HOW DOES A MATHEMATICIAN THINK - THE NATURE OF RESEARCH IN MATHEMATICS - The Remarkable Role of Evolution in the Making of Mathematics - Mathematics and the Real World

Mathematics and the Real World: The Remarkable Role of Evolution in the Making of Mathematics (2014)

CHAPTER IX. THE NATURE OF RESEARCH IN MATHEMATICS

What does a mathematician do when he gets to the office in the morning? • How does sleep help to solve problems in mathematics? • Does creativity in mathematics decline with age? • Why would a mathematician refuse to accept a million dollars? • Does pure mathematics exist? • Why did the engines of steamships start exploding? • Can aliens do the sum 2 + 2?

62. HOW DOES A MATHEMATICIAN THINK?

We will start with the bottom line: there is no difference between the thought process of a mathematician and those in other disciplines. Before we clarify this issue, we must explain what we mean when we use the word think. Thinking means activating the brain to analyze situations, reach conclusions, and propose courses of action or solutions. With regard to these functions, there are various levels of brain activity, and we will concentrate on two that are related but different. The first type of thinking takes place in situations in which you have to carry out an action about which you have received previous guidance on how to proceed. For example, you have been given a cake recipe, and now you have to make that cake. Or, someone explained to you how to use a road map, and now you have to find the route from one place to another. Or, you have been taught how to use a paintbrush and oil paints to paint a still life, and now you wish to paint flowers in a vase. Or, you have learned how to design a car engine, and now you have to design one. Or, you have been taught how to solve a certain type of exercise in algebra, and in an examination you have to solve an exercise of that sort. All of the above need thought, and the thought process is matching the required process to those learned or tried in one way or another. We will call such thinking thinking by comparison.

The second type of thinking occurs when we use our brain to react to an unfamiliar situation in which we have not learned how to operate, or in which we consciously want to deviate from the normal course of action. For example, you are on a desert island, you have to put together a meal from the vegetation you can find, and you do not even know if the various plants are edible. Or, you have reached an unfamiliar location and have to find your way without a road map. Or, you have artists’ materials in front of you, and you have decided to paint a picture in a completely new, unfamiliar style. Or, you have to design a vehicle that will be able to move on an asteroid, the nature of whose surface is unknown. Or, you are trying to find a mathematical characteristic of an unknown system that no one has researched. In such situations, creative thinking is needed.

The two types of thinking are not detached from each other. Even when comparative thinking is called for, in general there are differences between the situation previously encountered and the current one, and a measure of creativity will be necessary to match the solution properly to the new situation. Where new situations are encountered and creative thinking is required, creativity is not ex nihilo, and elements of comparative thinking will play a part. Comparative thinking focuses on searching and matching, which are essentially routine, sometimes automatic, operations and can even be performed by a computer. The other type of thinking, in contrast, is based mainly on intuition, feeling, hunches. When a person has to deal with a problem, the brain will “decide” whether to activate comparative thinking or to enlist creative intuition according to the person's familiarity with that or similar problems. The brain's “decision” is not generally a conscious one.

Thinking by comparison cannot be taught or learned in isolation from the subject; in other words, this method of thinking cannot be learned in the abstract. In every subject, from cooking, through engineering, to mathematics, more and more can be learned to a level such that additional problems will be within a person's knowledge so that he can make do with comparative thinking and shorten the route to an appropriate solution. Unlike comparative thinking, creative thinking cannot be taught or learned at all. Creative, intuitive thinking can only be encouraged and stimulated. Trying to teach creative thinking via examples is a positive step, but the effect of the examples is simply to enrich the collection of problems that can be solved by comparative thinking. With regard to intuition itself, the more you know, the more you will be able to widen the sphere in which you can exercise creativity, but we do not know enough about how human intuition works to be able to teach people how to develop their creative intuition.

Once we have understood what thinking is, we can repeat: there is no difference between the thinking of mathematicians and the thinking of those in other professions. Obviously they think about different subjects. Cooking, finding your way somewhere, painting, engineering, and mathematics are highly dissimilar, but the types of thinking and the methods of thinking are the same in all of them. The myth that to be a mathematician you need a special type of brain is just as correct, or incorrect, as saying that to be a chef you need a special type of brain, and the same applies to the ability to navigate or to introduce novel ways of painting. Research in mathematics for the most part deals with new developments and is more closely related to the creative type of thinking, but the same could be said of creative research in any subject. One difference between research in mathematics and other research is that the intuition required in mathematical activity needs more thinking time. That is because research in mathematics deals mainly with logical aspects, and these are less at home among the brain's abilities as formed by evolution. The expression “Wait a moment, give me time to think about it” comes up more often in mathematical discussions than in conversations about cooking or navigating a route. Therefore, for most of their research time, mathematicians are immersed in reflection and intuitive thinking. Only after they have reached a solution intuitively do they start work on recording the solution in the acceptable language of logic. Henri Poincaré is credited with having said, “It is by logic that we prove, but by intuition that we discover.” The proof stage itself, however, belongs essentially in the first, more technical, type of thinking.

This distinction between the logical formulation of a mathematical result and the intuitive thinking carried out before the formulation of the result is also reflected in the way information is transmitted between mathematicians. A conversation on a mathematical subject is unlike mathematical writing that we recognize from our studies, books, or academic articles. We are often asked about the utility of flying for several hours to meet a colleague in a research project or to participate in a mathematics conference in some distant land. Would it not be enough, in this age of electronic communications, to read articles or correspond via e-mail? The answer is that in the intuitive thinking stage there is no substitute for face-to-face meetings. Information, which in a certain sense is subconscious and as such is difficult even to define, can be transmitted only in such a meeting. This is not confined to mathematics alone but applies to other areas of research too. My reason for stressing this aspect of mathematics is that it is a common misconception that logical mathematics will be less in need of the intuitive, subconscious level of thinking. The opposite is the case: most mathematical research is based on that intuition.

The differences we have described are responsible for the fact that unexplained events related to thinking, which also exist in other disciplines, occur frequently among mathematicians. It is not unusual that a colleague asks me to help him check a mathematical development he has conceived and insists on showing me the problem, despite my objections that I do not understand anything about the subject. He starts covering the blackboard in my office with the development in question, and after a while, without my having uttered a word, he stops, thinks, and says, “Thank you very much. Now I understand what has happened. You have helped me a great deal.” And I have indeed helped him a lot, just as I have been helped in a similar manner by colleagues. The very fact that we have tried to explain our findings to someone who is likely to have at least some idea of what we are trying to achieve is of great help.

Another well-known fact is that intuitive thinking to a large extent takes place subconsciously, sometimes even during sleep. Both Poincaré, whom we have discussed previously, and Jacques Hadamard, another leading French mathematician, wrote in their books about mathematical problems that they tried to solve over a long period, and then, in the course of an event completely unrelated to mathematics, the complete solution suddenly dawned on them.

This phenomenon is known at all levels of the profession. Just recently I was deeply engrossed in trying to solve a problem but made hardly any significant progress in my daily work. One morning I woke up with the complete solution to the problem, or so I thought. Checking it in the office, I realized that the solution was incomplete. The complete solution required another night's sleep. The conclusion to be drawn is not to have a sound sleep every time you encounter a mathematical problem that you do not manage to solve. Investing effort and concentration are certainly essential. After you have invested time and much effort, having a break does not mean that your brain stops trying to solve the problem. Moreover, the break is likely to be beneficial.

Another aspect of subconscious thinking is the context in which the result of such thinking comes to light. I used to live about fifty minutes’ drive from my office in the Weizmann Institute. With a frequency that I cannot explain, at a certain point where the tree-lined road curved as I drove home, I would realize that I had made a mistake in my work that day. It would take a day or two to correct the error, but then, at the same spot on my journey home, another error would spring into my mind (obviously, when I moved closer to the office, the number of mistakes I had to correct in my articles declined significantly).

Colleagues tell me that quite often the solution to a problem, or a new idea leading to the solution, comes to mind while they are watching television (even today my wife finds it hard to believe that when I am watching a sitcom, I am actually working!). It may be that there is no significance in and no statistical explanation for these occasions, and the fact that we attribute importance to them is a result of some mental illusion. Such an illusion could derive from the fact that exceptional events, like discovering an error while driving past trees, are more memorable than more normal occurrences, such as discovering a mistake while working in the office. Yet it may also be that something in our brain causes solutions to spring to mind in just such unlikely situations.

Another surprising factor is the different degree of difficulty between solving a mathematical problem if we know that someone else has already solved it, say, as an exercise in a class, and solving a new problem that no one has solved yet. The latter are called open problems. The story is told of the mathematician John Milnor, that when he was a student he had dozed off in a class when the lecturer wrote on the board an open problem in mathematics. When he awoke from his nap and saw what was on the board, he thought that it was an assignment to be done for the following week. The next week he arrived with the solution. John Milnor is a top mathematician. In 1962, at the age of thirty-one, he won the Fields Medal (officially the International Medal for Outstanding Discoveries in Mathematics), awarded to mathematicians under the age of forty; in 1989 he won the Wolf Prize; and in 2011 he was the Abel Prize laureate. Of course you have to be a genius like John Milnor to solve an open problem as if it were simply an exercise to prepare for the next class, but this sort of thing is known to occur at all levels of research: you can work on an open problem for many months, and only after you have found the solution do you realize that the problem could have been solved much more simply, but that is being wise with the benefit of hindsight.

The fact that research problems, that is, open problems so far unsolved, are more difficult than other problems set in class frustrates many students setting out along the path of research. After investing great effort in solving an open problem, they discover that they could have solved it more simply, just as they had solved other problems in the various courses they took. Many of them draw the conclusion that they are not suited to research, which may be an incorrect conclusion. It is not clear why it is that the fact that a problem is an open problem makes it more difficult to solve. In my opinion, a reasonable explanation is that thinking by comparison and creative thinking may take place in different parts of the brain. The creative section of the brain works less efficiently than does the comparative part. When the brain has to solve a problem, it channels the problem to what it considers to be the appropriate section for handling the problem, and that is the cause for the difference in the level of difficulty in solving a problem presented as an exercise and one introduced as an open problem. I would even go so far as to say that the capability of creative thinking is what differentiates man from the rest of the animal world, but that of course is pure speculation.

Does creative thinking in mathematics decline with age? This is the place to dispel a myth about the relation between mathematical research and age. According to the myth, after a certain age, some claim as early as thirty, creative ability in mathematics declines and eventually disappears. That is not so, and I will explain why. Physicists, and I have heard this from top physicists, agree that creative ability in physics declines from the age of thirty-something. That does not mean that after that age physicists do not contribute to their profession, but rather that the breakthroughs and innovative developments are achieved by the younger physicists. Yuval Ne'eman claimed that this does not refer to chronological age but to the length of time in the discipline (Ne’eman himself completed his doctorate at a relatively late age), and the reason for the reduction in the ability to introduce innovation is that as people accumulate experience and become accustomed to certain truths and to a certain method of research, it gets harder for them to challenge accepted norms and practices and to disprove or change them. And indeed, the beginning of most breakthroughs in physics was the refutation of deep-rooted beliefs. That is not the case in mathematics. The fundamentals laid down by the Greeks are still applicable. Important discoveries in mathematics that contradicted previous basic approaches were few and far between. Knowledge expanded enormously, new areas of research were added, unexpected applications were discovered, but today's work method is the same as that of the Greeks. Moreover, the results achieved by the Greeks and their successors throughout the generations are still relevant. No other discipline in the natural sciences can claim to have such a stable, cumulative nature. Thus, knowledge and experience play a much more important role in mathematical research than they do in other sciences. That is why we see mathematicians creating and making new discoveries at a more advanced age, as long as their enthusiasm for the subject is maintained and their health allows.

So what does a mathematician do when he arrives at his office in the morning? The answer is obvious: he drinks a cup of coffee. And then? He drinks another cup of coffee. This description is not my idea. One of the more graphic mathematicians of the twentieth century, the Hungarian Paul Erdős (1913–1996), once defined a mathematician as a machine into which coffee is poured at one end, and at the other out come mathematical theorems. Most of a mathematician's research time is spent in thinking how to solve the problem he or she is working on. How to activate intuition varies from one mathematician to another. Some prefer to think via interaction with colleagues, others have to be alone in a perfectly silent room, others favor working while listening to classical music, and others think best while strolling around. One well-known example is that of Steve Smale of the University of California, Berkeley, a leading mathematician of the twentieth century. His many awards and prizes include the Fields Medal in 1966 and the Wolf Prize in 2007. He was given a grant so that he could devote his summer to research. The authorities discovered that he spent the summer lying on a beach in Rio de Janeiro and asked him to return the grant money he had received. Smale claimed that he was in fact working while lying on the beach, and indeed that was when he had some of his best mathematical ideas. He was able to prove his claim and convinced the committee investigating the case, which ruled in his favor. This was not just an excuse on Smale's part. I once participated in a conference at Luminy, near Marseilles in France. Steve Smale was among the participants, and he insisted that the conference timetable leave time for those who wished to do so to get inspiration on the beautiful Mediterranean beaches. Indeed, time spent on the beach did improve the quality of the lectures at the conference.