Chaos - James Gleick [41]
The École Normale and École Polytechnique were elite schools with no parallel in American education. Together they prepared fewer than 300 students in each class for careers in the French universities and civil service. Mandelbrot began in Normale, the smaller and more prestigious of the two, but left within days for Polytechnique. He was already a refugee from Bourbaki.
Perhaps nowhere but in France, with its love of authoritarian academies and received rules for learning, could Bourbaki have arisen. It began as a club, founded in the unsettled wake of World War I by Szolem Mandelbrot and a handful of other insouciant young mathematicians looking for a way to rebuild French mathematics. The vicious demographics of war had left an age gap between university professors and students, disrupting the tradition of academic continuity, and these brilliant young men set out to establish new foundations for the practice of mathematics. The name of their group was itself an inside joke, borrowed for its strange and attractive sound—so it was later guessed—from a nineteenth-century French general of Greek origin. Bourbaki was born with a playfulness that soon disappeared.
Its members met in secrecy. Indeed, not all their names are known. Their number was fixed. When one member left, as was required at age 50, another would be elected by the remaining group. They were the best and the brightest of mathematicians, and their influence soon spread across the continent.
In part, Bourbaki began in reaction to Poincaré, the great man of the late nineteenth century, a phenomenally prolific thinker and writer who cared less than some for rigor. Poincaré would say, I know it must be right, so why should I prove it? Bourbaki believed that Poincaré had left a shaky basis for mathematics, and the group began to write an enormous treatise, more and more fanatical in style, meant to set the discipline straight. Logical analysis was central. A mathematician had to begin with solid first principles and deduce all the rest from them. The group stressed the primacy of mathematics among sciences, and also insisted upon a detachment from other sciences. Mathematics was mathematics—it could not be valued in terms of its application to real physical phenomena. And above all, Bourbaki rejected the use of pictures. A mathematician could always be fooled by his visual apparatus. Geometry was untrustworthy. Mathematics should be pure, formal, and austere.
Nor was this strictly a French development. In the United States, too, mathematicians were pulling away from the demands of the physical sciences as firmly as artists and writers were pulling away from the demands of popular taste. A hermetic sensibility prevailed. Mathematicians’ subjects became self-contained; their method became formally axiomatic. A mathematician could take pride in saying that his work explained nothing in the world or in science. Much good came of this attitude, and mathematicians treasured it. Steve Smale, even while he was working to reunite mathematics and natural science, believed, as deeply as he believed anything, that mathematics should be something all by itself. With self-containment came clarity. And clarity, too, went hand in hand with the rigor of the axiomatic method. Every serious mathematician understands that rigor is the defining strength of the discipline, the steel skeleton without which all would collapse. Rigor is what allows mathematicians to pick up a line of thought that extends over centuries and continue it, with a firm guarantee.
Even so, the demands of rigor had unintended consequences for mathematics in the twentieth century. The field develops through