AN INTERVIEW WITH PROFESSOR GRAHAM HIGMAN ON PTV

Interviewer: Qaiser Mushtaq

Transcriber: Shayyan Qaiser

Pak Math Soc, Newsl 3(6), 2007.

Question         How did you become interested in group theory?

Answer           Yes certainly, of course I first become interested in mathematics in general when I was at school, in the way most school children choose a subject, it happened to be the subject that I was best at. While I was an undergraduate, I was interested in several branches of mathematics. When I became a research student under Henry Whitehead, who was of course a topologist, he put me on to a problem in group theory with the intention that it should be applied later on in topology, and I got as far as the group theory, but I got stuck there, so in a way you could say that I’m a failed topologist if you like and that’s really why I’m a group theorist. But of course, as a group theorist yourself you probably don’t want to see it in those terms.

Question         Why do you think that the study of group theory is important?

Answer           Well, like many other research mathematicians, I do what I do not only because I think its important, but because I think its interesting. The problems that one has to deal with in group theory are challenging. They are intellectually difficult and when 1 has got into a problem, its not so much as the question of how much it has an application to the real world that concerns you, its a question of exercising your mind, exercising the control that the mind has over the things it understands. Of course it is true that group theory has applications in the sciences, perhaps we should say for the benefit of most of the people who are likely to be listening to us and have no idea what group theory is, that group theory is in fact the study of symmetry. What a group theorist is concerned with is the possible amounts of symmetries that an object can have. Symmetry in the abstract is our study and because everything has some degree of symmetry, and because a physical scientist, for instance, who is studying a problem, finds it easier to deal with if he takes the symmetry into account. He needs to know what kinds of symmetries there are, and he therefore needs to know the results of group theory, and it is in that way that group theory acts as a servant of other sciences as well as being an interesting subject in its own right.

Question         The students would like to know what direct applications group theory has in some of the basic sciences.

Answer           To know the direct applications, you have to ask the scientists, you have to ask the physicists. A great many of the recent developments in group theory, in particular, in representation theory of certain groups, have been carried out not by people who regard themselves as mathematicians in the first place, but by people who regard themselves as physicists. They do them not because they are interested in mathematics in its own sake, but because it makes the work of physicist easier, and if your student asks you that sort of question, you point them to that kind of answer.

Question         In some countries, set theory is introduced to students at an early stage. What do you think about this?

Answer           It depends how much you do, and the sprit with which you do it. Set theory is probably a natural language for talking about things that occur in everyday life, provided you don’t try and develop a very abstract and harried sort of way. I think children enjoy it, and anything that children enjoy, there should be talk with a reason, and if they can be talked to enjoy mathematics, they will be more likely to take it up.

Question         Our students and teachers, they think, that set theory in those classes is quite abstract and axiomatic, and when the students don’t see any direct application, they find it very abstract.

Answer           This is natural, but there are many examples that you can give to show that it is a useful language. I think if you try to teach it as an axiomatic science, you will not get any response form children because the power and the attraction of an axiomatically developed subject, is something that comes to students later in life. For university students, it can very often be a very interesting and exciting thing but the younger child’s mind is much more concrete and therefore you have to begin with the more concrete things. We don’t begin with the axiomatic; we begin with the practical situations, which we activitise later when we want to form a theoretical subject.

Question         The most confusing thing for a student is the definition of “set”, as it is not well– defined as such.

Answer           The passion for exact definitions is another thing that comes later in life. The developing mathematicians or undergraduate knows he needs the exact definition if he is going to get exact proofs. But we don’t begin that way. That is the level of abstraction that you reach later on and you reach it because it is necessary to the subject, it makes the subject powerful when you get to it but to try and introduce it into the mind of a child to soon, then the child just curls up and runs away. So its not a good thing.

               

Question         In my opinion, advancement and perfection of mathematics are intimately connected with the progress and prosperity of the state. Do you think that a country can progress without giving due respect to fundamental research in mathematics?

Answer           If you are living in a country, which is blessed with fertile soil and comfortable climate, you can live without any intellectual life at all. The intellectual life is not in a way necessary to human existence. You may feel that as an unsatisfactory argument in many ways, and you may feel it would be a limited life. But if you happen to live in a country, which doesn’t have those benefits, you need it for various things, but u can’t give a perfectly satisfactory answer to such questions. We happen to face the fact that we don’t do mathematics solely, or even primarily because of its economic benefit, we do it primarily because it is, as one French mathematician told another, “the glory of the human mind, that is involved”, and if we are not prepared to do mathematics or other exact sciences for the glory of the human mind then we almost most certainly wouldn’t do them at all. The fact that they have economic benefits as consequences is incidental.

Question         There is school of thought, which believes that instead of developing your own fundamental sciences, why not to acquire technical know how from the developed countries and use them, rather than developing your own fundamental sciences and reaching to that point where you start producing things on your own.

Answer           This argument has more force if you are talking about technology than if you are talking about fundamental science. If you are talking about technology, it is important to consider the time factor. If you are insisting on developing your own technology, that’s going to take you a long time. But if you acquire technology from somewhere else, you can use it right away, but as far as fundamental research goes I think the case is different. You do fundamental research not in order to acquire the results solely but because as I said the process is such that it makes you more worthwhile than you were before, and if you cut yourself form all that, you are making yourself less human than you ought to be or so it seems to me, and I think we have to take that line.

Question         Did you find Pakistan any different form the picture that you had in mind before you arrived here?

Answer           Its always hard when you come to a new country and you wait there for about five days to remember what you thought it would be like before you get there as a matter of fact. I don’t think that I find it very different. I do remember when I first came here knowing that it was the end of a hot dry summer and that the monsoon hadn’t yet broken. I was a little surprised to find how green the place was and I was little surprised to find how near to the mountains Islamabad was, but these are sorts of small things, aren’t they? I would also say that concerning the workshop in algebra that we have just been through, I was pleasantly surprised by the enthusiasm and the standard of knowledge of the participants, and it went very well, and apart from those two things I am not sure that there is anything I would want to say that surprised me in Pakistan.