"A good teacher should understand and impress on his students the view that no problem whatever is completely exhausted. There remains always something to do; with sufficient study and penetration, we could improve any solution, and, in any case, we can always improve our understanding of the solution."
"We can scarcely imagine a problem absolutely new, unlike and unrelated to any formerly solved problem; but if such a problem could exist, it would be insoluble. In fact, when solving a problem, we should always profit from previously solved problems, using their result or their method, or the experience acquired in solving them." (George Polya, "How to Solve It", 1945)
"The answer to the question ‘Can there be a general method for solving all mathematical problems?’ is no! Perhaps, in a world of unsolved and apparently unsolvable problems, we would have thought that the desirable answer to this question from any point of view, would be yes. But from the point of view of mathematicians a yes would have been far less satisfying than a no is. […] Not only are the problems of mathematics infinite and hence inexhaustible, but mathematics itself is inexhaustible." (Constance Reid, "Introduction to Higher Mathematics for the General Reader", 1959)
"Some problems are just too complicated for rational logical solutions. They admit of insights, not answers." (Jerome B Wiesner, The New Yorker, 1963)
"A problem will be difficult if there are no procedures for generating possible solutions that are guaranteed (or at least likely) to generate the actual solution rather early in the game. But for such a procedure to exist, there must be some kind of structural relation, at least approximate, between the possible solutions as named by the solution-generating process and these same solutions as named in the language of the problem statement." (Herbert A Simon, "The Logic of Heuristic Decision Making", [in "The Logic of Decision and Action"], 1966)
"Deep in the human nature there is an almost irresistible tendency to concentrate physical and mental energy on attempts at solving problems that seem to be unsolvable." (Ragnar Frisch, "From Utopian Theory to Practical Applications", [Nobel lecture] 1970)
"In general, complexity and precision bear an inverse relation to one another in the sense that, as the complexity of a problem increases, the possibility of analysing it in precise terms diminishes. Thus 'fuzzy thinking' may not be deplorable, after all, if it makes possible the solution of problems which are much too complex for precise analysis." (Lotfi A Zadeh, "Fuzzy languages and their relation to human intelligence", 1972)
"A great many problems are easier to solve rigorously if you know in advance what the answer is." (Ian Stewart, "From Here to Infinity", 1987)
"A common mistake in problem solving is to encompass too much territory, which dilutes any solutions chance of success. [...] However, the opposite error occurs more frequently." (Terry Richey, "The Marketer's Visual Tool Kit", 1994)
"[…] the meaning of the word 'solve' has undergone a series of major changes. First that word meant 'find a formula'. Then its meaning changed to 'find approximate numbers'. Finally, it has in effect become 'tell me what the solutions look like'. In place of quantitative answers, we seek qualitative ones."
"Theorems are fun especially when you are the prover, but then the pleasure fades. What keeps us going are the unsolved problems." (Carl Pomerance, MAA, 2000)
"Heuristics are rules of thumb that help constrain the problem in certain ways (in other words they help you to avoid falling back on blind trial and error), but they don't guarantee that you will find a solution. Heuristics are often contrasted with algorithms that will guarantee that you find a solution - it may take forever, but if the problem is algorithmic you will get there. However, heuristics are also algorithms." (S Ian Robertson, "Problem Solving", 2001)
"Solving a problem for which you know there’s an answer is like climbing a mountain with a guide, along a trail someone else has laid. In mathematics, the truth is somewhere out there in a place no one knows, beyond all the beaten paths. And it’s not always at the top of the mountain. It might be in a crack on the smoothest cliff or somewhere deep in the valley." (Yōko Ogawa, "The Housekeeper and the Professor", 2003)
"Knowing a solution is at hand is a huge advantage; it’s like not having a 'none of the above' option. Anyone with reasonable competence and adequate resources can solve a puzzle when it is presented as something to be solved. We can skip the subtle evaluations and move directly to plugging in possible solutions until we hit upon a promising one. Uncertainty is far more challenging."
"Mathematical good taste, then, consists of using intelligently the concepts and results available in the ambient mathematical culture for the solution of new problems. And the culture evolves because its key concepts and results change, slowly or brutally, to be replaced by new mathematical beacons." (David Ruelle, "The Mathematician's Brain", 2007)
"The beauty of mathematics is that clever arguments give answers to problems for which brute force is hopeless, but there is no guarantee that a clever argument always exists!" (David Ruelle, "The Mathematician's Brain", 2007)
"Every problem has a solution; it may sometimes just need another perspective." (Rebecca Mallery et al, "NLP for Rookies", 2009)
"Don't mistake a solution method for a problem definition - especially if it’s your own solution method." (Donald C Gause & Gerald M Weinberg, "Are Your Lights On?", 2011)
"The really important thing in dealing with problems is to know that the question is never answered, but that it doesn't matter, as long as you keep asking. It's only when you fool yourself into thinking you have the final problem definition - the final, true answer - that you can be fooled into thinking you have the final solution. And if you think that, you're always wrong, because there is no such thing as a 'final solution'." (Donald C Gause & Gerald M Weinberg, "Are Your Lights On?", 2011)
"I have not seen any problem, however complicated, which, when you looked at it in the right way, did not become still more complicated." (Paul Anderson)
"It is efficient to look for beautiful solutions first and settle for ugly ones only as a last resort. [...] It is a good rule of thumb that the more beautiful the guess, the more likely it is to survive." (Timothy Gowers)
"One is always a long way from solving a problem until one actually has the answer." (Stephen Hawking)
"The best way to escape from a problem is to solve it." (Brendan Francis)
"The worst thing you can do to a problem is solve it completely." (Daniel Kleitman)
"This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by reason, for in mathematics there is no ignorabimus." (David Hilbert)
"We can not solve our problems with the same level of thinking that created them." (Albert Einstein)
"When the answer to a mathematical problem cannot be found, then the reason is frequently that we have not recognized the general idea from which the given problem only appears as a link in a chain of related problems." (David Hilbert)
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