The nature of numbers raises questions of divinity.
Sep 14, 2009, Vol. 14, No. 48 • By DAVID GUASPARI
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
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."