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See Inside Scientific American Volume 310, Issue 4

A Brief History of Infinitesimals: The Idea That Gave Birth to Modern Calculus

Learning to see the infinite in lines, planes and solids changed math forever
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Sometime in the 5th century B.C. the Greek philosopher Hippasus of Metapontum, a member of the secretive Pythagorean brotherhood, left his home in southern Italy and boarded a seagoing ship. We do not know why Hippasus was traveling or where he was journeying, but we do know he didn’t make it. According to the legend, once the ship was far from shore the poor philosopher was set upon by his fellow Pythagoreans and tossed into the sea.

The Pythagoreans had good reason to turn on their brother. Following the teachings of their founder, Pythagoras, they fervently believed that everything in the world could be described through whole numbers and their ratios. But Hippasus had proved that the diagonal of a square is incommensurable with the square’s side, or, as we would say today, that the square root of 2 (the length of the diagonal relative to the side) is irrational. This means that no matter how many times the side is divided and how many times the diagonal is divided the resulting magnitudes would never be equal.

Hippasus’ discovery changed the course of Western mathematics. For one thing, it showed that the proportion of a square’s side and diagonal could not be described as a simple ratio, dooming the Pythagorean enterprise. For another, it showed that lines could not be described as a sequence of tiny points strung together, or else these points would serve as a common measure for all magnitudes. Discrete numbers and points, Hippasus proved, could never fully capture a world comprising continuous entities such as lines and surfaces. The only proper mathematical science, it followed, was geometry—the study of relations between continuous magnitudes.

For the next two millennia the lesson of Hippasus remained largely unchallenged and geometry reigned supreme. It was not until the 16th and 17th centuries that a new generation of mathematicians in the Netherlands (Simon Stevin), England (Thomas Harriot, John Wallis) and especially Italy (Bonaventura Cavalieri, Evangelista Torricelli) began to probe the strict separation between discrete points and continuous magnitudes. What would happen, they wondered, if we assumed that a line is a string of infinitesimals—of tiny, or infinitely small, points? And similarly that a plane is composed of lines placed side by side, and a solid of planes stacked on top of one another?

The results, they quickly found, were spectacular. Aided by this problematic assumption, they were able to easily calculate the lengths of geometrical curves and their slopes, the areas of geometrical figures and the volumes of solids—results that would either be extremely difficult or simply impossible using traditional geometry. By 1700 Isaac Newton and Gottfried Leibniz had turned this approach into the powerful algorithm we know as “the calculus,” capable of being applied to anything from the motion of the planets to the vibrations of a string and the flight of cannonballs.

The pioneers of the new infinitesimal methods knew full well that their approach rested on precarious logical foundations, but for the most part they didn’t care. As long as their method led to correct results, they reasoned, it must be fundamentally sound. Others, however, were not so sanguine. Critics from Jesuits in Italy to the philosopher Bishop George Berkeley in England charged that infinitesimals undermine mathematics and even rationality itself, and would inevitably lead to serious errors. And so the debate raged.

In the end, it fell to the French mathematician Augustin-Louis Cauchy to put the matter to rest in the early decades of the 19th century. Cauchy realized that the problem with the new mathematics arose from the fact that it was supposed to correspond to material reality. This, Hippasus had shown, will never work. And so, in his Cours d’Analyse of 1821, Cauchy recast the calculus without resorting to the intuitive idea that a line is composed of infinitesimal points. He rigorously defined the core concepts of “derivative” and “integral” as the limits of infinite series, making no reference materialist notions of the slope of a curve or the area of a figure.

By transforming the calculus into a rigorous mathematical system, Cauchy ended a conflict that had lasted more than two millennia. In the 5th century B.C. Hippasus had shown that mathematics could never fully describe the world. In the 19th century A.D. Cauchy showed that it didn’t have to: Mathematics would survive, and thrive, on its own, freed from the shackles of material reality. Modern mathematics was born.

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