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2 The Deep Connection between Science and Mathematics |
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.
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:
The Miracle of Mathematics in the Natural Sciences
The Deep Connection between Science and Mathematics
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 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:
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
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