The Unreasonable Effectiveness of Mathematics in the Natural Sciences

Abstract

Why does mathematics work so well in describing some parts of the natural world? This question is profound, ancient, far-reaching and compelling. It seems to become more so in each respect as time goes by, at least for some people. For them it is an intellectual catalyst, serving as stimulus for further thought and questions at many levels without ever being significantly resolved itself. It was put in a particularly evocative form by the physicist Eugene Wigner as the title of a lecture in 1959 in New York: ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’. He was well-qualified for the task having discovered in the 1930s that the well-established mathematical theory of groups was just what he needed to make important progress in atomic physics. He received a share of the Nobel Prize in Physics in 1963 ‘for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles’. His 1959 lecture was published in 1960 (presumably with a minimum of editorial attention, hence its rather informal style). His paper and the themes around it, being re-visited fifty years on, form the main subject of this issue of ISR. Wigner’s central thesis was that mathematical concepts are often defined and developed in one context and then, perhaps much later, turn out to have a completely unanticipated but highly effective application in another context. Instead of citing the example of group theory and particle physics he mentions the way in which complex Hilbert space (developed as a natural part of functional analysis around 1900) turned out to be invaluable in the formulation of quantum mechanics a few decades later. In reference to such unexpected application he says, ‘It is difficult to avoid the impression that a miracle confronts us here’. Under the term ‘effectiveness’ he includes the fact that a mathematical formulation ‘leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena’, with accuracy ‘beyond all reasonable expectations’. He describes the usefulness of mathematics in the sciences as ‘bordering on the mysterious’ and declares that ‘there is no rational explanation for it’.