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.
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?
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.
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.
