Introduction
I'm going to make a number of observations about the art of mathematics, the people who practice that art, and the ways such people differ from other people. Perhaps enough such observations, here and in other articles in this volume, will add up to some semi-coherent picture of mathematics and mathematicians.
Despite some existing stereotypes, I don't think mathematicians as a class are all that different from the general run of humanity. But it is true that any small group, selected on the basis of the ability to perform some special task, is not unlikely to be a bit different, at least statistically. Here are three differences which I'll discuss later in more detail: Mathematicians learn faster than most people and are better able to cope with novelty. Mathematicians are better able to concentrate deeply for extended periods of time. Some mathematicians have the ability to carry out thought processes at an “intuitive” level, in order to reach conclusions that would be unattainable, or attainable only very slowly, using linear thought processes.
Some of these tendencies, when carried to extremes, can result in behavior associated with the stereotypical absent-minded professor or mad scientist. To the extent that the more random person on the street has any picture of a typical mathematician (and most are totally unaware of mathematics and mathematicians), the picture is probably of a schizophrenic à la A Beautiful Mind or Proof, or an absent-minded eccentric. Of course it must be admitted that mathematics does have its share of eccentrics, but to my mind that speaks well of the tolerance of the mathematical community and its priorities; the eccentric who produces good mathematics is welcome in the community.
Here are some other things I'll do. I'll describe how students move through the U.S. educational system to become mathematicians. I'll discuss the international community of mathematicians. I'll give my own take on how a mathematician produces mathematics. I hope that this loosely connected collection of small essays will give at least a partial picture of mathematicians and their community.
Mathematics and Mathematicians
To understand mathematicians, it is perhaps useful to begin with a definition or description of mathematics; here is one description with which you may, or may not, agree.