How to think like a mathematician

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Hello, I'm Brian Cox.

And I'm Robin Intz, and this is the Infinite Monkey Cage.

We are back from our Australian holiday.

All of us, as you can see, heavily bronzed by the old'em rain that you managed to bring to Australia.

How, what are the chances of us having a month of rain in Australia when you promised it was going to be all lovely and like spring?

Well,

all I'm saying is don't travel with him.

He brings the Moors with him.

He's like a terrible Heathcliff figure.

And I refer that Heathcliff figure, not Laurence Olivier playing him, Cliff Richard in the awful musicals.

Anyway.

Today's show is

basically it's a rebuttal to, I'm sure everyone here knows someone who goes, I don't even know, what was the point in learning maths at school?

All of those kind of stupid things where you had to learn about, you know, Euclidean Euclidean geometry and algebra and quadrilateral equations

quadrilateral equations yeah yeah yeah I remember I'm playing the part of someone who doesn't like maths yeah it was very convincing

you don't need to ask someone who doesn't like maths because I bloody hate maths so let's just focus on me do you do you know do you genuinely not like that no I didn't really like it I think it was my first maths teacher he just would go on and on and on and we would want to finish the lesson, and then he'd just go off at a tangent.

Mathematicians are often seen as otherworldly types, living in an alternative universe of numbers.

It's not this otherworldly eccentricity that we're looking at today.

On today's show, we're looking at pragmatic mathematics.

Can thinking like a mathematician make life more livable?

Is there better living through mathematics?

Today, we're joined by two mathematicians, one cartoonist, and someone who definitely knows how to divide up the cake, and they are.

I'm Professor Hannah Fry, I am a professor of mathematics at UCL.

And the elusive solution that I am looking for is a way to tell people that without them immediately asking me a very hard sum.

And I'm Randall Monroe.

I'm a cartoonist.

I draw XK C D and write books where I answer people's ridiculous questions using science and math.

And the elusive solution that I'm looking for is I've been texted a lot of six-digit codes that I am not allowed to tell anyone, and I want to know when I can finally share them because there are some really cool numbers in there.

Hi, I'm Dr.

Eugenia Cheng.

I'm a professor of maths and scientist in residence at the School of the Art Institute of Chicago.

And the elusive solution I'm looking for is how to fit an infinite amount of chocolate in my finite stomach.

Oh, that was mine.

Hello, I'm Joe Browns.

I do a bit of catwalk modelling

and I'm hoping to be the next Prime Minister.

And the elusive solution I'm looking for is how to stop people treating me like a half-wit.

Now I'm getting older.

This is our panel.

Hannah, before we get into the how do we live a better life and those kind of things, I wanted to start rigorously at the beginning.

So what is mathematics?

Oh, an easy one then.

Sure, thank you.

Okay, so I think that mathematics is a way of thinking rather than a thing itself.

So, I think it's about searching for patterns.

I think it's about building absolute truths on one another.

And I think it's about being playful with ideas.

I think it's those things together.

Or, the other thing that it could be, if none of that works as a definition, I think it's also what's that phrase?

A bit like pornography, difficult to define, but you know it when you see it.

We can't be building absolute truths on one another, can we?

Because that implies that you can derive.

You don't worry about the bottom too much.

Russell and Wallet,

all the axioms.

So, okay, they did try and do this, right?

At the end, the turn of about the 1900s or so, the mathematicians were like, hang on a second, we've got all of these truths built on truth, built on truths.

What's at the bottom?

Have we really made sure that we've got

solid foundations here?

So they went right back to basics.

There was one guy who decided to try and define the concept of two-ness.

He spent seven years and then just gave up, couldn't manage it.

Bertrand Russell also tried to define, or tried to prove, sorry, that one plus one equals two.

And if I remember rightly, it took about 300 pages of like very dense mathematical notation.

And then when they wrote the proof, said, once

arithmetic addition has been defined, then it will follow that one plus one equals two.

Basically, it's really, really hard to do.

You can't do this stuff.

And then, you you know, some other people came along and said, actually, I think you really can't prove this.

There's always going to be some things that you just can't prove.

And then so we just sort of gave up and don't worry about it too much.

Does that give you confidence, Joe, that basically it's not possible to prove one plus one equals two, and so we give up at that point?

I kind of went into a coma half later

because

I wouldn't even think you'd need to prove it, you know.

But as I've already said, I'm not a mathematician.

I found it, I've really struggled with it at school.

I mean, my maths teacher said to me I was very average, which I thought was quite mean of him.

Don't worry, I've got bloody loads of those.

I can always tell when you've had dinner with Giles Brandry.

It always just picks up on it, doesn't it?

You know what?

I'm proud to say I've never had dinner with Giles Brandford, but some well I might do now.

You see

following on,

how can then such an abstract logical framework, and as you said, even the foundations of that framework we can argue about for 300 pages.

But how can that be applied to everyday life?

Oh, in all kinds of ways.

Because the thing is that then you have this series of rules that are sort of separate from people that you know to be true, and then you can play with them and manipulate them in any way that you like.

And then there are certain things that are just absolute facts, right?

Like, for example, okay, let's say you're at the North Pole and it's zero degrees, right?

And then somewhere on the equator, it's like 40 degrees.

You know as a fact that there's going to be somewhere

along that line

from the North Pole to the equator, at every possible temperature in between the two, right?

It has to be the case, right?

And that is like a mathematical fact called the intermediate value theorem.

But then, once you have that as a fact, that if you have something that sort of changes slowly and incrementally,

and you've got these two points, zero and 50 in that example, and you kind of move through it, there has to be a point in the middle, then you can take that idea and you can apply it to different situations.

For example, if you're with a mathematician in a pub, right?

I mean, unusual events.

I wanted to, I don't understand it.

I said, he'll have a pint and he'll have a pint, and you bought no pint.

Oh, no, no.

He drank them both.

You see why these things are important.

Okay, so you know, when you're in a pub and like there's a bit of a wobbly table, and then people sort of get a bit of card and they like to shove it underneath the bottom.

You don't need to do that.

You don't need to do that because the intermediate value theorem tells you a way to mathematically fix a wobbly table, right?

So if you imagine you've got four legs, okay, so you can set up the table so that three of the legs are touching the ground and one of them is floating in the air.

And then if you imagine rotating that table so there's three legs stay attached to the ground, if you rotate it, then either the free leg is going to stay in the air or at some point it's going to hit a bump.

So if you just sort of imagine ignoring the laws of physics for a moment, right, so that

sorry,

try, Brian, try.

If you imagine that it could like pass through that bump, well, then that means if you carried on rotating the table, there'd be a point where the tip of that leg would be below the ground, right?

So, at one point, it's above the ground, at another point it's below the ground, which means there has to be a perfect point in between the two where that fourth leg is touching the ground.

And so, now you never have to have wobbly tables.

Only slight caveat with this is that there's no guarantee that the table will actually be flat.

But it works every time.

So it's a completely idealized.

No, I mean, it actually, no, it properly works.

It properly works.

Does it?

So you just rotate the table and it will find.

So you don't have to put beer mats underneath the leg or anything, just rotate it.

Yep.

Can I give you another application of the intermediate value theorem?

I'm so glad you mentioned it.

It's one of my favorite theorems.

Mine is much more ridiculous, though, because I once used it to settle an argument with an American about whether baby carrots exist.

Because in America, they don't have baby carrots, right?

They have baby cut carrots.

They use machines to cut carrots down into a baby size.

But the intermediate value.

You can have baby carrots, because it grows, right?

That's the intermediate value theorem.

You did it.

You applied the intermediate value theorem.

So at some point,

so there has to be some point in the middle before it became a big carrot when it was a little carrot.

It's the intermediate value theorem.

Can you just state the intermediate value theorem

without

precisely?

Let me see if I can, because you have to get the assumptions right.

And the assumption is that the function is continuous.

That's the key.

And that's actually

quite difficult definition.

And so

as long as it's continuous and it hits a certain point

and then another point that's higher up, then it will have to hit every point in between at some point.

There's an example in your book, which is a more

serious example where you can use mathematics to try to think more clearly about problems.

And it was a famous case in 2018 of the bakers that refused to bake a cake for a same-sex wedding.

Yes, so I'm a pure mathematician, which is usually taken to mean that I do stuff that's so abstract it has absolutely nothing to do with real life.

Well, first of all, what's real anyway?

What is real life?

Do we exist?

But what to be sure we do, yeah.

I agree with Hannah that maths is really about thinking more clearly.

And for me, having abstract things means that I can find it's about spotting patterns, and it can include patterns between different ways that people think.

And often, there's an analogy between someone's argument and some much more ridiculous argument.

And that if we can spot that analogy, then we can figure out what is going on with those arguments.

And the example I think you're referring to is where there's some lawsuit in America where some Christian bakers refused to bake a cake for a same-sex wedding.

And someone on Twitter, because everyone on Twitter is great, someone on Twitter said, said, that's like

forcing Jews to bake a cake for a Nazi wedding.

Now, there is very Twitter.

There is an analogy.

And so here's the thing, it's very tempting to just yell at people and go, oh my God, that's so dumb.

And then they yell at you and go, oh my God, you're so dumb.

And then that's the internet.

But I think it's more productive to acknowledge the very, very violent sense in which there actually is an analogy between those two things, which is at a very high level, saying,

yes, it's forcing people to bake a cake for people they disagree with.

But on the one hand, it's people who committed mass genocide against them and tried to kill off their entire people.

And on the other case, it isn't.

And so for me, abstract thinking is about very carefully

being able to construct arguments and find the sources of differences between arguments, which can actually help us to empathise with people.

See, there's something that often doesn't get put in the same sentence: abstract maths and empathy.

And so I'm trying to show, because I really believe it's true, that abstract maths helps me to understand what other people really mean.

So it's a mode of thought which so you're saying that it's it's useful because it disconnects you from the the the the emotional sort of uh in a way yes abstract maths is about making comparisons between different situations finding out what is similar about them but at the same time seeing what is different about them and so that when we have arguments with people instead of just going you're wrong you idiot and then they go you're wrong you idiot you go okay here is a sense in which you have a point and here is a sense in which I have a point and now we can go a bit deeper and find out where the root of those differences is, rather than just kind of yelling at each other.

I mean, Randall, Sindra, we're dealing there talking also about ideas of the abstract.

And you get, I mean, some of the things that you've dealt with in the Wadif books and also online is a wonderful way of seeing that I think very often people hide that abstract thinking.

And you have kind of opened this fantastic door of saying, whatever the most wonderful or strange quandary is that you have,

I'm going to try and work out how we can get through it.

Yeah, One of the things that to me is really fun about these tools of math and abstract thinking is that they don't care whether the question you're asking them is ridiculous or not.

And so

you can apply the same math to the question of what is the tidal force created by the moon to what is the tidal force created by a large wheel of cheese sitting nearby.

And you can use the same equations, the same math to figure that out.

That to me is really exciting.

That I can just take these tools and not just apply them to like the things you're supposed to in school, but to anything.

So what's the give us some examples of the rather surreal examples of that mode of thought?

The intermediate value theorem, someone asked a question that applies it in a way that I had never thought about before.

Eventually, all of the stars are going to burn out.

Sorry if that's a spoiler.

And I knew this.

I did a physics degree and they talked about this.

And then I also knew that if you go really far forward into the the future, all of the stars will be at absolute zero at some point.

Everything will cool down, all of the remnants that are left behind of these dead stars.

And what had never occurred to me was that meant at some point in between, there would be a point when the stars were about room temperature.

And so someone asked, when will that time be?

And can I go touch one?

And I had never thought about that before.

And that led me to think, okay, how on earth would you even approach this problem?

How do you figure out what year that's going to be?

And how would you get near one of these stars?

What would it be like?

None of these are things that are going to happen.

But it turns out

you can take scientific papers and demand these answers out of them.

And if you can figure out how to do the math right,

they give them to you.

It's not going to be for a while.

There's some maths in this, actually, because you can't cool something down to absolute zero in a finite number of steps.

No, the universe may go on forever, though.

So it's going to get closer and closer.

So is it possible to cool something down to absolute zero when you can't do it in a finite number of steps if the age of the universe is infinite?

That's not right.

That's not what I'm saying.

Each your muscle sounds.

So here's the thing, but we can actually even do an infinite number of things in a finite length of time.

We don't even need to go all the way to infinity because

you all did it today.

So this is one of Zeno's paradoxes.

How did you you all get here?

Well, first, you had to cover half the distance to get here, and then you had to cover half of the remaining distance, and then you had to cover half of the remaining distance of that, and then you had to go,

and so it sounds like that's an infinite number, that is an infinite number of things, and it sounds like you wouldn't be able to do that in a finite number of time, but you did, even if you got stuck on the train, you did all get here in a finite amount of time.

So, we go around doing infinite numbers of things every day.

I mean, is this partly, and I'll ask you, actually, you're under this bit when you're dealing with sometimes these abstract ideas and sometimes these ideas which we have have as you know the laws of physics, but is this a bit like when you're philosophically looking at these things, that you need to know the laws of physics in the same way that someone to play the piano badly in an amusing way needs to know how to play it well?

So you need to have the rules to then allow your imagination to go wild within there and create something truly satisfying?

Yeah, I think that sometimes you need to know which things are important so you know which ones you can ignore.

I think that's definitely true.

Like you need to understand like what is the measurement error in this, you know, in this quantity, like how much could this vary, how would it affect the solution, and then you can know, okay, it doesn't vary enough to matter for the solution.

And then you can ignore that.

But you can't ignore it until you have first figured it out and known, okay, yes, I didn't need to know that after all.

Well, that's what I wanted to ask you, Hannah, because you deal with the mathematics of cities.

And sometimes we would imagine that the idea of using maths when you're also dealing with emotional creatures, creatures that are seen as making irrational decisions, that the two might be very hard to join together to take the mathematics of the city.

So, how do you start out?

Can you give us an example of the mathematics of cities?

Yeah, of course.

Okay, so I think that when you try and use mathematics with its like really rigorous rules and like

hard-coded into the universe, and then you try and apply it to people it's really difficult especially when you're dealing with just like one person but but something happens when you zoom out when you start to look at big groups of people together it's like the noise in one direction ends up sort of cancelling out the noise in another direction and so while you know so you can do things like for example you can say with a really high degree of accuracy how many people are going to use the tube on a Monday morning right or how many people are going to turn up at AE on a Friday night and the thing is is that each one of those people in AE has had their own own life story, has had their own accident, their own reason to be there.

They shouldn't, in theory, be connected to one another, and they're not.

But something happens when you zoom out that actually, our behavior collectively becomes really predictable.

Is that in some ways disheartening?

When you, you know, this idea that people are, you know, there are a lot of people out there, you ain't the boss of me, et cetera.

And people like to imagine that they are moving, you know, with a level of freedom away from the context of everyone else.

But actually, the truth is that we are more of a mass as we move around, and those patterns.

So, when people first discovered this, that was genuinely how people thought about it.

So, it was a guy called a French guy called Ketelet.

This is in like maybe 17, 1800s.

And he was looking at crime across all of France.

He had like all of the crime stats.

And then he looked and saw that, like, the number of people who got murdered every year in all these regions of France was basically completely unchanging.

And not only that, it was like the type of murder.

So, there was, I mean, it's like France in whenever it was, 1800.

So there were some excellent ways of dying, right?

So like, there was lots of sort of the same number of jousting deaths.

I'm making up.

But there was definitely bludgeoning by a stone, that was one, and swords, that kind of thing, poisoning.

And those numbers were unchanging year on, year on year in every region.

And so then there was this big debate because it was like, well, hang on a second.

If there is this pattern that is really clearly there, how can you punish people for committing murder if, like, are they definitely doing it under their own steam?

Because, because otherwise, why would this sort of universal law appear?

And it's like, how do you take away an individual person's freedom for committing something if that pattern has to be there because of the universe?

So, the argument is with the murder, statistically, someone has to do it.

So,

if you get to the end of the year and you've had like five too few murders, were they like, who's going to do it?

Pick your stone to bludgeon with.

Yeah, that's for volunteers.

It's interesting, isn't it?

It's basically the idea behind Asimov's foundation, isn't it?

That given a large enough population, you could in principle predict how the future will play out.

Is that even conceivably a possibility?

I mean, Asimov being Asimov said, you know, you need a galaxy of billions and billions and trillions of people perhaps, but then statistically you can start to understand how the future will unfold.

I mean, that's the idea behind those books.

Yeah, so, okay, I don't think you can.

I don't think you can, basically,

in short.

And I think the reason why you can't is because I'm sort of sitting here saying, Yeah, you can make these predictions, but there's still, as Randall said, there's still error.

And the thing is, is that like when you start taking those predictions, you start taking them too seriously, then I think that that's where things start to become really problematic.

Because then you start applying them to individuals as though it's like an absolute cold hard fact, like somebody must murder.

Like, for instance, I've seen this one academic who claims that he can take somebody when they're born and tell you whether or not they'll be a criminal by the time they're 18, right?

And that's like seriously problematic.

But I think that the other reason why you just can't do this stuff, there's just too much noise, really.

I think you can do it in order to design a better city.

Yeah, but I mean, if you don't take it too seriously, which I think would work is you could make men stay in after six o'clock at night,

and there won't be any murders at night at all, apart from maybe a pissed woman.

So,

it's true, though, you do have statistics where huge numbers of, say, you know, one group are committing crimes more than another, or crimes are happening at particular times.

Or in a city, I would think, mathematically speaking, crimes are happening more often in particular areas of a city where it's easier for people to commit crimes.

So, I mean, I would imagine that you can actually predict and change

on that basis.

But I don't even know if

town planners use that sort of information, do they?

Okay, so it's true, you can.

You can.

You can look at a city and you can see where the crime hotspots are and you can see how the road network influences that.

Because, of course, you know, different types of crimes can happen on a cul-de-sac versus a really busy high street.

You can do all of those things.

The problem is, then what do you do about it?

Because what people have tried, and this is about five or six years ago, there were a few mathematicians who were like, hey, we can do this.

So they set up this company that would tell the police where the crime was going to happen that night.

Right.

And the thing is, well, okay, so there's two big problems here.

The first is that somehow in that translation, it's changed from being a probability, really tiny probabilities, too, here, by the way, to like being a cold, hard fact.

And that sort of uncertainty gets lost in translation.

So suddenly you have police turning up to an area being like, right, you know, thinking they're in minority report.

You know, where's the crime?

So that's kind of one problem.

But the second problem is that the way that those models work is that they work on the basis of how much crime has been reported from an area.

And so if you flood an area with loads of police because you think there's going to be more crime that happens there, then more crime is going to be spotted.

So more crime will be reported.

Yeah, because the police will be doing it.

So there's this thing they do in Chicago where they think there's going to be crime, which is that they play classical music into the street.

And all the criminals are like, oh, classical music.

Oh, I'm not going to hang around here.

And then they do that.

Yeah, they do.

Well, I don't know about the criminals, but they do play classical music in the street.

They play classical music in the street.

All the time, or just on particular crime time.

I suppose I haven't been there all the time, so I don't know, but I think it's all the time.

They just kind of pipe it into the street.

But I think that is, I mean, I can see that even you know, when one of the train stations, tube stations just up from here, yeah, maybe it's Euston Square, they quite often play classical music.

And you can literally see by the fact that if you are moving in a world that is frenetic, but there is something beautiful around you,

I can see how that would also have, you know, that effect of not being.

Oh, I love Beethoven's Fifth, I'm not going to murder my grandma tonight.

I think that's much more difficult.

Not Beethoven's Beethoven's Fifth.

That's the irony of the deaf grandmother, and that's how it was managed to get there.

You know, but it is kind of, but I can see that, you know, certain unique, you've got some LGAR going on there.

Although, although, do you remember that experiment where they took, what's his name?

Is it Joshua Bell?

And they stood him in a

Washington, D.C., yeah, in a tube station, and they gave him a Strado Barrier, and he stood there.

And he'd been like playing to this sold-out crowd, like night after night after night.

You know, could not get a ticket for love no money.

And there he was in the tube station with like a hat, a bucket hat, waiting for coins.

And basically, no one stopped at all.

No one gave him any money.

Every child who walked past stopped and listened.

Isn't that beautiful?

That gives me so much hope for the future.

And one of the things that I think it's like when we talk about maths, I don't know if you find this when you go around doing events, that children, little children, aren't afraid of maths yet.

And they're so excited by these ideas and they're so excited by thinking about infinity.

And they're not bothered by the fact that they don't understand it.

They love the fact that they don't understand it.

There's all these possibilities there for becoming more intelligent when you don't understand something.

And then somehow we get them through education, and all we do during the education process is get them to hate maths, worry about not understanding things, and then run away when something seems difficult.

It's really tragic.

And then the adults also won't stop and listen to the music, and all the children do.

I think it's beautiful.

Yeah, sorry.

Sorry, we'll move away from beauty.

Don't worry, Brian.

Let's go back to the cold mathematics universe.

It seems to me there are two

different uses of mathematics.

What you're describing, Hannah, is a very specific use of statistics.

For example, the way that crowds move through a city, you can design a city.

And I think what you're describing, Eugenia, is

a transferable skill that emerges from studying pure mathematics.

So it's a way of thinking that you derive from the study itself.

One of the things I think is really important is to stress that those are both really important aspects of maths.

But there's one, the transferability I think is really crucial because so many people maybe even appreciate the fact that somebody else does maths.

Do you appreciate the fact that somebody else does maths, Joe?

Very much.

Right.

So you appreciate the fact.

Few, right on.

Yay.

But then they go, well, I don't have to do it because someone else is doing it.

It's true.

And the thing is that they are actually in some sense correct.

I want to validate them, even if Joe doesn't like being validated.

I still want to validate everyone who thinks that they don't have to do maths because someone else will do it for them.

That is correct.

But that's because often the kinds kinds of maths that you're told are useful are not the ones you need to do.

It's like, oh, yeah, maths can send a spaceship into space, and maths can fly a plane, and maths makes your phone work, which is all true, but you don't have to understand maths to use your phone.

And so it can seem that that's just for other people.

But that's not the point.

The point is being able to use your brain in a way that's kind of better.

It's a core strength.

It's like doing core exercises so that you can use the rest of your muscles better or just not fall over when you're walking up the street.

And so for me, abstract maths is a core strength inside my brain that enables me to use the rest of my brain in a more efficacious way.

So Randall, equally importantly, you worked out what would happen if you shrunk Jupiter down to the size of a house.

Yeah, there's a few different aspects of mathematics.

There's like more kind of building models and taking theoretical structures and finding ways that they are analogous to everyday life.

And then there's the kind that I think about as more like counting and measuring.

And so a lot of the figuring out,

for example, the question someone gave me was if you shrunk Jupiter down to the size of the house, someone's house, and moved it into a neighborhood in place of a house, and then just kind of let it go, what would happen?

And

so my first thought, I think, when I saw that, was, is there a homeowners association?

Or is there, like, you're probably going to get in trouble.

But what I think is fun about these questions is like I don't know the answer I could imagine it being really really catastrophic, but then I could also imagine it not being it's just a ball of gas.

It's you know about the size of a house.

How bad could that be?

And

So if you took the Sun and shrunk it down to three kilometers in radius

then it would form a black hole inexorably.

If you took the earth and shrunk it down to about, I think it's just under under a centimetre in radius,

it would form a black hole.

Wait, you giving the same density, or are you just taking all of that mass and shrinking it down?

Yes, you take the whole thing,

all the matter in the Earth, and shrink it down.

So, what I'm trying to work out in my head is: so I know that the Sun, because I've got the numbers in my mind, I know the Sun down to three kilometers in radius will form a black hole.

I know the Earth down to just under a centimetre in radius will form a black hole.

Jupiter, Jupiter,

in between those.

So, So it's a good question, actually.

So what I want to know is the Schwarzschild radius of Jupiter, basically.

I should be able to work that out, but I need to know the mass.

Yeah, yeah.

So the real question here is

which kind of shrinking are we talking about?

Because I think of it, there's the compression shrinking, where you keep all the mass there and fit it into a smaller space.

And then there's,

I think of it as the honey, I shrunk the kids style of shrinking, where you're making the things smaller, but it seems like you're taking away some of the mass.

So it's it's made up of the same stuff, there's just less of it.

And you know, like if a character gets shrunk down in a movie, they don't usually

stay so heavy that they just immediately punch through the floor.

And so that's the way I interpreted this for Jupiter.

I was saying, well, what if you shrink it down?

It's made out of all the same stuff, same temperatures, same pressures, same chemicals.

There are just fewer of them, so it's the size of a house.

No, but then it would just sort of dissipate, right?

Because

then it's just a small bowl of microphone.

How bad can it be?

Yeah.

Well, but it's

together.

It is, it is.

It's gas that's really, really hot.

Yeah.

And so because it's really small and really, really hot and under high pressure, but it doesn't have all that mass that it had when it was big before we started messing around with it,

the gravity isn't there to hold it together, which means that it would start to expand.

And it would expand rapidly, which when you hear a physicist talking about that, often means what a normal person would call exploding.

And so, this, I found if you have a Jupiter-sized ball of gas with the temperatures and pressures of Jupiter, it turns out it would be enough to obliterate not just like your block of houses, but the entire surrounding neighborhood.

And you'd end up with a sort of a mushroom cloud rising over your portion of the city.

But what I thought was really fascinating about this is the more I thought about it, the more I realized this is actually just the process that formed Jupiter in reverse.

Because the reason Jupiter is really hot is because there was a big ball of gas in space that was cool and diffuse, and then it fell together under the force of gravity.

And falling together heated it up.

And that heat is still there.

There's some contribution from radioactivity,

but it's just it got squished together, and the solar system is only four or five billion years old, so it hasn't had time to cool down yet.

And so if you took all that heat and then removed the gravity holding it together, it would go from being a small hot ball of gas back to being a large, diffuse, cold one like it was when the universe formed.

So you'd just be watching Jupiter's formation in reverse.

It's an interesting thing, because all I'm thinking is that, you know, you're talking about the issue is what if Jupiter was the size of a house?

And Brian's so successful, all he's thinking about is, I need a house the size of Jupiter.

So it's kind of an interesting thing to.

There's a couple.

There's another one.

I mean, there's one more.

You've got to deal with the soup thing, right?

But the soup, you were asked by someone, what if the atmosphere, I hope this is right, the atmosphere of the solar system, rather than being made of what it was, was made of soup.

And I weren't, were they specific about the kind of soup?

No, this question came from a five-year-old named Amelia, which is my favorite questions come from little kids because I think, I don't know that little kids are necessarily like more creative than adults, but I do think they are much more nervous about asking questions that sound silly or make them look like they don't know what they're talking about.

So adults will try to ask very scientific questions, like sessions that have a lot of science words in them.

And little kids just ask questions questions like, what if I filled the solar system with soup?

Which turns out to be like a much more scientifically fun question than a lot of the adult ones.

Yeah, and so they, I don't, she asked specifically, what if she filled the solar system with soup out to Jupiter?

And

this

would create a black hole situation.

That is a lot of mass in one place, and the gravity would be so strong that not even light would be able to escape.

You can calculate where the event horizon of this supermassive black hole would be, and it's somewhere between Uranus and Neptune.

So everything inside of that zone

would be doomed, falling toward the center, forming probably a singularity.

Also, everything outside that zone would be doomed too, because it would soon fall inside that zone.

So this would completely destroy the solar system and then start in on sucking in nearby solar systems as well.

What type of soup are we talking about?

Yeah, so I looked at this.

I mean, the nice thing, this is one of those things where, because you can always try to figure out how much does the input matter, how much does the exact density of the soup matter.

And if you find a really diffuse soup, something that's like, you know, cotton candy density,

you know, maybe you could come up with something where it wouldn't immediately form a black hole.

But if it's a soup that has any kind of a water base, a wet soup, I don't know about cooking,

then you're going to have something that's about the density of water.

I did try to figure out the calorie count.

I tried the Campbell's tomato soup.

It came somewhere around 10 to the 41, 10 to the 42 calories.

You could maybe cut that down a little bit if it was more of a chicken.

It's gone from a black hole to a war hole now, which I think is

this is actually.

Two cultures there divided.

There were seven people who liked art.

The rest of them, yeah.

So Hannah, I mean, you know, Randall goes looking for

some of the, you know, the more peculiar questions, some of the things which can start off as surreal or abstract, but then you actually do find out the science.

I presume that when you're doing public lectures as well, you have moments where someone will ask a question, that moment they lose their shame.

And they think, I really want to know this.

What's the most kind of unusual quest that people have been on?

Okay, so a few years ago, I did a

talk on the mathematics of love, right?

But specifically, I had a whole thing about divorce.

And I was doing this talk and I used it as an example.

And then, at the end, someone came up to me and said, Okay, I was really interested in that example about the dynamics of arguments in a couple just before they break up.

Because I have this girl that I want to get with, and she's married, so how do I

think you're right that the question's from kids, Abella?

The other one I loved was you'd worked out how many people you have to date before, statistically speaking, you should choose your

I got in lots of trouble for this, right?

Because,

okay, it's true that you can use something called optimal stopping theory and you can move.

Optimal what, sorry?

Oh, sorry.

Optimal stopping.

Right.

So, okay, the idea, Joe, let's say that you're single and you're like, okay, I bloody wish I was.

Yeah, okay, let's go back to those happy days in my head.

Okay, so let's say you're like, okay, you know what?

I want to be settled down in a year.

So, what you do is you take that time window, and then the first 37% of it, you just go wild.

You just do whatever you want, you just enjoy yourself, just get a sense of what's available to you.

And then, after that window has passed, you then, the next person who comes along that's better than everyone else that you've met before, that's the person.

If you want to maximise your chance of getting someone good.

Now, the problem with this.

So, specifically, so what do you mean?

You said the first 37%,

yeah.

Right.

Yeah, so that's

of the time, exactly.

So you

have a lot of liaisons.

Yes, you just, you just

have to be a little bit more.

What decade did we

look?

He suddenly appears in a frock coat here.

It's radio four.

I don't want to offend anyone.

So yeah, so I mean you've had all of these liaisons.

After the liaison.

Sometimes they've been a little bit flirty perhaps.

Yeah.

Someone's lost a lace neckerchief, etc.

Right, and exactly.

How many do you need to have?

Oh, it's all about cutting up in the time window, right?

So it doesn't matter about how many.

Because what you don't want to do is you don't want to just like the first person who comes along, be like, yes, you.

But then equally, you don't want to leave it too long if you actually want to, if that's your main goal.

So this is like the way to maximize your chances of getting the best person.

But the thing is, it doesn't guarantee it, right?

It definitely doesn't guarantee it because there are lots of ways that it can go wrong.

So for starters, your perfect person could appear in that first 37% liaison window

when you're like goodbye and off into the sunset with like a swish of your petticoat.

And then they're gone forever.

Or the other thing is, it could be that it just so happens that all of the people that you meet in your liaison window are just like really boring people.

And then the next person that comes along is still really terrible, but just maybe marginally better than everyone that you've met before.

And then, if you're following the rules, you're like, okay, great, it's you, we're done.

Which means, you know, you only end up with somebody who's marginally better than the first 37%.

It's not kind of a good situation.

And overall, your chances of ending up, right, of getting the one, I think are about a third, right, if you do it this way.

Which means that two-thirds of the time, you're better following another strategy.

After this talk, I got in lots.

I got so many emails from people,

the two-thirds of of all the people who watched it, telling me how wrong I was.

And, Joe, what was your strategy then?

Just to do the liaison window for about 20 years.

Well, I think we've covered it, we've managed to do love, soup, and social justice in one show.

So,

I think we've covered a lot of ground with mathematics.

So, first of all, can we just say thank you to our panel, who are Dr.

Eugenia Cheng, Professor Hannah Frye, Randall Munro, and Joe Brand.

There, we asked our audience if you could find an equation to solve one problem of living.

What is the problem of living you would like to be dealt with?

And let's see, the first

question I got is: How long to keep an odd sock in the vain hope its sister sock will turn up?

So, that is the three-sock problem there.

So, we're getting towards that again.

Let's be inclusive towards socks.

Why do they have to match anyway?

Then you can just include your sock already and not wait.

And how are they?

How do we know know they're really three socks anyway?

How do I know I'm wearing a pair of socks?

Let's set and truck.

Belinda said, an equation to figure out how long can you hold a biscuit in a cup of tea before the end breaks off.

Which sounds like one of yours, Randall.

That's a good question.

You probably can't.

That sounds like a question for an experimentalist.

It does, doesn't it?

I actually suggest you do that, Belinda, because it is an experimental question.

You get a lot of cups of tea, a lot of biscuits, and do it.

Yeah.

I'm sure someone has.

there has been, I'm sure there was a paper on the

biscuits.

They do do that all the time with biscuits try and sort of check out how long they last because it's obviously really important.

Hobnobs, in my experience, they're the best ones for they last about half an hour.

This is a good one here.

It's from I'm not going to say whether it's from a man or a woman.

How do you calculate the minimum acceptable amount to spend on a spouse's Christmas present?

I suspect that might be from my husband.

So, thank you very much, everyone.

And next week, we are going to be doing a show about how to commit the perfect murder.

And then, a week after that, I'll be back with my co-host, Jim Al Khalili,

Or I'll be back with Dara O'Breen.

Well,

let us wait and see.

Thank you very much.

Goodbye.

In the infinite monkey cage.

In the infinite monkey cage.

Till now, nice again.

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