24 quotes
Analogy pervades all our thinking, our everyday speech and our trivial conclusions as well as artistic ways of expression and the highest scientific achievements.
How to Solve It (1945)
The best of ideas is hurt by uncritical acceptance and thrives on critical examination.
How to Solve It (1945)
As for the human race, so for the human child.
Mathematical Methods in Science (1977)
We need heuristic reasoning when we construct a strict proof as we need scaffolding when we erect a building.
How to Solve It (1945)
People tell you that wishful thinking is bad. Do not believe it, this is just one of those generally accepted errors.
In those days editions did not run to thousands or hundreds of thousands of copies as modern books—especially, bad books—do.
Mathematical Methods in Science (1977)
Good approximations often lead to better ones.
Mathematical Methods in Science (1977)
Simplicity is worth buying if we do not have to pay too great a loss of precision for it.
Mathematical Methods in Science (1977)
If you cannot solve the proposed problem, try to solve first a simpler related problem.
Mathematical Methods in Science (1977)
[M]athematicians and physicists think alike; they are led, and sometimes misled, by the same patterns of plausible reasoning.
Even if without the Scot's proverbial thrift, the difficulty of solving differential equations is an incentive to using them parsimoniously.
Mathematical Methods in Science (1977)
Euclid's manner of exposition, progressing relentlessly from the data to the unknown and from the hypothesis to the conclusion, is perfect for checking the argument in detail but far from being perfect for making understandable the main line of the argument.
How to Solve It (1945)
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.
Mathematical Methods in Science (1977)
Mathematics succeeds in dealing with tangible reality by being conceptual. We cannot cope with the full physical complexity; we must idealize.
Mathematical Methods in Science (1977)
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.
Induction and Analogy in Mathematics (1954)
The general or amateur student should also get a taste of demonstrative reasoning... he should acquire a standard with which he can compare alleged evidence of all sorts aimed at him in modern life.
Induction and Analogy in Mathematics (1954)
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.
In plausible reasoning the principal thing is to distinguish... a more reasonable guess from a less reasonable guess.
Induction and Analogy in Mathematics (1954)
An important step in solving a problem is to choose the notation. It should be done carefully. The time we spend now on choosing the notation carefully may be repaid by the time we save later by avoiding hesitation and confusion.
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.
Mathematical Methods in Science (1977)