George Pólya Quotes (1887-1985)

“Why should the typical student be interested in those wretched triangles? ...Without trigonometry we put back the clock millennia to Standard Darkness Time and antedate the Greeks.”

“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.”

“The examples of plausible reasoning collected in this book... may throw light upon a much agitated philosophical problem: the problem of induction. The crucial question is: Are there rules for induction? ...the question should be... treated... in closer touch with the practice of scientists. ...older writers, such as Euler and Laplace, clearly perceived... that the role of inductive evidence in mathematical investigation is similar to its role in physical research. ...so the door opens to investigating induction inductively.”

“[M]athematicians and physicists think alike; they are led, and sometimes misled, by the same patterns of plausible reasoning.”

“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.”

“We could call it "proof from n to n + 1" or still simpler "passage to the next integer." Unfortunately, the accepted technical term is "mathematical induction." This name results from a random circumstance. ...Now, in many cases... the assertion is found experimentally, and so the proof appears as a mathematical complement to induction; this explains the name.”

“We could call it "proof from n to n + 1" or still simpler "passage to the next integer." Unfortunately, the accepted technical term is "mathematical induction." This name results from a random circumstance. ...Now, in many cases... the assertion is found experimentally, and so the proof appears as a mathematical complement to induction; this explains the name.”

“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.”

“There is a traditional story about Newton: as a young student, he began the study of geometry, as was usual in his time, with the reading of the Elements of Euclid. He read the theorems, saw that they were true, and omitted the proofs. He wondered why anybody should take pains to prove things so evident. Many years later, however, he changed his opinion and praised Euclid. The story may be authentic or not...”

“A string, of which both extremities are fixed, passes through a heavy ring. Find the position of equilibrium. Let A and B denote the fixed endpoints of the string and X any position of the ring [on the string]. ...The ring must hang as low as possible. (In fact the ring is heavy; it "wants to come as close to the ground, or to the center of the earth, as possible.) Both parts of the inextensible string, AX and BX, are stretched, and so the ring, sliding along the string, describes and ellipse with foci A and B. Obviously, the position of equilibrium is at the lowest point M of the ellipse where the tangent is horizontal.”

“It was in Galileo's time that firearms were invented... What is the path of a cannonball? ...Characteristically, Galileo was engrossed with the problem; characteristically, he solved it. The outcome of his ingenuity we know today as the method of superposition.”

“Many elementary textbooks of geometry contain a few remarks about analysis, synthesis, and "assuming the problem as solved." There is little doubt that this almost ineradicable tradition goes back to Pappus, although there is hardly a current textbook whose writer would show any direct acquaintance with Pappus. The circumstance alone that it is restricted to textbooks of geometry shows a current lack of understanding...”

“There is a traditional story about Newton: as a young student, he began the study of geometry, as was usual in his time, with the reading of the Elements of Euclid. He read the theorems, saw that they were true, and omitted the proofs. He wondered why anybody should take pains to prove things so evident. Many years later, however, he changed his opinion and praised Euclid. The story may be authentic or not ...”

“The only student of mine I was ever intimidated by. He was so quick. 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.”

“Bachet de Méziriac remarked that any number (that is, positive integer) is either a square, or the sum of two, three, or four squares. He did not pretend to possess a proof. He found indications pointing to his statement in certain problems of Diophantus and verified it up to 325. In short, Bachet's statement was just a conjecture, found inductively. ...his main achievement was to put the question: How many squares are needed to represent all integers? Once this question is squarely put, there is not much difficulty in discovering the answer inductively.”

“If the flow is steady, the field velocity vectors and the system of streamlines remain unaffected by the progress of time. Looking at the vector field and its streamlines we do not notice any change. Yet if we could distinguish the different particles of fluid from each other, we could observe incessant change... We have here two aspects of a steady flow, one of unchanging persistence, the other of incessant change. ...Heraclitus was called the "Dark Philosopher"; his views of human affairs were sombre and his sayings obscure. ... "You cannot look twice at the same river; for fresh waters are ever flowing in." "We look and do not look at the same rivers; we are, and we are not." What is the intended meaning of these sentences? I do not venture to find out. Yet I think that the originator of these senteces came pretty close to formulating the concept "steady flow of a fluid."”

“Euler's step was daring. In strict logic, it was an outright fallacy... Yet it was justified by analogy, by the analogy of the most successful achievements of a rising science that he called... "Analysis of the Infinite." Other mathematicians, before Euler, passed from finite differences to infinitely small differences, from sums with a finite number of terms to sums with an infinity of terms, from finite products to infinite products. And so Euler passed from equations of a finite degree (algebraic equations) to equations of infinite degree, applying the rules made for the finite... This analogy... is beset with pitfalls. How did Euler avoid them? ...Euler's reasons are not demonstrative. Euler does not reexamine the grounds for his conjecture... only its consequences. ...He examines also the consequences of closely related analogous conjectures... Euler's reasons are, in fact, inductive.”

“Aristarchus... was a contemporary of Euclid. His fame rests on his heliocentric theory... Perhaps "theory" is too strong a word, for his proofs were weak; yet it was a great idea, an idea redeveloped centuries later by Copernicus. ...an observer on Earth sees a half-moon only when ∠EMS [the angle between the Earth and Sun, from the persective of the Moon] is a right angle. ...sometimes both Sun and Moon are visible when the phase of the Moon is half-moon. So? Measure ∠MES [the angle between the Moon and Sun, from the persective of earth]... This is what Aristarchus did. ...the third angle is determined [since the sum of the three angles of any triangle is 180º], so the shape but not the size of  ⃤ EMS [triangle formed by Earth-Moon-Sun] is known. Consequently, although the actual length of any side is not determinate, the ratio of any pair is. ...[Moon-Earth distance/Sun-Earth distance] ME/SE = cos α [where ∠MES = α].”

“Kepler also thought of the Inverse Square Law; he thought of it first. ...Kepler regarded gravitational attraction as analogous to propagation of light... Consider now the intensity of light falling on a planet P at a distance R from the Sun. Let S be the total amount if light emitted by the Sun. ...the intensity will be the same at all points distance R from the Sun. But these points constitute a spherical sheet (with center the Sun) whose radius is R and whose surface area, therefore, is 4πR2. Consequently, intensity of radiation =frac {S}{4pi}cdotfrac {1}{R^2}i.e., the intensity is inversely proportional to the square of the distance between the planet P and the Sun. ...Kepler thought carefully about the possibility, but was dubious... to his credit; he mistrusted the idea for a very good reason. ...although during a solar eclipse the Moon blocks the Sun's radiation to part of the Earth, there is no discontinuity in the Earth's motion. If gravitational attraction were radiated as light is radiated, this too would be temporarily blocked by the Moon, so that during the eclipse it would discontinue its eliptical orbit...”

“If we could be any mathematician in the history of the world (besides ourselves), who would we rather be? ...we narrowed the choice down to Euler and Pólya, and finally settled on George Pólya because of the sheer enjoyment of mathematics that he has conveyed by so many examples.”

“For mathematics education and the world of problem solving it marked a line of demarcation between two eras, problem solving before and after Polya.”