Math, Nature, and Knowledge

This paper by Eugene Wigner entitled “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” gets too little play in the faith/science discussions. He begins:

THERE IS A story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. “How can you know that?” was his query. “And what is this symbol here?” “Oh,” said the statistician, “this is pi.” “What is that?” “The ratio of the circumference of the circle to its diameter.” “Well, now you are pushing your joke too far,” said the classmate, “surely the population has nothing to do with the circumference of the circle.”

Perhaps a little note to preface this is appropriate. Wigner is adamantly not an uncredentialed crackpot, far from it. Of him, and a select few others, a science historian might write a paper on the “unreasonable effectiveness of Hungarian mathematicians” in 20th century physics and mathematics … and Mr Wigner would be a prime example.

Symmetry is a key principle in our modern physical understanding of the nature and as well is often closely connected to beauty in artistic settings. Two of the key insights driving the usefulness of symmetry are the prevalence of gauge theories to explain physical phenomena and the “deep theorem” of Emmy Noether’s which in a fundamental way connects continuous symmetries with conserved quantities.

In the latter part of the 19th century, the mathematicians Lagrange and Hamilton formalized and restructured Newton’s equations of motion. These methods recast the equations of motion for systems of particles and forces into elegant mathematical forms which structure them in a way in which all the modern geometrical methods and tools might be applied to them. Noether’s theorem applied to any generic problem which could be cast in the form in which Lagrange and Hamilton had developed. Her theorem stated that any continuous symmetry that those descriptions of systems possessed meant that a related (conjugate) “current” was conserved. In layman’s terms this means that, for example, because the equations of motion describing the motion of particles is the same where you’re reading this as where I wrote it, means that momentum is conserved. Because those equations of motion are the same today as next week, that means energy is conserved.

Mr Wigner’s essential observation is that in the first place starting from a number of relatively imprecise measurements a great mathematical structure (Lagrangian and Hamiltonian mechanics) is built. Ms Neother’s theorem is but one elegant and precise result that falls out from that mathematical structure. The quantity of results and their precision far exceeds the precision and quality of the experimental data going into the formation of those theories. Or as Mr Wigner suggests:

A possible explanation of the physicist’s use of mathematics to formulate his laws of nature is that he is a somewhat irresponsible person. As a result, when he finds a connection between two quantities which resembles a connection well-known from mathematics, he will jump at the conclusion that the connection is that discussed in mathematics simply because he does not know of any other similar connection. It is not the intention of the present discussion to refute the charge that the physicist is a somewhat irresponsible person. Perhaps he is. However, 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. This shows that the mathematical language has more to commend it than being the only language which we can speak; it shows that it is, in a very real sense, the correct language.

Filed under: Intelligent Design • Mark O. • Science

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