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‘In Mathematics, It May Be That We Don’t Discover Anything; We Just Make Things Up’

‘In Mathematics, It May Be That We Don’t Discover Anything; We Just Make Things Up’

Oxford Mathematics / Youtube

In 2018, the HSE International Laboratory of Mirror Symmetry and Automorphic Forms established a new school in Moscow called ‘Hodge Theory: Old and New’ to facilitate cutting-edge research by young and promising mathematicians. The school has served as a venue for a great number of prominent scientists from all over the world to give lectures on various topics. Among the speakers is Nigel Hitchin, a professor emeritus of mathematics at Oxford University. Professor Hitchin is one of the most distinguished experts in differential geometry, a Fellow of the London Royal Society and a recipient of numerous awards. Nigel Hitchin discusses the recent relations between mirror symmetry and the non-abelian Hodge theory.

We interviewed Prof. Hitchin to find out more about his career, the methods of his mathematical research, and his collaborative work with Russian colleagues.

At what point did you decide you might become a scientist?

In my younger years, I liked physics. I think that the people who influenced my choice was my father (he was a chemist) and my brother. If you’ve got an elder brother, you always look up to him, trying to do everything he does. My brother was keen on mechanics, and he wanted to be an engineer. We spent time together building mechanisms. My brother was good at engineering complex things, while my designs looked better than they worked. That’s about when I realized that I would be more interested in pure sciences, particularly, in mathematics.

I also loved reading books, and I was fond of fine art, so at school I was quite good in humanities subjects, too. However, I had never been particularly attracted to those subjects, not as much as I was to math. Once I fell ill and had to miss a lot of classes. That’s when I dropped physical engineering and really focused on mathematics.

What were your first scientific achievements?

During my first year at university, I had to get used to doing research. Fresh out of secondary school, I found research somewhat frustrating. You can find all necessary information in books to solve a school mathematical problem, and you can consult the textbook any time you want to make sure your solution is correct. However, this doesn’t seem to work when you conduct research. You have to take something in good faith to get a better insight into the problem.

I finally managed to find a topic that attracted my attention. I carried out some research on Dirac operator, which afterwards I incorporated into my dissertation. This research gave me confidence to continue my studies, rather than just graduating with a master’s degree and going on to work in the engineering computer department of the large corporation where I worked part-time during holidays. As a student, you cannot assess your achievements adequately. Things look trivial to you the moment you have understood them. I wouldn’t have now called the research I did a scientific achievement.

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Do mathematicians use any specific methods or approaches in their research?

Of course, they do. There are great advantages to joint projects in which scientists use different approaches. However, I have written most of my works on my own, without any coauthors. One of my well-known articles was co-written with three of my colleagues, but I didn’t see any of them while working on it. It was a collective work done remotely. The main thing for me is to solve a problem. I have never tried to prove a big theorem. I don’t even have an idea of how to tackle it. Some problems are so complicated that one needs to break symmetry to solve them. This requires intuition.

Another method is solving problems by analogy. Mathematics is highly metaphorical. Mathematical intuition relies on comparison with a physical space, so by solving a mathematical problem, you develop your geometric intuition as well. It might often happen that you suddenly realize that the problem you are stuck on is very similar to something else, and you could solve it by borrowing a method from another field of science.

I have never liked spending too much time on the same subject. These was a subject though that I dealt with for about thirty years. I wanted to give it up on many occasions, but I always came across new problems that I had to solve.

I find it difficult to focus on a problem if I don’t care about it. As a matter of fact, the process of proving is technical and not particularly interesting. But sometimes you feel that the problem itself is interesting and that it will lead to an interesting result. If so, I agree to take it on. However, if I’m not interested in the problem, I will not waste my time on it. 

Is mathematics an objective science? Will there still be the number π after humankind ceases to exist?

We often feel as if we have discovered something new, but it seems to be just a chemical reaction in our brain. It may be that we do not discover anything; rather, we just make things up. For example, I have an inkling that zero is a make-believe number. It is a convention. Similarly, negative numbers are a result of a convention. After all, you ask yourself, what are real numbers? Are they just axiomatic objects?

I began asking myself such questions more often after Andrew Wiles proved Fermat’s Last Theorem. He once said in an interview that Fermat’s Theorem was true, but it had always been true. His words aroused my interest. It seemed like Fermat’s Theorem was true even before it was proven. How far can we go back into the past to claim its truth? Does it mean it was true even before Fermat was born? Was it true before the first man appeared on Earth? And what about the fine-structure constant? Is it a number similar in this respect to the mathematical constant π? Or it is something physical existing in reality? I don’t think there are exact answers to these questions.

In an interview, you compared your method with Miss Marple’s forensic method. Can you explain?

I meant the analogy method of problem solving. When she investigates a crime, Miss Marple often recollects an event that happened in her native town. On the face of it, there is no connection between the event and the crime, but eventually they turn out to be quite similar, which helps her move forward in the investigation.

Did other fictional detectives actually employ any methods other than this one?

Sherlock Holmes said that ‘once you exclude everything that is impossible, what remains necessarily has to be the truth, no matter how implausible it may seem’. This observation is partly true for mathematics. It often happens here: you obtain true results which seem counterintuitive. You just need to take them for granted.

What is the relationship between mathematics and physics today?

Today, the main area of cooperation between physics and pure mathematics is string theory. An opponent of this theory would tell you that this is because physicists who are dealing with the theory are incapable of proving anything if they remain within this discipline. Physicists have an intuition that is different from the intuition we mathematicians use. Physicists’ intuition helps us look at our problems from a different angle and solve them more efficiently. Whenever we can solve a problem, it does not matter to us that string theory has no application.

seminarseries.muni.cz / Masaryk University (the Czech Republic)

I’ve heard you like jazz. How did you come to like it?

What I like about jazz is improvisation as long as it doesn’t deviate too much from the original music piece. When I listen to some jazz musicians, I catch myself thinking that I would like to do mathematics in the same way. Some mathematical research seems to me too linear. You start by establishing the problem, then you develop it, you search for the proof, etc. Personally, I love improvising by putting the problem in different contexts.

I grew up in the 1960s, the era of rock stars. That was the time when music poured on you from everywhere. I disliked most of it. Jazz became a way of escape for me. It was the most intellectual music.

Can we say that mathematics is an art?

Yes, we can, but it’s not always true. Sometimes, we need to adhere to facts only, irrespective of how we manage to reach those facts. Some mathematical theories are not aesthetically attractive, but we have to prove them. This is just mechanical work. It is like making paint brushes for an artist.

Can mathematics be an objective science and an art at the same time?

Yes, it can. You need to be able to present the findings of your research, get your message across. If you can make an elegant and aesthetically attractive description of your theory, people will find it easier to accept it. It will be a horrible day when computers do all mathematical research. There are some math proofs (e.g., the four colour theorem) that were generated by computers. Of course, it is good that we were able to get a result, but it would also be good if we had a better way of proving. This would enable us to put the result into the context of research. It is nice to see when a part of mathematics is used to prove another part. This demonstrates and confirms the integrity of mathematics.

You say in your interviews that you like watching films, although you haven’t watched any recent sci-fi movies. Why?

When I was a teenager, I liked sci-fi books because I wanted to learn more about the future and read the contemplations about the contemporary society those books contained. I was particularly fascinated by short stories, in which the author takes an everyday situation and extrapolates it to the future instead of revealing an entire new world for the reader. That is what Ray Bradbury did. I was keen on thinking over what I read and drawing conclusions.

As for movies, I prefer watching films somehow related to mathematics. For instance, ‘The Man Who Knew Infinity’ (2015) about the Indian mathematician Srinivasa Ramanujan.

msc.tsinghua.cn / Tsinghua University (China)

You visited the Soviet Union in 1979. What would you say if you compared the past and the present generations of Russian mathematicians and their research?

I didn’t know much about Soviet mathematicians in the 1970s. Back then, you could hardly meet those people outside the Soviet Union. Once a mathematician from Rostov-on-Don visited us, and it was just, wow—a Soviet mathematician! I visited annual international meetings of mathematicians that were held in Bonn and founded by Friedrich Hirzebruch. Hirzebruch invited many Soviet mathematicians to the event every year. Some scientists went, but they were not the ones he had hoped would come. The people he actually wanted to participate in the venue were not allowed to come for political reasons.

When I was in Moscow, I met Yury Manin, with whom I coauthored an article. I was also able to attend a seminar by Igor Shafarevich and talk with some other people. I only spent a couple of days in Moscow. On the day of my departure, I was given about a hundred of hand-written mathematical papers to have them published in Western European journals. I think those people were afraid that the works wouldn’t reach the destination if they sent them by mail. When the customs officers were checking my luggage, I thought they would seize the papers. Luckily, the only thing they confiscated was a copy of the Economist.

Moscow has certainly changed a great deal over the past thirty years. It has become brighter, more diverse, and more colorful.

Have you noticed any changes in how mathematics is practiced now compared to in the past?

The key change is the internet. Today, we can publish our articles online, and they are immediately accessible to scientists not only at Oxford, but in India and all over the world. We, mathematicians, have become a global community. However, there is another problem: in this massive flow of information, it is more difficult to find the things that are of interest to you personally.

In this situation, can we say there is a ‘national science’?

Yes, certainly. National universities are different in terms of their approaches to research and teaching students. When students become professors, they usually continue using methods they learned as students. For example, the seminars of Israel Gelfand were very influential in this respect. Highly interactive, they have been inspiring seminars of Russian colleagues ever since.

Is there any difference between how science is taught in British and American universities?

British professors choose students, while in the USA, it is the other way around, because students choose professors there. In the UK, we have a notion of ‘graduate course’. The student chooses a course of advanced learning to study together with his or her supervisor. British professors review applications from students and invite some of the applicants for interviews before selecting the final candidates. American students do not rely so much on their mentors. Mentors only suggest areas of study for their students, but the latter have to come to their mentors with ready-made ideas.

What are the most amazing things happening in mathematics now?

I am retired now, so it is not always easy for me to follow all the recent news. It is like fashion. Perhaps, I should mention category theory, where great discoveries are still in store for us. Most of the articles on this topic are quite long, so I don’t have the patience to read them to the end. This theory doesn’t look amazing now, but there is something hidden behind it—although the way it is described is not my taste.

Another intriguing thing are the possible connections between theoretical physics and number theory. We might arrive at something that will help us solve classical problems, or it might give rise to new problems. On the whole, there is great potential in the connections between different areas of mathematics.