George Polya Quotes

If the proof starts from axioms, distinguishes several cases, and takes thirteen lines in the text book ... it may give the youngsters the impression that mathematics consists in proving the most obvious things in the least obvious way.

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

Even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else. Doing so, they miss an important and instructive phase of the work. ... A good teacher should understand and impress on his students the view that no problem whatever is completely exhausted.

The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing.

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.

Euclid 's manner of exposition, progressing relentlessly from the data to the unknown and from the hypothesis to the conclusion, is perfect for checking the argument in detail but far from being perfect for making understandable the main line of the argument.

The elegance of a mathematical theorem is directly proportional to the number of independent ideas one can see in the theorem and inversely proportional to the effort it takes to see them.

Success in solving the problem depends on choosing the right aspect, on attacking the fortress from its accessible side.

The open secret of real success is to throw your whole personality into your problem.

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.

My method to overcome a difficulty is to go round it.