On the Reasonable and Unreasonable Effectiveness of Mathematics in Classical and Quantum Physics

The point of departure for this article is Werner Heisenberg’s remark, made in 1929: “It is not surprising that our language [or conceptuality] should be incapable of describing processes occurring within atoms, for . . . it was invented to describe the experiences of daily life, and these consist only of processes involving exceedingly large numbers of atoms. . . . Fortunately, mathematics is not subject to this limitation, and it has been possible to invent a mathematical scheme—the quantum theory [quantum mechanics]—which seems entirely adequate for the treatment of atomic processes.” The cost of this discovery, at least in Heisenberg’s and related interpretations of quantum mechanics (such as that of Niels Bohr), is that, in contrast to classical mechanics, the mathematical scheme in question no longer offers a description, even an idealized one, of quantum objects and processes. This scheme only enables predictions, in general, probabilistic in character, of the outcomes of quantum experiments. As a result, a new type of the relationships between mathematics and physics is established, which, in the language of Eugene Wigner adopted in my title, indeed makes the effectiveness of mathematics unreasonable in quantum but, as I shall explain, not in classical physics. The article discusses these new relationships between mathematics and physics in quantum theory and their implications for theoretical physics—past, present, and future.