George Pólya
30 quotes
Biography
George Pólya was a Hungarian-American mathematician. He was a professor of mathematics from 1914 to 1940 at ETH Zürich and from 1940 to 1953 at Stanford University.
"Mathematics is the cheapest science. Unlike physics or chemistry, it does not require any expensive equipment. All one needs for mathematics is a pencil and paper."
"[M]athematicians and physicists think alike; they are led, and sometimes misled, by the same patterns of ."
"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."
"Analogy pervades all our thinking, our everyday speech and our trivial conclusions as well as artistic ways of expression and the highest scientific achievements."
"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."
"To write and speak correctly is certainly necessary; but it is not sufficient. A derivation correctly presented in the book or on the blackboard may be inaccessible and uninstructive, if the purpose of the successive steps is incomprehensible, if the reader or listener cannot understand how it was humanly possible to find such an argument...."
"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."
"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."
"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."
"The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing. If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference."
"In plausible reasoning the principal thing is to distinguish... a more reasonable guess from a less reasonable guess."
"The general or amateur student should also get a taste of demonstrative reasoning... he should acquire a standard with which he can compare alleged evidence of all sorts aimed at him in modern life."
"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."
"I shall often discuss mathematical discoveries... I shall try to make up a likely story how the discovery could have happened. I shall try to emphasize the motives underlying the discovery, the plausible inferences that led to it... everything that deserves imitation."
"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."
"For many of the stories told the final form resulted from a sort of informal psychological experiment. I discussed the subject with several different classes... Several passages... have been suggested by answers of my students, or... modified... by the reaction of my audience."
"In my presentation I... follow the genetic method. The essential idea... is that the order in which knowledge has been acquired by the human race will be a good teacher for its acquisition by the individual. The sciences came in a certain order; an order determined by human interest and inherent difficulty. Mathematics and astronomy were the first sciences really worth the name; later came mechanics, optics, and so on. At each stage of its development the human race has had a certain climate of opinion, a way of looking, conceptually, at the world. The next glimmer of fresh understanding had to grow out of what was already understood. The next move forward, halting shuffle, faltering step, or stride with some confidence, was developed upon how well the [human] race could then walk. As for the human race, so for the human child. But this is not to say that to teach science we must repeat the thousand and one errors of the past, each ill-directed shuffle. It is to say that the sequence in which the major strides forward were made is a good sequence in which to teach them. The genetic method is a guide to, not a substitute for, judgement."
"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.<!--p.10-->"
"Good approximations often lead to better ones."
"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."
"Mathematics succeeds in dealing with tangible reality by being conceptual. We cannot cope with the full physical complexity; we must idealize.<!--p.161-->"
"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.<!--p.164-->"
"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.<!--p.164-->"
"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.<!--p.164-->"
"If you cannot solve the proposed problem, try to solve first a simpler related problem.<!--p.164-->"