George Pólya: Use

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

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

George Pólya

Induction and Analogy in Mathematics (1954)
Context: 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.

Learning

“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

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

George Pólya

Mathematical Methods in Science (1977)
Context: 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

“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

“"Groping" and "muddling through" is usually described as a solution by trial and error…. a series of trials, each of which attempts to correct the error committed by the preceding and, on the whole, the errors diminished as we proceed and the successive trials come closer and closer to the desired final result…. we may wish a better characterization…"successive trials" or "successive corrections" or "successive approximations."… You use successive approximations when… looking for a word in the dictionary… A mathematician may apply the term… to a highly sophisticated procedure… to treat some very advanced problem… that he cannot treat otherwise. The term even applies to science as a whole; the scientific theories which succeed each other, each claiming a better explanation… may appear as successive approximations to the truth.
Therefore, the teacher should not discourage his students from using trial and error—on the contrary, he should encourage the intelligent use of the fundamental method of successive approximations. Yet he should convincingly show that for… many… situations, straightforward algebra is more efficient than successive approximations.”

George Pólya

George Pólya, Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving (1962)

Truth , Error , Intelligence , Problem

“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.”

George Pólya

Mathematical Methods in Science (1977)

Understanding , Problem , Chance

“\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