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call “active.” When physicists wander through nature’s labyrinth, they light their way by mathematics—the tools they use and develop, the models they construct, and the explanations they conjure are all mathematical in nature. This, on the face of it, is a miracle in itself. Newton observed a falling apple, the Moon, and tides on the beaches (I’m not even sure if he ever saw those!), not mathematical equations. Yet he was somehow able to extract from all of these natural phenomena, clear, concise, and unbelievably accurate mathematical laws of nature. Similarly, when the Scottish physicist James Clerk Maxwell (1831–79) extended the framework of classical physics to include all the electric and magnetic phenomena that were known in the 1860s, he did so by means of just four mathematical equations. Think about this for a moment. The explanation of a collection of experimental results in electromagnetism and light, which had previously taken volumes to describe, was reduced to four succinct equations. Einstein’s general relativity is even more astounding—it is a perfect example of an extraordinarily precise, self-consistent mathematical theory of something as fundamental as the structure of space and time.

But there is also a “passive” side to the mysterious effectiveness of mathematics, and it is so surprising that the “active” aspect pales by comparison. Concepts and relations explored by mathematicians only for pure reasons—with absolutely no application in mind—turn out decades (or sometimes centuries) later to be the unexpected solutions to problems grounded in physical reality! How is that possible? Take for instance the somewhat amusing case of the eccentric British mathematician Godfrey Harold Hardy (1877–1947). Hardy was so proud of the fact that his work consisted of nothing but pure mathematics that he emphatically declared: “No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.” Guess what—he was wrong. One of his works was reincarnated as the Hardy-Weinberg law (named after Hardy and the German physician Wilhelm Weinberg [1862–1937]), a fundamental principle used by geneticists to study the evolution of populations. Put simply, the Hardy-Weinberg law states that if a large population is mating totally at random (and migration, mutation, and selection do not occur), then the genetic constitution remains constant from one generation to the next. Even Hardy’s seemingly abstract work on number theory—the study of the properties of the natural numbers—found unexpected applications. In 1973, the British mathematician Clifford Cocks used the theory of numbers to create a breakthrough in cryptography—the development of codes. Cocks’s discovery made another statement by Hardy obsolete. In his famous book A Mathematician’s Apology, published in 1940, Hardy pronounced: “No one has yet discovered any war-like purpose to be served by the theory of numbers.” Clearly, Hardy was yet again in error. Codes have been absolutely essential for military communications. So even Hardy, one of the most vocal critics of applied mathematics, was “dragged” (probably kicking and screaming, if he had been alive) into producing useful mathematical theories.

But this is only the tip of the iceberg. Kepler and Newton discovered that the planets in our solar system follow orbits in the shape of ellipses—the very curves studied by the Greek mathematician Menaechmus (fl. ca. 350 BC) two millennia earlier. The new types of geometries outlined by Georg Friedrich Bernhard Riemann (1826–66) in a classic lecture in 1854 turned out to be precisely the tools that Einstein needed to explain the cosmic fabric. A mathematical “language” called group theory, developed by the young prodigy Évariste Galois (1811–32) simply to determine the solvability of algebraic equations, has today become the language used by physicists, engineers, linguists, and even anthropologists to describe all the symmetries of the world. Moreover, the concept of mathematical symmetry patterns