I'd like to begin, if I may, with your childhood, and I'm going to ask a too obvious question.

Was it a scientific family that you were born into? Basically, no. It was more of a musical family than a scientific family, but my father had done mass 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. Another maybe obvious question. Were you pushed in any direction, or were you allowed to find your own way? I would say that I was strongly encouraged in the direct...

No, not in a sort of pushy way. No, no, no, I understand. 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 sort of distilled gradually down 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 a prodigy? Were you reaching for books at three? Were you solving problems at eight? What was the tenor of your intellectual interest at that point? I think I was a sort of early developer, but not a super early developer.

I wasn't a sort of remarkable prodigy that journalists would write about or something like that. 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. What was your birth order? When did you come?

I was the first born. You were the first born. And the only boy? Yes, I've got two sisters. Two sisters. Either of them mathematicians as well? No, far from it actually. So one is a novelist and writer more generally, and the other is a violinist.

Not too shabby. If we were to look for the moment of a spark that began your direction, when would it have been? As early as primary school? Do you mean a spark that made me think, right, I want to be a mathematician?

Yes. It didn't really work like that for me. It was 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 I had a succession of rather unusually enlightened teachers

who did sort of things that weren't 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 at any rate that you were being recognised as good in that by some of these exceptional teachers.

Well it wasn't just me, I wasn't being singled out. Ah, say this, I was just in a class with these very good teachers. And I can think of at least three different teachers at different stages in my schooling who played this sort of role.

I think teachers are important, as do you clearly. So, may we have their names of the ones that… So there's one at primary school called Mrs. Gazzard. She was the person who first told me about Pi for example. I think I was sort of five or six at the time.

And then a teacher called Mrs. Briggs was when I was at prep school. She was the wife of the headmaster and had a degree in mathematics from Cambridge and had very interesting ideas for problems to think about. And then I had a teacher called Norman Routledge who had been a fellow of King's 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 university mathematics perspective. And he too set really interesting hard problems regularly that were absolutely not needed for A-level

but just developed us in such a way that when A-level came around it was not much of a challenge because we'd been doing much more different things. And might have set your own approach very broadly to not being rulebook oriented.

I mean is it fair to give that particular teacher credit for that? I'm not sure exactly in what sense I'm not rulebook oriented. I mean one sense is clear. I very much regret the way exams have become so much the sort of thing that people focus on.

So my attitude to teaching is definitely one where I think it's a pity that the rulebook is so prescriptive. But in my own life I'm reasonably obedient.

We'll get to your ongoing education, but I'm going to stop and ask a question that may actually not be very important. Did you find that there was 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 one doesn't see so many women in the mathematical field even today.

Well that's an important question certainly, but the question of whether, well two of the three teachers I mentioned were women. One at primary school I think, because I was very young then I don't know all that much about women except that 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 mathematician.

So I think she had missed out that she was a 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 her talents were underused or whether that was what she wanted.

I don't quite know. But these days it is troubling that there aren't more women in the upper echelons of mathematics, although there certainly are some.

And it's a question that a lot of people are very clear 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 wound up doing a very important textbook in mathematics but couldn't be at the university level.

So this business of tracked 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 we'll just leave it at that 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, when I came to Cambridge to read mathematics, even then I wasn't set on a mathematical career. I didn't have any conception really then of what a mathematical career was and what it was like. I remember once saying to somebody whose father was a don in mathematics and physics,

that an academic career looked quite a nice idea because you'd have such long holidays. Break it to me that that was the holidays when you did your research. It wasn't quite as it seemed.

But gradually over the course of my degree I became clear in my mind that I wanted to do a PhD. It also became clear that what I most enjoyed during my degree years was working on hard problems.

We were working on hard problems that weren't exactly part of the service, weren't focused on the exams I'd be taking in the summer. I'm hardly unique in this but revising for exams in the summer wasn't something that gave me a huge amount of pleasure. So if I'd left after graduating or after doing one further year as a part three course,

I would have just missed out on what mathematics is really about, which is for me, which would have been doing research, thinking about hard problems. So it was clear that I wanted to do research and once I was set on that path,

I did indeed find that I got much more pleasure out of just sitting and staring at a blank sheet of paper for two weeks or scribbling on it, but just being sort of stuck on a really hard problem. Then I had learning lots of material for my exams before that. So then I think it was sort of, I really felt in my element and it was clear that that was what I would ideally like to do.

Of course, early in my 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, you manage to do something and then you sort of realise that it's not actually in principle impossible

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, is the opposite of a hard problem a soft problem?

I mean, what do you mean by a hard problem? I actually find that a very interesting question in the abstract, which I don't have a full answer to because, I mean, there's an obvious answer, which is that a hard problem is one that a lot of people have tried and not managed to do,

and a lot of good people, but I feel there's something more objective than that. I mean, I could imagine a situation where just by coincidence the people who've tried it went on the wrong track and actually it was quite an easy solution that sometimes happened, quite an easy solution that people had overlooked, then you wouldn't want to say that the problem was hard.

So I feel there is something objective about problems that makes them harder or easier, and I suppose very roughly it would be that there are toolboxes, sort of reasonably standard toolboxes that experts in an area have, and if by wheeling out the standard tools you reach a solution without too much extra thought,

then it would come to an easy problem, whereas if your tools you seem to keep running up against a brick wall, then you really need some idea to come out from left field or something, then that would make it a hard problem. But you also get a sense, and this is a bit more mysterious I think,

you get a sense with unsolved problems that this looks like a pretty hard problem and this one looks like the kind of one that you should be able to do if you really try in about three weeks or something. You have a sense of that, which is not totally reliable, but it's not completely unreliable and exactly how we make these assessments is something I'd like to understand better.

I'm next going to ask, quite naturally I think, how did you choose the hard problem to address for your PhD? Well, initially it was my research supervisor, Beto Bolivash, and I should say that this sequence of wonderful teachers continued at the University of London,

and he was the one who really stood out as inspiring me as an undergraduate, so that's why I wanted to be his research student and he accepted me, I was very pleased about that. 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 then I thought about it harder and managed to get a stronger bound for this particular quantity, and that led on in a natural way to some further questions, which I suppose I kind of thought of them myself in a sense,

or they were generated naturally out of this project and gave me more things to do. So in a way nearly all my PhD was an outgrowth of a project that he set me,

and then after that I gradually started being interested in problems that he hadn't necessarily suggested, but they just seemed like really nice problems. Can you just briefly explain the field within mathematics in which the hard problem lay?

Well, I suppose the first area I worked in was the geometry of Banach spaces, and so that, if you know just the bare definition of a 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 complex body, a symmetric complex body,

multi-dimensional, sometimes infinite dimensional, in a very straightforward way, and so questions about Banach spaces can be rephrased as questions about the geometry of convex 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 flavour, 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 specialising very much in the geometry of Banach spaces,

I sort of moved, this combinatorial interest was already there, but it became the dominant interest of mine, and the Banach spaces slightly sort of drifted away, although my combinatorial work has been somewhat informed by the way I think as a result of having worked in the geometry of Banach spaces.

I'm interested in, I don't mean it, the word that came to mind is the sociability of mathematics, what I really mean is, I'm very struck by the computer theorists I've spoken to,

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, or what I believe is often called pure mathematics, it's happening within you and perhaps between you and your mentor.

Is that fair to say, this process of inquiry? It sort of varies from person to person, but it is I think more private, so most people, I was, all my work as a PhD was just my solo work, I would have occasional discussions with Breda Polovac,

sometimes little 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 gone on 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 usually there's three, but more normally two or one,

so I think it probably is less, you don't have an idea that you have sort of teams of people working on problems, and that's it. 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 pure 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, but I think in the end if I had to, and also it's a little bit of a noble simplification to say applied mathematics is mathematics that is applied, because quite a lot of applied mathematics I think is probably done mainly out of reasons of theoretical interest,

even if in principle it's closer to applications than a lot of pure mathematics. But I think maybe the biggest sort of cultural difference between pure and applied mathematics is that pure mathematicians are more focused on the idea of rigorous proofs,

so we care, our sort of standard for when a statement is established as true is that there is a rigorous proof of that statement. An applied mathematician, this is again I'm sure I'm oversimplifying 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'd have to be very unreasonable not to accept that that statement was,

let's say, almost certainly true, and almost certainly true is good enough for many purposes, especially if you're just interested in applying it in some field, you don't have to wait for that extra 0.1% of certainty. But pure 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, and it's not so much because we're obsessed with certainty, it's because the quest for proofs turns out to be extraordinarily fruitful,

and if you look 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 a proof forces you to come up with ideas that you wouldn't have had to come up with if you weren't trying to find a proof, and those ideas often lead to solutions of other classes of problems and so on,

so the endeavor of finding proofs is an extraordinarily interesting one. Yes, and with some of your colleagues, I found something I'm sensing in you, which is the joy of the pursuit, and probably a tolerance for frustration.

That's essential. When it doesn't come. Well, we need to get you launched on your career. I think you published even before the dissertation was completed. I can't remember now whether the first paper came out before or after.

Anyway, it did go into the world, and it was well received, the work you had done. Yes, and it wasn't earth-shattering work, but it was a sort of, you know, I suppose it was signaling that there was another person who was joining the community who was capable of producing reasonable work.

In a way, it's what the dissertation does. Exactly what it's for. You seem to have, that's how I regard it really. I'm not looking at a PhD level for people who overturn their field in some way, but just somebody who can show that they can do a variety of problems.

They're technically strong, they have interesting ideas and so on. And then you hope that that's just the first step, and there will be 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 after what I've had.

Roughly 25 or 26, something like that. Okay, well then that's an opening to ask a question that is about a cliché about mathematics, and I think you have actually addressed it in various places, so I'm going to ask you directly about youth and mathematical insight,

because it's the only field 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 this decretion 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 see that assumption? Well, one thing is true, which is that very young people, people very early in their career can do, and sometimes do do,

remarkable things that are sort of not just results that are one step along the way, but are major results that will be remembered forever. 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 raw ability that tails off with age, perhaps there is when you get sort of, the older I get the more I want to believe that there isn't such a thing. Yes, of course.

And an 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 have to build this accumulated wisdom, but it saves you a lot of time if you can do what I was talking about before, and size up how difficult a thing is likely to be, what techniques are likely to be used for, and that sort of thing.

You can work more efficiently as a result of the experience that you build up over the years. But 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 maths faculties will tend, on average,

to be those people who did have success when young, you get to a situation where you become established at quite a young age and then the incentive, the sort of hunger, at least from a career point of view, 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 faculties in the first place. So with many people that's not a problem, they just love mathematics enough that the wish to solve problems that they've been preoccupied by is strong enough,

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 various 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 older, even though they aren't always. Isn't the bias built into the idea of the field medal? I mean, I can't think of another field where such an important award

is actually restricted to those, is it under 40? Yes, you're supposed to be under, or supposed to turn 40 at the latest, latest or earliest. Anyway, you mustn't, you must turn 40, you must, can't be older than having turned 40.

Which from outside the field looks eccentric as a condition. I'm not sure what I think about that. It's hard to, as a beneficiary of the current system, it's hard to... Well, we won't condemn it, but we'll just notice it. Yes, I mean, I think if, I mean, the extreme example

that I suppose people would cite is Andrew Wiles not being able to qualify for a field medal, which, but in that case they sort of slightly apologised for the rules by inventing this special award. I mean, that could be a comparable situation,

whether that makes, I think in a way the right attitude, I think the attitude that many mathematicians take is that yes, it's our 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 that field medalists have done. I think my own sort of, it's easy to say this, again, as a recipient, but my own sort of feeling was that when I was younger it seemed like

the sort of pinnacle of all possible achievements, and now I sort of feel whether it's slightly arbitrary and sort of a better problem to be right. Well, we need to get you to the insight that actually led to the field medal. So, you've gotten your degree,

you've published soon after, certainly. What's next in your career? Well, after my PhD I got a research fellowship at Trinity, so that was, at the time, that seemed like the pinnacle of what I'd been aiming for, and I was absolutely thrilled to have that.

So, the four years were stretching out where I could just do more research. Although, as it happens, after two years I got a, a job came up at University College London, and I got a letter inviting me to apply to it, and I went to talk to Bader Bolivar,

and she said, do you think I should apply? Yes, of course. And he said, well, 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 four year thing,

I went to UCL, I was very happy there at UCL, I think it was a very good department. Was it a department oriented in a particular way? Well, at the time, it's changed quite a lot over the years, but at the time there were quite a lot of kindred spirits of people who worked in complexity, various and functional analysis, and things that were quite close to my own interest,

which was part of the reason that they approached me, I think. And then, I was there for four years, and then I got a job back in Cambridge. So, the exile wasn't so long? No, and in fact, for most of the time, I actually lived in Cambridge, and I commuted to London. But the reason I got a job back in Cambridge,

my big break, so to speak, came while I was at UCL, and I'd always been interested in a problem in the geometry of banner spaces called the distortion problem, which I spent, even before I went to UCL, I spent a long, long time trying to solve that,

and this is actually something that's happened to me several times in my career. I never did solve it, but the thoughts that I had... You never did solve it? No, no, I eventually got solved by two other people, Ted Adell and Thomas Schumpfrecht. Oh. But the ideas that I had in struggling to solve that problem Yes. were useful in another problem, which was an old, famous problem in the area,

which I did manage to solve. And also, actually, as it happens, somebody else, Bernard Moret in France, solved it independently, and we published together. And that was a big step up from anything I'd done before, and that led to, kind of...

This was constructing a rather pathological example of a banner space, and somehow the expertise that I'd built up there allowed me to construct some other pathological spaces, and in fact, also led to a theorem about those spaces that was a positive theorem, not just a counter-example. So I had a body of work that was suddenly

a whole lot more important than what I'd done before, and that was enough to get me a job back in Cambridge. Excuse me for a very naive question, which is, is there a eureka 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. After that, I sort of thought, well, okay, there's still quite a lot to do, but basically, I know it's in the bag now. Yes. So I can remember, usually I can remember where I was, so one of these problems, I mentioned a positive result, I was on a train coming back from UCL,

when it suddenly became clear that this problem was within my sight. Are you ever not working on a problem? I mean, in a way, it seems like an all-consuming intellectual investment, and so your energy is always somewhat in that problem.

Yes, I mean, mathematicians are notorious, 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. It's fine, I'm just thinking about some mathematics. That happens to me quite a lot. Might be dangerous, crossing the streets and so forth.

Yeah, I'm not as, yeah, I'm not as sort of vague as that, but... Yeah. But nevertheless, I can be quite... Consumed. Relaxing and enjoying myself, and having a drink with friends and things like that. Right, right. Maybe going to the cinema is something I could do

and I wouldn't be always thinking in the background about a maths problem or watching a film. But, I mean, 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. I mean, it's something that we're just extremely interested in, and by some miracle,

this thing that gives us so much pleasure and somebody else is ready to pay us a decent salary to do. And then, because of that nature, that sort of hobby nature, yes, it's quite hard. I often spend evenings and weekends when I get the chance for thinking about problems as well.

Having done that and having had the luck of a society that considers what you're doing valuable, you then go back to Cambridge. You just said, mostly as a result of this work. Was it, and by the way, this work that in the end got you the Field Medal?

Yes, well, it was certainly a very important component, but in the run-up to the Field Medal, I did something else, which was to find a new proof of a famous result with Szemeredi's Theorem, and that was much more in the sort of

material direction and involved coming up with some reasonably new techniques or using some existing quite new techniques in a new way and that sort of thing. I actually got told by an indiscreet member of the committee that it had been absolutely essential

to have this other component to my retirement, to my work. And that really, that result, this new proof of Szemeredi's Theorem, changed the course of my research in a big way and 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 focused on this one result,

but the sort of broad area, which is now called additive combinatorics, which is this sort of area that's grown up in the last 20 years or so, out of, in part because of your work, of course.

It's certainly had an influence, but I'm by no means the only person to have contributed to that, but there's been a big sort of explosive growth in that area, that's been the area where I've been mainly doing my work since then. 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 and direction? 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 rather than that area. There are 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 where we're particularly strong. We cover everything, but we're not quite so good in this particular part or that part.

Right, right. So it's quite a big faculty. If you want to do a PhD in Cambridge, there are a lot of different things you might do. How much teaching came into your work when returning to Cambridge?

Was it a big chunk, or are you left mostly alone to prove? Well, I had a teaching position when I first started. I was a lecturer and then a fellow at Trinity, and the lecturership required me to do typically one course per, what Americans would call one-on-one,

so I'd do one course in the midterm, one course in the length term. These terms are eight weeks, and a typical course would either be 16 or 24 lectures. So it might be, say, 40 hours of lecturing a year, and then it would be six hours of supervising per week.

So that would be one to two small teaching classes, which was a fairly significant additional. Yes, it was a lot. But somehow it felt manageable, and after a while I became a,

no I didn't, sorry, after I became a professor, and at that point, I'm just trying to remember whether, I think initially I, in those days one didn't have to teach as a professor, and one was sort of expected not to do college supervising. Certainly my obligation to do so went down,

but then I, at some point I, either immediately or a couple of years afterwards, I reached a stage where I was teaching in college as a professor. But only four hours a week, and that 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 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're talking specifically about graduate students, actually things have slightly changed over the years. My way of supervising has changed, and now I do a lot more joint work with my graduate students. And so now, I mean it's just an essential part of my own research,

the projects I have with graduate students that are, I'm not really doing very much work at the moment other than work with graduate students. So, that gives a rather strong answer, it's certainly not a distraction from my research, it isn't. When I started,

it wasn't a, I mean these would be people who would come and talk to me, we'd 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, just in order to be able to have intelligent conversations and offer advice, and so, none of that is,

none of that would be, could be called a distraction, it's all great. And that applies actually even I've found to undergraduate teaching, so there you would have thought all I'm doing is just telling people about things that I know very well because I've known them for years and years, but I found particularly with lecturing that,

even when I lecture a subject I know well, I find that I organise things a little bit differently from how I would have expected before I started preparing the lectures. I learn 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. Certainly I've found that if I do no teaching, it doesn't, it's more productive, in fact slightly the reverse. Right. I've spent recently a fair amount of time

with the problem of the question of research as it applies to habits of 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 Asian societies for

developing the habits of research as we know it and even the habits of the challenge of the student. 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

students who challenge? Is it the habit in the researcher, himself or herself, of always asking difficult questions and 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. That is, yes. It may be too big for me to do an intelligent answer. Well, it really is about the habits of research as you've come to understand them. It's something I'm very interested in how,

sort of questions like how many hours a day do you spend on it and that sort of thing. Those are not what I'm talking about, but the processes of research, how one goes about 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 when I solve a problem? And I think actually it's beneficial to think about it because the better you understand it, the better chance you have of doing it more efficiently in the future.

And I suppose one thing that is not a particularly surprising thing, it's just the emphasis I would put on it is very strong, and I certainly emphasize this to my own research students, that what you're looking for

when you first start a logical problem is not so much a chain of steps, each connected with very simple logical procedures to the steps before, something like a formal definition of a proof, but you're looking for something slightly different, you'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. The best sort of question would be one that you can just about answer, so it's not easy, because if it's too 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'll help you solve that question. So you want to find a question that's somehow in between where you are and where you want to go, and then this is a process that can be iterated. If you find this question in between, then you might want to find another question that's in between where you are and that question,

and another question that's between the question you've just asked and the question you originally wanted to solve. So this process of asking questions, there are ways we have of coming up with these questions as well, and so I think this,

I'm going to use a footballing analogy, I'm not taking root one and hoofing the ball down the pitch, but it's passing the ball about or something, and just trying to build up a kind of feel for the area around the problem. That's a very important part of research,

sort of having a nose for a question. Sometimes also you might be wanting to solve a problem, but you find some question gets generated and you say well actually I don't care whether I ultimately solve the problem I was 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, and that's something where, going back to the question about experience, that comes in to recognizing that some subsidiary question is actually a pretty interesting question in its own right,

is something that, well it helps to have some experience to sort of judge what's interesting and what's not. It's absolutely essential but it's useful. Something I've read about your thinking by you and about you gives me the courage to ask 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 are computers going to take over the field of mathematics? I don't think that's a silly question. Well good. I take a slightly minority view on this, I think the answer is yes. The only question is how long it will take.

Ah. 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, I spent a small portion of my time working on automatic theorem proving, is that we'll get programs that can do quite easy problems,

problems that mathematicians would find easy. So you might think what's the point in writing a program that does problems that mathematicians find easy? And there is an answer to that, which is that different mathematicians find different things easy. So depending on what your background knowledge is and your expertise.

So if you had a computer that could answer easy questions for you in areas that you weren't yourself comfortable in but you sort of needed to know about for some purpose, that would be a fantastically 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. 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 a human to have some really insightful, creative ideas. 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 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, if you have a program that's really good

at solving easy problems, you can't then build on that. Another program that uses the first program to do slightly less easy things and then... Well, my response to that, again, very much as a layman, is will we be able to program curiosity? I think so, yes. So that sounds like a sort of thing, again,

it sounds like a particularly human thing. When I think of my own reasons in mathematics for getting curious about problems, it's not sort of just some aesthetic sense of what I'd like for this problem, it's more I sense a relationship between this question that I would like to know the answer to and various other questions that are 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 natural, internal mathematical reasons.

I don't see that there's in principle an obstacle to computers being able to work with those sorts of reasons as well. It's funny you use the word aesthetic, that gives me the opportunity to ask a question that I've enjoyed as a cultural historian looking at your field, and that is often the use of the word elegant for a solution.

Can you give me your sense of why the word, why the aesthetic words are used, particularly the word elegance, and what is an elegant solution? Maybe this will be an incomplete answer,

but a sort of 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 becomes straightforward. So if those ideas can be expressed in a very concise way, in the sense in which a 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 build some black boxes, you build those in stages, you maybe build a black box quite easily, and then out of those black boxes you build a slightly bigger black box. And then if you can just build up that way and then end up with an argument that proves some very non-trivial statement

using a couple of black boxes at the top, each of which was built in this very nice way, then that would count as a fairly extreme example of an elegant proof in most people's minds, I think. Somehow one of the things about,

one of the features that makes a proof elegant is what we would call conceptual, that somehow you're not just doing some calculations and then saying, look, it came out as we wanted. You want something that, I mean, there are lots of different,

slightly difficult to formalize concepts in which elegance is one, and this conceptual thing of another and 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 sort of suddenly noticed that you've ended up at your destination.

And one of the reasons for that is that, one of the reasons we like elegant proofs, I think, is that they're much easier to carry in your head afterwards than a long sequence of calculations. So it means, for example,

it's not just carrying your 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 and so on. You have all sorts of advantages. So it's tied up not just with some aesthetic sense

but also with the efficiency of doing research itself. So that would separate it from elegance in the artistic world, where the usefulness almost never comes into it. It would just be the balance of it

or the look of it or the satisfaction of it. I think there is a difference, but there is something about this notion of, there's a sort of balance and economy and so on that is common to the two things, which is why we use the word, but it is quite specific. I've said this before,

but aesthetic terminology in mathematics, it sounds very mysterious, but I think if one then looks at how we use aesthetic words in other domains, then 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 we use the word beautiful in those two instances,

and I have to say beauty in mathematics is another kind of beauty that's got very characteristically mathematical aspects to it that has something of the same feel about it, or 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 unfathomable for society at large, the notion it's something that five people do in beautiful places, which would be one of those,

but nothing that really can be grasped by the society at large, and you've challenged that. Well, I've challenged it to some extent, but I'm realistic about it. I know that if somebody asks me at a party, what do you do when I say I'm a mathematician? And then I get the follow-up question,

so can you tell me a little 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, but that's usually not socially acceptable and not particularly what to do anyway. But I think one thing that can be challenged, which I do challenge,

is the idea that mathematics is the preserve of people with very, very unusual special brains. Yes. I think as a normal level of academic, I think you have to be a pretty good academic to be a successful mathematician probably, or you're more likely to be anyway.

But you don't have to be a child prodigy, and you don't necessarily even have to have shined in a huge way at school. There are plenty of examples of people who develop a bit later on. And I think that the 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 it, a persistence, a tolerance, what you said earlier, a tolerance for frustration. Yes. If you're enthusiastic enough about it,

then when you get the hurdles that inevitably come, 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 enthusiasm I'm talking about.

What it is that makes people enthusiastic about mathematics is maybe slightly mysterious, but the point I want to emphasize there is just if you feel as though there are other people around who are 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 if that love is genuine, then you are the sort of person who could do mathematical research. Yes. You just have to put in the hours, and you may find you have to put in slightly more hours than that other whiz kid who could do things very quickly, but in the end, there are a lot of tortoises in mathematics

who beat the hairs, and so I think that's an important message to get out there because so many people get put off mathematics at a young age, and that's a real shame. I don't think everybody should be doing mathematics, but I think there are almost certainly people who have got put off who would in fact have really enjoyed it

and thrived at it. So a little reduction in the mystique might be useful for all of us. Yes, that is something I've tried to do quite often. On my blog I sometimes try to write what I think is

exactly as demystifications of arguments, so magic-looking arguments that I try to demonstrate when I can. It's not always easy, but sometimes if I find a natural route to the discovery of a magic-looking proof,

then I want to publicise that, just to emphasise that these clever ideas do actually come from somewhere, come out of a sort of process that anybody can... That is comprehensive. You don't need to have some light bulb going off in your brain because you had a brain that was unlike anybody else's brain. You just have to go through certain processes.

I think that will be the last word. Thank you very much.