The Unreasonable Effectiveness of Mathematics in the Natural Sciences

"The Unreasonable Effectiveness of Mathematics in the Natural Sciences" is a 1960 article by the physicist Eugene Wigner. In the paper, Wigner observes that a physical theory's mathematical structure often points the way to further advances in that theory and even to empirical predictions. Wigner begins his paper with the belief, common among those familiar with mathematics, that mathematical concepts have applicability far beyond the context in which they were originally developed. Based on his experience, 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 then invokes the fundamental law of gravitation as an example. Originally used to model freely falling bodies on the surface of the earth, this law was extended on the basis of what Wigner terms "very scanty observations" to describe the motion of the planets, where it "has proved accurate beyond all reasonable expectations". Another oft-cited example is Maxwell's equations, derived to model the elementary electrical and magnetic phenomena known as of the mid-19th century. The equations also describe radio waves, discovered by David Edward Hughes in 1879, around the time of James Clerk Maxwell's death. Wigner sums up his argument by saying that "the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it". He concludes his paper with the same question with which he began: The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.