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."
Livio travels many byways, but the central thread of his story lies in the accounts of non‑Euclidean geometry and mathematical logic. Non‑Euclidean geometry introduced and explored a realm of ideas once believed literally unthinkable--alternative geometrical axioms that contradict Euclid's. Euclid's axioms had always seemed inevitable, but the discovery of new geometries suggested that mathematics might be not a special domain of fundamental truths after all: It might only be a kind of game based on arbitrarily chosen assumptions.
The most famous result of mathematical logic, Gödel's Incompleteness Theorem, turned up the heat. It implies that no attempt to formulate axioms that capture all of mathematics can succeed. There will always be questions that cannot be answered without making further assumptions, so analogs to non‑Euclidean geometries must arise in all branches of mathematics. The foundational works of mathematical logic, by Frege and Russell and Whitehead, had undertaken to show that mathematics is, ultimately, nothing but logic. (A side effect: Mathematics is effective because the laws of mathematics are the laws of thought.) That attempt failed but, as is often noted, asked so deep a question that it led to amazing, if distressing, results.
The argument continues: So what if we can't capture all of mathematics in one go? If we lacked a map of all China, would it mean that China wasn't there? Marco Polo had to face facts, and so did Newton. Newton's math fit his physics because he tailored it to do so--he invented the mathematical game he needed.
Non‑Euclidean geometry was a product of intellectual and aesthetic interest--but turned out half a century later to be exactly what Einstein needed to replace Newton's universe with the general theory of relativity. It's eerie that so much mathematics pursued solely for its beauty finds applications. It's no more eerie than a dream that comes true: We remember that one, and forget the many more that don't.
The pleasure in this succession of arguments and counterarguments is that each makes a serious point--but to cut deep they must be sharp. My gripe is that, as presented, they sometimes are not.
Consider this lazy comparison of Newton's gravitational theory with Einstein's. Newton's gravitational forces are transmitted instantaneously, which is incompatible with Einstein's postulate that effects cannot propagate faster than the speed of light. To illustrate why this "contradiction" would be "disastrous" to our "perceptions of cause and effect," Livio offers the following example: If the sun suddenly disappeared, according to Newton, the earth's motion would immediately change from (roughly) circular to (roughly) straight, but we on earth wouldn't see the sun's disappearance for the eight or so minutes it takes the sun's light to reach earth. The effect would seem to precede the cause.
To which a reader who is at all awake can only say: So? A bullet strikes its target before we hear the gun's report. Big deal. Bullets that travel faster than sound aren't disastrous to my perception of cause and effect. So what if gravity turns out to travel faster than light?
Nontechnical discussions can also be slack. Is mathematics, Livio asks rhetorically, the "hidden textbook" of the world? Or, he continues, "to use the biblical metaphor, is mathematics in some sense the ultimate fruit of the tree of knowledge?"
This sounds grand but comes up empty. The meaning of the tree of knowledge of good and evil is, to say the least, disputed: What kind of knowledge it represents; whether that knowledge is genuine, or something humans should want or have; why it's forbidden. The comparison sheds no light in either direction. It offers the reader nothing but a fist bump over a shared crumb of "cultural literacy."
Livio's conclusion suggests that we might resolve one question by agreeing that mathematics is both invented and discovered: We invent mathematical concepts and then discover things about them. But to leave it there is to leave the hard question unasked: whether the concepts we invent are in some sense forced on us. If beings from another galaxy have a science of numbers, does "prime number" have to play the central role it does in ours? More radically, if they have "science," do they have to have numbers?
Here we approach the realm of Borges's famous Celestial Emporium of Benevolent Knowledge, an imaginary encyclopedia from an imaginary society whose categories of thought have nothing in common with ours. Its classification of animals includes "those that belong to the emperor," "stray dogs," and "those that from a long way off look like flies."
Livio has chosen a fascinating subject and tackled it with enthusiasm (if a few too many exclamation points). Laymen will find much to chew on, and will encounter illustrative examples, such as logic and knot theory, that may be new to them.
Those undeterred by technical talk might prefer Wigner's short, sharp essay.
David Guaspari is a writer in Ithaca, New York.