George Pólya Quotes

you should be grateful for all new ideas, also for the lesser ones, also for the hazy ones, also for the supplementary ideas adding some precision to a hazy one, or attempting the correction of a less fortunate one.

People tell you that wishful thinking is bad. Do not believe it, this is just one of those generally accepted errors.

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

The differential equation of the first order
<math>\frac {dy}{dx} = f(x,y)</math>
...prescribes the slope <math>\frac {dy}{dx}</math> at each point of the plane (or at each point of a certain region of the plane we call the field"). ...a differential equation of the first order... can be conceived intuitively as a problem about the steady flow of a river: Being given the direction of the flow at each point, find the streamlines. ...It leaves open the choice between the two possible directions in the line of a given slope. Thus... we should say specifically "direction of an unoriented straight line" and not merely "direction."

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.

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

<math>\frac {dy}{dx} = \frac {\omega^2x}{g}</math>...The first derivative, the result of the differentiation of <math>y</math> with respect to <math>x</math>, was written by Leibniz in the form
<math>\frac {dy}{dx}</math>...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 <math>\frac {dy}{dx}</math> was considered as the ratio of two "infinitely small quanitites", of the infinitesimals <math>dy</math> and <math>dx</math>. ...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. ...<math>\frac {dy}{dx}</math> is the limit of a ratio of <math>dy</math> to <math>dx</math>... Once we have realized this sufficiently clearly, we may, under certain circumstances, treat <math>\frac {dy}{dx}</math> so as if it were a ratio... and multiply by <math>dx</math> to achieve the separation of variables. We get
<math>{dy} = \frac {\omega^2x}{g}xdx</math>

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

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.

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.

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

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.

Good approximations often lead to better ones.

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.