# Frases de George Pólya

## George Pólya

**Data de nascimento:** 13. Dezembro 1887**Data de falecimento:** 7. Setembro 1985

George Pólya ; Budapeste, 13 de dezembro de 1887 — Palo Alto, 7 de setembro de 1985 foi um matemático húngaro e professor de matemática de 1914 a 1940 no ETH Zürich na Suíça, e de 1940 a 1953 na Stanford University. Pólya permaneceu como professor emérito de Stanford o resto de sua vida e carreira. Trabalhou com uma variedade de tópicos matemáticos, incluindo séries, teoria dos números, análise matemática, geometria, álgebra, combinatória e probabilidade. Também é notável sua contribuição para a heurística em educação matemática.

### Citações George Pólya

### „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.“

Mathematical Methods in Science (1977)

### „Mathematics succeeds in dealing with tangible reality by being conceptual. We cannot cope with the full physical complexity; we must idealize.“

Mathematical Methods in Science (1977)

Contexto: Mathematics succeeds in dealing with tangible reality by being conceptual. We cannot cope with the full physical complexity; we must idealize.<!--p.161

### „If you cannot solve the proposed problem, try to solve first a simpler related problem.“

Mathematical Methods in Science (1977)

Contexto: If you cannot solve the proposed problem, try to solve first a simpler related problem.<!--p.164

### „In those days editions did not run to thousands or hundreds of thousands of copies as modern books—especially, bad books—do.“

Mathematical Methods in Science (1977)

Contexto: 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.

### „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“

Mathematical Methods in Science (1977)

Contexto: 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]....

m\frac {dv}{dt} = mg - Kv

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 m, rate of change of velocity \frac {dv}{dt}, gravitational force mg, and velocity v. 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 mg due to gravity; the flow of the current is caused by the electromotive force or tension E 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 v; electrical resistance is proportional to the current i. And consequently rate of change of velocity \frac {dv}{dt} corresponds to rate of change of current \frac {di}{dt}.... The electromagnetic induction L opposes the change of current... And doesn't the inertia or mass m..? Isn't L, so to speak, an electromagnetic inertia?

L\frac {di}{dt} = E - Ki

### „Anything new that we learn about the world involves plausible reasoning“

Induction and Analogy in Mathematics (1954)

Contexto: 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.

### „The efficient use of plausible reasoning is a practical skill and it is learned… by imitation and practice.“

Induction and Analogy in Mathematics (1954)

Contexto: 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… present also examples of historic interest, examples of real mathematical beauty“

Induction and Analogy in Mathematics (1954)

Contexto: 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.

### „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.“

Mathematical Methods in Science (1977)

Contexto: 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.“

Mathematical Methods in Science (1977)

Contexto: 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

### „Even if without the Scott's proverbial thrift, the difficulty of solving differential equations is an incentive to using them parsimoniously.“

Mathematical Methods in Science (1977)

Contexto: 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.<!--p.224

### „Simplicity is worth buying if we do not have to pay too great a loss of precision for it.“

Mathematical Methods in Science (1977)

Contexto: Simplicity is worth buying if we do not have to pay too great a loss of precision for it.<!--p.215

### „As for the human race, so for the human child.“

Introduction

Mathematical Methods in Science (1977)

Contexto: 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.

### „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, livro How to Solve It

Fonte: How to Solve It (1945), p. 219

### „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.“

Induction and Analogy in Mathematics (1954)

### „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.“

Induction and Analogy in Mathematics (1954)

### „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….“

— George Pólya, livro How to Solve It

Fonte: How to Solve It (1945), p. 207

### „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.“

— George Pólya, livro How to Solve It

Fonte: How to Solve It (1945), p. 70