"Difficulties in identifying problems have delayed statistics far more than difficulties in solving problems." (John W Tukey, Unsolved Problems of Experimental Statistics, 1954)
"We call a problem well-defined if there is a test which can be applied to a proposed solution. In case the proposed solution is a solution, the test must confirm this in a finite number of steps." (John McCarthy, "The Inversion of Functions Denned by Turing Machines", 1956)
"A problem that is located and identified is already half solved!" (Bror R Carlson, "Managing for Profit", 1961)
"No mathematical idea has ever been published in the way it was discovered. Techniques have been developed and are used, if a problem has been solved, to turn the solution procedure upside down, or if it is a larger complex of statements and theories, to turn definitions into propositions, and propositions into definitions, the hot invention into icy beauty. This then if it has affected teaching matter, is the didactical inversion, which as it happens may be anti-didactical." (Hans Freudenthal, "The Concept and the Role of the Model in Mathematics and Natural and Social Sciences", 1961)
"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)
"Definitions, like questions and metaphors, are instruments for thinking. Their authority rests entirely on their usefulness, not their correctness. We use definitions in order to delineate problems we wish to investigate, or to further interests we wish to promote. In other words, we invent definitions and discard them as suits our purposes." (Neil Postman, "Language Education in a Knowledge Context", 1980)
"Define the problem before you pursue a solution." (John Williams, Inc. Magazine's Guide to Small Business Success, 1987)
"There are many things you can do with problems besides solving them. First you must define them, pose them. But then of course you can also refine them, depose them, or expose them or 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." (David Hawkins, "The Spirit of Play", Los Alamos Science, 1987)
"Most people would rush ahead and implement a solution before they know what the problem is." (Q T Wiles, Inc. Magazine, 1988)
"An internal model corresponds to a specific concrete situation in the external world and allows us to reason about the external situation. To do so you used information about the problem presented in the problem statement. The process of understanding, then, refers to constructing an initial mental representation of what the problem is, based on the information in the problem statement about the goal, the initial state, what you are not allowed to do, and what operator to apply, as well as your own personal past experience." (S Ian Robertson, "Problem Solving", 2001)
"Problem solving starts off from an initial given situation or statement of a problem (known as the initial state of the problem). Based on the problem situation and your prior knowledge you have to work towards a solution. When you reach it you are in the goal state of the problem. On the way from the initial state to the goal state you pass through a number of intermediate problem states."
"The way a problem is defined determines how we attempt to solve it. […] If the definition is wrong, you will develop the right solution to the wrong problem." (James P Lewis, "Project Planning, Scheduling, and Control" 3rd Ed., 2001)
"Understanding a problem means building some kind of representation of the problem in one's mind, based on what the situation is or what the problem statement says and on one's prior knowledge. It is then possible to reason about the problem within this mental representation. Generating a useful mental representation is therefore the most important single factor for successful problem solving." (S Ian Robertson, "Problem Solving", 2001)
"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 fledgling problem solver invariably rushes in with
solutions before taking time to define the problem being solved. Even
experienced solvers, when subjected to social pressure, yield to this demand
for haste. When they do, many solutions are found, but not necessarily to the
problem at hand."
"Framing the right problem is equally or even more important than solving it." (Pearl Zhu, "Change, Creativity and Problem-Solving", 2017)
"A problem well-defined is a problem half solved." (John Dewey)
"The greatest challenge to any thinker is stating the problem in a way that will allow a solution." (Bertrand Russell)
"The mere formulation of a problem is often far more essential than its solution. To raise new questions, new possibilities, to regard old problems from a new angle, requires creative imagination and marks real advances in science." (Albert Einstein)
"To ask the right question is harder than to answer it." (Georg Cantor)
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