Principle of Explosion (Ex Falso Quodlibet)

THAT DOES NOT COMPUTE

And yet even mathematicians don’t, except when cracking wise, try to make themselves beings of pure logic. It can be dangerous to do so! For example: If you’re a purely deductive thinker, once you believe two contradictory facts you are logically obliged to believe that every statement is false. Here’s how that goes. Suppose I believe both that Paris is the capital of France and that it’s not. This seems to have nothing to do with whether the Portland Trail Blazers were NBA champions in 1982. But now watch this trick. Is it the case that Paris is the capital of France and the Trail Blazers won the NBA championship? It is not, because I know that Paris is not the capital of France.

If it’s not true that Paris is the capital of France and the Trail Blazers were the champs, then either Paris isn’t the capital of France or the Trail Blazers weren’t NBA champs. But I know that Paris is the capital of France, which rules out the first possibility. So the Trail Blazers did not win the 1982 NBA championship.

It is not hard to check that an argument of exactly the same form, but standing on its head, proves that every statement is also true.

This sounds weird, but as a logical deduction it’s irrefutable; drop one tiny contradiction anywhere into a formal system and the whole thing goes to hell. Philosophers of a mathematical bent call this brittleness in formal logic ex falso quodlibet, or, among friends, “the principle of explosion.” (Remember what I said about how much math people love violent terminology?)

Ex falso quodlibet is how Captain James T. Kirk used to disable dictatorial AIs—feed them a paradox and their reasoning modules frazzle and halt. That (they plaintively remark, just before the power light goes out) does not compute. Bertrand Russell did to Gottlob Frege’s set theory what Kirk did to uppity robots. His one sneaky paradox brought the whole edifice down.

But Kirk’s trick doesn’t work on human beings. We don’t reason this way, not even those of us who do math for a living. We are tolerant of contradiction, to a point. As F. Scott Fitzgerald said, “The test of a first-rate intelligence is the ability to hold two opposed ideas in the mind at the same time, and still retain the ability to function.”

Mathematicians use this ability as a basic tool of thought. It’s essential for the reductio ad absurdum, which requires you to hold in your mind a proposition you believe to be false and reason as if you think it’s true: suppose the square root of 2 is a rational number, even though I’m trying to prove it’s not. . . . It is lucid dreaming of a very systematic kind. And we can do it without short-circuiting ourselves.

In fact, it’s a common piece of folk advice—I know I heard it from my PhD adviser, and presumably he from his, etc.—that when you’re working hard on a theorem you should try to prove it by day and disprove it by night. (The precise frequency of the toggle isn’t critical; it’s said of the topologist R. H. Bing that his habit was to split each month between two weeks trying to prove the Poincaré Conjecture and two weeks trying to find a counterexample.)

Why work at such cross-purposes? There are two good reasons. The first is that you might, after all, be wrong; if the statement you think is true is really false, all your effort to prove it is doomed to be useless. Disproving by night is a kind of hedge against that gigantic waste.

But there’s a deeper reason. If something is true and you try to disprove it, you will fail. We are trained to think of failure as bad, but it’s not all bad. You can learn from failure. You try to disprove the statement one way, and you hit a wall. You try another way, and you hit another wall. Each night you try, each night you fail, each night a new wall, and if you are lucky, those walls start to come together into a structure, and that structure is the structure of the proof of the theorem. For if you have really understood what’s keeping you from disproving the theorem, you very likely understand, in a way inaccessible to you before, why the theorem is true. This is what happened to Bolyai, who bucked his father’s well-meaning advice and tried, like so many before him, to prove that the parallel postulate followed from Euclid’s other axioms. Like all the others, he failed. But unlike the others, he was able to understand the shape of his failure. What was blocking all his attempts to prove that there was no geometry without the parallel postulate was the existence of just such a geometry! And with each failed attempt he learned more about the features of the thing he didn’t think existed, getting to know it more and more intimately, until the moment when he realized it was really there.

Proving by day and disproving by night is not just for mathematics. I find it’s a good habit to put pressure on all your beliefs, social, political, scientific, and philosophical. Believe whatever you believe by day; but at night, argue against the propositions you hold most dear. Don’t cheat! To the greatest extent possible you have to think as though you believe what you don’t believe. And if you can’t talk yourself out of your existing beliefs, you’ll know a lot more about why you believe what you believe. You’ll have come a little closer to a proof.

This salutary mental exercise is not at all what F. Scott Fitzgerald was talking about, by the way. His endorsement of holding contradictory beliefs comes from “The Crack-Up,” his 1936 essay about his own irreparable brokenness. The opposing ideas he has in mind there are “the sense of futility of effort and the sense of the necessity to struggle.” Samuel Beckett later put it more succinctly: “I can’t go on, I’ll go on.” Fitzgerald’s characterization of a “first-rate intelligence” is meant to deny his own intelligence that designation; as he saw it, the pressure of the contradiction had made him effectively cease to exist, like Frege’s set theory or a computer downed by Kirkian paradox. (The Housemartins, elsewhere in “Sitting on a Fence,” more or less summarize “The Crack-Up”: “I lied to myself right from the start / and I just worked out that I’m falling apart.”) Unmanned and undone by self-doubt, drowned in books and introspection, he had become exactly the kind of sad young literary man who made Theodore Roosevelt puke.

David Foster Wallace was interested in paradox too. In his characteristically mathematical style, he put a somewhat tamed version of Russell’s paradox at the center of his first novel, The Broom of the System. It isn’t too strong to say his writing was driven by his struggle with contradictions. He was in love with the technical and analytic, but he saw that the simple dicta of religion and self-help offered better weapons against drugs, despair, and killing solipsism. He knew it was supposed to be the writer’s job to get inside other people’s heads, but his chief subject was the predicament of being stuck fast inside one’s own. Determined to record and neutralize the influence of his own preoccupations and prejudices, he knew this determination was itself among those preoccupations and subject to those prejudices. This is Phil 101 stuff, to be sure, but as any math student knows, the old problems you meet freshman year are some of the deepest you ever see. Wallace wrestled with the paradoxes just the way mathematicians do. You believe two things that seem in opposition. And so you go to work—step by step, clearing the brush, separating what you know from what you believe, holding the opposing hypotheses side by side in your mind and viewing each in the adversarial light of the other until the truth, or the nearest you can get to it, comes clear.

As for Beckett, he had a richer and more sympathetic view of contradiction, which is so ever-present in his work that it takes on every possible emotional color somewhere or other in the corpus. “I can’t go on, I’ll go on” is bleak; but Beckett also draws on the Pythagoreans’ proof of the irrationality of the square root of 2, turning it into a joke between drunks:

“But betray me,” said Neary, “ and you go the way of Hippasos.”

“The Akousmatic, I presume,” said Wylie. “His retribution slips my mind.”

“Drowned in a puddle,” said Neary, “for having divulged the incommensurability of side and diagonal.”

“So perish all babblers,” said Wylie.

It’s not clear how much higher math Beckett knew, but in his late prose piece Worstward Ho, he sums up the value of failure in mathematical creation more succinctly than any professor ever has:

Ever tried. Ever failed. No matter. Try again. Fail again. Fail better.