…How do mathematicians solve problems? There have been few rigorous scientific studies of this question. Modern educational research, based on cognitive science, largely focuses on education up to high school level. Some studies address the teaching of undergraduate mathematics, but those are relatively few. There are significant differences between learning and teaching existing mathematics and creating new mathematics. Many of us can play a musical instrument, but far fewer can compose a concerto or even write a pop song.
When it comes to creativity at the highest levels, much of what we know – or think we know – comes from introspection. We ask mathematicians to explain their thought processes, and seek general principles. One of the first serious attempts to find out how mathematicians think was Jacques Hadamard’s The Psychology of Invention in the Mathematical Field, first published in 1945. Hadamard interviewed leading mathematicians and scientists of his day and asked them to describe how they thought when working on difficult problems. What emerged, very strongly, was the vital role of what for lack of a better term must be described as intuition. Some feature of the subconscious mind guided their thoughts. Their most creative insights did not arise through step by step logic, but by sudden, wild leaps.
One of the most detailed descriptions of this apparently illogical approach to logical questions was provided by the French mathematician Henri Poincaré, one of the leading figures of the late nineteenth and early twentieth centuries. Poincaré ranged across most of mathematics, founding several new areas and radically changing many others. He plays a prominent role in several later chapters. He also wrote popular science books, and this breadth of experience may have helped him to gain a deeper understanding of his own thought processes. At any rate, Poincaré was adamant that conscious logic was only part of the creative process. Yes, there were times when it was indispensable: deciding what the problem really was, systematically verifying the answer. But in between, Poincaré felt that his brain was often working on the problem without telling him, in ways that he simply could not fathom.
His outline of the creative process distinguished three key stages: preparation, incubation, and illumination. Preparation consists of conscious logical efforts to pin the problem down, make it precise, and attack it by conventional methods. This stage Poincaré considered essential: it gets the subconscious going and provides raw materials for it to work with. Incubation takes place when you stop thinking about the problem and go off and do something else. The subconscious now starts combining ideas with each other, often quite wild ideas, until light starts to dawn. With luck, this leads to illumination: your subconscious taps you on the shoulder and the proverbial light bulb goes off in your mind.
This kind of creativity is like walking a tightrope. On the one hand, you won’t solve a difficult problem unless you make yourself familiar with the area to which it seems to belong – along with many other areas, which may or may not be related, just in case they are. On the other hand, if all you do is get trapped into standard ways of thinking, which others have already tried, fruitlessly, then you will be stuck in a mental rut and discover nothing new. So the trick is to know a lot, integrate it consciously, put your brain in gear for weeks . . . and then set the question aside. The intuitive part of your mind then goes to work, rubs ideas against each other to see whether the sparks fly, and notifies you when it has found something. This can happen at any moment: Poincaré suddenly saw how to solve a problem that had been bugging him for months when he was stepping off a bus. Srinivasa Ramanujan, a self-taught Indian mathematician with a talent for remarkable formulas, often got his ideas in dreams. Archimedes famously worked out how to test metal to see if it were gold when he was having a bath.
Poincaré took pains to point out that without the initial period of preparation, progress is unlikely. The subconscious, he insisted, needs to be given plenty to think about, otherwise the fortuitous combinations of ideas that will eventually lead to a solution cannot form. Perspiration begets inspiration. He must also have known – because any creative mathematician does – that this simple three-stage process seldom occurs just once. Solving a problem often requires more than one breakthrough. The incubation stage for one idea may be interrupted by a subsidiary process of preparation, incubation, and illumination for something that is needed to make the first idea work. The solution to any problem worth its salt, be it great or not, typically involves many such sequences, nested inside each other like one of Benoît Mandelbrot’s intricate fractals. You solve a problem by breaking it down into subproblems. You convince yourself that if you can solve these subproblems, then you can assemble the results to solve the whole thing. Then you work on the subproblems. Sometimes you solve one; sometimes you fail, and a rethink is in order. Sometimes a subproblem itself breaks up into more pieces. It can be quite a task just to keep track of the plan.
I described the workings of the subconscious as ‘intuition’. This is one of those seductive words like ‘instinct’, which is widely used even though it is devoid of any real meaning. It’s a name for something whose presence we recognise, but which we do not understand. Mathematical intuition is the mind’s ability to sense form and structure, to detect patterns that we cannot consciously perceive. Intuition lacks the crystal clarity of conscious logic, but it makes up for that by drawing attention to things we would never have consciously considered. Neuroscientists are barely starting to understand how the brain carries out much simpler tasks. But however intuition works, it must be a consequence of the structure of the brain and how it interacts with the external world.
Often the key contribution of intuition is to make us aware of weak points in a problem, places where it may be vulnerable to attack. A mathematical proof is like a battle, or if you prefer a less warlike metaphor, a game of chess. Once a potential weak point has been identified, the mathematician’s technical grasp of the machinery of mathematics can be brought to bear to exploit it. Like Archimedes, who wanted a firm place to stand so that he could move the Earth, the research mathematician needs some way to exert leverage on the problem. One key idea can open it up, making it vulnerable to standard methods. After that, it’s just a matter of technique.
My favourite example of this kind of leverage is a puzzle that has no intrinsic mathematical significance, but drives home an important message. Suppose you have a chessboard, with 64 squares, and a supply of dominoes just the right size to cover two adjacent squares of the board. Then it’s easy to cover the entire board with 32 dominoes. But now suppose that two diagonally opposite corners of the board have been removed, as in Figure 1. Can the remaining 62 squares be covered using 31 dominoes? If you experiment, nothing seems to work. On the other hand, it’s hard to see any obvious reason for the task to be impossible. Until you realise that however the dominoes are arranged, each of them must cover one black square and one white square. This is your lever; all you have to do now is to wield it. It implies that any region covered by dominoes contains the same number of black squares as it does white squares. But diagonally opposite squares have the same colour, so removing two of them (here white ones) leads to a shape with two more black squares than white. So no such shape can be covered. The observation about the combination of colours that any domino covers is the weak point in the puzzle. It gives you a place to plant your logical lever, and push. If you were a medieval baron assaulting a castle, this would be the weak point in the wall – the place where you should concentrate the firepower of your trebuchets, or dig a tunnel to undermine it.
Mathematical research differs from a battle in one important way. Any territory you once occupy remains yours for ever. You may decide to concentrate your efforts somewhere else, but once a theorem is proved, it doesn’t disappear again. This is how mathematicians make progress on a problem, even when they fail to solve it. They establish a new fact, which is then available for anyone else to use, in any con-
text whatsoever. Often the launchpad for a fresh assault on an age-old problem emerges from a previously unnoticed jewel half-buried in a shapeless heap of assorted facts. And that’s one reason why new mathematics can be important for its own sake, even if its uses are not immediately apparent. It is one more piece of territory occupied, one more weapon in the armoury. Its time may yet come – but it certainly won’t if it is deemed ‘useless’ and forgotten, or never allowed to come into existence because no one can see what it is for.
Fig 1 Can you cover the hacked chessboard with dominoes, each covering two squares (top right)? If you colour the domino (bottom right) and count how many black and white squares there are, the answer is clear.