An Evening with the Cambridge Professor

Two months ago, after a class in algebraic geometry, taught by Professor Alan Beardon at the African Institute for Mathematical Sciences, South Africa, I had the privilege of interacting with the professor and picking his brain on variegated issues in mathematics. Issues such as the aesthetic nature of mathematics, the interest in mathematics among students, effective communication of the concepts in mathematics, the development of mathematics in Africa and of course we delved into his career as a mathematician and his life outside mathematics. So sit back and I hope this interview proves to be insightful and inspirational to you as it was to me.

Can you please tell us about your background in mathematics?

I have been interested in mathematics ever since I was a tiny boy. When I was very small, my father used to give me addition sums in an exercise book. I can remember being very excited when he gave me an addition sum. I went all the way down one page and all the way down the next page completing the sums.

What was exciting about this?

What excited me I think about it was the fact that it's very precise. There is no emotion involved, or maybe there is later when you get excited. Initially, everything can be done accurately and precisely and that's what I find exciting and good about mathematics. If it's right everybody has to agree it is right. There is no dispute

So you understood this right from when you were young?

Yeah, I was three or four years old.

...and you never had any doubts making a choice to study mathematics?

No! I never had any any doubt at all. As far back as I can remember, I wanted to study mathematics.

Do you think the kind of background you had in loving mathematics and your understanding of its precision at a very young age was helpful?

I think to some extent you are born with it. You're born with a sense of precision, a sense of logic. You can certainly develop it and it needs to be developed. But I also think I have a very geometric mind. I see things in geometric terms and I have always seen them as such.

Why do you think most brilliant students from high school often time choose to study courses like medicine, law, engineering etc. But not mathematics?

I think there are several reasons for that. But I think one reason is that some people want to do something in their lives that actually clearly helps other people, some sort of social contribution. But another reason I think is that mathematics is actually a very difficult subject. I think it's by far the most difficult of all subjects. Because in many subjects you can get some success by just remembering things. But I don't think in mathematics you can get far just by remembering things. You need to understand them as well.

Do you consider mathematics to be a creative field?

Certainly, it is very creative. You have to think for yourself, you have to create things yourself in order to understand.

Why then is it difficult for people to relate with its creativity? People can relate with a beautiful painting, a melodious song. But not a beautiful proof?

Mathematics is creative in the mind, which means you have to create thoughts about things. You are right, most people in the world have no idea what mathematics actually is. People throughout the world have an idea of what arithmetic is. But arithmetic is not mathematics.

Don't you think this in itself is a problem?

But mathematics in a sense is a logical game. It's a game that you have rules of logic and you play the game by rules of logic and you see where you can get to by those rules of logic. But most people are not logical (chuckled), but it is true. Without logic you cannot do mathematics.

Can you help someone who is passionate about mathematics think logically?

You can teach logic, you can make people think logically. I think the problem is not so much that they can't think logically, it is that they have never been taught what logic really is and how to think; that is a problem. I think that every school in every country should have lessons in logic from a very early age. I think students should for example read sections of the newspaper and discuss it and decide what is true and what is not true and decide whether the conclusions really are correct conclusions from the article and so on. It's not just mathematics, people should be taught to think constructively in logic and be critical. The problem is that in most societies, the people in control don't want that. That's the problem. But if you want to be a successful mathematician, you have to be critical. You have to check every fact, you have to ask every question until you are satisfied with the answer.

Can you give us insight into your career as a mathematician?

My career ... I left school at sixteen and for two years I studied for A level which is the pre university exam and I went out to work and did Metallurgy: research into metals for two years, while I studied in the evenings. After two years I had my A levels and went to the University of London, England for three years for a bachelor's degree. I went to America for a year to Harvard University for a year of graduate study, which was very different and I enjoyed that very much. I came back to London and did my PhD, then I went back to America and taught for two years in Maryland and then back to England for two years at Canterbury, University of Kent; which was a brand new university. It was very interesting because all the faculty had come from other universities, the university had no history, no precedent and everybody thought they knew what they were doing, but everybody was doing things differently. So that was an interesting time. I was there for two years, that was in 1968. Then I got a post in Cambridge and have been there ever since until I retired seven years ago. I've been retired for seven years and I'm still doing as much mathematics as I ever did...(laughter)... because it's a lovely subject.

Is there a point in the life of a mathematician where he attains his climax in terms of productivity?
It's usually said that the best age for a mathematician is about 25. Because at that time, you know some mathematics, some deep mathematics and your brain is perhaps at its best. I don't know! But experience makes up for a lot as you get older. I mean I've always thought as a mathematician I should be learning new subjects all my life and I am still learning new subjects. So I'm now nearly seventy-seven. By the time you get to that age, you've managed to learn quite a lot and that makes up for the brain that is now slowly getting worse....(laughter)..... so there is a balance between knowledge, experience and activity. Certainly when you're younger the brain is more active, that's for sure. But as you get older, you build up experience and knowledge.

In your classes, you have been teaching about mathematics as a whole, without any boundaries...

I believe mathematics is one subject and you should not distinguish between any of these things. Almost any two subjects in mathematics are linked in some way or the other. I think that is an extremely important lesson to learn. Unfortunately, in universities we teach seperate subjects because we are forced to examine.

Do you think this in itself is faulty?

I think it's bad, I think we should not do this. We should mix up the subject: we should learn about Groups applied to other branches of mathematics and so on. But we don't do it and I think it's bad. It holds back the potential of a mathematician. If you go to conferences of top great mathematicians, you will find that almost all lectures involve a lot of different mathematics. That's the way it is and that is the way you make progress. If you are working in an area and stay in a certain area of knowledge, where knowledge is bounded by a certain set of ideas, there is only a certain distance you can go to make progress. To make further progress, you need to bring in new ideas and almost always, those new ideas come from a different part of mathematics. So you get a different view enough to make progress. The best way to change your view is to look at a problem from different points of view.

When you compare recent advancement in mathematics, how far will you say we have come?

What used to be the case in the days of Isaac Newton and say in the 1700 and 1800 is that most mathematicians would know almost all that there was to know. And then from, say roughly 1900, perhaps a bit earlier than that, mathematicians began to specialize, and specialize more and more. So people's interest generally speaking became narrow. I think in the 1940's and 1950's, usually mathematicians were very specialized and since then I think the attitude has changed and now we're moving back to an era where successful mathematicians are by and large the ones who know a lot of different fields and combine them together. I think that is the way forward.

What idea do you think will constitute a dynamic change in mathematics in this century?

Hmmm...that is a difficult question...... I don't know the answer to that. But there was Erdos, who was a fantastic mathematician, extremely well known, who died a few years ago. He used to provide problems that nobody could solve. For some problems, he used to say that mathematics isn't ready yet for this kind of problem. In other words, he was really implying that we need something completely new. Not a strong advancement or something about which we already know. We need a brand new kind of subject somehow within mathematics. Who knows what that is? But I think that's probably right. I mean computers have had a huge impact on mathematics and things like coding theory have developed enormously since we've had computers and they will continue to develop. Partly because it's a practical issue that we need to understand. I think the progress being made in the last fifty years, that is since I have been a mathematician as it were. Most of the progress has been made by people linking ideas from different subjects. Really, I do not know what that idea may be. If I did, I would be famous.

What do you think can be done in Africa to help the development of mathematics so that Africa can be on par or exceed other other continents of the world?

I'm not sure this just applies to Africa. I think that we need to teach students to think and explore mathematics without worrying about the right answers. I think the idea of doing exercises to get correct answers is not a good idea under any circumstances and in any country. If you give a student 10 exercises of the same sort and ask them to do them following a set of rules, that is just like cooking a meal and looking up the recipe in a book and follow the instructions.

We don't want that, we want people to understand it. If you understand mathematics you do not need a recipe. You do not need to be told how to do it. I think that students will learn more if they are given open ended problems. Sometimes problems that cannot be solved. There is no harm in trying and seeing where you get with them. You learn mathematics by experimenting, trying things out yourself and seeing whether something works. If it does not, you have still learnt something. I mean getting the answer should not be the goal. Rather, understanding what you're doing whether you succeed or fail. It's better to try something out, understand it even if you fail, than it is to follow a recipe and get it right.

What is a normal day for you without mathematics involved?

A normal day...I think about mathematics when I wake up and I usually think about it when I go to bed. I do take some time off. I do have some relaxation. But, if I had nothing else to do then I would do mathematics. If I am not doing mathematics, I used to do a lot of woodwork, make furniture, make toys for my children and grandchildren. I like photography! I like to take some photographs of wildlife in Africa when I come to Africa and I like walking in the countryside in the mountains....and playing chess. I don't read very much and I'm not interested in fiction at all. If I read, it's a book about somebody who's travelled somewhere or some science experiment. If I read at all, I read factual things, I am not interested in fiction at all, I am not interested in music at all. I like to be involved in thinking and being creative in a logical kind of way. I understand painting a picture or composing music is creative, I understand that completely. But it's of no interest to me.

...Thank you very much Professor Alan Beardon for your time.