The HLF Portraits: Sir W. Timothy Gowers

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The HLF Portraits: Sir W. Timothy Gowers
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Point (geometry) Axiom of choice Multiplication sign Direction (geometry) Moment (mathematics) 1 (number) Perspective (visual) Element (mathematics) Mathematics Goodness of fit Many-sorted logic Different (Kate Ryan album) Universe (mathematics) Order (biology) Energy level Mathematician Musical ensemble Table (information) Family Social class
Geometry Trail Complex (psychology) Euler angles Direction (geometry) Mereology Rule of inference Theory Element (mathematics) Force Mathematics Goodness of fit Many-sorted logic Term (mathematics) Energy level Mathematician Modulform Cuboid Convex set Series (mathematics) Körper <Algebra> Angular resolution Symmetric matrix Set theory Combinatorics Area Focus (optics) Standard deviation Process (computing) Generating set of a group Forcing (mathematics) Chemical equation Projective plane Physicalism Basis <Mathematik> Sequence Numerical analysis Degree (graph theory) Category of being Universe (mathematics) Right angle Family Resultant Spacetime
Point (geometry) Variety (linguistics) State of matter Multiplication sign 1 (number) Frustration Student's t-test Parameter (computer programming) Open set Theory Goodness of fit Mathematics Many-sorted logic Natural number Average Different (Kate Ryan album) Term (mathematics) Mathematician Energy level Körper <Algebra> Social class Standard deviation Weight Forcing (mathematics) Projective plane Cartesian coordinate system Faculty (division) Proof theory Order (biology) Right angle Musical ensemble Pressure Resultant Spacetime
Axiom of choice Statistical hypothesis testing Euler angles Multiplication sign Axonometric projection Rule of inference Food energy Wave packet Mathematics Many-sorted logic Natural number Mathematician Convex set Theorem Körper <Algebra> Position operator Physical system Condition number Area Process (computing) Fields Medal Counterexample Moment (mathematics) Algebraic structure Ultraviolet photoelectron spectroscopy Funktionalanalysis Wiles, Andrew Degree (graph theory) Proof theory Family Resultant Spacetime Distortion (mathematics)
Point (geometry) Group action Divisor Direction (geometry) Connectivity (graph theory) Multiplication sign Student's t-test Mereology Mathematics Roundness (object) Many-sorted logic Term (mathematics) Different (Kate Ryan album) Extension (kinesiology) Cluster sampling Combinatorics Position operator Social class Area Focus (optics) Moment (mathematics) Projective plane Numerical analysis Faculty (division) Order (biology) Right angle Resultant
Area Process (computing) Routing Student's t-test Mereology Time domain Product (business) Mathematics Many-sorted logic Logic Analogy Chain Right angle Körper <Algebra>
Point (geometry) Computer programming Multiplication sign 1 (number) Routing Frustration Parameter (computer programming) 2 (number) Product (business) Mathematics Many-sorted logic Term (mathematics) Different (Kate Ryan album) Reduction of order Mathematician Modulform Energy level Theorem Körper <Algebra> Extension (kinesiology) Area Time zone Dependent and independent variables Process (computing) Chemical equation Computability Counting Sequence Time domain Proof theory Arithmetic mean Calculation Order (biology) Normal (geometry) Family Resultant
Mortality rate
[Music] I'd like to begin if I made me good childhood and I'm gonna ask a too obvious question was it a scientific family that you were born into basically no it was more a musical family and a scientific family but my father had done maths a-level and always retained an interest in mathematics and science so there was that element there but he was a musician and my mother was as well I know there may be obvious question Oh will you pushed in any direction or were you allowed to find your own way I would say that I was strongly encouraged not in a sort of pushy wait no no no anything but just in the directions that I naturally wanted to go which in my early childhood was both music actually and mathematics and in fact quite a few other subjects but it's sort of distilled gradually time to mathematics over the course of my childhood and adolescence only because this is true of some of your colleagues would you have been called the prodigy were you reaching for books at 3 were you solving problems at 8 where it was what was the the tenor of your intellectual interest at that point I think I was a sort of early developer but not a super any sort of remarkable prodigy that journalists would write about or something like that right definitely wasn't like that but I was a sort of early reader and was young for my class at school but not three years younger than everybody else or anything like that so you wish your birth order when I was first humor the first one and the only boy worse I've got two sisters two sisters mysticisms I do mathematicians as well and no far from it actually so one is a novelist and writer more generally and the other as a violinist not too shabby if if we were to look for the moment of a spark that began your direction when would have been but as early as primary school I would just give me a spark that made me think right I want to be a mathematician yes it didn't really work right now from there's more that mathematics was always in the mix it was always something I enjoyed so there were various reasons for that one of the most important being that had a succession of rather unusually enlightened teachers who did sort of things that went beyond the syllabus and that sort of thing so that meant that at each stage where I had to make a choice to specialise a bit more mathematics was always one of the things that was involved which suggests that any way that you were being recognized as good in that by some of these exceptional teachers it wasn't just me I was being singled out of classes so yeah I was just in a class with these very good teachers and I don't think of at least three two different teachers at different stages of my schooling who played this sort of role I think teachers are important as to you clearly so maybe we have their names of the ones that says well at Primary School for mrs. gasps odd uh-huh she was a person who first told me about PI for example oh good I think I was sort of five or six at the time and then the teacher called mrs. Briggs perhaps cause she was the wife of the headmaster and had had a degree in mathematics from Cambridge and had very interesting ideas for it which were problems to think about and then I had a teacher called Norman ravage who had been a fellow of Kings for a while King's College Cambridge and really was quite extraordinary in his lack of sort of deference to the official syllabus and did things in a very interesting way that was very much informed by a sort of universe the mathematics perspective and he too said really interesting hard problems regularly but absolutely not needed to a level but just developed us in such a way that when a table came around it was not much of a challenge but that would be another thing and might set your own
approach very broadly to not being ruled local oriented I mean is it fair to give that particular teacher credit for that you don't have I'm not sure exactly what sense I'm not rulebook oriented out yeah I mean one sense is clear I very much regret the way exams have become so much the thing that people focus on some attitude teaching is definitely one well I think it's a pity that the rule book is so prescriptive and but in my own life I'm of reasonably obedience well we'll get your ongoing education but I'm gonna stop and ask a question that may actually not be very important um did you find that there was an a rather strong number of women teaching math at the earlier level I'm really wondering about the generation of teachers who taught you and whether the women were is it fair to say stuck beneath the university level or is this really not an important question to ask because what doesn't see so many women in the mathematical field even today well that's an important question certainly yeah but the question of whether well for two of the three teachers I mentioned but women yes while at Primary School I think because it's I was very young then I don't know all that much right of course she was a wonderful teacher my guess is that she was not you know wouldn't have been in the running to be an academic mathematicians right we should have found fantastic teacher at that level I don't know about mrs. Briggs the one who had a degree from Cambridge I don't know what sort of forces in her life led to her whether she whether her talents were underused or whether that was what she I don't quite know but these days it is troubling that there aren't more we're going up in the upper echelons of mathematics although certainly our son and it's a question of the most people who write about we won't answer it today but I'll tell you why I ask you in other interviews I've had in this series a few people were privileged with having a really quite remarkable tutor in mathematics was a woman who one of doing a very important textbook in mathematics but it couldn't be at the university level so this business attract women at a certain generation who were very helpful to say the least but whose own career was stopped and occasionally the boy of the family even with the daughter being good in mathematics was the one who got the attention so well we'll just leave it at that and in terms of your own development you are now having to choose a path at what age where you really have to announce a direction in mathematics well I think even when I applied to university which when I came to Cambridge to read mathematics even then I wasn't sort of set on a mathematical career I didn't have any conception really then of what a mathematical career was and what it's like I remember once saying to somebody whose father was a Don in mathematics in physics hmm so one idea Demick career looked quite a nice idea because you'd have such long holidays and yeah break it to me that was you know the holidays when you did your research that wasn't quite as it seemed and but gradually over the course of my degree I became clear in my mind that I wanted to do a PhD yes it also became clear that what I most enjoyed during my degree years was working on hard probably the same thing we're working on hard problems that weren't exactly part of desilva's what what sort of focused on the exams I'd be taking in summer intended and hardly unique in this but revising for exams and someone wasn't something that gave me a huge amount of pleasure and so if I'd left after graduating or after doing one further goes a part three course I would have just missed out on what mathematics is really about which is for me which would be it would have been doing research thinking about my problem so it was clear that I wanted to do social once I was set on that path I did indeed find that I got much more pleasure out of just I'm sitting here staring at a blank sheet of paper for two weeks or quickly I was good but just being sort of stuck on a really hard problem then I had learning lots of material if you my exams before that and then I think it was sort of I really felt in my element and it was clear that I that was what I would ideally like to do of course early on a PhD you have no idea how it's going to go and the idea of solving problems that nobody else even these great experts have managed to solve seems very daunting and impossible but you sort of you build up gradually manage to do something on any sort of realize that it's not actually in principle impossible and so I think you may have just answered the question I was next going to ask but you'll help me clarify that I was going to say what is the opposite of a hard problem soft problem I mean what what do you mean by a hard problem um I actually find out a very interesting question in the abstract which I don't have a full because you know other than I mean
there's an obvious answer which is that our problem is one that a lot of people have tried and not managed to do a lot of good people but I feel there's something more objective than that I mean you could imagine a situation where just by coincidence the people who tried it went on the wrong track and I thought it was quite an easy property solution that sometimes happened quite an easy solution that people with overlooked no and you wouldn't want to say that the problem was hard because so I feel there is something objective about problems that makes them harder or easier and I suppose very ruffly it would be that there are two boxes sort of reasonably standard toolboxes that experts in an area have and if by weeding out the standard tools you reach a solution without too much extra thought then it would count an easy problem whereas if your tools you seem to keep running up against a brick wall and you really need some idea to come out from left field or something then that would make it a hard problem mm-hmm but you also get a sense and this is a bit more more mysterious I think we get a sense with unsolved problems that this looks like a pretty hard problem than just one looks like the kind of one that you should be able to do if you're really trying about three weeks or something you have a sensor that was not totally reliable but it's not completely unreliable and exactly how we make new assessments is something I like to understand better but mm-hmm next word is quite naturally I think how did you choose the hard problem to address for your PhD well initially it was my research supervisor paidipala Bashan I should say that this sequence of wonderful teachers continued at the University now with he was the one who really stood out as inspiring as an undergraduate and so that's why I wanted to be here so substitute and he accepted me I focused on that and so initially he suggested problems and it was over the course of my PhD well I solved one problem which then first of all I solved it in a weak way and I thought about it harder and managed to get a stronger stronger band for this particular quantity and that led on to a natural way to some further questions which I suppose I I can't thought of them myself in a sense all day well I generated naturally out of this project and gave me more things to do and so in a way nearly all my PhD was an outgrowth of a project that he set me mm-hmm and then after that I gradually started being interested in problems that he hadn't necessarily suggested but they just seem like really nice forms can you just briefly explain the field within mathematics in which the hard problem why well I suppose it's the first well the first area I worked in was the geometry of Banach spaces and so that if you know just a bare definition of Banach space that sounds a bit like an oxymoron because it's not a very geometric sounding thing but with every Banach space you can associate a convex body a symmetric convex body multi-dimensional or sometimes infinite dimensional in a very straightforward way and so questions about Banach spaces can be rephrased as questions about the geometry of complex bodies and so that's why we call it the geometry of Banach spaces and I was interested in problems that had a somewhat combinatorial flavor and some of the techniques that I used were inspired directly by results in combinatorics and later on in my career after about ten years or so specializing very much in advance Rabanne spaces I've sort of moved this combinatorial interest was already there but it became the dominant interests of mine and the balance basis slightly sort of drifted away although my comment or work has been somewhat informed by the way I think as a result of that we worked in their balance pieces I mean I'm interested in I don't mean it where the word they came to mind is the sociability of mathematics but I what I really mean is I'm very struck by the computer theorists I've spoken to who of course many of them began in mathematics but how collegial a context their graduate work was how they really found something together rather than alone but I'm sensing that in mathematics are what I believe is often called pure mathematics it's it's happening within you and perhaps between you and and your your mentor is that fair to say this process of inquiry it
shouldn't perish from person to person but it is I think more private so most people I was all my workers PhD was just my it was so like hmm I would have occasional discussions with bed of automation sometimes there's no sort of conversations with others of his research students but not who didn't seriously work together but and then my research students have tended to work either on solo projects or with me but sometimes they've then had gone off and gone out and talked to other people and that has led to joint work with other people so I think it varies quite a lot but it's I would say it's usually groups of at that stage most three so I think it probably is less you don't have an idea that you have sort of teams of people working upon yes that's quite an that's it um does that have anything also to do with the question of the
applicability of the work as opposed to the pure nature of the research because I have heard the distinction between or mathematics and applied mathematics and I'd appreciate some clarification about that too because I certainly sense your work was in pure mathematics yes it certainly is there's quite a lot of debate about that topic and some people say that the distinction is very exaggerated and but I think in the end if I had to and also it's a little bit of an oversimplification to say applied mathematics is mathematics that is applied the first years of applied mathematics I think is going to be done mainly out of reasons of theoretical interest mm-hmm even if in principle is closer to applications and a lot of pure mathematics but I think maybe the biggest sort of cultural difference was being purified mathematics is that pure mathematicians are more focused on the idea of rigorous proof so we can our sort of standard for when something when a statement is established as true is that there is a rigorous proof of that statement hmm applied mathematician this is again I'm sure I'm over sitting like somewhat but in broad terms I think an applied mathematician would be satisfied with an argument that any reasonable person would accept so supposing you had a very convincing heuristic argument backed up with a lot of extremely compelling numerical evidence from a computer or something then I think you have to be very unreasonable not to accept that that statement was that say almost certainly true and all those years true is yes it's good enough for many purposes especially if you're just interested in applying it in some absolute weight for that extra naught point one percent certainty but your mathematicians we like proofs sometimes we just have to accept that we don't have a proof and yet we still very strongly believe in a statement but we regard that as an unsatisfactory state of affairs there's not so much because we're obsessed with certainty it's has the quest for boost that to be extraordinarily fruitful and if you moon for a proof of something even if you know in advance that it's true you know in this sense of beyond reasonable doubt finding the proof forces you to come up with ideas that you wouldn't depend to come up there that you weren't trying to find proof and those ideas often lead to solutions of whole other classes and problems and selves an endeavor of finding proofs is an extraordinarily interesting one yes and some of your with some of your colleagues I found something I'm sensing in you which is the joy of the pursuit and it probably a tolerance for frustration that's as such when it doesn't come well we need to get to launched in your career I think you've published even before the dissertation was complete I can't remember now whether well the first paper came out before or after outside anyway it's any submitted it did go into the world and it was well received the work you had done yes I mean wasn't path shattering work but it was sort of you know I suppose it was signaling that there was another person who was joining the community right Ravana spaces who was capable of producing reasonable between of ways but to dissertation secular sports of this you seems that I that I regard it really a lot looking at the PhD level for people who overturn their fields in some way but just somebody who can show that they can do a variety of problems technically strong and they have interesting ideas and so on and then you hope that that's just the first step and help you further steps up as they progress through their early career how old were you when you got your PhD just to place you in your life to what roughly twenty five or six I like that okay well then that's an opening to ask a question that is as about a cliche about mathematics and I think you have actually addressed in various places so I'm gonna ask you directly about youth and mathematical insight because it's the only fuel I'm a historian by background there's no chance to expect anything great from a 25 year old I mean it happens but it's not expected there's the accretion of experience and insight but mathematics has maybe only in the lay world the reputation of a place where the brilliance comes young and then you sort of live with that insight for a while how do you how do you see that an assumption well one thing is true which is that very young people people very early in their career can do and sometimes do too remarkable things that are sort of not just results that sort of one step along the way but their major results that will be remembered forever but there is a trade-off between well something like raw mathematical ability I don't know whether there's any sort of meaningful concept of war ability that tails off with age perhaps there is when you get sort of the right feels the older I get the more I want to believe that there isn't yes yes of course but and then experience does count for quite a lot actually I mean it's not just not as much as in a subject like history where you actually build this accumulated wisdom but in it saves you a lot of time if you can do what I was talking about before is size up how difficult a thing is likely to be what techniques are likely to be useful and I think you can work more efficiently as a result of the experience that you build up over the years that I think also one of the reasons that older mathematicians sometimes produce less than younger ones is just that because you can establish yourself when you're younger and then the people who end up in math faculties will tend on average to be those people who did have success when young dare you get to a situation where people can become established at quite a young age and then the incentives are sort of hunger at least from a career point of view did the sort of incentive to produce more top class research is lower than it is for somebody who's younger who needs that in order to get those yes of course so with many people that's not a problem they just love mathematics enough to wish to solve problems that they've been occupied by is strong enough to that's a strong enough incentive on its own but you can add on other things that create disincentives like getting administration heaped on your shoulders and being asked to write lots of letters of reference and there are things like that so I think those sort of natural pressures that come from the academic
career structure have quite a lot to do with why people can sometimes be less productive when they're old or even know that as always isn't the bias built in to the idea that the field metal I mean I can't think of another field where such an important reward is actually restricted to those is it under 40 yes you're supposed to be on the course must turn 40 at the latest latest or earliest anyway you mustn't use turn 40 you must can't be older than you having turned 40 which from outside the field looked eccentric as a condition as I I'm not sure what I think about that it's hard to sort of acid the beneficiary of the current system it's hard to well we won't condemn objective but we'll just notice it there's no I mean I think if I mean the extremely example that's where Google Website is Andrew Wiles not be able to qualify for appeals mail which but in that case they sort of slightly apologized for the rules by inventing this special award immunogen that an open that could be comparable situation whether that makes I think in a way the
right attitude I think attitude that many mathematicians take is that yes it's a sort of headline award but we all know that there are plenty of people who've done work that is at least as good as or better than work you know this have done and I think my sort of it's easy to say this again as the recipient but I my own sort of feeling was a but when I was younger it seemed like a little pinnacle of all possibility yes now I got back I sort of feel whether it's a slightly arbitrary well we need to get you to the insight that actually led to the field medals so um you've gotten your degree you've published soon after certainly um what's next in your career well I go to the after my PhD I got a research fellowship at Trinity okay so I would say that was at the time that seemed like the pinnacle of what I've been aiming for and I was absolutely thrilled to have that and so there are four years were stretching out where I could just do more research well that as it happens after two years I because a the job came up at University College London hmm I got a letter inviting me to apply to it and I went to talk to bethe Bala Bosch and said that do you think I should apply it's only two years and he said well that's up that'd be a pretty good job to have I think really you don't have that much choice so with some slight regret at only doing two years of this for you yes yes thing I went to UCL I was very happy that you see I think it was a very good department was a new department oriented in a particular way well at the time has changed quite a lot over the years but at the time they were quite a lot of kindred spirits of people who worked in convexity Maha Varys and functional analysis and things that were quite close to my interest was possibly Giunta they approached me I came closer and then I was there for four years and then I got a job back in Cambridge so the Exile wasn't so long now in fact for most of the time I actually lived in caves I commuted to London ah but the reason I got a job back in Cambridge had my big break service years Kay and while I was at UCL and I had always been interested in a problem in job shabana spaces called the distortion problem which I spent even before our to usually I spent a long long time trying to solve that and this is actually something has happened to me several times in my career I never did solve it but the thoughts that I had he never did solve it no no they actually got solved by two other people tell Adele and Thomas one fact well but the idea is that I had in struggling to solve that problem yes were useful in another problem which was an old famous problem in in the area which I did manage to solve and also actually as it happens somebody else Banwari in France solved it independently every published together and that was a big step up from anything I had done before and that led to kind of this was constructing a rather pathological example of a Banach space and somehow the expertise that I built up then allowed me to construct some other ethological spaces and then in fact also led to a theorem about those spaces that was positives there I'm not just a counter example and so I had a body of work that was suddenly a whole lot more important than what I've done before and that was enough to get me a job back in Cambridge mm-hmm I mean excuse me for a very native question which is is there a you Rica moment sometimes yes sometimes there is a moment I can point to where before that moment I wasn't sure I'd be able to solve the problem and that has sort of well I can i still might a lot to do but basically i know it's in the bag yes so I can remember usually I couldn't remember where I was something and so one of these problems I mentioned the positive results I was on a train coming back from Museo and when it suddenly became clear that this problem was within my size are you ever not working on a problem I mean in a way it's and it seems like an all-consuming intellectual investment and so your energy is always somewhat in there yes I mean I've mathematicians know Taurus I think for looking a little bit vacant or I'm sure I'm not the only one whose family members have sort of said there's anything wrong maybe dangerous crossing streets and so forth yeah I'm not I'm not I'm not a sort of Vegas at that yeah but then the less I can be quite consumed I'm not that good at just relaxing and enjoying myself yeah of having a drink with friends and graduate maybe going to the silver or something I could do and I wouldn't be always thinking in the background about math problem or watching a film but I mean that we're in a sort of odd position a lot of us that we are being paid to do what feels like a hobby really and it's something that we're just extremely interested in and by some miracle this thing that gives us so much pleasure somebody else is ready to pay us a decent salary to do all right because of that nature or that's a hobby nature it's yes it's quite as often spend evenings and weekends while I get the chance Thank You about problems as well having done that and having had the luck of a society that considers what you're doing path for you then go back to Cambridge you just said mostly as the result of this work was it was it and by the way this work in the end got you the field now yes well it was certainly a very important component but in the run-up to the to the Fields Medal I did something else which was to find a new proof of a famous result when summer eighties theorem and that was
much more in the sort of conv toriel direction and involved coming up with reasonably new techniques or using some existing quite new techniques in a new way and that sort of thing that's I actually got told by an indiscreet member of the committee that it had been absolutely essential to have this other component of my retirement to my work and that really is that result this new group some readers have changed the course of my research in a big way and become became that and things related to it became the focus of my research to this day well pretty much yes I mean it's not just focus on this one result but it was a little broad area which is now called additive combinatorics which is sort of area that's grown up in the last twenty years or so mmm-hmm I would have in part because of your word of course it's happening to suddenly said yeah have an influence but I by no means the only person sitters contributed to that but there's been a big explosive growth in that era that's been the area I've been mainly doing my work so soon and as I asked you to characterize broadly the department in London can you characterize the department you returned to in terms of its mathematical interest in direct yes I would think the main characterization I would give is that it's pretty broad you wouldn't say Oh Cambridge specializes mainly in this area runs a meadow there three or four or five really strong clusters in Cambridge I wouldn't say it's got perfect coverage there are some areas where we're a little weak but it's more that way round it's not sort of we have two areas I'm a particularly strong it's more like we cover everything but we're not quite so good and this particular part of that right okay so it's you know it was quite a big faculty and yeah if you wanted to catch deer kyndra's and lots of different things you might do it instead how much teaching came in to your work when returning to Cambridge was a big chunk or you left mostly alone two to one I'm gonna had a teaching position in Mike when I first started I had I was a lecturer and I'm a fellow of Trinity of the lecturer ship required me to do typically one course per I want America called one and once I did one course in but it must have one course in you and then term these terms are eight weeks and a typical course will either be 16 or 24 lectures so it might be saved 14 hours of lecturing mmm yeah and then there would be six hours of supervising we we wanted to small teaching classes which was a fairly significant additional guests who are lit but somehow it felt manageable and after a while I became a dude so I probably became a professor and at that point I'm just trying with whether I think initially I in those days one didn't have to teach as a professor and one sort of expected not not to do college rising suddenly my obligation to do so went but then I at some point I either immediately or a couple of years afterwards I reached us thanked where I was teaching at college as a professor but only four hours a week and I mean fit into one afternoon rather than two and that made quite a big difference I'll ask you in a minute about your broader notion of of actually teaching mathematics as a factor in society but most specifically I want to begin with is having a number of graduate students a distraction or is it a contribution to your own intellectual development if you look specifically about graduates yes graduates are actually things that slightly changed over the years my way of supervising has changed and now I do a lot more joint work with my graduates do Oh and so now I mean it's just an essential part of my own research the projects I have a graduate students I'm not really doing very much work at the moment other than work with graduate students so that gives a nice floor so I'm not a distraction with my research it is massive right great when I started it wasn't I mean these would be people who had come and talked to me we have pretty interesting conversations and I would have to think about the things that they were to some extent about the problems they were working on them just in order to have intelligent conversation - not for advice and so none of that is none of that would be could could be called a distraction it's all great and that applies actually even I've found to undergraduate teaching so they would have thought all I'm doing it's just telling people about things that I know very well because I've known for years and years but I found particularly were lecturing that even when I lecture a subject I know well I find that I organize things a little bit differently from how I would have expected money before I started preparing their lectures I learned things as a result of teaching not sort of something that feeds immediately directly into my research but just different ways of thinking about things which I think does have a cumulative effect and does benefit my research as well suddenly I've found that if I do no teaching is it doesn't I don't feel I'm suddenly more productive in fact slightly the reverse right mm-hmm I've spent recently a fair amount of time with the problem of the question of research as it applies to happen to teaching in the West and habits of teaching in the East in Asia because I think there's a growing both question and hunger in in Asian societies for developing the habits of research as we know it and even the habits of the challenge of the student and I wonder if you could give some sense of what you
think leads to innovation and thinking at least in the tradition you've been involved in I mean is it is it students who challenge is it habit in in the researcher himself or herself of always asking difficult questions what do you see that has led to this extraordinarily productive innovation tradition in mathematical thinking and technological thinking it's a big question I understand is I may be too big well it really is really about the habits of research as you understand them is something I'm very interested in how is it not maybe exactly the right question not so much a habits of research a sort of questions like how many hours a day do you spend on this and that sort of thing that does I know not what I'm talking about but the processes of research how one goes about yes solving a problem that initially seems impossible that's something that I've always been very interested not just very interested in this sort of meta question what is it that I'm doing yes yes the exam I saw the problem yes and I think actually it's something that's very beneficial to think about because the better you understand it the better chance you have of doing it more efficiently in the future yes and I suppose one thing that it's not a particularly surprisingly persistent emphasis I would put on it it's very strong that I'd certainly emphasize this to my own so students that what you're looking for when you first start trying to solve a problem is not so much a chain of steps each connected with very simple logical procedure sort of steps before there's something like a formal definition of a proof but you're looking for something slightly different they're looking for a kind of fruitful exploration of the mathematical domain around that problem so one of the key things that you want to do you've got a question you can't answer try and find a question you can answer you know the question a better sort of question if you want that you could just imagine so it's not easy to easy it won't really help you very much and that it's also sufficiently relevant to the question you initially wanted to answer that if you can answer this question it will help you solve that question hmm so you want to find a question that's I mean between where you are and where you want to go and then this is process that can be iterated you if you want to get this we find this question in between and you might don't find another question that's in between where you are in that question and the question is between the question you've just asked and the question you originally wanted to know it's also difficult process of asking questions there are ways we have of coming up with these questions as well and so I think this sees a footballing analogy not to take you route 1 and moving the ball down the pitch but his passing the ball back yes trying to build up a kind of feel for the area around a problem that's a very important part of research sort of having a nose for a question but sometimes also you might be wanting to
solve the problem but recognize it you find some question gets generated and you say what actually I don't care whether I ultimately solve the problems originally trying to solve this is a pretty interesting question on it by itself and could well be useful for lots of other things and so that's something that happens that something like we're going back to the question about experience that comes into recognizing that some subsidiary question is actually a pretty interesting question its own right is something that well it helps to have some experience to sort of judge what's interesting what's not absolutely essential it is useful something I've read about your thinking by you and about you gives me their Kurt's ass the next question which I'm going to ask in a very silly way but I think it's a serious one and that is our computer is going to take over the field of mathematics I don't think that's a silly question okay well you're very seriously good I take a slightly minority view on this I think the answers yes the audition is how long it will take but I think it will happen in
stages and the first stage which I've actually worked to try to bring about to some extent so I spend a small portion of my time working on automatic theorem proving is so we all get programs that can do quite easy point forms and mathematicians would find easier so you might think what's the point in writing a program that does problems and that petitions find easy and there is an answer to that which is that different mathematicians find different things easy mm-hmm so depending on what your background knowledge is your expertise so if you had a computer that could answer easy questions for you in areas that you weren't yourself comfortable in yeah useful needed to know about for some purpose that would be a fantastic and useful tool and another way in which it might be useful sometimes when you're working on a problem you generate one of these subsidiary questions I was talking about and sometimes that turns out to have an easy answer which it takes you several days to spot mm-hmm if you had a computer that was able to find easy answers it would find it in ten seconds instead of you having taken three weeks so it could be a tool that would speed up the processes of research without initially sort of threatening this romantic picture we have of needing to be the humans that have some really insightful creative ideas right but my own view is that once you have a program that can do easy things it's only a matter of time before you can build some superstructure over that program that can generate potentially creative interesting ideas and check whether they get and get you anywhere and so on so I think this stage where you have I don't see any reason not to have programs that can solve easy problems and I also don't see any reason I feel in the program that's really good at solving easily from that you can't then build on that hmm another program that uses the first program to do slightly less easy things and then provides my response to that again very much as laymen is will we be able to program curiosity I think so yes that sounds like it's sort of saying you know again it sounds like a particularly human thing yes what I think of my own reasons in mathematics for getting curious about problems it's not sort of just some aesthetic sense of I'd like this problem it's more I sense a relationship between this question that I would like to know the answer to and various other notions that have already established as ones that we would like to solve and the reasons for those relationships are not sort of wishy-washy human emotional things they are sometimes quite concrete things questions generate other questions for quite quite natural eternal mathematical reasons I don't see that there's in principle an obstacle to computers to get to work with those sorts of reasons as well it's funny use the word aesthetic that gives me the opportunity to ask you a question that I've enjoyed as a cultural historian looking at your field and that is the often the use of the word elegant to look for a solution can you give me your sense of why the word where the aesthetic words are used particularly the word elegance and what is an elegant solution maybe this is a little bit incomplete answer but that's a partial answer is that an elegant solution so sometimes you get answers to questions where what you need is basically one or two nice ideas and once you've got those ideas that sort of generates the rest of the argument becomes straightforward and you've got these intermediate steps and filling in the rest of the gaps become straightforward so if those ideas can be expressed in a very concise way that is one sense of what proof can be elegant although it may be that in order to get to implement those nice ideas you've got a lot of complicated technicalities in which case that would reduce I suppose the level of elegance so another notion of elegance I think is if you can build yourself a toolbox that somehow hides all the sort of technical work under the hood somehow you get you build some black boxes yeah these builders in stages you maybe build a black box quite easily and then out of those black boxes you build a slightly bigger black box yes yes and then if you can just build up that way and then end up with an argument some very non-trivial statement using a couple of black boxes at the top each of which was little built-in that's very nice way then that would have done count as a fairly extreme example and a little proof in most people's minds I think so somehow one of the things about one of the features that makes it prove elegant is what what we would call conceptual this and you're not just doing some calculations and then saying look it came out as we wanted you want something that I mean all these are lots of different slightly difficult terms and formalized concepts which elegance is one but in this conceptual thing of another an explanatory is another you want proofs that leave you with a feeling that you understand why the result is true and you haven't just followed a collection of arbitrary steps and then suddenly notice that you've ended up at your destination and one of the reasons for that is that while the reasons we like elegant proofs I think is that they're much easier to carry in your head yeah afterwards there was a long sequence of calculations so it
means for example there's not just carrying a head if you want to say give a lecture on them well that's true as well but also much easier to notice on some future occasion that you can use an idea that's related to this elegant proof to do something else until you have all sorts advantages so tied up not just with some aesthetic sense but also was the efficiency of doing research itself so that would separate it from elegance in in the artistic world where the usefulness whatever comes into it it would just be the balance of it or the look of it or satisfaction of it I think there is a difference but there is something about this in balance and economy and zone that is common to the two things of why we use the work that wrote but it is are quite specific I know I've said this before but aesthetic terminology and mathematics it sounds very mysterious but I think if one then looks at how we use aesthetic words in other domains and we find that it's very different from domain to domain so saying that somebody's got a beautiful face is very different from saying that somebody's written a beautiful piece of me yes yes we use about beautiful of those two yes instances and I have to say beauty in mathematics is another kind of beauty that's very very characteristically mathematical aspects to that has something of the same feel about it well has the same family of concepts so to speak good to finish with just noting and if it's interesting to talk about to ask you to do so about the interest you take in the broader reputation about mathematics as unfashionable for society at large and the notion it's something that five people do in beautiful places so they came it would be one of those but nothing that really can be grasped by the society at large and you challenge that well I challenge to some extent but I'm realistic about it I know that if somebody asks me to party what you do when I say a mathematician yes and then I get the follow up question so can you tell me this what about your work I have to say well not really unless you want to sit down with a piece of paper for half an hour yes but that's usually not socially acceptable and particularly what to do anyway but I think one thing that can be challenged is which I do challenges the idea that mathematics is to preserve with people with very very unusual special brains and yes I think I think that's on normal level of academic I think you have to be a pretty good academic to be a successful mathematician probably but we're more likely to be anyway but you don't have to be a child prodigy or you don't necessarily even have to have shined in a huge way mmm at school plenty of examples of people who develop a bit later on and I think it's a sort of genius idea is quite damaging to mathematics because actually what you need I think sort of what you really need is just a love of the subject a real wish to do to persistence Torrence what you said tolerance for frustration yeah if you want to Z astok enough about it then when you get the hurdles or to never to become you will make the effort to surmount them and I think a lot of what seems like quite extraordinary mathematical ability is in fact the product of many many hours spent as a result of the sort of into zeal and I'm talking about what it is that makes people enthusiastic about mathematics it may be slightly mysterious but the point I want to emphasize there is just if you feel as though you're there are other people around to a sort of quicker than you at solving problems and yet you would absolutely love to be somebody who could do mathematical research I would say the answer is love is genuine mmm then you are the sort of person yeah just have to put in the hours and you may find you put in slightly more hours than that other whiz kid who could do things very quickly but in the end a lot of tortoises in mathematics who bleed the hairs and so I think that's an important message to get out because so many people get put off mathematics at a young yes that's a real shame I think I don't think everybody should be doing mathematics but I think are almost certainly people who've got that off who would in fact have really enjoyed it and thrived at it so a little reduction in the mystique might be useful for for all of us si that is something I've tried to do quite often on my blog app sometimes try to write what I think of the team but exactly those demystification of arguments so magic looking arguments dear writer demonstrate when I care it's not always easy but sometimes I find a natural route to the discovery of a magic looking proof then I want to publicize that just to emphasize that these clever ideas do actually come from somewhere come out of a sort of process that anybody can that is comprehensive yeah there's anyway you don't need to have some light bulb going off in your brain because you have a brain that is unlike anybody else's breath you just have to go through certain processes I think that'll be the last word thank you very much
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