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Sacred Division

The nature of numbers raises questions of divinity.

Sep 14, 2009, Vol. 14, No. 48 • By DAVID GUASPARI
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Is God a Mathematician?

by Mario Livio

Simon & Schuster, 320 pp., $26

Fifty years ago Eugene Wigner, soon-to-be Nobel laureate in physics, published a soon‑to‑be‑famous essay titled "The Unreasonable Effectiveness of Mathematics in the Natural Sciences." It is remarkable, he wrote, that there should be laws of nature, remarkable that we should be able to discover them, and a "miracle" that they can be formulated in the language of mathematics. Unable to find a satisfactory explanation, Wigner could conclude only that this was "a wonderful gift which we neither understand nor deserve."

Questions about the nature and significance of mathematics have been central to Western thought since at least the time of the semi‑legendary Pythagoras--whose followers, said Aristotle, "fancied that the principles of mathematics were the principles of all things." According to one (suspect) tradition, a sign above the entrance to Plato's Academy read, "Let no one ignorant of geometry enter here." Plato regarded mathematics not as the highest kind of knowledge, but as a model: It grasped a reality beyond the world presented to our senses, and of which that perceptible world is an imperfect copy. Galileo famously declared that "the grand book [of the] universe .  .  . is written in the language of mathematics."

How can that be? Why can that be? Can such questions have answers?

Is God a Mathematician? offers to guide the general reader through this vertiginous intellectual terrain. We can hardly demand a rigorous reply to that metaphorical title--though some thin concluding thoughts are put on offer. The book should be judged by its success in making the question vivid, even unsettling, by how well it awakens the wonder in
Einstein's famous dictum that "the eternal mystery of the world is its comprehensibility."

By that standard I'd say it's, well, okay. (Gripes to follow.)

Chapter one introduces Wigner's puzzlement and some of the large themes into which it opens, such as the relation between the mind and the world and the closely related question of whether mathematics is invented or discovered. Most of the book is devoted to building up the mental muscles needed to enjoy a battle royal in which no side seems able to land a knockout blow.

How could Galileo and Newton have such success if there did not exist a world of mathematical truths that captured the structure of the world around us? But what could this supposed realm of immaterial objects have to do with the physical world? Mathematics is a human construct, invented for our own purposes.

But why should pure imaginings, human constructs, be able to describe the world? They allow us to think in the only way we can. Evolution has given us certain cognitive abilities. We have numbers because our distant ancestors found counting useful.

That may account for counting (with small numbers) but could not possibly explain the ability to invent calculus or non-Euclidean geometry .  .  .

Building up the muscles means learning some mathematics, which Livio presents historically. Popularizations seem to get much of their history by recycling the anachronisms of their predecessors. Like so many
others, for example, Livio describes the celebrated Pythagorean discovery that the side and diagonal of a square have incommensurable lengths as a discovery about "the number √2"--even though "the number √2" was a concept unavailable to Pythagoras (or Plato or Euclid), for whom a number always meant something to count with.

So the reader is advised to trust, but verify.

After sketching the lives and works of Pythagoras and Plato, Livio does the same for a series of mathematical heroes: Archimedes, Descartes, Galileo, and Newton. With the triumph of Newtonian science, his title question assumes its modern form. Plato constructed a mythical mathematical cosmology that symbolized the intelligibility of a world that lies behind appearance. Newton's System of the Universe postulated a set of laws that made detailed predictions about the perceptible world. It could have been wrong in ways that Plato's imagery could not. Astonishingly, it wasn't--despite originating in, as Wigner points out, "a single, and at that time very approximate, numerical coincidence."