Posts Tagged ‘George Pólya’
Quote of the Day
The result of the mathematician’s creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing. If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference…In plausible reasoning the principal thing is to distinguish a guess from a guess, a more plausible guess from a less reasonable guess…A serious student of mathematics, intending to make it his life’s work, must learn demonstrative reasoning; it is his profession and the distinctive mark of his science. Yet for real success he must also learn plausible reasoning; this is the kind of reasoning on which his creative work will depend.
Our fearful symmetry
Yesterday, I spent an hour fixing my garage door, which got stuck halfway up and refused to budge. I went about it in the way I usually approach such household tasks: I took a flashlight and a pair of vise-grips and stared at it for a while. In this case, for once, it worked, even if I’m only postponing the inevitable service call. But it wouldn’t have occurred to me to tackle it myself in the first place—when I probably wouldn’t have tried to fix, say, my own car by trial and error—if it hadn’t been for two factors. The first is that the workings were all pretty visible. On each side, there’s basically just a torsion spring, a steel cable, and two pulleys, all of it exposed to plain sight. The second point, which was even more crucial, is that a garage door is symmetrical, and only one side was giving me trouble. Whenever I wasn’t sure how the result should look, I just had to look at the other half and mentally reflect it to its mirror image. It reminded me of how useful symmetry can be in addressing many problems, as George Pólya notes in How to Solve It:
Symmetry, in a general sense, is important for our subject. If a problem is symmetric in some ways we may derive some profit from noticing its interchangeable parts and it often pays to treat those parts which play the same role in the same fashion…Symmetry may also be useful in checking results.
And if I hadn’t been able to check my work along the way using the other side of the door, I doubt I would have attempted to fix it at all.
But it also points at a subtle bias in the way we pick our problems. There’s no question that symmetry plays an important role in the world around us, and it provides a solid foundation for the notion that we can use beauty or elegance as an investigative tool. “It seems that if one is working from the point of view of getting beauty in one’s equations, and if one has a really sound insight, one is on a sure line of progress,” Paul Dirac famously said, and Murray Gell-Mann gave the best explanation I’ve ever found of why this might be true:
There’s a quotation from Newton, I don’t remember the exact words but lots of other physicists have made the same remark since—that nature seems to have a remarkable property of self-similarity. The laws—the fundamental laws—at different levels seem to resemble one another. And that’s probably what accounts for the possibility of using elegance as a criterion [in science]. We develop a mathematical formula, say, for describing something at a particular level, and then we go to a deeper level and find that in terms of mathematics, the equations at the deeper level are beautifully equivalent. Which means that we’ve found an appropriate formula.
Gell-Mann concludes: “And that takes the human being, human judgment, out of it a little. You might object that after all we are the ones who say what elegance is. But I don’t think that’s the point.”
He’s right, of course, and there are plenty of fields in which symmetry and self-similarity are valuable criteria. Yet there’s also a sense in which we’re drawn to problems in which such structures appear, while neglecting those that aren’t as amenable to symmetrical thinking. Just as I was willing to take apart my garage door when I wouldn’t have done the same with my car—which, after all, has a perfectly logical design—it’s natural for us to prefer problems that are obviously symmetrical or that hold out the promise of elegance, much as we’re attracted to the same qualities in the human face. But there are plenty of important questions that aren’t elegant at all. I’m reminded of what Max Perutz, who described the structure of hemoglobin, said about the work of his more famous colleague James Watson:
I sometimes envied Jim. My own problem took thousands of hours of hard work, measurements, calculations. I often thought that there must be some way to cut through it—that there must be, if only I could see it, an elegant solution. There wasn’t any. For Jim’s there was an elegant solution, which is what I admired. He found it partly because he never made the mistake of confusing hard work with hard thinking; he always refused to substitute one for the other.
In The Eighth Day of Creation, Horace Freehand Judson calls this “the most exact yet generous compliment I have ever heard from one scientist to another.” But there’s also a wistful acknowledgement of the luck of the draw. Both Perutz and Watson were working on problems of enormous importance, but only one of them had an elegant solution, and there was no way of knowing in advance which one it would be.
Given the choice, I suspect that most of us would prefer to work on problems that exhibit some degree of symmetry: they’re elegant, intuitive, and satisfying. In the absence of that kind of order, we’re left with what Perutz calls “thousands of hours of hard work, measurements, calculations,” and it isn’t pretty. (As Donald Knuth says in a somewhat different context: “Without any underlying symmetry properties, the job of proving interesting results becomes extremely unpleasant.”) When we extrapolate this preference to the culture as a whole, it leads to two troubling tendencies. One is to prioritize problems that lend themselves to this sort of attack, while overlooking whole fields of messier, asymmetrical phenomena that resist elegant analysis—to the point where we might even deny that they’re worth studying at all. The other is to invent a symmetry that isn’t there. You can see both impulses at work in the social sciences, which tend to deal with problems that can’t be reduced to a series of equations, and they’re particularly insidious in economics, which is uniquely vulnerable to elegant models that confirm what existing interest groups want to hear. From there, it’s only a small step to more frightening forms of fake symmetry, as Borges writes in “Tlön, Uqbar, Orbis Tertius”: “Any symmetry with a resemblance of order—dialectical materialism, anti-Semitism, Nazism—was sufficient to entrance the minds of men.” And the first habit has a way of leading to the second. The more we seek out problems with symmetry while passing over those that lack it, the more likely we become to attribute false symmetries to the world around us. Symmetry, by definition, is a beautiful thing. But it can also turn us into suckers for a pretty face.
Solving for X
According to the poet Robert Earl Hayden, W.H. Auden once said: “Writing a poem is like solving for X in an equation.” More recently, a similar analogy was employed by the journalist and podcaster Alex Blumberg, who explains:
I’ve developed a mathematical test to tell whether you’re on the right track. It’s called the “and what’s interesting” test. You simply tell someone about the story you’re doing, adhering to a very strict formula: “I’m doing a story about X. And what’s interesting about it is Y.” So for example, again, taking the homeless story, “I’m doing a story about a homeless guy who lived on the streets for 10 years, and what’s interesting is, he didn’t get off the streets until he got into a treatment program.” Wrong track. Solve for a different Y.
And while this might seem to make the art of poetry or storytelling feel unbearably dry, it’s really quite the opposite. As Jakob Einstein famously told his nephew Albert: “[Algebra is] a merry science. When the animal that we are hunting cannot be caught, we call it X temporarily and continue to hunt until it is bagged.”
Thinking of writing, or any creative endeavor, as a subcategory of this “merry science” clarifies many of the issues that confront the aspiring solver. The typical problem in mathematics or geometry consists of an unknown, some data, and a condition, and the same can be said of many of the narrative issues that a writer is compelled to address. When you’re first plotting out a story, particularly a novel, the number of individual decisions you have to make can seem overwhelming, but you usually have more information than you realize. Once you’ve spent even a modicum of time mulling over an idea, you wind up with at least an initial premise, a location, some primary characters, and a few of the major story beats—although, as I’ve noted before, many of these seemingly fundamental units are also the result of working backward from an earlier problem. When you line them all up, you generally find that they also imply other scenes or ideas: to get your characters from point A to point C, it doesn’t take a genius to see that you should pass through point B first, at least in your initial outline. (Point B often ends up being omitted in the rewrite, but it helps to lay it out blandly at first, if only in hopes that it generates some useful material.) And by the time you’ve laid all the obvious scenes from end to end, along with the connective tissue that they suggest, you often discover that you’ve got most of what you need. The hard part is solving for the remaining unknowns.
And you can’t do this until you’ve suitably arranged the pieces that you have, which can be easier said than done. Just as the first step in solving a linear equation is to get the variable by itself on one side, the unknown in any story can only be found once you’ve isolated it as much as possible from the surrounding elements. Hence the charts, graphs, and lists that writers produce in such quantities: once you’ve got everything down on paper in some kind of rough order, you start to see where the gaps exist. George Pólya, in his classic book How to Solve It, advises:
If there is a figure connected with the problem [the student] should draw a figure and point out on it the unknown and the data. If it is necessary to give names to these objects he should introduce suitable notation; devoting some attention to the appropriate choice of signs, he is obliged to consider the objects for which the signs have to be chosen.
And this last point is crucial. The outline isn’t the story, any more than an equation is the physical object that it represents, but by giving names or signs to the component parts, you can see through to the reality beneath for the first time.
In On Directing Film, David Mamet says much the same about identifying the beats of a story: “Here is a tool—choose your shots, beats, scenes, objectives, and always refer to them by the names you chose.” Once you’ve named the unknown, you can start to hunt for it more systematically, using some of the methods that Polya describes:
Look at the unknown. This is old advice; the corresponding Latin saying is: “respice finem.” That is, look at the end. Remember your aim…Focusing our attention on our aim and concentrating our will on our purpose, we think of ways and means to attain it. What are the means to this end? How can you attain your aim? How can you obtain a result of this kind? What causes could produce such a result? Where have you seen such a result produced? What do people usually do to obtain such a result? And try to think of a familiar problem having the same or a similar unknown. And try to think of a familiar theorem having the same or a similar conclusion.
Pólya compares this “same or similar unknown” to a stepping stone, and he adds drily: “The new unknown should be both accessible and useful but, in practice, we must often content ourselves with less.” It’s a system of successive approximations, or good hunches, converging at last on an answer that fits. And if we’re lucky, we’ll find that X, for once, marks the spot.
Quote of the Day
Certainly, let us learn proving, but also let us learn guessing.
Proverbs for problem-solvers
It could be an interesting task to collect and group proverbs about planning, seeking means, and choosing between lines of action, in short, proverbs about solving problems. Of the space needed for such a task only a small fraction is available here; the best we can do is to quote a few…
A good idea is a piece of good fortune, an inspiration, a gift of the gods, and we have to deserve it: Diligence is the mother of good luck. Perseverance kills the game. An oak is not felled at one stroke. If at first you don’t succeed, try, try again. It is not enough however to try repeatedly, we must try different means, vary our trials. Try all the keys in the bunch. Arrows are made of all sorts of wood. We must adapt our trials to the circumstances. As the wind blows you must set your sail. Cut your coat according to the cloth. We must do as we may if we can’t do as we should. If we have failed, we must try something else. A wise man changes his mind, a fool never does. We should even be prepared from the outset for a possible failure of our scheme and have another one in reserve. Have two strings to your bow. We may, of course, overdo this sort of changing from one scheme to another and lose time. Then we may hear the ironical comment: Do and undo, the day is long enough. We are likely to blunder less if we do not lose sight of our aim. The end of fishing is not angling but catching.
Quote of the Day
Pedantry and mastery are opposite attitudes toward rules. To apply a rule to the letter, rigidly, unquestioningly, in cases where it fits and in cases where it does not fit, is pedantry…To apply a rule with natural ease, with judgment, noticing the cases where it fits, and without ever letting the words of the rule obscure the purpose of the action or the opportunities of the situation, is mastery.
Quote of the Day
When you have satisfied yourself that the theorem is true, you start proving it.
Alec Nevala-Lee 