George Pólya Quotes

If we deal with our problem not knowing, or pretending not to know the general theory encompassing the concrete case before us, if we tackle the problem "with bare hands", we have a better chance to understand the scientist's attitude in general, and especially the task of the applied mathematician.

There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain theorem, and I said it is not proved and it may be difficult. Von Neumann didn’t say anything but after five minutes he raised his hand. When I called on him he went to the blackboard and proceeded to write down the proof. After that I was afraid of von Neumann.

Good approximations often lead to better ones.

Perseverance kills the game.

A great discovery solves a great problem, but there is a grain of discovery in the solution of any problem. Your problem may be modest, but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery.

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

Beauty in mathematics is seeing the truth without effort.

The advanced reader who skips parts that appear too elementary may miss more than the less advanced reader who skips parts that appear too complex.

teaching of calculus to engineers and physicists, could be essentially improved if the nature of heuristic reasoning were better understood,

Demonstrative reasoning penetrates the sciences just as far as mathematics does, but it is in itself (as mathematics is in itself) incapable of yielding essentially new knowledge about the world around us. Anything new that we learn about the world involves plausible reasoning, which is the only kind of reasoning for which we care in everyday affairs.

Here is a typical story about Mr. John Jones. Mr. Jones works in an office. He had hoped for a little raise but his hope, as hopes often are, was disappointed. The salaries of some of his colleagues were raised but not his. Mr. Jones could not take it calmly. He worried and worried and finally suspected that Director Brown was responsible for his failure in getting a raise. We cannot blame Mr. Jones for having conceived such a suspicion. There were indeed some signs pointing to Director Brown. The real mistake was that, after having conceived that suspicion, Mr. Jones became blind to all signs pointing in the opposite direction. He worried himself into firmly believing that Director Brown was his personal enemy and behaved so stupidly that he almost succeeded in making a real enemy of the director. The trouble with Mr. John Jones is that he behaves like most of us. He never changes his major opinions. He changes his minor opinions not infrequently and quite suddenly; but he never doubts any of his opinions, major or minor, as long as he has them. He never doubts them, or questions them, or examines them critically — he would especially hate critical examination, if he understood what that meant. Let us concede that Mr. John Jones is right to a certain extent. He is a busy man; he has his duties at the office and at home. He has little time for doubt or examination. At best, he could examine only a few of his convictions and why should he doubt one if he has no time to examine that doubt? Still, don’t do as Mr. John Jones does. Don’t let your suspicion, or guess, or conjecture, grow without examination till it becomes ineradicable. At any rate, in theoretical matters, the best of ideas is hurt by uncritical acceptance and thrives on critical examination. 2. A mathematical example. Of all quadrilaterals with

Life is full of surprises: our approximate condition for the fall of a body through a resisting medium is precisely analogous to the exact condition for the flow of an electric current through a resisting wire [of an induction coil<nowiki>]</nowiki>. ...
<math>m\frac {dv}{dt} = mg - Kv</math>
This is the form most convenient for making an analogy with the "fall", i.e., flow, of an electric current.
...in order from left to right, mass <math>m</math>, rate of change of velocity <math>\frac {dv}{dt}</math>, gravitational force <math>mg</math>, and velocity <math>v</math>. What are the electrical counterparts? ...To press the switch, to allow current to start flowing is the analogue of opening the fingers, to allow the body to start falling. The fall of the body is caused by the force <math>mg</math> due to gravity; the flow of the current is caused by the electromotive force or tension <math>E</math> due to the battery. The falling body has to overcome the frictional resistance of the air; the flowing current has to overcome the electrical resistance of the wire. Air resistance is proportional to the body's velocity <math>v</math>; electrical resistance is proportional to the current <math>i</math>. And consequently rate of change of velocity <math>\frac {dv}{dt}</math> corresponds to rate of change of current <math>\frac {di}{dt}</math>. ...The electromagnetic induction <math>L</math> opposes the change of current... And doesn't the inertia or mass <math>m</math>..? Isn't <math>L</math>, so to speak, an electromagnetic inertia?
<math>L\frac {di}{dt} = E - Ki</math>

Facing any part of the observable reality, we are never in possession of complete knowledge, nor in a state of complete ignorance, although usually much closer to the latter state.

One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else. We have a natural opportunity to investigate the connections of a problem when looking back at its solution.

The cookbook gives a detailed description of ingredients and procedures but no proofs for its prescriptions or reasons for its recipes; the proof of the pudding is in the eating. … Mathematics cannot be tested in exactly the same manner as a pudding; if all sorts of reasoning are debarred, a course of calculus may easily become an incoherent inventory of indigestible information.