[WoahNoun]

This article reminds me of the story of Fourier. It's hard to tell the history of math without the way in which Fourier series and the heat equation shaped the development and increasing rigor in calculus. The mathematicians of the day were extremely skeptical of infinite sums of trigonometric functions because it didn't fit their "a-priori" model of calculus.

>Here was the heart of the crisis. Infinite sums of trigonometric functions had appeared before. Daniel Bernoulli (1700-1782) proposed such sums in 1753 as solutions to the problem of modeling the vibrating string. They had been dismissed by the. greatest mathematician of the time, Leonhard Euler (1707-1783). Perhaps Euler scented the danger they presented to his understanding of calculus. The committee that reviewed Fourier's manuscript: Pierre Simon Laplace (1749-1827), Joseph Louis Lagrange (1736-1813), Sylvestre Francois Lacroix (1765-1843), and Gaspard Monge (1746-1818), echoed Euler's dismissal in an unenthusiastic summary written by Simeon Denis Poisson (1781-1840). Lagrange was later to make his objections explicit.

>Well into the 1820s, Fourier series would remain suspect because they contradicted the established wisdom about the nature of functions. Fourier did more than suggest that the solution to the heat equation lay in his trigonometric series. He gave a simple and practical means of finding those coefficients, the ai, for any function. In so doing, he produced a vast array of verifiable solutions to specific problems. Bernoulli's proposition could be debated endlessly with little effect for it was only theoretical. Fourier was modeling actual physical phenomena. His solution could not be rejected without forcing the question of why it seemed to work.[0]

I picture this in my head as Fourier setting some shit on fire, hand calculating Fourier coefficients, then just pointing and yelling "SEE! SEE!" at Poisson and Lagrange.

[0]: A Radical Approach to Real Analysis - David Bressoud

[chalst]

This is a curious take on the state of calculus before Cauchy: mathematicians were interested in infinite series, but aware of paradoxes around convergence: for example, the set {1,-1/2,1/3,-1/4,1/5,...} doesn't have a particular sum, but instead you can arrange the elements into series to converge to numbers of any size or even not converge at all. Fourier's work didn't threaten an established notion of calculus, rather mathematicians were having difficulty sorting sense from nonsense in the fertile but chaotic subfield.

It wasn't until half a century later, after Cauchy, that mathematicians had a powerful and coherent foundation for calculus. It's true that then interesting ideas such as inginitesimals were rejected because they lacked comparable rigour: was Bressoud conflating these two time periods?

[Koshkin]

Yeah, it's kind of ironic that the modern - and entirely rigorous - direct treatment of infinitesimals comes under the name "nonstandard analysis."

As to the foundations, what Fourier's work shattered was the sufficiency of then established notion of what a function is, which had been limited to what we now call analytic functions; this eventually lead to the abstract definition of a function that we have today.

[WoahNoun]

There's a lot of historic detail in the book that is left out of this snippet, but his point was that mathematicians barely understood the convergence of infinite series and considered an infinite sum of trig functions to sort of be unfathomable. A big part of Cauchy's work made rigorous fourier series.

[Koshkin]

From [0]:

> calculus was given a new name: Analysis

"Mathematical analysis of the infintesimal", peaked in the works of Euler, was just that: an attempt to mathematically analyze, among other things, the notion and applications of the infinitely small (and infinity in general); for some reason the words "mathematical" and "infinity" were subsequently dropped, and we are left just with a generic term "analysis" which now requires context to be understood properly.