The original version of this story appeared in Quanta Magazine. Calculus, a significant mathematical tool, was developed in the 17th century but initially relied heavily on intuition and informal arguments without precise formal definitions, resulting in an unstable foundation for centuries.
Two distinct schools of thought emerged in response, as noted by Michael Barany, a historian of math and science at the University of Edinburgh. French mathematicians predominantly focused on the practical application of calculus to physical problems, such as calculating planetary trajectories or investigating electrical currents. Meanwhile, by the 19th century, German mathematicians sought to critically examine and challenge existing assumptions, aiming to solidify calculus on a more stable basis through the identification of counterexamples.
One influential mathematician in this effort was Karl Weierstrass. Although he demonstrated early mathematical talent, his father urged him to pursue public finance and administration, preparing him for a career in the Prussian civil service. Disillusioned by his university studies, Weierstrass reportedly spent much of his time engaging in leisure activities like drinking and fencing. After failing to obtain his degree, he became a secondary school teacher, instructing students in various subjects, including mathematics and physics.
Weierstrass did not embark on a professional mathematical career until nearly 40. He eventually revolutionized the field by introducing what came to be known as a mathematical “monster.”
In 1872, Weierstrass published a function that challenged foundational calculus concepts, particularly alarming proponents of the French mathematical school. Henri Poincaré criticized Weierstrass’ function as “an outrage against common sense,” while Charles Hermite labeled it a “deplorable evil.”
To comprehend the unsettling nature of Weierstrass’ contribution, it is essential to understand two fundamental calculus concepts: continuity and differentiability. A continuous function is one without gaps or jumps, allowing a path to be traced from any point to another without needing to lift a writing instrument. Calculus primarily seeks to determine the rate of change in continuous functions, often by approximating them with straight, nonvertical lines.
Illustration by Mark Belan for Quanta Magazine.
