George Pólya Quotes

44 Sourced Quotes

Mathematics presented with rigor is a systematic deductive science but mathematics in the making is an experimental inductive science.

George Pólya
We wish to see... the typical attitude of the scientist who uses mathematics to understand the world around us.... In the solution of a problem... there are typically three phases. The first phase is entirely or almost entirely a matter of physics; the third, a matter of mathematics; and the intermediate phase, a transition from physics to mathematics. The first phase is the formulation of the physical hypothesis or conjecture; the second, its translation into equations; the third, the solution of the equations. Each phase calls for a different kind of work and demands a different attitude.

George Pólya
Analogy pervades all our thinking, our everyday speech and our trivial conclusions as well as artistic ways of expression and the highest scientific achievements.

George Pólya
The differential equation of the first order

... prescribes the slope at each point of the plane (or at each point of a certain region of the plane we call the field").... a differential equation of the first order... can be conceived intuitively as a problem about the steady flow of a river: Being given the direction of the flow at each point, find the streamlines.... It leaves open the choice between the two possible directions in the line of a given slope. Thus... we should say specifically "direction of an unoriented straight line" and not merely "direction."

George Pólya
I... present also examples of historic interest, examples of real mathematical beauty, and examples illustrating the parallelism of the procedures in other sciences, or in everyday life.

George Pólya
We need heuristic reasoning when we construct a strict proof as we need scaffolding when we erect a building.

George Pólya
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.

George Pólya
If you cannot solve the proposed problem, try to solve first a simpler related problem.

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

George Pólya
It is hard to have a good idea if we have little knowledge of the subject, and impossible to have it if we have no knowledge. Good ideas are based on past experience and formerly acquired knowledge.

George Pólya
You should not put too much trust in any unproved conjecture, even if it has been propounded by a great authority, even if it has been propounded by yourself. You should try to prove it or disprove it...

George Pólya
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.

George Pólya
A great discovery solves a great problem but there is a grain of discovery in the solution of any problem.

George Pólya
It may be more important in the mathematics class how you teach than what you teach.

George Pólya
In order to solve a differential equation you look at it till a solution occurs to you.

George Pólya
This principle is so perfectly general that no particular application is possible.

George Pólya
In our personal life we often cling to illusions. That is, we do not dare to examine certain beliefs which could be easily contradicted by experience, because we are afraid of upsetting our emotional balance.

George Pólya
The traditional mathematics professor of the popular legend is absentminded. He usually appears in public with a lost umbrella in each hand. He prefers to face a blackboard and to turn his back on the class. He writes a, he says b, he means c, but it should be d.

George Pólya
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.

George Pólya
Geometry is the art of correct reasoning on incorrect figures.

George Pólya
Mathematics is being lazy. Mathematics is letting the principles do the work for you so that you do not have to do the work yourself.

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

George Pólya
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.

George Pólya
Even if without the Scott's proverbial thrift, the difficulty of solving differential equations is an incentive to using them parsimoniously. Happily here is a commodity of which a little may be made to go a long way.... the equation of small oscillations of a pendulum also holds for other vibrational phenomena. In investigating swinging pendulums we were, albeit unwittingly, also investigating vibrating tuning forks.

George Pólya
The efficient use of plausible reasoning is a practical skill and it is learned... by imitation and practice.... what I can offer are only examples for imitation and opportunity for practice.

George Pólya
The volume of the cone was discovered by Democritus... He did not prove it, he guessed it... not a blind guess, rather it was reasoned conjecture. As Archimedes has remarked, great credit is due to Democritus for his conjecture since this made proof much easier. Eudoxes... a pupil of Plato, subsequently gave a rigorous proof. Surely the labor or writing limited his manuscript to a few copies; none has survived. In those days editions did not run to thousands or hundreds of thousands of copies as modern books—especially, bad books—do. However, the substance of what he wrote is nevertheless available to us.... Euclid's great achievement was the systematization of the works of his predecessors. The Elements preserve several of Eudoxes' proofs.

George Pólya
Everyone knows that mathematics offers an excellent opportunity to learn demonstrative reasoning, but I contend also that there is no other subject in the usual curricula of the schools that affords a comparable opportunity to learn plausible reasoning.... let us learn proving, but also let us learn guessing.

George Pólya
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.

George Pólya
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.

George Pólya
Why should the typical student be interested in those wretched triangles?... He is to be brought to see that without the knowledge of triangles there is not trigonometry; that without trigonometry we put back the clock millennia to Standard Darkness Time and antedate the Greeks.

George Pólya