The Rest Is Science - Paradoxes Of Infinity (Infinity Part 1)

Episode Date: March 24, 2026

Is infinity actually a real number, or just a brilliant mathematical hallucination? Professor Hannah Fry and Michael Stevens (VSauce) tumble down the numerical rabbit hole to explore the mind-bending origins of infinity. They unpick exactly how humanity managed to trap the endless void. From ancient paradoxes to endless hotel rooms, they dive into the bizarre history of our universe's most impossible idea. ------------------- For more information about Cancer Research UK, their research, breakthroughs and how you can support them, visit ⁠⁠https://cancerresearchuk.org/restisscience⁠⁠ Cancer Research UK is a registered charity in England and Wales (1089464), Scotland (SC041666), the Isle of Man (1103) and Jersey (247). A company limited by guarantee. Registered company in England and Wales (4325234) and the Isle of Man (5713F). Registered address: 2 Redman Place, London, E20 1JQ. ------------------- Find The Rest Is Science all over the internet by ⁠⁠clicking here.⁠⁠ ------------------- Video Producer: Adam Thornton + Oli Oakley Video & Social: Bex Tyrrell Assistant Producer: Imee Marriott Researcher: Lucy Lipscomb Producer: Simona Rata Senior Producer: Lauren Armstrong-Carter Head Of Digital: Samuel Oakley Exec Producer: Neil Fearn Learn more about your ad choices. Visit podcastchoices.com/adchoices

Welcome to the rest of science. I'm Hannah Frye. And I'm Michael Stevens. Okay, Michael, I'm going to start with an easy question for you today. Okay. Would you want to live forever? No. No, me neither. I think life only has meaning because it's finite. Yeah, I agree.

Here's another question for you, though, that's related. If you had the choice between dying in the next five minutes or living another week, what would you, what would you choose? This episode is brought to you by Cancer Research UK. If you wanted to type out the entire human genome, you would have to type at 60 words a minute for eight hours a day for about 50 years. Okay, that's the scale of the DNA rulebook inside each one of your cells, telling it when to grow, when to divide, and when to stop. And different tissues read that same rule book in different ways. So a skin cell doesn't behave like a lung cell. And cancer can begin when those instructions change.

Not one dramatic moment, but through small, gradual edits over time. Now, cancer isn't one disease. It is more than 200 types shaped by where those changes to the rule book happen and how cells respond. Cancer Research UK is the world's largest charitable funder of cancer research, backing studies across all types of cancer. Work that takes years of very careful, steady progress to deliver each breakthrough. For more information about Cancer Research UK, their research, breakthroughs, and how you can support them, visit Cancer Researchuk.org forward slash the rest is science. Amazon presents Laura versus fruit flies.

Swarming your fruit and terrorizing your kitchen, these little freaks multiply at a rate that would make a rabbit say, yo, chill. But Laura shopped on. Amazon and saved on cleaning spray, countertop wipes, and fly traps. Hey, fruit flies, your baby boom ends here. Save the Everyday with Amazon. This episode is brought to you by Defender. With the towing capacity of 3,500 kilograms and a weighting depth of 900 millimeters, the Defender 110 pushes what's possible. Learn more at land rover.ca. Well, both of those are finite, but I will pick the large of the two, I would pick a week. You're right.

So now let's imagine I ask this question again in a week's time. Would you rather live another five minutes or survive for another week? I imagine that a week from now, I would still prefer one more week. What about the week after that? Another week, please. You may see where I'm going with this. Right. Because here's the thing, if you continue on with the argument forever, this is like a classic philosophical

experiment. This is originally put forward by Thomas Nagel in The Viewer, from nowhere in 906. And he said, given the simple choice between living for another week and dying in five minutes, I would always choose to live for another week. Thus, I conclude, I would be glad to live forever. I don't agree with that conclusion. I mean, I'm not, I'm not saying that Nagel's a liar. I just think that eventually you will not prefer a week over five minutes. Eventually, in my life, I imagine I will reach a point where I say, yeah, give me just five more minutes. That's all I need and that's all my loved ones and friends need.

of me. Like, I'm done, and it's their turn. You're happy with a finite life, effectively. I'm happy with it, and I desire it, yeah. I think that if I was granted immortality right now, my first emotion would not be, whoa, it would be an immense anxiety and claustrophobia, a feeling of being so trapped. I'm trapped here in this universe. And that would be terrifying. It would not be freedom at all. No. The concept of infinity is just a bit too much to bear. And so today, we're going to bear it. We're going to wade into and grasp infinity.

Or at least try. First things first, let's set the ground rule about what infinity is. When I say infinity, and I want to hear what you think, too, infinity to me is a number. It can be a number. It can be an amount. And the amount that infinity means is unending. Okay, there's there's no end to it

There's no final member And so it's not on the number line But if you come if you think of the entire number line And you ask how many numbers are there The answer is infinity Yeah, see I think I'm going to slightly disagree with you Because I don't think it's a number In the traditional sense purely because

You know, you can't do number like things with it You know, you can't like It doesn't really lend itself to additional multiplication or like subtraction. I think that infinity, and by the way, we're going to do two episodes on infinity, but I think that for the purposes of this episode, I think it is something that you can approach but never reach. I think it is something that is boundless, that is endless, that is larger than any other

measurable quality or quantity. Excellent. And I completely disagree. I think that we reach it every day multiple times. I think that infinity is an amount. I think that there are different kinds of numbers. Yes, it's hard to do certain kinds of arithmetic with infinity, but try taking the square root of a negative number, you know?

I think that we just have to admit that there are different kinds of numbers. There's rational, there's irrational, there's irrational, there's imaginary, and there's infinite. They're all numbers, though. Can we just talk a little bit about how weird infinity is? Because it is so much weird than any of the other numbers that you've described. Yeah, I grant that. So let's talk about it. Okay, so I think one of my favorite explanations about how weird infinity is comes from the mathematician Hilbert's German mathematician. He imagined this hotel, okay, called Hilbert's Hotel,

and it's a wonderful demonstration. So essentially, this hotel, it's got an infinite number of rooms, all right? It's just sort of wild to imagine. But also, every single one of those rooms right now happens to be fully booked. It is an infinite hotel that is infinitely full. You turn up to the hotel looking for somewhere to stay for the night. And how do you get in? I mean, the hotel's completely full. It's booked. Yeah, it's booked. And I just want to say, just for clarity's sake, this hotel does not have a lot of rooms. It has an unindy number of rooms. You think you found the last room. There's always one after it. Always. And yet it's fully booked. So like, there's no vacancy.

or is there. There is no vacancy or is there because you're very clever so you say what's fine. You can make room for me in this infinite hotel because all you need to do is everybody needs to leave the room that they're currently in

and then go to the room that is one along. So if you're in room two, go to room three, room seven, go to room eight. And then all of a sudden there's always a room that you can move down to even though it's infinitely full. But now the very first room, room one, is now vacant because nobody has gone into film.

So you as the additional guest have suddenly found space in an otherwise completely, completely booked hotel. So, okay, addition doesn't really work in the same way for infinity, but there's, I mean, it gets way weirder than this, because now imagine that a bus with an infinite number of seats on it, which is also full, turns up to your infinite hotel, which is also completely full. I mean, this is much more of a puzzle. How else do you fit infinity inside infinity? This is just infinity plus one. How do you do it? But there is, in fact, a way to do this. It's a similar trick to before. All you do is you say, okay, if you are in room N, you double the number of your room. You go to 2N. So if you're in room two, you move to room four. If you're in room seven,

you move to room 14. If you're in room 25, you move to room 50. But because everybody has doubled their room number, suddenly all of the odd numbers are completely vacant. And there is an infinite number of odd numbers, which means you can fit the infinite bus load of people into this infinitely full, infinite hotel. Yeah, infinity plus infinity is just still infinity. It's still a fully booked hotel. You had room. You can always make room. You can always make room for one more because infinity is unending. There are layers to this, right? Because, I mean, you can go further with the weirdness of this infinite hotel.

And we should. And we shall, no less. Because now imagine that you've got this infinite car park outside. And now, instead of one person or one bus or one bus with an infinite number of people, now this infinite car park is full of an infinite number of buses, which are full of an infinite number of people. How can you fit an infinity of individuals? infinities inside of infinity.

And it turns out you can do that too. That's not even a problem. You can do it because you start off with the same trick as before. Every person doubles their room number, leaving all of the odd number rooms completely free and open. There's this infinite number of prime numbers. So the first bus you say, okay, you can be bus number three, all right? The first real useful prime, you know, the first odd prime.

You'll bus number three. The first person goes in to three to the power of one, the third room. The second person goes in three to the power of two, the ninth room. The third person goes in three to the power of three and so on and so on and so on. You go through and you fit that infinite number of people, all into odd rooms, by the way. All powers of three. All powers of three. But you have completely left untouched.

Room five, room seven, room 11, all of the other prime numbers, you can repeat this trick for all of those. So the second bus is bus five. The first person goes in five to the one. The second person goes in five to the two and so on and so on and so on. I mean, a bit difficult to follow this on a podcast, I'll admit. But you can trust me, you can trust me, that this works perfectly. An infinite number of buses with an infinite number of people can fit into an infinite hotel with an infinite number of rooms that are all books.

This is extremely strange. So, yes, this is all very strange behavior. but it's still numbery behavior in my opinion. I mean, infinity. What does it even mean? It just means in, which means not, finity, finite. So not finite, not ever coming to an end. And, you know, today we're very familiar with the concept of infinity.

The word is very popular, you know, and the symbol. It's a flock to over eight, let's be honest. The limniscate, yeah, it's a flopped over eight. It's a lazy eight. And it represents infinity, but we actually don't know why that symbol came to mean without end. The very first, the very earliest use of the limniscate to mean unending was by John Wallace in 1655. And he just uses it, but does not explain what it means or why he, chose that symbol. So maybe people were

already talking about using this symbol. I think the best guess is that, my favorite at least, is that it came from Roman numerals. Because yes, there was a one of the earliest forms of Roman numerals did this

thing where parentheses were used to change the value of a number. And so 500 in Roman numerals we all think of as being the letter D, the capital letter D is what it looked like. But before it was a D, it was an I followed by a backward C, which is, I'll do it for the mirror.

Yeah, which you combine them, that's a D. So the D for 500 may have come from an I followed by a backward C and they got squished together. So before the whole D thing, the parenthetical arrangement of Roman numerals worked where for every backward C you added the value went up by a thousand more. So I, backward C, backward C, wasn't 500. It was 5,000. But if you put a C on either side of the I, it didn't represent one. It represented 1,000.

So C, I, backwards C, represented 1,000. And it's believed that that might have been where the M came from that eventually came to be. And today, we usually think of as being the Roman numeral for 1,000, that it was originally regular C, I, backwards C, and then they not only melded together, but the bottom broke open and it became an M, all right? But this, this like I that's encapsulated in a circle gets us really close to that sideways eight. In John Wallace's time, a thousand was often used hyperbolicly to mean you can't even count. Massive. Yeah.

So a thousand in Roman numerals could mean never ending, is what I'm really saying. Not exactly 1,000, not, you know, 10 hundred. but just, you know, infinite. I really like that a lot. I really like that. I'd always sort of assumed that it was related to the number zero. Zero being this symbol from Indian mathematicians that's very related to this idea of like a state of nothingness

and like no beginning, no end. So I'd always, I think, assumed that it was something like that, like a twisted zero. But, you know, if you're saying 1655, right, this simple, I mean, zero wasn't even, I mean, was it even widely used in as far away as Britain at that point? I think it's around about the same time, right? That makes much more sense that it comes from the Roman. It could have. It could have come from Greek, right?

Another theory is that the letter Omega, which in lowercase looks like a little W, it could have gotten closed up and made the Lemonscape. And the Omega is the last letter, you know, the end. So, I don't know. It could be something like that. Thank goodness we started using zero. Like Roman numerals are a joke. I don't even care if the ghost of some Roman soldier comes to...

Julius Caesar comes to kill you. Right. I'll be like, dude, don't even try. You don't even understand positional notation. Like, give it a break. Because just yesterday I finished Ursula Le Guin's short story, The Masters. And it's this really... I don't know if you read it, but I'll tell you, the story is fantastic.

Because the premise is that there was some terrible apocalyptic. thing caused by human technology, right? You could imagine that it was like a nuclear war. They just call it the hellfire. And it happened 14 generations ago. Because it was so traumatic and all these famines occurred, there's been this taboo created around rational thinking, the scientific method, using numbers. No one's allowed to do them. But they still have all the technology that we have. They have steam engines and cars and all this stuff, but everything's built just with comparing sticks. You're not allowed to have numbers like Western Arabic numerals, you know, one, two, three, four, zero through nine, those things. You can't use those. Everyone's been

forced to revert back to Roman numerals because they're so hard to do math with. And you've got, you know, you've got the apprentices who are building the steam engines being like, hey, what's, what's XIV plus XXVI? And they can't do it mentally. They have to just memorize this stuff. But there's a guy who draws a circle in the sand. and he's like, guys, that's the symbol for nothing. And they're like, stop it. That's one of the black letters. We're not allowed to talk about that.

And it's just very cool. And all the examples of people trying to do math with Roman numerals is so hilarious that Julius Caesar come at me. I am not afraid. Yeah. I don't know. I think as we'll come to in the second part of this episode, Zero and Infinity, they're sort of sisters, you know, they really do go hand in hand.

And so I think it is quite interesting, actually, that it's around about the same time. that English mathematicians at least are writing about infinity as zero is coming into common usage. You know, I think there's probably not a coincidence. I mean, the folks on the rest of history can correct us on a wild, wild guesses as to what might have happened in the past. But I think that's probably not a coincidence. Okay, just picking up on that idea of omega, because the ancient Greeks, they did have this like inkling about infinity. They weren't, I mean, they didn't like it. It didn't feel comfortable.

They sort of wanted to avoid it as much as possible. But they definitely had this inkling that there might be something there. So the particularly famous story is about Pythagoras and his disciples. So by the way, Pythagoras is like, he's kind of survived through history as this like, this genius figure, this lone genius. Not true at all. He's basically a cult leader. He had some absolutely wild ideas about, well, Well, notably about beans.

I've mentioned this in this podcast before, but he was absolutely terrified of beans. He thought that they were essentially little humans. Do you know why he thought beans were little humans? I think he sort of thought that they looked like them. I mean, once you get this far back, the number of interpretations and original source materials that you have to work on, there's many, many-fold different ways to go. I think also that there's a few different reports that Pythagoras was trying to persuade bulls not to eat father beans.

You know, it's, it's a, it's commitment. The thing about Pythagoras, his cult, and actually the ancient Greeks more generally, is they had this really deep belief that the universe was made up of exquisite order, you know, that the movement of the planets in the sky, that, you know, the shape of circles, of triangles, it was just, you know, unending beauty in. fractions and whole numbers. So in music, for example, this whole idea of harmonics, it was the Pythagorean who worked out that frequency of certain notes, they sound good together when you mix frequencies that are an exact multiple of one another. You know, that's how you get sort of harmonic and beautiful sounding music. And so the Pythagoreans, they had this list, this table of opposites, they called it. In one side, there would be be good, the good things, the things they liked, things they were happy with. And on the other

side, there'd be sort of the bad things. So they had odd and even, one and many, right and left, male and female. See if you can guess which column female went in. That's the bad side. The evil side. Yeah, it was joined by good and evil. Evil's also in the bad column. Square and oblong. Oblong, disgusting. Oblongs and females. Get over there. Light and darkness. Light and darkness, exactly, but also finite and infinite, an infinite being like this horrible, disgusting thing that kind of belongs over there. There is this story that Hippasus, who was an early follower of Pythagoras, and he was playing around with triangles, and he had a right-angle triangle with two sides

that were both equal to one, right? So they're kind of equivalent to each other. And then he was working at how long the hypotenuse was, how long the diagonal that cut between those would be. Essentially, if you take a square and you split it across the diagonal, how long is that line? And it's equal to the square root of two, which Apatius was like, okay, hang on a second, the square root of two, there's no order to it. There's no natural beauty. There's these numbers that continue on indefinitely.

There is infinity contained within this beautiful geometric shape. And the story goes, I mean, we can never be quite sure. The story goes that they took Capacus out on the ocean and then just chucks him off the boat, drowned him at sea for arguing. Wait, for the sin of what entertaining an irrational number? I think he had a proof for it. I think he even managed to prove that it was irrational. And they were like, don't you come at me with this? With this sacrilegious nonsense.

This is despicable. Despicable, almost a little female. Almost a little female, exactly. What if it turned out that I was a closet Pythagorean and I believed all these things and I was part of his cult like a modern day neo-Pathagorean cult and I was like, look, Hannah, infinity is evil and so is darkness and so is, so are odd numbers and women. Did you not know that about me? There's a very easy way to test this. I'll just make you bake beans on toast to see what you do. Beans, more like human beans.

See, now the language at least matches our weird belief. I think we should bring back the idea of being a closet Pythagorean as an insult, you know? I think we should start throwing it around. Well, I know, but the problem is that they also had some really great things. I mean, I love their worship of numbers and ratios as being something really fundamental, something that was so timeless, it was outside of time. It didn't change. It was godlike.

Maybe numbers and math literally were God, right? I don't know why they also hated beans. But look, this is just, it's just too long ago, and we can't ask them, and we don't have a lot of what they wrote. Speaking of which, we don't have anything that that famous guy, Zeno wrote. Because he was also tackling infinity, right? Well, kind of. You know, it's unclear to what exactly. stint. Zeno was around in like the fifth century BCE and Zeno was a follower of a guy named,

first of all, here's a cool trick. People bring up Zeno and especially Zeno's paradoxes all the time. And whenever people bring them up, I like to be like, I act confused at first and I go, oh, Zeno Avelia. Right. Yes, of course, go on. As though there are like other Zenos I know. and I need the clarification. So the Xeno of Elia is the kind of the full name. And Elia is the Greek colony that he lived in. The mathematicians and, well, really the philosophers who lived there were the Iliatics. And they had a very, to me, impenetrable belief that motion was impossible.

Like, I cannot describe this in a way to convince you, but they believed that there was no change, that it was all an illusion, that everything was just one. Everything was the monad. That's it. When we think that something's changing or moving, we're being fooled. And so the righteous, the wise thing to do was just to sit and do nothing and chant. It is. It is. That's it. It, it. Everything. And of course, there were other thinkers at the time who were like, you guys are ridiculous. What does this even mean? Obviously, things can move. Obviously things can change. And it's real. And so Zeno was like, guys, I know you'd, think that we're silly, but you're just as silly. Listen, if I believed in motion, then here's a contradiction. Let's have Achilles race a tortoise. The tortoise is obviously slower. So Achilles says,

oh, I'll give you a head start. I'll let you, you know, go for, you know, a minute before I even start. And the question is, who wins? And you might think, well, I guess it kind of depends. Like, Achilles can clearly outrun the tortoise eventually. But Zeno says, logically no because sure when Achilles starts running the tortoise is already somewhere up ahead

and Achilles has to first run to where that tortoise was when he started but in that period of time the tortoise has already moved a little bit so the tortoise is now ahead of him and Achilles now has to run from where the tortoise used to be to where the tortoise is now

but by the time Achilles gets there the tortoise will be yet a little further ahead and Achilles has to close that gap. But by the time Achilles has closed that gap, the tortoise will be a little bit further still. So it's impossible since we can divide this out forever. It's impossible for Achilles to ever

outrun the tortoise. And the other thinkers were like, yes, but he does. It does happen. And Zeno was like, but you can't explain why. And so that's the paradox. And Zeno wasn't making a joke. He wasn't necessarily arguing, here's proof

that motion is impossible. He was just saying, that if you believe in motion, you have to embrace contradictions, just like we are perhaps embracing some as the Iliatics. Well, that was the conclusion that he was drawing, right? It was that, like, this idea that motion is real really obviously breaks down once you consider this paradox. Yeah. And thus, motion can't be real.

Motion must be this illusion. Yeah, how do you explain that? And Zeno and the people that we do have existing writings from never really used the word infinity to talk about what Zeno was describing here, but they did have to dismiss it in some way, usually just by literally getting up and walking and saying, eh, what do you think about this? How am I possibly doing this? But, you know, Zeno would say, guys, I don't, but how does it make sense that you can move? Because in order to move, you have to, you know, first cover like half the distance that you're

going to cover. But before you can do that, you have to cover a quarter of it. And an eighth of that first, but then a 16th of that first is, so wait, you have, there's How do you even start? How does a journey even begin? And they were like, yeah, where is the problem? At the heart of it is infinity there, right? But infinity is this philosophical monster that nobody could quite get their head around

that was, I mean, essentially breaking human logic. And that was the case for many, many hundreds of years. I mean, thousands of years, frankly, until... Until... Until... Until... Until... Newton, the big fat baby came along.

The big fat baby. And... And... And... And... ... started squealing about it. Sort of squealing about it. And, like...

seemingly resolved the paradoxes, right? Today we have a very powerful tool. I would argue, though, that really we haven't solved Xenos paradoxes. We have constructed a bunch of great answers about them, but I think we should look at those answers because they approach infinity in a really different way, where instead of just dancing around it, they hold it. Okay, I'll tell you what. Let's do this after the break. I am going to do a little story about why Newton is a big fat baby. what he did that resolved Zeno's paradox, and then Michael and I can argue till the end of the program as to whether it actually is resolved or not.

Sound good? Good. Let's do it. Okay. This episode is brought to you by Project Hail Mary, the new spectacular space adventure movie coming to cinemas from the author of The Martian, Andy Weir, and the directors of the Spider-Verse movies,

Phil Lord and Christopher Miller. But here's an even better combination, Teachers in Space. Hello. Thank you. Project Helmeri stars Ryan Gosling as science teacher of Rylan Grace, who is sent unexpectedly on an impossible mission into space to discover why the Sun and the Stars are dying, and he teams up with an unimaginable ally to defy all odds and save the universe from extinction. Okay, here's a question for you, Michael. What kind of prep would you hope Ryan Gosling had done for this role

in order to play the role of a science teacher? He should have spent a bunch of time with cool teenagers. and tried to teach things to them so that it wasn't just like a good explanation but also kept their interest and made them want to hear more and understand it so that they could share it to be cool too. See Product Helmery now in cinemas and IMAX everywhere. This episode is brought to you by Cancer Research UK.

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It shows what long-term research makes possible. For more information about Cancer Research UK, their research breakthroughs and how you can support them, visit cancerresearchuk.org forward slash the rest is science. Square note that in hospitality, efficiency is everything. That's why their system lets you take payments. Track sales, handle inventory, manage staff, send invoices, and keep up with finances all in one place.

Fly through orders with zero mistakes. Get the data you need and keep everything working together. So you're ready for whatever's next. Learn more about their customizable plans at squareup.com. Okay, the thing that we're hinting at here is calculus. You know, we don't, it's not mandatory in the UK, probably rightly so. You don't really learn it until your 17 or 18 and you're doing A-level.

There are huge swades of the British population who just have never come across this absolutely beautiful subject. I think it's my favorite area of mathematics, boy. I think calculus has become slang for difficult math. And that's somewhat unfair. I felt that way up until I was in my 30s and I got a copy of calculus by Michael Spivik, which is literally a textbook. But I don't know if you've read it. I once recommended this on Twitter that this was like the best way to understand calculus

and some of the most brilliant people that I follow on Twitter who are mathematicians were like, no, that's a terrible book to recommend to people. And I'm like, no, well, okay, so take this with a grain of salt, but it changed my life because Spivik doesn't just say, look, here's how to solve problems. He says, what the heck is change? And can we make sense of change in smaller and smaller increments or even change at an instant, he like breaks down what all these terms mean so fundamentally

that he's like, look, we need to define even what like an ordered pair is, what is a function? And suddenly it just unlocked an understanding of everything that mentions calculus. So I love that book. We should start from the beginning here then, because calculus, it is the mathematics have changed. Everything that came before it, you know, geometry, number theory, they're all sort of standing still, as it were. But in the real world, everything's constantly changing. Everything is moving, speeding up, slowing down, moving on curves, fluctuating. And so calculus was this really big idea that is what we use to measure and understand anything that is moving or changing

effectively. The idea of it is actually incredibly simple. And essentially what you say is, let's say that you've got this really wibbly wobbly line, right? Like a kind of really strange little curve. You can't measure it with a ruler. I mean, you can't really, can't really sort of say anything proper about it if all you've got to work on is a ruler. But what Newton and Leibniz, I'm going to tell you about in a second, what both of them realized is that if you zoom in closer and closer

on any section of this really curvy line, If you zoom in close enough, it will look straight. Yeah. And so this was the really big idea is that if you cut up any line into an infinite number of chunks, each one of those chunks you can handle yourself. You can handle it. There's no problem there. So if you want to look at the area under a curve,

if you want to work out the shape of anything, no matter how kind of crazy the outside of it is, you can use calculus to chop it up and then not actually have to do the infinite number of steps because what this method gives you is a shortcut of doing an infinite number of things all in one go, essentially. And this is by using the concept of a limit. Of a limit. Yeah, exactly. So describe for me what's happening here because if I am trying to walk from here to the door, I have to cover half the distance. You do.

You don't have to cover half of what's left and then half of what's left and then half of what's left. And this goes on forever. It's an infinite number of things. And yet, I cover them all in a finite number of time. How? Because the key thing that was missing from the original formulation of Zeno's paradox is that the amount of time that it takes for you to do that, it's a certain amount of time for your first step. It's a smaller amount of time for the next step, smaller and smaller and smaller. So you are doing an infinite number of things.

But once they get infinitely small, they take you an infinitely small amount of time too. And so you can sum up an infinite number of things and end up with a finite number. And calculus formalized this and made it not just an idea, but a mathematical, logical, demonstrable thing. Exactly. And one with which, I mean, at the heart of it, it's an incredibly simple idea, right? Zoom in close enough and everything is, you can handle everything. But the power of it is just phenomenal. I mean, you know, there's no, like Newton's orbital mechanics, roller coasters, right?

Like race cars. I mean, everything you can imagine, anything that moves or changes, the stop market, right? like anything at all is going to have calculus in there somewhere, this mathematics have changed. I've sort of been saying Newton so far, right? But the thing about Newton is he did come up with this. He wrote it down and then he stuck it in a drawer for 40 years and didn't tell anyone about it.

And then a little while later, this German diplomat, actually, this is a diplomat and philosopher called Gottfried Wilhelm Liebnitz or Leibniz. He completely independently, he can, came up with his own version of exactly the same thing. And Newton had been doing it looking at motion and time. And Leibniz had been looking at geometry and space. I mean, Newton's a big deal, right? So Leibniz knew he'd heard some rumours that maybe Newton had done a bit of this stuff already.

So he writes this letter to Newton saying, oh, I've heard this rumor that you're working on this. I don't want to tread on your toes. But Newton, who, if you've listened to our previous episodes, we know, was like a right little whiner. he was not happy about this at all. And so what he did is he sent back, leaving this really bizarre Latin anagram, where if you unscramble it and translate it, basically it was very cryptic.

It was essentially the kind of equivalent of sending a sort of very cryptic sub-tweet and timestamping it. So that when you look back at it later, it said something like, given any equation involving any number of fluent quantities to find the Fluxians and vice versa. Right.

That's just sort of what he was saying. He was proving that he knew how this thing worked without telling him. But he could prove later, look, I described it all in this letter that's got a date on it. So I deserve the credit. Totally. So anyway, leave this. It's just like, oh, okay, I'm not really sure what to do about Newton. This is a weird guy, but I'm going to publish it anyway.

Everyone loves it. It goes great. And then Newton forms one of the deep. most vengeful attacks on Leibnizabeth's for the rest of his life. So it starts off and Newton gets this Scottish mathematician called John Keel. He sets him off as his attack dog. And Keel writes his article publicly accusing Leibniz of being a thief. Leibniz is not a thief, to be absolutely clear.

What's the evidence that he's a thief? Is it that he received this cryptic letter and then solved it and stole the idea? No. So, I mean, this is, I mean, evidently not. He'd already come up with a theory by that point. But Newton had been sending letters to people in Europe that had inklings of this idea. And he had done it, he had done it a few decades earlier, right? We know now for sure that Newton did come up with it independently and Lieblitz came up with it independently. But Newton just didn't want, he wasn't happy that this, that this other guy had come in and got the credit for an idea that he'd already had. So it's just basically a smear campaign, bluntly. So there's all these public articles. Leibniz is furious about this, like, you know, this Scottish mathematician accusing him of whatever. So he writes a formal letter of complaint to the Royal Society, who's sort of the ultimate arbiter of science in the day.

And Newton, who was the president of the Royal Society, okay? It's like, I'm sorry, I'm not going to censor any of these members. They can say what they want to. But secretly was going behind everybody's back and whispering in the ear of John Keel being like here's what I need you to say next. Here's how you can attack him more. So this argument like extends and extends and extends until the early 1700s when Newton's like publicly accusing him of being a fraud and a plagiarist. And this fight gets so bad that the Royal Society decides that they're going to assemble this independent impartial committee of like the greatest minds

in order to work out once and for all who actually came up with the idea of calculus. Now, I mean, I've mentioned that Newton was the president of the Royal Society. So what he does is he secretly hand-fix every member of the independent jury when they released their official report, which destroyed Leibniz's reputation. We now know that actually Newton secretly wrote that report himself. Okay. And then to like really twist the knife, Newton then anonymously publishes the glowing review of his own secret report in the Royal Society's journal saying about how undeniable. undeniably correct and thorough it was. And then Leibniz, his reputation is genuinely ruined.

Like, he's over. And he ends up dying, you know, a few years later. He's impoverished. He's out of favor with the Royal Courts. And then Newton, in his private notes, writes how proud he was that he'd broken Leibniz's heart. Goodness, cranes. I mean, Newton, honestly, is not a nice guy.

He's horrible. This is like the whole time I've been listening. I've been imagining Tina Faye, if you're out there listening, let's do a prequel to mean girls and have it set in the time of Newton and Leibnids. And the Royal Society can be a little click. Yeah. Luckily, we got the knowledge. We got the knowledge. We did.

And here's the thing, right? So while Newton did come up with the idea before and Leibniz came up with it independently, the thing is, is that Newton's, the way that he writes it, Newton's, the way that he writes it, Newton's, notation, as we describe it, was kind of rubbish. It just doesn't, it was way more clunky. It's like the Roman numerals of calculus, effectively. Leibniz is one is way neater, makes way more sense. And because of this loyalty, sort of the national loyalty that the British mathematicians had to Newton, they refused to use Leibniz's notation. Whereas on the continent, in France, in particular, they're quite happy they were using Leibniz's notation. And because it was so much better, so much clearer, so much easier to work with, actually, British mathematics on that front kind of stalled for a few hundred years.

So now, if you do learn calculus in school or even in the Spivak book that you were recommending earlier, almost certainly they'll be using Liebenets's notation. And the word calculus itself was Leibniz, not Newton. He wanted to call it method of fluxions. Okay, so first of all, Flexians is a pretty cool word. It's a pretty cool. Like, I got to admit that. But I guess that's a nice little, like, happily ever after story that at least Leibniz got to name it and decide on the symbols and the way we learn it today. I mean, it's probably not worth the horrible death, but, you know.

Yeah, I bet if you asked him, he would have said, you know, hey, give me one more week, please. Yeah, exactly. By the way, to be absolutely clear, this is going to be a continual running theme in the rest of science about what a massive bitch Newton was. Yeah, as it should be. This is one of my favorite parts of the podcast because I think, yeah, it's like Yin and Yang. You know, Newton did a lot, gets a lot of accolades, gets an entire unit of force named after him. I think it's about time we applied some force too. Quite right.

He was still a human and had lots of human flaws. It did resolve Zeno, though. did resolve Zeno, or at least I think many generations of Mathematicians are comfortable that it resolves Zeno. That's right. We all feel like we've moved on from Zeno's paradoxes. We say, look, it's possible. And, you know, you can use the calculus concept of a limit to show that you arrive at the door,

even though you've got an infinite number of little pieces. This segment is brought to you by Cancer Research UK. In England, every year, around 4,300 people begin. treatment with a drug that is designed to switch off a very specific cancer driving signal that appears inside their cells. That, by the way, is enough people to fill 10 jumbo jets. And those drugs are now used worldwide, boosting the impact of existing cancer treatments and giving millions of people more time. And that idea of finding the signal that is driving a cancer and then blocking that specifically, it traces back to research that Cancer Research

UK was funding back in the 1980s. That's right. And so today we are asking, how did a clue from chickens lead to targeted cancer drugs? So a long time ago, like in 1916, scientists noticed that chickens that were infected with a specific virus developed cancer. That this virus produced cancer-causing molecules. But it was also noticed that this molecule called epidermal growth factor would cause cells to grow out of control in petri dishes. Now, epidermal growth factor, EGF, was pretty well understood. understood. But what we didn't know a lot about was the EGF receptor on the cells. And so that's what Julian Downward needed to collect a lot of. And as it turns out, placentas are a great place to collect EFGR, epidermal growth factor receptors. So then he started working out the

sequence of amino acids that made them up and defined it how they worked. And then he compared those sequences to other known proteins and was doing this late at night, just like, for the X to appear to say, oh, we found a match, we found a match. And suddenly, boom, a match was found with proteins created by a virus that was known to cause cancer in chickens. And so this was huge news, right? He famously called his boss middle of the night. The boss comes over and they spend all night working on this because this was the beginning

of much more targeted cancer drugs. Right, because you have to realize the idea that it could be your own internal cellular machinery that could just simply go a bit haywire and cause cancer. At this point in time, it was still a barely formed thought. But what Julian Downward had done here in looking at stuff that is naturally occurring in humans, this EGFR, this cell surface receptor that sits on the outside of your cells and helps to regulate cell growth and division, by noticing that it had this link to chicken cancer, it changed the entire game. It meant that people started realizing that it could be something in your own body that causes yourselves to just go completely

overactive. But having that knowledge, having that groundwork, that is what has led to this gigantic shift in treatment. Because if you know exactly which signal is faulty, which naturally occurring signal in the human body is going wrong, then you can design a drug to block just that, which then in turn, hopefully, results in far fewer side effects. And so by the early 2000s, the very first generation of EGFR inhibitors, things that block this protein, ended up reaching the clinic. And now there are 13 different drugs out there that targets EGFR, which are used to to treat six different types of cancers, including certain types of lung cancer, of bowel cancer, of head and neck cancers. And all of this can be traced to that exact moment, late night in the

lab, when Julian, the Cancer Research UK-funded PhD student, discovered this incredible link. And I mean, this is an entire wave of innovation that got sparked from that moment. In the UK today, more than 8 in 10 people who receive cancer drugs are receiving a treatment that was developed either by Cancer Research UK or with their involvement. Yeah, so the story here is about the question changing. It's not just about what causes cancer, but which signal inside the cell is driving it. And once you can identify that, that specific signal, you can try to interrupt it. And you can tailor treatments to the biology of each cancer, which makes treatments kinder and more effective.

And for the 11 people every day who are beginning, treatments in England and many more people across the world. That change is shaping the care that they receive. So for more information about Cancer Research UK, their research, their breakthroughs, and how you can support them, visit cancerresearchukuk.org forward slash rest is science. There still remain a lot of metaphysical puzzles around here. And I think we should kind of leave everyone with these. Let's imagine that I get up and I walk to the door. Yes, I have to cover half the distance And then I have to cover half of what's left

And then half of what's left and half of what's left forever But now imagine that I've got two flags Like I've got a green and a red flag And after each half of the journey that I cover I switch which flag I'm holding So I walk halfway there, put up the red flag Then I cover half of what's left put up the green flag Half of what's left put up the red

Half of what's left put up the green And I keep alternating Eventually I get to the door as we know I do, which flag am I holding up when I reach the door? Last one. That's right. Is the number of halves I've covered even or odd? I'm asking for the last number, the biggest number.

And how do you feel about these kinds of questions? Because they're not resolved. They're not resolved. Because I think that this is when it really becomes clear that infinity doesn't act like a normal number. We did that episode on really large finite numbers like Graham's number. that we know ends in a three, you know, infinity is not like that. We don't know what it ends in because it doesn't ever end. Right. And yet my flag question presupposes that there's an end. Mm-hmm. Because we reach the destination, so it ends. I should be able to figure out, you know,

what the state there is. Another famous, very similar thought experiment is called Thompson's Lamp, where I'd say you're going to play with the lamp for a minute. And what you do is you wait half a minute and you turn the lamp on. It starts off. And then you wait 15 seconds and you turn it off. And then you wait seven and a half seconds and turn it back on. So as you can see, after half of the time that's left has passed, you switch the state of the lamp. Okay.

So it's going off on, off, on, off on, in this accelerating rate. A minute eventually passes. When that happens, is the lamp on or off? And in exactly the same way, it's like, what's the last step? What is the last step of an infinite number of steps in a finite amount of time? Surely a minute can pass, but yet the lamp cannot be on because every time it's on, it's immediately turned off. Because infinity is unending. But every time it's off, it's turned back on right afterwards.

There can't be an end. And yet the sequence itself can end. So the best I've heard as a response to these paradoxes is, I don't even know if it totally answers them, but it basically says we don't have enough information. There's a trick to these questions where we're being asked to say something about the state of a lamp or a flag, but we've only been given rules about its behavior before that moment. You've told me when a certain flag will be raised or not, and I can tell you at which step, which flag will be up. I can always tell you that. But now you're asking me a question about when you reach the door.

But that's not part of the flag rules. So I can't know. I don't know if that just kind of sidesteps the problem. So what you would do in a mathematical sense with this, Because I think ultimately the problem here comes in when you are trying to make this ultimately imaginary concept of infinity fit into reality, right? Because planting the flag suddenly makes it real, right? Suddenly you're physically interacting with the world. Cutting something into an infinite number of pieces that you never actually do. You're just using that as a shortcut to get you to the answer. sort of works, right?

But once you're planting flags and then saying what's the last one, that's where you end up coming into this problem. So what you do in this situation, I'm thinking about like in fluid dynamics, for example, right? You have this concept of this perfectly smooth fluid that you can cut and cut and cut and cut and cut an infinite number of times. But once, of course, you actually get to reality.

Reality isn't made like that. You can't cut up reality an infinite number of times. or maybe you can. We'll discuss that more in the next episode, maybe. But, you know, there comes a point where you're down to atoms and then you're down to quarks. And so what you can do mathematically is you can say, okay, we're going to draw a line in the sand, right?

And anything that is below this order of magnitude can't be included. Because there comes a point where your infinitesimally small steps no longer match with reality. Yeah, they don't have any physical meaning anymore. So it's like a clash between our ability to think and reason and the world we've been given. The lamp, I can imagine turning it on and off in this accelerating fashion,

but yet at a certain point, and I'd love to know when, actually, at a certain point, I am having to push this lamp switch faster than the speed of light in order to continue following the rules. And so are we just actually prohibited from ever actually doing these? And so they're what, like little made-up fictitious paradoxes? If reason can create the paradox, why can't it also resolve it? What's, where's the problem? I get that the real world won't allow us to experiment and just observe the answer.

Doesn't feel very satisfying there, does it? It doesn't feel very satisfying. There's another famous one called the Ross Littlewood paradox. And this one is pretty fun because it doesn't even involve having to cut space up into small pieces. Here's the experiment. You think in your mind that you're going to do this. You're going to take a big jar and you put 10 balls in there. Let's say they're ping pong balls. Then you take 10 more balls and you put them in. But when you do that, you also remove one ball. Then you put in 10 more and remove one. You put in 10 more and you remove one. You put in 10 more and you

remove one. And you do this in infinite number of times, which because of the tasks we've already described, we can imagine doing in a finite amount of time. You know, you put the first balls in after half a minute, you put the second batch in after just 15 seconds and remove one. After you've done this process an infinite number of times, 10 balls in, one out, 10 balls in one out, how many balls are left in the jug? I think a lot of people out there listening are going to say, well, it's an infinite number because, yeah, you put in 10 and then remove one. That means you really just put in nine. And nine plus nine plus nine, forever is infinity. However, let me put it to you this way. Let's say that the balls have numbers on them. One, two, three, four, going all the way up, right? And I put in balls one to ten. And then I put in balls 11 to 20. And I remove ball number one. And then I put in the next ten balls and I remove ball number two. And then I put in another ten balls and I remove ball number four. Or three. I forget where I was. But you see what I'm saying here, right? If that's what I do, then the answer is at the end of an infinite

number of steps, the jug's empty. There are no balls in it because ball number one was removed at, you know, step one, ball number 10 was removed at step 10, ball number 18 bajillion was removed at step 18 bajillion. For every single ball I put in, there is a moment when it was removed. So, there are no balls? Or are there an infinite number of balls? There's that zero and infinity are kind of like twins thing again, where it's like it's one of those. It's either none or unending. Not anywhere in between. I mean, what you're describing here essentially is it does take us back to calculus. You're essentially describing what happens in the limit of something, but you're describing

sequences that have no limit that don't converge. You know, the sequence half plus a quarter plus an eighth, plus a 16th, and so on and so on, which is the Zeno's paradox one in one of the formulations that you described it. That's fine. That equals one. No big deal. We can deal with that. But on off, on off, 0101, 0101, it just carries on oscillating forever. It doesn't have a limit. You know, likewise, the thing that you're describing there, because there are other sort of strange paradoxes, which are similar to the one you're describing, that do have limits, that can have resolution. So one that I really like is, imagine that you have an ant that is on an elastic band. This elastic band is pretty special.

You can stretch it as much as you like, right? You can carry on stretching it forever. So this elastic band in the first second, it's 10 centimeters long. In the first second, the ant can crawl one centimeter along the length of the band. But at the end of one second, you stretch the elastic band to 10 kilometers long. Now, the ant crawls another one centimeter. You stretch it again to another 10 kilometers. And this goes on and on every second the ant traverses one centimeter.

meter and the rubber band gets stretched an additional 10 kilometers. The question is, does the ant ever reach the end of the elastic band? I reckon we should leave this one as something that the listeners can work on. I reckon we leave that because it does have a solution. It does have a solution. It has a solution. And I mean, this is a bit of a spoiler, but it's surprising. The answer.

It's surprising. It's really surprising. The ant covers one centimeter of distance every second, but the entire string or band becomes 10 kilometers longer every second. Can he reach the end? And I want to add, too, that in the next episode, we will tackle another question, which I really love, which is, okay, fine, Hilbert's Hotel.

To go back to the beginning, Hilbert's Hotel can always accept more guests. But we're having to move the guests from the front and on, right? but what if another hotel opened next door? And it needs to buy some number plates to put on its doors. But Hilbert's hotel, annoyingly, has bought all the plates that there are. It's got every numbered plate. What should the first room in this new hotel be numbered?

What number plates does it use? Or even better, let's imagine you're running a race and you're really slow. An infinite number of people finish the race before you do. What place did you get? Now we're not talking about rearranging infinity or approaching infinity or completing infinity. We're talking about after infinity. What numbers are there?

And that's where we're going to go next week. It certainly is. And we're going to come to, I think, the even more mind-boggling conclusion that some infinities are larger than others. It's going to hurt. Yeah. Prepare your brains for some more mind-bending infinity strain. And if you'd like to ask us a question, we might answer it in our Thursday field notes episode.

So send those questions to The Rest Is Science at Gollhanger.com. And in the meantime, you can also sign up to our free newsletter, therestis.com forward slash science. See you next time. Bye.