Amir Alexander’s latest book is Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World. He teaches at UCLA.

Contact Amir Alexander via email.

In 1842, when the famed German mathematician Carl Gustav Jacobi was invited to speak to a scientific meeting in Manchester, he had a surprise in store for his English hosts. “It is the glory of science to be of no use,” he announced to the startled gathering of physical scientists. The true aim of science is “the honor of the human spirit,” and whether it turns out to be of any practical use matters not at all.

Jacobi made few converts that day. His declaration, he reported to his brother with satisfaction, “caused a vehement shaking of heads,” which was only to be expected from a crowd of men who were devoting their careers to improving industrial processes in the manufacturing capital of Europe. But things were different among Jacobi’s mathematical colleagues, who increasingly came to share his view that mathematical truths stood for themselves, and needed no further justification.

To be sure, no one (including Jacobi) denies that some fields of mathematics have proven extremely useful, and had made modern technology possible. But other fields, including some of the greatest mathematical discoveries ever, seem to serve no practical purpose whatsoever.

It was so from the beginning. The ancient science of geometry, as its name suggests, had its origins in the practical art of land measurement, but by the time Euclid codified it around 300 BCE it had strayed far from its roots. Is there really a practical application, for example, for Euclid’s construction (Book IV, prop 16) of a* *15-angle equiangular equilateral figure inside a circle? And who, one wonders, ever made use of Archimedes’ ingenious method for calculating the area enclosed in a parabola?

In the modern era things only got worse. Jacobi’s contemporary Évariste Galois (1812-1832) won immortality for developing a method that could determine, for any given equation, whether it was solvable by standard algebraic operations. Impressive, except that the method, as Galois freely admitted, was so cumbersome that it might take a mathematician a lifetime to complete the calculations for even a single equation. Non-Euclidean geometry, another 19^{th} century invention, described fantastic worlds in which the shape of a figure depends on its size, and Georg Cantor’s discovery of different orders of infinity stirred up a storm in mathematical circles, but barely a ripple outside them.

It is sometimes noted that some mathematical fields, developed with no specific use in mind, ultimately prove useful in contexts that their inventors could never have imagined. But such cases are the exception: most fields of higher mathematics remain as they were conceived, with no practical application in sight. So is higher mathematics just an intellectual game played by exquisitely trained professionals for no purpose? And if so, why should we care about it?

One answer was given by the great English mathematician G.H. Hardy, who argued that “real mathematics must be justified as art if it is to be justified at all.” It is an answer that might have appealed to Jacobi, but is hardly satisfying for those of us who look for a more down-to-earth use for mathematics.

So here’s another answer: mathematics is the science of order, and throughout history people have tried to make use of mathematics to order their lives, their societies, and the world.

Consider for example Plato, the Greek philosopher who had the words “let no one ignorant of geometry enter” carved above the entrance to his Academy in Athens. His faith in geometry was such that he used it not only as a model for acquiring the highest truths, but also as the basis for the political order he favored. Just as everything in geometry had its precise, rational, and unchallengeable place, so everyone in Plato’s Republic would have his or her precisely assigned place in the state hierarchy. Plato’s prescription of a rigid oligarchy ruled by philosopher-kings might sound repulsive to us today, but from his own day into modern times it served reformers as a model of an enlightened and rationally ordered society.

The idea of ordering society and the state according to geometrical principles persisted as well. In the 1600s for example, the Jesuits tried to model their reform of the Catholic Church on geometrical principles, using it to support their case for papal supremacy and an unchallengeable hierarchical order. More grandly, Louis XIV of France created the dazzling geometrical gardens of Versailles as an emblem of his rule. Every stone, flower, and blade of grass in the world of Versailles was fixed in place by the power of geometry, and all were subject to the king’s palace, where all lines met.

On the other side of the ideological divide, the critics of fixed hierarchies were just as happy to harness mathematics to their causes, promoting the new “method of indivisibles” as an alternative to rigid geometry. A forerunner of calculus, the method was paradoxical and imperfectly understood, and yet produced beautiful and powerful results. To its adherents it was a model of what could be accomplished by putting dogmatism aside and working pragmatically and tolerantly for the greater good.

Mathematics played a part not only in politics, but also in shaping cultural trends. In the early 19^{th} century higher mathematics was integral to the Romantic Movement, turning away from the natural world and towards an alternate reality, governed solely by mathematical principles. Much like the Romantic painters, poets and composers of the age, mathematicians were reaching out to a pure realm of truth and beauty, away from our flawed and fallen world. And at the turn of the 20^{th} century the mathematical field of non-Euclidean geometry upended seemingly self-evident assumptions about reality. Our Euclidean world was revealed to be just one of an infinity of possible ones, a discovery that profoundly impacted modernist art and literature, with its multiplicity of perspectives and absence of a single, unifying narrative.

This is just a short sample of the ways mathematics has shaped people’s lives over generations, but I hope it is enough to show that mathematics does indeed matter. Not just because it results might one day be used to create powerful technologies. But because our search for order and meaning, which is the sum total of human history, always brings us back to mathematics.

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