Problem

Douglas, A. Vibert

It is a solemn thought that no man liveth unto himself. It is equally true that no star, no atom, no electron, no ripple of radiant energy, exists unto itself. All the problems of the physical universe are inextricably bound up with one another in the relations of space and time.

The Atlantic Monthly From Atoms to Stars (p. 165) Volume 144, August 1929

Einstein, Albert

There are so many unsolved problems in physics. There is so much that we do not know; our theories are far from adequate.

Scientific American In I. Bernard Cohen An Interview with Einstein (p. 69) Volume 193, Number 1, July 1955

Halmos, Paul R.

A teacher who is not always thinking about solving problems—ones he does not know the answer to—is psychologically simply not prepared to teach problem solving to his students.

I Want to Be a Mathematician Chapter 14 (p. 322)

Hawkins, David

There are many things you can do with problems beside solving them. First you must define them, pose them. But then of course you can also refine them, depose them, or expose them, even dissolve them! A given problem may send you looking for analogies, and some of these may lead you astray, suggesting new and different problems, related or not to the original. Ends and means can get reversed. You had a goal, but the means you found didn't lead to it, so you found a new goal they did lead to.

It's called play. Creative mathematicians play a lot; around any problem really interesting they develop a whole cluster of analogies, of playthings.

In Necia Grant Cooper (ed.) From Cardinals to Chaos The Spirit of Play (p. 44)

Hilbert, David

As long as a branch of science offers an abundance of problems, so long is it alive; a lack of problems foreshadows extinction of the cessation of independent development. Just as every human undertaking pursues certain objects, so also mathematical research requires its problems. It is by the solution of problems that the investigator tests the temper of his steel; he finds new methods and new outlooks, and gains a wider and freer horizon.

Bulletin of the American Mathematical Society Mathematical Problems (p. 438) Volume 8, July 1902

Kiepenheuer, Karl

For the astronomer, the inexhaustible store of problems in the world he has set out to conquer remains the real mainspring of all his arduous researches.

The Sun Conclusion (p. 158)

Unknown

What is the best you can do for this problem? Leave it alone and invent another problem.

In George Polya Mathematical Discovery Volume II

The Traditional Mathematical Professor (p. 36)

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