George Pólya quotes - page 2

Mathematics consists in proving the most obvious thing in the least obvious way. (George Pólya)
Anything new that we learn about the world involves plausible reasoning. View quote
George Pólya
The efficient use of plausible reasoning is a practical skill and it is learned... by imitation and practice. View quote
George Pólya
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. View quote
George Pólya
In those days editions did not run to thousands or hundreds of thousands of copies as modern books-especially, bad books-do. View quote
George Pólya
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. View quote
George Pólya
As for the human race, so for the human child. View quote
George Pólya
I... present also examples of historic interest, examples of real mathematical beauty. View quote
George Pólya
The differential equation of the first order \frac {dy}{dx} = f(x, y) ... prescribes the slope \frac {dy}{dx} 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." View quote
George Pólya
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. View quote
George Pólya
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. (George Pólya)
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. View quote
George Pólya