The unreasonable effectiveness of mathematics

In a famous article written in 1960, The Unreasonable Effectiveness of Mathematics in the Natural Sciences, the physicist Eugene Wigner (see sidebar) examines the relationship between mathematics and physics and raises two questions:

• How can the same mathematical concept arise in situations that seemingly have nothing to do with one another, yet describe each of them equally precisely?

• Is the unsettling effectiveness of mathematics in the natural sciences, particularly physics, evidence of a one-to-one correspondence between theories and what they describe?

Who was Eugene Wigner? -----------------------

Eugene Paul Wigner was a theoretical physicist born in Hungary in 1902. After becoming David Hilbert's assistant at the University of Göttingen in 1927, he took an interest in the theory of symmetry groups and its applications to quantum physics, whose algebra he formalized with Hermann Weyl. Following the Nazis' rise to power in Germany, he settled in the United States and became a US citizen in 1937. He became a professor of mathematics at Princeton University in New Jersey, where he remained for the rest of his career. His interests were then wide-ranging, but always brought mathematical theories to bear on current questions in physics. Among his fundamental contributions, besides his work in nuclear physics, which earned him the Nobel Prize in 1963—particularly the model describing the shell structure of the atomic nucleus—were representations of the Poincaré group; an explanation of the mechanism behind the nuclear chain reaction; a method for quantizing a fermion field, developed with Pascual Jordan; the Wigner effect, in which a high-energy particle displaces an atom in a crystal lattice; and the use of random matrices to model atomic nuclei, including an important limiting law known as Wigner's theorem. During the Second World War, from 1942 to 1945, Eugene Wigner also played a key role in the Manhattan Project, which developed the first atomic bomb. He died in Princeton in 1994.

Take the simple formula c / d = π, which relates a circle's circumference c to its diameter d. Remarkably, the same number π also appears in the sum of the reciprocals of the squares of all odd numbers:

\sum_{n=0}^{+\infty}\frac{1}{({2n+1})^{2}}=\frac{\pi^{2}}{8}.

The way mathematics enters physical theories can inspire the same sense of wonder: a single mathematical concept may prove extremely useful in widely disparate physical theories. This is all the more surprising when we consider theories in which mathematics is not merely a tool but plays a foundational role, tending to confirm the view generally attributed to Galileo that the laws of nature are written in the language of mathematics. This is true of the three great theories of modern physics: the theory of gravitation, quantum mechanics and quantum electrodynamics. Let us take a closer look at this "unreasonable effectiveness" of mathematics in action.