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.<p>>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.<p>>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]<p>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.<p>[0]: A Radical Approach to Real Analysis - David Bressoud
> The majority of mathematicians quickly became "Formalists", holding that pure mathematics could not be philosophically considered more than a sort of elaborate game played with marks on paper (this is the theory behind Robert Heinlein's pithy characterization of mathematics as "a zero-content system").<p>This isn't quite what Formalism is, at least not as Hilbert -- the originator of that philosophy -- described it. In short, Formalism says that <i>some</i> mathematical sentences <i>might</i> not have an external meaning, and those are the ones that are no more than a game with symbols.<p>More precisely, Hilbert divided mathematical formulas to "real," those that do have external meaning, and "ideal", those that do not. The real formulas are usually finitary, while the ideal ones usually deal with infinities. Formalism is the view that mathematics is allowed to contain ideal sentences provided that they do not yield contradictions with real ones.
After you have struggled with a category of good theories, it really gives you a vehicle of thought, sort of frees you from thinking about things in a certain context.
I'm not knowledgeable enough about their works to say anything interesting, but on the philosophy side of this there were a lot of developments in the 20th century, notably the logical positivists (e.g. Carnap) and later Quine.<p>Quine disagrees with the logical positivists in a way that I find a little tricky to pin down, despite his and their writing being much clearer than "continental" philosophers, but I have found everything I've read from either camp very thought provoking.
I think that in the age of autonomous robotics the reality gap between math and meat space is more relevant than ever. But the article does not really get into the hot and messy part. The closest match I know of is the varied field of cybernetics.
Mathematics is also surprisingly useful within mathematics itself: once thought of as two unrelated disciplines, algebra and geometry are now routinely used for the mutual benefit.