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Math’s Beautiful Monsters

How a destructive idea paved the way for modern math.

When Newton first developed calculus, he’d been inspired by the physical world: the trajectory of a planet, the swinging of a pendulum, the motion of falling fruit. This thinking led to a geometrical intuition about mathematical structures. They should make sense in the same way that a physical object would. As a result, many mathematicians concentrated on “continuous” functions. Conceptually, these are functions that can be drawn without taking pen from paper. Plot the speed of a falling apple over time and it will be a solid line; there will be no gaps or sudden jumps. A continuous function was, it was thought, a natural one.

Conventional wisdom held that for any continuous curve, it was possible to find the gradient at all but a finite number of points. This seemed to match intuition: A line might have a few jagged bits, but there would always be a few sections that were “smooth.” The French physicist and mathematician André-Marie Ampère had even published a proof of this claim. His argument was built on the “intuitively evident” fact that a continuous curve must have sections that increase, decrease, or remain flat. Which meant that it must be possible to calculate the gradient in these regions. Ampère did not think about what happened when the sections became infinitely small, but he claimed that he didn’t need to. His approach was general enough to avoid having to consider things that were “infiniment petits.” Most mathematicians were happy with his reasoning: By the middle of the 19th century, almost every calculus textbook quoted Ampère’s proof.

But during the 1860s, rumors started circulating about a strange creature, a mathematical function that contradicted Ampère’s theorem. In Germany, the great Bernhard Riemann told his students that he knew of a continuous function that had no smooth sections, and for which it was impossible to calculate the derivative of the function at any point. Riemann did not publish a proof, and neither did Charles Cellérier at the University of Geneva, who—despite writing that he’d discovered something “very important and I think new”—stuffed the work into a folder that would only become public after his death decades later. Yet if the claims were to be believed, it meant a threat to the very foundations of calculus was forming. This creature threatened to tear apart the happy relationship between mathematical theory and the physical observation on which it was based. Calculus had always been the language of the planets and stars, but how could nature be a reliable inspiration if there were mathematical functions that contradicted the central ideas of the subject?

The monster was finally born in 1872, when Karl Weierstrass announced that he had found a function that was continuous and yet not smooth at any point. He had constructed it by adding together an infinitely long sequence of cosine functions:

As a function, it was ugly and awkward. It was not even clear what it would look like when plotted on a graph. But that didn’t matter to Weierstrass. His proof consisted of equations rather than shapes, and this was what made his announcement so powerful. Not only had he created a monster, he’d built it from concrete logic. He had taken his new, rigorous definition of a derivative and shown it was impossible to calculate one for this new function.

Illustration by Shout

The result threw the mathematics community into a state of shock. The French mathematician Émile Picard pointed out that if Newton had known about such functions, he would have never created calculus. Rather than harnessing ideas about the physics of nature, he would have been stuck trying to clamber over rigid mathematical obstacles. The monster also began to trample over previous research. Results that had been “proven” started to buckle. Ampère had used the vague definitions favored by Cauchy to prove his smoothness theorem. Now, his arguments began to collapse. The vague notions of the past were hopeless against the monster. Worse, it was no longer clear what constituted a mathematical proof. The intuitive, geometry-based arguments of the previous two centuries seemed to be of little use. If mathematics tried to wave the monster away, it would stand firm. With one bizarre equation, Weierstrass had demonstrated that physical intuition was not a reliable foundation on which to build mathematical theories.

Established mathematicians tried to brush the result aside, arguing that it was awkward and unnecessary. They feared that pedants and troublemakers were hijacking their beloved subject. At the Sorbonne, Charles Hermite wrote, “I turn with terror and horror from this lamentable scourge of functions with no derivatives.” Henri Poincaré—who was the first to call such functions monsters—denounced Weierstrass’ work as “an outrage against common sense.” He claimed the functions were an arrogant distraction, and of little use to the subject.

“They are invented on purpose to show our ancestors’ reasoning is at fault,” he said, “and we shall never get anything more out of them.”

Many of the old guard wanted to leave Weierstrass’ monster in the wilderness of mathematics. It didn’t help that nobody could visualize the shape of the animal they were dealing with—only with the advent of computers did it become possible to plot it. Its hidden form made it hard for the mathematics community to grasp how such a function could exist. Weierstrass’ style of proof was also unfamiliar to many mathematicians. His argument involved dozens of logical steps, and ran to several pages. The trail of ideas was subtle and technically demanding, with no real-life analogs to guide the way. The instinct was to avoid it.

But monsters have a habit of finding their way in from the cold…


The full article appears in the Spring 2014 Nautilus Quarterly. Subscribe today!


Adam Kucharski is a research fellow in mathematical epidemiology at the London School of Hygiene & Tropical Medicine.

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