 I'd like to welcome everybody to another installation of math encounters and my thanks go out as always to the Simons Foundation for sponsoring this series for more than six years now. But I also want to call out a special thanks to someone sitting in the front row, David Eisenbud. You may not have heard his name, but it was David's idea to have this series more than six years ago. He said, wouldn't it be great if there were some place where people who loved math but weren't necessarily mathematicians could gather and learn and talk together about interesting math topics. So our gratitude to David for a wonderful idea. And finally, I will introduce the introducer briefly. Geva Pats has been working with MoMath for a number of years and if you love the exhibits here and you love when they're all working well and doing delightful things, Geva is a large part of the reason why. He's behind the scenes helping us make sure that everything works well. In fact, we have a new exhibit opening on June 22nd, Hoop Curves. On your way out, you can stop and read the blow up of the New York Times article. It will be opening June 22nd. It's our basketball and statistics exhibit. So we hope you'll all come out and enjoy that and Geva's been a large part of helping make that exhibit work. I also should mention that Geva was our champion this year at the MoMath Masters. That's our adult math tournament, stiff competition. It came down to the wire, but he was triumphant in the end. And with that, I give you Geva Pats. Thank you, Cindy. So there's a quote I enjoy, which in the usual formulation goes, the more our body of knowledge expands, the more our circle of knowledge expands, the greater the circumference of darkness around it. Now, it's the MoMath crowd and I can already hear some of you going, okay, so the pie is cancelled and there's that too down there. So the approximate ratio of what we know to the edge of what we don't know is about pie by two. That's not the point. What I found interesting and inspiring about that quote is this idea that we keep knowing more and more. You know, we can do amazing things. We can reverse engineer the brain to the point where we can plug a camera into it that speaks the code of the brain well enough that we can have an artificial camera give blind people sight. We can send probes to our neighboring planet Mars all the way through space that not only get there, but are able to accurately measure micro fluctuations in the gravitational field of Mars so that we have a density map of Mars, despite the fact we've never actually been there. This is amazing, but it also feels potentially a little alarming because you start wondering, well, are we ever going to run out of stuff to do? There was a lovely question at the last Q&A about, you know, is there a boundary to how fast someone could run a marathon? I hope there isn't because then what's left to inspire marathon runners. And the same thing here, I find this quote really inspiring because it says the more we know, the more we understand what questions there are, the more questions we have to ask. Now, this quote is usually attributed to Einstein. I've heard it attributed to Bertrand Russell. I've heard it attributed to Richard Feynman and a host of other sort of professional clever people. So, unfortunately, I can't give you with absolute certainty an attribution for that quote. And if that bothers you, then you should leave right now because tonight's speech is very much a journey into the unknown. We're not just going to the edge of that disk of light. We're not just going to the edge where the known meets the unknown. We're going to the edge of the unknown. We're going to the very edge of darkness, which may sound like a scary thing, but you've got a really good guide to get you there. And that would be Marcus de Stourtoy, who is a renowned mathematician and more. He's a professor of mathematics at the University of Oxford. He is a fellow of the Royal Society. He has an OBE. He's a fellow of the American Mathematical Society. But he's also the Simony Professor for the Public Understanding of Science at Oxford. And that's a post that allows him to wander far and wide through the breadth and depth of science and bring back to us amazing stories about what's going on at the very edge of science. In fact, those of you who are fortunate enough to make it to the book table fast enough to grab a copy of his new book before they all sell out tonight will see that the book is actually organised by edges. There's a number of them there. And the best person to tell you about them is Marcus de Stourtoy himself. As you always makes me laugh a little bit when I hear that introduction and I'm introduced as the Professor for the Public Understanding of Science because it's such a ridiculously grand title and people expect that I must know the whole of science and here I am to explain it to MoMath and to Manhattan. Certainly when I took over this job a few years ago I would get journalists just phoning me up, expecting me to know the answers to almost anything. So I remember the Nobel Prize for Medicine had just been announced when I took over and I had this journalist from the Daily Telegraph in London phoned me up and said, yes, just been announced for the discovery of telomeres. Could you tell me what a telomere is? Now I must admit that biology has never been my strong point. So I'm a little bit embarrassed to admit that I was in fact in front of my laptop at the time. So I quickly pulled up the Wikipedia page to telomeres and told the journalists very confidently, ah, it's the bit at the end of the DNA which controls how long the DNA lasts and the journalists seemed quite happy but it made me realise that there's no way that any scientist today can be expected to know it all because there is so much science that we do know. I mean if you think who was the last scientist that perhaps knew all the science that was on the books at the time? Maybe Newton, Galileo, but since Newton and Galileo somehow science has exploded and it's become so large that no one scientist could know at all. But it set me off on this journey thinking, okay, well obviously I can't know at all but could we come to a point where science, where scientists know it all where we've answered all the great unsolved problems there are about the universe or maybe there are some questions out there that will always be beyond the limit of science to know. So it set me off on this journey which I've been on for the last three or four years which culminated in the publication of this book, The Great Unknown, trying to identify whether there are questions that we might not be able to answer. I mean surely the universe isn't set up as a kind of exercise in the philosophy of science. There must be some questions that we as humans can't answer. It would be very anti-Copernican if actually we could answer everything. But what are those questions? So this is the journey I've been on and I've divided the kind of science up into kind of seven regions that I think what I've called seven edges of knowledge which I think might have unnerbles hidden inside them. So what I want to do is to take you on a journey of some of these unnerbles but I think that desire to know is really basic in us and you know as already been illustrated in the introduction the kind of things we have discovered over the last few decades is extraordinary. I just kept a little track of things that have been discovered over the last few years. For example we landed a spaceship the size of a washing machine on the side of a comet. It did bounce around a bit before it stabilized. We've managed to make robots that can create their own language using machine learning. We don't know what the robots are saying. We have to interact with them to learn this language. We've sequenced the DNA of a 50,000 year old cave girl and we're using stem cells to create artificial pancreases for diabetic patients so it's kind of extraordinary the pace of discovery and I think it has fueled this expectation in the public that yeah maybe we might come to a point where we can answer all the big unsolved questions and I think that desire to know is so basic in us we just want to know the answers to these things and I think it's driven our evolutionary development. It's the desire to know has helped us to survive in our world. Those who didn't want to know died out and I think I rather like this quote from Aristotle's metaphysics which I start the book with actually which is his statement that everyone by nature desires to know and I actually did a little analysis. The word to know is one of only about 100 words which have a universal translation across all languages. Even the word to eat does not easily translate across all languages. So I think that desire to know drives us as a species. But I realized it was probably a very dangerous move to stand up at any point in history and declare this is a question we'll never be able to answer. And so I kept in mind a few stories of scientists in the past who tried to do what I'm going to try and do now which is declare some questions that's potentially unanswerable. And so one of my favorite stories is astronomer Auguste Comte who in 1835 very confidently declared we shall never be able to study by any method the chemical composition of stars. Now that seemed a very fair comment in 1835 but today for example we have never visited a star and dug a bit of stuff out taking it back to the lab to see what it's made out of and yet some decades later scientists were able to give a very complete analysis of what a star is made out of because although we have not visited a star of course every night the stars visit us we look up there and the light from the star visits us and these scientists realize they can analyze the light and give a very complete description of what stars are made out of. So I kept this story in mind and a few others of scientists who declared we can never answer this only to be shown so wrong. But I thought as a mathematician surely I should be able to use my logical skills to analyze the nature of a question and show that there's nothing that you can do to show that this has any answer. So that was my role. Is there something by the nature of the question that makes it unanswerable? Now I'm not the first to talk about the kind of unknowables one of our great 21st century philosophers Donald Rumsfeld very famously made some statements about the nature of the unknown. This was a quote he gave in response to questions about weapons of mass destruction in Iraq and the journalist was slightly perplexed when he answered there are no knowns there are the things that we know that we know it's a bit dangerous to do an American accent if I agree in America so you'll bear with me. We also know that there are known unknowns that is to say we know there are some things we do not know but there are also unknown unknowns those are the ones we don't know we don't know. And he got given the foot in the mouth award by the Plain English Speaking Society for this very bizarre response to are there weapons of mass destruction in Iraq. But actually I think it was a little bit unfair because I think it's a beautiful description of different states of knowledge. The known knowns well the things that we know that we know well scientists love talking about those those are the things that we've discovered and in a sense my book is full of the things that we think we know but also how knowledge changes and adapts and evolves and we realize that our description is just an approximation of reality or a completely different story emerges so I think it's important if you're going into the unknown to understand the knowledge that we do have and why we believe that knowledge is a description of reality. The unknown unknowns I'd love to tell you about those but they are by their nature unknowns I can't tell you about those but I think these are very interesting because they're kind of the game changers they're the things which just transform science I would say if you look at the end of the 19th century people like Lord Kelvin were saying oh we've done science we understand it all it's a matter of decimal places now how wrong he was a few decades later suddenly you get Einstein and his generation coming along completely transforming our view of science so I like to call these like the black swan events if any of you have read Talib's book Black Swan he identifies these are the moments in history transformative moments when something happens unexpected which completely changes the game so we love these in science because the things that were discovered in the Large Hadron Collider well we kind of knew they were going to be there what we want is an unknown unknown which is going to make us totally have to reassess what we think about the physics that describes our universe so these are really exciting but I can't tell you about those what I want to go for are these known unknowns now there are lots of things we don't know we don't know what dark matter is for example but I want to try to identify the known unknowns that will always remain unknowable are there any or maybe we could answer every all of these big questions so I'm after these are there anything is there anything that will always remain unknowable actually I think Donald Rumsfeld missed a category here because there's also what about the unknown knowns and interestingly as our political commentator Slavoj Zizek pointed out these are quite important for a politician because they're kind of like your Freudian delusions the things you deny but actually they're the things which drive you as a human but you deny that you know them I think for example Donald Trump has a lot of these unknown knowns which he doesn't recognize but are really driving his character but I'm after these known unknowns so can we look at science and identify any questions that will by their nature always be unknowable so as I said I divided the book up into seven edges and what I'd like to do in this presentation is to take you in depth into a couple of those edges and then give you a tour around the last five so the first edge I'm going to look at the first edge that I consider in the book really goes to the heart of what mathematics my own subject is all about I don't know the mathematicians here have this but when I'm at a party and somebody asks me so what do you do I slightly look forward and dread this question because when I say oh I'm a mathematician you can see this kind of look of horror appear on their faces and they kind of back away and their glass suddenly becomes very empty and they run to the bar but I'm very persistent so I run after them and say no no no we actually do something really interesting and when I talk to them I suddenly realize what they think I'm doing is as a research mathematician is long division to a lot of decimal places surely I've been put out of a job by my computer by now to explain to them that a mathematician is really a pattern searcher we're kind of studying the science of patterns and that ability to spot patterns has really given us the power to kind of navigate our environment and perhaps know what's going to happen next it's our best tool of looking at the data that's gone to in the past spot a pattern you might know what it's going to do into the future so it's a very powerful tool for us to survive in this world but just how powerful is this tool of mathematics to be able to know the future can we genuinely know if we have the equations and the description of the universe now can we know what it's going to do into the future and this is the challenge of this edge how good a tool is mathematics to know what's going to happen next so at the heart of this is a challenge in mathematics called chaos theory which really challenges just how much we can know about the future now on my journey to the edges of knowledge I decided to take an object with me which I thought would help the reader and also me to kind of navigate this unknown so each edge I take a physical object which I think will kind of capture the challenge of this unknown so when I went to the edge of kind of trying to predict the future I thought so the ultimate symbol of kind of unpredictability and unnervability is a dice the reason we find a dice so exciting is that when I throw it on the floor I just don't know what it's going to land on and so I look and I'm excited to say well this time I've got a one but somehow the dice is an interesting object because of its unpredictability but just how unnervable is this you see Isaac Newton gave us the laws which govern all this dice falls from my hand what happens when it impacts on the floor he developed the idea of the calculus which is giving us a language to navigate a world in flux so post Newton scientists believe that well now we have the tools to be able to work out what the universe is going to do next if you know what the universe is doing now and you've got equations to describe its evolution we can predict the future so I certainly sort of spent many years as a mathematics undergraduate learning these tools that Newton had given us and I thought surely that's going to give me an edge so I went off to Vegas and thought I can use these tools to cash in on the craps table so I went off to Vegas unfortunately I lost a lot of money but they let me keep this dice so this is the dice that I lost a lot of money with and somehow it's really the challenge of well surely I should be able to know what this is going to do next given the tools that Newton gave us many scientists post Newton really believe that we should be able to know the future but if Newton is my hero in this desire to know what's going to happen next my nemesis in this story is this guy here this is Henri Poincaré French mathematician who at the beginning of the 20th century discovered this idea of chaos theory which puts limits on what we can know even if we have the equations which describe the universe you see our intuition was that okay I can never have a complete description of the present but I can get a very good approximation of what's the present state of the universe is then I run the equations and that should give me a very good approximation of the future state of the universe Poincaré showed that's not true that a very small error in the present can explode into a huge great inaccuracy in your prediction of the future and my favorite examples of these are very simple systems it's not a complicated system and the universe is very complicated but even very simple systems which we have equations for can be so sensitive that a small change in the starting conditions can cause a completely different outcome in the evolution of this system and my favorite example is a pendulum now a pendulum is usually so predictable we use it to keep track of time but this is a slightly different pendulum than you'll have on a clock it's two pieces of metal jointed together a little bit like a leg now the mathematical geometry of this is extremely straightforward I can write down a description of that the physics that controls how this will drop when I let it go is just simply gravity this is practically frictionless so very simple system yet trying to predict the behavior of this pendulum is almost impossible I mean it's crazy the sort of behavior of this thing I mean it's you know can you predict which way it's going to go next now but the real challenge of chaos theory is that so we saw one particular behavior of the pendulum now I have a little mark here and I try and replicate the behavior each time so I'm going to try and start it off so you remember what it just did yeah good so now is it going to do the same thing so okay that takes a little bit more time to get going almost went through there but this time it's a completely different behavior yet I started it in almost the same position and this is the signature of chaos theory that although you might have a very good approximation of how the thing starting that may not tell you anything about what it's going to do next this is my favorite desktop toys indulge me I'll just do one more because it's so beautiful this thing it's like a sort of circus performer so there you go I started it again quite similarly but now did a few rotations and now it's got into this kind of resonant mode and not really doing anything exciting at all so another of my desktop toys which illustrates this quite nicely is this one here I've brought it along I have this on my desk in my office in Oxford it's a pendulum again but it's got six magnets and the pendulum is attracted to the magnets now actually each magnet has a question answer to a question over it so it's got like yes maybe definitely ask a friend try again no way so I use this to make all my decisions in life so well let's ask it and I mean one of my great passions is the Riemann hypothesis it's a great unsold problem in mathematics I would love to know whether the Riemann hypothesis is going to be solved in my lifetime so let's set this off and see whether I'm going to be lucky and be alive when it's proved so now I've got a little example of this this is a slightly simple system that we did in the lab it's got three magnets you've got to try and predict which magnet do you think the pendulum is going to end up at so I'm going to set this off so this thing is going away it's kind of crazy doesn't know which way it wants to go is it going to be the top right hand one seems to be and then right at the last minute gets pulled into the bottom left hand one so this one's made up its mind it says try again Riemann hypothesis okay right try again let's see where I set it off here's a little experiment idea to where I did a simulation and I started the pendulum off in the top left hand corner and I've coloured the magnets blue yellow and red the first time I set it off the pendulum ended up at the blue magnet and then I changed just the sixth decimal place of one of the starting coordinates so you barely notice that it's starting in a different position it started off quite similarly but then ended up at the red magnet then I changed a little bit again and ended up at the yellow magnet so almost you can't tell that you've changed the starting position yet a completely different outcome Poincare discovered chaos theory because he was trying to predict whether our solar system was stable or not and he realized just with the if you have three planets in there that a very small change in the starting position of one of those planets can cause a stable system to completely fall apart so this is a little bit like an asteroid coming in and it's potentially going to knock out one of three planets now one of those is our planet and we were quite like to know is it are we are we going to get knocked out or our neighbouring red planet it seems like that's something we'll find very difficult to know because a very small change in the position of that asteroid could cause it to knock out a completely different planet so this seems to be so sensitive to small changes that you cannot know which planet will be wiped out oh it's saying definitely this time excellent so that's good news that well I'd like to know who's going to prove it next so good so this is a graph a picture which helps you to know what this pendulum is going to do so the way you read this picture is that if you start the pendulum over a yellow region that means that the pendulum is going to end up at the yellow magnet so you can see there's a large area of yellow in the top left corner that's where the yellow magnet is so of course if I start the pendulum near the yellow magnet the other two magnets aren't strong enough to do anything so it just wobbles and settles at the yellow magnet and a small change in the starting position is not going to change that so here is a region where we can know what the future has in store there's another way knowable region if you look at opposite that yellow region there's another big swathe of yellow there and if you start there and make a small change it doesn't matter basically the pendulum just swings backwards and forwards and ends up again at the yellow magnet but in those simulations I started the pendulum off in the top left hand corner and this top left hand corner is an example of a mathematical structure we call a fractal there are some lovely demonstrations of fractals here in the museum of math and the fractal has a quality that it has infinite complexity so as you zoom in on this region you might hope that suddenly it will simplify and just become a region of yellow and then you can know where the pendulum is going to go and as you zoom in it just retains complexity so you might say ok well I've got to the ninth decimal place surely it's all gone yellow now but no even here when you've zoomed in that much you see yellow, blue and red colours still and you zoom in a bit more hoping to see it just become a single colour so you know what the pendulum is going to do maybe I need a bit more data a bit more decimal places but this region tells you it doesn't matter how far you go there's always going to be three colours there and a small change can shift you even in the fiftieth decimal place onto a different result for where the pendulum is going to go so this is saying that chaos theory doesn't say that you can't know anything we've been able to use the equations of mathematics to land a spaceship on the side of a comet that was a region which was knowable in one of these regions which small changes didn't matter but it's also powerful enough to tell you when you can't know things there are regions where actually now a small change results in you having no control on what the future has in hold and this doesn't just affect the solar system it infects so many different things in nature that we would like to know so the weather very classically is a chaotic system which within five days however much data we seem to gather the weather seems very unpredictable beyond five days now as well as an object to the edge of knowledge I also took a person with me a person who spent their academic life at this edge to try and see how do you deal with the unnerval of something like chaos theory so I took a colleague of mine in Oxford this is Bob May who understood that these chaotic systems happen very often in biological systems so he understood that population dynamics can be very chaotic so small change in the population can cause a completely different population in the next generation next season actually I should call him Lord May because he's now a cross-party member at the House of Lords in the UK advising governments and I went and had lunch with him at the House of Lords to see how he was getting on explaining the importance of chaos theory to politicians and over lunch he said not only in research but in the everyday economics we would be better off if more people realised that simple systems do not necessarily possess simple dynamic properties and I asked him how he was getting on teaching the MPs a bit of mathematical chaos theory and he said Marcus they're mostly interested in their egos here not in high level mathematics but Bob May is now working with Andrew Haldane at the Bank of England looking at the banking crisis of 2008 to understand that maybe the equations which control the economy have some predictable regions the moments when people can make money out of the stock market but maybe there are regions which then go rather fractal and chaotic and then you know you have to have a very conservative investment policy because now you're in a region where small changes could send you on a high or a massive low so interesting that he's now applying it to economics something in economics we would love to know the future sometimes we can but maybe it's as important to know when you cannot know well I came back to my dice at the end of this edge and wondered well maybe the reason that I didn't win any money is because this dice is chaotic and very sensitive to small changes but I got a little bit of a surprise this dice is not as unknowable as you might think I discovered a piece of research done recently by some Polish mathematicians who've analysed using high speed cameras and also some high level mathematics how this dice is behaving as it hits the floor and rolls and I'm going to show you some pictures which are rather similar to the pictures that I showed you for the pendulum with the yellow, red and blue regions which showed us when we could know what the dice is going to do and when not so I'm going to colour the faces of my dice with different colours red, yellow, blue, green and then we're going to try and work out what are the changes and parameters that help me to predict whether it lands on say a yellow face or a blue face and this is the results that these Polish mathematicians discovered so you can see in the bottom right hand corner we have a picture which is very fractal it has no simplicity to it at all as you zoom in it just seems to be very just complexity of colours now the graph here what I'm doing is varying two parameters when I throw this dice so one is the height of the dice above the craps table and the other I'm varying is the angle at which the dice leaves my hand so what we want to know is does a small change in the height affect the outcome so in this bottom right hand corner this is describing craps tables which are very rigid where the dice does not lose much energy as it cascades across the table so here is very rigid and as I roll this thing it bounces a lot of times before it settles and this will be a chaotic described by this fractal picture a small change in the angle or the height will give me a completely different answer it's these tables here that you want to go and play on because here we see a picture which is not fractal the top left hand corner is a picture where a small change in the angle or the height seems to not make much difference you can be in that yellow region and alter the height a little bit and still you know it's going to land on the yellow face so what craps tables are this picture describing because these are the ones you want to go and play on in Vegas turns out that these are tables which are a little bit soggy which where the dice loses a bit of energy as it hits the table so what you've got to look for is a casino which is a little bit down on its heels it hasn't renovated its craps tables recently and then and then these are going to be much more knowable when you throw the dice that a small change is not going to alter very much the behavior of the dice and if you take anything away from this presentation hopefully a book afterwards but also the following fact which is that more often than not on these tables the height you are above the table and how the dice leads your hand more often than not it lands on the face that is face down on your hand so when you go on the opposite of what you can see on the top of the dice because more often than not it lands on that side so the message of this edge was really that mathematics there are regions where we cannot know but also regions where we can know and that's the important thing to know when you cannot know is as important as knowing when you can know now I think it's a tradition in these encounters that I set you a little challenge halfway through the presentation so I've got a little challenge about making some predictions a dice is pretty unpredictable but also the toss of a coin is traditionally something which we find it very difficult to predict we use it to decide the outcome of various things even the world cup soccer world cup a few years ago some years ago it was a draw at the end of the final and they didn't do penalties they decided it on the toss of a coin it was extraordinary so here's the challenge for you if you've got a coin great get it out of your pocket if you haven't got a coin MoMath are offering a sense for each of you so put up your hand if you don't have a coin and we will get one to you so listen to the challenge I'm going to set you I'm going to ask you to toss your coin 10 times and we're going to do a kind of, it's like a casino so if you get three heads or three tails in a row you're going to give me one dollar but if you don't do that I will give you one dollar so you're going to play at this casino it's a dollar for a dollar or you're going to go next door so let's up the how far would you go so the other casino is going to offer you two dollars if you don't get three heads or three tails in a row in your 10 tosses so would you go there or what about the casino next door which is offering three dollars if you don't get it you've got to pay a dollar to play but if you don't get it then they'll give you three dollars or if you get the next one is four dollars or five dollars so at what points will you so I've got this right so I'm going to give you five dollars so you've got one, two, three, four or five so I want you to now, before we do our experiment just think which of these casinos will I go to to yes so so which one would you not go to let's say that one of course you're going to go to the five dollar one although even that one case so yes the challenge is which of these would you not go to one dollar okay so which ones of these would you would you say one dollar what about the two dollar one who thinks that they would go to the two dollar one okay there's somebody put their hand there who thinks they'll go to the three dollar one okay a few more who thinks they'll go to the four dollar one and who thinks well of course you'll all go to the five dollar one okay so alright you think even the five dollar one okay so who's going to go to okay so we'll go up who's going to go to the five dollar one yeah who's going to go to the six dollar one well when are you going to flip okay you want certainty well I want it infant dollars okay let's do a little experiment now I want you to take your coin so I think there's a bit of uncertainty about quite what the odds are here so start doing it and keep track of whether in your ten choices you've got three heads or three tails in a row so off you go I'm not going to pay you double or no you have to pay me double I think I'm only getting tails I think this one's a I'm a bit worried that oh no there's a head so who's got three in a row already wow does anyone not got three in a row good okay keep going I think the challenge will be can we get not get three heads or three tails in a row okay is everyone done there ten tosses okay let's have hands up those people who did not get three heads or three tails so I'm going to pay you okay so we've got one two three four five six seven eight nine ten eleven twelve thirteen fourteen fifteen now I think there are over a hundred people here but I know I did the math and in a hundred people if you'd done this eighty two of you would have got a three heads or three tails in a row and only eighteen so actually we've got far even less amongst you who didn't get that so and I think that's quite surprising it is so high I can still pay you four dollars and expected the long run to make money now most people would probably take that bet thinking well three heads three tails that's probably quite rare obviously not that rare but I think it's quite unexpected that it's that likely that you're going to get the three heads or the three tails and if you want to work out the probability it's a really beautiful little bit of maths because you get a sort of Fibonacci like explanation to help you to work out that probability so if you think how many if I do one toss there are two possibilities and neither of them had three heads or three tails if I do two tosses there are four possible sums and both of those still don't have three heads three tails but if I now do three tosses two of those are things where I pay out three heads or three tails and there are six where I don't pay out so that's actually the Fibonacci rule I add two plus four I get six and if you want to work out the number of times in the next toss or with four tosses that you'll not get three heads or you'll get three heads or three tails it's this Fibonacci rule which tells you so if you'd like to find out a little bit more one of the other books I wrote has the story of how Fibonacci like rules can help you to know something so this is the number mysteries book which talks about even though the dice is and the coin is potentially unknowable we can over the long run know quite a lot about the behaviour of this coin actually even the coin has been analysed and it's been shown that there's a slight gyroscopic effect that happens when it goes up in the air because there's a slight difference in weight on each side which gives a slight edge again to the face that's down on your thumb being the one that turns up when you actually do the toss it's so small that it's not really worth betting on in a casino but it's interesting again that there is a little bit of a bias so that was the edge of pale theory I'm going to take you into another edge which actually says that the unpredictability of the dice is the heart of the way we must do science because that edge was under the whole conditions that the universe is deterministic that once you know the setup and you run the equations you can know the future but physics in the 20th century challenged that idea and says that unknowability is the heart of the way we must do science and this is the message of quantum physics that quantum physics says that physics by its nature has unknowability at its heart now the object I took with me on my journey to the edge of the quantum world to kind of tease out how unknowable is physics is this little pot of uranium that I bought on the internet it's amazing what you can buy on the internet these days so I bought this on Amazon I was very pleased with it and I thought about writing a review actually you know you can write those little reviews and I looked at the reviews that are already being written there and there was one wonderful review this guy had given it a five star I was really pleased with it and said I'm so glad I don't have to buy this from Libyans in the parking lot in the mall anymore it was very sweet but there was one guy who was completely dissatisfied this gave it a one star and said I've had this for several million years and it's half empty ah you got the half life too great excellent now for me the interesting thing about this pot is the number on the side because it says 984 counts per minute what this is telling me that is over a minute there will be 900 oh probably shouldn't do that it's amazing what you can get through airport security as well but anyway oh gosh blow up Manhattan no this is not weapons grade uranium so don't worry about this in fact a bunch of bananas is radiating more this little piece of uranium in here but this number is telling me over a minute there will be 984 bits of radiation kicked off this uranium but what sir this little pot can't tell me and what it seems like physics can't tell me is when it's going to do that this is on average but there seems to be no known mechanism to tell me when it's going to do that radiation and quantum physics says that is an unknowable but no when it's going to do that and the explanation that helps us to understand this is probably something you've heard of Heisenberg's uncertainty principle Heisenberg's uncertainty principle articulates in a mathematical formula what we cannot know about our universe so this is an equation if you're trying to do Newtonian physics for example and I've got an electron flying through the air I want to know where the electron is and how it's moving and how it can make a prediction of how it's going to evolve into the future Heisenberg says you can't know those two things simultaneously you actually cannot get the information to apply Newton's ideas and this is captured in this mathematical formula it's not some wishy washy statement it's a very precise formula which emerges from the mathematics of quantum physics so it says the more that you know about the position of this electron the more uncertainty of possibilities for the momentum increases and then you might say well I know where the electron is so now I'm going to measure its momentum but the more you know about the momentum suddenly there's an uncertainty in the position of the electron it can suddenly be in different places from where you're expecting it to be and this formula articulates this delta x is getting more and more precision in the error in where you think the electron is going to make this equation balance out the error in the momentum has got to get bigger and you try and get the momentum error smaller the error in the position also gets bigger so here's a little test you understand Heisenberg's uncertainty principle this is my favorite physics joke so Heisenberg is bombing down the motorway and it's BMW and a policeman comes over pulls him over, gets him out of the car and says sir do you know how fast he's going and Heisenberg goes no but I know exactly where I was a few titters there but that was a bit weak people were saying why is that funny I know you shouldn't have to explain a joke but the reason that was funny is because because Heisenberg knew exactly where he was he had no knowledge of his momentum, his speed he couldn't know, could be a whole range of different values so the police officer says to Heisenberg well you were going 200 kilometers per hour and Heisenberg goes now we lost yeah that's a bit better good, so now the point is now we know about the momentum he's lost knowledge about position and so they could be anywhere in the universe there's a follow up if you want to ask me in the Q&A Schrodinger happens to be in the back of the car we'll come to that a little later but some have said that one interpretation of quantum physics is that this electron does not have a position and momentum until you observe it in fact its position for example is kind of unknown and it could be anywhere in the universe and we have this thing called a probabilistic wave function which says what is the probability that you're going to find the electron in this particular place in the universe when I observe it the wave is very high it's kind of saying well I'm very likely to find it here but if it's low it says not very likely to find it here but for me the challenge about this whole theory is that every time you run this experiment you might have run it once and you think well I know where it is but you run this same experiment again and it can be somewhere else it's not determined by the setup it seems to be genuinely random this seems to be the one thing which might possibly be random but is that really true is it really true that we can't know when this the radiation is going to get kicked off this Heisenberg's uncertainty principle kind of explains this radiation because we know a lot about the bound state of the nucleus of the uranium so we know something about the momentum of the particles inside the nucleus which means we have uncertainty in the position that causes something called tunneling so suddenly the position we thought it was inside but sometimes randomly it can be outside and that's the radiation but is that really true is this truly something unknowable around our universe or might there be a kind of little mechanism that we don't know about that is determining when it's going to do that now I'm a kind of believer that this is not going to be an unknowable that it's always going to be there I'm not part of how we do physics at the moment but I don't think this is going to be where we'll end up and I'm not the only one to believe this Einstein also believed that there was something rather fishy about this kind of randomness and he very famously wrote quantum mechanics is very impressive but an inner voice tells me that it is not yet the real thing the theory produces a good deal and it certainly does with all tested theories on the books it gives such precise predictions about the behavior of matter that I think the quantum physics have almost given up thinking about an alternative because why do they need it but as Einstein said it hardly brings us closer to the secret of the old one I am at all events convinced that he does not play dice so is this true is the universe genuinely random and there are unknowables of the mechanism which is controlling when this thing is going to radiate we now know that if there is something it's a very weird mechanism our first intuition would be that it's something very small localized inside the uranium but we now know due to ideas of entanglement that if there is a mechanism it's a mechanism which spreads across the whole universe there's something on the other side of the universe can have an influence on when this thing might radiate so it's going to be weird but it doesn't mean it doesn't exist so I'm intrigued about this unknown because it's certainly at the heart of how we do physics but will it be there in 100 years time or are we ready for an unknown unknown a black swan a new way of looking at physics which will show us that although it was a very good approximation it isn't really the real thing so that was a kind of whiz through a couple of the edges let me take you on a whistle stop tour through the other five so what is my uranium made out of we used to think that uranium atoms were indivisible they were part of the periodic table and then we discovered no they'd pull apart into electrons protons and neutrons and then the proton and neutron pulled apart into quarks we think that's the bottom layer now but each generation thought they'd hit the bottom layer so how can we know that it won't just pull apart again and again and again maybe it's turtles all the way down so the infinitely small can we know that we've hit the last layer so digging inside matter but what about the infinitely big our universe is our universe infinite or does it somehow wrap up in some finite way and how could we ever know that we know thanks to Einstein that information travels no faster than the speed of light the universe has been going for 13.8 billion years that means there's a cosmic horizon surrounding us beyond which we cannot get any information so the universe is infinite we have no information reaching us which will tell us this so we are in this kind of finite bubble rather kind of Greek model of our idea of the universe this bubble is expanding our knowledge more and more stuff is coming in you might think well that's great because we're knowing more and more but it turns out we're knowing less and less because the accelerated expansion of the universe is pushing things over the cosmic horizon faster than it's expanding so weirdly we're losing information if we were evolved at some point in the distant future to do astronomy all of the other galaxies will have been pushed over and our model of the universe would be that we're a single galaxy surrounded by a void and we could not know about those other galaxies because they're beyond our cosmic horizon so extraordinary that what we know might depend on when we were evolved into the history of the universe we do know about these galaxies but future generations will only know about them because they will read them in books so the infinite universe is that something we'll never know what about time what happened before the big bang that's often a question which gets asked and mathematics kind of says well the big bang is the singularity singularity is a moment beyond which you cannot know what's going to happen you cannot extend your data to know beyond that so is time before the big bang a no-go area something we cannot know turns out there might be ways to know what happened before the big bang that might not be as unnerval as we think what about so those kind of big physics ones but what about our brain the idea of consciousness that's philosophers some believe that the hard problem of consciousness knowing whether another entity is truly having conscious experiences similar to mine I mean some of you looking a little bit tired and zombified you know are you really conscious inside there I mean I know it's late and well maybe you send your avatars along and you're at home enjoying the baseball or something and so how can I know whether you are really conscious inside there if I drink too much I have a pain in my head but is that pain anything like your pain we give it the same name but how can we ever know so many philosophers challenge whether the hard problem of consciousness maybe something that science will never be able to answer because it's somehow self-contained in our own system now we have this idea of homogeneity which says that well you're very like me so I believe that your experience will be quite like mine and you will have a similar consciousness but what about this thing we carry around in our pockets we don't believe it's conscious at the moment but can it get complex enough at some point that it suddenly start saying iPhone think therefore iPhone am and you know we have to declare it conscious it's an important thing we need to understand and this first thought I thought was going to be very much outside my comfort zone as a mathematician but the exciting thing was I discovered a mathematical formula by an Italian mathematician who works here in the United States in Madison a mathematical formula which describes the complexity of a network which might make it conscious so this is a formula that could be applied maybe to an iPhone and tell you when it actually passes a threshold we must consider it's having an internal world so I love this it's something even what makes me me is a piece of mathematics and actually the final edge that I end with is looking at my own subject of mathematics because mathematics has been able to turn on itself and understand that there are things in mathematics that we will never be able to prove are true now the object I took with me on my journey to the edge of mathematics was rather a weird one especially I wrote this book in the UK and when it was published here in America a few months ago I had to explain this thing because apparently at Christmas in America you don't have Christmas crackers now this is part of our culture in England these weird ritual over dinner that we have these things you pull them apart there's a big explosion and inside there's a joke and a little novelty toy and things so I clearly see there's a marketing opportunity here in America that if you miss out on these things they're wonderful but I'm very nerdy and I make my own mathematical crackers for my family they despair at this but so I put little mathematical joke in and a little mathematical kind of toy so they thought the jokes were very lame but I actually one of them I think is really funny so this one that I got in my cracker so Benoit B. Mandelbrot is the guy who one of the guys who came up with the idea of fractals these things with infinite complexity so the joke is well Benoit B. Mandelbrot so what does the B stand for in Benoit B. Mandelbrot Benoit B. Mandelbrot Benoit B. Mandelbrot exactly what does that mean? His name is a fractal get that so I thought that was quite funny but my family thought that was pathetic but quite like the little so I put a little paradox into this cracker as well because I always love these as a kid these kind of linguistic paradoxes so you have you know a little card and it says this statement is false and you go oh okay so let's suppose that statement's true but it says it's false so it can't be that so it must be false but if it's false that means the statement is true and then you get into this kind of infinite regress and so with linguistic statements we realize that you can have these paradoxes that doesn't necessarily have to have a truth value but in mathematics if you've got an equation it must be either true or false but a logician Kurt Girdle realized that you can make these self-referential statements in mathematics with devastating consequences for what we can prove in mathematics so Kurt Girdle proved something which is a kind of limitation to knowledge within my own subject of mathematics he proved this thing called the Girdle's incompleteness theorem which says that within any system for mathematics there are true statements within that system which you cannot prove are true and how did he do this well he used that idea of the little paradox I had in my cracker but instead of saying this statement is true he wrote this statement is unprovable but the clever thing he did was to use something we call the Girdle coding to change this statement into an equation of mathematics which must either be true or it must be false now suppose this equation is false so what does that mean so it means that the statement is actually provable but provable statements are true so we've got a contradiction again but because this is an equation of mathematics this equation must be either false or true so it means it must be true but that says there's a true statement now in mathematics this equation is true when we analyse what it means it says that this statement is unprovable so we now have a true statement which cannot be proved true but actually we have proved it's true it can't be proved true within this system we've pulled outside the system looked in and proved it's true but unprovable within the system but that new system has its own true statements which are unprovable now this is quite devastating for mathematicians because I'm working on a lot of conjectures what gets me up in the morning is the don't know that's what gets me to my desk things like the Riemann hypothesis or challenges of symmetry that I'm working on but my great fear is about this unknown what if the thing that I'm working on is actually something which is true but does not have a proof within the system that I'm working in and what I found when I went on this journey to the edges of knowledge and talked to many scientists many scientists would just would not admit that there was anything that they couldn't know because I think in order to be able to do science you must go in with maybe arrogant belief or just kind of like the innocent fool but one of my favorite operas is Siegfried part of the Ring Cycle where Siegfried is able to slay the dragon because he's not frightened of the dragon everyone else is frightened of that dragon and that's why I think it's important in a way to kind of believe that in science maybe we can know it all thank you alright so we have time for some questions so if you raise your hand I'll bring the mic to you I would just like to comment that's why there are more male scientists than female scientists that assumption that I must be right no I think that's an interesting kind of thought and you know I think that if you're doing biology for example I mean I think I got drawn to mathematics because I also hankered off that certainty and I found biology you have to be able to deal with the unknowable in biology because it's very messy I think I have a lot of respect for biologists because it's just this it's much more difficult than maths I think because maths there's a kind of way to work your way to that this kind of certainty so again it's interesting that is that quality as well that those people are able to deal with kind of uncertainty and things are drawn more to biology was I mean I was almost slightly aspergic and I mean my wife definitely thinks I'm aspergic but I think you know if you look there's quite a lot of evidence of maths departments attracting people with something like aspergers because of wanting that the certainty that mathematics gives you so actually I had to Simon Barankone not Sasha Barankone they are related so Simon Barankone is an expert in autism and aspergers but he gave and very often it's considered as as a negative thing but he's trying to and very often it is very incapacitating but he's trying to spin that there is a positive side to this and he's doing a research analysis at the moment to see whether those with aspergers and autism are actually much more represented within the mathematical community so I took part in a piece of research which put me way high up on the aspergic spectrum than an average male so it's interesting that I think that there seems to be some evidence that that that kind of need for I mean I think that's why social interaction is quite difficult for somebody with aspergers or autism because they can't deal with the unpredictability of human behavior so I think this kind of interesting issue, many interesting issues that your statement kind of sparks yes Does your shirt have anything to do with tonight's presentation? Very good so my shirt I think is quite gaudillion in it's statement because how can somebody say I already know nothing because surely you know something then if you know nothing so I thought it was quite nice but I'm also a big game of Thrones fans so this is Jon Snow Jon Snow's girlfriend keeps on telling him Jon Snow you know nothing but he never says himself so I quite like the shirt I'm a big game of Thrones fan and I felt it had a kind of girdle kind of nature to it as well yeah, good right, glad you got it Hi, can we have Schrodinger in the back seat now? Yeah, Schrodinger in the back seat, exactly so I have to do a little set up to tell you Schrodinger's paradox about the cat which is if an electron can be in two places at the same time doesn't that mean a cat can be both dead and alive at the same time because the electron could be in one position and sets off so you enclose in a box so it's unobserved, the electron here means that a poisonous gas is released and the cat gets killed but if the electron is here the cat is safe so it means before you observe this if the electron is in both places at the same time the cat is in two states what we call superposition and alive at the same time until I observe it okay, so now you know enough to find the joke funny hopefully which is, so the police officer goes round to the back of the car flips the boot open goes oh my gosh you've got a dead cat in the back of your car and Schrodinger goes well it's dead now okay because he's just observing he collapsed the weight function I haven't said so next question great presentation by the way I have a question talking back in the chaos theory, so for example we know like these for example molecules sometimes they behave like in Brownian motion and sometimes scientists are able to protect just by applying one dimension the way that these particles behave can we apply the same thing to chaos theory for example yes, your story is very interesting because it's the reason that we know that the world is not continuous but is made out of atoms and this is one of the stories I tell in the book about matter now how do we know that about atoms and it's a really recent kind of revelation it was Einstein who used this idea of the Brownian motion so you see the molecules being kicked around in a kind of random manner and he did the maths to be able to work out what the size of these small things must be and their mass in order to be able to create the effect of the pollen say on the surface of the water so yes I mean you will the behaviour of these things as soon as you've got many particles in this system are going to probably have a chaotic nature to them but again we will use this is the I mean what we did with this exercise with the coin was although that may be very sensitive to small changes yet in the long run we can get some knowledge about this thing so what we use is a statistical analysis of the behaviour of the atoms in the room here we may not know what an individual is doing but it's enough to have a kind of statistical knowledge to be able to know a lot about what the behaviour of a collection of atoms say in this room is doing so I know that if I go into the corner I'm not going to find a vacuum suddenly that would be very unusual so Thank you for the great discussion yesterday so I read that Darwin once made the statement that he felt that we would not be able to know everything because the brain actually Darwin so he made the statement that we will not be able to know everything because the human brain did not evolve to know all these things it basically evolved to find food and sex Yes I talk about this in the book because there's clearly limitations just on capacity of what our brains can know our brain is a finite piece of equipment which will a finite number of neurons a lot of them a finite number of operations within a lifetime so we can already know that one human will only know a finite number of things and mathematics has an infinite number of true statements and I often worry whether say something like the Riemann hypothesis maybe this will be provable but perhaps the proof has a complexity that will be beyond any human to be able to navigate so it might have a proof but maybe it's a capacity limitation of our brain and that's so you could use that to prove there are things we won't know but it won't tell you what they are and so that's why I wanted to push a bit further and not just use the limitations of the capacity of the brain but you know here's a question that by its nature it doesn't matter what you do even if you had infinite capacity or something or unlimited capacity let's say you still won't be able to answer this but consider the universe as one great big computer and there's scientists have analysed how much can the universe know at this point after 13.8 billion years and it's still a finite amount of information so if you consider the universe as a computer you might ask what's it computing it's actually computing its own evolution so but can only know again a finite amount yes great hi this is not a question this is just a kind of personal question about your what sort of mathematical research you do and your background in math yeah so I do research into something called group theory and number theory so group theory is the study of symmetry so actually the subject of my second book we actually had they have a wonderful book club here at the museum of maths which I really encourage you to come along and they chose quite randomly the symmetry book which is telling the story of my own research and the larger story of symmetry but I use tools from a number theory to actually try and understand symmetry so we try and understand prime numbers using something called a zeta function and it's a very powerful tool to try to navigate something which looks very random and unpredictable and so I use a zeta function to look at possible symmetries that might exist and very often they look very wild but I can use this tool from number theory to give me some find some patterns and order in this kind of unpredictable mess so that's kind of the math side I do excellent presentation thank you and I love Christmas crackers by the way great going back to the craps story that you told because I'm a craps player but I was yes I was interested in finding out more about why you pick those two variables of height and angle from the yeah and why like the power in which you throw especially at the table if the table makes a difference at how hard you throw it or how many times you know you roll it versus just kind of tossing it and why that doesn't come into the certainly does I mean but this was just to give you a two-dimensional picture so I just so there are many variables but then the pictures that are I wouldn't be able to show you those pictures so and I must say this is the decisions by as well by the Polish team so I took this so it's not my own research so they've got varying different parameters but in order to make these pictures you really want to just fix some things and just vary too and then you'll start to see pictures that you can sort of interpret but you're absolutely right there are many very more variables than just the angle and the height above the table which will impact on those and it's interesting which of those are going to have a similar sensitivity on which ones you know there might be systems where actually the strength that you throw is not fractal and but the height is so yeah it's important to analyze all of these things and know which ones are you sensitive to which are a bit more robust so with the understanding that Marcus will be available for questions in person afterwards I can take one more question hi just a quick question when you do your your research in your work I was just kind of wondering how you did it do you text message the entire time while you're doing math do you lock yourself in a cabin and isolate yourself from the rest of the world or do you watch The Telly while you do it I don't know I certainly don't watch The Telly the symmetry book sort of tells that story because I keep kind of diary of a month a year in my working life I think people just don't get what does a mathematician do all day and how do they do it so for me I use music a lot so I'm a trumpet player and I love classical music and I find that a very good environment and I almost feel is it stimulating a similar sort of pattern searching side of my brain but it also helps the anxiety of when things aren't going well that I allow my brain to go somewhere else and I don't feel oh I can't do this oh my god so I need a safe place to kind of go to when just nothing's happening as many mathematicians will talk about the role of the unconscious is very important so it's very you will sit at your desk and you'll try and think but very often the ideas will come when you move away from your desk and you allow the unconscious I almost feel like my brain I set things up and it's a bit like the brain is slamming away on the piano and it's all sounding really rubbish and then suddenly it hits a sequence of chords and you think oh oh listen you've got to listen to this and it throws it up into my conscious mind and suddenly says listen to this and then that's what we call that flash of aha and I've described one of those moments in the book which seems to come out of nowhere but obviously it didn't I had to set up with a lot of hard work this kind of unconscious piano player who's then suddenly found a sequence of chords which my brain told me I needed to be conscious about wonderful all right let's give one round of applause for Marcus