“Pi” is not My Easy Reader. In a typical sentence, Barrow “explains” that “the procedure FK(n) is true of the number n if and only if there is no number N for which the Nth possible proof in the formal system S is a proof whose Godel number is n.” Such eye-glazers are partially redeemed by a lively romp through math history, focusing on the priests and traders who first used symbols rather than notches on a stick to represent quantities, invented 0 (so 8,007 could be distinguished from 87) and created place value (so 369 is different from 936).
This time travel leads to the hoary question of whether math is discovered or invented. Does it exist Out There with Plato’s Forms, waiting for the mind to see it? Or do people make it up like, say, a song? There are intriguing hints that the world is profoundly, even eerily, mathematical–that Pi is in the sky. Pure math, mere squiggles of ink across a page, predicted the existence of a new subatomic particle. Only later did an atom smasher find it. Why should some bit of math, invented for esthetic reasons, describe the world so perfectly? “One could easily believe that, in some deep sense, the world is mathematical, and we discover its principles,” he says. “If math were a human invention, you’d not expect it to have much effect on arenas [like particle physics] far removed from human experience.”
Yet for all its success, mathematics is fundamentally flawed. In the 1930s Czech mathematician Kurt Godel analyzed systems with self-consistent rules–arithmetic, for instance. He found that they always contain statements that cannot be proved either true or false using those rules. Take someone who says, “I am lying.” If he is he isn’t, and if he isn’t he is. There is no way to determine, using the rules of logic, whether the statement is true or false. There are similar “undecidable” statements in every branch of math, from set theory to algebra to logic itself. Mathematicians cannot even in principle–this has nothing to do with how smart they are–decide whether the assertions are true.
The trouble is that some science is not just mathematical; it’s practically mathematics. Speculative theories about the birth of the cosmos and the world inside the atom are so abstruse they can barely be tested against reality. Instead they’re measured by the seeming truth of their mathematics. Dangerous idea’ it turns out to be impossible to prove that any system–geometry, arithmetic, logic-is self-consistent. That is, the math that physicists use might turn out to be internally contradictory: working out the implications of one set of rules might lead to the conclusion that, say, there are six sorts of particles smaller than atoms, while working out the implications of other rules gives eight. Unless scientists find the contradictions by dumb luck–there’s no way to discover them methodically–they may never know if their theories are so much fiction.
Just because the math that physicists use hasn’t tripped them up yet means nothing. It could just be a matter of time. Take, as Barrow does, the search for a Theory of Everything. A TOE would explain the fall of an apple, the smell of a rose, the ping of radioactivity and the force that keeps atoms together–all by a single set of equations (this was Einstein’s quest). “It is an article of faith that a Theory of Everything will be a mathematical theory,” writes Barrow. “[But] without a deep understanding of the meaning and possible limits of mathematics we run the risk of building our house upon the sand.”
It is also impossible to determine whether the mathematical laws of nature that scientists derive are the best possible. By trial and error one may discover a simpler one. Simplicity is the mark of the “right” theory. But there is no way to prove that a simpler one exists, or doesn’t exist. “One can never know whether or not one has discovered ’the secret of the universe’,” writes Barrow. Similarly, it’s impossible to test whether a particular computer program is the shortest possible. Most likely every program-whether controlling a washing machine or monitoring the skies for ICBMs–contains redundancies. Redundancy is the enemy of efficiency and could keep science from reaching “the ultimate powers of artificial minds.”
Although the fragility of mathematics has long been recognized by philosophers, the word hasn’t filtered down. Disciplines are still dubbed “rigorous” if, like quantum chemistry, they are shot through with equations; they are suspect if, like sociology, they have few mathematical statements at their core. If Barrow is right, the hard sciences would do well to lose their smugness and examine their foundations. And the “soft” sciences should stop suffering Pi-ness envy.
