How Not to Be Wrong: The Power of Mathematical Thinking

by Jordan Ellenberg

He was the first social scientist in the modern sense. (Condorcet’s term was “social mathematics.”) Condorcet, born into the aristocracy, quickly came to the view that universal laws of thought should take precedence over the whims of kings.

Condorcet lays out his theory in his 1785 treatise Essay on the Application of Analysis to the Probability of Majority Decisions.

“I must act not by what I think reasonable,” Condorcet wrote, “but by what all who, like me, have abstracted from their own opinion must regard as conforming to reason and truth.” The role of the jury is much like the role of the audience on Who Wants to Be a Millionaire? When we have the chance to query a collective, Condorcet thought, even a collective of unknown and unqualified peers, we ought to value their majority opinion above our own.

“social mathematician.”

Condorcet thought he could. He wrote down an axiom—that is, a statement he took to be so self-evident as to require no justification. Here it is: If the majority of voters prefer candidate A to candidate B, then candidate B cannot be the people’s choice.

Imagine what would happen if someone constructed a complicated geometric argument showing that Euclid’s axioms led, inexorably, to a contradiction. Does that seem completely impossible? Be warned—geometry harbors many mysteries.

We don’t talk about democracy that way now. For most people, nowadays, the appeal of democratic choice is that it’s fair; we speak in the language of rights and believe on moral grounds that people should be able to choose their own rulers, whether these choices are wise or not.

In 1820, the Hungarian noble Farkas Bolyai, who had given years of his life to the problem without success, warned his son János against following the same path: You must not attempt this approach to parallels. I know this way to the very end. I have traversed this bottomless night, which extinguished all light and joy in my life. I entreat you, leave the science of parallels alone. . . . I was ready to become a martyr who would remove the flaw from geometry and return it purified to mankind. I accomplished monstrous, enormous labors; my creations are far better than those of others and yet I have not achieved complete satisfaction. . . . I turned back when I saw that no man can reach the bottom of this night. I turned back unconsoled, pitying myself and all mankind. Learn from my example. . . . Sons don’t always take advice from their fathers, and mathematicians don’t always quit easily.

Gore is the candidate who Condorcet’s axiom declares the victor: a majority preferred him to Bush, and an even greater majority preferred him to Nader.

We define the public will to be that mark that appears most frequently on the pieces of paper collected at the voting booth. Even that number, of course, is open to argument: How do we count a partially punched ballot, the so-called hanging chad? What to do with votes mailed from overseas military bases, some of which couldn’t be certified as having been cast on or before Election Day? And to what extent were Florida counties to recount the ballots in an attempt to get as precise a reckoning as possible of the actual votes?

ontological stance ever made by a professional sports official: “It ain’t nothin’ till I call it.”

Legal realists, like judge and University of Chicago professor Richard Posner, argue that Supreme Court jurisprudence is never the exercise in formal rule following that Scalia says it is: Most of the cases the Supreme Court agrees to decide are toss-ups, in the sense that they cannot be decided by conventional legal reasoning, with its heavy reliance on constitutional and statutory language and previous decisions. If they could be decided by those essentially semantic methods, they would be resolved uncontroversially at the level of a state supreme court or federal court of appeals and never get reviewed by the Supreme Court. In this view, the hard questions about law, the ones that make it all the way to the Supremes, are left indeterminate by the axioms.

the same position Pascal was when he found he couldn’t reason his way to any conclusion about God’s existence. And yet, as Pascal wrote, we don’t have the choice not to play the game.

Formalism has an austere elegance. It appeals to people like G. H. Hardy, Antonin Scalia, and me, who relish that feeling of a nice rigid theory shut tight against contradiction. But it’s not easy to hold to principles like this consistently, and it’s not clear it’s even wise. Even Justice Scalia has occasionally conceded that when the literal words of the law seem to require an absurd judgment, the literal words have to be set aside in favor of a reasonable guess as to what Congress must have meant.

ouroboric,

Fun adjective

we are Hilbert’s children; when we have beers with the philosophers on the weekend, and the philosophers hassle us about the status of the objects we study,* we retreat into our formalist redoubt, protesting: sure, we use our geometric intuition to figure out what’s going on, but the way we finally know that what we say is true is that there’s a formal proof behind the picture. In the famous formulation of Philip Davis and Reuben Hersh, “The typical working mathematician is a Platonist on weekdays and a formalist on Sundays.” Hilbert didn’t want to destroy Platonism; he wanted to make the world safe for Platonism, by placing subjects like geometry on a formal foundation so unshakable that we could feel as morally sturdy the whole week as we do on Sunday.

Shortly after the above passage was written, in March 1794 (or, in the rationalized revolutionary calendar, in Germinal of Year 2), Condorcet was captured and arrested. Two days later he was found dead—some say it was suicide, others that he was murdered. Just as Hilbert’s style of mathematics persisted despite the destruction of his formal program by Gödel, Condorcet’s approach to politics survived his demise. We no longer hope to find voting systems that satisfy his axiom. But we have committed ourselves to Condorcet’s more fundamental belief, that a quantitative “social mathematics”—what we now call “social science”—ought to have a part in determining the proper conduct of government. These were “the instruments which increase the power and direct the exercise of [our] faculties” that Condorcet wrote about with such vigor in the Sketch. Condorcet’s idea is so thoroughly intertwined with the modern way of doing political business that we hardly see it as a choice. But it is a choice. I think it’s the right one.

As the philosopher W. V. O. Quine put it, “To believe something is to believe that it is true; therefore a reasonable person believes each of his beliefs to be true; yet experience has taught him to expect that some of his beliefs, he knows not which, will turn out to be false. A reasonable person believes, in short, that each of his beliefs is true and that some of them are false.”

There is a young man in China named Lu Chao who learned and recited 67,890 digits of pi. That’s an impressive feat of memory. But is it interesting? No, because the digits of pi are not interesting.

was for ever denying or distinguishing upon trifles, to the disturbance of all conversation.”

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.

Logic forms a narrow channel through which intuition flows with vastly augmented force.

The lessons of mathematics are simple ones and there are no numbers in them: that there is structure in the world; that we can hope to understand some of it and not just gape at what our senses present to us; that our intuition is stronger with a formal exoskeleton than without one. And that mathematical certainty is one thing, the softer convictions we find attached to us in everyday life another, and we should keep track of the difference if we can.

Every time you observe that more of a good thing is not always better; or you remember that improbable things happen a lot, given enough chances, and resist the lure of the Baltimore stockbroker; or you make a decision based not just on the most likely future, but on the cloud of all possible futures, with attention to which ones are likely and which ones are not; or you let go of the idea that the beliefs of groups should be subject to the same rules as beliefs of individuals; or, simply, you find that cognitive sweet spot where you can let your intuition run wild on the network of tracks formal reasoning makes for it; without writing down an equation or drawing a graph, you are doing math...