The Unreasonable Effectiveness of Mathematics in the Natural Sciences, published by physicist Eugene Wigner in 1960, argues that the capacity of mathematics to successfully predict events in physics cannot be a coincidence, but must reflect some larger or deeper or simpler truth in both.

Table of contents
1 The Miracle of Mathematics in the Natural Sciences
2 The Deep Connection between Science and Mathematics

The Miracle of Mathematics in the Natural Sciences

Wigner begins his paper with the belief, common to all those familiar with mathematics, that mathematical concepts have applicability far beyond the context in which they were originally developed. Based on his experience, he says “it is important to point out that the mathematical formulation of the physicist’s often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena”. He uses the law of gravitation, originally used to model freely falling bodies on the surface of the earth, as an example. This fundamental law was extended on the basis of what Wigner terms “very scanty observations” to describe the motion of the planets and “has proved accurate beyond all reasonable expectations.” Another oft-cited example is Maxwell’s equations, derived to model familiar electrical phenomena; additional roots of the equations describe radio waves, which were later found to exist. Wigner sums up his argument by saying that “the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it”. He concludes his paper with the same question he began with:

The 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. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.

The Deep Connection between Science and Mathematics

Wigner's work provided a fresh insight into both physics and the philosophy of mathematics. Specifically, it speculated on the relationship between the philosophy of science and the foundations of mathematics:

"It is difficult to avoid the impression that a miracle confronts us here, quite comparable in its striking nature to the miracle that the human mind can string a thousand arguments together without getting itself into contradictions, or to the two miracles of laws of nature and of the human mind's capacity to divine them."

Later, in What is Mathematical Truth, Hilary Putnam would explain "the two miracles" as being both necessarily derived from a realist (but not Platonist) view of the philosophy of mathematics. However, Wigner went further in a passage he cautiously marked as 'not reliable', about cognitive bias:
"The writer is convinced that it is useful, in epistemological discussions, to abandon the idealization that the level of human intelligence has a singular position on an absolute scale. In some cases it may even be useful to consider the attainment which is possible at the level of the intelligence of some other species."

The question of whether humans checking the results of humans can be considered an objective basis for observation of the known (to humans) universe was interesting and has been followed up in both cosmology and the philosophy of mathematics.

Wigner also laid out the challenge of a cognitive approach to integrating the sciences:

"A much more difficult and confusing situation would arise if we could, some day, establish a theory of the phenomena of consciousness, or of biology, which would be as coherent and convincing as our present theories of the inanimate world."

He further proposed that arguments could be found that might "put a heavy strain on our faith in our theories and on our belief in the reality of the concepts which we form. It would give us a deep sense of frustration in our search for what I called "the ultimate truth." The reason that such a situation is conceivable is that, fundamentally, we do not know why our theories work so well. Hence, their accuracy may not prove their truth and consistency. Indeed, it is this writer's belief that something rather akin to the situation which was described above exists if the present laws of heredity and of physics are confronted."

Some believe that this conflict exists in string theory, where very abstract models are impossible to test given the experimental apparatus at hand. While this remains the case, the 'string' must be thought either real but untestable, or simply an illusion or artifact of mathematics or cognition.

See also: Eugene Wigner, foundations of mathematics, quasi-empiricism in mathematics, philosophy of science, cosmology