Paradoxes Of Infinity (Infinity Part 1)

18:07Hannah Fry: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 different ways to go. I think also that there's a few different reports that Pythagoras was trying to persuade bulls not to eat fava beans. You know, it's a 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.

18:49Hannah Fry:You know, that the movement of the planets in the sky, 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 Pythagoreans who worked out that frequency of certain notes, they sound good together when you mix frequencies that are an exact multiple of one another. 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 good, the good things, the things they liked, things they were happy with.

19:34Hannah Fry: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.

19:47Michael Stevens:The evil side.

19:49Hannah Fry: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. And infinite being like this horrible, disgusting thing that kind of belongs over there. There is this story that Hypassus, who was an early follower of Pythagoras, and he was playing around with triangles. And he had a right-angled 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 out how long the hypotenuse was, how long the diagonal that cut between those would be.

20:31Hannah Fry: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 Hypatius 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, and we can never be quite sure, the story goes that they took Capacis out on the ocean and then just chugged him off the boat, drowned him at sea for arguing.

21:06Michael Stevens:Wait, for the sin of what entertaining an irrational number?

21:11Hannah Fry: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 sacrilegious nonsense. This is despicable. Despicable. almost a little female almost a little female exactly what if it what if it turned out that

21:31Michael Stevens:i was a closet pythagorean and i i believed all these things and i was part of his cult like a modern day neo-pythagorean cult and i was like look hannah infinity is evil and so is darkness and so is so are odd numbers and women just did you not know that about me there's um there's 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?

22:07Hannah Fry:I think we should start throwing it around.

22:08Michael Stevens: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 extent.

22:51Michael Stevens:Zeno was around in like the 5th 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 they're like other Zenos I know and I need the clarification. So the Zeno of Ilya is kind of the full name. And Ilya 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.

23:38Michael Stevens: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. 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 think that we're silly, but you're just as silly.

24:19Michael Stevens: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 go for 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.

25:03Michael Stevens: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.

25:43Michael Stevens:He was just saying that if you believe in motion, you have to embrace contradictions, just like we are perhaps embracing some as the Iliadics.

25:53Hannah Fry:Well, that was the conclusion that he was drawing, right? 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.

26:05Michael Stevens: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, 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 sixteenth of that first.

26:43Michael Stevens:So wait, you have, there's, how do you even start? How does the journey even begin? And they were like, yeah, where is the problem?

26:48Hannah Fry: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 Newton, the big fat baby, came along. The big fat baby.

27:10Michael Stevens:And started squealing about it. Started 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 Zeno's 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.

27:36Hannah Fry: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.

27:53Michael Stevens:Let's do it.

27:54Hannah Fry:Okay.

28:01Michael Stevens: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.

28:18Hannah Fry:Hello. Thank you. Project Hail Mary stars Ryan Gosling as science teacher 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?

28:45Michael Stevens: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.

29:03Hannah Fry:See Project Hail Mary now in cinemas and IMAX everywhere.

29:13Michael Stevens:This episode is brought to you by Cancer Research UK. Cancer drugs aren't developed overnight. They start as ideas in the lab, then move into testing to check they're safe and work effectively.

29:24Hannah Fry:In the late 1990s, Cancer Research UK scientists began exploring a bold idea. Could the antibodies that normally trigger allergic reactions be used to treat cancer?

29:35Michael Stevens:The lab results were promising, but allergic reactions carry real risks. After years of work, an early stage trial showed these antibodies could be used safely.

29:45Hannah Fry:And for one person on the trial, their tumour shrank. Research is ongoing, but this careful process is how treatments move from the lab into hospitals.

30:18Michael Stevens:to science. This episode is brought to you by Focus Features. Would you let AI pilot your plane, raise your child, decide your future? On March 27th, Focus Features presents The AI Doc, or How I Became an Apocaloptimist. Critics and audience at the Sundance and Southwest Film Festivals call it the most urgent movie of our time. The AI Doc, or How I Became an Apocaloptimist, Rated PG-13. Only in theatres March 27th.

30:55Hannah Fry: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 you're 17 or 18 and you're doing A-levels. There are huge swathes of the British population who just have never come across this absolutely beautiful subject. I think it's my favorite area of mathematics.

31:16Michael Stevens:I think calculus has become like 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.

31:52He says, what the heck is change?

31:56Michael Stevens: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.

32:20Hannah Fry:We should start from the beginning here then, because calculus, it is the mathematics of change. 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 wobbly line, right?

33:04Hannah Fry:Like a kind of really strange little curve. You can't measure it with a ruler. I mean, you can't really 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 study about in a second, what both of them realized is that if you zoom in closer and closer and closer on any section of this really curvy line, if you zoom in close enough, it will look straight.

33:31Michael Stevens:Yeah.

33:32Hannah Fry: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.

34:10Michael Stevens:And this is by using the concept of a limit.

34:13Hannah Fry:Of a limit. Yeah, exactly.

34:15Michael Stevens: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. But then I 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?

34:35Hannah Fry: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

35:11Michael Stevens:and made it not just an idea, but a mathematical, logical, demonstrable thing.

35:20Hannah Fry:Exactly. And one with which, I mean, at the heart of it, it's an incredibly simple idea, right? Zoom in close enough and you could 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 stock 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 he stuck it in a drawer for 40 years and didn't tell anyone about it.

36:06Hannah Fry:And then a little while later, this German diplomat, actually, this diplomat and philosopher called Gottfried Wilhelm Leibniz or Leibniz, he completely independently, he 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 Liebnitz 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 rumour that you're working on this. I don't want to tread on your toes.

36:42Hannah Fry: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 Liebniz this this really bizarre latin anagram where if you unscramble it and translate it basically it was uh it was very cryptic it was essentially the kind of equivalent of sending a sort of very cryptic subtweet and time stamping it so that when you look back at it later it said something like given any equation involving any number of affluent quantities to find the fluxions and vice versa. Right. That's just sort of what he was saying.

37:23Hannah Fry:He was proving that he knew how this thing worked without telling him.

37:27Michael Stevens: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.

37:36Hannah Fry:So anyway, Liebnitz is just like, oh, OK, I'm not really sure what to do about Newton. This is a bit of a weird guy, but I'm going to publish it anyway. Everyone loves it. It goes great. And then Newton forms one of the deepest, most vengeful attacks on Leibniz 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 this article publicly accusing Leibniz of being a thief. Leibniz is not a thief, to be absolutely clear.

38:10Michael Stevens: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?

38:17Hannah Fry:No. So, I mean, this is, I mean, evidently not. 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 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 Scottish mathematician accusing him of whatever.

38:57Hannah Fry: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.

39:31Hannah Fry:And this fight gets so bad that the Royal Society decides that they're going to assemble this independent, impartial committee of 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. And then to 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 undeniably correct and thorough it was.

40:20Hannah Fry: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 Liebniz's heart. Goodness gracious.

40:41Michael Stevens: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 Fey. If you're out there listening, let's do a prequel to Mean Girls and have it set in the time of Newton and Liebniz. and the Royal Society can be a little click. Luckily, we got the knowledge.

41:03Hannah Fry: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 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, 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.

41:47Hannah Fry: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 Leibniz's notation. And the word calculus itself was Leibniz, not Newton. He wanted to call it method of fluxions.

42:15Michael Stevens:Okay. So first of all, fluxions is a pretty cool word. It's a pretty cool word. I got to admit that. But I guess that's a nice little happily ever after story that at least Liebnitz got to name it and decide on the symbols and the way we learn it today.

42:36Hannah Fry:I mean, it's probably not worth the horrible death, but you know.

42:40Michael Stevens:Yeah, I bet if you asked him, he would have said, you know, hey, give me one more week, please.

42:43Hannah Fry: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.

42:53Michael Stevens: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.

43:10Hannah Fry:He was still a human and had lots of human flaws. It did resolve Zeno, though. It did resolve Zeno. Or at least I think many generations of mathematicians are comfortable that it resolves Zeno.

43:21Michael Stevens: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.

43:43Michael Stevens:This segment is brought to you by Cancer Research UK.

43:46Hannah Fry: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.

44:00Michael Stevens:And those drugs are now used worldwide, boosting the impact of existing cancer treatments and giving millions of people more time.

44:08Hannah Fry: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.

44:21Michael Stevens: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. 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.

45:10Michael Stevens:So then he started working out the sequence of amino acids that made them up and to find out how they worked. And then he compared those sequences to other known proteins and was doing this late at night, just like waiting 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.

45:43Hannah Fry: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 Damwood 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 your cells to just go completely overactive.

46:30Hannah Fry: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 target EGFR, which are used to treat six different types of cancers, including certain types of lung cancer, of bowel cancer, of head and neck cancers.

47:15Hannah Fry: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 eight in 10 people who receive cancer drugs are receiving a treatment that was developed either by Cancer Research UK or with their involvement. Yeah.

47:42Michael Stevens: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.

48:04Hannah Fry:And for the 11 people every day who are beginning these treatments in England and many more people across the world, that change is shaping the care that they receive.

48:13Michael Stevens:So for more information about Cancer Research UK, their research, their breakthroughs and how you can support them, visit cancerresearchuk.org forward slash rest is science.

48:31Michael Stevens: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.

49:04Michael Stevens: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.

49:28Hannah Fry: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,

49:49Michael Stevens:But my flag question presupposes that there's an end 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 you're going to play with a 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?

50:29Michael Stevens: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,

50:39Hannah Fry: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.

50:46Michael Stevens: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.

51:32Michael Stevens: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.

51:53Hannah Fry: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 because planting the flag suddenly makes it real right suddenly you're 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 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?

52:44Hannah Fry: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.

53:20Hannah Fry:Because there's a, comes a point where your infinitesimally small steps no longer match with reality.

53:29Michael Stevens: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?

54:08Michael Stevens:Where's the problem? I get that the real world won't allow us to experiment and just observe the answer.

54:15Hannah Fry:Doesn't feel very satisfying there, does it?

54:16Michael Stevens: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 an infinite number of times which because of the 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?

55:09Michael Stevens: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 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, for going all the way up, right? And I put in balls 1 to 10, and then I put in balls 11 to 20, and I remove ball number 1. And then I put in the next 10 balls, and I remove ball number 2. And then I put in another 10 balls, and I remove ball number 4. Or 3.

55:46Michael Stevens: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.

56:24Michael Stevens:Not anywhere in between.

56:25Hannah Fry: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 sixteenth and so on and so on and so on, which is the Xenos paradox one in one of the formulations that you described here. That's fine. That equals one. No big deal. We can deal with that. but on off on off 0 1 0 1 0 1 0 1 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 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 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 and the rubber band gets stretched an additional 10 kilometers.

57:55Hannah Fry: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.

58:11Michael Stevens:It has a solution. And I mean, this is a bit of a spoiler, but it's surprising. The answer.

58:17Hannah Fry:It's surprising. It's really surprising.

58:19Michael Stevens: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.

59:01Michael Stevens: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.

59:31Hannah Fry: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. Prepare your brains for some more mind-bending infinity strangeness.

59:47Michael Stevens: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 therestisscience at goalhanger.com.

59:56Hannah Fry:And in the meantime, you can also sign up to our free newsletter, therestisscience.com forward slash science. See you next time. Bye.