5. How Did Human Beings Acquire the Ability to do Math?

Stanford University okay so for today's final session well as usual our two lectures one is dealing with sort of mathematical cognition and how the brain got to be able to do mathematics at least a story that we can tell that we think is moderately plausible I think is plausible and then I'll finish with a sort of a grand finale by talking about the seven Millennium problems which means that other topics that I wanted to do as I said in that email I sent out will have to postpone in particular it occurred to me that I could do a whole course on infinity and that will be really cool and undecidability so probably next year I'll do a course on that so for those who wanted that I'll do it but it'll be next year now if my voice sounds a bit different the big question is whether I can make it through the next hour and a half and still have a voice I woke up this morning with the beginnings of a sore throat and it's been getting steadily worse all the time I'm now drugged off as much as a professional cyclist in order to get through the lecture so this is this is what you're going to see is is a drug enhanced performance tonight I have on I'm on something I'm actually I'm on set of thought lozenges and sudafed and all of that stuff okay so we'll see how it goes okay but the extra low tones in the voice is just the voice about to disappear already so this was a question it actually puzzled me for many years because it was really it's intriguing because mathematics is you know 2,000 years old to sort of formal mathematics ten thousand years since we invented numbers maybe thirty five thousand years max before we had any kind of counting with notches and things so it's fairly recent I mean way to a recent to have seen any major structural changes in the brain or the body although you got to be careful making statements like that because the brain is very plastic but with to all intents and purposes we've got a stone-age brain doing this modern stuff and so the question was how did it be an acquire that ability it's the capacities for doing mathematics must have been in the brain latent or do other things for many hundreds of thousands of years okay and that was puzzling because mathematics is clearly kind of an unusual use of the brain it takes a lot of training it takes effort it makes you tired and it's it's powerful but it's an interesting thing so there was this puzzle that I couldn't get my mind around as to how the brain acquired it what was going on how did our ancestors eventually come up with this way of thinking that's proved to be so so effective okay and I spent many years thinking about this off and on and finally in the year 2000 I wrote a book that it began as a very thick heavy sort of scholarly tome and then you know I by then in my career I wanted to make more people access to accessible to more people so I I ended up writing what's known as a popular book in mathematics although this is really a thesis and the this was the the American version with the subtitle how mathematical thinking involved and why numbers are like gossip and I'll get to the gossip part later on but interestingly enough at the same time as I was doing this as often happens two other people were writing very similar books there was Stanislas Dehaene and I'll mention later on in France who wrote a book that came out about a year before mine called the number sense where he focused on how the brain does sort of numerical work he's a cognitive psychologist and just sort of brain imaging research that came out at about a year beforehand and I actually pulled my book back from the publishers and added some material from his book and put some introduction at the beginning because I didn't I didn't want to me do what he'd done about number so I just reduced my coverage to a short amount and then when my book was actually the publishers I saw an announcement for a talk at Berkeley by George Lake off about a book he was writing with Rafael Nunez who's an old friend of man an old colleague of mine ELISA nanu George vaguely what's knowing better since but then I knew him as a linguist so there was George was writing a book which was actually came out called I forget what they call it now anyway how these how the brain does mathematics or something I forget the exact title now but in any case he'll bring in his book out and when I saw the announcement of the talk he was giving it Berkeley I thought whoa that's my book well it turned out it wasn't it turned out his book with nine years begins exactly where mine stops so we have these three books all in the same general area coming out at the same time and in fact Georgie's book with the same publisher it was weird and we had the same editor and at no point in the development have you told either of us that he had another book on the same topic so he was probably doing that knowingly to let us just go through our process and we ended up with different books and actually we ended up on a panel together talking about our books so these three books came out about the same time and that sort of cemented what became known as mathematical cognition which is the study of mathematics as a way of thinking and I've pursued that myself since then thinking of mathematics as a way of thinking rather than a product that you apply think of it as a thought process and if at a lot of my own applied research in the last ten years for the Defense Department and various branches of the military and private industry has really been focused on applying mathematical thinking rather than applying mathematics there's actually a significant difference between those two but that would be a that's another course basically okay the title in the book was was taking a very popular metaphor you know if there's a message in the book and I actually nailed this within the first few pages is there is no math gene you know I'd been a College Dean long enough to know that I wasn't going to listen to any students or parents trying to get exempt from a math requirement because their son or daughter didn't have the math gene you know and I had a book to back it up then I've written and if you think about it it clearly cannot be something like that I mean it's like it's a capacity you know just think of how natural selection works I mean if if some pit that sounds like math or if some people can do mathematics it'll stuff me what it my fun that sounds like Martha okay okay so feature of modern life okay it's also the ring that means this is a business call that's someone from the office calling me okay so if natural selection puts something in our gene pool then it's sort of there for everybody - what - greater or lesser degrees you know it's just in there at least for large groups of people so it's a capacity that somehow got into the human gene pool and that was the interesting part how did it get there interesting because you know it's done in the brain right which is the most expensive organ in the body you know it's about 2% of body mass and it consumes about 20% of your energy so the brain in new Volusia nary terms by all accounts is a non-starter you just shouldn't have these huge brains that require you know we're born without the brain fully grown it takes many years to reach maturity without parental control and all that gravity is you know masses amounts of time you don't reach maturity for the first 20 years I mean it's just not an efficient organizing an organ in terms of getting in the gene pool so the brain clearly offers huge advantages and we can all sort of guess what they are and recent research that was just published actually in the latest Proceedings of the National Academy is more evidence to suggest that what enabled the brain to grow was cooking it was a father that when we cook our food we get much more energy out of out of it and not at that fuel the growth of the brain that's a thesis bacin been around a while in any case I was looking at this sort of evolutionary development of the brain all of the advantage it conferred on our ancestors at great cost in terms of energy consumption and immaturity for many years and so forth and seeing if I could fit into that how mathematics got into the picture so that 10,000 years ago people could invent numbers and then two and a half thousand years ago they could invent geometry so how did that get to be you can ask a similar question about language and I suspected from the start that the answer had to be bound up with language because language is equally puzzling you know I mean hard to say how language got into the brain and people often say well it offers great facility for communication well you actually don't need language to do communication in your local environment right you can go to a foreign country and providing you congest you and you know a little bit of vocabulary you can get by fine you just need one or two words you know me Tarzan you Jane sort of stuff so I said what's not as proto-language gets you by you need a vocabulary and you can point but that's all that's all you need you only need language with all of his grammatical structure to talk about things in the future or the past or somewhere else in the world to form plans you have to talk about complex things you have to create models of mental models of the world okay so if language is not used for communication I mean issues for communicating but if that wasn't the driving force in terms of evolution what is okay I mean it clearly can't be if you want to need language to express complex thoughts then complex thoughts are a prerequisite for the language if you try to pull them apart at all and so language only becomes necessary when you have complex thoughts but when you start to think about it having complex thoughts begins to sound like forming mathematical models so I always had this suspicion that mathematics came out of the same mental roots as language and eventually after fooling around and reading a lot of stuff about evolution of language in which there wasn't a lot and still isn't a great deal looking around he eventually came up with the thesis that got written about in this book okay so the questions I asked and what this goes back at least ten years before the book got written how did the human brain acquire this ability when did it acquire it in evolutionary terms what else was going on how many thousands of years ago was it and what are you volution advantage did it confer okay you can Shawn would have enough water to get through tonight but we'll see how it goes and the approach I took was just the standard sort of scientific like approach I said well let's just split it up in this divide and conquer let's just split into some questions and say let's just not talk about mathematics mathematics is a broad subject split it apart into basic constituents simple mental capacities it's a level of simplicity where we can tell a plausible natural selection story ask what led to the human brain acquiring each capacity if I do it right most of these are going to be obvious just self-evident we can just easily answer them and if we can easily answer them we can sort of nail it from human evolutionary history we can sort of nail it in our history as to when it was acquired and then ask Lee the $64,000 question how and when they came together to give mathematical thinking because there was a sudden blossoming about 10,000 years ago where the whole bandwagon of mathematics went forward and so what was going on 10,000 years ago that drove this you know once you've got mathematics it's very useful but which is the simple language once you've got language by it's useful to communicate but something else has to be going on the brain to drive that acquisition of language and the most obvious thing wasn't constructing models complex thinking planning and a whole bunch of stuff around that ok so I came up with a list of basic capacities that go together to give mathematics there's number sense this is the one that Dehaene wrote about and are you had a whole book on this and you could write a whole book on lots of these but his book came out as I say a year before mine and it was actually called I think number since I'm not sure whether hack had the word number since I don't think I did I think I got though I think I got that phrase from him or from whoever came looking it numerical ability that's for sure I think I just has something like quantitative capacity or something I figure I mean numerical ability with numbers being different from from number sense number sense isn't about having numbers it's the sense of size his term actually is a little bit misleading in that sense because it makes you think of numbers but it really means sizes of collections numerical ability is really about numbers so you have to have numbers to have numerical ability but you don't necessarily need numbers for number sense spatial reasoning ability a sense of cause and effect that's what mathematical reasoning is all about the ability to construct and follow a causal chain sort of chaining things together that's believed by many people to be a driver of large brains as well spear-throwing was one of the theses that was put I had a ton of stuff on evolution spear-throwing was thought to be a something that drove you know when our ancestors got to the point there threw spears or rocks to cut to kill animals because they calculated how long the brain would have the given the speed of doing that there isn't enough time in sort of milliseconds for feedback mechanisms to correct the motion of the arm when you aim you have to sequence its instructions to the muscle and fire them off down the nerve so that it all happens instantaneously or virtually instantaneously so just the speed of it it's too fast for it to be done online in real time an algorithm has to be put together that's the sequences the muscles to go forward that was a story anyway and I don't have the training to really judge between them if it got published I tended to to pick it off the shelf okay so that was that was an important one algorithmic ability and some of these are related this is a sort of a cognitively reflective well not in necessarily colony reflecting but a sort of an almost an abstract variation of number five the ability to sequence together instructions like throwing for a spear the ability to handle abstraction mathematics is inherently abstract that was clearly a crucial thing on the on the world logical reasoning ability and relational reasoning ability and at that point I stopped now that this is not meant to be minimal some of these are related indeed part of the story are telling the book is about the relationships but I stopped because I thought there's nothing in mathematics that I know that isn't covered by combating these if you've got these nine ingredients you can bake the mathematical cake now you know this it takes a good chef to make a good cake out of ingredients so just mixing them in the mold isn't going to be enough there's going to be a lot in this is a souffle there's a lot in the art of the of the person mixing them together but if you put the nine together you've got the ingredients so that was the thesis that if we can explain first of all how these got into the gene pool and secondly gene pool and secondly what brought them together and how then we've told the story of where mathematical ability came from and we have to give a rationale for why that's it at a certain point in time about ten thousand years ago this stuff was put together okay so the question was when to each of these mental capacities evolve what survival value gated offer and what brought them together to give the ability for mathematical thinking that's really what the books about and it's a book because there's a story to tell about all of these okay so number sense the sense of the size of a collection you know eeveelution airy stories for this obvious you know you could waste your energy climate you'd use energy climbing the tree that's got more fruit on it if you're outnumbered by a threatened by a potential threat you hightail it out of there if you outnumber the potentials threat you steer where there's lots of research on contemporary by the way a lot of the evidence I've collected together was some contemporary work with animals people studying animal cognition and animal behavior many many creatures have number sense in fact like I collected together so much data on animal cognition that when I'd finished publishing the math gene I ended up writing another book called them the number I forgot what it's called now was it the number sense I think was a number sense no it kind of been the number sense that was wow I forgot what it's cover of articles about animal cognition to do with mathematics okay getting bad when you forget the titles of your own books sorry yeah anyway so so yeah tonight's homework is to find out what the title of that book was doesn't require numbers many of these creatures that exhibit number sense don't have going to have any real sense of numbers possessed by many creatures as I said and you can find any young babies within a few days of birth there's been tests done by various people that show that very small children exhibit this this number sense okay and some of the seminal work was actually done by Karen Wynn in a 1992 MIT dissertation PhD dissertation she was one of the first people that did this work for small children and Finland has were two years at two-year-old children but since then have been a lot of work in that area so this one is sort of fairly well trawled by the cognitive psychologists that look at number there's a bunch of them here at the medical center that do this kind think a new American ability that's the stuff that requires numbers other than very limited forms with some chimpanzees and bonobos and various kinds of birds and rats it's you can train them to recognize certain things power there was that there was a famous power that died recently that that was trained to do various things with numbers but it takes a lot of training they never get to be a hundred percent and it's clearly it's hard to say exactly what's going on I mean they certainly have behaviors that correlate with different numbers so in some sense they're responding to numbers but it doesn't you know that the best they are equivalent to a small two or three-year-old child beyond that the human just just race away and so it's really it's it's humans that really have numbers to the degree where we have accurate numbers we can be really precise with numbers very large numbers you know and it depends on language there's plenty of research des Haines research plying Butterworth's research and others somewhere some of Steven Pinker's work that links numbers to language research that shows that the two are really linked together bilinguals are faster doing arithmetic in the language that they learnt it in and then whatever the second language was in terms of the arithmetic they can never match the speed in fact Express if you look at the time differences almost certainly what they're doing is Sousa ventually subconsciously going back to the language in which they learnt it because the numbers are embedded in language and there's various things stroope tests and things that that have shown that so numbers are very much linguistic constructs and de Haines work the guy that what about the number sense a lot of his book is about the connections between language and number a spatial reasoning ability right by the way the reason for number well why do we why would we need numbers well to be more precise about collections of things if you look at the behaviors of the animals that learn it it's all about making fine distinctions between different collections of seeds and things and things of value so there's an advantage to having more precise numbers all right sir it's not clear that it's a sufficiently precise line one to get it into the gene pool but since it comes in on the tech eight on the coattails of language we don't have to tell too strong a story for that one you basically take number sense and language and put them together and that little connection of two ingredients will give you numbers spatial reasoning ability well moving you know swinging from tree to tree recognizing where an animal's going to be when it's running towards you all of this stuff about negotiate any any creature that moves must have a built-in spatial reasoning module of some kind relational reasoning ability how things are related together because we use that to understand human relationships you know one of the key things about humans is we are creatures of society we form societies we have extended families and that's all about having and tracking human relationships you know huge part of human beings we're I mean evilly they could these the terrorist negotiations I mean the trick is do you have to get establish a relationship with the terrorists in the moment you do with that things get diffused so human and that depends upon knowing about them okay so huge impact on our behavior by human relationships because that's what we do I mean here's a huge a huge advantage for humans right we we're not the fastest we're not the strongest we don't have sharp teeth we don't have long claws I mean we have weaklings physically there are creatures all around us that can beat the whatnot out of us and do evil things to us but we can cooperate we can be smart and we can cooperate and that turns out to be a at least of a moderately human lifespans turns out to be a good strategy I mean there's no way we can outlive the ants but that's another issue okay and they aren't going to be able to outlive the viruses so you know ultimately it's all viruses they're willing to hear it the earth the ability to handle abstraction that according to my story was the key to understanding mathematics how mathematics can well once you could abstract and collect those things together in an abstract form because you had to distract them to fuse them together and once you did that you got their sequences of cause and effect we know that there's a driving force in humans to understand that because it's an obvious survival technique to know what causes what and very young children get annoyingly obsessed with chains of reasoning you know they love chicken licken I mean we've probably always had kids we've all spent years of our life as parents reading chicken looking to the kids and kids just love that idea of one thing causing another causing another so it's learning about that's this huge part of becoming a human agent okay so the abstraction I think is the is the key because when you abstract cause and effect chains you get algorithms you get proofs you get deduction okay and that one and this is where I was lucky I just saw some work in a research monograph about the origins of language and it was written in such a way that when I read it having been thinking about it for many years he was talking about language in society and I just read into it mathematics I could have just gone through in fact I did go through and just replace all of the references with mathematics and mathematical thinking and he'd written the thesis for me so the chapter where I do this essentially was his just co-opted for my purposes okay so that mathematics comes out of the language capacity so the hypothesis are formed and talked about in the book that the key step was handling abstraction and it involved it involved the language module in the brain by the way the language module we think this is no more recent than 75,000 years and maybe as much as 250,000 years it depends differ as to when our brains acquired the capacity for language with a grammar a recursive grammar that allows you to construct potentially indefinitely long sentences so you know the proto-language having vocabulary in sequence in a couple of words me Tarzan you Jane danger they're hungry and that kind of stuff lion tiger that could be a couple of million years I mean like that was probably around a long time but forming and forming these mental constructs seems to be very recent and part of my thesis is the moment the brain got that ability it had the latent capacity for mathematics what it needed was a trigger no language was an easy one because once you got language boy you can use it to communicate interesting we saw that report was recapitulated with the internet you know the internet was not invented as a system for sort of communicating the way we do today the World Wide Web came later but boy that's how it's been co-opted okay but one of the key things that I liked was it's not a grid of complexity of thought processes you know I talked last week about mathematical modeling you just iterate this simple idea of abstract abstract abstract and you get all the shoes from mathematics okay so here was my how I described it and I'll summarize it here if you want to understand mathematics you should think of it as a fictional analog of parts of the real world it's about modeling the world if you model the world in terms of human actions and human motives then you've got language you've got grammar the grammatical structure is what allows you to chain words together to form complex descriptions so what you're doing is you're modeling the world and you're articulating that model of the world and communicating about it you're planning something you're hypothesizing how the world is going to be and how you're going to act in the world okay and if you do it one way that grammar gives you it's just it gives you language it gives you sentences and phrases and things that talk about the world in that way but if you do it in a different way in an abstract sort of quantitative logical way you get mathematical models of the world and once you've got one you've really got the other and the fact that we know that language a number at AG close together sort of buttresses this so you think about the two has been just two sides of the coin the trigger for language was instant because communication is a winning card mathematics you had to wait until there was a need for mathematics society had to get sufficiently complex that there was any need to do it well remember that the first lecture talked about the invention of numbers as banking people wanted to mediate trade and transactions society had to reach the level where that was necessary and that was some area 10,000 years ago so you get some area you've got this need for using this latent capacity and how it comes it's been waiting there all those years and so now you've just got it as a you should really think of it as that just like a novel is a fictional analog or a movie is a fictional analog of things in the world so mathematics is a fictional analog of something in the world but it's expressed in a in a different kind of language and to get it we take these mental capacities that develop to negotiate the physical and the social world and apply them to reason about fictional abstract objects numbers triangles circles lines and dots as we saw in a cute little video at the beginning again which means we actually know what the secret is to doing mathematics and it's that and that was deliberately an in-your-face statement to just make people realize that I was been serious about this a mathematician is someone of use mathematics literally as a soap opera if you think about what a soap opera is it's a fictional animal of the real world we all know that the real world isn't in detail anything like the soap opera right now idiom actually the we it was a weird thing after this book came out I went on this book tour and I was going around and I had this talk and I said you know we the people in the soap operas you know they're all beautifully turned about the world beautiful they were these wonderful clothes they're gorgeous and I'm talking tonight I'm giving my spiel and then it dawns on me I'm in Santa Monica and in fact half the audience it wasn't a huge audience but half the audience looked like that so I I realize that there are parts of the world where people do look like this soap operas but and so I part of the reason I well let me let me stay a little bit more and I'll give a little bit of flesh to this claim okay the characters I don't mean the sex lives of mathematicians I mean that's an interesting story in its own right though other disciplines seem to have a better time than mathematicians what I mean is the characters just as the characters in a soap opera fictionalization of abstractions from real people so the the characters in the mathematical so proper abstractions from the real world things like numbers geometric figures we've seen them all around us we can look up we can see circles we can see straight lines we can see right angles rectangles what I mean we could see the world around us in terms of these abstractions and you know we could replace this by a line drawing that represents the room and then we've got an abstraction of the room and various more complicated things so those are the characters and in mathematics when in a soap opera those fictional characters the soap opera is about the relationships between them well that's what mathematics is about in the retina in the TV soap opera the facts and relationships have interested births deaths marriages love affairs and business relationships but mathematical relationships are different at least they seem to be different until you really dig down about them almost all mathematics boils down to questions like are these two objects equal what is exactly the relationship between objects x and y find an object X that has property P and I solve this equation as an example do all objects of type T have the property P do all right angle triangles have the property that the square on the hypotenuse is the sum of the squares of the other two sides and so on how many objects of type Z are there those are the kind of questions we asking mathematics in fact pretty well all of mathematics boils down to one of those fact answer questions well that's what happens in an episode of a soap opera you find out about the weather to create the relationship between two people are they married on it whatever are they in love what exactly is a relationship with them does Fred love Susie and Susie not love Fred in other words is it commutative or not there's all of these relationships but this is really what soap operas are about you know the procedural criminal series find the crime find that find the person who committed the crime do all these people have that property you know that's the soup offer all the soap opera does is lays out and shows you weak bar all of the new things about the relationships well that's what mathematics do that's why mathematician does when they're when we do research we we establish these relationships at which point you said to yourself hang on I have never found it difficult to watch your soap opera I just sit there and it just flies by and I understand everything that's going on more or less everything depends on the soap opera but most of the things there was always Mary Hartman Mary Hartman but that was another special case do you remember that one yeah how many people remember Mary Hartman Mary Hartman yeah for the fix that was that this was this this was this in all of us middle-class intellectual trendies got hooked on this thing cause it was a count of soap opera this was a soap opera that was a spoof of soap operas so we just sit there for weeks on end thinking about I don't really watch soap operas but this is really clever and then you get hooked on it we all got hooked on it and we were watching another soap opera week in week out it's really kind of funny to see that happen I don't know if that was the intention of the guys who produced it but it was a very interesting experience and it was a ton of fun okay but we don't found watching a soap opera difficult and yet if you think about it the complexity of a soap opera is really quite high those characters they have love in tingly complicated relationships I mean it's just unbelievably complicated much more complicated than anything you'll find in most mathematics books up to first second year university level in fact I was like that there's something your connection you can do this if you want to do an interesting exercise you can you can watch a soap opera and you can write down you know how many characters are in that soap opera what kind of things do you need to know about human relationships in order to follow that soap opera episode you know so how many characters are there how complex are the relationships between them how many relationships are there how many different threads be connections between them you can ask yourself basic questions about how the plot unfolds in a soap opera then you can kind of note the numbers down and then you can get a page of a mathematics textbook up to maybe first second year university and you can ask the same questions how many objects are there what's the relationship between how complicated are the relationships what you need to know about this relationship to follow the proven and every time the mathematics comes out way simpler than the cell proper human relationships are much more complicated than the mathematical ones so by that simple metric reading appears your mathematics should be much easier than watching an episode of a soap opera and it's not it's quite the reverse you don't have to make any effort to watch the soap up well you have to make a lot of effort to do the mathematics so you've got to ask yourself it's not the logic it's not the complexity it's not the complexity of the relationships or anything in the TV soap opera everything's more complicated you need to know a ton of stuff about human relationships to follow why people act in that soap opera in a piece of mathematics you need to know three or four axioms commutativity associativity that's all you need a bit of logic I mean modus ponens and things so the only difference between the two is that the abstraction in the soap opera is a pretty thin abstraction those characters have only slightly abstracted from the real world so that soap opera pulls upon all of our knowledge of the world and human relationships it's part of us it just triggers things and we just immediately jump to those things mathematics you have to create those abstractions yourself they're not in the world you have to create them hold them in your mind have available all of their properties even though there are not so many you have to have them in the mind and then you have to reason with them on top of that the killer is the abstraction it's the fact that it's abstract there's evidence to read the book there's all sorts of evidence that I sort of pull in and refer to in favor of this but the main distinction between those two is the abstraction by any other measure TV soap operas are more complicated in other words what I say in that what I demonstrate in that book are the conclusion I guess is this that first I mean the book is about the story that I tell about how this came about the evolutionary story but what I tell the conclusion you draw is that we actually there's not a different brain we're just using a standard-issue brain in a novel way you know in terms of those evolutionary traits it's what Stephen Jay Gould used to call an exaptation we're taking something that was developed for one purpose and reusing it for another and we did it because 10,000 years ago society got a stage of complexity where we needed that but I will mention battle that the subtitle of the book was the subtitle of the book was why numbers are like gossip that was the the the publishers subtitle taken from the contents of my book and one of the pieces of evidence are brought in to talk about the importance of relationships and why relational thinking was really what drove Mathematica the list of those things but other than the abstraction the thing that was the most crucial one and I captured this in the soap opera analogy was relationships it's really the complexity of relationships and there's a lot of evidence that anthropologists of together and sociologists have gathered about relationships and human relationships for example you know the a snog refers follow you a lie around and they follow you and me around the television really influences you they follow you and me around okay the followers around and they keep track of our language use and they will tell us that every one of us this rich thing called language this really evolutionary expensive capacity in our brains that lets us communicate this this language organ two-thirds of its uses a gossip they're talking about the relationships of other human beings if you go to a serious newspaper like the Wall Street Journal that two-thirds drops to about 40% 40% is just pure gossip you know depending on it it's gossip sometimes about presidential candidates and things but it's just sheer gossip it's about human beings if you read USA Today it's about 80 percent even down at 40 percent in a so-called serious newspaper that's a lot of devotion to gossip but if all of that organ and is spent on gossip by that's got to be important to the human being and of course it is it's a it's the oil well it still mattifies the oil and the glue that fuse and smooth relationships between us right it's it's how we keep track of things with some other interesting web evidence that I collected was a ton of this around anthropologists have studied various groups of people around the world and there's various aboriginal groups in in Australia that by all reckoning don't have mathematics I mean II know the standard thing was they don't have any sort of mathematical capacity that was that was that was what you read in sort of earlier not books and things that they didn't have mathematics and they certainly didn't have anything we recognized in mathematics they didn't have geometry for example and they didn't have trigonometry because they didn't sort of look at space the same way what they did have was an incredibly rich vocabulary for talking about human relationships and they negotiated not in terms of geography but in terms of relationships to other people and then you can you can you can extrapolate from that to where they live themselves so it was all ended this very rich grammar of human relationships and if you look at that grammar as a mathematician that looks awfully like algebra these were fixed relationships and they had also the grammatical construct to them they could vary they could go down change of relatives and identify people so they had mathematics it just wasn't devoted to the physical world it was devoted to the social world they had a social mathematics and so you know the the gossip part is just that the key to mathematics is really the relational stuffy it was the complexity of sumerian society it was it was crucial because me and money only makes sense if you have commitments between people those little objects only have value because people endow them with value those little squiggles on the on the outside of the clay tablets they had value because people endowed them with value so it was all depending on the fact that society was complex and in many ways it was an abstraction from a society so this is I think what we're doing and indeed this is definitely the case if you look at in fact I forget who did this someone did this once they went to a math meeting and they filmed mathematicians talking to each other about mathematics and then it took out the sound and they showed it to people and you could not distinguish that from people just gossiping about other people getting excited getting you know arguing about things there's no way you could tell without the soundtrack that they went gossiping about people they were gossiping about mathematics with all the emotions that we have when we gossip you know the scurrilous clancy's and the chuckles and all of that kind of thing that's what we do we just tear that capacity and we know it sometimes mean we have trouble with our spouses anxiously it is the case that mathematicians when they're in the business when they're in the groove of doing mathematics they're devoting all those mental capacities to this mathematical world and the real world can sometimes get shortchanged and anyone who's been spend any time with a mathematician or even where's married to one will know that that can sometimes have fallout but my excuse is that's the price we pay for doing mathematics you know there's a limited capacity in that brain and if we devote it to one world we take it away from the other world but it's really the same capacities it's just pulled away to this abstract world we can be eight and that was the story I tell in there I've actually thought about this since then and I've written a couple of papers for conferences one corollary from this is that don't up some more water before it does explain at least to my satisfaction why it is that when we do mathematics we feel as though we're dealing with a real world it's not as platonism it's that this mathematician does not talk about inventing things they talk about discovering things does this sense that we have this overwhelmingly powerful sense that in mathematics we are discovering this world that was there long before people came along long before the universe was their eternal world and we are making discoveries about it lots of metafiles about going into a dark room and stumbling around and figuring out where the furniture is and then putting a small light on then putting more lights that's the one of the metaphors for doing mathematical research mathematicians always talk about discovery searching for the answer then all physical metaphors lick off or none years in their book go into detail in the as metaphors because that's us that's dust that's lay coughs thing he finds metaphors in everything to greater and less degree of convincing but that's what he does oh gosh not a good day okay so he saw you okay that's probably enough about that one but now let me let me finish that one because this feeling of concreteness why do we feel as the other thing is real when we're doing it well because we're using mental capacities that we evolved to deal with the real world so those very capacities are rooted in our evolutionary history to real things and so when we use them they're going to carry that all of that stuff with it so that sense of discovery I think is just a natural consequence of where those capacities came from so even those of us who don't think mathematics is platonic I got older and I think wiser I've come to think that mathematics is essentially a social construct a social psychological construct and then it tells us more about the human brain and about anything else it tells us how the brain encounters the world but that's be that as it may even though I think that's what's really going on the moment I start to do mathematics by things like discovery of real objects that are out there those numbers are just out there and waiting to be discovered pop it is waiting to be done with them but I think that's a consequence of the of the way it came about okay but if you want to sort of look for more deed there was I was gonna think of another point I was going to make but it's gone my head is beginning to get thicker as we as the evening goes on any questions about this part of the evening that start down here as I was reading that this next between or the social intelligence and rational especially probabilities and so there's an example say marriage has two attributes and there's a hard-worker and please passionately move writers man this is like what are the odds are that she's a bank teller oh yeah what about it you think oh I had a feminist and you know didn't say what people would say that there's more I don't want to get automatically yeah but the counter to that which I was interesting I've always heard that for Phoenix all these people are just service screwed up know that no I don't know right it's like why why did the person who was giving this information about marry choose to include that extra detail yeah yeah so that's sort of like up inside of you any thoughts about yeah I thought that was one of the give that was one of the questions in my online course that just finished about those kinds of reasoning because I actually that course was structured on literally in that fabric cost took them with their everyday intuitions about language and insured them how you could reinterpret it in terms of mathematics and get very different answers there's it's been observed actually this came up on an earlier lecture I mentioned ahead to me on the NPR that this guy from NASA who was looking at how people reason are having complex situations at the level of societies we screw up badly because we are conditioned to think in these simple metaphors about stereotypes and so forth and it just doesn't work in terms of the real world I mean we literally have to rely upon the probability in the mathematics even though we don't believe the answers and there's the Monty Hall problem for those who know about it is a notorious example of a probability question that many people get wrong and some people can never see that they're wrong even though it's a sort of elementary argument to show that they're wrong it's it's just very problematic for the brain to wrap itself around it really somewhere that neuroscientists think that function the the cortex really was to keep track of complex social relationships among primates to be still an evolutionary advantage so that females could align in some more powerful males yep and then that would entail some abstract oh yeah that's just part of my part of my data this is this all of that stuff fed into this I mean it if you can judge in a scientific explanation by how well the evidence fits then I've got a winner but it's still speculative I mean they did any evolutionary story is just rational reconstruction so there have been something that there are some you can perform you can make some influences from my thesis and there was a cognitive psychologist in in Canada whose name I'm now forgetting who did some research with young children and and the results she got with the ones that my thesis predicts so there's been some complimentary evidence but you know there's really no such thing as confirmatory evidence in science it's just there's evidence and you can see ok that's consistent oh yeah there's a problem about filming in that corner ok so that's a piece of clip that we cut you just forgot about that sorry yeah yeah yeah the most recent it's a big between 75,000 years maybe 250,000 years I mean it's sometime in their respect to go buy artifacts and you've got to infer that people had language to do things it was really tricky there's been a lot written about interest when I started the working on this which was called it 25 years ago when I started thinking about this it was hard to find a linguist who would talk about the evolution of language it was regarded as a not a respectable subject but it actually over the next over the last 15 years or so it's got to be very respectable lots of linguist now talk about this has been books and papers and articles written about it so all of a sudden linguist began to talk about the evolution I mean you can it's understandable why it's sort of sometime regarded is not respectable because it it is speculative I mean you've got this you've got this anthropological evidence you've got other stuff and you've got to infer that people are language and we're doing things you say well how else could they have done things so it's there's no artifacts they didn't have books or things it's it's very I mean you get scratchings on things so it has to be rational reconstruction you have to tell a couple of convincing story and you know the timespan is huge as to as to when that might have come it's also tricky to explain it in terms of the brain structure so it's it it's clearly a phase transition in physics terms all of a sudden you've got the on one plane because you've got words lots of words and vocabulary in your doing rich things with them and suddenly you've got recursive grammar now there's no such thing as half a recursive grammar either you can rehearse and iterate something or you can't so but there has to have been a first brain or a set a set of fair's brains that had the ability to do this recursive structure and not because there's no halfway measures it can't possibly be gradual now physically its gradual maybe it could be gradual but you have to tell a story that says lots of small changes in the brain son at one point led to this phase transition and suddenly the brain delete left to a different level of capacity that actually is how technology advances but it seems that the brain that way too now I'm getting to the edge of my I mean this is not my domain I mean I read a lot of stuff to tell my story but at this point I'm really the limits of my my training is to what's going on complex don't go the ability to figure out how to get this thing to making is insist yeah all of the evidence is when people are various kinds of tools where they've actually done have view the certain so this point things yeah and they seem like a language center yeah and watching various creatures that use tools you know these these creatures that sort of figure out that you can put a stick and get the amps out of the thing that's the other creatures have tools as well so looking at tools is an interesting one you basically like one of the things I already know that the stone axe probably wasn't a linguistic thing because you just it's a sharp thing but if you think about tying that acts on a piece of wood and making an axe like that that's kind of more complicated that you've really got to think if I touch I mean if I tied this to that then I've got something that I mean that's that's mathematics heaven say that satsang guys they didn't have the three equations but they're thinking mathematically so when tools got complex but there's then you into the question of when did this happen bad Spears was another example when did they first put barbs on Spears that takes but that's sort of language in the sense of complex thinking and repeat up yeah so yeah there's a part there's a moment when you've got recursion but it's got to be just a sort of phase transition thing that small change you just suddenly slow you up but we do see that with if you look at these technology curves they're sort of SK and then there's a jump then is an S cam there's a jump I don't know how many people have seen draw that picture in the last year and Silicon Valley because they speculate I mean basically that's a pitch to the VCS you know I'm about to make another leap so so give us 20 million Billups it takes it took venture capital to do those leaps yeah able to access the part of their brain that allows them to grasp the abstraction there may be some neuroscience that sort of gives some evidence but I'm not aware of it not be I would be very surprised if there's really convincing stuff I mean you can tell what I think of plausible stories you know there's trials I can see the most important thing to been able to harness that capacity is wanting to the only distinction I'm aware of between those of us have become mathematicians and those of it that don't is at some stage in our history we decided we wanted to and we devoted a lot of time to it you know this certainly mathematics like everything else is subject to this 10,000 hours business you know anything you do for 10,000 hours you tend to get good at and mathematics is no different if you look at the notebooks of very famous mathematicians or ordinary mathematicians from when they were children growing upwards they leave behind if they leave anything copious notebooks full of arithmetic and algebra and doing mathematics they've got to be good by doing hundreds and hundreds and hundreds of examples so I think it's just a case of something tends you want you are to have a relative or you see someone who's like a mathematician and it just looks cool yet for some some reason you decide you want to do it and then you just devote a lot of time to it if I had to put my money onto it and then you know there's some way of being able to cash out the bet that's where I'll put my money on it's just wanting to otherwise otherwise it seems like a complex story about different brains mathematics brains yeah they're both about formals yeah the thought about formal patterns there yeah they're both about formal patterns that resonate with the brain music is patterns in time it's fundamentally muted these are patterning time but mathematics is patterns in space and relationships but it's formal patterns I don't know if you came in when I will studied the pie mover at the beginning before everyone came in I had the playing pie where you take pie and you take the numerals you convert them into notes and at first it's really discordant when you start piling them on and putting the harmonies and things then eventually it turns out to be music but by then it's really hard to tell where pie is gone because it's got into this mess yeah yeah yeah there's a whole branch of mathematics called ethno-mathematics which looks at mathematics at artists societies and there were whole books written about the innate mathematics in weavings of these different societies in the Alhambra tilings everywhere many societies have actually discovered through their art fundamental theorems of geometry about different translation patterns and so forth in the seventh or only 17 distinct types of wallpaper pattern well wallpaper pattern lists people dual design wallpaper I've actually discovered them all to be wallpaper it has to be something that you can repeat over and over again so you can put one wall next to the other and the pattern repeats well there's only 17 ways of putting a pattern together you know forget the squiggles and things there were 17 different structures that allow you to create wallpaper and wallpaper desire mathematicians have proved that years ago and wallpaper designers implicitly discovered that because they've come up with all seventeen and no more because there aren't any more and lots of things in in the natural world and in different societies that are artistic or just animal creatures that do things they turn out to be mathematical it's just not done as mathematics it's it's looking for patterns and making use of patterns still can't remember the title of that book the way I do all this stuff for the album's like this is really embarrassing Laurie in the break I'm going to look and see what what the title of one of it's one of our best-selling books actually it still sells very well and I can't remember the title is animation work what mathematics is a very slick object and the relationship center so I think computer programming or this programming language this follows similar approach and directing the the they apply the mathematical principles also yeah now I thought that yep I think I got a high when you mention was the soap opera a book came into my mind is called bleeding at the keyboard or anybody is interested here and they use the soap opera concept to actually explain explain Java programming language yeah it's a book written by a professor at Indiana University yeah when I was like I did most of my heavy-duty programming when I was a teenager and I bought her actually a text editor for a firm machine a machine with nothing with a complete 8k of memory so I had to be very careful with the memory but I literally wrote a text editor initials in machine language and what I was doing and I feeling was I was literally writing a drama because I was saying this text is coming in you know this you know they're going to show this event and I was saying how you do stuff it was and I was in that world for weeks literally I was in that world for weeks I mean I don't have the capacity for doing that anymore I got older but it was very much that feeling that feeling of been in the flow actually working on something you're telling a story and by the way that thing about talking mathematicians talking you can do that with other professionals you just film them that they're compressional meetings and cut the sound off you've no idea whether they're talking about some scurrilous relationship of a couple of people that you in your university or what are they talking about the profession if they're professionals all of their gestures and their facial reactions and everything its engagement its passion it's lots of passion and you might just as well be talking about people they're using the social brain because it is a social brain you know all of the evidence we've got suggests that that brain got complex and big to deal with social structures that was the that was the killer app and so if you're doing mathematics the more you can harness that the better that's why I matter machines to make it look easy because they've just got to the stage where they can engage in the same way as a person will engage with this favorite soap opera and then it gets to be easy in a sense easy now the difficult thing is you're trying to draw conclusions in the soap opera the conclusion comes very quickly usually at the beginning of the next episode in mathematics you're left on your own to come up with a conclusion interestingly enough one project I did not long ago and I'll finish this will be the last one before the break was for the Defense Department and they were interested in how they could how much information can they get from surveillance video because it needs video that's it then and the intelligence analyst has the job this is non classified so I can tell you this intelligence has the job of looking at lots of video and seeing how much can you how can you recreate a story out of these distinct pieces of video from from surveillance cameras in different parts of the world from satellites you get on this video and you have to be able to tell a story which in a sense means it's that their task is like the movie maker it was trying to put together the sequences so we actually took as our test case the movie memento which has two time sequences folding backwards and the question was that was a movie with an underlying story and at the last episode the last scene you do find out what happened the question we had was can we develop a system a protocol well as you went through the movie and watched every every sequence could you predict the end the answer was no we couldn't do it it's a surprising end and even if you can do what you need to know have a lot of meta knowledge about things but it was a fun project and then I got paid by the military to watch my mentor lots and lots of times it loses its attraction after a while but not as muddly as you think it's still one of our favorite movies okay we better take a break now I won't give my voice a rest and ten minutes time well we'll hit the Millennium problems it is it the math instinct it was the math instinct yeah I couldn't it seemed so hacked but I'm yeah it's a math instinct it's got a picture of a dog on the front yeah the math instinct is the book whose title I couldn't remember it's one of our best-selling books and it's mathematics of warm and fuzzy things like cats and dogs and that kind of thing okay I wanted to finish with the Millennium problems for a couple of reasons one is it's still a by design it's meant to sort of encapsulate the frontiers of mathematics today and it's fairly topical it's it's a new topic these are problems that only recently have been thrust into the spotlight as major problems tool to attack so for a course like this one that's meant to give an overview of mathematics a lot of which has been about applied mathematics using mathematics in the world applying it in different ways thinking of mathematics and relationship with how the brain works this is all about essentially mathematics for its own sake some of its got a plaque some of it is applied some musical applications but this is about the pursuit of mathematics purely for its own sake the stuff that I did for 20 years my early part of my career just doing mathematics for its own sake and there's a 7 million dollars of prizes available technically now 6 million dollars because one of them problems has since been solved but we'll come to that but there were 7 million dollars was set up as a prize and you get 7 million dollars by taking 7 problems this is the part of the talk where you'll understand the math from here on you won't understand it and quite frankly I have very limited understanding of limited understanding of some of it and zero understanding of other parts of it these are 7 problems that are undoubtedly hard and in many cases really really complicated and here's what happened back in May 2000 in France Paris France there was a conference and at the conference Sir Michael a tier of UK and John tier to the USA made the following announcement a prize of 1 million dollars will be awarded to the person or persons who first solves any one of seven of the most difficult to open problems of mathematics these problems will henceforth be known as the Millennium problems so this was a millennium you know if you remember 2000 everything was millennium notwithstanding the fact that it wasn't clear whether it was watney over the next year I mean useless you know arguably it should have been the next year but anyway that's another issue and this was the conference but the ecology Francis was the art therefore the fly for the thing there was the usual things there was keynote speeches and so forth and this was one of the unusual things about this at the time I was editing a mathematical journal called focus which was the monthly newsletter of the mathematical Association of America and it was a mathematics magazine I never got press releases I mean back then I mean I just didn't but I got one for this week's in advance I got an embargoed press release telling me about this conference that already was kind of unusual they'd hired a sort of PR firm to sort of hide this thing because this was meant to be a big public relations exercise in favor of pure mathematics and what had happened was London clay was a Harvard graduate he went to university wants it like in mathematics wanting to study mathematics found it in his words too hard but he went off and became a sort of investment banker and whatever made millions of dollars at least many millions anyway so he actually made a lot of money in in in the markets of stock markets or derivatives or whatever it was it made a ton of money but always remained fond of mathematics and realized that mathematic recognizing that mathematics was getting a bad rap people just went kids were no longer getting interested in universities weren't getting seen of math majors and it just wasn't getting in it wasn't sexy enough for the for the modern age and so he recognized the power of money and says if I puts if we have seven problems with a million dollar prize the press will take note and he was absolutely right because this was a story all around the world he actually set up a foundation or an institute at Cambridge called a clear mathematics Institute endowed it with 90 million dollars a friend of mine Arthur Jeff was the founding director and he established a committee of very famous mathematicians world famous mathematicians pinnacle mathematicians - Andrew Wiles was the guy who sold firmers last year by the way their job was to come up with a list of seven of the most difficult challenging and important on solve problems of mathematics problems which were unlikely to be solved easily and within a few years these were meant to be really gold ring problems that represented the future of mathematics slightly beyond reach so there came up and there spent some time coming up with a list of seven problems and they produced a list of seven problems it was some of the problems everyone agreed should have been on the list there was some other problems that were sort of idiosyncratic but what was clear was all seven problems were justifiably on the list people might have chosen different ones to some of them but they were all clearly major problems here's some of the committee that put them together there's the Clay's at the front interesting that the he's the lawyer this this is a guy with a lot of money and very unusual to get a group of mathematicians posing with a lawyer but there was there were a lot of issues about seven million dollars how do you decide when to award it who would you award it to who certifies whether it's correct and so there was there was a lot of legal wrangling to get this thing going and there he was on the on the photograph and then I got after Jeff on the back Andrew Wiles ed Witten and Alan Kahn's at the the left okay so they put together this London clay put together this institute and the prize and the inspiration for the prize was what had happened 100 years earlier in the previous millennium 1900 because at the meeting of the the world mathematicians the international mathematics meetings that didn't happen every actually they happened every four years they had to move it to be near 1900 so twice professional mathematicians have agreed that the millennium is the year with two zeros so it enough of this nonsense about what which is the Millennium Year mathematicians have spoken it is the year that has two zeros which is just as well because that's with all the journalists the people who don't like it are the math teachers because they have to teach arithmetic to kids and you have to say when you get to 2,000 but that's another issue okay what he did and this was very unusual at a mass meeting if you're giving a plenary you describe the greatest things that have been discovered recently or you maybe have a historical survey he looked forward this was a futuristic talk he was a very respect to a mathematician very famous so he could get away with it and he got up and he said listed problems that he said were really problems that mathematicians should work on and they will mark the future of the discipline if by looking at these problems this is the way mathematics is going to go forward now it turned out that some of them were not well formulated in one way or another all but one of them were disposed of some were actually solved so relatively recently within the last 30 or 40 years but over the years most of them were eventually solved or else it was realized that there wasn't really a neat solution okay so they were resolved maybe I should say resolved or solved all but one and of that and the one that remained was a natural candidate to be one of the seven and indeed it is one of the new ones okay so I will come to that one in a minute but Hilbert had done this and so it was a natural thing to do the same thing a little bit later a hundred years later and that's what the claims that you did and this is London Clare's own explanation of why he put this thing together let's read what he says curiosity is part of human nature unfortunately the established religions no longer no longer provide the answers that are satisfactory and that translates into a need of certainty and truth and that is what makes mathematics work makes people commit their lives to it it's the desire for truth and the response to the beauty and elegance of mathematics that drives mathematicians so he was still I mean II had mathematicians among his friends including Arthur Jeff and he who became as I say the first president of the clear Institute and he Healy expressed that sort of love for mathematics in this statement and here they are if you order them in the right way you get a nice Christmas tree and this was the way they were ordered in within the poster at the conference and so you get this nice diagram thing and I'll go through them one by one the first one is from computer science the second one is from let's just say algebraic varieties or something I mean I'll come to what that's about that the third one is about well-taken it's about sort of surfaces in a general sense the Riemann hypothesis is the one that was on Hilbert's list from 100 years earlier if you ask almost any mathematician at random what's the most important unsolved problem in pure mathematicians we're all conditioned to say the riemann problem or the Riemann hypothesis for various reasons about its level of fundamentality how long it's been around how it seems to resist a solution I think we all agree pretty well that that is the problem that will be we've always regarded that as more important in Fermat's Last Theorem for example this won't always rank much higher than that yang-mills existence of mass cap that's one from physics which is after Jeff subject theoretical physics I mean it's basically why do things have mass it turns out that most of the basic fundamental questions in physics can't be answered by physicists you know why do things I've masks I mean this you could have to do some mathematics as healer well it's part of the picture it's part of the picture but it still doesn't really answer the why it sort of gives you the mechanism okay navier-stokes equation SAS about fluids I mean you have to solve those if you want to design how performance yachts or airplanes and things like that and then there's another one from number theory to do with numbers they've even hypotheses number theory it's about crime numbers essentially and this is a sort of a generalized question sort of related to the Riemann hypothesis at least in the branch of mathematics ok so let me go through and just see what these are because these are a survey of what makes mathematicians excited today and let me start with this one this goes back to 1859 bernhard riemann this is the only one that was on Hilbert's list it hasn't been solved everyone's favorite all solved problem in mathematics and part of the reason is it comes from very simple considerations about prime numbers the atoms you know the physicists of atoms chemists sub molecules and mathematicians have prime numbers they're the building blocks out of which you can ultimately build all numbers and the way you put the prime numbers together to form a other whole numbers tells you a lot about their property the so-called plam decomposition of a number is like finding the atomic the structure of an atom observe a certain element or finding the atomic molecular structure of something if you understand a number in terms of its primes you can answer a lot of questions about it okay now the definition is a very simple one so we're at the level of school mathematics now our numbers prior if it's only divisors are one and n so if you're going to look for the primes less than twenty here they are except for the first one no more odd because everything else will be divisible by two and on it and so on on our dealers will be prime 3 5 7 the first odd number that's not is 9 because it's divisible by 3 11 13 you have to skip 15 because it's divisible by 3 and 5 then 17 and 19 you'd have to skip 21 because it's divisible by 3 and 7 and then you get to be 23 and so forth so those are the prime less than 20 and the ones that are not prime are these there were more of them that's evident so Prime's look as though they're the sort of the the smaller collection more numbers seem to be not primes all right number 4 beador 10 2 3 5 and 7 there were 25 below 100 so we increased by going up a factor of 10 but we haven't gone up a factor of 10 in the found you didn't go to 40 just went to 25 if we go up to a thousand we go up 268 they've got to get bigger but they're slowing down the number of primes nevertheless even though the primes seem to thin out and they do if you go farther out along the numbers the primes get fewer and fewer and fewer they get hard to find and yet in 350 ECE Euclid proved by a rather elegant mathematical argument that there were infinitely many of them so we know that there are infinitely many of them if you want to see the proof log on to my math course actually there's a YouTube video a rather low grade low fidelity and low resolution YouTube video of me proving this results on the course so if you just typed or if you just Google if you went into the YouTube and your Google Keith Devlin primes you can watch the little video it's a little five-minute video it was just a sort of demo for the course but it's kind of low res so it's hard to make it out but it's an extant elegant proof okay so let's do the following the primes go on forever but they're thin out so you can ask yourself but what weight do they thin out do they thin out in just some sort of haphazard way or do you thin out in a somewhat orderly way well here's one way you could look at it you could say let's go out on the numbers go up to 10,000 and count how many Prime's are and then we can talk about the density of the primes how many Prime's are there in 10,000 so if you divide the number of primes up to some point by the number of numbers all together then you've got the density how what's the frequency if you like of the of the primes so if we look at these various numbers we can go up to 10 100 well we know that 10 therefore less than that so the density here is point 4 we knew that there were 25 Prime's less than 100 so 25 divided by 100 is 0.25 there were 168 pounds less than a thousand so the density is 168 divided by a thousand and if you keep iterating those are the numbers so now we've got some numbers that show how the primes are thinning out the question is can we discern any pattern in the numbers now you or I could write a little more many of us can write a little computer program I'll just do it in a spreadsheet with a prime generator and we could we could currently we could generate these four large numbers and we get a long sequence of numbers chances are high that there's none of us in this room including myself that could eyeball that in a sequence of numbers and say I've got a good idea what that sequence is I don't think I could do it even if I could have done it with spite when I was a kid because teacher told me what it was okay so the question is is there a pattern the answer is yes but it's not an easy one to spot unless you're one of the world's best mathematicians like carl friedrich gauss who at a very young age in his early 20s i maybe he was still a teenager i forget he said that that sequence is like the reciprocal of the natural logarithm now if you've spent a lot of your childhood this goes back to the question about how people who become mathematicians do you spend a lot of your childhood as he did working out the values of the logarithm function and looking at the reciprocals you become really familiar with that it's like a character in your favorite soap opera so he can look at that secret of numbers and see it I've seen that before I seen versus the natural logarithm so he had a very early age conjectured that the only way I could have done that was by doing a lot of calculations and becoming familiar with it and that turned out to be correct he couldn't prove it he just recognized the sequence and said I know what that is I think I know what that is it wasn't until sometime later over a hundred years later that it was finally proved by I think they were Belgians jacques hadamard and charles de laval apasa the prime number theorem at the least one of them was Belgian maybe there was a version of the Frenchman but they spoke French and you have French names okay when they finally prove this it's now known as the prime number theorem it's quite a difficult seminal proof in mathematics okay so we do know that there's an order to the way the prime numbers turn out it's the inverse of the natural logarithm function but I'm going to be mentioning lots of things in this particular lecture that you know you don't you probably got no what they are it doesn't matter there's a function that's of interest to the mathematicians and it's got definitions and in the second part of this lecture I'll be talking about things I don't understand either so which was kind of interesting cuz I actually got a book on this that's how I got to know about and when they millennium problems came out they approached me and said would you write but when they putted into things they approached a team of professional mathematician world famous mathematicians in the different disciplines and asked them to each to write a chapter of a book written by the expert on that domain so they got world experts on each of the seven problems to write chapters of the book and bring that book out it was many years before that finally came out but they also said we need a version of the book that journalists and high school students and people like that could read and they came to me and said will you write it for us and I said well I don't know what most these problems are so they said well we'd like you to write it will you do it so I went away and I started to do a bit of research into what some of these problems were well I knew the Riemann hypothesis and you P equals NP there were three of them I could have just sat down and written a chapter about and then there were another four and I thought oh dear this is not going to be easy eventually I decided to do it because it just seemed like a big challenge but it wasn't easy and I really had to rely upon other people to help me out on a lot of occasions okay so we've got the prime number theorem we know how the primes thin out the proof that the steps that got us to that point with the first proof depended upon some work that Riemann had done is this guy dreamin and he'd excuse me I think I'm about to sneeze nope sneeze isn't coming but let's tie some water maybe that doesn't work that sure that that density function the thing that you're trying to approximate you know what what function is it he found that that was closely related to the solutions of what's known as the Riemann zeta function he denoted it by the Greek letter Zeta so it's now known as Riemann zeta function and this is a function of complex numbers okay well you'll just bear with me these are just words this is a soap opera you've come into a so pop you don't know the backstory but you sit back and relax and you can still get something out of the soap opera each of these characters had an interesting past scoreless passes all sorts of things you don't know about them but they've got an interesting story to tell and you're just now auditing this soap opera for the first time so of course you don't know the background but it doesn't matter they're just mathematical things just like the characters in the film soap opera you just follow it on the surface these are characters in the mathematical soap opera that play their own roles so you've got this character called a density function which I've explained and then you've got this other character this function of complex numbers so it's some sort of mysterious character and he's sure that they're related and then Riemann knew that that function had that equation let's go back to it no can I do it yeah it was this equation this thing equals zero now Riemann you there comes up sneeze I hope you're not listening to my is my mic feeding is he oh yeah yeah because if I sneeze I could I could kill somebody oh yeah okay so you got an equation and Riemann you some of the solutions of this equation he knew he knew okay I got about 20 minutes to go I think I'll make it he knew that all of the negative even integers solve that equation if you plug those numbers into there into that function that Zita function you get zero coming out plug - - when you get zero plug - flow and you get zero plug those numbers in you get zero out he knew that and then he just pulls this rabbit out of a hat conjecture and says there's only two kinds of solutions there's either these negative integers and if that's fairly straightforward if you look at the if you look at the way that the zeta function is constructed yeah any first-year graduate student can see immediately what's going on well officially a graduate student of Stanford okay this however was bizarre he said there's only one of the collection of solutions and they're complex numbers of the form 1/2 plus the square root of -1 times a real number ready to get it from well when people at night after his death when historians look at these papers they discovered a bunch of calculations but not many out of a small number of calculations he made this guess about all the other solutions all of K which were completely many - all of these infinitely many ones who actually have their real part equal to 1/2 so that was the Riemann hypothesis quite an amazing leap of faith to think this is probably true one of the consequences of this because it deals with prime numbers is internet security the public key cryptography codes that keep all of our communications with our banks and so forth secure this relates to that you're in you're into internet security when you start talking about the Riemann hypothesis there are all sorts of algorithms to do with numbers that we know will work very efficiently on the assumption that this is true okay this conjecture it's been verified you can you can do computations about it it's been verified at least to the first 1.5 billion solutions we still don't know whether it's true or not it's a kind of problem for some problems if you can sort of verify it for billions you think it's probably true for various reasons mathematicians think this may well be true but not because there's a lot of evidence like that that evidence is really worth noting in in assessing this thing that's a question it's been around since Riemann's time 1859 most important result in mathematics unsolved problem so that's one of the Millennium problems if you go away tonight and you can prove it there's a million dollars waiting for you yeah I actually think I've probably made more money out of the millennium problems and most other people's because I wrote a book and it did not make a million dollars by any means but Sir it made more than most people did it's also been calculated that if I pick these pubs are so difficult that if someone eventually solves them if you calculate how long it took them in their lives to get to the point of been able to solve them and you divide a million dollars by the number of hours they've spent this is this is a slave labor you know this would be pennies I mean it's very er it's not a lot of money if you think of it in those terms okay if you want to make money you found a company called Instagram and get bought out by Facebook but that's another story so the question is this thing through you have no idea prove it's either to or false and the millennia you're get a million dollars okay look at the next one the know of your Stokes equations this is about fluid flows under vase we talked about this a little bit when wind up to the aircraft we didn't talk about this particular set of equations but when you've got fluids or gases or anything flowing you get any some particularly fluids you get one that was different they were sold these equations were formulated by Navy Aaron Stokes in the early 19th century you can solve them numerically and people do all the time and these are fundamental equations in fluid flow but there's no solution other than computational solutions there is no one sat down and come up with a formula that solves these things the millennion problems ask you to come up with a formula solution you know in the same way that a quadratic equation is solved by the formula x equals minus b plus or minus the square root of yadda-yadda-yadda come up with a nice little formula that solves these thing well these are differential equations but it's the question is can you come up with a formula that gives you the solution no one's managed to do it this one's modern this is computer science the only computer science problem that got in there and it was somewhat controversial though I mean anyone who knew about computer science said this is probably a really very difficult problem it's not clear why it's a difficult problem but it's been around and ended up the real problem was it hunt been around that long it's relatively recent problem so when this came out it was only less than 30 years old but this is another question that has potential impact on today's world of communication security it's about how fast can you solve problems on computers the details aren't that important again we're just getting an overview of the computers of the the domain but essentially what it's saying is well this is a distinction there are many problems where if someone gives you a potential solution you can quickly check whether it's correct you could just do a computation on the other hand a finding that solution in the first place is not obvious so there's a difference between checking a proposed solution and finding it the classic example of this is known as a travelling salesman problem can you find a route where the salesman needs to visit 50 cities and minimize the travel time or minimize the distance in an efficient way visiting one city H if someone produces a candidate route you can quickly see whether it's optimal you can just see how far I didn't see whether what's going on but finding the food of optimal note in the first place is there's no known really fast way of doing it there were where's it get very close to being the fastest the point is it's easy to check if something's optimal but it's very difficult to find it essentially the only way you can find it would be by enumerating all the routes and comparing them and that's computationally long so the reason this is important in security is a lot of internet security depends upon having computations where if you have a secret key you can very quickly unlock some information but if you had to find the key in the first place it could take you a long time now if that equation up there was true it would essentially mean there's not much distinction between those two if you can check its if you can check that the key works you could find it quickly as well whether that would cash out in practice is another issue but in terms of theoretical computer science this is a big big question and it does have implications and it's been around long enough and it's been attacked a lot by some of the smartest people around and this is a problem that's locked up by computer scientists and mathematicians it really is a mathematical problem it's not about practical computers it's about theoretical computing devices ok that's another one of them so we've got one about prime numbers we've got one about fluids we've got one about computations this was one that was sold recently this is your under one that's been solved and we're doing on time much like I said a little bit about this one okay here's the issue this came from when Korea was trying it on the question that the big question prong Korea was asking was can we understand the shape of the space we live in with a universe what's its geometry what is topology how is this space configured for example are we in the inside of a big sphere like thing we know some infinitely far out you can't get to the edges but is it essentially a sphere like the universe or is it the universe like a like an inner tube a torus and we are actually on the inside of this big infinite torus and so we don't know edges are there but in fact the ailee's tallest hedges and if we wandered around in this universe we might be going around this Taurus and how could you tell what the shape is because she's the distinction is from the outside you're on the inside all you know is a space all around you because see the boundaries are way out there the question is are those boundaries infinitely far aware that they are whatever other's boundaries essentially a sphere or are they essentially like a toes you don't know because you're never going to come to the boundaries well the question was hypothetically can a creature that's locked inside the space sorry the other day that the way they would describe it is that God is our God is on the outside and God looking down can see that you're inside a Taurus or that you inside a sphere can you as a creature inside with a brain figure out what the shape is without going outside and looking that was the question it's the limitations of physics is there a way we could in principle figure out the shape of the thing and he came up with a hypothesis and said here's the idea imagine it so this is it this is a thought experiment you said imagine you get a space rocket a space rocket you can leave this space rocket you go out on its long journey and on the as you go out you're going to splay out a string you're going to stir out a string this is a sort of Hansel and Gretel thing so you're going through and this string is being split up behind you and you're going to go off and round and round and after a long time you're going to come back and so you've now look put this loop of string through thought through the universe and now you start to sort of do a little loop when you turn it into Alice ooh when you start to pull the string through the LA Zoo so you're tightening the noose if you're on the inside of a sphere eventually the noose will just come right down but if you're on the inside of an inner tube and you've gone all the way around the inner tube you're not going to be able to do that because it's going to be stuck in that inner tube so in principle you could tell the shape of the universe by this process of extending strings and seeing if the suit is up well that was the hypothesis the question is does it work in a theoretical level and the Condor Poincare conjecture and they said this is actually you ask the same question in any number of dimensions the question essentially was does this really where does this distinguish space one thing I didn't tell you was those images I gave you were in a three-dimensional universe where the surface was two-dimensional but we're not wearing a three-dimensional universe where the boundaries would be high dimension so you're at least in four dimensions so the story I told you was fine when you're talking about inner tubes and things but does it work if we go one dimension up the conjecture was yes it does so if this was true it means we can in principle figure out what shape our universes from the inside it was a conjecture and then it was a millenium problem and now it's a theorem so Planck arrays idea turned out to be right on the ball we can in principle I mean not in practice because that's a completely infeasible activity but in practice in principle we can figure out the space they were solved by this person Grigori Perelman in 2002 only a couple of years after although it took several years to verify the proof and it is an interesting story about this he's a reclusive guy lives in st. Petersburg didn't really publish the result he just put it upon a website for new mathematical results very few of details it was just a sketch of a poof and many years elapsed before mathematicians agreed that it was correct many years and then when they did they said they had award him two prizes the the Fields Medal which is the biggest prize for mathematicians and they were going to order him a ward him the millennium prize at the million dollars he turned him but he said no to both of them he didn't accept either the Fields Medal all the million-dollar prize for reasons best known to himself he still lives in with his mother in a small flat in st. Petersburg it's a very interesting story in some ways in other words it's probably not an inch story he's just a guy that doesn't want to be part of the world of our world but in any case this was a brilliant piece of mathematics so one of the sevens been solved and we can we know now with cut with certainty that we can figure out the space of our universe from the inside this one's from physics it's this is what I read it on but much about it so far the problems are things I know something about this one I have no real understanding of this thing it goes back to the 50s if you can find a solution to this thing it will say that particles have a minimum mass but that's about as much as I know about this thing it's deep heavy modern theoretical physics this one a little bit more about it's related to that Riemann zeta-function it's a sort of generalization and it's to do with distribution of primes and all that kind of jazz but again there's no need to sort of go into this except to mention that these things called elliptic curves they also play a role in Internet security most of these problems turn out to have relevance to Internet security in one way or another okay and finally the Hodge conjecture apart which handles 0 and as far as I can tell there is not a single living mathematician who fully understands what this thing says how do I know because I talked to some of them and they all said it says something different it's an extremely high level general statement that this was a statement I think I sort of after talking to people a long time I said is it ok if I put this in my book and this sort of first of all said no and then after a long time they said well suppose it's sort of ok and I took that as authorization to put it in there I have no idea really what it means and most mathematicians don't either including people very close to this it sort of says that there's this general thing that links to pieces of 2 different areas of mathematics those are the same millennium problems when I wrote the book so that there they are and one of them has now been solved so we put a tick against that when I wrote the book and this is the book what was cut there were several things that were interesting about writing them first of all I had to write a book about something that I really didn't understand I knew three of them and the others increasingly became more problematic and moreover since I was writing for a general audience I was faced with the problem about what what what can I say about these things it's of interest to anybody I mean other than these are big problems these are Mount Everest problems these are the k2 problems the Mount Everest seasons the big challenges go to the moon go to Mars these are the big big big mathematical challenges how can I describe them to a general audience or even having a mathematically sophisticated general audience in the end what I decided to do was focus on their very non comprehensibility to the experts I mean it I thought that what was really interesting that mathematics had reached a stage where mathematicians were able to formulate questions that were not only difficult to solve but almost impossible for most mathematicians to understand so we've got this degree of complexity now where groups of mathematicians have reached this great advanced stage that other groups of mathematicians equally expert in their own fields really don't know what they're doing they can't understand what this other groups doing it's got that intricate and so this stone-age brain that 10,000 years ago started with inventing banking is now able to produce this mathematics which is beyond most people's comprehension beyond most experts comprehension and yet there's this great sense that if you solve any one of these problems it will change the course of human history these problems really are on the tops of mountains and if you dislodge something on the top of a snowy mountain you're likely to start an avalanche that's going to make a lot of damage down below and so these problems if any of them solved an avalanche is going to come down and it's not clear which way it's going to go on while it's going to affect but if I had to put money on anything and I don't have the money because I haven't won one of these prizes but if I had one of these prizes I'd put the money to try and increase it on the fact that any one of these problems is solved it will change the world in ways that we probably can't understand now either because the solution itself has a big impact or more likely because the way it's solved will require insights that failure to have ramifications down the line from everybody and I think it was really remarkable at least that this committee put together these problems that when you get in father and I got into I got inside some of the menu as I said some of them are managed to get inside by talking to experts some I just really couldn't try them except I mean I could get a general vague sense but what was clear was the these really are big problems not just within mathematics but solving them will really change the world in in fundamental ways and they managed to sort of pick problems like that so the fact that you don't understand much of what I've been saying is sort of irrelevant because I don't understand it either and that was the point that really was the point that we've got the stuff out there that will affect the lives of our children and grandchildren when it's solved or even if it's not solved just by people trying to solve them and yet it's at a level so far out that not only can not ordinary people understand it most professional mathematicians can't understand it either that was not the case a hundred years ago that's different that's such something very different in complexity well that's a good place to end I think that's the future and as I said I'm almost certain now that I'll give a course on infinity in related matters sometime next year I hope you've enjoyed the ride and look forward to seeing you again if anyone watch have questions we can answer questions from people will have to leave always happy to answer questions one of two things is either I'll answer the question while my voice will completely give out or oh I'll keel over but the first one is more likely well there will be something that the people who formulated it obviously had I mean Hodge obviously had some kind of insight that nobody else has quite grasped into there are other experts but now I think our filters which I mean even from my level you develop these intuitions even for my level have been remote no no you there's a good sniff tense that you develop you really can when these results get announced these do is one recently what has been a couple of recently one was something called the ABC conjecture that seemed rather wild but the sniff test from what we've heard seems okay the sniff test is pretty good you know experts can usually do that they're not always right but those seven have been subject to enough scrutiny that they're not stupid if it's stupid you could have it could have been overturned quite quickly yeah yeah but but you there's an aspect of that there's an aspect of truth of trust I'm saying the things I do largely because people I trust and respect tell me that you know the old idea that Euclid had the old idea that a proof convinces you that something's true well now know what convinces you it's true is person X head it form person Y who heard it from person Z that it's true and you trust all of that chain of people you know I believe Fermat's Last Theorem is true because people are known respects have read Andrew Wiles is proof and said it's true but yeah it's got it halogens I mean look what they're saying is true on the television every day they're saying this is true and you look at that you go yes yeah what but we are subject to some some ground truth in mathematics there is a truth detector down there like why is there mass no that was that's just an application of it yeah you've got basically all you've got to find a solution to these equations the one of the consequences in order to what what happened was there was this equations that these equations had to be solved the yang-mills equations can you come up with a solution in it's almost like navier-stokes in that you can solve these things computationally but can you solve them with a formula and in order to make it a solution that would have the question was what would classify as a solution to that thing it's such a big question in many senses that it wasn't clear what would classify as a solution and they had to decide whether to give a prize and so they said to be accepted as a solution it has to be able to tell you why particles have mass that was the acid test of whether it was an acceptable solution maybe the lawyer said you need to put more in there but it was to make it it was to give it this sense of because what happened to some of the Riemann questions are one the nineteen hundreds is it wasn't clear what a solution was going to be now to say that something's got a solution that might sound like a mout sound as odd as some sort of hard and fast way of deciding it but a solution is simply expressed in terms of formulas and expressions that already have meaning somewhere else now what they wanted to eliminate was the possibility that someone would come up with a solution that sort of gave names to things that weren't understood and the question is has it really solved it you know it's like the solution nice like someone on us on an exam paper who says I've solved this equation but I'm keeping the answer secret well you know the teacher would usually say al you tell me the answer are you docking any marks well for this one it was a case of if you've solved it you might not show me the actual answer but at least you show me how your answer gives the father particles of mass so it was a it was a reality check if you like yeah yeah I read recently the mathematicians this is a generals if mathematicians are deathly afraid of being wrong from in public under promote and then physicists don't mind so much and they tend to err on the side of yes absolutely absolutely there is yeah yeah I mean I've sort of I've sort of grown up our gas speculates about the stuff all the time but but I'm not trying to solve million-dollar problems it is part of the discipline it actually goes back to the question about whether things are true or that are they just BS like let the stuff on the news there is this arbitrary of truth there's this arbiter of truth these things called proofs to get something published in mathematics it has to be you have to prove it and it has to be peer-reviewed and the proof has to be iron cast and so it's part of the discipline that you don't talk about something until you know it's true you know riemann did something unusual he gave a lecture about speculations but mathema to a if you put some with a paper to a conference it's not going to be accepted if there's not proven results in there it's all about results that approved it really in a sense it's not mathematics till it's been proved he kept use for some years he kept a secret because he had to it's a culture is the culture of mathematics if Andrew Wiles had said I'm working on fair most people people would have gone oh said poor Andrew I mean I know he's got married he's got kids what a shame you know what a waste and obvious thing I mean his life would have been made miserable people would have started making fun of him now maybe you know he had a good reputation so maybe not but I can't understand I'm not I'm not his culpable of mathematicians by any means but I can certainly understand why he would he would do that you keep it a secret then other mathematicians I've known have been working on big problems they don't tell you unless you get really close to them physics yeah you matter you can make connection in mathematics but no one gives you that much credence for doing it it's not that big a deal there now and now to be made between see Einstein it's not he looked at the world differently we have looked at gravity differently so improving it or the sort of the equations were sort of discipline it's just the way he viewed phenomena that was actually the important work and that actually solved it like for some these things maybe it's a reef rate it's not the solution it's the choosing of the viewpoint that would actually lead to the insight that will lead to formalize yeah no it's it certainly comes down a lot to how the wit to how you look at something you've got to look at it in the right way many cases the key to solving a problem is to just look at it in the right way and if you get trapped you just go down the tunnel you get you get trapped on the tool yeah you've got off some sort of annealing process where you just jump out of that trough and try again there is a whole bunch of heuristics for solving mathematical for doing mathematics but you know sometimes it takes someone who's prepared to develop their base entually their whole career to one problem like Andrew Wiles did you know there was good reason why he thought he could solve it but to sustain that for seven months or seven years that's kind of unusual and I don't know how long Pearlman was working on the Poincare conjecture well it was a long time and nobody knew that until he did it yeah you don't go around telling people you're working on these big problems I don't think I've ever heard anyone saying they're working on one of these millennium problems you just wouldn't do it you just it's just not part of the discipline your respect among your community would be just shot if you said you were working on that so that means you're going to work on it in secret and if you're lucky it's a problem because some of these problems that means you're not going to be able to do global collaboration a one thing what interesting phenomenon that arose was John Frey who owns Fry's Electronics lives vaccine martinis and has his office in Palo Alto and things he was a math graduate at Santa Clara and a few years ago he established something called the American Institute of mathematics I'm on their advisory board and a good friend of mine is the executive director and they're based actually next to Phi's just in Palo Alto yeah yeah they're building a complete life-size replica of the Alhambra down in Morgan Hill and that's going to be the end that's the American Institute of mathematics when it comes out and so that's but he set up this this Institute and one of its sort of missions or like we thought it wasn't totally stated was to apply a sort of a Manhattan Project approach to the Riemann hypothesis he said why don't we just try for once to get all the world's experts together and bring them together regularly to Palo Alto and eventually to Morgan Hill in the hope that that kind of Manhattan Project assault will eventually to a solution of the Riemann hypothesis it was an interesting idea I still an interesting idea it goes against the grain of mathematicians and the culture and I worry that when they come together they don't they keep their had accounts hidden it's tricky because the person who gets the credit for proving a theorem is the one who sort of puts the period at the end you don't want to be that close so when Andrew Wiles announce his proof of Fermat's Last Theorem and there was a mistake found he really was in panic mode because he he wanted to be the one that found that solution and he refused to talk to anybody and show them his proof except his former student and he worked with his former student and there were two papers published there was Wildish paper and then there was a pair another paper with his former student that established some of the results but he wasn't going to show people what he had in case they could finish the proof off it's part of the culture of mathematics the culture of all these disciplines is important you know it's not political so much but it is cultural and it's powerful so that the yeah yeah there was this huge debate about who did it and it was not not as far as well not so much between the two individuals although Newton apparently was very cantankerous but their supporters you know I mean first of all they were English and German and the English and Germans have always had this love-hate relationship and yeah that's good so did he goes on the line there's Koreas and that kind of thing it's there's the usual petty decision for rivalries you get in a dealer discipline but the secrecy that they're not the secret so much as the unwillingness to make conjectures that really is a part of it well if it's a stuff if it's a substantial conjecture people will make it like the Riemann conjecture or smell like you make those kind of things if they're if they're bold and backed up by some data but just sort of making speculations about this might be proved or that might be proved you just don't see it it really isn't like physics in that respect yeah interesting that the one problem of all these money problems are solved this all by somebody who didn't want it it was kind of interesting it was yeah I mean you can speculate all sorts of things about the kind of mind it takes to prove that kind of thing well I think it's just you know people are different and that person happen to be the one that sold under wild salt Fermat's Last Theorem and he's a nice friendly personable guy who gave a public lecture here at Stanford and went on TV they made a nova documentary about him he's a very shy private individual but you know he recognized he'd done something big and just accepted the accolades and then went away quietly did his own thing though other people who just accept stardom really they do there was a guy it got a Fields Medal at Stanford in the 60s called Paul Cohen he died a couple of years ago the moment he got his field mate got he saw he saw the hilbert problem and when he solved it he just left Stanford for that and said I'm going to take a break and he went on a world tour and gave lectures and became a superstar that was just the way he reacted I didn't resign from Stanford I mean and he didn't didn't matter anyway because they were going to have him back he'd solved the Hilbert problem but he just became a celebrity mathematician people just react differently I don't think you can draw inferences we're all nerds in the business yeah yeah well yeah but they don't live in Palo Alto yeah no yeah yeah yeah well we don't know you got a pocket protector you just have little phones on your hip how many people are just making sure we don't see the phones on it yeah I'll slide really that was that was a nice badge you've been and being a techie yeah okay thanks everyone see you see you next year maybe for more please visit us at stanford.edu

Channel: Stanford

Views: 97,353

Rating: 4.8694234 out of 5

Keywords: mathematics, cognition, brain, society, science, evolution, numbers, humanity, technology, physics, animal biology

Id: NnVubBrATIU

Length: 114min 24sec (6864 seconds)

Published: Tue Dec 11 2012