"Banishing fundamental facts or problems from science merely because they cannot be dealt with by means of certain prescribed principles would be like forbidding the further extension of the theory of parallels in geometry because the axiom upon which this theory rests has been shown to be unprovable. Actually, principles must be judged from the point of view of science, and not science from the point of view of principles fixed once and for all." (Ernst Zermelo, "Neuer Beweis für die Möglichkeit einer Wohlordnung", Mathematische Annalen 65, 1908)
"Our science is like a store filled with the most subtle intellectual devices for solving the most complex problems, and yet we are almost incapable of applying the elementary principles of rational thought. In every sphere, we seem to have lost the very elements of intelligence: the ideas of limit, measure, degree, proportion, relation, comparison, contingency, interdependence, interrelation of means and ends." (Simone Weil, "The Power of Words", 1937)
"A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery." (George Polya, "How to solve it", 1944)
"The great progress in every science came when, in the study of problems which were modest as compared with ultimate aims, methods were developed that could be extended further and further." (John von Neumann & Oskar Morgenstern, "Theory of Games and Economic Behavior", 1944)
"The more the rate of change increases, the more the problems that face us change and the shorter is the life of the solutions we find to them. Therefore, by the time we find solutions to many of the problems that face us, usually the most important ones, the problems have so changed that our solutions to them are no longer relevant or effective; they are stillborn." (Russell Ackoff, 1981)
"Not only the mathematical way of thinking, but also simulations assisted by mathematical methods, is quite effective in solving problems. The latter is utilized in various fields, including detection of causes of troubles, optimization of expected performances, and best possible adjustments of usage conditions. Conversely, without the aid of mathematical methods, our problem-solving effort will get stuck most probably [...]" (Shiro Hiruta, "Mathematics Contributing to Innovation of Management", [in "What Mathematics Can Do for You"] 2013)
"As long as a branch of science offers an abundance of problems, so long is it alive." (David Hilbert)
"Nothing stimulates great minds to work on enriching knowledge with such force as the posing of difficult but simultaneously interesting problems." (John Bernoulli)
"Science is bound, by the everlasting vow of honour, to face fearlessly every problem which can be fairly presented to it." (Lord Kelvin)
"Science starts from problems, and not from observations." (Karl Popper)
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