Interdisciplinary Science Reviews

I am going to argue that one unfortunate problem with Wigner's paper is its pervasive vagueness, indicate some of the problems this vagueness causes, and then analyse the problem on largely historical grounds insofar as we can see what it is. I will conclude that the paper does not help with the questions of what it means for physics to have laws and how we can know them, and I shall argue on historical grounds that the effectiveness of mathematics in physics may be less unreasonable than Wigner suggested but still be a profitable matter for investigation.

When Wigner claimed that the effectiveness of mathematics in the natural sciences was unreasonable it was due to a dogmatic formalist view of mathematics according to which higher mathematics is developed solely with a view to formal beauty. I shall argue that this philosophy is not in agreement with the actual practice of mathematics. Indeed, I shall briefly illustrate how physics has influenced the development of mathematics from antiquity up to the twentieth century. If this influence is taken into account, the effectiveness of mathematics is far more reasonable.

The source of human applications of mathematics to natural phenomena is very likely to be found in the perceptual processes necessary for most animals to survive, either as prey, predator, or forager. Much of the resources of the brain are used to transform the patterns of energy that impinge on sensory nerve endings into representations more or less isomorphic to the structure of some portion of the world. It is one of the intellectual triumphs of Ancient Greek thought, especially by Aristotle, to have recognized for the first time systematically the centrality of such isomorphisms for perception. The move from visual perception to intuitive geometry, and then to systematic Euclidean geometry seems psychologically natural. The Greek application of this geometric viewpoint to the motion of the heavens constitutes the first great work in mathematical physics. The progress from the Stone Age to Ptolemy is surely a very much bigger step than the move from Ptolemy to Newton and Einstein.

Philosophical discussion of applied mathematics has tended to focus on a cluster of related ‘problems of applicability’ that are often conflated. In this paper, I argue that several of these problems raise no significant philosophical issues. I then identify two particular aspects of the involvement of mathematics in science: that mathematics is indispensable for science, and that mathematics sometimes plays an explanatory role in science. I argue that there are genuine questions surrounding these two features of applied mathematics, and in particular that the issue of the role of mathematics in scientific explanation has ramifications for the debate between realist and anti-realist views in the philosophy of mathematics.