What Fourier proved is that every function could be written as a sum of sines and cosines. However his proof worked for "functions" like step functions, which were not at the time accepted as functions, and which could not be sensibly analyzed with the infinitesmal techniques of the day.
It is hard to overstate the shock that came from a sum of nicely behaved functions turning into the pathological step function. If that was possible, what else could go wrong? And if infinitesmals could not be trusted, how could Calculus be put on a rigorous footing?
The question of how to put mathematics on a secure footing were central to 19th century mathematics, and the issues lead directly to set theory, and the unavoidable dead end discovered by Goedel. (Ironically the work in logic that came out of that eventually lead to nonstandard analysis, which in turn justified the infinitesmal approach and most of the infinitesmal arguments. But by then mathematics didn't much care.)
Fascinating! I know about that story from the point at which the search for rigorous foundations had already begun (i.e. late 19th century), but no awareness of the connection to Fourier. Thanks.
Have you read The Mathematical Experience? Some of its essays address Fourier specifically and what it meant for math. The whole book is fascinating, even if you think you know the material.
It is hard to overstate the shock that came from a sum of nicely behaved functions turning into the pathological step function. If that was possible, what else could go wrong? And if infinitesmals could not be trusted, how could Calculus be put on a rigorous footing?
The question of how to put mathematics on a secure footing were central to 19th century mathematics, and the issues lead directly to set theory, and the unavoidable dead end discovered by Goedel. (Ironically the work in logic that came out of that eventually lead to nonstandard analysis, which in turn justified the infinitesmal approach and most of the infinitesmal arguments. But by then mathematics didn't much care.)