To Infinity and Beyond

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Hello, I'm Robin Ince.

And I'm Brian Cox.

And welcome to the podcast version of the Infinite Monkey Cage, which contains extra material that wasn't considered good enough for the radio.

Enjoy it.

Hello, Pi by Two Radians to my left is Brian Cox.

A further Pi Radians counterclockwise, Robin Ince.

This is the Infinite Monkey Cage, and today we're talking about mathematics, which requires precision, hence the precision introduction.

Mathematics is a subject that has a reputation for confusing people, even though it lies at the very foundation of our understanding of the universe.

So, for instance, when I look at Maxwell's wave equations, and I do often,

I'm told anyway that it's all about the unerring precision and elegance of the behaviour of light itself.

But how do I know?

It's just squiggles.

Is it just some kind of mathematical game?

Are the mathematicians up to something and coding everything, making their Turing machines?

Our panel today.

Colva Roney-Dougal is a senior lecturer in pure mathematics and has been working on solving word problems via generalizations of small cancellation.

As she has said, it is well known that the word problem for finitely presented groups is algorithmically unsolvable in general, and I have no idea what I'm saying.

John Lloyd is a producer, writer and host of Radio 4's Museum of Curiosity and co-author with Douglas Adams of the Lyft Books.

Simon Singh is journalist, campaigner for the use of rational thought in journalism, and authors of books, including Furmit's Last Theorem, Furmit's Next Theorem, and Furmit's Best Theorem.

And three of those were lies, by the way.

Or was it two?

Well, you'll have to work that out later on.

Having wondered how to stop his wife saying, Are you going to sit there all day watching The Simpsons, or are you going to help me?

His new book is The Simpsons and Their Mathematical Secrets, in some ways more an alibi for laziness than a publication.

And this is our panel.

Colbert, I'm going to come first to you because I've read a little bit about your work.

I have already admitted that I have a fear of mathematics and I'm intrigued by language.

So, when I was talking about the fact that it's well known that the word problem for finitely presented groups is algorithmically unsolvable in general,

what was I saying?

Well, Alan Turing and Gödel back in the 1930s proved the rather surprising fact.

So, Gödel did the initial proof that there exist true mathematical statements which are impossible to prove using our standard axioms of mathematics.

Turing then brought that forward forward and showed that there are mathematical problems where you'd like to design an algorithm for them where no algorithm can exist.

Now, my research is on the study of symmetry.

Word problems are about working with ways of representing collections of symmetries that might be infinitely many symmetries or might be finitely many, and you represent each symmetry as a little word in a bunch of letters.

So I could be writing symmetries as letters in A and B, and my word might be ABBA.

Now, the word problem is to decide whether or not one of these little words is equal using a bunch of rules to another of the little words.

And what Turing was able to show, well, what follows from Turing's work is that there can exist no general algorithm to solve all of these problems.

However, a mathematician called Gromov about 30 years ago showed that although there exist

word problems which no algorithm could ever decide, if you take a random word problem, then it's easy.

So, what my research project is about, solving word problems by generalizations of small cancellation, is trying to

find ways to develop fast algorithms for the easy cases.

We know that a random case is easy, but we don't actually know how to even do those easy cases all that well.

So, we're trying to write programs which will solve these supposedly easy random problems in less than the time it takes for me to go and get a cup of coffee, which is my definition of a reasonable amount of computing time.

So, I suppose as it's a science show, we need to be specific.

I should ask, how far away is the kettle?

Two floors.

Right, okay, I've got some perspective now.

People often ask, is there any use for this kind of research?

Which is a terrible problem to ask for an academic.

People ask it a physicist as well.

This particular one, no, but previous work I've done on algorithms for symmetries.

But in general, understanding how to work on a computer with symmetries has lots and lots of applications.

For example, for a while, I was in a research group that was talking to the NHS, and we were interested in solving search problems where there's lots of symmetry around.

And so, for example, if you're trying to schedule staff to work in a hospital, now this isn't true, but you might want to think that any two nurses of a particular grade are interchangeable, and any two doctors of a particular grade might be interchangeable.

So, if you try your schedule with nurse A and nurse B in one set of slots and you can't find a solution that way around, then there's no point trying nurse B and nurse A simply swapped over because there's not going to be a solution that way around either.

And that's what you mean by a symmetry in this case.

It's this symmetry of swapping

people around, symmetric nurses are symmetric doctors.

John, you have, I suppose, later on, after post-school, post-university, you studied law, but you seem to become more fascinated in things like, you know, ideas of numbers, ideas of science.

What is it that kind of has drawn you further into this world?

Well, the QI philosophy is that everything is interesting if looked at for long enough, closely enough, or from the right angle.

And maths was my particular hate subject at school.

I thought I was going to have a heart attack at 13 when I took O-level.

So I was 40-something before I got interested in maths.

And I read a book by the great Cambridge mathematician G.

H.

Hardy called A Mathematician's Apology.

He was the guy who discovered Ramanujan, the wonderful Indian, self-taught Indian mathematician, genius.

And in the introduction to this book, Hardy writes that a mathematician, like a poet or a painter, is a maker of patterns, and that there is no permanent place in the world for ugly mathematics.

And this completely changed my mind about what maths actually was.

And I think that's the way that maths is defined now.

It's the study of patterns.

I thought it was about number crunching.

And I think that I've never been any good at that.

I can still not read a balance sheet without feeling faintly sick.

But the idea that maths underlies the universe and is imaginative and a way of describing what actually is, is amazing to me.

And this goes right back to Pythagoras, who said, all is number, one of his great dictums.

I wondered actually, Colbert, when you got that bug for mathematics, being a professional mathematician now.

Actually, really late.

I mean, it was in a lecture lecture on.

Now, the technical term is functions that are continuous but nowhere differentiable.

The non-technical term, which my lecturer was using at the time, was infinitely kinky functions.

He actually explained to us that the deal with infinitely kinky functions is no matter how closely you examine them, they're kinkier than you can possibly imagine.

What these are is these are functions that it would take you a while to draw, but they are just a straight line on the page.

However, when you look at them, they're turning everywhere.

At no point do they have a slope, because at every point they're doubling back on themselves.

And I found this notion that somebody, in this case, it was Weierstraus in the 19th century, could cook up one of these things from nowhere when it was generally believed that no such thing existed, was just beautiful.

And that, I think, was the moment where I thought, being a professional mathematician, there's the chance to actually make beautiful objects in a way which

doing various other sciences, you have to deal with what's out there.

You can't just make something up from your head.

Galileo Galileo once said nature's great book is written in the language of mathematics.

So, Simon, to what extent can we make the claim that mathematics is not only in an abstract sense, but really is the language of nature?

Well, it just seems to be incredibly useful.

You know, when Newton and Galileo started applying mathematical equations and started plugging in some numbers into physics, that's when physics really blossomed, and that's when chemistry really blossomed.

So, my background, your background as well, we have both backgrounds in physics.

And I think probably, I don't know how you felt about maths, but for me, maths was just a tool.

So you studied mathematics in order to do physics.

But it was only later for me when I started writing books about maths that I realized: okay, it's a useful tool for doing sciences, but it's inherently beautiful.

And then I began to understand why people do pure mathematics.

Coming up with a great mathematical theorem is like coming up with a great symphony, it's like coming up with a great work of art.

You do it for its purity and its beauty, and then down the line you find out it has some tremendous application.

So most of the mathematics that's in modern cryptography dates back to pure mathematical ideas hundreds of years ago.

Pure research at the time when things are discovered often doesn't seem important.

For example, when Hertz discovered radio waves in the 1890s, and people said he called them radio waves because they radiated.

And people said to him, Heinrich, what's the point of these radio waves?

He said, I have no idea, but somebody will think of a use for them.

I agree with you.

I would say, listening to a lot of commercial radio, I'm still uncertain of what the use is of them.

But I read something only yesterday that when they first invented petrol, nobody could think of a use for it except for use as a stain remover.

Isn't that great?

And that was during the particular period of time where a lot of messy people also suffered spontaneous combustion, which looking back, we seem to be able to link up.

It is, I mean, it's like the whole thing, is it?

You know, of what use is a newborn baby?

And as we have discussed already, yes, there is use for a newborn baby if you run out of clothes pegs.

Fantastic grip.

So it does work.

Did we have any insight

into why this should be the case?

Why the language of nature should be mathematics?

I think the answer is just simply no.

I think it just works.

The universe seems to be governed by numbers.

Going back to Pythagoras and the relationships of musical notes that Pythagoras noticed, right up to modern-day string theory.

In science, ideas change.

You come up with a theory and then somebody tries to break it down, you do an experiment, you come up with a better theory, and theories keep on evolving and changing from Newton to Einstein and so on.

But in mathematics, once you've proven something, that's it.

It's absolutely logically secured.

And even the Nobel Prize-winning physicist Leon Lademan said, physicists will only ever defer to mathematicians, and mathematicians will only ever defer to God.

So, mathematics is universal.

We would expect, would we, if we found some civilization out there tomorrow, that they would share an understanding of certain things, pi, for example, these numbers, e.

Yeah, and for example, when we sent out the Voyager space probes, and there was a little engraving on them, which was designed by Carl Sagan, and there's a picture of a man and a picture of a woman not holding hands in case aliens thought they were one single creature with two arms and four legs.

But what I thought was also wonderful about that is I think the images, the man and woman, had no genitalia in case it offended the aliens.

Honestly, it's really.

No, you've got it.

No, the men have genitalia, but the women may have a secret.

That's actually true, yeah.

Speaks of the US in the 70s, doesn't it?

But together with all of that, we sent them some examples of our kind of maths.

We sent the incredibly deep expression two times three is six, and we sent them two plus three is five, on an understanding that if they had any ability to understand this thing that had just crash-landed on their planet at all, they would understand that we were mathematically numerous and they would have the same kind of maths as we do.

I like the story that Douglas Adams told about that.

They sent some music as well, a range of human music, and somebody suggested sending a bit more bach because it was so nice.

And they thought, no, because that would look like showing off.

And another take on the universality of maths is the fact that we're becoming increasingly clear now that lots of animals can count as well.

This isn't just a human construct.

One of the things I found recently was a biologist who'd been out in China on the Lee River.

So they have a strange way of fishing out there.

I gather that now it's mostly done for the tourists, but the fishermen go out with cormorants, birds, and they put a ring round the cormorant birds' neck so they can't actually swallow.

But down near the bottom of their neck, the cormorants go out and they dive into the water and they catch fish.

Then they come back to the boat and they regurgitate the fish, which are then the fish that you've caught.

People eat these.

But the deal that persuaded the cormorants to carry on fishing with the fishermen was that every eighth fish they were allowed to eat.

So after they'd caught seven fish, the ring would be taken off their neck and they would go off again to catch their eighth one.

And it was observed by this biologist that cormorants, after catching seven fish, would sit grumpily on the side of the boat, waiting for the ring to be taken off before they'd go and catch the eighth one.

So, cormorants can count up to eight.

That's quite strong because I know that there were stories of crows or something.

There was a tower, and the crows would fly to the top of the tower,

and then the hunter would approach the tower, and the crow would fly away.

So, then one hunter would go into the tower, and the crow wouldn't come back until one hunter had left.

Then the crow would come back.

And then, two hunters would enter the tower, but the crow wouldn't come back until two hunters had left.

Three hunters would go in, the crow still wouldn't come back until three hunters had left.

But if four hunters went in, but only three came out, the crows couldn't differentiate, they'd fly back and they'd get shot.

So there seems to be that they, whether I don't know whether they're counting or whether they're almost looking at something like a pile of hunters, that we can we can tell the difference between maybe a pile of three teaspoons of sugar as opposed to a pile of four teaspoons of sugar.

But are we counting teaspoons of sugar or are we counting the size of the pile?

One of the beginnings of mathematics, if you look back at ancient Babylonian times, so maybe 5,000 years ago, the very beginning of the birth of the concept of written numbers is happening before there's even writing.

But what the early numbers are is they're literally just collections of dots.

And so that very, very early maths must have been literally just the physically matching one dot per jar of oil.

And then after a while, they start realizing that maybe I could draw a big circle which could indicate, I forget, maybe 10 jars of oil.

And so you can see the very birth of this idea of numbers.

But it took a long while before they started realizing that they could use the same 10 for 10 jars of oil as they would use for 10 baskets of grain.

They would have a different 10 for the 10 baskets of grain from the 10 for the 10 jars of oil.

So you can see this gradual abstraction from just matching up one to one to a number as an abstract thing.

And that's an incredibly brilliant thing, isn't it?

That jump of insight to think that two bees and two turtles are the same in a certain kind kind of way.

In the Andamanese language of the Andaman Islands and Southeast Asia, they've only got two words for numbers, one and more than one.

So it would be a bit difficult being an accountant in the Andaman Islands.

And

this idea of building mathematics up from those early beginnings, so we get the idea of a number,

and then this abstract idea, I suppose you'd call it a set, wouldn't you?

So there's a quality of the three bees are the same in some sense as three piles of sand.

How do you then build up levels of abstraction from that?

I'm thinking possibly about Bertrand Russell's attempt to do such a thing, to actually say, right, I'm going to start from numbers,

some very simple axioms, and build the whole edifice.

Well, Bertrand Russell actually started from sets and started building the whole edifice of numbers.

A set is just a collection of objects.

And so there was a belief that you could construct all of the whole counting numbers by starting with the empty set.

So that's a set containing no objects.

And for mathematical purposes, there's only one of them.

It doesn't matter whether you have no bananas or no apples or no cherries, you've got no nothings.

You can then, given just that, say, well, what can I make?

Well, I can make a set containing the empty set.

So I'm going to identify this empty set with the number zero.

I'm going to identify the set containing the empty set with the number one.

And so slowly by taking more and more sets containing only the things that I've made so far, you can sort of maybe imagine how you might be able to make all the counting numbers.

So that was one of the things that Bertrand Russell was doing, and along the way, proving that one plus one equals two.

So, the idea was to build this logical edifice that's absolutely pure, and you can start from a very basic idea, like the set of empty things, and build the lot from which all mathematical statements could be stated, and within which all mathematical statements could be proven.

At the same time, in the 1910s, 1920s, you're going to the other end of the scale, and for the first time, really beginning to understand what infinity is.

Georg Cantor would come along, he would say, Okay, well, what is this thing, infinity?

Is infinity plus one bigger than infinity, or is it the same thing?

And it was his friend David Hilbert, another great German mathematician, who said, Well, okay, here's one way to think about infinity plus one.

Let's imagine we've got an infinite hotel.

People talk about Hilbert's hotel.

Infinitely large hotel, an infinite number of rooms.

Business is good, you've got an infinite number of guests, the hotel's full.

And then somebody arrives at reception, they're desperate for a room that you really want to squeeze them in.

And you think, oh, I know how I'll get them in.

I'm going to ask everybody to move up one room.

So, the person in room 10 goes to 11, the person in room 100 goes to 101, and so on.

And obviously, the person in room 1 goes to room 2.

Now, if you want to ring up anybody in this hotel, I can tell you exactly where they are.

They were in their old room plus one, and the new guest can slip into the first room.

So, the infinite infinity can also include infinity plus one.

And then it gets even more interesting because an infinitely large coach turns up, and you had it-you know, you've got an infinite hotel, you've already got an infinite number of guests, in fact, you've got an infinity plus one number of guests, and then an infinite number more turn up.

And you think, hang on, I can solve this.

You ask everybody in the hotel to move to the room that's double the one they're in.

So somebody in room three goes to six, somebody in room ten goes to twenty.

All the even rooms are full, all the odd rooms are empty.

So all the infinite number of people, I've got an infinite number of odd rooms, all the infinite number of people can take all the odd rooms.

So infinity plus infinity equals infinity.

So Gay or Cantor was proving all of these things.

So basically, any kind of infinities are all the same, except they're not.

See, John, this is where my brain buckles.

And I heard a certain amount of murmuring in the audience as well.

There were some people pretending they'd understood it.

And

no,

I didn't say that I understood mathematics.

I just said it was interesting.

Oh, no, I'm not saying it.

No, I'm.

I always think that is the best option out, isn't it?

That's very interesting.

Just by saying it's interesting.

But it seems to me that infinity plus one and so on is just brain-bending, and cannot see the use of it apart from a piece of intellectual cleverness.

Whereas for example fractals, which I came across ten or so years ago, you think this is astonishing.

I mean, I remember reading a book called Chaos by James Gleich and rushing off to see a friend.

I said, I suddenly understand how a tree works.

I understand why you recognize a tree as the same species, although they're all different.

These two kinds of infinities that we're talking about, which are the two smallest kinds of infinities, actually they do have.

Oh,

to be incredibly important because the very first one, the rooms in Hilbert's Hotel that Simon was talking about, that's the infinity that you can list.

That's the infinity where I can start saying one, two, three, four, five, and I'll never get to the end, but I'm getting somewhere.

The next kind of infinity is one you can't list anymore.

So, this is the difference between things you can count

and things where there's so many of them you can't even begin making a list of them.

So, for example, if you draw a straight line on a piece of paper, the number of points along that straight line, you can't even begin counting them.

I have to say, Simon, your Simpsons, their mathematical secrets, the main thing that's just done to me is all I had in my head during that, about two minutes in, was meow, meow, meow, meow, meow, meow, meow, meow.

Everything went homer at that point.

Again, though, when we talk about mathematics as a language,

this is the trouble that a lot of us have, which is it seems to be sometimes impenetrable.

It seems to be as if we are approaching Sanskrit.

The way I see it is, I know nothing about music.

So, you show me a score of music, it's just scribbles.

I've got no idea what these little squiggles and blots mean.

Now, you can play me some music, and I can begin to appreciate what this bit of paper is all about.

And the sad thing about mathematics is that it just looks like musical scribbling.

And unfortunately, you can't play the music of mathematics, so it's harder for people to maybe grasp what this mathematics is all about.

Music is mathematics.

That's what one of the things Pythagoras said, is the relationships between the notes that make the music.

And there are certain harmonious versions.

He got that from hearing a blacksmith hit different lengths of metal in his blacksmith shop.

So

you should.

Yeah, I'm one of those very thoughtful things.

My argument about infinity is this, Simon.

Infinity is a word.

That belongs to the wordy people, like me and Robin, the word creatures, and not to the numbery people.

Oh, you can have two as well then.

Two is a word.

No, no, no.

Two is a numeral.

But the point is, you cannot place a numerical value on infinity, and therefore you cannot have a plus one to it or a minus one.

No, but

it was just proved that you could, so we can stop this.

It's been proved.

That's not

a problem with you.

That's just word-free.

It's gameplay.

But you absolutely take infinity as something which is, you know, could be bigger.

But you're right, you're right in that whenever I say.

No, he's not.

No, no, no.

He's right in as much as whenever I say, so I'm going to talk about the number infinity, somebody always corrects me, and absolutely right to correct me that infinity is not a number.

It's a concept.

And we can play with that concept using mathematical ideas and so on.

But as you say, in a concrete way, it's not a number.

There are either an infinite number of numbers or there aren't.

There are.

I don't see

where the problem is.

No, there aren't.

There aren't an infinite number of numbers, right?

Because you can always have more than infinity, and therefore infinity is a meaningless concept.

See, I'm annoyed now because I'm thinking about the infinite number of monkeys, and you get them all together, and then you go, what have they written?

And you go, they've written the Jew of Malta.

But that's by Marlowe.

We didn't have enough monkeys.

You know, the whole kind of thing is a disaster.

But this is like saying

the word zero, for example.

Another brilliant concept.

Who came up with the.

Was it the Indians?

The Indians.

Amazing idea that you can describe nothing.

It's like saying there is a more nothingy-nothing.

There's not just zero, there's even less than zero.

I know minus one, but that's a different thing.

It's a zero that's more zero.

Well, it's not, it's less than zero.

I mean, to be fair,

minus two is less than so you claim.

And there are an infinite number of those negative integers, and there are more negative infinite decimals.

For a long time, lots of mathematicians didn't accept the notion of a potential infinity versus a completed infinity.

So the Greeks did not accept completed infinities.

They didn't think that you could have a thing that was infinite.

They did accept potential infinities.

So they did accept that given any number, there was a number one bigger than it.

And you're right to say that infinity is not a well-defined term, and mathematicians, when they're working with infinity, would use more technical names for the different sizes.

We wouldn't just call them all infinity.

Again, one of the intriguing things, I mean, there have been a few mathematicians who have got to the point of madness in terms of seeking perfect patterns.

And I wonder, you know, this is, in terms of mathematics, is something that, unlike perhaps, you know, sciences, which at one point you have to go into the world, you have to leave your head.

You know, whether it's talking about particle physics, where you can then, you know, throw bundles of electrons underneath Switzerland around, whether it's kind of looking through telescopes, whether it's looking at the behavior of monkeys, whatever it might be,

you can do it all in your head.

You don't have to depart from your head and you don't have to see anyone else.

It can just keep going around there.

And I just wonder, you know, is that, in terms of the language of mathematics, is it something that sometimes is almost more in your mind than to discuss it with other mathematicians?

Do you use the language of mathematics, or is there almost a mental one?

And then there's the kind of verbal one I think mathematicians really vary, actually.

I have a very pictorial imagination, but I know other people who think far more symbolically.

And what lots of people don't realise is the extent to which maths is a collaborative exercise.

Most of the new maths I do will be done by a bunch of us sitting in a room consuming a frankly excessive quantity of coffee, writing down symbols on a whiteboard and getting overly excited at each other.

And the math that we will do during that short period is somehow just the essence of what we're trying to prove.

And then afterwards, we'll go go away and I'll sit in my room and try and turn that into cogent sentences that are going to make sense to somebody else.

There's a chap called Tim Gowers, who's one of our greatest mathematicians, he's a Fields medalist.

So there's no Nobel Prize in maths, but you get a Fields Medal every four years.

So they're much harder to get.

And Nobel Prize is easy, a Fields Medal.

That's really something.

Tim Gowers, he started something called Polymath.

So Polymath is a blog where he will just throw down a problem, he'll throw down a gortnut, and other mathematicians will then add comments to his blog, saying, Well, look, I tried this approach and it seemed to offer this, and then somebody else will build on it.

And the idea is that eventually there may be 10, 20, 50, or 100 mathematicians who will contribute comments in order to resolve this problem, and they will all get their names on the mathematical paper when it's eventually published.

So, this is a radically new way of doing mathematics.

Although, as you say, math has always been collaborative.

There's a great mathematician called Paul Erdős.

Some people say Paul Erdős was crazy, but I think he just found the thing he loved.

He would just travel around the world.

He would say, Oh, somebody's got an interesting problem in Belgrade.

I'm going to go and live with them for a month.

Someone's got an interesting problem in Sydney, I'm going to go and live with them for a month.

His entire life was in two suitcases, and he never took a salary.

He just traveled around the world solving problems and collaborating with all these people.

If you've worked with airdish, you're said to have an airdish number of one.

And if you've worked with somebody who's worked with airdish, you have an airdish number of two.

It's like the bacon number in Hollywood.

If you've been in a film with Kevin Bacon, you have a Bacon number of one.

But there are a few mathematicians now who actually have an Airdish Bacon number.

So

I actually met the man who was Russell Crowe's hand double in A Beautiful Mind.

Gets a screen credit because he was actually got a small role in one of the scenes.

So he's been in a film with Russell Crowe, who's been in a film with Gary Sinisi, probably, who's been in a film with Kevin Bacon, and he's also published papers with people that have published papers

with Paul Airdish, not Kevin Bacon.

This idea of playfulness

is important in mathematics, isn't it?

I noticed in your book there's this idea of narcissistic numbers.

Yeah, so the thing about The Simpsons, a lot of their writers are mathematicians.

I mean, people with degrees in math, master's degrees, PhDs.

One guy was even a professor at Princeton before he became a writer for The Simpsons.

And they smuggle the math into the show when no one's looking.

And there's an episode called Marge and Homer Turner Couple play.

And there's a baseball scoreboard, essentially, a jumbo vision screen at Springfield Stadium.

And it's got some numbers on it.

And you kind of think, oh, they're just random numbers relating to some score or the size of the crowd.

But one of the numbers is a narcissistic number, 8208.

And the reason that's a narcissistic number is 8208 has got four digits.

So you raise each of those digits to the fourth power.

So 8208 becomes 8 to the power 4 plus 2 to the power 4 plus 0 to the power 4 plus 8 to the power 4.

And when you add all of that up, you get back to 8208.

So the number regenerates itself from its own digits.

The number's kind of in love with itself.

Yeah.

And the number's a narcissistic number.

That's the mass I like.

Well, those cute things.

The thing I found most remarkable about these narcissistic numbers is the largest one, which has been proved to be the largest one that exists, is 115132219018, 763, 992, 565, 095, 597, 973, 971, 522, 401.

Big number.

There are only 88 of these things, and that's the largest.

No others exist.

How do you discover that number?

You can't just sit there having a go, can you?

The reason we know it's the large number, well, I say we know, I don't know, some

reason they know, these mathematicians, that number is 39 digits long.

Now, if you imagine that you had 99999 39 digits long,

so 9, 39 times.

Now, if we tried to see if that was narcissistic, we'd raise each of those nines to the 39th power.

So you've got 9 to the 39 plus 9 to the 39 plus, da da da, 39 times.

And when you add all of that up, it doesn't quite make 9999999.

So it's deficient.

And if you make the number bigger, it just gets more and more deficient.

So it's like it's a race between the additive side of of it and the number itself.

And after that 39-digit number, you just lose that race forever.

So that's why we know it.

I remember when I first came across non-Euclidean geometries,

Riemannian geometry, for example, and the idea that we all taught at school that parallel lines can never meet, that's the rule.

And Riemannian geometry says that they can, and it's one of the reasons

we can build an atomic bomb for some reason.

And I thought

this is a weird idea.

How can that be true?

We know that parallel lines don't meet.

That's why they're called parallel.

But when you think of lines of longitude or latitude, and they go to two people start, you know, 25 miles apart and they walk in a parallel in a parallel line towards the North Pole and they end up next to each other.

So they do.

In the real world, parallel lines meet all the time.

All the way through this discussion, I think there's been this notion of mathematical beauty as well.

And you hear mathematicians say that a lot.

You hear physicists say it a lot about different theories, a beautiful theory, a beautiful equation.

Could you just describe in a little bit more depth what a mathematician means by something that's beautiful?

So, a beautiful proof probably has to be proving something surprising.

It probably has to be proving it using only very simple steps,

and it probably has to prove it surprisingly quickly.

So, Hardy, in a mathematician's apology, included the proof that root 2 is irrational as his idea of a beautiful proof.

You imagine that it's a fraction, You imagine that you can write the square root of 2 as a over b.

And then you say, okay, well I'm going to square it.

So on the one side I'm going to have a squared over b squared because that's what happens when I square the fraction.

And on the other side I'm going to have 2 because I just squared root 2.

So then I can rewrite that as a squared is 2b squared.

And then you look at that and you go, oh, well if a squared is twice b squared then a squared had better be an even number.

Things that are going to square to even numbers are even themselves.

So that means there must be some number half a.

Let's call that z for a brand new thing.

So now I can write a as 2z.

So now I've got 2z squared is 2b squared, so I square the 2z, and I get 4 times z squared is 2 times b squared.

So then I can divide by 2, and I get 2 times z squared is b squared.

And now I've got that b is an even number as well.

And I thought I had written this thing as a nice little fraction.

That's beautiful.

That's wonderful.

And Simon, you made it famously made a programme on the proof of Fermat's last theorem, which was actually maybe in the way that Culver described it, not beautifully in the sense it's extremely complicated.

The proof of Fermat's last theorem is massive.

Originally, it ran to 200 pages.

Perhaps you could state the problem.

Oh, yeah,

so x squared plus y squared equals z squared has lots of solutions.

3 squared and 4 squared is 5 squared.

9 and 16 is 25.

Infinitely many.

Infinitely many.

But if we change the 2 to a higher power,

3, 4, 5, 6, 7, all the way up to infinity, those infinite number of equations have no solutions whatsoever.

And that's astonishing.

You've got an infinite number of equations with no solutions, and then one equation with an infinite number of solutions.

And Fermat said he could prove that.

He wrote in the margin of his book, he wrote, I have a truly marvelous proof, but this margin is too narrow to contain that proof.

And then he drops dead.

And

for 300 years, they wrestle over it.

But the final proof, the beauty of that final proof, is probably in the intricacy of it and the beautiful way it brings in so many tools and techniques.

I always think of E equals MC squared, which is again an extraordinary insight to think that the equivalence of matter and energy can be exactly described in this tiny little

idea.

It's not an idea, an equation, it's a thing that is true, isn't it?

And I think the fact that Fermat's last theorem is intricate doesn't

rule out its beauty because lots of buildings, for example, are incredibly, a landscape is incredibly intricate, but doesn't rule out its possibility of beauty.

And the secret, I suppose, being remembered forever as a mathematician is always leave a note at the back of an exercise book saying, I think I've got the solution, but I can't quite fit it here.

And then when you wake up alive the next day, go and write, throw away that one and fill it in again.

So we asked the audience as usual a question, and today's question was: if the answer is 42, what could the question be?

So I have one here.

According to Boris Johnson, it's the average IQ of a member of the working class.

How many units of alcohol does it take to get through Christmas Day?

What is the level of the best band ever?

What is the average age of Brian's female fans?

Average, though it's a very wide Gaussian.

So there we had hundreds of answers.

We have

almost an infinite number of answers.

So this brings us to the end of the show.

Thank you very much to our guests, John Lloyd, Colveroni Dougal and Simon Singh.

Next week we ask, is it time we said to hell with the pandas just because they're pretty?

Should we be caring more about the uglier animals such as the blobfish, which is undoubtedly the most anthropomorphic of any animals?

Jitter Brobfish goes, oh no, it hasn't happened to me again, is it?

Am I

impossible?

Have a look at a blobfish, you will find out.

So that's the end of it today.

Thank you very much for listening and goodbye.

That was the Infinite Monkey Cage podcast.

I hope you enjoyed it.

Did you spot the 15 minutes that was cut out for radio?

Hmm.

Anyway, there's a competition in itself.

What do you think?

It should be more than 15 minutes.

Shut up.

It's your fault.

You downloaded it.

Anyway, there's other scientific programmes also that you can listen to.

Yeah, there's that one with Jimmy Alca-Seltzer.

Life Scientific.

There's Adam Brother Fiddy.

His dad discovered the atomic nucleus.

That's Inside Science.

All in the Mind with Claudia Hammond.

Richard Hammond's sister.

Richard Hammond's sister.

Thank you very much, Brian.

And also Frontiers, a selection of science documents on many, many different subjects.

These are some of the science programs that you can listen to.

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