Sometimes a mathematical paper is just a crock, putting forth an unsound argument to support an incorrect conclusion. in a very few painful cases, this has left nothing behind at all once the error has been corrected, or at least nothing new. Today, you will know about the worst mistake ever done in history or at least by a famous mathematician.
Maybe the wrongest mathematical paper ever, that wasn’t simply a complete mistake, was Fourier’s paper introducing, in effect, the continual Fourier transformation. it had been followed by an unclear torrent of counter-examples. Fourier had tripped over an armed and dangerous version of plenty of issues with infinitesimals which had been dodged or swept beneath the carpet.
Up to that time, mathematicians had tended to contemplate solely particular functions, or particular very restricted categories of functions, that were well-behaved nearly all over of interest. If a particular manipulation of a particular function led to odd results, one would merely quit and take a look at something else. currently, Fourier gave a general mechanism for manufacturing functions out of different functions, that wasn’t itself something new, however, it had a new ability, to begin with, a well-behaved function and switch it into something badly-behaved, without any warning.
Thanks to Fourier and his ambiguous accomplishment, mathematicians had to re-think what they meant by “function”, what they meant by “continuous function”, and what they were doing with those infinitesimals anyway. This led tolerably to a very deepening of the foundations of mathematical analysis. It took a period of agony before Cauchy ushered within the new world together with his Cours d’Analyse of 1821, followed up (and considerably corrected) by others, especially Weierstraß, over future few decades.
Unlikely as it sounds, it was really investigating Fourier-style representations of functions with these new tools and this new rigor that led Cantor to open up his separate paradise of infinite sets.