Why Are (Most) Laws of Nature Mathematical?
Dorato (Mauro)
Source: J. Faye, P. Needham, U. Scheffler and M. Urchs, (eds.), Nature’s Principles, 55–75, 2005, Springer
Paper - Abstract

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Author’s Introduction

  1. In a frequently quoted but scarcely read paper, the Hungarian physicist Eugene Wigner rediscovered a question that had been implicitly posed for the first time by the Transcendental Aesthetics of the “Critique of Pure Reason”. More precisely, rather than asking, in the typical style of Kant, “how is mathematics possible”, Wigner was wondering how it could be so “unreasonably effective in the natural sciences” (Wigner, 1967).
  2. The effectiveness in question refers to the numerous cases of mathematical theories, often developed without regard to their possible applications, that later have played an important and unexpected descriptive, explanatory and predictive role in physics and other natural sciences. A frequently given example is that of the conic sections, already known by the Greeks before Christ and used by Kepler many centuries after their discovery to describe the orbits of celestial bodies. Even more striking is the case of non-Euclidean geometries, applied by Einstein to describe how heavy matter bends the structure of spacetime in his general theory of relativity: the theory of curved, non-Euclidean spaces had already been built a century earlier by Gauss, Lobacevski and Riemann.
  3. A literary quotation addressing the role of complex numbers, due to the German writer Robert Musil, will conclude my necessarily short list of examples: “The strange fact is that with these imaginary or even impossible numbers one can anyway make perfectly real calculations which end in a concrete result”. Ironically, at the time of “The Confusions of the Young Torless” (1906), from which this passage is taken, Musil could not be aware at the fact that the most successful theory of the atomic structure of matter – quantum mechanics – would have been using imaginary numbers to calculate the probability of measurements.
  4. In this paper, I want to raise once again Wigner’s question (to which I will be referring as ‘WQ’) in order to shed light on the related issue of the nature of scientific laws. Namely, my main purpose is to show that typical questions of the philosophical literature on laws, like
    1. What laws are, and
    2. How we come to know them,
    can be fruitfully approached afresh if we pay due attention to their mathematical character.
  5. Note first of all that if we replied to WQ by saying that “nature itself is mathematical”, we would trivialize the question only in appearance, since such a metaphysical answer should itself be explained: if, say, mathematics is a creation of ours, why are laws of nature itself mathematical? On the other hand, the fact that the laws of science are mathematical poses the question of the mathematizability of nature, which provides the clue for a correct understanding of WQ.

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