As far as I can remember, I have always found mathematics a thoroughly enjoyable, if challenging, endeavour. It carries the same intellectual challenge and sense of competition as chess and gives the same sense of understanding and accomplishment as art. Of course it is demanding. It requires both work and experience to appreciate a good theorem, but enjoying music or painting also is largely an acquired taste, greatly enhanced by the effort one puts into it.
Not many people share this rosy picture. It is a common opinion that mathematics is useless, if honourable, and that its overuse in education may turn students' attention away from practical situations towards abstract problems. Such people are fine if they stay within mathematics, but if they stray outside, they turn out as the wrong kind of theoreticians, the ones who pay more attention to their own ideas than to hard facts.
We have here all the elements of a heated debate, which is stirred up at regular intervals, most recently in France by Claude All gre, in a book called Plato's Defeat.
Claude All gre is not just anybody. At the time of writing, as prime minister Lionel Jospin puts together his Socialist cabinet, he stands poised to become minister for higher education and/or research. He is also a world-renowned geologist, and an outspoken personality. His book is a panorama of modern science in historical perspective. Plato is seen as the archetypal mathematician, trying to submit the whole world to a single mathematical theory. His ultimate defeat lies in the fact that this approach "from the top down" has proved ineffective: according to All gre, history proves over and over again that progress comes from experiments, not from theories, and that mathematics has never been a significant factor in scientific discoveries.
By so doing, he is picking up on an argument started a few years ago. The French Nobel prize winners, de Gennes and Charpak, have been saying for some time that French education gives students too much theory and too little practice, and that current programmes in physics were no more than mathematics in disguise. More hours should be spent in the lab and less in the teaching room. Students should experiment, and discover physical reality by themselves, instead of ingesting ready-made theories, and deriving practical consequences as exercises in mathematics. They should understand the questions before they learn the answers.
The school programmes have been adapting to this general mood. Programmes have been drifting away from the top-down approach to the bottom-up approach: "if you perform this experiment, this happens, and some day you will learn why". Even in mathematics, this approach is becoming prevalent, and professors are complaining that students enter universities with little idea of what a proof actually is.
High school mathematics is now mainly centred on teaching pupils "facts", meaning mathematical statements and operating rules, without backing them up with proofs. This would not be surprising in North America, where high school teaching gave up on proofs a long time ago, but it is quite a new situation in France, where the particular benefits of teaching mathematics were traditionally seen as training students in logic and precision.
As always in France, this phenomenon should be set against the general background of vastly increasing access to college and universities. The baccalaureat is traditionally divided into specialisations. Fifty years ago, 5,000 pupils registered as specialising in mathematics. Nowadays there are 200,000. The perspective has changed correspondingly, from a time where the majority of these students were aiming for the elite schools (Polytechnique, Ecole Normale Superieure) to now when their sheer number forces most of them into universities. The change in the mathematical programmes can thus be seen as a response to this shift in the population, away from an elitist idea of weeding the happy few, and towards a broad idea of giving the many something to hold on to.
The movement seems set to continue. Its proponents see it as a factor in the breakdown of some of the stereotypes of French society. People who hold public office, and high-ranking civil servants (they are often the same), are seen as remote from practical considerations, prone to forming general rules without due consideration to the variety of situations and finding out too late the difficulties of implementing these rules on the ground. Over and over again, French research is diagnosed as more adept at making scientific advances than in technological progress, French industry as better at making technological advances than in capitalising on them. All this is attributed (partly) to the education system, which supposedly puts too much emphasis on mathematics (and on theory in general), and too little on experimental science (and on fact-finding in general).
One can surely agree with the new emphasis on experiment, and the importance for the student of discovering the world by himself. Theory must be taught, because it saves enormous amounts of reflection and stores immense amounts of knowledge, but students must also learn to use the theory to good purpose, that is, they must learn to reason properly. One can point to the use of computer programs, which lead some students to accept without question numerical answers which can be incomplete, misleading or downright wrong. No kind of science can live without some kind of internal criticism, without raising at every step the question: what kind of numbers am I getting, and what do they actually mean?
Going back to my opening statement, it seems strange to me that, while mathematics is gaining an even wider acceptance in fields where it used to be quite marginal, it is now being challenged in fields which have used mathematical models for a long time. The financial industry now routinely does numerical computations of a level of complexity that only theoretical physics or chemistry used to reach. All this requires an inside knowledge of the way mathematics operates.
Perhaps physicists are now beyond mathematics, in the sense that they have assimilated it so well as a technique that they feel that it is no longer a crucial factor for further progress. Economists and financiers are still a long way from reaching that stage, and students in these domains need a strong dose of old-fashioned mathematics, the ones with the proofs and the nasty equations.
Ivar Ekeland is a former president of the University of Paris Dauphine.