How Not to Be Wrong: The Power of Mathematical Thinking

by Jordan Ellenberg

It’s easy to lose sight of the importance of work, because mathematical inspiration, when it finally does come, can feel effortless and instant.

Here’s the French mathematician Henri Poincaré’s famous account of a geometric breakthrough he made in 1881: Having reached Coutances, we entered an omnibus to go some place or other. At the moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry. I did not verify the idea; I should not have had time, as, upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty. On my return to Caen, for conscience’s sake I verified the result at my leisure.*

Mathematics, mostly, is a communal enterprise, each advance the product of a huge network of minds working toward a common purpose, even if we accord special honor to the person who places the last stone in the arch. Mark Twain is good on this: “It takes a thousand men to invent a telegraph, or a steam engine, or a phonograph, or a telephone or any other important thing—and the last man gets the credit and we forget the others.”

Terry Tao writes: The popular image of the lone (and possibly slightly mad) genius—who ignores the literature and other conventional wisdom and manages by some inexplicable inspiration (enhanced, perhaps, with a liberal dash of suffering) to come up with a breathtakingly original solution to a problem that confounded all the experts—is a charming and romantic image, but also a wildly inaccurate one, at least in the world of modern mathematics. We do have spectacular, deep and remarkable results and insights in this subject, of course, but they are the hard-won and cumulative achievement of years, decades, or even centuries of steady work and progress of many good and great mathematicians; the advance from one stage of understanding to the next can be highly non-trivial, and sometimes rather unexpected, but still builds upon the foundation of earlier work rather than starting totally anew. . . . Actually, I find the reality of mathematical research today—in which progress is obtained naturally and cumulatively as a consequence of hard work, directed by intuition, literature, and a bit of luck—to be far more satisfying than the romantic image that I had as a student of mathematics being advanced primarily by the mystic inspirations of some rare breed of “geniuses.”

Genius is a thing that happens, not a kind of person.

Gödel, whose theorem ruled out the possibility of definitively banishing contradiction from arithmetic, was also worried about the Constitution, which he was studying in preparation for his 1948 U.S. citizenship test. In his view, the document contained a contradiction that could allow a Fascist dictatorship to take over the country in a perfectly constitutional manner.

And yet Condorcet’s belief in the inexorability of progress guided by reason and math didn’t desert him. Sequestered in a Paris safe house, knowing he might not have much time left, he wrote his Sketch for a Historical Picture of the Progress of the Human Mind, laying out his vision of the future. It is an astonishingly optimistic document, describing a world from which the errors of royalism, sex prejudice, hunger, and old age would be eliminated in turn by the force of science. This passage is typical: May it not be expected that the human race will be meliorated by new discoveries in the sciences and the arts, and, as an unavoidable consequence, in the means of individual and general prosperity; by farther progress in the principles of conduct, and in moral practice; and lastly, by the real improvement of our faculties, moral, intellectual and physical, which may be the result either of the improvement of the instruments which increase the power and direct the exercise of those faculties, or of the improvement of our natural organization itself?

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.

People like that, the quibblers and the naysayers and the maybesayers, don’t make things happen. When one wants to denounce those people, it’s customary to quote Theodore Roosevelt, from his speech “Citizenship in a Republic,” delivered in Paris in 1910, shortly after the end of his presidency: It is not the critic who counts; not the man who points out how the strong man stumbles, or where the doer of deeds could have done them better. The credit belongs to the man who is actually in the arena, whose face is marred by dust and sweat and blood; who strives valiantly; who errs, who comes short again and again, because there is no effort without error and shortcoming; but who does actually strive to do the deeds; who knows great enthusiasms, the great devotions; who spends himself in a worthy cause; who at the best knows in the end the triumph of high achievement, and who at the worst, if he fails, at least fails while daring greatly, so that his place shall never be with those cold and timid souls who neither know victory nor defeat.

You can find issues there we’ve discussed elsewhere in this book, as where Roosevelt touches on the diminishing utility of money— The truth is that, after a certain measure of tangible material success or reward has been achieved, the question of increasing it becomes of constantly less importance compared to the other things that can be done in life.

the main theme, to which Roosevelt returns throughout the speech, is that the survival of civilization depends on the triumph of the bold, commonsensical, and virile against the soft, intellectual, and infertile.*

Roosevelt’s goal was just the opposite: he pays lip service to the accomplishments of the French academics, but makes it clear their book learning is of only secondary importance in the production of national greatness: “I speak in a great university which represents the flower of the highest intellectual development; I pay all homage to intellect and to elaborate and specialized training of the intellect; and yet I know I shall have the assent of all of you present when I add that more important still are the commonplace, every-day qualities and virtues.”

And when Roosevelt sneers at the cold and timid souls who sit on the sidelines and second-guess the warriors, I come back to Abraham Wald, who as far as I know went his whole life without lifting a weapon in anger, but who nonetheless played a serious part in the American war effort, precisely by counseling the doers of deeds how to do them better. He was unsweaty, undusty, and unbloody, but he was right. He was a critic who counted.

Against Roosevelt I place John Ashbery, whose poem “Soonest Mended” is the greatest summation I know of the way uncertainty and revelation can mingle, without dissolving together, in the human mind. It’s a more complex and accurate portrait of life’s enterprise than Roosevelt’s hard-charging man’s man, sore and broken but never doubting his direction. Ashbery’s sad-comic vision of citizenship might almost be a reply to Roosevelt’s “Citizenship in a Republic”: And you see, both of us were right, though nothing Has somehow come to nothing; the avatars Of our conforming to the rules and living Around the home have made—well, in a sense, “good citizens” of us, Brushing the teeth and all that, and learning to accept The charity of the hard moments as they are doled out, For this is action, this not being sure, this careless Preparing, sowing the seeds crooked in the furrow, Making ready to forget, and always coming back To the mooring of starting out, that day so long ago.

But mathematics is also a means by which we can reason about the uncertain, taming if not altogether domesticating it. It’s been that way since the time of Pascal, who started by helping gamblers understand the whims of chance and ended up figuring the betting odds on the most cosmic uncertainty of all.*

Math gives us a way of being unsure in a principled way: not just throwing up our hands and saying “huh,” but rather making a firm assertion: “I’m not sure, this is why I’m not sure, and this is roughly how not-sure I am.” Or even more: “I’m unsure, and you should be too.”

If it’s September 2012 and you ask a bunch of political pundits, “Who’s going to be elected president in November?” a bunch of them are going to say, “Obama is,” and a somewhat smaller bunch are going to say, “Romney is,” and the point is that all of those people are wrong, because the right answer is the kind of answer that Silver, almost alone in the broad-reach media, was willing to give: “Either one might win, but Obama is substantially more likely to win.”

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.”

The journalist Charles Seife included in his book Proofiness a very funny and mildly depressing chronicle of the similarly close contest between Democrat Al Franken and Republican Norm Coleman to represent Minnesota in the U.S. Senate.

the digits of pi are not interesting. As far as anyone knows, they’re as good as random. Pi itself is interesting, to be sure. But pi is not its digits; it is merely specified by its digits, in the same way the Eiffel Tower is specified by the longitude and latitude 48.8586° N, 2.2942° E. Add as many decimal places to those numbers as you want, and they still won’t tell you what makes the Eiffel Tower the Eiffel Tower.

Benjamin Franklin wrote cuttingly of a member of his Philadelphia set, Thomas Godfrey: “He knew little out of his way, and was not a pleasing companion; as, like most great mathematicians I have met with, he expected universal precision in everything said, or was for ever denying or distinguishing upon trifles, to the disturbance of all conversation.”

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?)

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.

when you’re working hard on a theorem you should try to prove it by day and disprove it by night.

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.

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.

To the greatest extent possible you have to think as though you believe what you don’t believe.

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 h...

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...

as any math student knows, the old problems you meet freshman year are some of the deepest you ever see.

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.

What’s true is that the sensation of mathematical understanding—of suddenly knowing what’s going on, with total certainty, all the way to the bottom—is a special thing, attainable in few if any other places in life. You feel you’ve reached into the universe’s guts and put your hand on the wire.

To do mathematics is to be, at once, touched by fire and bound by reason.

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 ...

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 fo...

You’ve been using mathematics since you were born and you’ll probably ne...

perhaps it’s best not to complain too loudly about the incorrect use of “exponential” to mean simply “fast”—I recently saw a sportswriter, who had no doubt been scolded at some point about exponential, refer to sprinter Usain Bolt’s “astonishing, logarithmic rise in speed,” which is even worse.

see E. Roy Weintraub’s How Economics Became a Mathematical Science,