Cliff’s Edge - The Unreasonable Effectiveness of Mathematics in the Natural Sciences
Writing for Communications in Pure and Applied Mathematics (vol. 13, no. 1, Feb. 1960), physicist Eugene Wigner told about a statistician who showed a friend a complicated paper on population trends. Pointing to a mathematical symbol in the paper, the friend asked what it was.
“Pi,” answered the statistician, the ratio of the diameter of a sphere to its circumference.
Already skeptical, the friend said that the joke went too far, because surely “the population has nothing to do with the circumference of a circle.”
Apparently, however, it does: another example of “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” the title of Wigner’s famous article, in which he asked: How can mathematics—abstract concepts that arise from within the human mind—so powerfully describe the world outside the mind? Or, even if numbers were already there, preceding the prime ingredients of existence, and thus baked in with those ingredients at the creation (as opposed to having been pasted on the surface of things by us after the fact), the question remains. Why are numbers, which might not even exist (have you ever tripped over the number 2; what does -4 weigh; how much space does .1415962 occupy?), so “unreasonably” effective in describing what exists, the world that natural science studies?
Why are numbers, which might not even exist, so “unreasonably” effective in describing what exists?
Open a physics textbook and two pages past the Table of Contents you will find almost nothing but numbers until you reach the inside back cover. The closer physicists poke and probe around the edges of existence, the more complicated the numbers become that either get them to those edges, or the numbers they find once there, or both. Why didn’t Isaac Newton title his famous 1687 work on gravity “On the Law of Gravity” or the like, instead calling it “The Mathematical Principles of Natural Philosophy”?
This belief in the primacy of numbers goes back to at least 500 BC, with the Greek Pythagoras, who taught that all reality was fundamentally numbers, whole numbers, and he founded a religious sect based on that belief. (Legend has it, however, that when the Pythagoreans stumbled upon the existence of irrational numbers, which upended their theology of reality being only whole numbers, they drowned Hippasus of Metapontum for having divulged that secret to outsiders.)
A few thousand years later, Galileo, influenced by the Pythagoreans, but working in a more rational, experimental, and scientific manner, argued that mathematics, including Euclid’s geometry, was the language of nature itself.
About 200 years after Galileo, some mathematicians in the nineteenth century created new geometries in which (contrary to Euclid) the shortest distance between two points was not a straight line; in which (contrary to Euclid) the angles of a triangle were not 180°; and in which (contrary to Euclid) parallel lines did, in fact, intersect. All this, even though for 2,100 years everyone knew that Euclid’s geometry was not only a true depiction of the world but a mathematical expression of logical and rational thought as well. Thus these new geometries, even if internally as consistent and deductive as Euclid’s, were deemed mere intellectual exercises and of no more practical value than using a map of seventeenth century Japan to navigate around twenty-first century Stockholm.
Then in the early twentieth century, Albert Einstein found that Non-Euclidean geometry was just what he needed for his General Theory of Relativity. The math that these men pulled out of the air turned out to precisely describe the curvature of space-time, which, according to Einstein’s theory, explained what gravity was, and why, among other things, it holds us to the earth.
The most reasonable explanation for the “unreasonable effectiveness” of mathematics is that the Lord “created the heavens and the earth”.
However dramatic, the use of Non-Euclidean geometry was not the first nor the last time a mathematician conjured up some theoretical math that, years later, ended up describing an aspect of reality—the swirl of a conch shell, the arrangement of artichoke petals, complicated knots—that math geeks at the time never imagined.
Another example of the “unreasonable effectiveness of mathematics in the natural sciences” occurred in 1928, when British physicist Paul Dirac published mathematical equations that, if correct, predicted the existence of a new kind of sub-atomic particle hitherto unknown. Five years later, physicists discovered what became known as the positron, a particle that Dirac’s math predicted. How could some guy, scribbling lines and marks on paper, predict the existence of a sub-atomic particle found only with sophisticated (for its time) technology?
No wonder Wigner in his article called the link between mathematics and physics a “miracle” and a “wonderful gift.” A gift, of course, implies a Giver. So unless one wants to attribute to chance alone the miraculous way mathematics describes reality, the most reasonable explanation for the “unreasonable effectiveness” of mathematics is that the Lord who “created the heavens and the earth” (Gen. 1:1), did so with such precision that sometimes only complicated math best describes how they, and all the wonders in them, work.
Clifford Goldstein is editor of the Adult Sabbath School Bible Study Guide. His next book, Baptizing the Devil: Evolution and the Seduction of Christianity is set to be released this fall by Pacific Press.