The Unreasonable Effectiveness of Mathematics in the Natural Sciences

"The Unreasonable Effectiveness of Mathematics in the Natural Sciences" was the title of the 1959 Richard Courant Lecture in Mathematical Sciences, delivered at New York University by the physicist Eugene Wigner, and published in Communication in Pure and Applied Mathematics in 1960. In it, Wigner observes that pure mathematical concepts that have been developed and studied independently of physics, often so mathematicians can show their “ingenuity and sense of formal beauty”, nevertheless often find applications in physics. The mathematical structure of theoretical physics often points the way to further advances in that theory and has empirical predictive power in describing nature. Albert Einstein and the philosopher Alfred North Whitehead had previously thought that the success of abstract mathematics in physical science was a mystery that required explanation, but Wigner's lecture became the classic statement of the problem.

Observations and arguments

Wigner argues that mathematical concepts have applicability far beyond the context in which they were originally developed. He writes: "It is important to point out that the mathematical formulation of the physicist's often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena." He adds that the observation "the laws of nature are written in the language of mathematics," properly made by Galileo three hundred years ago, "is now truer than ever before."

Wigner's first example is the law of gravitation formulated by Isaac Newton. Originally used to model freely falling bodies on the surface of the Earth, this law was extended based on what Wigner terms "very scanty observations" to describe the motion of the planets, where it "has proved accurate beyond all reasonable expectations." Wigner says that "Newton ... noted that the parabola of the thrown rock's path on the earth and the circle of the moon's path in the sky are particular cases of the same mathematical object of an ellipse, and postulated the universal law of gravitation on the basis of a single, and at that time very approximate, numerical coincidence."

Wigner's second example comes from quantum mechanics: Max Born "noticed that some rules of computation, given by Heisenberg, were formally identical with the rules of computation with matrices, established a long time before by mathematicians. Born, Jordan, and Heisenberg then proposed to replace by matrices the position and momentum variables of the equations of classical mechanics. They applied the rules of matrix mechanics to a few highly idealized problems and the results were quite satisfactory.