In the late 19th century, Karl Weierstrass invented a fractal-like function that was decried as nothing less than a “deplorable evil.” In time, it would transform the foundations of mathematics.

Calculus is a powerful mathematical tool. But for hundreds of years after its invention in the 17th century, it stood on a shaky foundation. Its core concepts were rooted in intuition and informal arguments, rather than precise, formal definitions.

Two schools of thought emerged in response, according to Michael Barany(opens a new tab), a historian of math and science at the University of Edinburgh. French mathematicians were by and large content to keep going. They were more concerned with applying calculus to problems in physics — using it to compute the trajectories of planets, for instance, or to study the behavior of electric currents. But by the 19th century, German mathematicians had begun to tear things down. They set out to find counterexamples that would undermine long-held assumptions, and eventually used those counterexamples to put calculus on more stable and durable footing.

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