20:14):
about infinity, we kind of think of like I don't
know the infinite distance, you know, or infinite time, And
what we're really thinking there is we're thinking of it
is like a limit is something that's just just beyond
our finite realm. That that that's you know, if you
keep on counting forever, you know, it's kind of the
at the end of that, you're sort of almost beyond
the end of that. Now, that's in some sense thinking
(20:36):
of infinity as not a number, as a limit of saying,
you know, the whole numbers. But what Cancelor Judge cancel that,
you know, the great German mathematician from the late Victorian times,
what what what he did was actually taught us how
to count beyond infinity. So literally, using really smart ideas
(20:57):
associated with something called set theory, he was able to
show that actually you can have all the sort of
finite numbers, and beyond that you can have infinity. But
there's that's just one layer of infinity. You can have
the infinity, which is all the whole numbers, but you
can also have a different layer of infinity, which is
all the numbers between zero and one. So think of
(21:18):
the continuum of the numbers between zero and one. That's
you think there's an infinite number of numbers between zero
and one, but that's actually a different infinity to all
the whole numbers. So you've got, you know, these discreet infinities,
continuum infinities, and they they have different sizes, and they
have you have many layers of what can be an
(21:38):
infinite number. And this is what Cancel really really began
to explore and and and develop. And he met a
lot of resistance when he was doing it. He actually
people thought he was crazy. He sort of fell into
a lot of depression. Um, you know, he was in
battles with with someone called Chronicker, who was kind of,
you know, the big guy in Berlin at the time,
the elite university, and Jereman. He he thought that Cancer
(22:01):
was just delving into sort of witchcraft and he was
a shot. He called him as Charlatan, a corruptor of youth.
And this really bothered Cancer and actually quite a sad story.
I mean, Cancer actually sort of really fell into into
quite a bad depression. Whether it's because of this or
whether he was he was predisposed anyway, it's not clear.
But he actually ended his day sort of very sort
(22:21):
of emaciated in a in a sanatorium, essentially starving because
of the effects of the First World War at the time.
And I'm not having enough food, so it's quite a
tragic tale in the end, but he was certainly a
tremendous mathematician, and now all his ideas are really you know,
I think people acknowledging for the genius that he was. Yeah,
of course brings to mind those um like the infinity
(22:44):
hotel discrete scenarios that are used to describe infinity. I've
always found those to be super interesting and and men
mind blowing. Yeah, I mean that's so, that's that's what
I mean. So, so, as I said, cancer sort of
had these these different layers. So you can sort of
imagine the first infinity, which he called alve zero, which
is he defined as the set of all of all
(23:05):
the whole numbers, essentially all the natural numbers you know,
one to three for all the way up to well infinity,
all of them basically, so that that's what he called
the sort of first infinity. But then you can have
these higher infinities, which are the you know, things like
the the the set of the continuum, essentially the continuum
between zero and one, so not just all the all
(23:25):
the fractions and irrational numbers, but also the irrational numbers
numbers like one over the square out of tow that
kind of thing um and and this is a new
letter that he actually proved that they're actually got a
bigger infinity. And it's not immedially obvious, but but he
did show it, and it's it's it's remarkable, and and
there's so many sort of things about infinity. There's so
(23:46):
many paradoxes associated with them. For example, one one thing
you can say is you think about the number of
the more square numbers or whole numbers, and you think, well,
you think naively, obviously there are more whole numbers and
square numbers because one is a square, but but two
isn't a square, and three isn't a square, Okay, four
is So it seems that there's obviously more whole numbers
(24:08):
than square numbers. But actually it's not true. And the
reason you know that's not true because you just take
a square number and you can map it to its
square roots and you get the whole numbers. So so
the number of whole numbers is actually exactly the same
as the number of square numbers. It's completely crazy. And
these these parents and it's the same. There are the
same number of even numbers, there are even at odd numbers,
(24:29):
and there's all these one there's the same number of
numbers between zero and one as there are between zero
and two. There's all these paradoxes that are merged just
the minute you start to think about infinity. And that's
why most mathematicians for a long time just stayed away
from it. But Cancer was brave enough to climb into
this infinite heaven and explore it. Now. One of the
(24:54):
numbers that that comes up a lot in your in
your book, and you've done your videos on this as well. Um,
I'm also afraid to ask about it because it just
seems kind of I get confused anytime I read anything
about it. And that's this idea of and I'm not
even sure what I'm saying it correctly? Is it? Do
we say tree three? Yeah? That's right, Yeah, yeah, three three?
(25:14):
So yeah, what is this? What is tree three? So?
So there's a particular game that was that was developed
involving some trees, right, So, so the details aren't too important,
it's just but basically you draw these little stick trees
and you have some seeds, you have some lines which
are kind of like the branches, and you and you
build these trees. Right, So, so what are the rules
(25:36):
of the game is is that you know, for example,
you can't have a tree that's got a bit of
a tree that that's has appeared before. So so if
I draw, like, you know, one particular tree, then later
on you can't draw a bigger tree that's got my
tree stuck in it somehow. It's it's just not allowed.
That would end the game. And so there's a bunch
of rules in how you draw these trees and build
up this this particular game, which I call the game
(25:56):
of trees. Now, how long the game depends on how
many different types of seeds you have. So you could have,
for example, just black seeds, okay, or maybe you can
have black seeds and you're also got white seeds, or
maybe you've got black seats, white seeds and yellow seeds.
You know, there's a there's whole bunch of possibilities. How
(26:16):
many seeds you play with sort of of changes how
long the game can last. For now, if you've just
got one seed, the game can only last one move.
You can just write down one seed and that's it.
You can't write down anything else because anything else that
follow is going to contain a tree that went before. Okay,
you've got two seeds, like it's like a black and
(26:37):
a white seed. The game can last up to you
can draw up to three trees and the game will
automatically and after just three moves, it can't go beyond
three moves. So you've got this this sort of sequence.
So we've got one seed, you can play only one move.
You've got two seeds, you can play three moves. And
so then you go to three seeds, and you might think, well,
(26:59):
I can I start for one and it went to
three and I've got three seeds, Maybe maybe I can
play ten moves or something called fifteen moves something. It's
not gonna be some it shouldn't be anything crazy. Well
it is. So this sequence just goes bang. It just
goes from one. So just from one seed you get
one move, two seeds, you get three moves, and then
three seeds you get tree. Three moves is where the
(27:20):
game will last too. And this is a number which
just blows everything else. So we talked about Google and
a Google place, Well that's just nothing compared to three three.
Talk about Graham's number which will collapse, your heads fall
and black hole, that's nothing compared to three three Tree
three is just it's it's impossible. I mean, I actually
think it is impossible to imagine how ridiculously big this
(27:42):
number is. And it's just so mundane. Where it comes
from is this game. He starts off you. So you're
playing this game with two seeds, as game keeps ending
after three moves, and then somebody comes along and adds
a different color of seed, and you're like, okay, how
long is the how long can the game last? Now?
And somebody says tree three, and this is tree three,
just and not but it's actually too big for the universe.
(28:03):
Just whoa where did that leap come from? The leap
should not be that big, but that that's that's so
that's in essence what what tree three is. And it
is too big for the universe. So one of the
things I worked out was suppose you're playing this game
involving these trees. So you're writing drawing these trees, right,
So you play one, go draw a tree. Play next,
go draw a tree. And so you've got three seeds,
(28:24):
three different colors of seeds. So we know the limit
of the game is tree three moves treating three three
different trees in the forest. How long could you finish
the game. And one of the thing I imagine is,
you know, you're playing this game at high speed, so
you're playing it as fast as spacetime will allow. So
you literally, if you play any fastest, spacetime will break
(28:45):
due to quantum effects. So you play it's super super fast.
And so you played again, You played again, You play
it through a lifetime. You'll get nowhere near tree three.
After you die, and maybe you replace yourself with some
artificial intelligence. You've got two AI machines playing to each other,
you know, powered by the light of the sun. They'll
keep playing the game at this crazy pace, and they
keep going and keep going. The sun gets bigger, you know,
(29:07):
he goes to a red giant. All these things happen.
Eventually it falls back forms a white dwarf. Over many
billions of years, and still this these two ais are
still playing the game because they have got nowhere near
tree three, and they're playing at breakneck speed as well,
and so eventually they lose power. They can't get any
power because the sun dies, right, so they need to
(29:27):
somehow develop some new technology which gets energy from I
don't know, the cosmic by background radiation and the game
goes on, and the game goes on, and the game
goes on. In fact, the game will go on way
beyond the sort of heat death of the universe, and
still you will not get to the end of tree three.
And actually there's a phenomenal called Puan career recurrence which
(29:48):
says that in any system, in any finite system, you'll
eventually get back to where you started. And that applies
to our universe too. So you can imagine a pack
of cards. You know, if you shuffle a pack of cards,
he off times. You're a lot of times, but enough
times you'll eventually get back to the point where all
the cards are in order. It'll take a long time,
but it will happen eventually. It's the same with our universe.
(30:11):
You shuffle the universe enough times, you allow it to
evolve for long enough, eventually you'll get back to where
it started. It will reset, And that reset time for
our universe is actually shorter than the time it would
take to play this game of trees all the way
up to three three moves, playing as fast as you
possibly can. And so even even if you could do it,
(30:31):
even if you could live past all these you know,
gargantia and time scales. The universe is just going to go. Now, mate,
game over, we're resetting. You ain't going to get to
the game which you ain't gonna end this game. So
three three is actually a number that's that's actually too
big for the universe. That's how big it is. It's
just so astounding that it as you describe it, it's
just it's such a short step to reach that point.
(30:52):
Because because a lot of these names, like when you're
talking about the Googles and the google Plexes, it's easy
to think, well, those those big numbers live out there
like they're like in the d deep water. But then
this seems to illustrate that the deep water is is
far closer than you think, and it's not I wouldn't
even call it deep water. It's it's water. That's you know,
you're sort of like, yeah, you're just sort of tiptoeing
(31:12):
across the you know, through the shallows and they're doing
and then bang, it just gives away underneath you. And
and there's just it's a it's bottomless as far as
you're concerned. You know, I wouldn't even got a d
what it's beyond deep it's too it's it's too deep
for the universe. So ultimately, what do fantastic numbers reveal
(31:33):
about the cosmos? Like what is the what I guess,
what is the lesson of big numbers, fantastic numbers, etcetera.
So so for me, I think all the ideas that
I talk about in the context of the big numbers
in the book, they all come back to the same
thing which we've talked about, which is the holographic truth,
the idea that that a lot of the ideas associated
with black holes and and how much information you can
(31:55):
fit inside a black hole. Where that information stored for example,
is it stored inside the black call or is it
stored on the edges of the black hole? And these
are ideas which which which leads you to to to
the to the holographic truth, to the idea that actually,
maybe the information in our world isn't stored inside the world.
Maybe it's stored on the boundary of the world, at
(32:17):
the edge, on the walls that surround it. And in
that sense, that's why it's it's holographic. All the ideas,
all the limits that we're talking about, you know, counting
how much information you can store in ahead, you know,
and when it's going to turn into a black hole,
you know, counting how long it takes for our universe
to reset itself. All these ideas come back to the
question of of how our universe stores its information, does
(32:40):
its story inside and if so, how does it story? Well,
actually no, it turns out it's seemed like it stores
it on the edge of space, and that allows you
to count how much information there is in that space
and how many different ways you can combine things. But
it all comes back to that holographic truth. Um. I
have to ask about this because I again am ad
is um his versed in mathematics. Uh, there's a lot
(33:03):
of people out there. And one of the things that
I kept thinking about reading the book was just a
one quick joke from the season one episode of the
British comedy look Around You, in which the narrator that
the episode is about math, and the narrator tells us
that the largest known number is around forty five million,
but that larger numbers might exist and they like speculated
(33:26):
forty five million in one could be another number and
you know, of course that's absurd and that's absurdist humor.
But Um, there's something about that that seems to sort
of ring true with with a lot of these uh,
these these these concepts. And I was wondering what you
thought about the role of absurdity in contemplating big numbers. Yeah, absolutely, no,
(33:48):
I really do think so. When you think of something
like three three, at least within our universe, you can't
fit it in. It cannot fit in. There's nothing that could,
you know, you could describe because it's that's too big
for anything that we can talk about in our universe. Now,
you might imagine other universes which could accommodate it. And
in a you know, a sort of multiverse scenario, like
(34:10):
maybe you get from something like string theory, could you
get universes that can contain three three? Well maybe we
don't know, right, we don't know enough about about the
multiverse of string theory. But but it's not inconceivable potentially so,
but certainly in our world you can't. It's interesting. One
of the things I did a video quite quite recently
actually about the biggest number that nobody will ever think of.
(34:33):
And I did these sort of quite a bunch of
estimates based on a bunch of dubious sort of you know,
sort of assumptions, which I acknowledge with quite dubious. But
but I think I came up with an estimate that
if you think of a random seventy three digit number,
um also something of that order, then probably nobody's going
(34:54):
to ever ever think of it other than you. I mean,
you know, so I'm not saying, like, just think of
a one followed by SEO is clearly not something like that,
but just completely random, random seventy seventy three digit number
something like that. Chances are nobody in the history of humanity,
either before or to come, we'll ever think of that number.
(35:16):
And it's kind of that kind of mind blowing. I
think it's kind of yours. Just think of it, and
that's yours forever. So just everybody should just write down
a seventy three digit number and name after themselves. Well
that's wonderful. Well, Tony, thanks for taking time out of
your day to chat with us. I want to make
sure we're we're hitting all the plugs here. The book
(35:37):
which which is? Which? Is out? I believe it's out now, correct? Yeah, yeah,
it's actually released today in the US. I probably should
say today, should I guess it'll be it'll all will
have been released two days ago when we published this,
so yeah, it's it's out. It's Fantastic Numbers and Where
to Find Them? Um. And then the YouTube series is
number File, correct, Yes, so I appear on Number File.
(35:58):
There's another channel I appear on which is physics Base
called sixty Symbols. Um. So they're both made by by
Brady Harron and yeah, so so I pay regularly on
both of those so it's a lot of fun. But yeah,
it's um. I hope people enjoy enjoy the book. It's
and just don't think too recklessly about Grave's number, because
what's gonna have. Yeah, we don't want anybody's heads to
(36:22):
collapse into black holes absolutely. All right, Well, well thanks
for coming on the show. Have I hope you have
a great day. Thanks all right, Well, thanks once again
to Tony for taking time out of his day to
chat with me here. The book again is Fantastic Numbers
and Where to Find Them? Highly recommended for anyone who
was at all intrigued by what we were talking about
(36:43):
here today. As always, if you want to reach out
to us and ask any any questions, share your relationship
with fantastic numbers. Well, you can find us in a
number of ways. Let's see if you email us and
I'll give you that email. On a second, you can
have access to the discord where you can discuss show
matters with with with other Stuff to Blow your Mind listeners.
(37:06):
There's also the Stuff to Blow your Mind discussion Mondule
that is on Facebook. You can find that and seek
access to that as well. And of course thanks as
always to Seth Nichols Johnson for producing the show here
and yeah, if you want to get in touch with us,
you can simply email us at contact at stuff to
Blow your Mind dot com. Stuff to Blow Your Mind
(37:34):
is production of I Heart Radio. For more podcasts from
my heart Radio, visit the iHeart Radio app, Apple Podcasts,
or wherever you're listening to your favorite shows. No
Welcome to stot to Blow Your Mind, production of My
Heart Radio. Hey, welcome to Stuff to Blow Your Mind.
My name is Robert Lamb. My co host Joe is
away from work today, so I am conducting an interview
here with Professor Antonio Padella, author of the new book
(00:28):
Fantastic Numbers and Where to Find Them, a fascinating read
about big numbers, fantastic numbers, black holes, and more. This
is a really fun chat. I think you're all going
to enjoy it, So go ahead and jump right in
with me right now. Hi, Tony, Welcome to the show. Hi,
(00:49):
Hi up, how you doing? Oh? Pretty good? Pretty good? Um?
Really excited to talk about the new book Fantastic Numbers
and Where to Find Them? A wonderful read, and it's
a book that get into some pretty wonderful mind rending
cosmological territory as well. Note I'll discuss here, but first
I wanted to start with just a really basic sort
(01:10):
of grounding question. I guess we encounter numbers every day,
and you discuss some numbers that most of us don't
encounter really every day. If we could back up a
whole lot, I guess, and just ponder the basics here,
what exactly is a number? Well, I mean, this is
this is an idea I sort of, you know, delve
into him in my book, because of course, when you
go really back in into history, back to sort of
(01:31):
the ancient Sunarians or something like that, you know, obviously
they really began to use numbers to talk about And
while I've got five jars of oil, I've got five
loaves of bread. But then it sort of begs the question,
is that five the same five is the five that
describes the jars of oil, the same five that describes
the loaves of bread. And then you really sort of
(01:52):
when you sort of make that disconnected too, and you
start to build the idea of like what I call
an emancipated number, but it's independent and of the thing
that it's describing, then you're really sort of making quite
quite an intellectual leap. So that, for me is what
is what a number is. It's kind of emancipated from
from the thing that it's describing. Whether such a thing
really exists in a philosophical sense, is a whole new
(02:15):
debate that you can have. But yeah, that for me
is is the key mathematical leap that I think was
made you know, a long time ago, and and yeah,
it's really important now getting back into that sort of
philosophical territory. This is one that I know that you
you tackle a lot. Uh, it's pretty standard sort of
philosophical math question. But is mathematics more of a human
(02:37):
discovery or more of a human invention? Yeah, I mean
I don't think there's a straightforward answer answer to this.
Of course. This this sort of you know, boils down
to like a sort of alluded to whether numbers exist,
whether whether maths exist, and it's kind of I mean,
I'm not a philosopher, but but but I know that
philosophers talk about this, and sort of this kind of
three different angles that you can take on it. So
(02:58):
so on the one hand, you've got the platonists who
will say that numbers and mathematics is true and it exists,
but it exists outside of space time. Is as like
an abstract concept. It's not something that can affect the
things in space time. It can't affect the material objects
that we have arounders. You also have the nominalists, who
(03:19):
says basically that numbers and maths only exist to sort
of understand stuff. So so in some sense, we talked
about the five, you know, five jars of oil, the
five five loaves of bread. That's the only reason that
the number five exists to describe the jars of oil,
to describe the loaves of bread. And then, of course
you've got the third sort of you know school, which
(03:41):
is perhaps in some sense the most extreme, which just
says the numbers that exist at all, that they're just
a useful tool, that that that we used to describe
the universe arounders. And I guess the analogy people use
here is it's like saying, well, you could be an atheist,
but you can still believe with some of the sort
of moral messages that you read in the Bible or
the Koran. It doesn't mean that, you know, you can't
(04:02):
be inspired by them, but you just don't have to
believe in every element of it. And I guess as
a physicist, for me, it's kind of hard to sort
of go with that fictionist idea and yet see a
universe that is so amazingly described by mathematics. Now, is
that something that's embedded in the universe or not. I
guess it's really difficult to know. We've certainly not seen
(04:24):
any evidence that it is. And yet now your book
deals with, as the title indicates, fantastic numbers. Uh, what
what defines for you a fantastic number? And are there
categories of categorizations of numbers other than that that we
need to have in our heads before we can get
to the idea of what it's truly fantastic. Yes, it's
(04:44):
so for me and my own relationship with numbers kind
of and it comes from, on the one hand, you
have a number, whatever that number might be, and if me,
I always want to bring that sort of personality alive,
the sort of real spirit of the number, sort of
to the four. And so it's always been physics for
me that that does that. So when you know you
can have these wonderful mathematical concepts ideas like Graham's number
(05:07):
three three, these truly bizarre and wonderful numbers to have
a wonderful place in mathematics, but then you really bring
them to life when you try to sort of squeeze
them into our physical world. So that for me is
what what makes a number fantastic. It's almost like, what
makes a number fantastic is the fantastic physics that it
can lead you towards and lead you to imagine and
whatever that might be. You also talk about I believe
(05:29):
specifically you're talking about Graham's number pretty early on in
the book, and you point out that if you if
you try and actually picture it in your head, your
head collapses into a black hole. And this this made
me wonder, like, what what are the largest numbers roughly
speaking then an average person can fit into their head
by one definition or another, Like, at what point does
it just become this this other enterprise entirely? Yes, So
(05:52):
it's a good question. So, I mean, it kind of
depends on how how you sort of define the question.
In some sense, we're just thinking about your rs. How
many neurons have you have you got in your in
your brain as about a hundred billion neurons, and so
you might say that you can use them if you
managed to clear your mind of every other thought to
imagine a hundred billion digit number. Okay, that might not
(06:16):
be particularly practical, it might be quite challenging for most
of us. But but in principle you might say that
that that that would be the limit. And of course,
if you then go beyond that and start to say, well,
what if I could somehow get my head to find
a way to actually store information store concepts more efficiently
than just the usual idea of neurons firing on and off.
(06:37):
Let's suppose that it could do that somehow. Then then
the numbers get get much bigger, and you start to
the things that limited are literally preventing your head collapsing
to form a black hole. Because black holes what they
do is that they're they're the best thing at storing information.
So if you want to get something the size of
a head of a human head, and you want to say,
(06:58):
what's the best thing the size of human head that
can store information, it's a black hole the size for
human head that that's the nothing can do it better,
and so so so that that places a new limit,
and you can ask, well, again, what what is that
limit would be? But he's certainly way below grains. No,
but you're not gonna get anywhere near the magnificence of
Graham's number. And you could probably get a digit that's
(07:21):
that's about tens of the seventy and number that's about
ten to the seventy digits long, and so less than
a Google digits long. Having said that, you could imagine
a number like a google plex. A google plex has
a Google digits. Now I've just said that you can't
imagine a Google Digit's not possible, but a google plex
(07:41):
you could, because what you know about a google plex
is that it's a one followed by a Google zero.
So you know that all the numbers that come later
on as zeros, and so there's not much information in that.
So it doesn't cost as much as many bits. You
don't have to put as many bits in your head
to imagine that. So what we're really talking about now
(08:02):
are really a random assortment of digits. Are completely random
assortment digits the kind that would appear in Graham's number.
And I don't think you can get passed around tens
of the seventy, which is a one with seventies zero.
You couldn't get past that many digits completely randomly sort
of allocated. At that point, your head is going to
collapse into a black hole. Now, now backing up to
(08:23):
the Google and the google plex. Can't can you? Can
you walk as briefly through the difference between a Google
and google plex, and and and maybe realms beyond that?
This is this is about the only area of fantastic
numbers that i'd I'd really heard anything about prior to
reading your book. Yes, So, so Google is is um
it's a number which is which is a one followed
(08:45):
by a hundred zero. So I think everybody would agree
that sounds like quite a big number. And it goes
back to to a physicist called Edward Kasner who is Columbia,
and he was writing a popular science book and he
was trying to sort of know convey He really wanted
to show how big infinity he was, and so he
wanted to quote with numbers that we all think are
(09:05):
really big, like a one followed by a hundred zeros.
And he said, well, okay, that's really small compared to infinity, right,
even though something really big is actually really small compared
to infinity. So he came up with this one with
a hundred zeros. He wanted a name for this number,
so at the time he asked his nephew who was
(09:26):
nine years old at the time. He was called Milton Serata.
He said, can you come up with a name for this?
And and Milton said, well, a Google, which is an
absolute stroke of genius, right, It's such a great name.
And and then so they wanted to then develop things further,
so then they wanted an even bigger number, again building
on this idea that it's nothing compared to infinity and
(09:48):
and so so he said, well, okay, I'm gonna quote
with the idea of a Google plex. It's going to
be an even bigger number. Well how big? So Kasna
then goes to to Milton. He says, well, how big
should it be? And Milton's like, well, it should be
a one, not followed by a hundred zeros, but zeros
until you get tired. But Kasner is like, you know,
(10:09):
a sort of you know, steamed academic at Columbia and
all that. That's just not precise enough for him. So
he went with it which are much more sort of
well defined idea, which is a google plex should be
a one followed by a Google zeros. So a Google
is already massive. That's a one followed by a hundred zeros.
A google plex is a one followed by a Google zero.
(10:32):
So it's a whole new level of big compared to
what we normally used to. And then it just it
keeps building on that. Right, there's there's even like what
a google plexian is that the next level. So so yeah,
I mean this is this is a really nice, nice idea.
You can really now start to to really build very
big numbers, very very quickly using this this mathematical technique
called recursion. So for example, you can develop the idea
(10:55):
of a Google duplex. What a Google duplex, Well, it's
a one followed by a Google plex zeros. And then
you can go to a Google triplex. Well, you can
probably guess what it's gonna be. It's gonna be a
one followed by a Google duplex zeros. And then a
Google quadruplex is a one followed by a Google triplex zeros.
(11:15):
And you can see each time you're growing the number
just by so much, by such an unimaginably large amounts,
And that's what. You're not just adding zero every time,
You're kind of really ballooning the number of zeros on
the end of this number in Gargangian proportions. And that's
what and it's this power of mathematical riccasion that allows
(11:37):
you to do that. Now you also talk about fantastic
numbers that are I guess you would say smaller than
the main example that comes to mind. You refer to
this several times in the book is a number associated
with Olympic sprinter Hussain Bolt. Would you tell us a
(12:00):
little bit about this number? Yeah, yeah, so so well,
actually this is one of my big numbers. Actually, even
though it doesn't seem that it's it's actually it's one
of my big numbers. So I can read out the number.
What it is one point, I think it's fifteen zero
eight five eight, So it's just a number just slightly
(12:22):
north of one. So it's it doesn't seem like a
big number. But but in my book, I say it
is a big number. And the reason is it's it
measures the amount by which Usain Bolt managed to slow
down time. And when he was he was running in
the World Championships and I think Berlin Um and he
set this the world record. And this is due to
(12:43):
the effects of relativity, so that when when somebody actually
moves quickly, they actually slow time actually slows down for them.
And actually this is the amount by which Usain Bolt
was actually able to slow down time due to the
effects of Einstein's theory. And it's compared to the people
in the stadium, for example, this is this was the
(13:05):
difference that that he experienced. So it's one of the
one of the weird consequences of it is that you
can actually it's not that you say, but actually, even
though he slowed down time, it's not that he that
he actually ran the race any quicker. He still runs
the race at roughly ten ms per second. It's actually
an even more strange consequence. He actually the track also
(13:25):
shrinks for him a little bit, so so he actually
runs it in less time, but in the same speed. Therefore,
the track shrinks because the relative to him, the tracks moving,
And this is another effect of relativity, one of the
remarkable things. And and yes, you could perhaps argue that
he didn't actually finish the race because the tracks rank
so he didn't run quite a hundred Wow. I was
(13:47):
really blown away with this, because you know, you often
hear the standard analogies concerning airplanes and pyramids and so
forth when it comes to time dilation and so forth.
But but I hadn't. I hadn't heard this particular example
for this is great. Yeah, I mean, it's true. It's true.
If like taxi drivers, if you imagine a taxi driver
that's driving around I don't know any city in New
York wherever, you know, sort of forty fifty years of
(14:10):
their life because of that extra extra speed that they're
picking up. That's going to accumulate over time. And actually
they can probably leap forward in time by probably I
think about a micro second over the course of their career.
It's not a lot, but it's still fairly amazing when
you think about it. So they've got the knowledge, and
then they have that as well, right, Oh yeah, of course,
yeah exactly, not just the knowledge, Yeah, they actually got it.
(14:30):
They actually get a little bit younger. So your book
makes makes use of written numbers, um. And and then
of course you have this wonderful YouTube series number file,
and in that you benefit not only from some fantastic
descriptions and pop culture tie ends as you do in
the book, but you also have a lot of helpful illustrations.
Uh So I was I was curious since you are
(14:51):
a regular communicator of of this this topic. Um is
it is? It? Is? It? Is? It more challenging or
in some cases almost too challenging to describe some of
these numbers without the visual aids or the actual numerals
to like visually present somebody with. Yeah, I think you
so this is where the physics comes in in some respects. Right.
So on the one hand, if you really want to
(15:13):
describe the number, like I said, a number like Grams number,
you do need those visual aids because it's not a
number that you're going to sort of stumble across in
any kind of normal environments. Right, It's not a number
you're going to see on on a price tag, at
least you'd hope not. And you know, so these are
you need new notation, new sort of symbolism to to
(15:33):
sort of actually even describe the number. So you've got
to introduce that. There's just no getting away from it.
But I guess what you can do is described the
physics associated with it, and and that you can certainly do,
you know, just just just with words. And you know,
in the case of a number like Grahams number, you
can talk about how you just can't picture in your
(15:53):
head because your head will will collapse to for a
black hole. And that's already going to make people think, wow,
that number. There's something big of something big and crazy
about that number. Or a google plex, you know, when
you can talk about a universe that's that's a google
plex meters across. And then you can ask, well, if
the universe is that big, if the universe is literally
that large, then it's likely that you would find multiple
(16:16):
copies of yourself, like literally exact apple Gangers elsewhere in
this ginormous universe. Yeah, I wasn't. I was. I wasn't
prepared for dopple gangers to enter into the scenario. So
there was there was another great part about the book
for me um and another thing that that comes up
in the book that I was very intrigued by. I
was wondering if you might talk about is the the
(16:37):
idea of the of the holographic truth? Yes, so the
holographic truth is I mean, it's an idea. It's probably
the most important idea I would say that's emerged from
theoretical physics in in the last thirty years. And it's
it's actually mind blowing when you really think about what
it pertains to. It's it's it's this following statement that
essentially one of the dimensions of space that we experience
(17:00):
around it. So we normally talk about say three dimensions
of space, Well one of them could well be an illusion.
It might not exist, and it's really remarkable. So what
we're saying is that there are two ways in which
you can describe the physics that we see around as
On the one hand, we can imagine three dimensional world
(17:20):
with a gravitational force and the force of gravity doing
its thing, with planets around the Sun and so on
and so forth. On the other hand, there's a completely
equivalent description of the same phenomena which just uses two
dimensions and no gravity. So think of it a bit like,
you know, on the one hand, somebody's you know, in English,
(17:41):
we say if we see a place of meat balls,
we call them meat balls, but a Spaniard might call
them album the gas. They're both describing the same things,
they're just using a different language. And that's kind of
what what the holographic truth says. It says that you
can have a theory like a three dimensional world with gravity,
and you can use that to describe all the physical
phenomena you see, or use this different language which has
(18:05):
no gravity and only requires two dimensions of space. So
is it true of our world? We don't know. It's
a conjecture. It's a conjecture that has sort of evidence
coming from from the physics of black holes. There are
actually concrete examples that we know of of sort of
toy universe is so not our universe, but but but
(18:26):
space times that may be that the higher dimensional there
may be warped in weird and wonderful ways. And you
can think about gravity in these in these simple toy universes,
and you can show that there's an equivalent description in
one dimension less like a holographic description, and it's called
a hologram because that's essentially what what holograms do? Right?
If you think of a hologram, what have you got?
(18:46):
You've got an image on a that's stored on a
holographic plate. You know, it's just some light and dark
bands on a holographic plate, a two dimensional plate. It's
stores a bunch of information that way, But that's just
one way of looking at the information. You can decode
it in a different way by shining monochromatic light through
it and creating a three damage. You're not creating any
(19:08):
new information. It's the same information, just stored either in
two dimensions or three. And it's that seems to be
that that seems to be a fundamental property of gravitation,
of gravitational worlds that you can think of them as
as like as I said, a three D world with gravity,
or you just forget about gravity and consider a world
with one dimension or less and you can describe exactly
(19:31):
the same physical phenomena. Now here's another question that that
that came came up reading the book that that I
don't know if of all our listeners are necessarily would
have thought of this question. I think some of them
would have. And that comes to infinity um infinity, Like
sometimes it's easy to think of like, okay, infinity is
the it's it's if we think of it as a number.
(19:52):
We think it's the eight on its side representing infinity.
Is infinity a number? And if it's not a number,
like what do we think of it at? How do
we classif high infinity? So I love this question because
the answer is that it's both not a number and
lots of numbers. This is the wonderful thing about infinity.
So it depends how you want to think about infinity.
And I think most of us when we intuitively think
(20:14):
about infinity, we kind of think of like I don't
know the infinite distance, you know, or infinite time, And
what we're really thinking there is we're thinking of it
is like a limit is something that's just just beyond
our finite realm. That that that's you know, if you
keep on counting forever, you know, it's kind of the
at the end of that, you're sort of almost beyond
the end of that. Now, that's in some sense thinking
(20:36):
of infinity as not a number, as a limit of saying,
you know, the whole numbers. But what Cancelor Judge cancel that,
you know, the great German mathematician from the late Victorian times,
what what what he did was actually taught us how
to count beyond infinity. So literally, using really smart ideas
(20:57):
associated with something called set theory, he was able to
show that actually you can have all the sort of
finite numbers, and beyond that you can have infinity. But
there's that's just one layer of infinity. You can have
the infinity, which is all the whole numbers, but you
can also have a different layer of infinity, which is
all the numbers between zero and one. So think of
(21:18):
the continuum of the numbers between zero and one. That's
you think there's an infinite number of numbers between zero
and one, but that's actually a different infinity to all
the whole numbers. So you've got, you know, these discreet infinities,
continuum infinities, and they they have different sizes, and they
have you have many layers of what can be an
(21:38):
infinite number. And this is what Cancel really really began
to explore and and and develop. And he met a
lot of resistance when he was doing it. He actually
people thought he was crazy. He sort of fell into
a lot of depression. Um, you know, he was in
battles with with someone called Chronicker, who was kind of,
you know, the big guy in Berlin at the time,
the elite university, and Jereman. He he thought that Cancer
(22:01):
was just delving into sort of witchcraft and he was
a shot. He called him as Charlatan, a corruptor of youth.
And this really bothered Cancer and actually quite a sad story.
I mean, Cancer actually sort of really fell into into
quite a bad depression. Whether it's because of this or
whether he was he was predisposed anyway, it's not clear.
But he actually ended his day sort of very sort
(22:21):
of emaciated in a in a sanatorium, essentially starving because
of the effects of the First World War at the time.
And I'm not having enough food, so it's quite a
tragic tale in the end, but he was certainly a
tremendous mathematician, and now all his ideas are really you know,
I think people acknowledging for the genius that he was. Yeah,
of course brings to mind those um like the infinity
(22:44):
hotel discrete scenarios that are used to describe infinity. I've
always found those to be super interesting and and men
mind blowing. Yeah, I mean that's so, that's that's what
I mean. So, so, as I said, cancer sort of
had these these different layers. So you can sort of
imagine the first infinity, which he called alve zero, which
is he defined as the set of all of all
(23:05):
the whole numbers, essentially all the natural numbers you know,
one to three for all the way up to well infinity,
all of them basically, so that that's what he called
the sort of first infinity. But then you can have
these higher infinities, which are the you know, things like
the the the set of the continuum, essentially the continuum
between zero and one, so not just all the all
(23:25):
the fractions and irrational numbers, but also the irrational numbers
numbers like one over the square out of tow that
kind of thing um and and this is a new
letter that he actually proved that they're actually got a
bigger infinity. And it's not immedially obvious, but but he
did show it, and it's it's it's remarkable, and and
there's so many sort of things about infinity. There's so
(23:46):
many paradoxes associated with them. For example, one one thing
you can say is you think about the number of
the more square numbers or whole numbers, and you think, well,
you think naively, obviously there are more whole numbers and
square numbers because one is a square, but but two
isn't a square, and three isn't a square, Okay, four
is So it seems that there's obviously more whole numbers
(24:08):
than square numbers. But actually it's not true. And the
reason you know that's not true because you just take
a square number and you can map it to its
square roots and you get the whole numbers. So so
the number of whole numbers is actually exactly the same
as the number of square numbers. It's completely crazy. And
these these parents and it's the same. There are the
same number of even numbers, there are even at odd numbers,
(24:29):
and there's all these one there's the same number of
numbers between zero and one as there are between zero
and two. There's all these paradoxes that are merged just
the minute you start to think about infinity. And that's
why most mathematicians for a long time just stayed away
from it. But Cancer was brave enough to climb into
this infinite heaven and explore it. Now. One of the
(24:54):
numbers that that comes up a lot in your in
your book, and you've done your videos on this as well. Um,
I'm also afraid to ask about it because it just
seems kind of I get confused anytime I read anything
about it. And that's this idea of and I'm not
even sure what I'm saying it correctly? Is it? Do
we say tree three? Yeah? That's right, Yeah, yeah, three three?
(25:14):
So yeah, what is this? What is tree three? So?
So there's a particular game that was that was developed
involving some trees, right, So, so the details aren't too important,
it's just but basically you draw these little stick trees
and you have some seeds, you have some lines which
are kind of like the branches, and you and you
build these trees. Right, So, so what are the rules
(25:36):
of the game is is that you know, for example,
you can't have a tree that's got a bit of
a tree that that's has appeared before. So so if
I draw, like, you know, one particular tree, then later
on you can't draw a bigger tree that's got my
tree stuck in it somehow. It's it's just not allowed.
That would end the game. And so there's a bunch
of rules in how you draw these trees and build
up this this particular game, which I call the game
(25:56):
of trees. Now, how long the game depends on how
many different types of seeds you have. So you could have,
for example, just black seeds, okay, or maybe you can
have black seeds and you're also got white seeds, or
maybe you've got black seats, white seeds and yellow seeds.
You know, there's a there's whole bunch of possibilities. How
(26:16):
many seeds you play with sort of of changes how
long the game can last. For now, if you've just
got one seed, the game can only last one move.
You can just write down one seed and that's it.
You can't write down anything else because anything else that
follow is going to contain a tree that went before. Okay,
you've got two seeds, like it's like a black and
(26:37):
a white seed. The game can last up to you
can draw up to three trees and the game will
automatically and after just three moves, it can't go beyond
three moves. So you've got this this sort of sequence.
So we've got one seed, you can play only one move.
You've got two seeds, you can play three moves. And
so then you go to three seeds, and you might think, well,
(26:59):
I can I start for one and it went to
three and I've got three seeds, Maybe maybe I can
play ten moves or something called fifteen moves something. It's
not gonna be some it shouldn't be anything crazy. Well
it is. So this sequence just goes bang. It just
goes from one. So just from one seed you get
one move, two seeds, you get three moves, and then
three seeds you get tree. Three moves is where the
(27:20):
game will last too. And this is a number which
just blows everything else. So we talked about Google and
a Google place, Well that's just nothing compared to three three.
Talk about Graham's number which will collapse, your heads fall
and black hole, that's nothing compared to three three Tree
three is just it's it's impossible. I mean, I actually
think it is impossible to imagine how ridiculously big this
(27:42):
number is. And it's just so mundane. Where it comes
from is this game. He starts off you. So you're
playing this game with two seeds, as game keeps ending
after three moves, and then somebody comes along and adds
a different color of seed, and you're like, okay, how
long is the how long can the game last? Now?
And somebody says tree three, and this is tree three,
just and not but it's actually too big for the universe.
(28:03):
Just whoa where did that leap come from? The leap
should not be that big, but that that's that's so
that's in essence what what tree three is. And it
is too big for the universe. So one of the
things I worked out was suppose you're playing this game
involving these trees. So you're writing drawing these trees, right,
So you play one, go draw a tree. Play next,
go draw a tree. And so you've got three seeds,
(28:24):
three different colors of seeds. So we know the limit
of the game is tree three moves treating three three
different trees in the forest. How long could you finish
the game. And one of the thing I imagine is,
you know, you're playing this game at high speed, so
you're playing it as fast as spacetime will allow. So
you literally, if you play any fastest, spacetime will break
(28:45):
due to quantum effects. So you play it's super super fast.
And so you played again, You played again, You play
it through a lifetime. You'll get nowhere near tree three.
After you die, and maybe you replace yourself with some
artificial intelligence. You've got two AI machines playing to each other,
you know, powered by the light of the sun. They'll
keep playing the game at this crazy pace, and they
keep going and keep going. The sun gets bigger, you know,
(29:07):
he goes to a red giant. All these things happen.
Eventually it falls back forms a white dwarf. Over many
billions of years, and still this these two ais are
still playing the game because they have got nowhere near
tree three, and they're playing at breakneck speed as well,
and so eventually they lose power. They can't get any
power because the sun dies, right, so they need to
(29:27):
somehow develop some new technology which gets energy from I
don't know, the cosmic by background radiation and the game
goes on, and the game goes on, and the game
goes on. In fact, the game will go on way
beyond the sort of heat death of the universe, and
still you will not get to the end of tree three.
And actually there's a phenomenal called Puan career recurrence which
(29:48):
says that in any system, in any finite system, you'll
eventually get back to where you started. And that applies
to our universe too. So you can imagine a pack
of cards. You know, if you shuffle a pack of cards,
he off times. You're a lot of times, but enough
times you'll eventually get back to the point where all
the cards are in order. It'll take a long time,
but it will happen eventually. It's the same with our universe.
(30:11):
You shuffle the universe enough times, you allow it to
evolve for long enough, eventually you'll get back to where
it started. It will reset, And that reset time for
our universe is actually shorter than the time it would
take to play this game of trees all the way
up to three three moves, playing as fast as you
possibly can. And so even even if you could do it,
(30:31):
even if you could live past all these you know,
gargantia and time scales. The universe is just going to go. Now, mate,
game over, we're resetting. You ain't going to get to
the game which you ain't gonna end this game. So
three three is actually a number that's that's actually too
big for the universe. That's how big it is. It's
just so astounding that it as you describe it, it's
just it's such a short step to reach that point.
(30:52):
Because because a lot of these names, like when you're
talking about the Googles and the google Plexes, it's easy
to think, well, those those big numbers live out there
like they're like in the d deep water. But then
this seems to illustrate that the deep water is is
far closer than you think, and it's not I wouldn't
even call it deep water. It's it's water. That's you know,
you're sort of like, yeah, you're just sort of tiptoeing
(31:12):
across the you know, through the shallows and they're doing
and then bang, it just gives away underneath you. And
and there's just it's a it's bottomless as far as
you're concerned. You know, I wouldn't even got a d
what it's beyond deep it's too it's it's too deep
for the universe. So ultimately, what do fantastic numbers reveal
(31:33):
about the cosmos? Like what is the what I guess,
what is the lesson of big numbers, fantastic numbers, etcetera.
So so for me, I think all the ideas that
I talk about in the context of the big numbers
in the book, they all come back to the same
thing which we've talked about, which is the holographic truth,
the idea that that a lot of the ideas associated
with black holes and and how much information you can
(31:55):
fit inside a black hole. Where that information stored for example,
is it stored inside the black call or is it
stored on the edges of the black hole? And these
are ideas which which which leads you to to to
the to the holographic truth, to the idea that actually,
maybe the information in our world isn't stored inside the world.
Maybe it's stored on the boundary of the world, at
(32:17):
the edge, on the walls that surround it. And in
that sense, that's why it's it's holographic. All the ideas,
all the limits that we're talking about, you know, counting
how much information you can store in ahead, you know,
and when it's going to turn into a black hole,
you know, counting how long it takes for our universe
to reset itself. All these ideas come back to the
question of of how our universe stores its information, does
(32:40):
its story inside and if so, how does it story? Well,
actually no, it turns out it's seemed like it stores
it on the edge of space, and that allows you
to count how much information there is in that space
and how many different ways you can combine things. But
it all comes back to that holographic truth. Um. I
have to ask about this because I again am ad
is um his versed in mathematics. Uh, there's a lot
(33:03):
of people out there. And one of the things that
I kept thinking about reading the book was just a
one quick joke from the season one episode of the
British comedy look Around You, in which the narrator that
the episode is about math, and the narrator tells us
that the largest known number is around forty five million,
but that larger numbers might exist and they like speculated
(33:26):
forty five million in one could be another number and
you know, of course that's absurd and that's absurdist humor.
But Um, there's something about that that seems to sort
of ring true with with a lot of these uh,
these these these concepts. And I was wondering what you
thought about the role of absurdity in contemplating big numbers. Yeah, absolutely, no,
(33:48):
I really do think so. When you think of something
like three three, at least within our universe, you can't
fit it in. It cannot fit in. There's nothing that could,
you know, you could describe because it's that's too big
for anything that we can talk about in our universe. Now,
you might imagine other universes which could accommodate it. And
in a you know, a sort of multiverse scenario, like
(34:10):
maybe you get from something like string theory, could you
get universes that can contain three three? Well maybe we
don't know, right, we don't know enough about about the
multiverse of string theory. But but it's not inconceivable potentially so,
but certainly in our world you can't. It's interesting. One
of the things I did a video quite quite recently
actually about the biggest number that nobody will ever think of.
(34:33):
And I did these sort of quite a bunch of
estimates based on a bunch of dubious sort of you know,
sort of assumptions, which I acknowledge with quite dubious. But
but I think I came up with an estimate that
if you think of a random seventy three digit number,
um also something of that order, then probably nobody's going
(34:54):
to ever ever think of it other than you. I mean,
you know, so I'm not saying, like, just think of
a one followed by SEO is clearly not something like that,
but just completely random, random seventy seventy three digit number
something like that. Chances are nobody in the history of humanity,
either before or to come, we'll ever think of that number.
(35:16):
And it's kind of that kind of mind blowing. I
think it's kind of yours. Just think of it, and
that's yours forever. So just everybody should just write down
a seventy three digit number and name after themselves. Well
that's wonderful. Well, Tony, thanks for taking time out of
your day to chat with us. I want to make
sure we're we're hitting all the plugs here. The book
(35:37):
which which is? Which? Is out? I believe it's out now, correct? Yeah, yeah,
it's actually released today in the US. I probably should
say today, should I guess it'll be it'll all will
have been released two days ago when we published this,
so yeah, it's it's out. It's Fantastic Numbers and Where
to Find Them? Um. And then the YouTube series is
number File, correct, Yes, so I appear on Number File.
(35:58):
There's another channel I appear on which is physics Base
called sixty Symbols. Um. So they're both made by by
Brady Harron and yeah, so so I pay regularly on
both of those so it's a lot of fun. But yeah,
it's um. I hope people enjoy enjoy the book. It's
and just don't think too recklessly about Grave's number, because
what's gonna have. Yeah, we don't want anybody's heads to
(36:22):
collapse into black holes absolutely. All right, Well, well thanks
for coming on the show. Have I hope you have
a great day. Thanks all right, Well, thanks once again
to Tony for taking time out of his day to
chat with me here. The book again is Fantastic Numbers
and Where to Find Them? Highly recommended for anyone who
was at all intrigued by what we were talking about
(36:43):
here today. As always, if you want to reach out
to us and ask any any questions, share your relationship
with fantastic numbers. Well, you can find us in a
number of ways. Let's see if you email us and
I'll give you that email. On a second, you can
have access to the discord where you can discuss show
matters with with with other Stuff to Blow your Mind listeners.
(37:06):
There's also the Stuff to Blow your Mind discussion Mondule
that is on Facebook. You can find that and seek
access to that as well. And of course thanks as
always to Seth Nichols Johnson for producing the show here
and yeah, if you want to get in touch with us,
you can simply email us at contact at stuff to
Blow your Mind dot com. Stuff to Blow Your Mind
(37:34):
is production of I Heart Radio. For more podcasts from
my heart Radio, visit the iHeart Radio app, Apple Podcasts,
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