George Polya Quotes about Math

It is better to solve one problem five different ways, than to solve five problems one way.

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

An idea which can be used only once is a trick. If one can use it more than once it becomes a method.

Solving problems is a practical skill like, let us say, swimming. We acquire any practical skill by imitation and practice. Trying to swim, you imitate what other people do with their hands and feet to keep their heads above water, and, finally, you learn to swim by practicing swimming. Trying to solve problems, you have to observe and to imitate what other people do when solving problems, and, finally, you learn to do problems by doing them.

Beauty in mathematics is seeing the truth without effort.

To teach effectively a teacher must develop a feeling for his subject; he cannot make his students sense its vitality if he does not sense it himself. He cannot share his enthusiasm when he has no enthusiasm to share. How he makes his point may be as important as the point he makes; he must personally feel it to be important.

Mathematics is not a spectator sport!

If you have to prove a theorem, do not rush. First of all, understand fully what the theorem says, try to see clearly what it means. Then check the theorem; it could be false. Examine the consequences, verify as many particular instances as are needed to convince yourself of the truth. When you have satisfied yourself that the theorem is true, you can start proving it.

I am too good for philosophy and not good enough for physics. Mathematics is in between.

Mathematics consists in proving the most obvious thing in the least obvious way.

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.

When introduced at the wrong time or place, good logic may be the worst enemy of good teaching.

What is the difference between method and device? A method is a device which you use twice.

In order to solve this differential equation you look at it until a solution occurs to you.

Mathematics has two faces: it is the rigorous science of Euclid, but it is also something else. Mathematics presented in the Euclidean way appears as a systematic, deductive science; but mathematics in the making appears as an experimental, inductive science. Both aspects are as old as the science of mathematics itself.

A mathematician who can only generalise is like a monkey who can only climb up a tree, and a mathematician who can only specialise is like a monkey who can only climb down a tree. In fact neither the up monkey nor the down monkey is a viable creature. A real monkey must find food and escape his enemies and so must be able to incessantly climb up and down. A real mathematician must be able to generalise and specialise.

The principle is so perfectly general that no particular application of it is possible.

Geometry is the science of correct reasoning on incorrect figures.

A GREAT discovery solves a great problem but there is a grain of discovery in any problem.

I am intentionally avoiding the standard term which, by the way, did not exist in Euler's time. One of the ugliest outgrowths of the "new math" was the premature introduction of technical terms.