Historians don’t know much about the Pythagoreans or their founder, except what was said about them by Greek writers centuries later. It’s unlikely that Pythagoras was the first to formulate the famous observation about right triangles which made his name immortal. But there’s no doubt that his followers placed their faith in the divine nature of numbers, a religious conviction that starkly illustrates one answer to the question astrophysicist Mario Livio asks in Is God A Mathematician?, his latest wide-eyed peregrination through the philosophy of science. If Pythagoras discovered that the sum of the squares of two sides of a triangle equals the square of the hypotenuse, that would indicate that mathematical truths and relationships are improbably woven deep into the structure of reality. Einstein, on the other hand, insisted that mathematics is a creation of the human mind, abstracting from messy reality to invent a language whose implications can be unspooled in an imaginary realm of perfection. What would account, then, for what Livio calls “the unreasonable effectiveness of mathematics” in describing and predicting the physical world?
On the side of the discoverers, Livio ranks Archimedes, Galileo, Descartes, Newton, Bernoulli, and innumerable others, from the classical age to the Enlightenment, who considered their mathematical codification of time, space, and motion to be literal description. Bodies displace water, the planets move, apples fall, and missiles arc in ways that can be precisely delineated by numerical formulae. Even the science of probability, which regards single chance actions as unpredictable, can make true statements about a sufficiently large number of chance events. But with the invention of non-Euclidean geometries, a rift appeared in mathematicians and physicists’ confidence about their subjects’ real existence. Suddenly the Platonic maxim “God ever geometrizes” seemed positively parochial, as it became clear that replacing some of Euclid’s foundational axioms would result in consistent, useful descriptions of spaces that don’t happen to exist, as far as we know. Georg Cantor, the late-19th-century creator of set theory, spoke for the new spirit of invention when he insisted that mathematics be freed from relevance to the world of human senses or intuitions. Our minds’ power to understand completely imaginary inventions—like numbers—in a rigorous, defined, and way is proven by the discipline of mathematics.
Yet just as the formalists were celebrating their victory over the Platonic realists in the 20th century, Kurt Gödel showed that their ideal of complete, self-consistent systems based on axiomatic first principles is impossible to achieve. He believed that some axioms and results “force themselves upon us as being true,” meaning that we perceive them in reality somehow. And Livio is particularly interested in the ways that pure mathematical play turns out to solve problems in physical reality, from Brownian motion to computer-chip design. A mathematical model of the atom that turned out to be wrong turned into the pure mathematics of knot theory, which then yielded the key to understanding the structure of DNA. As Livio surveys the field, he exhibits a charming sense of surprise at each unexpected turn in the problems he’s describing. Like a beloved professor, he guides readers downstream from the source of an unresolved dilemma, pausing to twirl in interesting eddies of personality and history along the way. And his conclusion betrays a passionate faith in the importance of the philosophical deliberations that are needed to consider his titular question. After all, as he provocatively proposes in his introduction, it’s as far from a moot point as the question of whether God has been discovered or invented by his followers.