But why should, or how can, mathematics possibly play these sorts of roles? Scientists and philosophers have sometimes asked, what accounts (in Eugene Wigner’s words) for the apparently “unreasonable effectiveness of mathematics in the natural sciences.” The question can also be phrased metaphorically in terms of “God,” as in the book reviewed here.

This review of Is God a Mathematician? by Mario Livio, appeared originally in the Raleigh News & Observer. It is reprinted here from Science Book Reviews.

The magical marriage of math and science

by Phillip Manning

Pick up any high school physics textbook. Turn a page or two and you will find a mathematical equation. College texts have equations on virtually every page. And advanced texts are almost nothing but math, dense with symbols and letters that are as foreign to most of us as Sanskrit. Why are the laws of physics, the laws that govern the universe, written in the language of mathematics?

Mario Livio, an astrophysicist and author of several previous books about mathematics and science, tackles this question and others in his oddly named book “Is God a Mathematician?” Some readers will no doubt react to the title the way one of his students did, exclaiming, “Oh God, I hope not!”

Although its aims are philosophical, much of the book is a nicely written romp through the history of mathematics. Beginning with simple counting 1, 2, 3, …, simple arithmetic 1 + 1 = 2, 1 + 2 = 3 and geometry, generations of thinkers have produced more and more sophisticated mathematics. The hairy stuff found in today’s journals, such as the the calculus of variations or matrix algebra, all sprouted from the same root: counting, arithmetic, geometry.

This history reveals that mathematics has not been long wed to science. The elegant math developed by the Greeks was rarely applied to the real world of science. The use of mathematics in physics did not blossom until 1687, when Isaac Newton’s “Principia” was published. The book contained mathematical expressions of his laws of motion and universal gravitation. Until then, Livio writes, “the motions of the planets had been regarded as one of the unmistakable works of God.” Like most others of that era, Newton was a staunch believer in God. He was also convinced that the world was governed by mathematical laws. So, Livio concludes, “to Newton, God was a mathematician.”

Mathematicians and scientists — even doubting ones — have never hesitated to offer opinions on how God arranged the universe. Two of the twentieth century’s greatest physicists, Albert Einstein and Niels Bohr, had heated philosophical discussions about quantum mechanics. Where is an electron in an atom? There is no definite answer, only a probability that it is here or there. Einstein could not tolerate this uncertainty. He famously told Bohr that “He [God] does not throw dice.” To which, Bohr is supposed to have replied, “Einstein, do not tell God what to do with his dice.”

Mathematicians have had and continue to have similar arguments today about the role of God. Who created mathematics, God or man? Livio quotes partisans on both sides. One distinguished mathematician writes that “man has created mathematics by idealizing and abstracting elements of the physical world.” Nonsense, says Martin Gardner, a prominent writer on subjects mathematical. According to him, mathematics would exist even if we humans did not. He sums up his argument as follows: “If two dinosaurs joined two other dinosaurs in a clearing, there would be four there, even though no humans were around to observe it, and the beasts were too stupid to know it.” Yet a third mathematician straddles the fence. “God created the natural numbers,” he proclaims, “all else is the work of man.”

Although mathematicians argue about the role of God and the source of mathematics, Livio — a scientist — is more interested in a related issue: Why is mathematics is so central to the physical sciences? To him and many other working scientists, mathematics describes the physical world so well it seems magical. Eugene Wigner, a Nobel laureate in physics, addressed this extraordinarily tight relationship between science and math in a famous lecture titled “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” In it, Wigner pointed out “that mathematical concepts turn up in entirely unexpected connections.”

This is absolutely true. One of the most abstract branches of mathematics is number theory. G.H. Hardy, the famous British number theorist, once bragged that no purpose would ever “be served by the theory of numbers.” A few decades later, his work on number theory led to a breakthrough in cryptography. Today, that breakthrough is used to encrypt your credit card number when you send it over the Internet.

Another celebrated case of finding a use for useless mathematics arose when Albert Einstein was searching for a mathematical way to describe the warping of space-time by matter. He needed to develop a new kind of math to complete his theory of relativity. After many frustrating false starts, Einstein was amazed to find that the math already existed, a curious form of geometry developed 50 years earlier by the German Bernhard Riemann.

In other cases, scientists have developed new branches of mathematics to solve a specific problem. Newton, for instance, developed the calculus to solve problems of motion and change. But no matter when the math was developed — before, during or after a scientific advance — it is certainly true that mathematics is necessary to understand and predict how the physical world works. And that brings us back to Wigner’s question: What is behind the unreasonable effectiveness of mathematics in the natural sciences? Why are physics texts written in the language of mathematics rather than the language of literature?

Contrary to Newton’s belief, the effectiveness of mathematics is likely not due to God being a mathematician. In chasing down the reasons why mathematics is indispensable in science, Livio quotes the distinguished British mathematician Sir Michael Atiyah. His opinion is straightforward common sense. ”The brain evolved in order to deal with the physical world, so it should not be too surprising that it has developed a language, mathematics, that is well suited for the purpose.”

Livio then goes a step further. He points out that “mathematical tools were not chosen arbitrarily, but precisely on the basis of their ability to correctly predict the results of … experiments or observations.” In other words, since Newton’s groundbreaking insights, mathematics has evolved to solve real-world scientific problems, as scientists used the math that worked and ignored what did not. So the unreasonable effectiveness of mathematics is not unreasonable at all. It is exactly the result one would expect, the end product of a process of evolution guided by scientists and mathematicians.

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