The original version of this story appeared in Quanta Magazine.
Calculus is a significant mathematical tool that, for centuries following its development in the 17th century, was built on uncertain foundations. Its fundamental concepts were initially based on intuition and informal reasoning rather than on precise formal definitions.
According to Michael Barany, a historian of mathematics and science at the University of Edinburgh, two distinct schools of thought evolved in response to this issue. French mathematicians were largely content to continue using calculus, focusing on its application to solve problems in physics, such as calculating planetary trajectories and examining electric currents. By contrast, in the 19th century, German mathematicians began challenging established assumptions. They sought counterexamples to build a more stable and enduring framework for calculus.
Karl Weierstrass was among these pioneering mathematicians. Despite demonstrating an early talent for mathematics, his father directed him towards studying public finance and administration with the intention of joining the Prussian civil service. Disenchanted with his university studies, Weierstrass reportedly spent much of his time engaged in drinking and fencing, eventually failing to earn his degree. Subsequently, he became a secondary school teacher, instructing students in a range of subjects including mathematics, physics, penmanship, and gymnastics.
Weierstrass did not embark on a professional career in mathematics until he was nearly 40 years old. Nevertheless, he significantly impacted the field by introducing a groundbreaking mathematical concept.
In 1872, Weierstrass unveiled a function that challenged the prevailing understanding of calculus. His work was met with a mix of indifference, anger, and fear, particularly from the influential members of the French school of mathematics. Henri Poincaré criticized Weierstrass’ function as an affront to common sense, while Charles Hermite described it as a “deplorable evil.”
To comprehend why Weierstrass’ findings were so unsettling, it is important to first grasp two fundamental concepts in calculus: continuity and differentiability. A continuous function, as the name suggests, is one without gaps or jumps, allowing a smooth path from any point to another without lifting a pencil. A major focus of calculus is on understanding how these continuous functions change, typically by approximating them with straight, nonvertical lines.
