Aphorisms by George Polya, Part II
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.