Author: James Geary

I don’t know much about mathematics (as the saying goes), but I know what I like … and I like George Polya’s style. A lot of the mathematics in his books is way over my head, but nearly every page has some memorable phrase or keen insight into human nature, the psychology of invention, or the process of discovery. Here’s Polya describing, with great perceptiveness and mathematical precision, the experience of wrestling with a problem:

“Your mind becomes selective; it becomes more accessible to anything that appears to be connected with the problem, and less accessible to anything that seems unconnected. You eagerly seize upon any recollection, remark, suggestion, or fact that could help you to solve your problem, and you shut the door upon other things. When the door is so tightly shut that even the most urgent appeals of the external world fail to reach you, people say that you are absorbed.”

Polya’s own sayings may be ostensibly about problem-solving in mathematics, but his shrewdness and humor make them easily adaptable to solving any kind of problem …

To be a good mathematician, or a good gambler, or good at anything, you must be a good guesser.

If you can’t solve a problem, then there is an easier problem you can solve: find it.

Examine your guess. Many a guess has turned out to be wrong but nevertheless useful in leading to a better one.

No idea is really bad, unless we are uncritical. What is really bad is to have no idea at all.

If you cannot solve the proposed problem, try to solve first some related problem.

The more ambitious plan may have more chance of success.

More questions may be easier to answer than just one question.

Look at the unknown. Try to think of a familiar problem having the same unknown.

The first rule of discovery is to have brains and good luck. The second rule of discovery is to sit tight and wait till you get a bright idea.

Look around you when you have got your first mushroom or made your first discovery; they grow in clusters.

When you have satisfied yourself that the theorem is true, you start proving it.

George Polya was born in Budapest, but left Europe for America in 1940. He became a professor of mathematics at Stanford University and remained at Stanford for the rest of his life. He worked on a wide variety of mathematical subjects but is best remembered for his classification of “wallpaper patterns,” the 17 symmetry groups found in any given plane. Polya’s description of these patterns fascinated the artist M.C. Escher, who based many of his works on these tessellations.

Polya is also remembered for his ideas about problem-solving and how it should be taught and carried out. In his book How To Solve it, Polya created a set of heuristic strategies for solving mathematical problems. He even cited a small collection of proverbs that he felt represented his approach.

“Solving problems is a fundamental human activity,” he wrote. “In fact, the greater part of our conscious thinking is concerned with problems … Some people are more and others less successful in attaining their ends and solving their problems. Such differences are noticed, discussed, and commented upon, and certain proverbs seem to have preserved the quintessence of such comments … It would be foolish to regard proverbs as an authoritative source of universally applicable wisdom but it wold be a pity to disregard the graphic description of heuristic procedures provided by proverbs.”

But Polya was more than anthologist; he was a talented aphorist himself. His problem-solving strategies are couched in concise, memorable sentences. First, some of Polya’s collected proverbs are appended below; tomorrow, some of his own aphorisms.

Polya’s Proverbs
Being a collection of wise words on the subject of solving problems

Who understands ill, answers ill.

Think on the end before you begin.

A wise man begins in the end, a fool ends in the beginning.

Try all the keys in the bunch.

Do and undo, the day is long enough.

A wise man will make more opportunities than he finds.

If you will sail without danger you must never put to sea.

We soon believe what we desire.

Do it by degrees.

He thinks not well who thinks not again.

The next time you’re in South Tyrol, consider dropping in on Pension Remichofin St. Josef am See. And when you do, make sure to have a look at the Schaukasten, which features a regular series of aphorisms, this year from German-language aphorist Elazar Benyoetz.

Rosmarie Maran, who runs Pension Remichoff, didn’t plan to have an aphorism showcase at the entrance to her hotel. But when that part of the house was renovated some years ago, the Schaukasten was installed, too. At first, it just featured advertising information about the hotel. “But it soon lost its appeal for us to always say more or less the same thing about our house,” Maran says. “So we thought about a new way to use this little advertising area. And this is what came out.”

“Aphorisms,” Maran admits, “are not at the center of my private interests. I do not really look out for them excessively, and do not try to produce some myself, because too many aphorisms in series always have a somewhat demotivating and paralyzing effect on my mind, even making me a bit sick, like too much chocolate at once.” Still, Maran finds the aphorism postings rewarding: “It really takes time and attention to always find something that is interesting, astonishing, thought-provoking, or at least makes our passers-by smile for a moment.” Guests find it rewarding, too; some regulars make the Schaukasten their first stop on arrival.

Individual aphorisms remain on display for at least one week, sometimes two, before being replaced by a new one. This year, Elazar Benyoetz was the showcased aphorist. Choosing him was not an easy thing, Maran says, “because he is not really funny and many of his aphorisms are deeply philosophical, even religious. He is a real addict of the German language and it is often more the language itself he brings to speech rather than making use of it to express his thoughts.”

You can read a selection below of some of the Benyoetz aphorisms Maran has posted recently. Or better yet, head down to Pension Remichhof and see for yourself…

You have the choice you make. (Man hat die Wahl, die man trifft.)

Your world is only as large as the window you open onto it. (Deine Welt ist nur so groß wie das Fenster, das du ihr aufreißt.)

The time we have is just enough for the moment. (Unsere Zeit reicht für den Augenblick genau aus.)

All understanding is understood correctly, all misunderstanding misunderstood correctly. (Alles Verstehen ist richtig verstanden, alles Missverstehen richtig missverstanden.)

Don’t waste the opportunity your weaknesses offer. (Verscherze dir nicht die Gunst deiner Schwächen.)

You can take any place, but fill only one. (Du kannst jeden Platz einnehmen, nur einen einzigen ausfüllen.)

Thanks to Ursula Sautter, who stayed at Pension Remichoff and told me about the aphorism Schaukasten, who translated the Benyoetz aphorisms in this post, and who also translated from German, Italian, and Spanish for Geary’s Guide.

I was struck by something Rahm Emanuel, President-elect Obama’s White House chief of staff, said the other day: “You don’t ever want a crisis to go to waste; it’s an opportunity to do important things that you would otherwise avoid.” With a little editing (”Never let a crisis to go to waste”) this is not only a great aphorism but a great summation of what aphorisms can do for you. There are plenty of crises to go around these days, whether they be personal, financial, professional, psychological, ecological, or otherwise. Aphorisms are designed for use in just such emergencies but, in contrast to the trite platitudes that too often pass for wisdom, aphorisms do not offer solutions.

Aphorisms don’t even make you feel better. Instead, as Emanuel’s quotation demonstrates, they urge you deeper into crisis as the only way to get out of crisis. Aphorisms don’t offer any false sense of hope that things will be easy. It is impossible to be complacent in the face of a good aphorism. What they do offer is a burst of strength, an invigorating injection of confidence and determination to meet whatever challenges face you.

I came across this aphorism via the AfriGadget website:

When you have nothing, anything is possible.

That is not by any stretch of the imagination a comforting saying, but it is inspiring. And in an emergency, inspiration is more valuable than consolation. This saying made me think of perhaps the greatest crisis management aphorism of all time. When you’re feeling hopeless, just break the glass and pull this lever, supplied by the inimitable Winston Churchill:

When you’re going through hell, keep going.

SRL enjoys hyperbole, which makes him well-suited to writing aphorisms. The initials SRL are not merely an abbreviation, nor are they a nom de plume. They are, he says, his “shadow”: “The shadow of a name is its initials.” His aphorisms “tend towards bludgeoning critical targets and exposing hollow idols,” SRL says, and he typically addresses psychology, politics, economics, and quantum physics. Propaganda is also a recurrent theme, and the subject of one of his most pointed sayings:

Propaganda: One cannot write about it without producing it.

SRL runs a blog, Maximum Advantage in All Things, and the following aphorisms are from his current work-in-progress, Backlash Papers:

Introversion is not withdrawal; it just looks that way.

Subversion is not advocacy.

Decadence can never be undermined, only eliminated.

Reality represents intrusion by those cocooned within anti-natural concerns: a hungry stomach focuses attention even as it drains the mind.

“Why get my knuckles sore?” I asked the hammer.

One can be well educated and still know nothing.

I blame La Rochefoucauld, Chamfort, Vauvenargues, et al… Ever since the golden age of the French aphorism, which lasted roughly from the 16th to the 18th centuries, aphorisms have had an aristocratic reputation, as if the only people who could write and understand them were wealthy noblemen and disaffected dukes. I’ve always thought that was complete hogwash. Aphorisms are, in fact, the most democratic of all written art forms. They have been written and understood for millennia by absolutely everybody, everywhere, at every time in history. And they can still be found in the most unexpected places—on billboards, in pop songs, and out of the mouths of beloved family members. So I am grateful to Ingrid Hunter for sharing some of the sayings of her “late, great Aunt Della,” who is living proof that aphorisms are alive and well and probably regularly spoken by someone near and dear to you:

A drunk mind speaks a sober thought.

Never count on a dead man’s shoes.

Opportunity has long hair in the front and short hair in the back.

Anything is easy if someone else is doing it.

Very seldom does a leopard change its spots.

At a recent appearance at the Falmouth Festival of Literature and Arts, I was asked how you go about writing aphorisms. So I explained the two aphoristic writing methods I had observed: the spontaneous combustion method (inspired impromptu aphorisms scribbled on napkins, receipts or anything else that’s handy, as practiced by aphorists like Stanislaw Jerzy Lec) and the formal composition method (whittling down a much longer piece into one sparkling sentence, as practiced by aphorists like Francois Duc de la Rochefoucauld). All the aphorists I have encountered seem to practice one form or the other. But then the woman who asked the question said that she was actually asking howyou (i.e. me) write aphorisms. I was a bit surprised by the question since, though I still consider myself an aspiring aphorist, I’ve spent most of my time recently writing about aphorisms rather than writing the aphorisms myself. But yesterday I had a moment of spontaneous combustion and realized a few things about the practice of aphorism composition that might suggest the beginnings of an answer to her question.

I was out running on Hampstead Heath. It was a crisp, early autumn afternoon. The trees were turning orange, the leaves already on the ground puckering up like skin that’s been too long in the bath. The sun was slung low on the horizon. I was running past a group of about five people, all standing in a row next to one another, their hands raised to their brows and squinting into the sun. As I jogged past, the following line came to me:

People tend to salute anything that is unnaturally bright, at least until the shade from their hands reveals what it really is.

As aphorisms go, this isn’t great. It’s a bit flabby, a bit too verbose, but it will serve for the purposes of this anatomy. This was a classic case of spontaneous combustion. The line came to me whole and complete. Apart from tinkering a bit with “shows what it really is” versus “reveals what it really is”, I didn’t revise or edit the sentence at all. (I went with “reveals” because of the internal rhyme with “really”.) I looked at those people and the line appeared. That was it. So the first step in the process of composition was observation: seeing something in the world that I could use as an image. Running past those five people all strung out in a line with their hands to their brows reminded me instantly of a group of soldiers standing at attention, saluting as some officer struts past. It seemed to me a really comical sight. I knew, of course, that they were just shielding their eyes against the low-slung sun to see what was going on in the park, but the military image stuck with me. And that led to the second step in the process: making the link between the image and the moral, psychological or philosophical “lesson” aphorisms contain.
So the second stage of composition involved using the image of those people as a metaphor for some other observation; in this case, a sort of psychological comment on how people tend to react to authority figures. When you’re squinting into the sun, objects can appear larger, more luminous, more impressive than they really are. I remember driving through central Spain years ago and seeing what appeared to be enormous black bulls in the distance. Through the heat waves rising from the highway, these huge silhouettes seemed magnificent and menacing. When we got closer, though, and the glare was gone, they turned out to be just raggedy old billboards in the shape of bulls advertising some kind of Spanish beer or something. The same thing often happens with famous people and authority figures: the spotlights that come with their positions make them seem intensely bright and larger than life. But when you see them in ordinary illumination, shorn of spotlights, they turn out to be far less impressive. So the second stage in the process was: making the metaphor that gives the aphorism its psychological point.

Again, I’m not making any grandiose claims about my little aphorism. It is what it is, a shard of reflected light from a brief moment of observation and inspiration. But it seems to me that this must be what happens when aphorisms are composed. Even those who practice the formal composition method must start with some sudden revelation or insight. And it’s amazing how immediately that observation becomes entwined with language. You see some image in the world and less than a nanosecond later your brain has processed it into some clever little sentence. The observation and the insight seem to arrive together, inextricably linked in the mind. For the aphorist, I think, seeing something and saying something are the same thing.

I ruined at least one person’s breakfast yesterday when, during an appearance on The Takeaway to discuss aphorisms (or the lack thereof) in the second presidential debate, I illustrated my point that political slogans are,  by design, almost content-less by citing the Obama campaign slogan “Change you can believe in.” At least one listener found this slanted, further noting that most of the successful political aphorisms I cited during the program were from Republicans. I realize now that, during this time of crisis when what America needs is true bi-partisanship, I should have reached across the aisle and stated explicitly that the McCain campaign slogan, “Country first,” is more or less meaningless, too. My point is, ALL political slogans, from any political party, are deliberately designed to be empty vessels that voters can fill with whatever they please. Political slogans are meant to cast a wide net and sweep up as many people as possible in the context of an uplifting, engaging yet completely vague sentiment. Aphorisms, on the other hand, are unsettling, provocative and intended to make you question assumptions; hence, they are not very popular with politicians.

Neither McCain nor Obama seem, to me at least, to be natural aphorists. Some great politicians of the past have been, though. Lincoln was a brilliant aphorist:

Give me six hours to chop down a tree, and I will spend the first four sharpening the axe.

And so was Truman:

It’s a recession when your neighbor loses his job; it’s a depression when you lose yours.

The aphoristic phrases that candidates use say a lot about them, I think. In both debates so far, Obama has said, in reference to proposed budget cuts, a variation on the theme that McCain wants to

take a hatchet to the budget when what you need is a scalpel.

That’s a very professorial, parsing approach, in keeping with Obama’s cool and cerebral take on issues. McCain, in contrast, takes a much more tough-talking, straight-shooting, rough-riding approach. In the first debate, he cited former Secretary of State George Schultz:

If you point a gun at somebody, you better be prepared to pull the trigger.

That’s an aphorism worthy of a true “maverick,” in keeping with the McCain’s invocation of the spirit of Teddy Roosevelt, a hero of McCain’s and another great presidential aphorist. (In the interests of bi-partisanship and familial comity, may I also add here that the Roosevelts, Franklin and Eleanor both, were also great aphorists.)

The best aphorism of the campaign so far, though, comes not from a politician but from Peggy, the woman in New Hampshire who came up with the last question of the second debate:

What don’t you know and how will you learn it?

This is exactly the question aphorisms ask of us! Aphorisms inspire as they challenge, and it is the ‘challenging’ part that politicians find so, well, challenging. These are extraordinarily challenging times; aphorisms pry open our minds by slipping some healthy doubt, skepticism, and self-reflection into our thinking. It is in this mental space that aphorisms open up that new ideas, new solutions can form. Neither candidate really answered Peggy’s question. A shame, really, because as Mark Twain (was he a Democrat or a Republican?) said:

It ain’t what you don’t know that gets you into trouble, it’s what you know for sure that just ain’t so …

Multatuli (see pp.163–165 of Geary’s Guide) was the pseudonym of Eduard Douwes Dekker, the son of a Dutch sea captain. Dekker seemed destined for a career as an obscure colonial bureaucrat until he uncovered corruption in Dutch–administered Java and decided to expose it. When Dekker brought the exploitation of local labor to the attention of his superiors, they fired him. He returned to Europe and roamed around the Continent for a while, trying to earn enough money gambling to survive. He believed he had invented a foolproof system for winning at casinos, but he always lost. In 1860, he published Max Havelaar, a fictionalized account of the colonial abuses he had witnessed in the Dutch East Indies.

The book caused a sensation throughout Europe, though it initially did nothing to stop the exploitation of the Javanese. After the success ofMax Havelaar, Dekker made a career out of polemical writing, becoming an early supporter of women’s rights, an impassioned lobbyist for educational reform and a fierce critic of religion. He was the first Dutchman to be cremated, in Germany, because at the time of Multatuli’s death it was illegal to be cremated in the Netherlands. His pseudonym is Latin and means, “I have suffered much.” I recently visited the home in which he was born, a tiny house in a narrow lane between the Singel and the Herengracht in central Amsterdam, now a museum. The museum houses an extensive collection of Multatuli books in many different languages, as well as a few scattered personal effects: his desk, an urn for ashes (though his aren’t in it), and the tattered, tasselled red divan on which he died. Some of his aphorisms:

Faith is the voluntary incarceration of the mind.

No one has a high enough estimation of what he could be, nor a low enough one of what he is.

A standpoint reached as the result of an ascent has a different meaning from the same standpoint reached as the result of a fall.

One does not advance the swimming abilities of ducks by throwing the eggs in the water.

He who has never fallen has no true appreciation of what’s needed to stand firm.