4th HLF - Laureate Lectures: Sir Michael Atiyah

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4th HLF - Laureate Lectures: Sir Michael Atiyah
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Sir Michael Atiyah: “The Soluble and the Insoluble” What do we mean by a solution to a problem? This is both a philosophical question, and a practical one, which depends on what one is trying to achieve and the means, time and money available. The explosion in computer technology keeps changing the goal posts. I will reflect on these issues, primarily from the viewpoint of an elderly mathematician. The opinions expressed in this video do not necessarily reflect the views of the Heidelberg Laureate Forum Foundation or any other person or associated institution involved in the making and distribution of the video.
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okay and now for the second half of the doubleheader we have a legend of mathematics Sir Michael Atia he is winner of both the Fields Medal 50 years ago which is amazing and the Abba prize which he could not have won 50 years ago because it hasn't existed for very long in 2004 and in both of these awards they cited his revolutionary work with his singer on bringing together topology geometry and analysis in the proof of the Atia singer index theorem which was central in unifying mathematics and theoretical physics I came of age in a world that was permeated by the Atia singer index there and my PhD is in mathematical physics and if you know I I met Sir Michael either in 82 or 83 I don't even know if you were a sir at that point I think he became a sir right around that point but we when I was a postdoc at Harvard we had weekly seminars on the Atia finger index theorem and its ramifications in physics which you know was a huge area of research and it's still impacting physics tremendously Michael has won many other awards notable among the in 1968 he won the Royal medal of the Royal Society in 1988 he won the Copley medal of the Royal Society in 1983 he was made a knight bachelor in 1992 he won the order of merit something which is not listed in your programs but we should be there is that he has been named and I'm sorry my French accent is not very good by the gound office you do Li Zhong oh no which is a very very high honor in France he has also done tremendous service for the community he was president of the Royal Society he was master of Trinity College in Cambridge which is a you know that just the absolute top of mathematics and other fields and physics and other fields and he was also a central and establishing the Newton Institute which many of you have probably been to conferences at the Newton Institute and many of the students here I'm sure during your careers will be impacted by workshops at the Newton Institute so let us welcome Sir Michael Atia
my nose at my house sorry I know disappeared cool got to get them I should just say I totally enjoyed the previous lecture I came in knowing practically no computer silence as you'll see and by the end I answered it why because those hidden structure what was hidden structure actually called mathematics yeah and you'll find this institution that perhaps as I go on is to become what evidence now right so I've chosen as my title by the way I should say another thing when you were here yesterday so you explain to you how new this hall was some of you may remember the date anyway I'm as new as the hall exactly the exact date that was here I was born so it's very very good timing now I've taken with my title famous topic really the soluble in the insoluble I think if you go through the history of mathematics and everything else understanding what is a soluble problem wasn't in sir is it the heart or everything that's done it's fundamental in everything you think about in your own field and you'll you'll see that so I've told title that I think is covered every everything so old and young good scientists mathematicians or whatever you are now I'm gonna be your tour guide I'm gonna take you a rapid tour of the last century let's say and in that with a tour guide you know don't believe everything to a guide says it's all guys trying to show off he serves you names out telling us his stories so it may be true a lot of them run embellished you know you had a bit of tidbits making more interesting for the Torah so I will poly do because you're the same don't believe everything I say okay warning red light so I any of your guys and listen to my guide but you can ignore what I say as well if you like so now I'm gonna I'm supposed to press a few buttons and see what my chance my first slide is Heidelberg in 1904 I don't think many of you were here this is photograph of the actual brochure given to those who attended Heidelberg home exactly so photography in the first picture is not perfect but this was 1904 photography had me going that long first but not bad pictures of Heidelberg you know four of the printing of the day underneath my German is not very good you quality better than that but it's tells you that david hilbert from goodman was there and talked about logic and the Rizzoli now you probably know the famous talk that Hilbert gave at International Congress back seen Paris in 1900 that were the one where he laid out a list of problems famous problems you solved one a couple problems he became famous amongst those well questions like provide foundations mathematics and for physics small questions and there's a certain sort of course well this was Heidelberg in 1904 and Hilbert spoke then very precisely he followed up so he's talking Paris by deciding to focus on the easiest part we knew the foundations of logic and arithmetic and they went hand in hand you might explain what the logical foundations were and your pride in the same time to see everybody and that was the simplest part of the program which rumbled runs the next century going from more and more complicated things this was a beaming of a whole century of development of mathematical logic and it's theoretical foundations Hilbert well of course the big man without question the leading mathematician of his time recognized as such by everybody now the beginning I think you want to think of he was talking about the foundations of arithmetic he was concerned with the integers you know you can't get more fundamental of that integers and invidious have two properties they are discrete and if you look at only finance at a time your finiteness infant integers are actually infinite of course we know that you can go on but any given moment you look at the finite set so you start with finite things and it's frequencies and you build your structures step by step from that from a finite one way to the infinite discrete by the way eventually getting down to the continuum that is history of the development of mathematics and logic and computer science think of it as lifelines that is a lot in to start with one man now the next big era in history was associated with the name of Alan Turing here we have all these people who won cheering Awards they will know more about ensuring than I do and during when celebrate his sentence this year see essentially and big big name during films about during he was a great man undoubtedly genius had a big influence on serious theory practice of computing and much beyond that time he dies very sadly of tragic reasons he was on his way to doing crazy things still so cheering is that dr. day one of the heroes everybody in his room now he worked during the war very famously at a place called Bletchley Park you can go visit it now it's a museum this is where the decode is the German Enigma machine and save millions of lives hopefully shorten the war and he had in there a team of people working with him he was in charge that question not only of the theory but of the practice and they built the machine called the Colossus there's a picture of the closest means you noticed two young ladies doing all the work and there were lots of young ladies doing all the work I even there one now she's 95 but she still remembers working Alan Turing so these were the great days of the early history of computing as created by curing and the people who fought them of course there are many other names I can't an instant them all but Turing was undoubtedly a very big figure now another place that played a role in history of well this was a place called the Institute for Advanced Study in Princeton where I went the graduates per stock in 1955 and over the years spent many time many years there later it is without doubt the first occasion of an so what they would call a think tank means Nicholson it was unique at the time now there are many coffees and some of the people went there were similarly unique I haven't shown a picture of Einstein he was there I picked only those who had names was on computer science john von neumann the ultimate whiskey kurt g?del man who turned the world upside down logic and Hermann vile who was a gourd mining mathematician of all types I heard vile and von Neumann lecture the amsterdam congress in 1954 when i was a graduate student and i became a faculty member of the Institute for Advanced Study later that makes a Girdler dinner we'd have talked to him so know the young people I personally connect with but they are without doubt the biggest names in history and a subject there are many others we can mention but these certainly have to be included in the first team now well somebody else who undoubtedly named we come to do is is that an NEA bra mom Saddam no he was about that a very forceful personality as you can see from this picture you wouldn't want to have an argument with him you you would lose he took over every single combats I don't mean with a sword but intellectually and he always won he was without doubt the most powerful brain turned his attention to this whole issue at that time he actually played a big role all the way through in his early life but first of all he started life as a famous psychologist he proves theorems till sick point theorems later on he's all rubbish because you couldn't find the message' find the fixed-point what you what useful it to boot existing they became a constructivist I don't believe anything I can't construct then after that later on in life he moved to skin he became an intuitionist he did I have an intuition that tells me I can do
things and somehow after work out how to join my intuition without my logic and it was a big battle and that battle is still going on how do we combine our intuition without logical faculty how does it affect what we do Brahe was the heart of all that discussion and by that time the theory of logic he moved on from what Hilbert started with about the discrete finite gradually getting towards the continuum the integers are hard enough with the continuum the real numbers are much much harder and that's what bra wanted to keep didn't want to throw away the the real numbers but he knew that is very hard to justify it so the battle how you go from the integers the real numbers is history of whole theory of logic and computation let's just meet the surface don't think you have to call your computer scientists you can ignore it these are your these are the grandfather's these are the prophets on which the original computer scientist based you have to bow before them they are almost gods so these are the great figures of the past and I this is my Grand Tour don't believe over here I say of course but they won't know no question they're the great figures now that pings up to about 1945 these these things happened in the 1930s pennies when you were there I was a schoolboy I worked out that had I been ingenious and I had been a prodigy halloween' born in Germany I would be able to sit at the feet of Hilbert when he died 1942 I was 13 so you know if I'd been a prodigy and I wasn't you have been German I wasn't yeah I could have said other things Hilbert and learned all but I missed out on the chance that's life okay so I'm not counting the practical questions in 1945 after all the rumpus was over sorry war and people got back and they will be tired of all this discussion of Hilbert's and talking about the foundations of mathematics here one of the Guerlain do things they weren't too bothered about no logical foundations some people were but most people like myself included getting on doing things we one of the two theorems it results if you're practical man you want to build things we didn't care about these logical foundations which was worried the great philosophers of past we were we were packed your people so I belong to that generation who had their careers after the war and 1945 wrong words things I would describe that as a period which is enormous growth in the fifty years from what 45 seconds half of the 20th century was the growth of the computer industry you know it started war and by the year 2000 look when it is reached and now it's going beyond that into the future 50 years computer life was fantastic and you know much more about that me so let's have a quick review of that and I think that will affect most people in the audience I imagine well they're young students computer scientist old people who worked and feel mathematicians they're already innocent in this area and which is what I'm going to talk about here that come to my proper topic what is a soluble was it what is the solution you got a problem you want to solve it what does that mean okay secondly if you solved it is it computable can you can you get an answer even I've got a solution and you say well what is the solution well I that's not good enough people want to know can you calculate the odds can you give me a number so two questions is what is the solution secondly is it computable so first level is you ask yourself it is a solution who is the answer for who is asking you the question who is your audience who is to solution well the whole range of people I put them up here in some order but don't pay too much attention to the order but roughly speaking with top level are the theoreticians on the bottom level of the practitioners so we start with a magician's that's where it all started power during Roman brothers they were really interested in logic because foundations of logic and don't forget the outside outside at least I gave you Bertrand Russell a famous people and logic was the root of the study and then mathematics was very close to that of course these decisions were also mathematicians and the physicists alongside a little bit labeling physicists then of course on physics you go to astronomers people's will be telescopes Galileo NASA and then you get onto the India's people building machines who make the telescope so make the spacecraft at the bottom you have the computer scientist who as you know do everything you see today there everybody look in the all new computers so I haven't there's not one category computer scientist themselves from a whole Caterpie then after that we get the fact users who are the people who use well we have to have the mention of the bankers people who have money they are they are the people trade on some exchange who gamble billions and his fraction a second these are guys who use mathematics and I've known bankers and one bank came in one day to my office my share they went down by 30% yesterday I'm going bankrupt he was a backbend bust it wasn't our fault just don't make seasons they just give you information your fault what you do that's true no and then of those traders traded as our people to trading all sorts of things money goods site information they can become spies trading information spying you can become a hacker you can break into things though you can become an Antioch I mean the hackers I'm not sure we to the bad guy who said the good guys take your choice hackers make information available to the public antagonists try to close them down then the Apple the word general but for general needs Admiral they'll commander no Secret Service KGB CIA at all there and there are all the active users of information and nowadays that means new science so whether you like it or not and I'm sure up there I don't mean up in heaven but I mean have the seedings there are buggers you can't avoid them I probably my pocket so now these groups are not disjoint they both vertically and horizontally many people are both computer scientists they're all more than changeable one very notable example you might think there aren't many bankers or mathematicians okay how many people know a real bankers a real math listen Lee sir I know one his name is Jim Simons he runs the Science Foundation he's now funds more fellowships than metal but if he's fantastic he's successful hedge fund that employs math listen the physicists Glor to make guaranteed profits soon billions and billions and he puts all that money back into the sport of young people hope you notice that many of you may be supported by Tim salmon himself he's a great man he's a great man and he's now retired from banking making money he's gone back to solving problems as has he always wanted to do but yet to wait he's got a few millions he once told me that you know he went his life from mathematics and I said I was successful I mean for money baby modest JEP and Becky making money was easy the difficult part was giving it away because he's very interested in giving money away when you were usefully spent so he examines their body carefully that's very hard work so making money easy give you the way part that's a horse's mouth now so these people are all interchangeable in engineers astronomers little there you want to position yourself here you can say I do everything or you say I'm only a mere computer scientist who's interested in money or any mixture you like whether at all we're all in this together and nature of what a solution is depends on the customer volition wants a logically correct proof magazine wants something approximately correct physicists doesn't care too much long as it's expects missing data your son must get even less astronomy it's well known you know if you get within ten orders of magnitude you're okay engineers ah that's different there's a plane crashes they're in trouble computer scientists well I won't
speak to you and the others all have their own requirements bankers and traders they need to know the answer within a fraction of a second I'll come to timing in a moment they need though for the day-to-day business of their lives so the answer depends what is the solution it entirely depends on the customer there's no single answer very important point to bear in mind you work for anybody secondly how much you have how much you have a what well that's all how much time you have when do you want the solution by zero do you want God the on top answer by tomorrow morning or yesterday that's what happened in the real world they often if you want to somebody wants to gamble on stock market even know the answer within microseconds well not only if you're an astronomer cosmology is in season origins of the universe you have more expensive view of time you were about to wait you don't mind reading a few billion years you know what was a few billion years in comparison when the age of the universe nothing so time is very elastic quantity and the customer was also the problem it's very different requirements depending on what his time scale is I put in 10 0 to 10 to the 40 seconds I thought 10 to the 40 was a big number well are they up there it gets missed in powers of 2 well you have little conversion factor to give path tell me to work it out 10 to the 40 is a pretty big number okay it's the size of the universe and actually when you get to numbers and size doesn't matter what you need to use if you tend to the 40 seconds one in 40 years many differ by a small number they're sensitive these big numbers are independent of units except it doesn't matter what what you counted in and the rosy light well it's still finite so you can't go faster than light even in your computer okay you commute as fast they can't actually tell faster than light that's nice if you believe in Einstein so time is a very important notion but it depends very much who asked the question when he wants the answer by and you have to tailor your work to suit the customer the customers pays the bill you have to fulfill your contract as required and that answer can be anything as I mentioned and magicians physicists bankers all have different timescales secondly money now money doesn't mean know what already you know pounds shillings and pence these days or dollars or Euros actually doesn't matter what units you use again when you talk I put ten to the twenty here because that's smaller than the amount of money fort knox by a longshot and it doesn't matter what units you use this scale they're all equivalent but money don't necessarily mean just cash or credit card money you mean seventeen is in terms of things like effort energy brainpower things that use resources resources get converted in the money money by sources money is a useful word to use what is money mean anyway I need a bit of paper or even nowadays a notion money is now a chip on the computer so the money just is a word to describe the resources and the losses can be intellectual resources so don't think I'm being merciful every form of information is governed by behind it the brain power thinking algorithms money buys your machinery these individual devote to it these are all resources put in to make any without resources nothing happens so I've given you month time and money it is more broad since then of course I mention computer power now as you know there's something called Moore's Law its I can't even exactly what it says but within every 10 years thinking by factor by you know the exponential growth and Moore's law means that in 50 years we've gone from sending starts to you know almost the velocity of light so this what I call the moving frontier it really play games in English Remmy nology there's something called keep moving the goal posts movies the gopis means that you think you reach the end of the football field no no the gopis I hope you shoot me down to further down you think that's cheating well some people think sports works but actually in this game you start playing football before you know where you are I find your ending playing ping pong and the Chinese usually win the ping pong so moving frontier doesn't just change the golf course it changes the nature of the game and you'll play a different game when you come out so while you put in that's that's a bit you've got to be prepared for all sorts of changes so Moore's Law moving frontier means that things are constantly on the change and you know that very well and obviously they will take even more in the generation years to come all the young people are back watch it see the moving fast get on that train quickly or you'll be left behind well what do we do after that that's that was the house solve problems how much time it costs what is the solution and so on now let me go back a bit having moved so fast up to the front there is no back a little bit in time back to Plato anybody here Tonopah who later was well well the word Plato means many things but the pathan ik ideal is the world in which no points are things no no dimensions lines things ever no widths everything is perfect sphere is a perfect sphere well these things don't exist in the real world there are idealization idealize means but what does it mean Maya I think it means that it exists but it exists only in human imagination we can imagine every sphere you never see a perfect sphere you can't make up every sphere penny see you do not exist in the real world have reality exist in the human imagination and don't think he may engage is unimportant human imagination is the most important thing that is so the Platonic world is the world of the human imagination and Plato please ideal the word by word philosophy but ideally ideal things the world of their ideas ideas and ideals live up there outside the real world the messy real world where we play around actual hardware and software the Peter planning idea is the ultimate abstraction and so the world in the imagination and the magma is very important now I mentioned I think that that's meant to be a picture of Gordon Moore some of you probably know him I hope it's great picture I I rely of my friends to produce these pictures by the way I dunno about that what about the that's the painting by Raphael of what's called school bathrooms of course it's a it's a Renaissance painting it shows absence is what looked like in brothels time and coming down the steps leading the way I think it's meant to be Socrates and Plato or something like that and down on the steps cribbing in a corner meant to be Euclid and somewhere on the side there's actually a picture of raffle himself it's a very famous painting oddly enough you know where you can buy this painting I bought it you buys in the Vatican a lot of space you think we'll be celebrating the science is anything the way that she did Galileo but that's where you buy the this painting of Plato I had to go second time the Vatican to get my first coverage CDs that's the real picture and this Gordon Moore and there's you know that he's the one who invented Moore's law about things being moving very fast but I claim plate have infinite time and money and he can unwrap ha unknown more it was funny by the you know those Tortoise and the hare well play-doh because he's not restricted we all really live worlds idea well time has no boundaries we can travel faster than light in the ideal world because you see when you say science is based on experiments things that you reproducible that's false see I can imagine you go on television you click it on say man you're back on the Big Bang this is what the Big Bang looks like how do they know that who was there a Big Bang you can't travel back to the Big Bang floss the reasons you take you for long but you long we've got the burnt or shrivel so there's no possible way but of course we can go back in our imagination we do that imagine you back of the Big Bang here is he it looks like this we have an ideal picture television approximation
and you do it every day without realizing you break the barriers of space and time in the idea of the world of the imagination there are no limits you can go back faster than light you can't go back the other universe and you can certainly I'm not run Gordon Moore so completers move fast but both are bounded by the speed of light the imagination is not the imagination is faster now in the in the Platonic world they're also very important thing besides imagination there is there is beauty in the past people had a great appreciation of beauty arts art forms all kind the Greeks were masters of sculptor paint they draw me at Renaissance paintings and mathematics as form of beauty too we madison's and some of you appreciate think is beautiful and it's same fan of beauty as we find in the autistic world that I once paper with a friend of mine published and general neuroscience showing that he acted true neuroscience shows you the in fact the part of the brain that appreciates beauty deals of mathematics I think that with art and music everything else so we don't have to be apologetic about saying that we like beautiful things now here I'm going to talk about the future and the future well that's the you people the young people's back maybe you straight to the front future is the paper talks about it what is the future quantum computers around the corner people are investing billions make quite a few because the first person who makes a condom pewter can do things in which now are impossible because they take too long and of course as soon as the first person or company makes the cognate pewter you will of course not hear about it because well they're trying to see the you money for the bank they're trying to spy on you this will be an enormous advance and he will not be announced and maybe three happened but we haven't been told because obvious reasons who wants to tell the other guy then he can make in his bank account by logically everybody's heard about that Balaji you see biology is much longer history but even computers on earth there's been life for about three billion years your take and Balaji has had time to evolve and try out almost everything something's working something don't work a throw away the ones that don't work Darwinian theory natural selection flights or ideas biological system involved and there's no chance that you know scientist I have a clever will be cleverer than evolution evolution is you know eons ahead the bubbliest can't possibly do better than evolution in terms of treating a brain let me try packed have a network or two poor poor man's brain but they were but of course they might like to piggyback they take a piece of biology somebody's they'll write something else and they'll say here is a bit of real biology we'll feed this in and we build three silicon chips we'll use little bits of biological tissue oh that's actually not unrealistic you're not beyond the bounds of practical say you'll build a computer those component parts are biology because biology solve the problems of information processing and speed much faster than yes so you'll piggyback sit on silicon tips on biological chips this is beginning to happen and you'll do that but you under got a fully fledged being you got a little poor imitation of large organism because all you got is a few little bits of skin and do the whole thing well I think Santa's have a hard time doing better now robotic computers I put down below because well I'm not quite sure what a robotic computer is is it one base of silicon chip it was biology is it based on hardware well I leave it there is a question for you we don't know what a robotic computer is but whatever the future may be called robotic computers they be computers that design themselves these automating things that people are worried about on the science-fiction stories well they're so bizarre that we can't quite conceive them so won't you find them there's leave it as a question what is the robotic computer they'll you'll find out now hey I got that's slide 9 I think I was like ten yes well I've talked about that a bit already evolution and complexity evolution Darwinian evolution and Darwin against Google Darwin wins I'm sorry I know that appeal from Google here who reject that but I think against biological evolution even Google doesn't send a chance and Darwin's stands for not of course Darwinian theory in his sixth form of Darwinian theory as itself has evolved the modern theory but he's not quite as a state forward as it was Darwin realized when he put forward his theory he was just taking the first step in a complicated process people keeps covering new things he didn't know all just in your DNA there's also other information sword conveyed in other ways it's more confident but fundamentally Darwinian Theory still there that's natural selection is still there although it now operates the more complicated way so it evolves itself the theory is not static and Darwinian Theory evolution is I think one of the great successes of intellectual creativity of the human brain as represented by Darwin who has of course this magnificent beard that shows he was great thinker I know two young people in the audience and trying the same trick well good luck to them now I should actually I once wrote a poem which I in the end with I'll keep it as a final excitement things look forward to em but it was a few Jetson remarks yeah I had this morning you know anyway at 6:00 in the morning you say I got to give a talk what am I going to talk about oh yeah and I've got some slides but that's not good enough so I've got some additional material here not on the slide because it's the results of early morning thought now in the 21st century which we are now will embark on I never thought I'd live to the 21st century I remember the days when it seemed by when 1984 George Orwell's yeah I've seen a long way in the future we were there I need any for was so far ahead that you don't always chose it because seems a long way ahead now it's on we past we live actually in the purse Orwellian world and when you hear some of the things said on television you realize that George always here with us now yeah American television in particular sorry I'm sure the Americans the others will agree with me now the the after all the changes didn't happen in logic computation I think in many ways we've some of us tend to say we've gone back we forgotten about the past we all the battles about foundations of mathematics are irrelevant to us practitioners we do things we get answers but suddenly beginning in 21st century life is caught up on us and I'll tell you why you see you can you can if Polly know that the very famous thinker was axiom of choice we just worry about excellent choice essentially also says it's also called the law of the excluded middle you can prove a theorem by contradiction you say suppose this is true you get a contradiction therefore it's negation is true proof by contradiction now people like bra didn't accept that you thought middle shouldn't be experienced in other words you couldn't prove my conviction been approved by construction that was a strict constructivist you proof my conviction was not allowed well that's a bit awkward because most of our proofs met nice well by contradiction so that really was a awkward point and struggling for years to square the circle on that one um believe you do accept the action a choice then you know Wells oyster you can do things I still miss choice leads you to you know the balls on a fire starts there of all the foundations of analysis everything follows and but of course then you when you get that action of choice at the same time girl came along and said okay you got the axe I'm a choice but you've lost computability you can say something it's true but unfortunately you can't compete it not unless you use a few more axioms I've got a few more necks with my pockets the sale and I can sell them to
you and then you can go a bit further and if that's not enough I got more accident I it so Google started this whole idea of more and more accidents to do more and more computation well that turns out I think I've lose myself recently that's a long period of skepticism there's a lot of the important problems mathematical problems of the present time all of that kind and just to show that how learning I am and I know you are I mention a few if you don't believe me fine friendly expert in the audience who will tell you what they mean let us sake see what's called number theory Steve recently we have Andrew while somewhere in the audience if you don't ask questions ask him and number theory inside number theories a lot of very famous unsolved problems you want to make a million bucks well they're solve what he follows use your money's yours but leaving aside that there are some very famous things in number theory one of which is a really unusual one this is only mathematics symbol which is taken from the Russian alphabet its second is from the letter sha which if you know any Russian Jesus all horrible with three vertical lines that stands for the take separate each group so I can do after to tell you what teacher for every group is anyway whatever it is is a very important animal that's wouldn't this deserve rush letter otherwise and but he's a very mysterious animal because lots of beautiful formulas appear in mathematics you means the order of the teacher for every Jacobi's part of the formula so P part of formula is all to be Jamie number but nobody knows all the time with that number is finite or not and they can't prove it the biggest unsolved problem in number theory in some things will this takes every of each group he's actually a finite or you found the generative or not and nobody knows that is I think I'm told by my Excel friends that he's in some sense the hardest unsolved problem in number theory and it has to do with the mystery of where the thing is computable or not deep level let us say geometry which is my field and the fields go back to Euclid of course you boy know that you've heard about it that thing's called Sears sphere is two-dimensional sphere this is a three-dimensional sphere there's a four dimensional you can imagine more but the interesting ones are dimensions two three and four the two-dimensional sphere to be known for a long time we play football world cup I think of that what's more important work at football well the the 3-sphere was the next one up and there was a famous prize it was one of these famous millennium prizes which the prey foundation offered a million dollars for if you could solve and it was the only one that's been solved and the famous is solved by my rational Cartman who turned it down and I see you can't get more famous than that you win a prize worth I mean you're not and then you turn it down you get much bigger publicity so another nice shows that math editions are brilliant he says they don't care about money oh isn't that great we appeal integrity we saw problems that we sit don't take the million dollars off at night that you can't get a better example of you know integrity than that I don't know where the next problem will go the same way but let's hope it's a good example and now same for I'm gonna say that's C dimension that was solved but it's a bigger problem in four dimensions and four dimensions the problem is still open in fact what is called a prank Eric injection dimension for for smooth for manacles is the biggest unsolved problem mathematics and its problem is a bizarre and in some sense very parallel for thinking number theory because if the four-dimensional sphere if it's not you need to determine there are exotic one things that look like sphere art these are all these exotic see is you can simply glue them together to make make you smell safe a group out of them you get something called the group sonic spheres but lo and behold nobody knows anything about that group this is finite group a symphony generated group it looks suspicious smells like the tape chef for Amy's group in disguise exactly it is it's cousin it is the geometrical cousin of the takes every image group in a rather precise sense and it's not a famous unsolved problems you begin to suspect something is wrong with our basic groundwork but somehow he's a problem which we can't solve and we may have to go back through our foundations to see why we cancel them I think that's true when we come to physics the hardest one in Phoenix don't believe Gilley gave a talk later this morning next week I think I'll tell you about the universe well but how does Pauline feelings are called black holes and not just small back / big black holes and you know what happens when you actually go inside about bit the black hole is marvelous phrase invented long time ago and it's really the core of the hardest questions in astronomy what is a black hole that is a similar problem any computer science well I think the corresponding thing is the notion of proof what is the lesson of proofer computer scientists people try to get algorithms to get proven truth checking pooling you know everywhere every talk here in computer science is about the nature of proof I think all these specials I mentioned our same common frontier they allow the frontier between soluble insoluble there are things which you don't know the answer to and in some things they all have to do with the question input/output you put in something comes out and what it's actually happening in that process so let me I've got just ceased 30 seconds in he was my poem I wrote this poem long I don't write closed he very often you know math lessons are well sorry we would even riveting so when I don't know him he was a short poem don't worry and it was written for the iacs in Paris with John Pope you know it was the directive many years and it says this about mathematicians well it could be applied in the board light of day mathematicians check their equations and their proofs leaving no stone unturned in the search rigor that's the hard guys but at night under the full moon a dream they flick amongst the Stars and wonder of the miracle of the heavens they are inspired without dreams there is no art no matter attics no life season hands out there thank you perhaps I misunderstood your characterization of platonic ideals as creatures of the imagination I think that misses an important distinction certainly Mickey Mouse and Beethoven's fifth symphony and perhaps even the gods are creatures of imagination but the integers and plain triangles they are ideals that await our discovery and we know this because we're confident that if aliens were to visit us from some distant galaxy they would recognize integers and triangles though they may have different names for them but they would be surprised by Beethoven's fifth and then there is a third category somewhere in between these are the moral laws are the Ten Commandments something to be discovered or are the inventions that's a harder question the vaulters you've all seen secrets an exact answer the easiest one of these is taking one's out because to talk not have time they're all profound questions you ask I have thought about one quite deeply and that is about the integers I think critical said in God made engineers all the rest is made by man and I can dispute that because think of it this way suppose the evolution had taken different course science fictiony is full of people so if evolution had led the first intelligent life not being bodies like us a small beam gigantic
protoplasm filling oceans there's been one big jellyfish in its and egos and one big jelly faces the Pacific Ocean and some small jellyfish ever the said you can ignore the jellyfish would not have bothered to count you can still meet many other big jelly fishes around but what of course known all about pressure and volume they'll work have the laws of hydrodynamics got niggas ethically yeah probably well but put in a for me so it didn't use integers so yes you would have discovered the integers would have been lasting on a mind for this gigantic creature in the ocean because if there is so another integer another object individual objects don't exist until you have lots of individuals if you are the one person in the universe why bother to count so it is not inevitable that infinite existed God didn't say that integers exist evolution was such that the first intelligent person who started think about it were mammals humans the monkeys they were individuals they were not gigantic sea animals but he even lucien could have been different says not a fundamental law of logic within does it exist Kenickie was wrong it could have been otherwise its evolution and saw to it that the first people had brains or individuals at odds these equation that's that's i was wondering what you make of heightens theorem which you didn't mention and i think is appropriate that a system of actions that has a information content of K bits does not allow you to prove any statements that has an information content that is K plus some constants so any set of actions finally I referred the girl I'm a girl said that is much more profound way I mean we know that hierarchical systems to get the next assume you need more power more bits more thought so that's that is the part of history logic yes I like I gave full credit to that okay thank you