The Applicability of Mathematics to Physical Reality

The evidence explained

In a well-known 1960 essay, physicist Eugene Wigner described what he called 'the unreasonable effectiveness of mathematics in the natural sciences,' pointing to cases where abstract mathematical structures, developed by mathematicians pursuing purely internal, formal interests with no application in mind, later turned out to precisely describe physical phenomena discovered or better understood only afterward, such as the use of complex numbers in quantum mechanics or Riemannian geometry, developed decades earlier for its own sake, in Einstein's general relativity.

Wigner's puzzle is genuinely philosophical rather than merely a report of an empirical curiosity: it raises the question of why an abstract formal system, whose development is guided by considerations of elegance, consistency, and internal mathematical interest rather than empirical observation, should so often turn out to map accurately onto a physical world it was not designed to describe, and why simple, aesthetically compelling equations so often prove to be the ones nature obeys.

Several philosophical responses exist. A minority of realist philosophers of mathematics, following a Platonist tradition traceable to Plato and defended in some form by Kurt Godel, argue that mathematical objects and structures exist independently of human minds and that physical reality is itself in some sense mathematically structured, making the applicability of mathematics less surprising since both mathematics and physics are describing (or partaking in) the same underlying structure, a view echoed in physicist Max Tegmark's more radical Mathematical Universe Hypothesis, that physical reality simply is a mathematical structure.

Other philosophers offer more deflationary responses: some argue that mathematics is simply a very large toolbox of possible structures, and that scientists naturally select, sometimes after considerable trial and adaptation, whichever structures happen to fit observed phenomena, meaning the fit is less surprising once one accounts for how much mathematics exists to choose from and how much post-hoc fitting occurs; others, in a broadly Kantian vein, argue mathematics is in part a product of the structure of human cognition itself, applied to a world that human cognition is adapted to navigate, making some degree of fit expected rather than mysterious.

There is no settled philosophical consensus on how to explain the applicability of mathematics, with live debate between mathematical Platonism, structuralism, fictionalism (which denies mathematical objects exist at all while explaining their usefulness instrumentally, defended by Hartry Field), and more deflationary or naturalistic accounts, with Wigner's original puzzle still frequently cited and discussed by both physicists and philosophers of science and mathematics.