Experimentation and Proof in Mathematics

670 NOTICES OF THE AMS VOLUME 42, NUMBER 6 T he English word “prove”—as its Old French and Latin ancestors—has two basic meanings: to try or test, and to establish beyond doubt. The first meaning is largely archaic, though it survives in technical expressions (printer’s proofs) and adages (the exception proves the rule, the proof of the pudding). That these two meanings could have coexisted for so long may seem strange to us mathematicians today, accustomed as we are to thinking of “proof” as an unambiguous term. But it is in fact quite natural, because the most common way to establish something in everyday life is to examine it, test it, probe it, experiment with it. As it turns out, much the same is true in mathematics as well. Most mathematicians spend a lot of time thinking about and analyzing particular examples. This motivates future development of theory and gives one a deeper understanding of existing theory. Gauss declared, and his notebooks attest to it, that his way of arriving at mathematical truths was “through systematic experimentation”. It is probably the case that most significant advances in mathematics have arisen from experimentation with examples. For instance, Prove: to try or test or to establish beyond doubt the theory of dynamical systems arose from observations made on the stars and planets and, more generally, from the study of physically motivated differential equations. A nice modern example is the discovery of the tree structure of certain Julia sets by Douady and Hubbard: this was first observed by looking at pictures produced by computers and was then proved by formal arguments. It is disturbing that such considerations are usually totally excluded from the published record. What one generally gets in print is a daunting logical cliff that only an experienced mountaineer might attempt to scale, and even then only with special equipment. Is this the best thing for the research community? Is it fair to graduate students? Should we give the impression that the best mathematics is some sort of magic conjured out of thin air by extraordinary people when it is actually the result of hard work and of intuition built on the study of many special cases? In our educational institutions, we spend too much time revealing these almost unscalable logical edifices instead of giving others