George Pólya: Quotes about mathematics

George Pólya was Hungarian mathematician. Explore interesting quotes on mathematics.
George Pólya: 70   quotes 2   likes

“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.”

George Pólya

[Jon Fripp, Michael Fripp, Deborah Fripp, Speaking of Science: Notable Quotes on Science, Engineering, and the Environment, https://books.google.com/books?id=44ihCUS1XQMC&pg=PA45, 2000, Newnes, 978-1-878707-51-2, 45]

Mathematics , Science

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

George Pólya

Mathematical Methods in Science (1977), p.161

Mathematics , Reality

“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.”

George Pólya

Mathematical Methods in Science (1977)
Context: 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

World , Mathematics , Understanding , Problem

“I… present also examples of historic interest, examples of real mathematical beauty”

George Pólya

Induction and Analogy in Mathematics (1954)
Context: 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.

Beauty , Mathematics

“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.”

George Pólya

Induction and Analogy in Mathematics (1954)

Mathematics , Learning

“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

Induction and Analogy in Mathematics (1954)

School , Mathematics , Learning

“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.”

George Pólya

Induction and Analogy in Mathematics (1954)

Mathematics , Motivation

“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 book How to Solve It

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

Mathematics

“\frac {dy}{dx} = \frac {\omega^2x}{g}…The first derivative, the result of the differentiation of y with respect to x, was written by Leibniz in the form
\frac {dy}{dx}…Leibniz's notation …is both extremely useful and dangerous. Today, as the concepts of limit and derivative are sufficiently clarified, the use of the notation… need not be dangerous. Yet, the situation was different in the 150 years between the discovery of calculus by Newton and Leibniz and the time of Cauchy. The derivative \frac {dy}{dx} was considered as the ratio of two "infinitely small quanitites", of the infinitesimals dy and dx. …it greatly facilitated the systematization of the rules of the calculus and gave intuitive meaning to its formulas. Yet this consideration was also obscure… it brought mathematics into disrepute… some of the best minds… such as… Berkeley, complained that calculus is incomprehensible. …\frac {dy}{dx} is the limit of a ratio of dy to dx… Once we have realized this sufficiently clearly, we may, under certain circumstances, treat \frac {dy}{dx} so as if it were a ratio… and multiply by dx to achieve the separation of variables. We get
{dy} = \frac {\omega^2x}{g}xdx”

George Pólya

Mathematical Methods in Science (1977)

Mathematics , Time

“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.”

George Pólya

Donald E. Knuth, comments at Pólya's 90th birthday celebration quoted by Gerald L. Alexanderson, The Random Walks of George Polya (2000)

World , Mathematics , History