The Miracle of Applied Mathematics 1. the Unreasonable Effectiveness of Mathematics

Mathematics has a great variety of applications in the physical sciences. This simple, undeniable fact, however, gives rise to an interesting philosophical problem: why should physical scientists find that they are unable to even state their theories without the resources of abstract mathematical theories? Moreover, the formulation of physical theories in the language of mathematics often leads to new physical predictions which were quite unexpected on purely physical grounds. It is thought by some that the puzzles the applications of mathematics present are artefacts of out-dated philosophical theories about the nature of mathematics. In this paper I argue that this is not so. I outline two contemporary philosophical accounts of mathematics that pay a great deal of attention to the applicability of mathematics and show that even these leave a large part of the puzzles in question unexplained. 1. THE UNREASONABLE EFFECTIVENESS OF MATHEMATICS The physicist Eugene Wigner once remarked that [t]he 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. (Wigner 1960, 14) Steven Weinberg is another physicist who finds the applicability of mathematics puzzling: It is very strange that mathematicians are led by their sense of mathematical beauty to develop formal structures that physicists only later find useful, even where the mathematician had no such goal in mind. [ . . . ] Physicists generally find the ability of mathematicians to anticipate the mathematics needed in the theories of physics quite uncanny. It is as if Neil Armstrong in 1969 when he first set foot on the surface of the moon had found in the lunar dust the footsteps of Jules Verne. (Weinberg 1993, 125) Mark Steiner also believes that there is a problem here worthy of attention: [H]ow does the mathematician – closer to the artist than the explorer – by turning away from nature, arrive at its most appropriate descriptions? (Steiner 1995, 154) Indeed, this puzzle,1 which Wigner calls “the unreasonable effectiveness of mathematics”, is often remarked upon by physicists and applied Synthese 127: 265–277, 2001. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.