Many scientists and philosophers have remarked on the powerful role played by mathematics as the language of the exact sciences. The use of mathematics reveals unsuspected connections in nature and generates new discoveries and new concepts. The best scientists, from the days of the Greeks, and then to the leading figures of the Renaissance or the pioneers of modern physics talked with each other using mathematics. Today, investigations in the most advanced (i.e., well-codified) branches of science cannot be carried out without a very good command of abstract mathematics as well as computational mathemativs.

The question prompted Eugene Wigner to write an article entitled 'The Unreasonable Effectiveness of Mathematics in the Natural Sciences' [227]. He wrote:

...The first point is that the enormous usefulness of mathemativs in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it. Second, it is just this uncanny usefulness of mathematical concepts that raises the question of the uniqueness of our physical physics, it will be useful to say a few words on the question, "What is mathematics?", then, "What is physics?", then, how mathemativs enters physical theories, and last, why the success of mathematics in its role in physics appears so baffling...

Wigner was a scientist and savant who formulated the abstract theory of symmetry in nature, made major contributions to particle physics as well as solid state physics. He justified most of his work in terms of aesthetic considerations. For him, mathematics was an exploration of beautiful abstract forms that challenged one's ingenuity. Discussing the nature of physics. Wigner points out that:

"it is not at all natural that 'laws of nature' exist, much less that man is able to discover them. Schrodinger, in his book What Is Life? says that this second miracle may well be beyond human understanding... All the laws of nature are conditional statements which permit a prediction of some future events on the basis of the knowledge of the present,...the overwhelming majority of the determinants of the present state of the world, are irrelevant from the point of view of the prediction..."

Thus we see that Wigner is simply stating that the laws of physics apply within the 'conditional statements' that we referred to as 'boundary conditions' of the reductionist physical model that we study. He notes that 'the overwhelming majority of determinants are irrelevant'. That is, the reduction is valuable in many circumstances and hence widely applicable.

In discussing the role of mathematics in physics, Wigner notes the secondary need of applying mathemativs to do calculations, and proceeds to point out that mathematics plays a far more 'sovereign role', in that when the laws of physics are formulated in the language of mathematics, we seem to get 'more than we put in'. Wigner points out that Newton's formulation of planetary motion, based on very flimsy empirical evidence, yielded a rich harvest of new results and explanations that are demonstrated in Newton's Principia (Sec. 2.2.4). Discussing a similar bounty in the quantum theory, Wigner points out how the application of mathemativs to the problem produced a bumper harvest.

...Max Born notived that some rles of computation, given by Heisenberg, were formally identical with the rules of computation with matrices, established a long time before by mathematicians. Born, Jordan, and Heisenberg then proposed to replace by matrices the position and momentum variables of the equations of classical mechanics. They applied the rules of matrix mechanics to a few highly idealized problems and the results were quite satisfactory. However, there was, at that time, no rational evidence that their matrix mechanics would prove correct under more realistic conditions....the first application of their mechanics to a realistic problem, that of the hydrogen atom, by Pauli...gave results in agreement with experience. This was satisfactory but still understandable because Heisenberg's rules...included the old theory of the hydrogen atom. The miracle occurred only when matrix mechanics, or a mathematically equivalent theory, was applied to problems for which Heisenberg's calculating rules were meaningless. Heisenberg's rules presupposed that the classical equations of motion had solutions with certain periodicity properties; and the equations of motion of the two electrons of the helium atom, or of the even greater number of electrons of heavier atoms, simply do not have these properties, so that Heisenberg's rules cannot be applied to these cases. Nevertheless, the calculation of the lowest energy level of helium, as carried out a few months ago by Kinoshita, at Cornell and by Bazley at the Bureau of Standards, agrees with the experimental data within the accuracy of the observations, which is one part in ten million. Surely in this case we 'got something out' of the equations that we did not put in ...physics as we know it today would not be possible without a constant recurrence of miracles similar to the one of the helium atom,... The quantum theory of the Lamb shift, as conceived by Bethe and established by Schwinger, is a purely mathematival theory and the only direct contribution of experiment was to show the existence of a measurable effect. The agreement with calculation is better than one part in a thousand.

Thus we see that Wigner leaves the matter as a mystery, but a clarification is possible. We look at problems where the application of mathemativs with the same vision as in physics has led to no progress. For example, Frederich von Hayek in his 1974 Nobel address, as well as in his earlier writings [101] pointed out that methods borrowed from the physical sciences, as applied to economics have led to a mere 'pretense of knowledge', where there is none. Attempts to understand financial markets using models borrowed from mathematical physics have not helped, as discussed in a popular work by Nicholas Taleb [205]. Leaving aside stock markets, more elementary complex systems (see Part-II of this book), e.g., simple protein molecules, have conformations that defy prediction even though such systems are governed by deterministic laws. The key to this problem was given by Poincare in the latter part of the 19th century, when he showed that there are dynamical systems whose motion is essentially unpredictable since they follow chaotic dynamics. The trajectory is extremely sensitive to initial conditions (or boundary conditions) and no useful predictions can be made.

In contrast, in the physical sciences we choose to deal with simple reductionist models where the dynamics is well behaved. As discussed in this chapter (Ch. 3) and other chapters, most of the laws of physics can be extracted from simple symmetry principles and conservation laws. The conservation of particle number leads to the 'equation of continuity', while the need to make the laws of physics appear the same to all observers leads to Galilean or Einsteinian relativity and various 'gauge conditions'. When Paul Dirac attempted to derive a relativistic equation for the electron, he looked fro a linear equation that satisfied continuity, relativity and energy conservation (see Sec.6.10.2). Out of that effort, he not only obtained a theory of the electron, but also a multiple bonus of explaining the abstract up-spin and down-spin features of the electron, a prediction of the existence of the anti-particle of the electron (i.e., the positron), and a new vision of the vacuum state that led to the development of quantum field theory!

Thus the 'unreasonable effectiveness' of mathematics in its application to physics is probably anchored in the fact that nature at its most basic level turns out to be that of a homogeneous, space-time symmetric (isotropic) plasma with 'integrable' dynamics that obeys various symmetry principles and conservation laws. The metastable, highly compressed homogeneous initial state of the universe that gave rise to inflation and the big bang may be the 'reason' why such simplicity in inherent to basic physical phenomena.