Where does new mathematics come from? The great French mathematician Henri Poincaré, a brilliant expositor of the scientific method, described how he grappled for months with an arcane problem in function theory.
Exasperated by lack of progress, he went on holidays and forgot about the problem. But, as he was boarding a bus, the answer came to him in a flash. He was later able to return to his office and complete a proof of the result.
This is characteristic of the sudden spurts of progress that often occur. New mathematical ideas do not arise according to some pre-determined plan; they seem to pop up in random fashion, but this invariably happens following sustained efforts and deep thought, which lay the groundwork for discovery.
Some mathematicians keep regular and limited working hours. The renowned English mathematician GH Hardy began his research after breakfast each day, working until 1pm. Then he would have lunch and head for the cricket ground. But, of course, progress was not confined to the four morning hours. The intense conscious mental effort triggered unconscious activity, with sudden insights breaking through at any time, day or night.
Where do the ideas come from?
Ideas come from a multitude of sources, but the researcher needs time to think about them, follow up hunches and explore various options. No holds are barred at this stage and intuition is given free rein. A mathematician may have a feeling that a certain result is valid but cannot prove it.
Intuition is a valuable guide, but it can also mislead us to believe what appears obvious but is false. For this reason, mathematicians insist on a cast-iron proof, understandable to their colleagues and removing all doubt. A result does not become a theorem until the conditions for its validity are established and it is shown to follow logically from these conditions.
In the past, most research was done by mathematicians working alone. They might maintain contact with colleagues and attend meetings and conferences, but published papers usually had a single author. This is no longer the case: today, most papers have two or more authors and collaborative work is the norm.
Where does it all happen?
The conventional notion is that research is carried out at an office desk or in a library. A mathematical idea may arise at a meeting, a seminar, or a conference. There are also research centres dedicated to mathematics: INI in Cambridge, IHP in Paris, MFO in Oberwolfach, IMA in Minnesota and many more, where mathematicians gather for a week or two to work together.
Walking provides a great opportunity for thinking and many researchers have written about how their best ideas came to them during a ramble. René Descartes did his mathematical thinking while still in bed in the morning. The Indian genius Srinivasa Ramanujan was inspired by Mahalakshmi of Namakkal, a family Goddess when, as he slept, complex mathematical results unfolded before his eyes.
American topologist Stephen Smale used to do his research on the beach in Rio de Janeiro, and André Weil proved one of his most famous results, the Riemann hypothesis for curves over finite fields, while in prison. Other likely places for inspiration are on a train or plane, in a café or church, at a concert or in dreamland.
Peter Lynch is emeritus professor at UCD school of mathematics & statistics – he blogs at thatsmaths.com