{"version":"1.0.0","segments":[{"startTime":5.465,"endTime":11.602,"body":"So today we are going to look at puzzles, riddles, conundrums and paradoxes, and how"},{"startTime":11.602,"endTime":15.285,"body":"they can lead to fascinating mathematical ideas and discoveries."},{"startTime":16.065,"endTime":21.577,"body":"We as humans are inherently curious and playful, so it's no surprise that puzzles date"},{"startTime":21.577,"endTime":24.885,"body":"back as far as recorded history can take us."},{"startTime":24.905,"endTime":30.125,"body":"And probably further, we just love entertaining our minds with a little conundrum."},{"startTime":30.545,"endTime":35.336,"body":"So some of these puzzles I'm gonna talk about today, date back hundreds, even thousands"},{"startTime":35.336,"endTime":35.655,"body":"of years, and we'll see that they can morph and change as they cross from"},{"startTime":35.655,"endTime":41.725,"body":"different languages to one another."},{"startTime":41.915,"endTime":45.325,"body":"They're moving over, uh, uh, through cultures across time."},{"startTime":45.625,"endTime":48.445,"body":"We can see little changes creeping into some of these puzzles."},{"startTime":48.475,"endTime":49.765,"body":"It's quite interesting to observe."},{"startTime":50.255,"endTime":55.88,"body":"Let's start with a puzzle that dates back quite a while, and it's called the"},{"startTime":55.88,"endTime":57.005,"body":"17 Camels Problem."},{"startTime":58.375,"endTime":62.479,"body":"The Gresham College lecture that you're listening to right now is giving you knowledge and"},{"startTime":62.479,"endTime":62.752,"body":"insight from one of the world's leading academic experts, making it takes a lot of"},{"startTime":62.752,"endTime":63.026,"body":"time, but because we want to encourage a love of learning, we think it's well"},{"startTime":63.026,"endTime":71.235,"body":"worth it."},{"startTime":72.055,"endTime":75.315,"body":"We never make you pay for lectures, although donations are needed."},{"startTime":75.815,"endTime":77.275,"body":"All we ask in return is this."},{"startTime":77.825,"endTime":80.115,"body":"Send a link to this lecture to someone you think would benefit."},{"startTime":80.495,"endTime":84.378,"body":"And if you haven't already, click the follow or subscribe button from wherever you are"},{"startTime":84.378,"endTime":85.155,"body":"listening right now."},{"startTime":86.215,"endTime":87.795,"body":"Now, let's get back to the lecture."},{"startTime":90.645,"endTime":92.785,"body":"So the story is, there's lots of variants of this."},{"startTime":92.785,"endTime":95.585,"body":"It's not always camels, but the, this is the basic idea."},{"startTime":96.085,"endTime":100.756,"body":"So a father in his will leaves provision, um, for his, his herd of camels"},{"startTime":100.756,"endTime":102.625,"body":"to be left to his sons."},{"startTime":102.965,"endTime":106.305,"body":"So a bit unfairly, half the herd goes to the eldest son."},{"startTime":106.475,"endTime":107.465,"body":"There are three sons."},{"startTime":107.725,"endTime":111.593,"body":"One third of the herd goes to the next son, and the youngest gets one"},{"startTime":111.593,"endTime":112.625,"body":"ninth of the herd."},{"startTime":112.855,"endTime":113.145,"body":"Okay?"},{"startTime":113.145,"endTime":117.382,"body":"So after the father dies, the sons gather together to divide up this herd of"},{"startTime":117.382,"endTime":117.665,"body":"camels."},{"startTime":117.665,"endTime":118.425,"body":"And there's a problem."},{"startTime":119.075,"endTime":123.615,"body":"There are only 17 camels in the herd, and if you try and divide 17,"},{"startTime":123.615,"endTime":123.918,"body":"uh, into half even, um, you hit a problem and half a camel isn't much"},{"startTime":123.918,"endTime":129.065,"body":"good for anyone."},{"startTime":129.125,"endTime":130.145,"body":"So what are they going to do?"},{"startTime":130.145,"endTime":135.011,"body":"They're standing around perplexed, and then a traveler passes by and says, I can help"},{"startTime":135.011,"endTime":135.985,"body":"you with this."},{"startTime":136.365,"endTime":138.385,"body":"Uh, I'm gonna lend you my camel."},{"startTime":138.595,"endTime":138.945,"body":"Right?"},{"startTime":139.605,"endTime":144.188,"body":"So now there are 18 camels and everything's easy 'cause 18 is divisible by all"},{"startTime":144.188,"endTime":145.105,"body":"of these things."},{"startTime":145.565,"endTime":148.985,"body":"So now the eldest son, half of 18 is nine, so that's his bit."},{"startTime":149.685,"endTime":152.145,"body":"And the next one, he gets a third of the herd."},{"startTime":152.165,"endTime":155.905,"body":"So a third of 18 is six, so he's happy, he's got his six camels."},{"startTime":156.085,"endTime":159.132,"body":"And then the youngest son, he gets one ninth of the herd, and one ninth"},{"startTime":159.132,"endTime":159.945,"body":"of 18 is two."},{"startTime":160.165,"endTime":162.225,"body":"So he gets two camels, they're all happy."},{"startTime":162.575,"endTime":166.113,"body":"That just leaves the traveler's camel leftover, and he gets back on it and off"},{"startTime":166.113,"endTime":166.585,"body":"he goes."},{"startTime":167.255,"endTime":168.225,"body":"Problem solved."},{"startTime":168.725,"endTime":173.366,"body":"But of course, the, this is a bit confusing because we moneyed we couldn't solve"},{"startTime":173.366,"endTime":173.985,"body":"the puzzle."},{"startTime":174.365,"endTime":177.065,"body":"Uh, we couldn't divide up the herd with 17 camels."},{"startTime":177.495,"endTime":181.91,"body":"Then the 18th camel allowed us to do it, but we didn't use the 18th"},{"startTime":181.91,"endTime":182.205,"body":"camel."},{"startTime":182.905,"endTime":184.645,"body":"So what on earth is going on?"},{"startTime":185.185,"endTime":190.476,"body":"And, and this kind of problem, problems like this where you are trying to divide"},{"startTime":190.476,"endTime":190.829,"body":"up, um, some amount into, into a, a collection of fractions, um, in various guises"},{"startTime":190.829,"endTime":197.885,"body":"go back a long, long time."},{"startTime":198.465,"endTime":203.824,"body":"Uh, the expert on recreational mathematics, David Singh master, who's most famous actually for bringing"},{"startTime":203.824,"endTime":206.325,"body":"the Rubik's cube to Britain, uh, 1980s."},{"startTime":206.665,"endTime":212.527,"body":"Uh, he has analyzed this, this problem and, and tracked it back through history, and"},{"startTime":212.527,"endTime":212.917,"body":"it dates back in, in, with these fractions about 600 years, but with other fractions"},{"startTime":212.917,"endTime":221.125,"body":"much, much further, even almost 4,000 years."},{"startTime":221.745,"endTime":224.605,"body":"So what's going on behind this, behind this puzzle?"},{"startTime":225.115,"endTime":229.175,"body":"Well, you are hoodwinked right at the beginning because if you actually look at those"},{"startTime":229.175,"endTime":232.965,"body":"fractions and add them up, you find that they don't add up to one."},{"startTime":233.985,"endTime":238.541,"body":"You can't divide, um, an amount into a a half and a third and a"},{"startTime":238.541,"endTime":238.845,"body":"ninth."},{"startTime":239.225,"endTime":239.845,"body":"Uh, exactly."},{"startTime":239.875,"endTime":244.765,"body":"There's some leftover, the sum of these fractions is in fact 17 eighteenths."},{"startTime":245.305,"endTime":250.011,"body":"So that's why we could divide up in this way using 17 camels out of"},{"startTime":250.011,"endTime":250.325,"body":"18."},{"startTime":251.305,"endTime":255.91,"body":"But each brother actually gets a little bit more than the amount to which he's"},{"startTime":255.91,"endTime":256.524,"body":"entitled, right?"},{"startTime":256.545,"endTime":260.201,"body":"So the first brother should really get eight and a half camels, but he gets"},{"startTime":260.201,"endTime":260.445,"body":"nine."},{"startTime":260.745,"endTime":261.644,"body":"So of course they're happy."},{"startTime":261.644,"endTime":265.1,"body":"They've all got a bit more than than the will provided for, and the traveler"},{"startTime":265.1,"endTime":267.405,"body":"can go away on, on his way with his camel."},{"startTime":267.705,"endTime":272.34,"body":"So this kind of question about how do we divide things up into fractions that"},{"startTime":272.34,"endTime":273.885,"body":"don't add up to one?"},{"startTime":274.405,"endTime":277.325,"body":"I mean, that potentially is a, is a challenging thing to deal with."},{"startTime":277.705,"endTime":281.253,"body":"And if you go back far enough, this is really, you know, a problem that"},{"startTime":281.253,"endTime":284.565,"body":"has to be addressed if you've got to divide things up in certain ways."},{"startTime":285.225,"endTime":289.344,"body":"So seeing the answer to this problem, and even the fact that it is being"},{"startTime":289.344,"endTime":289.619,"body":"asked as a puzzle or a little conundrum, tells you something about the mathematical knowledge"},{"startTime":289.619,"endTime":296.485,"body":"that is, that is there in that society at that time."},{"startTime":297.345,"endTime":302.262,"body":"And that's one of the great things about, uh, about mathematical puzzles, just like fairytales"},{"startTime":302.262,"endTime":304.885,"body":"or folk tales or, you know, creation myths."},{"startTime":305.185,"endTime":310.55,"body":"We can see stories like this moving across different cultures and getting changed and developing,"},{"startTime":310.55,"endTime":314.485,"body":"but if it's a mathematical puzzle, we get the extra thing."},{"startTime":314.585,"endTime":318.352,"body":"We are not only seeing knowledge being transferred, we're also seeing what the mathematics is"},{"startTime":318.352,"endTime":318.603,"body":"at the time and how the, and how the mathematics of the time deals with"},{"startTime":318.603,"endTime":323.125,"body":"the questions like these."},{"startTime":323.785,"endTime":328.579,"body":"Um, so for example, if you had a, a puzzle whose solution requires you to"},{"startTime":328.579,"endTime":328.898,"body":"do something involving, you know, a quadratic expression, something with X squares in it, then"},{"startTime":328.898,"endTime":329.218,"body":"that tells you that, um, at that time and in that place there was some"},{"startTime":329.218,"endTime":340.085,"body":"knowledge of something like quadratic expressions."},{"startTime":340.185,"endTime":341.525,"body":"And that goes back quite a long time."},{"startTime":342.065,"endTime":346.047,"body":"Uh, for example, there's a, there's a puzzle on this Babylonian, I was gonna say"},{"startTime":346.047,"endTime":349.765,"body":"tablet, I don't know if that's the right word for it, prism or something."},{"startTime":350.305,"endTime":353.845,"body":"Um, that dates back as you can see, nearly 4,000 years."},{"startTime":354.425,"endTime":357.9,"body":"And the puzzle is about finding the dimensions of a mysterious rectangle."},{"startTime":358.585,"endTime":362.542,"body":"And so you are told some things, you are told that to the area of"},{"startTime":362.542,"endTime":365.445,"body":"this rectangle, um, was added the excess of length over width."},{"startTime":365.465,"endTime":370.285,"body":"So in other words, that's area plus length minus width equals 183."},{"startTime":370.705,"endTime":374.593,"body":"And you're also told that length plus width is 27, and you have to then"},{"startTime":374.593,"endTime":377.445,"body":"find the length, the width, and the area of that rectangle."},{"startTime":378.025,"endTime":380.605,"body":"Now you can solve this puzzle if you like, on your way home."},{"startTime":380.905,"endTime":384.465,"body":"Um, if you do, you are gonna find that you end up probably with a"},{"startTime":384.465,"endTime":386.365,"body":"quadratic equation to solve, uh, at some point."},{"startTime":387.495,"endTime":393.027,"body":"Babylonian mathematics did not have quadratic equations in the way we think of them now,"},{"startTime":393.027,"endTime":397.085,"body":"but they had an algorithm that would solve problems like this."},{"startTime":397.865,"endTime":402.325,"body":"So in some sense, they could solve certain kinds of quadratic equation."},{"startTime":402.425,"endTime":403.685,"body":"That's very interesting to know."},{"startTime":405.875,"endTime":411.962,"body":"Sometimes we find that when puzzles move between different, uh, different cultures and different traditions,"},{"startTime":411.962,"endTime":417.645,"body":"they can change in subtle ways that actually affect the solution to the problem."},{"startTime":418.025,"endTime":419.565,"body":"So I'm gonna tell you a, a puzzle."},{"startTime":419.715,"endTime":421.125,"body":"This is a modern version of it."},{"startTime":421.425,"endTime":424.08,"body":"Uh, it's the one that my dad told me when I was eight years old"},{"startTime":424.08,"endTime":424.965,"body":"and I got it wrong."},{"startTime":424.985,"endTime":428.485,"body":"So, you know, this is now my rehabilitation finally complete."},{"startTime":428.785,"endTime":432.165,"body":"Um, the snail in the well might have seen a puzzle a bit like this."},{"startTime":432.745,"endTime":437.362,"body":"So a snail, uh, Paul thinks at the bottom of a, well, two meters deep,"},{"startTime":437.362,"endTime":437.669,"body":"and starting first thing Monday morning, sunrise on Monday, it starts to climb up very"},{"startTime":437.669,"endTime":445.365,"body":"slowly and it climbs up half a meter during each day."},{"startTime":445.505,"endTime":448.973,"body":"But then at night it kind of slides back down again, one fifth of a"},{"startTime":448.973,"endTime":449.205,"body":"meter."},{"startTime":450.385,"endTime":454.125,"body":"And the question is, um, what day is it gonna escape from this?"},{"startTime":454.125,"endTime":458.305,"body":"Well, so, okay, first thing, Monday, um, it's at the bottom and then it is"},{"startTime":458.305,"endTime":458.584,"body":"going up half a meter, but then it slides back down a fifth of a"},{"startTime":458.584,"endTime":462.765,"body":"meter."},{"startTime":462.865,"endTime":466.525,"body":"So kind of the net progress each day from sunrise one day to sunrise."},{"startTime":466.665,"endTime":469.885,"body":"The next is a half minus a fifth, which is three tenths."},{"startTime":470.355,"endTime":470.645,"body":"Okay?"},{"startTime":470.665,"endTime":474.896,"body":"So, so on Tuesday morning, first thing it'll be n 0.3 meters above the bottom,"},{"startTime":474.896,"endTime":478.845,"body":"and then it's gonna make n 0.3 meters progress in every 24 hour period."},{"startTime":479.265,"endTime":485.605,"body":"So we can mark off those point threes and Wednesday, Thursday, Friday, Saturday, Sunday, uh,"},{"startTime":485.605,"endTime":488.565,"body":"but look by Sunday, 1.8 meters up."},{"startTime":488.905,"endTime":492.005,"body":"And so it's only got 20 centimeters left to climb."},{"startTime":492.265,"endTime":495.045,"body":"So it's gonna do that at some point on Sunday, right?"},{"startTime":495.265,"endTime":497.805,"body":"So it escapes sort of Sunday teeter, okay?"},{"startTime":498.305,"endTime":498.965,"body":"No, it doesn't."},{"startTime":500.075,"endTime":504.216,"body":"That is, that is the mistake little Sarah made, and that's okay because I learned"},{"startTime":504.216,"endTime":505.045,"body":"something that day."},{"startTime":505.725,"endTime":508.943,"body":"\u003claugh\u003e, I learned that 30 or 40 years, hence I would be able to get"},{"startTime":508.943,"endTime":510.445,"body":"it right in front of an audience."},{"startTime":510.785,"endTime":516.703,"body":"Um, so rewind, rewind, Saturday morning, 1.5 meters, that's where it's, but during the day"},{"startTime":516.703,"endTime":521.044,"body":"on Saturday it goes up, it can climb half a meter."},{"startTime":521.825,"endTime":524.765,"body":"So it actually escapes right at the end of the day on Saturday."},{"startTime":525.385,"endTime":529.214,"body":"It doesn't get to the point of sliding back at night because it's already out"},{"startTime":529.214,"endTime":529.725,"body":"by then."},{"startTime":529.945,"endTime":533.845,"body":"So actually the answer is that the snail will escape from this well, last thing"},{"startTime":533.845,"endTime":534.365,"body":"on Saturday."},{"startTime":535.945,"endTime":541.375,"body":"But the reason I'm telling you this puzzle is that it started off or puzzles"},{"startTime":541.375,"endTime":546.805,"body":"a bit like this began, uh, life probably around about the sixth century in India."},{"startTime":547.785,"endTime":551.628,"body":"And then they traveled, you know, along trade routes with other knowledge that was being"},{"startTime":551.628,"endTime":551.885,"body":"transmitted."},{"startTime":552.035,"endTime":558.792,"body":"They got translated into different languages and they made their way to Italy, uh, in"},{"startTime":558.792,"endTime":561.045,"body":"maybe the 13th century ish."},{"startTime":562.465,"endTime":567.755,"body":"And in the Italian books that had this puzzle in, or puzzles like it, the"},{"startTime":567.755,"endTime":573.045,"body":"solution was for the first few decades wrong, it was the Sunday tea time solution."},{"startTime":573.505,"endTime":575.125,"body":"But there, there's a reason for this."},{"startTime":575.515,"endTime":581.374,"body":"It's not that they were stupid, there's the, the reason is that something's lost in"},{"startTime":581.374,"endTime":581.765,"body":"translation."},{"startTime":582.185,"endTime":586.402,"body":"The original puzzles of this form had this a half minus of fifth or something"},{"startTime":586.402,"endTime":586.965,"body":"like that."},{"startTime":587.825,"endTime":593.205,"body":"But that's because it was a common way to express more complicated fractions."},{"startTime":593.545,"endTime":597.205,"body":"You write them in terms of simpler unit fractions with a one on the top."},{"startTime":597.865,"endTime":602.332,"body":"So actually originally the puzzle puzzles like this would say something like the snail or"},{"startTime":602.332,"endTime":602.63,"body":"whatever animal climbs a half minus a fifth meters per day, not meters, but whatever"},{"startTime":602.63,"endTime":608.885,"body":"unit was it at the time, right?"},{"startTime":609.185,"endTime":614.24,"body":"And then this is all happening in one go, so you don't have this progressing"},{"startTime":614.24,"endTime":615.925,"body":"and then separately falling back."},{"startTime":616.465,"endTime":621.405,"body":"So actually that it's simply, it climbs three tenths of a meter every day."},{"startTime":621.405,"endTime":624.445,"body":"And so you don't have this sort of end special case at the end."},{"startTime":625.105,"endTime":631.45,"body":"So those puzzles, as they originally were, the solution was the Sunday tea time solution"},{"startTime":631.45,"endTime":633.565,"body":"when these puzzles were translated."},{"startTime":634.225,"endTime":639.031,"body":"Um, and the pe people who were doing the translations and copying out the, the"},{"startTime":639.031,"endTime":639.352,"body":"puzzles, they saw this a half minus a fifth type expression and kind of thought,"},{"startTime":639.352,"endTime":648.005,"body":"oh, it must be because first it goes and then it comes back."},{"startTime":648.625,"endTime":652.36,"body":"And therefore they started to say, that doesn't seem like much of a change, but"},{"startTime":652.36,"endTime":653.605,"body":"it totally changes the puzzle."},{"startTime":654.385,"endTime":658.871,"body":"So it's very interesting to see, yeah, actually it took, you know, decades for the"},{"startTime":658.871,"endTime":660.965,"body":"solutions to catch up with the puzzle."},{"startTime":661.105,"endTime":664.333,"body":"So maybe if you're doing, if you find an old puzzle book and you dunno"},{"startTime":664.333,"endTime":664.549,"body":"why the solution given is the correct one, maybe it isn't \u003claugh\u003e, it's always, always"},{"startTime":664.549,"endTime":669.285,"body":"worth like keep it, keep your hope alive."},{"startTime":669.665,"endTime":673.645,"body":"Uh, okay, so we can see knowledge moving and transmitting and sometimes changing."},{"startTime":674.455,"endTime":675.725,"body":"Let's look at another old puzzle."},{"startTime":676.705,"endTime":681.055,"body":"Uh, this diagram is a very misleading in terms of how you actually shove to"},{"startTime":681.055,"endTime":681.925,"body":"solve this puzzle."},{"startTime":682.465,"endTime":687.523,"body":"Um, so this is one of a, a, a whole collection of puzzles around this"},{"startTime":687.523,"endTime":687.86,"body":"kind of theme that are often called decanting problems, where you have, uh, one or"},{"startTime":687.86,"endTime":688.197,"body":"more jugs, uh, of fixed volumes and you have to create a different volume of,"},{"startTime":688.197,"endTime":699.325,"body":"of water or sometimes wine."},{"startTime":699.785,"endTime":703.895,"body":"So in this case, you have a, a jug that measures exactly five liters and"},{"startTime":703.895,"endTime":708.005,"body":"another one that measures exactly three liters, and there's no markings or anything on them."},{"startTime":708.225,"endTime":712.856,"body":"Uh, but you have to use those jugs to create a volume of exactly one"},{"startTime":712.856,"endTime":713.165,"body":"liter."},{"startTime":714.115,"endTime":714.405,"body":"Okay?"},{"startTime":714.405,"endTime":719.266,"body":"So, so you've gotta somehow work out how to do this now, solving this problem"},{"startTime":719.266,"endTime":719.59,"body":"and problems a bit like it, um, turn out that to have important applications in"},{"startTime":719.59,"endTime":727.045,"body":"kind of the real world, not in puzzle world."},{"startTime":727.345,"endTime":732.245,"body":"Um, obviously the most important one, uh, was discovered by noted mathematician."},{"startTime":732.435,"endTime":736.976,"body":"John McClain in the film Die Hard with vengeance \u003claugh\u003e, which I'm sure you remember"},{"startTime":736.976,"endTime":737.885,"body":"being highly mathematical."},{"startTime":738.025,"endTime":742.615,"body":"So in, I'm being a bit silly, but in this film, uh, Bruce Willis and"},{"startTime":742.615,"endTime":742.921,"body":"Samuel L. Jackson are somehow pitted against this mean, uh, bad guy, Jeremy Irons, who"},{"startTime":742.921,"endTime":743.227,"body":"sets them these problems to do against the clock or, you know, and if they"},{"startTime":743.227,"endTime":754.245,"body":"don't solve them, then everything explodes or something."},{"startTime":754.505,"endTime":759.888,"body":"Um, and Bruce Willis noted, mathematician manages to solve this puzzle, \u003claugh\u003e and you know,"},{"startTime":759.888,"endTime":760.965,"body":"save the day."},{"startTime":761.385,"endTime":763.485,"body":"So I wonder how he solved it."},{"startTime":763.485,"endTime":768.885,"body":"Maybe he, he did what I'm about to show you, um, just a bit of"},{"startTime":768.885,"endTime":769.245,"body":"terminology."},{"startTime":769.355,"endTime":772.845,"body":"I'll call this the five three, uh, Walter Jug problem."},{"startTime":773.305,"endTime":777.186,"body":"Uh, but actually as mathematicians of course, we immediately want to know can we solve"},{"startTime":777.186,"endTime":777.445,"body":"this?"},{"startTime":777.465,"endTime":781.137,"body":"But can we solve all the problems like it, can we solve the XY jug"},{"startTime":781.137,"endTime":781.382,"body":"problem where the volumes of the, the jugs that we have are X and y,"},{"startTime":781.382,"endTime":786.525,"body":"we're still trying to make one liter."},{"startTime":787.785,"endTime":789.325,"body":"So how do we approach a question like this?"},{"startTime":789.325,"endTime":792.445,"body":"Something we could trial and error will get us some sort of distance."},{"startTime":792.985,"endTime":795.845,"body":"Um, we've got our five liter jug and our three liter jug."},{"startTime":795.935,"endTime":798.765,"body":"We've gotta fill and empty them in some way to end up with one liter."},{"startTime":800.065,"endTime":805.368,"body":"So the crucial observation is it, this is the same as finding some combination of"},{"startTime":805.368,"endTime":809.965,"body":"fives and threes that gives you one, just the numbers five and three."},{"startTime":810.345,"endTime":815.634,"body":"So for instance, uh, two times three take away one times five is one six"},{"startTime":815.634,"endTime":817.045,"body":"minus five is one."},{"startTime":817.075,"endTime":817.365,"body":"Okay?"},{"startTime":817.625,"endTime":819.325,"body":"Now, why should that help us with the problem?"},{"startTime":819.795,"endTime":826.07,"body":"Well, you can turn an expression like this into a, a rule for filling and"},{"startTime":826.07,"endTime":827.325,"body":"emptying the jugs."},{"startTime":827.745,"endTime":829.965,"body":"You've got a positive multiple of three."},{"startTime":830.145,"endTime":835.2,"body":"So that's going to mean filling the three liter jug two times and you've got"},{"startTime":835.2,"endTime":836.885,"body":"minus, uh, one outta five."},{"startTime":837.315,"endTime":839.805,"body":"That means we're gonna empty the five liter jug once."},{"startTime":840.625,"endTime":844.085,"body":"So minus sign means empty, plus sign means fill."},{"startTime":844.745,"endTime":845.565,"body":"So let's do this."},{"startTime":845.625,"endTime":848.365,"body":"So we're gonna fill the three liter jug for the first time."},{"startTime":848.675,"endTime":849.685,"body":"What are we gonna do with that water?"},{"startTime":849.685,"endTime":851.325,"body":"We'll just pour it into the five liter jug."},{"startTime":851.945,"endTime":855.56,"body":"Um, then we fill three to jug again and pour as much as we can"},{"startTime":855.56,"endTime":856.765,"body":"into the five liter jug."},{"startTime":856.985,"endTime":859.725,"body":"Um, there's room for two more liters, so we'll pour those in."},{"startTime":860.185,"endTime":863.885,"body":"Now the five liter jug is full, we'll empty it, aha, wonderfully."},{"startTime":863.905,"endTime":866.725,"body":"We have exactly one liter left in the three liter jug."},{"startTime":866.795,"endTime":867.085,"body":"Okay?"},{"startTime":867.085,"endTime":870.535,"body":"And you see we've used plus two, lots of three and minus one lots of"},{"startTime":870.535,"endTime":870.765,"body":"five."},{"startTime":871.025,"endTime":875.335,"body":"So you can translate this, you know, expression just in terms of numbers into a"},{"startTime":875.335,"endTime":876.485,"body":"solution to the problem."},{"startTime":877.265,"endTime":880.645,"body":"Uh, we can do it from a slightly bigger problem version."},{"startTime":880.945,"endTime":883.125,"body":"So this is the seven five jug problem."},{"startTime":883.585,"endTime":887.23,"body":"So first step we have to try and find an expression, a combination of sevens"},{"startTime":887.23,"endTime":888.445,"body":"and fives that make one."},{"startTime":888.505,"endTime":893.109,"body":"If you look around for a bit, you might notice that three lots of five"},{"startTime":893.109,"endTime":895.565,"body":"take away two lots of seven is one."},{"startTime":895.775,"endTime":896.125,"body":"Great."},{"startTime":896.465,"endTime":898.285,"body":"So let's use our little process."},{"startTime":898.905,"endTime":903.165,"body":"So we are adding five, so that means we are filling the five liter jug."},{"startTime":903.165,"endTime":906.577,"body":"So we fill it, pour it into the seven liter jug, fill the five liter"},{"startTime":906.577,"endTime":906.805,"body":"jug."},{"startTime":906.805,"endTime":910.78,"body":"Again, there's room for two liters in the top there so that they'll fit in"},{"startTime":910.78,"endTime":911.045,"body":"there."},{"startTime":911.465,"endTime":913.925,"body":"Now the seven liter jug is full, let's empty it."},{"startTime":914.505,"endTime":917.805,"body":"And now we can keep going and fill the five eth jug one more time."},{"startTime":918.385,"endTime":921.565,"body":"Now look, there's room for four liters left in that seven liter jug."},{"startTime":921.745,"endTime":922.925,"body":"So we'll pour those in."},{"startTime":923.185,"endTime":926.605,"body":"Now it's full empty it and we're done hurrah."},{"startTime":926.985,"endTime":928.005,"body":"So solve the problem."},{"startTime":928.625,"endTime":932.975,"body":"So any, any bad guys who want to make us solve these things at speed,"},{"startTime":932.975,"endTime":933.265,"body":"we now have a process that will allow us to do it if we can"},{"startTime":933.265,"endTime":938.485,"body":"find expressions like this."},{"startTime":939.185,"endTime":940.805,"body":"So how do we find expression like this?"},{"startTime":940.945,"endTime":942.685,"body":"Do we, how can we be sure there even is one?"},{"startTime":943.145,"endTime":944.485,"body":"Um, I mean there might be many."},{"startTime":945.185,"endTime":947.045,"body":"It will work for any expression you like."},{"startTime":947.065,"endTime":950.605,"body":"So three times seven minus four times five is one as well."},{"startTime":950.955,"endTime":954.963,"body":"That would give us a solution involving filling the seven liter jug and emptying the"},{"startTime":954.963,"endTime":955.765,"body":"five liter jug."},{"startTime":957.385,"endTime":959.245,"body":"But how do we know there's even one of these?"},{"startTime":960.315,"endTime":964.92,"body":"Well, what we've said so far is that if we want to be able to"},{"startTime":964.92,"endTime":969.525,"body":"do these problems in general, we, it's the same as finding an expression like this."},{"startTime":970.025,"endTime":973.88,"body":"So au minus bv, it's one, uh, U and V are the, the volumes of"},{"startTime":973.88,"endTime":975.165,"body":"the jugs in some order."},{"startTime":975.625,"endTime":977.885,"body":"I'm not going to care which one's bigger or small at this point."},{"startTime":978.005,"endTime":979.925,"body":"I just wanna find some expression like this."},{"startTime":980.465,"endTime":982.205,"body":"And, and A and B are are whole numbers."},{"startTime":982.985,"endTime":984.645,"body":"So if we can do that, we are happy."},{"startTime":985.635,"endTime":989.765,"body":"There's one instance where we definitely are never going to be able to do that."},{"startTime":989.915,"endTime":992.205,"body":"This, we are never gonna be able to find an expression like this."},{"startTime":992.705,"endTime":997.165,"body":"And that's when the jug volumes, um, have a common factor bigger than one."},{"startTime":997.425,"endTime":1002.063,"body":"So if for example, they're both U and V, if they're both uh, divisible by"},{"startTime":1002.063,"endTime":1002.372,"body":"three say then the whole of the left hand side of that expression that a"},{"startTime":1002.372,"endTime":1002.681,"body":"U minus bv, it'll also be divisible by three 'cause U as a factor of"},{"startTime":1002.681,"endTime":1012.885,"body":"three and so does V."},{"startTime":1013.025,"endTime":1016.512,"body":"So that combination will, but that's bad news if you want the answer to be"},{"startTime":1016.512,"endTime":1018.605,"body":"one, 'cause one definitely does not divisible by three."},{"startTime":1019.665,"endTime":1024.27,"body":"So we know if, if UNV have a common factor, common divide are bigger than"},{"startTime":1024.27,"endTime":1025.805,"body":"one, we are in trouble."},{"startTime":1026.665,"endTime":1030.402,"body":"So that means if their greatest common advisor is anything other than one, we can't"},{"startTime":1030.402,"endTime":1032.645,"body":"do this, we absolutely, definitely cannot solve this problem."},{"startTime":1033.785,"endTime":1035.405,"body":"So it's impossible in that situation."},{"startTime":1036.233,"endTime":1039.568,"body":"Well, okay, what about if they are co prime, if they greatest common divides really"},{"startTime":1039.568,"endTime":1041.125,"body":"is one, can we do it there?"},{"startTime":1041.464,"endTime":1045.165,"body":"So then the question is, is it enough if they're co prime these numbers?"},{"startTime":1045.223,"endTime":1049.16,"body":"So like five and seven or three and five, can we then find an expression"},{"startTime":1049.16,"endTime":1049.685,"body":"like this?"},{"startTime":1050.505,"endTime":1053.405,"body":"And the answer is brilliantly that we can, we always can."},{"startTime":1053.575,"endTime":1057.547,"body":"We're guaranteed to be able to do it and there's a way that we can"},{"startTime":1057.547,"endTime":1060.725,"body":"do it, a simple process that allows us to find these expressions."},{"startTime":1061.025,"endTime":1065.902,"body":"And it comes from something called the Euclidean algorithm, as in Euclid who uh, devised"},{"startTime":1065.902,"endTime":1069.805,"body":"it a couple of thousand years ago, but it's still very useful."},{"startTime":1070.505,"endTime":1074.925,"body":"So actually this algorithm is really for finding the greatest common divisor of numbers."},{"startTime":1075.665,"endTime":1079.148,"body":"Now, the numbers we are looking at, you can instantly spot what the greatest common"},{"startTime":1079.148,"endTime":1079.845,"body":"DI advisor is."},{"startTime":1080.005,"endTime":1082.125,"body":"'cause you can just factorized them and know what the factors are."},{"startTime":1082.585,"endTime":1083.885,"body":"Um, so we'll just do it."},{"startTime":1083.885,"endTime":1088.256,"body":"An example with, uh, 35 and 11, obviously they're co prime, so their greatest common"},{"startTime":1088.256,"endTime":1092.045,"body":"DI advisor is one, but we're still gonna go through the little calculation."},{"startTime":1092.045,"endTime":1096.778,"body":"It's very quick because then we're going to do something with that process to allow"},{"startTime":1096.778,"endTime":1100.565,"body":"us to get this, this equation that solves the water jug problem."},{"startTime":1100.985,"endTime":1107.072,"body":"So what you do, you do the same thing repeatedly just yeah, um, you take"},{"startTime":1107.072,"endTime":1110.725,"body":"the bigger number, divide it by the smaller one."},{"startTime":1110.745,"endTime":1114.401,"body":"So we've got 35 and 11 and you'll get a quotient and you'll get a"},{"startTime":1114.401,"endTime":1114.645,"body":"remainder."},{"startTime":1114.785,"endTime":1118.725,"body":"So 35, 3 times 11 plus two, fine."},{"startTime":1119.505,"endTime":1121.125,"body":"Now there's something interesting we can say."},{"startTime":1121.225,"endTime":1127.375,"body":"Now this 35, we've written it in terms of 11 and two, any number that"},{"startTime":1127.375,"endTime":1130.245,"body":"divides any factor of 11 and two."},{"startTime":1130.265,"endTime":1135.11,"body":"So a common factor of those two things divides the whole right hand side of"},{"startTime":1135.11,"endTime":1136.725,"body":"this and therefore divides 35."},{"startTime":1137.105,"endTime":1140.005,"body":"So it's also a common factor of 35 and 11."},{"startTime":1140.785,"endTime":1143.485,"body":"And the other thing is true, the other way round is true as well."},{"startTime":1144.005,"endTime":1148.245,"body":"Anything that divides both 35 and 11 also must divide two."},{"startTime":1149.145,"endTime":1154.109,"body":"So what that tells you is, um, after a bit of thought, it tells you"},{"startTime":1154.109,"endTime":1154.44,"body":"that these pairs 35 and 11 and 11 and two have the same greatest common"},{"startTime":1154.44,"endTime":1159.405,"body":"divider."},{"startTime":1160.245,"endTime":1163.645,"body":"Anything that divides both 35 and 11 must also divide two."},{"startTime":1164.125,"endTime":1166.925,"body":"Anything that divides both 11 and two must also divide 35."},{"startTime":1166.945,"endTime":1168.845,"body":"And that gives you the solution."},{"startTime":1171.065,"endTime":1173.365,"body":"So what do we do?"},{"startTime":1173.555,"endTime":1174.805,"body":"Well, we, we are gonna carry on."},{"startTime":1174.805,"endTime":1178.005,"body":"So we take the 11 and two, we've made ourselves a smaller problem, and that's"},{"startTime":1178.005,"endTime":1179.285,"body":"something we often do in mathematics."},{"startTime":1179.345,"endTime":1182.005,"body":"We take hard problem and reduce it to a simpler one."},{"startTime":1182.145,"endTime":1183.645,"body":"So we now have a simpler challenge."},{"startTime":1183.905,"endTime":1186.045,"body":"Can we find the greatest common advisor of 11 and two?"},{"startTime":1186.235,"endTime":1189.405,"body":"Well, yes, of course we can, but let's do this little process."},{"startTime":1189.985,"endTime":1190.805,"body":"We do the same thing."},{"startTime":1190.805,"endTime":1194.405,"body":"Divide the bigger number by the smaller 11 is five times two plus one."},{"startTime":1195.505,"endTime":1199.346,"body":"And now we can say the same, uh, thing about these three numbers here, that"},{"startTime":1199.346,"endTime":1199.602,"body":"the greatest common DI advisor of 11 and two is equal to the greatest common"},{"startTime":1199.602,"endTime":1204.725,"body":"DI advisor of two and one."},{"startTime":1204.905,"endTime":1209.352,"body":"And at this point, we really can be absolutely sure that that is one because"},{"startTime":1209.352,"endTime":1211.725,"body":"one, the o the only divisors of one."},{"startTime":1212.105,"endTime":1214.685,"body":"Uh, well it's just one if you're talking about positive divisors."},{"startTime":1215.145,"endTime":1219.581,"body":"And so any divisors of the greatest common di advisor of one with anything is"},{"startTime":1219.581,"endTime":1220.765,"body":"always gonna be one."},{"startTime":1221.265,"endTime":1224.765,"body":"So, okay, well we already knew the greatest common DI advisor of 35 and 11."},{"startTime":1224.995,"endTime":1226.485,"body":"That wasn't, that isn't why we're doing this."},{"startTime":1226.985,"endTime":1232.722,"body":"But we've obtained now two uh, equations that are gonna be very useful to us"},{"startTime":1232.722,"endTime":1236.165,"body":"because what we do is we're gonna rearrange them."},{"startTime":1237.185,"endTime":1242.285,"body":"So those two equations there, 35 equals something 11 equals something, we're gonna rearrange them."},{"startTime":1242.585,"endTime":1244.925,"body":"So 11 equals five times two plus one."},{"startTime":1244.925,"endTime":1249.045,"body":"We are rearranging it to be one is now 11 minus five times two."},{"startTime":1249.705,"endTime":1253.365,"body":"And then the top one, we've rearranged that to make two the subject."},{"startTime":1253.505,"endTime":1255.845,"body":"So two is 35 minus three times 11."},{"startTime":1256.545,"endTime":1261.864,"body":"And the reason we are doing that because we want to find an expression that"},{"startTime":1261.864,"endTime":1265.765,"body":"looks like something times 35 and something times 11 equals one."},{"startTime":1266.665,"endTime":1271.191,"body":"So we take that first one there, one is equal to something and it's equal"},{"startTime":1271.191,"endTime":1273.605,"body":"to 11 times 11 minus five times two."},{"startTime":1274.265,"endTime":1275.325,"body":"We don't really want a two."},{"startTime":1276.265,"endTime":1278.885,"body":"We want instead to involve 30 fives and elevens."},{"startTime":1278.885,"endTime":1281.085,"body":"So we've got elevens and twos in that expression."},{"startTime":1281.505,"endTime":1286.573,"body":"I'm gonna just replace that two with this expression for it that I've got on"},{"startTime":1286.573,"endTime":1287.925,"body":"the right hand side."},{"startTime":1288.875,"endTime":1289.165,"body":"Okay?"},{"startTime":1289.185,"endTime":1293.507,"body":"So if you do that, you've now got an expression for one, which is just"},{"startTime":1293.507,"endTime":1295.525,"body":"in terms of 30 fives and elevens."},{"startTime":1295.745,"endTime":1296.645,"body":"That's what we're after."},{"startTime":1297.345,"endTime":1302.957,"body":"And if you just gather up your terms, there you go, we've got that now"},{"startTime":1302.957,"endTime":1306.325,"body":"16 times 11 minus five times 35 equals one."},{"startTime":1306.585,"endTime":1311.006,"body":"And that's right because 16 times 11 is 176 and five times 35 is a"},{"startTime":1311.006,"endTime":1313.365,"body":"hundred thirty five, a hundred and seventy five."},{"startTime":1313.665,"endTime":1318.368,"body":"So we are happy, uh, we've got this expression that allows us to solve the,"},{"startTime":1318.368,"endTime":1322.445,"body":"I'm sure we all wanted to solve it, the 35 11 jug problem."},{"startTime":1322.635,"endTime":1322.925,"body":"Okay?"},{"startTime":1322.945,"endTime":1328.15,"body":"So now hopefully that little example shows you there is this technique, um, which allows"},{"startTime":1328.15,"endTime":1329.885,"body":"you to solve these problems."},{"startTime":1330.145,"endTime":1335.032,"body":"So we can solve any XY water jug problem as long as X and Y"},{"startTime":1335.032,"endTime":1337.965,"body":"are Cori, that their greatest common advisor is one."},{"startTime":1339.665,"endTime":1345.515,"body":"Now this is all very nice if we are presented suddenly with emergency drug, uh,"},{"startTime":1345.515,"endTime":1346.685,"body":"solving \u003claugh\u003e conundrums."},{"startTime":1347.425,"endTime":1354.261,"body":"But this algorithm, this process to find combinations of numbers that equal one turns out"},{"startTime":1354.261,"endTime":1354.716,"body":"to be absolutely vital in something many of us do almost every day, if not"},{"startTime":1354.716,"endTime":1355.172,"body":"every day, which is online transactions because it's a crucial component of the so-called RSA"},{"startTime":1355.172,"endTime":1368.845,"body":"encryption algorithm."},{"startTime":1369.345,"endTime":1373.876,"body":"And this is used a lot online for kind of, uh, banking securities that prevent"},{"startTime":1373.876,"endTime":1375.085,"body":"your details being stolen."},{"startTime":1375.185,"endTime":1379.698,"body":"So if you wanna send, uh, some message, you know, please transfer X amount of"},{"startTime":1379.698,"endTime":1381.805,"body":"money from my account to your bank."},{"startTime":1381.825,"endTime":1384.485,"body":"You don't want anyone, anyone intercepting your bank details."},{"startTime":1385.105,"endTime":1390.524,"body":"So of course that information is encrypted, and I won't go into the details of"},{"startTime":1390.524,"endTime":1390.885,"body":"the algorithm of how this is done, but to encode, encrypt your message, um, you"},{"startTime":1390.885,"endTime":1391.246,"body":"use a number called the public key to do something to your message that encrypts"},{"startTime":1391.246,"endTime":1402.085,"body":"it mathematically."},{"startTime":1402.945,"endTime":1406.886,"body":"And then you don't want just anyone, anyone ought to be able to encrypt and"},{"startTime":1406.886,"endTime":1407.149,"body":"send messages, but only the person to whom the message is really sent, uh, should"},{"startTime":1407.149,"endTime":1414.245,"body":"be able to decrypt and, and work out what the message actually says."},{"startTime":1415.025,"endTime":1419.851,"body":"So at a crucial moment in the process, in order to do this decryption, what"},{"startTime":1419.851,"endTime":1420.173,"body":"needs to happen is this Euclidean algorithm process has to be applied, um, for two"},{"startTime":1420.173,"endTime":1425.965,"body":"numbers, K and x."},{"startTime":1426.265,"endTime":1431.142,"body":"Now, K is this public encryption key that you know, and everyone can know it's"},{"startTime":1431.142,"endTime":1435.045,"body":"public, but this number X is not, that is a private key."},{"startTime":1435.195,"endTime":1437.965,"body":"Only the bank or whoever it is, will will have that."},{"startTime":1437.965,"endTime":1440.245,"body":"And so only they will be able to apply this."},{"startTime":1440.265,"endTime":1443.325,"body":"You plead in algorithm process to decre the message."},{"startTime":1443.465,"endTime":1448.933,"body":"So this little cute little algorithm that helps you solve the water jog problem actually"},{"startTime":1448.933,"endTime":1451.485,"body":"also helps, uh, online banking stay secure."},{"startTime":1454.425,"endTime":1457.005,"body":"But this is very nice."},{"startTime":1457.025,"endTime":1458.605,"body":"It links to a nice little puzzle."},{"startTime":1459.155,"endTime":1463.325,"body":"However, the, this algorithm was not developed in order to solve the puzzle."},{"startTime":1463.385,"endTime":1466.403,"body":"It helps to solve it, but wasn't, it was not developed in order to solve"},{"startTime":1466.403,"endTime":1466.605,"body":"it."},{"startTime":1466.785,"endTime":1471.602,"body":"So you might say, well, okay then why should we be interested in what's called"},{"startTime":1471.602,"endTime":1472.245,"body":"recreational mathematics?"},{"startTime":1473.225,"endTime":1475.325,"body":"Um, it's just a bit of fun."},{"startTime":1475.715,"endTime":1477.005,"body":"Well, that's actually my answer."},{"startTime":1477.115,"endTime":1478.405,"body":"It's a bit of fun, \u003claugh\u003e."},{"startTime":1478.545,"endTime":1479.365,"body":"What's wrong with that?"},{"startTime":1479.365,"endTime":1479.685,"body":"Right?"},{"startTime":1480.165,"endTime":1483.405,"body":"Actually, for a mathematician, the word recreational here is completely redundant."},{"startTime":1483.845,"endTime":1484.885,"body":"'cause what do, what's recreational?"},{"startTime":1484.885,"endTime":1485.685,"body":"It's something that's fun."},{"startTime":1485.785,"endTime":1488.405,"body":"We do 'cause we enjoy it, and that's what mathematics is."},{"startTime":1488.625,"endTime":1493.966,"body":"So, okay, it is mathematics and we don't need that word recreational, but if you"},{"startTime":1493.966,"endTime":1494.322,"body":"insist, if you insist, there are plenty of examples of puzzles that have given rise"},{"startTime":1494.322,"endTime":1501.445,"body":"to genuinely new and exciting mathematics."},{"startTime":1501.905,"endTime":1506.696,"body":"Um, the most famous of these, which I feel sort of, there's a bylaw of"},{"startTime":1506.696,"endTime":1507.016,"body":"mathematics that you cannot give a lecture involving recreational mathematics without mentioning it, is the"},{"startTime":1507.016,"endTime":1513.405,"body":"famous, uh, bridges of burg problem."},{"startTime":1513.945,"endTime":1519.426,"body":"So I'll just briefly mention this, um, the situation was there's a real life, uh,"},{"startTime":1519.426,"endTime":1519.792,"body":"town burg, uh, and there's a river through the center kind of splits and joins"},{"startTime":1519.792,"endTime":1526.005,"body":"back together again."},{"startTime":1526.625,"endTime":1531.156,"body":"And, uh, there were, there aren't anymore, but there were seven bridges, uh, at various"},{"startTime":1531.156,"endTime":1532.365,"body":"places in this river."},{"startTime":1533.025,"endTime":1536.325,"body":"And the townsfolk had this little pastime on the weekend."},{"startTime":1536.585,"endTime":1540.365,"body":"They'd go for a walk around the town and they wanted an interesting fun walk."},{"startTime":1540.745,"endTime":1545.582,"body":"So the question was, is it possible to design a route that will cross each"},{"startTime":1545.582,"endTime":1548.485,"body":"of these bridges but not go back on itself?"},{"startTime":1548.585,"endTime":1552.773,"body":"So you don't wanna go over the same bridge twice, so you want to cross"},{"startTime":1552.773,"endTime":1555.845,"body":"every bridge once and only once is this possible to do."},{"startTime":1556.225,"endTime":1561.085,"body":"And it seemed somehow it didn't seem possible, but they couldn't quite work out really."},{"startTime":1561.085,"endTime":1563.845,"body":"Is it impossible or are we just not clever enough to think of the answer?"},{"startTime":1564.505,"endTime":1567.765,"body":"Um, or, or, you know, can it be done?"},{"startTime":1568.305,"endTime":1573.444,"body":"So enter the great mathematician, Leonard Oiler, who we, we have mentioned a few times"},{"startTime":1573.444,"endTime":1573.787,"body":"in in Gresham lectures, uh, because he was, uh, just an amazing genius and he"},{"startTime":1573.787,"endTime":1581.325,"body":"was involved in many, many developments in mathematics."},{"startTime":1582.145,"endTime":1585.565,"body":"If you came to my lecture on Sudoku puzzles, we talked about him briefly then."},{"startTime":1586.065,"endTime":1591.383,"body":"Uh, so he came along and he did what mathematics, um, does very well, which"},{"startTime":1591.383,"endTime":1591.737,"body":"is to get to actually the real structure hidden underneath this problem about, you know,"},{"startTime":1591.737,"endTime":1597.765,"body":"towns and rivers."},{"startTime":1598.105,"endTime":1602.509,"body":"He said, what's really going on here, and this is, uh, his diagram is you've"},{"startTime":1602.509,"endTime":1605.445,"body":"got four regions of land, A, B, C, and D."},{"startTime":1605.445,"endTime":1605.885,"body":"There they are."},{"startTime":1606.865,"endTime":1609.485,"body":"And connecting them are these bridges."},{"startTime":1609.905,"endTime":1614.245,"body":"So each bridge gives you a path between two regions."},{"startTime":1614.585,"endTime":1617.365,"body":"So we can draw those in, uh, that's where those bridges are."},{"startTime":1618.025,"endTime":1622.765,"body":"And now in fact, that's the information we need to deal, deal with this problem."},{"startTime":1622.945,"endTime":1625.925,"body":"That's all the information that is required to solve the problem."},{"startTime":1626.585,"endTime":1628.245,"body":"Um, so we can throw away everything else."},{"startTime":1628.745,"endTime":1630.845,"body":"We get that and we can make it look a bit pretty."},{"startTime":1631.465,"endTime":1635.901,"body":"Uh, so now it looks like this and that gives, that is the essence of"},{"startTime":1635.901,"endTime":1637.085,"body":"the problem right there."},{"startTime":1637.985,"endTime":1643.028,"body":"We are now asking, translate that, that the problem into a problem involving this, this"},{"startTime":1643.028,"endTime":1643.365,"body":"network."},{"startTime":1643.475,"endTime":1648.55,"body":"Here we are asking, is there a way you can walk around this, this network"},{"startTime":1648.55,"endTime":1648.889,"body":"or graph where you go along each edge exactly once those edges correspond to the"},{"startTime":1648.889,"endTime":1653.965,"body":"bridges."},{"startTime":1654.545,"endTime":1661.228,"body":"And once you've got it in this format, um, it's becomes easier to see how"},{"startTime":1661.228,"endTime":1662.565,"body":"we might proceed."},{"startTime":1662.865,"endTime":1666.615,"body":"And all it came up with this brilliant argument to prove that it is impossible"},{"startTime":1666.615,"endTime":1667.365,"body":"to do this."},{"startTime":1668.385,"endTime":1673.34,"body":"Uh, so what you have to do, if you're walking around this, this graph from"},{"startTime":1673.34,"endTime":1673.671,"body":"the nodes traveling along, along the edges to each node or vertex, every time you"},{"startTime":1673.671,"endTime":1674.001,"body":"come into a node and go out of it, you are using up two edges,"},{"startTime":1674.001,"endTime":1685.565,"body":"an in edge and an out edge."},{"startTime":1685.705,"endTime":1687.045,"body":"So you can sort of cross those off."},{"startTime":1688.265,"endTime":1692.36,"body":"And except for possibly the starting point where you're just leaving and the finishing point"},{"startTime":1692.36,"endTime":1692.633,"body":"where you're just arriving, that must mean that for all the other points, the edges"},{"startTime":1692.633,"endTime":1692.907,"body":"can be paired off with each other, an in edge, in an outage, an in"},{"startTime":1692.907,"endTime":1701.645,"body":"edge, in an outage."},{"startTime":1701.705,"endTime":1703.285,"body":"So you've taken 'em away in pairs."},{"startTime":1703.825,"endTime":1709.055,"body":"And that means that apart from possibly the start and the end points, all the"},{"startTime":1709.055,"endTime":1714.285,"body":"other vertices or nodes have to have an even number of edges join to them."},{"startTime":1714.965,"endTime":1717.325,"body":"'cause half of them are in edges and half of them are out edges."},{"startTime":1717.625,"endTime":1721.608,"body":"But in this picture, unfortunately, you can see that none of them have an even"},{"startTime":1721.608,"endTime":1722.405,"body":"number of edges."},{"startTime":1722.665,"endTime":1726.249,"body":"Uh, so most of 'em have three edges, and one of them, that one up"},{"startTime":1726.249,"endTime":1727.205,"body":"there has five edges."},{"startTime":1727.385,"endTime":1728.445,"body":"So you cannot do it."},{"startTime":1728.555,"endTime":1732.685,"body":"It's impossible to traverse all the bridges of kosberg exactly once."},{"startTime":1733.185,"endTime":1738.179,"body":"And converting the problem into this really simple graph actually was the way to see"},{"startTime":1738.179,"endTime":1738.845,"body":"the solution."},{"startTime":1739.825,"endTime":1746.285,"body":"But graphs like this are incredibly useful and important in many, many areas of life."},{"startTime":1746.705,"endTime":1750.995,"body":"The map of the London underground or, um, the, the connections between webpages online, you"},{"startTime":1750.995,"endTime":1751.281,"body":"know, Google's search engine relies on this, uh, kind of concept of a graph like"},{"startTime":1751.281,"endTime":1751.567,"body":"this where things are joined together, uh, pages are joined together, if there's a link"},{"startTime":1751.567,"endTime":1761.005,"body":"from one to the other."},{"startTime":1761.865,"endTime":1766.546,"body":"So this oiler oiler idea actually kickstarted a whole new area of mathematics called graph"},{"startTime":1766.546,"endTime":1766.858,"body":"theory, uh, which is incredibly important and useful today, but it came from trying to"},{"startTime":1766.858,"endTime":1772.165,"body":"solve a puzzle."},{"startTime":1773.145,"endTime":1776.845,"body":"Uh, I'll mention another little puzzle, which, because I like it."},{"startTime":1777.185,"endTime":1779.525,"body":"Uh, this is, uh, called the Puzzle of the Birds."},{"startTime":1780.145,"endTime":1786.042,"body":"And the example I'm giving is from a book by Fibonacci, but actually it dates"},{"startTime":1786.042,"endTime":1786.435,"body":"back probably a thousand years before, uh, this, um, puzzles of this form probably originated"},{"startTime":1786.435,"endTime":1797.445,"body":"in China, um, in maybe 500 ad this is about constraints in a problem."},{"startTime":1797.905,"endTime":1802.605,"body":"So you've got 30 birds, you buy 30 birds for 30 pennies in the market,"},{"startTime":1802.605,"endTime":1806.365,"body":"and there are three kinds of birds, and you're given the prices."},{"startTime":1806.665,"endTime":1808.965,"body":"And then the question is, how many of each bird did I buy?"},{"startTime":1809.665,"endTime":1815.808,"body":"So in order to solve this, you have to take that information and use the"},{"startTime":1815.808,"endTime":1819.085,"body":"constraints that you've got, 30 birds, 30 pennies."},{"startTime":1820.145,"endTime":1824.726,"body":"And to do that, what you end up doing is something that we now call"},{"startTime":1824.726,"endTime":1825.031,"body":"linear programming or linear optimization that again, has loads and loads of applications in terms"},{"startTime":1825.031,"endTime":1831.445,"body":"of, you know, manufacturing or transport links."},{"startTime":1831.745,"endTime":1836.311,"body":"Um, and it's, you know, a whole area of mathematics on its own, but you"},{"startTime":1836.311,"endTime":1839.965,"body":"can also solve, these are other charming, uh, bird problems with it."},{"startTime":1841.075,"endTime":1848.791,"body":"Okay, the final example of problems or puzzles that have very interesting mathematical ramifications, uh,"},{"startTime":1848.791,"endTime":1854.965,"body":"and kind of lead to new mathematics is our segue into paradoxes."},{"startTime":1855.665,"endTime":1857.725,"body":"So what's the most famous paradox ever?"},{"startTime":1858.425,"endTime":1862.405,"body":"Um, well, I don't know, but \u003claugh\u003e, but zeno's paradoxes are probably up there."},{"startTime":1862.545,"endTime":1862.965,"body":"The list."},{"startTime":1863.785,"endTime":1867.005,"body":"And the most famous of those is probably Achilles and the tortoise."},{"startTime":1867.605,"endTime":1868.925,"body":"Actually, I'm, I'm not sure this is Achilles."},{"startTime":1868.925,"endTime":1871.885,"body":"I think it might be Mercury this in this, but let's pretend it's Achilles."},{"startTime":1872.305,"endTime":1873.045,"body":"No one will tell me."},{"startTime":1873.505,"endTime":1878.165,"body":"Uh, so somebody who can move very fast is in a race against a tortoise."},{"startTime":1878.465,"endTime":1881.8,"body":"Of course we know they can beat the tortoise, okay, we'll give the tortoise a"},{"startTime":1881.8,"endTime":1882.245,"body":"head start."},{"startTime":1882.425,"endTime":1885.302,"body":"But even if we give the tortoise a head start, you know, Achilles is so"},{"startTime":1885.302,"endTime":1887.605,"body":"fast, he's gonna catch up to the tortoise only there's a problem."},{"startTime":1887.865,"endTime":1891.965,"body":"And the problem is that there's achilles, there's a tortoise."},{"startTime":1892.305,"endTime":1896.447,"body":"So he runs along very fast where the tortoise has been, but in that time,"},{"startTime":1896.447,"endTime":1899.485,"body":"the tortoise will have moved on a little bit, okay, fine."},{"startTime":1899.545,"endTime":1903.25,"body":"So he's just gonna now run to that point, but then in that little time,"},{"startTime":1903.25,"endTime":1903.497,"body":"the tortoise has moved on again and then so he catches up, but then the"},{"startTime":1903.497,"endTime":1903.744,"body":"tortoise still moved, however small the distance is that he has to cover to catch"},{"startTime":1903.744,"endTime":1903.991,"body":"the tortoise in the time it takes him to cover it, the tortoise has moved"},{"startTime":1903.991,"endTime":1904.238,"body":"on still a little tiny bit so he can never catch the tortoise, which is"},{"startTime":1904.238,"endTime":1921.285,"body":"sort of paradoxical because we all know he, he can catch the tortoise."},{"startTime":1921.285,"endTime":1922.245,"body":"So what is going on?"},{"startTime":1922.785,"endTime":1924.325,"body":"And a, a similar problem."},{"startTime":1924.465,"endTime":1929.325,"body":"So I think there were nine paradoxes of Xeno, um, similar flavor."},{"startTime":1929.945,"endTime":1932.165,"body":"So a similar one, uh, is about lanter."},{"startTime":1932.665,"endTime":1937.376,"body":"Um, so she, again, very, very fast runner, uh, but she can't ever get between"},{"startTime":1937.376,"endTime":1938.005,"body":"two points."},{"startTime":1939.105,"endTime":1939.325,"body":"Why?"},{"startTime":1939.325,"endTime":1942.801,"body":"Because to get from A to B first she has to get to the midpoint,"},{"startTime":1942.801,"endTime":1943.033,"body":"uh, and then she has to get to, to the midpoint of those two, and"},{"startTime":1943.033,"endTime":1948.365,"body":"then half again and half again and half again."},{"startTime":1948.985,"endTime":1952.913,"body":"Um, but each one of those, we know she can do it very quickly, but"},{"startTime":1952.913,"endTime":1954.485,"body":"it's still a thing to do."},{"startTime":1955.025,"endTime":1956.605,"body":"She still has to cover that distance."},{"startTime":1956.785,"endTime":1960.045,"body":"So there's an infinite number of small distances that she has to cover."},{"startTime":1960.545,"endTime":1965.53,"body":"So anything you do infinitely, an infinite number of however small time it takes you,"},{"startTime":1965.53,"endTime":1967.525,"body":"it's surely gonna take infinitely long."},{"startTime":1967.585,"endTime":1968.805,"body":"So she can't get anywhere."},{"startTime":1969.385,"endTime":1974.041,"body":"Um, both of these problems really boil down to the, the question of how do"},{"startTime":1974.041,"endTime":1976.525,"body":"we add an infinite number of things together?"},{"startTime":1977.385,"endTime":1980.685,"body":"How do we sum that infinite sequence?"},{"startTime":1981.665,"endTime":1986.607,"body":"Uh, now we can sort of think, well, a half plus a quarter plus an"},{"startTime":1986.607,"endTime":1987.925,"body":"eighth, plus 16th, we.dot."},{"startTime":1987.955,"endTime":1992.026,"body":"That will eventually the limiting case of that, if we did do it to infinity,"},{"startTime":1992.026,"endTime":1992.298,"body":"we kind of get one, um, and we can convince ourselves that might be true,"},{"startTime":1992.298,"endTime":1999.085,"body":"but it's still a bit, maybe feels a bit dodgy, right?"},{"startTime":1999.225,"endTime":2000.805,"body":"We really, how do we really, really know?"},{"startTime":2000.805,"endTime":2005.155,"body":"We're adding these infinitely many things and you know, how can we be sure that"},{"startTime":2005.155,"endTime":2007.765,"body":"it really is going to stop at some point?"},{"startTime":2008.305,"endTime":2011.055,"body":"For example, you could say, well, these numbers are getting smaller and smaller, so it's"},{"startTime":2011.055,"endTime":2011.605,"body":"gonna be fine."},{"startTime":2011.635,"endTime":2015.748,"body":"They'll stop at some finite place, but these numbers are also getting smaller and smaller,"},{"startTime":2015.748,"endTime":2018.765,"body":"a half a third, a quarter, a fifth, and so on."},{"startTime":2019.225,"endTime":2024.215,"body":"Uh, if you try to add all of those up, it can actually be shown"},{"startTime":2024.215,"endTime":2024.548,"body":"that however big of a number you can think of, eventually the sum of this"},{"startTime":2024.548,"endTime":2030.205,"body":"will overtake it."},{"startTime":2030.665,"endTime":2034.848,"body":"So this goes to infinity, this second one, and the top one doesn't, it only"},{"startTime":2034.848,"endTime":2035.685,"body":"stops at one."},{"startTime":2035.685,"endTime":2039.723,"body":"It's sort of limited above by one, but discerning which is which and how we"},{"startTime":2039.723,"endTime":2042.685,"body":"can really be sure that took a long, long, long time."},{"startTime":2042.685,"endTime":2046.925,"body":"It took until really the, uh, 18th, 19th century to, to get that sorted out."},{"startTime":2047.345,"endTime":2053.476,"body":"Um, well beyond Zeno's time, let's mention just one more paradox of Xeno, the arrow"},{"startTime":2053.476,"endTime":2053.885,"body":"paradox."},{"startTime":2054.385,"endTime":2059.138,"body":"So this is an arrow flying through the air very fast, but if you actually"},{"startTime":2059.138,"endTime":2059.455,"body":"try and look at its motion and how it can move, um, if we just"},{"startTime":2059.455,"endTime":2068.645,"body":"freeze frame at any point, it's still, it's motionless, just fix time at an instant."},{"startTime":2068.745,"endTime":2070.445,"body":"At that instant, the arrow isn't moving."},{"startTime":2071.025,"endTime":2073.885,"body":"So how do we then put all the, what is time?"},{"startTime":2073.955,"endTime":2077.804,"body":"It's an infinite number of instance, but at each instant there's no motion."},{"startTime":2078.225,"endTime":2079.804,"body":"So how on earth can there be movement?"},{"startTime":2080.425,"endTime":2083.085,"body":"And then, you know, your brain sort of melts and dribbles out of your ears."},{"startTime":2083.725,"endTime":2084.764,"body":"\u003claugh\u003e, I think, how do we solve this?"},{"startTime":2085.264,"endTime":2089.214,"body":"Uh, if we think, how would we really think work out how far say the"},{"startTime":2089.214,"endTime":2090.005,"body":"arrow had gone?"},{"startTime":2090.505,"endTime":2093.575,"body":"Um, if it's going at a constant speed, well, you could draw, you could draw"},{"startTime":2093.575,"endTime":2094.804,"body":"a graph of speed against time."},{"startTime":2095.385,"endTime":2096.605,"body":"Um, constant speed there."},{"startTime":2096.605,"endTime":2098.925,"body":"I've just randomly said it's going at six meters per second."},{"startTime":2099.385,"endTime":2101.245,"body":"How far has it traveled in 10 seconds?"},{"startTime":2101.635,"endTime":2103.205,"body":"Well, speed is distance over time."},{"startTime":2103.545,"endTime":2105.085,"body":"So distance is speed, time is time."},{"startTime":2105.545,"endTime":2109.803,"body":"So you take that really what you want is the area under the graph then"},{"startTime":2109.803,"endTime":2110.087,"body":"in that, in that, uh, ten second time interval and the distance travel will be"},{"startTime":2110.087,"endTime":2115.765,"body":"six times 10, 60 meters fine."},{"startTime":2116.425,"endTime":2118.405,"body":"But what do we do if the speed is not constant?"},{"startTime":2118.595,"endTime":2119.805,"body":"What if the speed is like this?"},{"startTime":2120.795,"endTime":2122.845,"body":"Then how are we gonna find the area under that graph?"},{"startTime":2123.315,"endTime":2127.144,"body":"Well, you can kind of, you can make approximations, you can say I'll divide it"},{"startTime":2127.144,"endTime":2127.399,"body":"up into, into rectangles and that, you know, we'll say, we'll sort of pretend the"},{"startTime":2127.399,"endTime":2127.654,"body":"speed is constant over each of these little rectangles and then add up all of"},{"startTime":2127.654,"endTime":2136.845,"body":"those areas over the width times the height."},{"startTime":2137.235,"endTime":2138.485,"body":"Fine, that's not very accurate."},{"startTime":2138.585,"endTime":2140.365,"body":"So let's make the rectangles a bit thinner."},{"startTime":2141.025,"endTime":2142.125,"body":"Um, and we can do that."},{"startTime":2142.125,"endTime":2143.285,"body":"We'll get a better estimate."},{"startTime":2143.705,"endTime":2147.565,"body":"And as we make the rectangles thinner and thinner, the estimate gets better and better."},{"startTime":2148.225,"endTime":2151.325,"body":"But in the limit, the width of those rectangles is zero."},{"startTime":2152.265,"endTime":2154.805,"body":"And, and so the area of each rectangle will be zero."},{"startTime":2155.225,"endTime":2159.541,"body":"So then they're somehow supposed to add up to make something that isn't zero, that's"},{"startTime":2159.541,"endTime":2160.405,"body":"a bit confusing."},{"startTime":2161.025,"endTime":2164.205,"body":"And dealing with these infinite decimals, right?"},{"startTime":2164.215,"endTime":2168.913,"body":"These, these tiny quantities that close, closer and closer to zero is exactly what, uh,"},{"startTime":2168.913,"endTime":2172.045,"body":"Newton and liveness still at the calculus to cope with."},{"startTime":2172.865,"endTime":2175.845,"body":"But that again, took many, many centuries after Z know."},{"startTime":2176.105,"endTime":2179.576,"body":"Uh, and really it wasn't until then that we could say these paradoxes, we sort"},{"startTime":2179.576,"endTime":2180.965,"body":"of understood what was going on."},{"startTime":2181.195,"endTime":2185.374,"body":"It's the difference between the discreet and the continuous and the mathematics of the continuous"},{"startTime":2185.374,"endTime":2187.325,"body":"is what calculus is so good at."},{"startTime":2188.355,"endTime":2188.645,"body":"Okay?"},{"startTime":2188.665,"endTime":2193.279,"body":"So for the rest of the time today, we are gonna think about some more"},{"startTime":2193.279,"endTime":2195.125,"body":"paradoxes, but they are logic paradoxes."},{"startTime":2196.425,"endTime":2200.266,"body":"Now, this is in honor of all those times I sat an exam when I"},{"startTime":2200.266,"endTime":2200.522,"body":"was at school and I got very cross because there would be a page and"},{"startTime":2200.522,"endTime":2206.925,"body":"it would say on it, this page is left intentionally blank."},{"startTime":2207.925,"endTime":2208.125,"body":"Remember that?"},{"startTime":2208.725,"endTime":2212.98,"body":"I used to think I should write a very sniffy letter to the examiner saying"},{"startTime":2212.98,"endTime":2213.264,"body":"that due to extreme emotional distress caused by malfunctioning logic in your exam, I was"},{"startTime":2213.264,"endTime":2220.925,"body":"unable to complete it and you should give me an a star immediately."},{"startTime":2221.145,"endTime":2225.36,"body":"Um, you know, this kind of sentence, it, it, it, it has an antecedent that"},{"startTime":2225.36,"endTime":2226.765,"body":"go back a long way."},{"startTime":2227.505,"endTime":2232.605,"body":"Um, it refers to itself and that's, that's really the essence of what's going on."},{"startTime":2233.345,"endTime":2240.645,"body":"Um, the classical version of, of something like this is the, so-called liar paradox."},{"startTime":2241.145,"endTime":2245.525,"body":"So epi amenities says all credences are liars."},{"startTime":2246.585,"endTime":2252.678,"body":"Um, okay, well it's not very nice of him, but you know, no paradox seemingly"},{"startTime":2252.678,"endTime":2258.365,"body":"until you know that epi amenities came from knossos and knossos is on Crete."},{"startTime":2258.945,"endTime":2262.565,"body":"So epi amenities, the Creon is saying that all Creon are liars."},{"startTime":2263.465,"endTime":2268.296,"body":"So if he's telling the truth, so then it would be right that true that"},{"startTime":2268.296,"endTime":2272.805,"body":"all cretins are liars, but then he himself as a creon would be lying."},{"startTime":2273.025,"endTime":2275.445,"body":"So if this statement is true, then it's also false."},{"startTime":2275.985,"endTime":2277.405,"body":"So that's a bit confusing."},{"startTime":2277.475,"endTime":2281.565,"body":"However, this isn't really a paradox because the statement could just be false."},{"startTime":2282.545,"endTime":2286.125,"body":"He could lie being a cretin."},{"startTime":2286.125,"endTime":2288.365,"body":"He could lie and say that all cretins are liars."},{"startTime":2288.795,"endTime":2290.125,"body":"That makes him a liar."},{"startTime":2290.545,"endTime":2292.405,"body":"But it doesn't mean that all cretins are liars."},{"startTime":2292.745,"endTime":2296.12,"body":"Now, the, the opposite of all cretins are liars is, you know, at least one"},{"startTime":2296.12,"endTime":2299.045,"body":"cretin tells the truth at least some of the time, something like this."},{"startTime":2299.465,"endTime":2303.11,"body":"So it's not absolutely a paradox, but we can say this, we are pretty sure"},{"startTime":2303.11,"endTime":2304.325,"body":"this is a false statement."},{"startTime":2305.165,"endTime":2307.765,"body":"'cause if it is true, that would imply its own falsehood."},{"startTime":2308.585,"endTime":2310.085,"body":"But try this one."},{"startTime":2311.985,"endTime":2313.565,"body":"Is this sentence true or false?"},{"startTime":2314.385,"endTime":2317.005,"body":"It asserts that it is false, right?"},{"startTime":2317.345,"endTime":2318.725,"body":"Is that a true statement?"},{"startTime":2319.235,"endTime":2320.325,"body":"Suppose it's true."},{"startTime":2321.265,"endTime":2324.325,"body":"Um, then it would be true that it's false, so it'd be false."},{"startTime":2325.235,"endTime":2325.525,"body":"Okay?"},{"startTime":2325.665,"endTime":2326.885,"body":"On the other hand, suppose it's false."},{"startTime":2327.425,"endTime":2329.405,"body":"Um, so it's false to say this sentence is false."},{"startTime":2329.785,"endTime":2333.022,"body":"Um, but then the sentence is actually true because it's saying that it's false, which"},{"startTime":2333.022,"endTime":2334.965,"body":"it, it's, so, if it's true, then it's false."},{"startTime":2334.965,"endTime":2336.045,"body":"And if it's false, then it's true."},{"startTime":2336.425,"endTime":2339.156,"body":"Um, the third way out of this is to notice, I forgot to put full"},{"startTime":2339.156,"endTime":2339.885,"body":"stop at the end."},{"startTime":2339.885,"endTime":2343.565,"body":"So it isn't even a sentence \u003claugh\u003e, but you know, there, that's, that's cheating."},{"startTime":2344.065,"endTime":2348.355,"body":"Um, there are lots of ways to think about this and people have had various"},{"startTime":2348.355,"endTime":2352.645,"body":"theories or, or ideas almost philosophically about how to deal with a statement like this."},{"startTime":2353.025,"endTime":2358.941,"body":"Um, I quite, so you could say it's both, it's true and it's false, maybe"},{"startTime":2358.941,"endTime":2360.125,"body":"I'm not sure."},{"startTime":2360.165,"endTime":2363.573,"body":"I like that because I think especially if we want to do mathematics, we don't"},{"startTime":2363.573,"endTime":2365.845,"body":"want to have statements that are ambiguous in that way."},{"startTime":2366.125,"endTime":2369.901,"body":"I mean, it's one thing not to be able to prove something, but if you"},{"startTime":2369.901,"endTime":2370.153,"body":"can prove it's true and then the next day your friend comes along with their"},{"startTime":2370.153,"endTime":2370.405,"body":"equally valid proof that it's false, then you know, it's very difficult to imagine mathematics"},{"startTime":2370.405,"endTime":2378.965,"body":"being put at a firm foundation."},{"startTime":2379.075,"endTime":2382.485,"body":"We'd like, we'd like phagocyte either to be true or to be false."},{"startTime":2382.905,"endTime":2384.765,"body":"Um, not both, and not neither."},{"startTime":2385.425,"endTime":2390.855,"body":"So things being both true and false at the same time, we perhaps want to"},{"startTime":2390.855,"endTime":2396.285,"body":"avoid, uh, there was, uh, one idea put forward by the logician, uh, Arthur Pryor."},{"startTime":2396.825,"endTime":2402.14,"body":"And he thinks you can get round it by saying or noticing that every sentence,"},{"startTime":2402.14,"endTime":2402.495,"body":"you know, if we're just talking normally, every sentence that we say implicitly asserts its"},{"startTime":2402.495,"endTime":2408.165,"body":"own truth."},{"startTime":2408.865,"endTime":2412.933,"body":"So we are saying something and that kind of implies that we're saying, and it's"},{"startTime":2412.933,"endTime":2413.205,"body":"true."},{"startTime":2413.825,"endTime":2418.336,"body":"So if you say the sky is blue, that is logically equivalent to says Arthur"},{"startTime":2418.336,"endTime":2421.645,"body":"Pryor saying this sentence is true and the sky is blue."},{"startTime":2422.665,"endTime":2428.196,"body":"Uh, if we agree with that, then the sentence that we have here, this sentence"},{"startTime":2428.196,"endTime":2428.565,"body":"is false, is logically equivalent to this sentence is true and it is false, and"},{"startTime":2428.565,"endTime":2428.933,"body":"that is isn't possible because we believe things can't be both true and false at"},{"startTime":2428.933,"endTime":2440.365,"body":"the same time."},{"startTime":2440.625,"endTime":2444.278,"body":"So if you look at it that way, then you can say what it, it's"},{"startTime":2444.278,"endTime":2447.445,"body":"false because something can't be both true and false at the same time."},{"startTime":2447.945,"endTime":2450.725,"body":"So I think this is one we don't have to absolutely decide on."},{"startTime":2450.725,"endTime":2452.365,"body":"We can argue about it and discuss it."},{"startTime":2452.785,"endTime":2458.349,"body":"Um, what we think about this sentence, as long as we don't try and have"},{"startTime":2458.349,"endTime":2458.72,"body":"things like this at the heart of, you know, at the mathematics probably to be"},{"startTime":2458.72,"endTime":2459.091,"body":"avoided, um, since I enjoy finding mathematics everywhere, and especially I enjoy finding it in,"},{"startTime":2459.091,"endTime":2459.462,"body":"uh, literature, I wanted to just mention a paradox a bit like this called of"},{"startTime":2459.462,"endTime":2479.125,"body":"liar paradox or self-referential paradox that crops up in, uh, donte."},{"startTime":2479.985,"endTime":2486.214,"body":"So, uh, here they are, there's donte and there's his loyal faithful Sancho Panza, uh,"},{"startTime":2486.214,"endTime":2487.045,"body":"beside him."},{"startTime":2487.465,"endTime":2491.112,"body":"And at one point in the, in the story, uh, some, some not very nice"},{"startTime":2491.112,"endTime":2491.355,"body":"people are kind of teasing Sancho panza and they're saying, we're gonna make you the"},{"startTime":2491.355,"endTime":2498.165,"body":"governor of this province and you are gonna have to make all these decisions."},{"startTime":2498.165,"endTime":2502.398,"body":"So just to make sure you can cope with the responsibility, what should we do"},{"startTime":2502.398,"endTime":2503.245,"body":"in this situation?"},{"startTime":2503.465,"endTime":2506.974,"body":"So there's a town where, you know, there's a bridge and you have to, to"},{"startTime":2506.974,"endTime":2507.208,"body":"get into the town, uh, you are questioned by the, by the guards, and you"},{"startTime":2507.208,"endTime":2513.525,"body":"have to say what you're going to do in the town that day."},{"startTime":2514.545,"endTime":2519.288,"body":"So if you speak the truth, fine, off you go, it's good, you can get"},{"startTime":2519.288,"endTime":2519.605,"body":"enter."},{"startTime":2519.825,"endTime":2523.645,"body":"But if you lie, then you'll be hanged, you know, in the town square."},{"startTime":2524.025,"endTime":2526.503,"body":"So one day a traveler arrives and says, and they say, what are you gonna"},{"startTime":2526.503,"endTime":2527.165,"body":"do in the town?"},{"startTime":2527.425,"endTime":2528.965,"body":"And he says, I will be hanged."},{"startTime":2530.825,"endTime":2532.165,"body":"So should you hang him?"},{"startTime":2533.825,"endTime":2538.28,"body":"Um, if you do hang him, then actually he was telling the truth, so you"},{"startTime":2538.28,"endTime":2539.765,"body":"should have let him go."},{"startTime":2540.585,"endTime":2544.991,"body":"But if you don't hang him, then his assertion that he would be hanged well's"},{"startTime":2544.991,"endTime":2545.285,"body":"false."},{"startTime":2545.665,"endTime":2547.485,"body":"So what he actually, he should be hanged."},{"startTime":2547.865,"endTime":2549.405,"body":"So what are you gonna do?"},{"startTime":2549.705,"endTime":2553.385,"body":"Um, well, if you want to know what Sancho Panza decided, you can either read"},{"startTime":2553.385,"endTime":2556.085,"body":"the book or you can ask me at the end, \u003claugh\u003e."},{"startTime":2556.755,"endTime":2558.965,"body":"Alright, onto another, another sentence."},{"startTime":2559.465,"endTime":2562.365,"body":"So we've dealt with this sentence is false, sort of what about this?"},{"startTime":2562.365,"endTime":2564.005,"body":"This is not, this is at the moment, this is fine, right?"},{"startTime":2564.005,"endTime":2565.165,"body":"This sentence has seven words."},{"startTime":2565.165,"endTime":2566.445,"body":"Well, it doesn't, okay, fine."},{"startTime":2567.065,"endTime":2569.605,"body":"So it's a false statement, nothing to see here."},{"startTime":2570.145,"endTime":2573.664,"body":"Um, well, but we know if something is false or we believe if, if, if"},{"startTime":2573.664,"endTime":2576.245,"body":"a statement is false, then it kind of, it's opposite statement."},{"startTime":2576.245,"endTime":2578.405,"body":"The negation ought to be true, right?"},{"startTime":2578.985,"endTime":2580.565,"body":"If something's false, the opposite is true."},{"startTime":2580.565,"endTime":2581.725,"body":"If something's true, the opposite is false."},{"startTime":2582.275,"endTime":2582.565,"body":"Okay?"},{"startTime":2583.385,"endTime":2587.285,"body":"So, um, this sentence does not have seven words, right?"},{"startTime":2587.915,"endTime":2590.165,"body":"Only unfortunately, uh, it does."},{"startTime":2590.545,"endTime":2593.405,"body":"So that is also a false statement, \u003claugh\u003e, right?"},{"startTime":2593.705,"endTime":2598.13,"body":"So how, what we've got a a statement and its negation both being false, this"},{"startTime":2598.13,"endTime":2599.605,"body":"ought to not be possible."},{"startTime":2600.065,"endTime":2605.92,"body":"And so this kind of problem is really good for helping our brains think very"},{"startTime":2605.92,"endTime":2610.605,"body":"clearly about what do we mean by negation and true and false."},{"startTime":2612.185,"endTime":2616.565,"body":"What's really going on, I think here is that we haven't got the correct negation."},{"startTime":2617.835,"endTime":2620.405,"body":"This sentence at the top, this sentence has seven words."},{"startTime":2620.715,"endTime":2622.085,"body":"It's talking about its self."},{"startTime":2622.745,"endTime":2628.157,"body":"If we say that that's sentence A for example, uh, let's call that sentence a"},{"startTime":2628.157,"endTime":2631.405,"body":"then the top sentence A is asserting that itself."},{"startTime":2631.565,"endTime":2633.365,"body":"IE sentence A has seven words."},{"startTime":2634.265,"endTime":2638.348,"body":"So the opposite of that, the negation of that is not, this sentence does not"},{"startTime":2638.348,"endTime":2639.165,"body":"have seven words."},{"startTime":2639.345,"endTime":2640.445,"body":"That's a different sentence."},{"startTime":2641.105,"endTime":2644.165,"body":"The negation is that sentence."},{"startTime":2644.445,"endTime":2646.805,"body":"Sentence A has seven words, right?"},{"startTime":2648.395,"endTime":2652.965,"body":"That deals with it because it's true that sentence A does not have seven words."},{"startTime":2653.705,"endTime":2656.325,"body":"So we have to be really careful with self-referential statements."},{"startTime":2656.325,"endTime":2657.365,"body":"They're talking about themselves."},{"startTime":2657.825,"endTime":2661.654,"body":"But then the negation is a different statement that it should be talking about that"},{"startTime":2661.654,"endTime":2662.165,"body":"over there."},{"startTime":2663.145,"endTime":2664.405,"body":"So it's quite a fun little thing."},{"startTime":2664.535,"endTime":2667.525,"body":"Takes a while to reason out what's gone wrong in that situation."},{"startTime":2668.145,"endTime":2672.586,"body":"Um, so this can be resolved by being really careful about the meanings of words,"},{"startTime":2672.586,"endTime":2672.882,"body":"kind of semantics, um, thinking not just about the statements, but what the statements are"},{"startTime":2672.882,"endTime":2680.285,"body":"saying and being sure we can distinguish between those two things."},{"startTime":2680.785,"endTime":2687.085,"body":"But sometimes, uh, these kind of paradoxical self-referential statements can really have a, a genuine"},{"startTime":2687.085,"endTime":2689.605,"body":"deep effect on, on mathematical thought."},{"startTime":2690.945,"endTime":2695.461,"body":"And one instance of this, it's not quite this, but it's a, a statement very"},{"startTime":2695.461,"endTime":2696.365,"body":"like this one."},{"startTime":2696.625,"endTime":2702.528,"body":"And actually, um, mathematician Bertrand Russell and philosopher, um, used this example to illustrate a"},{"startTime":2702.528,"endTime":2707.645,"body":"mathematical argument that he, he had made, which we'll mention in a moment."},{"startTime":2708.185,"endTime":2713.965,"body":"So it's the barber paradox and it says in a certain town there's just one"},{"startTime":2713.965,"endTime":2718.205,"body":"barber and he shaves only those men who don't shave themselves."},{"startTime":2718.785,"endTime":2720.965,"body":"So, um, you shave yourself, okay?"},{"startTime":2720.965,"endTime":2725.285,"body":"If you don't, then the barber will shave you question, who shaves the barber?"},{"startTime":2726.875,"endTime":2727.165,"body":"Okay?"},{"startTime":2727.305,"endTime":2734.063,"body":"So if the barber shaves himself, um, then he's one of the men who do"},{"startTime":2734.063,"endTime":2734.965,"body":"shave themselves."},{"startTime":2735.665,"endTime":2738.565,"body":"But the barber only shaves the men who don't shave themselves."},{"startTime":2738.905,"endTime":2741.325,"body":"So the barber definitely does not shave the barber."},{"startTime":2742.475,"endTime":2742.765,"body":"Okay?"},{"startTime":2744.505,"endTime":2747.845,"body":"Oh dear, all, let's suppose the barber doesn't shave himself."},{"startTime":2748.395,"endTime":2752.485,"body":"Well then we know that the barber precisely shaves the men who don't shave themselves."},{"startTime":2752.705,"endTime":2754.685,"body":"So then the barber does shave the barber."},{"startTime":2755.225,"endTime":2758.651,"body":"So do you have a way you start from, you get into this kind of,"},{"startTime":2758.651,"endTime":2761.165,"body":"um, doom spiral of logic from which there is no escape."},{"startTime":2761.465,"endTime":2764.485,"body":"If he does shave himself, then he doesn't, and he doesn't, then he does."},{"startTime":2765.145,"endTime":2768.597,"body":"So we dunno what to do with this quandary and actually what you do, I"},{"startTime":2768.597,"endTime":2768.828,"body":"mean, there's, so there's ways around it to like try and sort of wriggle out"},{"startTime":2768.828,"endTime":2769.058,"body":"of it semantically and say, well, perhaps the barber, you know, is bald or something,"},{"startTime":2769.058,"endTime":2776.885,"body":"but \u003claugh\u003e, let's not do that."},{"startTime":2776.885,"endTime":2778.445,"body":"Let's take it in spirit in which it's intended."},{"startTime":2778.745,"endTime":2782.365,"body":"The only conclusion really is to say there can be no such talent."},{"startTime":2782.955,"endTime":2783.765,"body":"This cannot happen."},{"startTime":2784.045,"endTime":2787.645,"body":"'cause if it did, we'd get into this logical, um, vicious cycle."},{"startTime":2789.075,"endTime":2796.337,"body":"However, unfortunately, well, or interestingly, there was a serious bit of mathematical research, um, that"},{"startTime":2796.337,"endTime":2796.822,"body":"came up against something very like the Bible paradox in the late 19th, early 20th"},{"startTime":2796.822,"endTime":2804.085,"body":"century."},{"startTime":2804.585,"endTime":2810.362,"body":"Mathematicians were really trying very hard to put mathematics on a very, very firm foundation,"},{"startTime":2810.362,"endTime":2810.747,"body":"a rigorous logical footing, and do a bit like what Euclid had had done with"},{"startTime":2810.747,"endTime":2816.525,"body":"geometry."},{"startTime":2816.525,"endTime":2818.165,"body":"You know, you start off with ground rules."},{"startTime":2818.165,"endTime":2822.995,"body":"You say what points and lines are you say, um, some, some basic facts about"},{"startTime":2822.995,"endTime":2823.317,"body":"them, the axioms that, uh, you know, I dunno, you can, you can draw a"},{"startTime":2823.317,"endTime":2830.725,"body":"circle from any point of any radius, for instance."},{"startTime":2831.025,"endTime":2835.565,"body":"Um, these basic rules can then be combined and you can make deductions from them."},{"startTime":2835.565,"endTime":2838.445,"body":"And then you create the whole beautiful edifice of geometry."},{"startTime":2838.915,"endTime":2841.645,"body":"Well, it, wouldn't it be nice if we could do that for all of mathematics?"},{"startTime":2842.505,"endTime":2846.387,"body":"And in particular, um, there was a guy called Got Love frager who tried to"},{"startTime":2846.387,"endTime":2846.646,"body":"do this and come up with a theory that would cover all of, um, everything"},{"startTime":2846.646,"endTime":2851.565,"body":"you could do with sets."},{"startTime":2851.785,"endTime":2855.495,"body":"So if you take, you know, sets of numbers or sets of anything you like,"},{"startTime":2855.495,"endTime":2856.485,"body":"you could combine them."},{"startTime":2856.485,"endTime":2857.965,"body":"You can make the union of two sets."},{"startTime":2857.985,"endTime":2859.045,"body":"You can make the intersection."},{"startTime":2859.385,"endTime":2863.565,"body":"Uh, you can look at subsets, you can define subsets according to various rules."},{"startTime":2863.945,"endTime":2866.765,"body":"So you might think, you know, the prime numbers are a subset of all numbers."},{"startTime":2867.785,"endTime":2869.325,"body":"You can have sets, offsets."},{"startTime":2869.705,"endTime":2873.88,"body":"Uh, you might look at all the possible subsets of a particular set, all the"},{"startTime":2873.88,"endTime":2874.159,"body":"possible subsets of numbers or, or all the possible potential friend groups in a \u003claugh\u003e"},{"startTime":2874.159,"endTime":2880.005,"body":"in a class or something like this."},{"startTime":2880.385,"endTime":2888.311,"body":"So he frager, um, wrote a huge book actually of two volumes about his idea"},{"startTime":2888.311,"endTime":2894.125,"body":"of how to do this really rigorous, uh, theory of sets."},{"startTime":2894.985,"endTime":2902.399,"body":"Um, but unfortunately the kind of sets he allowed, made way for a problem, and"},{"startTime":2902.399,"endTime":2905.365,"body":"this is called now Russell's Paradox."},{"startTime":2906.275,"endTime":2907.965,"body":"It's basically the barber paradox."},{"startTime":2908.425,"endTime":2909.685,"body":"Um, but in the form of sets."},{"startTime":2910.345,"endTime":2914.51,"body":"So if you define this set, but a strange set, let's call it s it's"},{"startTime":2914.51,"endTime":2917.565,"body":"the set of all sets that are not elements of themselves."},{"startTime":2917.825,"endTime":2918.925,"body":"So you can have a set of sets."},{"startTime":2919.425,"endTime":2922.901,"body":"So then you can think of the concept of a set that's actually a member"},{"startTime":2922.901,"endTime":2923.365,"body":"of itself."},{"startTime":2924.675,"endTime":2927.965,"body":"This set S is all of those sets that are not members of themselves."},{"startTime":2929.865,"endTime":2935.965,"body":"Now is s an element of s and this is just like the Barbara paradox."},{"startTime":2936.745,"endTime":2941.204,"body":"If S is inside s then the entry criterion for being in s is that"},{"startTime":2941.204,"endTime":2943.285,"body":"you are not a member of yourself."},{"startTime":2943.705,"endTime":2947.578,"body":"So that doesn't work because if it's in, if it's a member of itself, that"},{"startTime":2947.578,"endTime":2949.645,"body":"would imply it's not a member of itself."},{"startTime":2950.105,"endTime":2950.325,"body":"Bad."},{"startTime":2950.875,"endTime":2951.165,"body":"Okay?"},{"startTime":2951.235,"endTime":2954.723,"body":"What about if s is not inside s Well then it fits the bill to"},{"startTime":2954.723,"endTime":2954.956,"body":"be a member of S because s is all the things that are not inside"},{"startTime":2954.956,"endTime":2958.445,"body":"themselves."},{"startTime":2959.465,"endTime":2961.165,"body":"So either way we get this."},{"startTime":2961.265,"endTime":2963.045,"body":"If it's in itself, then it isn't."},{"startTime":2963.045,"endTime":2965.908,"body":"And if it isn't in itself, then it is round and round and round and"},{"startTime":2965.908,"endTime":2967.245,"body":"round forever till we fall over dizzy."},{"startTime":2968.065,"endTime":2973.645,"body":"But unfortunately, uh, Rieger's lovely setup allowed this for this set."},{"startTime":2973.865,"endTime":2977.707,"body":"It, it said this set exists, not in so many words, but it sort of"},{"startTime":2977.707,"endTime":2979.245,"body":"implied by the work he'd done."},{"startTime":2979.945,"endTime":2984.507,"body":"And so Ho and Russell wrote to Frago about this and the poor guy, he"},{"startTime":2984.507,"endTime":2987.245,"body":"had to horribly put an appendix in his book."},{"startTime":2988.065,"endTime":2991.645,"body":"Um, it was just kinda at the printers and he wrote hardly anything."},{"startTime":2992.165,"endTime":2992.965,"body":"I feel so bad for him."},{"startTime":2993.075,"endTime":2997.255,"body":"More unfortunate can fall a scientific writer than to have one of the foundations of"},{"startTime":2997.255,"endTime":2999.485,"body":"his edifice shaken after the work is finished."},{"startTime":2999.995,"endTime":3002.085,"body":"This was the position I was placed in by a letter of Mr."},{"startTime":3002.365,"endTime":3005.685,"body":"Bert and Russell, just when the printing of this volume was nearing its completion."},{"startTime":3006.105,"endTime":3011.205,"body":"So yeah, it like it all just fell apart because there was a paradox."},{"startTime":3011.205,"endTime":3016.216,"body":"Something that couldn't work was at the heart of, you know, it it implied by"},{"startTime":3016.216,"endTime":3016.885,"body":"his setup."},{"startTime":3017.345,"endTime":3019.645,"body":"So that was obviously he was devastated, of course."},{"startTime":3020.225,"endTime":3024.893,"body":"Um, but the problems for this kind of great idea of putting mathematics on a,"},{"startTime":3024.893,"endTime":3025.204,"body":"on a, on a ideal footing, where there'd be one wonderful system that within which"},{"startTime":3025.204,"endTime":3025.515,"body":"you could express every mathematical idea, and that if you just worked long enough, you"},{"startTime":3025.515,"endTime":3037.965,"body":"could prove anything that was true and there wouldn't be any contradictions."},{"startTime":3038.195,"endTime":3042.203,"body":"That itself, and that was something that Bert and Russell was involved in as well,"},{"startTime":3042.203,"endTime":3043.005,"body":"was ultimately doomed."},{"startTime":3043.945,"endTime":3047.845,"body":"Um, that worry, there is good news at the end, \u003claugh\u003e doom is, is featured"},{"startTime":3047.845,"endTime":3048.885,"body":"a bit too much."},{"startTime":3049.345,"endTime":3054.368,"body":"So the question was, can we come up with some really kind of great system"},{"startTime":3054.368,"endTime":3054.703,"body":"of, of mathematical logic within which in theory, I mean they didn't have computers at"},{"startTime":3054.703,"endTime":3055.038,"body":"this time, but in theory you could start off with the basic rules, the axioms,"},{"startTime":3055.038,"endTime":3067.765,"body":"and you could say, what are the rules of inference?"},{"startTime":3067.835,"endTime":3070.165,"body":"What kind of rules of logic are you allowed to use?"},{"startTime":3070.825,"endTime":3073.365,"body":"And you plug it all in and you kind of crank the handle."},{"startTime":3073.625,"endTime":3078.2,"body":"And eventually, um, every kind of statement that you could make about numbers, you get"},{"startTime":3078.2,"endTime":3082.165,"body":"either a proof that it's true or proof that the opposite is true."},{"startTime":3082.185,"endTime":3083.605,"body":"So you know that it was false."},{"startTime":3084.105,"endTime":3085.645,"body":"So that's what would be great to do."},{"startTime":3086.465,"endTime":3091.424,"body":"Um, so what we want from this, the three Cs, what we want from this,"},{"startTime":3091.424,"endTime":3091.755,"body":"this putative wonderful system that's gonna solve all our problems and we want it to"},{"startTime":3091.755,"endTime":3097.045,"body":"be consistent."},{"startTime":3097.465,"endTime":3100.165,"body":"So we don't want anything to be both true and false."},{"startTime":3100.225,"endTime":3101.645,"body":"We don't want any contradictions."},{"startTime":3102.065,"endTime":3105.205,"body":"Um, the system should not be able to prove a false statement."},{"startTime":3105.205,"endTime":3108.877,"body":"In other words, you shouldn't be able to start from the axioms and use the,"},{"startTime":3108.877,"endTime":3111.325,"body":"the allowed rules and end up with something that's false."},{"startTime":3111.985,"endTime":3114.445,"body":"So only true things ought to be provable."},{"startTime":3114.485,"endTime":3115.765,"body":"I think we could all agree that's a good thing."},{"startTime":3116.385,"endTime":3119.085,"body":"Um, it also should be complete."},{"startTime":3119.705,"endTime":3123.769,"body":"So not only, uh, are the only things that you can prove true things, but"},{"startTime":3123.769,"endTime":3124.04,"body":"actually if a thing is true, if a statement that you can make in that"},{"startTime":3124.04,"endTime":3131.085,"body":"system is true, then it's got to be provable within that system."},{"startTime":3132.585,"endTime":3136.994,"body":"So in principle, every kind of statement you could make about numbers, uh, whether that's,"},{"startTime":3136.994,"endTime":3137.288,"body":"I don't know, the gold back conjecture, the Raymond hypothesis or any one of these"},{"startTime":3137.288,"endTime":3137.582,"body":"things, um, if it's true, there should be, there will be a proof of it"},{"startTime":3137.582,"endTime":3148.165,"body":"within this, within this system you're setting up."},{"startTime":3148.705,"endTime":3153.553,"body":"And so this is the final C that it, this big edifice we're making has"},{"startTime":3153.553,"endTime":3153.876,"body":"to be complex and sophisticated enough to be able to make any mathematical statement about"},{"startTime":3153.876,"endTime":3158.725,"body":"numbers."},{"startTime":3159.625,"endTime":3164.27,"body":"And then with those three things, it really would be, you could kind of start"},{"startTime":3164.27,"endTime":3164.58,"body":"it going and, and eventually outward come a proof of everything that was true and"},{"startTime":3164.58,"endTime":3171.085,"body":"nothing that was false could be proved."},{"startTime":3171.905,"endTime":3172.885,"body":"So wouldn't that be lovely?"},{"startTime":3173.875,"endTime":3174.645,"body":"Only there's a problem."},{"startTime":3175.425,"endTime":3179.782,"body":"And the problem is in the form of a, uh, a young man called Kurt"},{"startTime":3179.782,"endTime":3181.525,"body":"Girdle who did this amazing thing."},{"startTime":3181.945,"endTime":3187.925,"body":"He managed to come up with a proof that said, actually this is impossible."},{"startTime":3188.275,"endTime":3194.13,"body":"Because if you've got a system like this that is sufficiently complex enough to be"},{"startTime":3194.13,"endTime":3194.52,"body":"able to make any statement that you might want about numbers and it's consistent and"},{"startTime":3194.52,"endTime":3194.91,"body":"complete, as I've said, then it would be possible to make a statement within this"},{"startTime":3194.91,"endTime":3210.525,"body":"system that says, essentially this statement cannot be proved within this system."},{"startTime":3211.905,"endTime":3214.845,"body":"Now the the hard, this is really hard thing to do."},{"startTime":3215.045,"endTime":3217.645,"body":"'cause you have to prove that such a thing is possible, always gonna be possible."},{"startTime":3218.225,"endTime":3223.042,"body":"Um, but if you have got a statement like this that asserts that, it cannot"},{"startTime":3223.042,"endTime":3223.685,"body":"be proved."},{"startTime":3223.745,"endTime":3225.845,"body":"So this statement cannot be proved within the system."},{"startTime":3226.425,"endTime":3227.485,"body":"Now let's think about this."},{"startTime":3228.225,"endTime":3233.501,"body":"Uh, if, if this statement can be proved right, either can be or it can't"},{"startTime":3233.501,"endTime":3234.205,"body":"be proved."},{"startTime":3234.205,"endTime":3235.765,"body":"Because in this system, it's a complete system."},{"startTime":3235.935,"endTime":3237.525,"body":"Every statement is either true or false."},{"startTime":3237.525,"endTime":3240.965,"body":"And every statement therefore can either be proved or can not be proved."},{"startTime":3240.965,"endTime":3242.205,"body":"If it's false, it can't be proved."},{"startTime":3242.225,"endTime":3243.445,"body":"If it's true, it can be proved."},{"startTime":3243.625,"endTime":3246.725,"body":"So this statement, suppose it can be proved."},{"startTime":3248.385,"endTime":3252.87,"body":"So you can prove that the statement cannot be proved, but that then we can"},{"startTime":3252.87,"endTime":3254.365,"body":"only prove true things, right?"},{"startTime":3254.365,"endTime":3255.445,"body":"This is supposed to be consistent."},{"startTime":3255.665,"endTime":3256.725,"body":"We can't prove a false thing."},{"startTime":3257.025,"endTime":3258.805,"body":"So if you can prove it, then it's true."},{"startTime":3259.825,"endTime":3263.619,"body":"Uh, but that would mean it cannot be proved 'cause it's asserting that it can't"},{"startTime":3263.619,"endTime":3264.125,"body":"be proved."},{"startTime":3264.125,"endTime":3265.445,"body":"So if you can prove it, then it's true."},{"startTime":3265.465,"endTime":3266.725,"body":"And that means you can't prove it."},{"startTime":3266.825,"endTime":3267.645,"body":"So that doesn't work."},{"startTime":3267.705,"endTime":3269.205,"body":"If you can prove it, then you can't prove it."},{"startTime":3269.345,"endTime":3271.485,"body":"That's, that, that is a contradiction that doesn't work."},{"startTime":3272.035,"endTime":3272.325,"body":"Okay?"},{"startTime":3272.325,"endTime":3274.365,"body":"So that means you can't prove it within the system."},{"startTime":3274.985,"endTime":3277.045,"body":"But hang on a minute, that's exactly what it's telling you."},{"startTime":3277.235,"endTime":3281.242,"body":"It's saying I cannot be proved within the system, and we've just deduced that it"},{"startTime":3281.242,"endTime":3282.845,"body":"can't be proven in the system."},{"startTime":3283.585,"endTime":3287.595,"body":"So it's a true statement, but it can't be proven in the system."},{"startTime":3288.495,"endTime":3292.663,"body":"So what this tells you is, and this is called girdles incompleteness theorem, it tells"},{"startTime":3292.663,"endTime":3292.94,"body":"you that there will always be however brilliant you, the system that you set up,"},{"startTime":3292.94,"endTime":3293.218,"body":"is there are always gonna be things that are true that you can't prove within"},{"startTime":3293.218,"endTime":3301.555,"body":"that system."},{"startTime":3301.575,"endTime":3305.715,"body":"You're gonna have to think of other ways and other mathematical arguments to prove them."},{"startTime":3306.135,"endTime":3308.435,"body":"So that's where, this is the good news we're finishing on."},{"startTime":3308.435,"endTime":3311.475,"body":"Mathematicians are not going to be out of a job anytime soon, \u003claugh\u003e."},{"startTime":3312.425,"endTime":3316.475,"body":"Okay, well, I will stop there next time, coincidences, but thank you for listening today."},{"startTime":3321.575,"endTime":3327.395,"body":"How do you think mathematic mathematicians are getting on board with this idea of things"},{"startTime":3327.395,"endTime":3331.275,"body":"like superposition or things being simultaneously happening and yet contradictory?"},{"startTime":3331.895,"endTime":3333.715,"body":"How is mathematics coping with that?"},{"startTime":3334.105,"endTime":3340.451,"body":"Yeah, so we can address things like that with, with ideas of probability really, and"},{"startTime":3340.451,"endTime":3340.875,"body":"statistics."},{"startTime":3341.215,"endTime":3345.229,"body":"So we can, you know, if you roll a dice or something, um, you don't"},{"startTime":3345.229,"endTime":3345.497,"body":"know exactly what's going to happen, \u003claugh\u003e, but you could say there's, there's a probability"},{"startTime":3345.497,"endTime":3345.765,"body":"of each different event and you might not know until you look, which it's going"},{"startTime":3345.765,"endTime":3353.795,"body":"to be."},{"startTime":3354.095,"endTime":3358.55,"body":"So I suppose my feeling on that is, yeah, something has happened, but we don't"},{"startTime":3358.55,"endTime":3360.035,"body":"know what it is yet."},{"startTime":3360.095,"endTime":3363.966,"body":"So I'm still clinging to things are either true or they're false \u003claugh\u003e, but we"},{"startTime":3363.966,"endTime":3365.515,"body":"may not know and that's okay."},{"startTime":3365.775,"endTime":3369.962,"body":"But I of course, in the quantum world, yeah, things can be they, you know,"},{"startTime":3369.962,"endTime":3372.475,"body":"until you collapse the wave packet, you don't know."},{"startTime":3372.935,"endTime":3377.893,"body":"And, and things are sort of how in this state of unknowability until, until you"},{"startTime":3377.893,"endTime":3378.555,"body":"look though."},{"startTime":3378.575,"endTime":3383.44,"body":"So I suppose at the point at which you observe, then things become yes or"},{"startTime":3383.44,"endTime":3383.765,"body":"no."},{"startTime":3384.185,"endTime":3387.845,"body":"But, you know, we have to allow our math, mathematics does allow for uncertainty."},{"startTime":3387.845,"endTime":3388.085,"body":"Yeah."},{"startTime":3388.395,"endTime":3388.845,"body":"Very clever."},{"startTime":3388.965,"endTime":3392.106,"body":"I I thought that was a horrible thing to chuck you, but that was a"},{"startTime":3392.106,"endTime":3392.525,"body":"fantastic answer."},{"startTime":3393.185,"endTime":3394.965,"body":"Do infinitesimal exist?"},{"startTime":3395.185,"endTime":3396.005,"body":"Uh, \u003claugh\u003e?"},{"startTime":3397.795,"endTime":3400.205,"body":"Well, it depends what you mean by infinite decimal."},{"startTime":3400.325,"endTime":3405.445,"body":"I, I mean, I would say not, I don't think so, right?"},{"startTime":3405.685,"endTime":3410.205,"body":"'cause ultimately, kind of calculus lets you take the limit all the way to zero."},{"startTime":3410.625,"endTime":3412.045,"body":"And I'm, I'm happy with that."},{"startTime":3412.385,"endTime":3416.301,"body":"But there are alternative viewpoints of how it should all work that do allow for"},{"startTime":3416.301,"endTime":3417.085,"body":"these infinite SALs."},{"startTime":3417.625,"endTime":3422.138,"body":"Um, so it's different schools of thought, but kind of ultimately when you're done, you"},{"startTime":3422.138,"endTime":3424.245,"body":"end up differentiating in the same way."},{"startTime":3424.305,"endTime":3428.565,"body":"So maybe it's not \u003claugh\u003e, maybe it's not terribly problem, but problematic for real life."},{"startTime":3428.905,"endTime":3429.845,"body":"But I don't think so."},{"startTime":3430.635,"endTime":3434.881,"body":"Like, not in that sense, not, not infinitely small numbers that are nevertheless bigger than"},{"startTime":3434.881,"endTime":3435.165,"body":"zero."},{"startTime":3435.425,"endTime":3435.845,"body":"Worked"},{"startTime":3435.945,"endTime":3436.605,"body":"On this for"},{"startTime":3436.605,"endTime":3436.965,"body":"200."},{"startTime":3437.475,"endTime":3438.605,"body":"Well, yes, yes, yes."},{"startTime":3439.105,"endTime":3442.565,"body":"We should, we should have a lecture just about, just about calculus, I think."},{"startTime":3443.255,"endTime":3443.805,"body":"There you go."},{"startTime":3443.805,"endTime":3444.685,"body":"You heard it here first."},{"startTime":3444.905,"endTime":3448.325,"body":"Infant decimals, possibly not too worrying for everyday life."},{"startTime":3448.645,"endTime":3453.645,"body":"\u003claugh\u003e"},{"startTime":3449.105,"endTime":3449.885,"body":"Yes, we heard."},{"startTime":3450.385,"endTime":3453.788,"body":"I'm very sorry if you didn't get a chance to ask a question possibly Sarah"},{"startTime":3453.788,"endTime":3454.015,"body":"might be willing to, to, um, have a bit of a chat at the, at"},{"startTime":3454.015,"endTime":3457.645,"body":"the end."},{"startTime":3457.665,"endTime":3460.885,"body":"But in the meantime, please join me in thanking professor Sarah."},{"startTime":3460.945,"endTime":3461.165,"body":"Ha."}]}