On Problem Solving V: Approaches

"He who seeks for methods without having a definite problem in mind seeks for the most part in vain." (David Hilbert, 1902)

"The materials necessary for solving a mathematical problem are certain relevant items of our formerly acquired mathematical knowledge, as formerly solved problems, or formerly proved theorems. Thus, it is often appropriate to start the work with the question; Do you know a related problem?" (George Pólya, "How to Solve It", 1945)

"We acquire any practical skill by imitation and practice. […] Trying to solve problems, you have to observe and to imitate what other people do when solving problems and, finally, you learn to do problems by doing them." (George Pólya, "How to Solve It", 1945)

"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, 1945)

"We have to find the connection between the data and the unknown. We may represent our unsolved problem as open space between the data and the unknown, as a gap across which we have to construct a bridge. We can start constructing our bridge from either side, from the unknown or from the data. Look at the unknown! And try to think of a familiar problem having the same or a similar unknown. This suggests starting the work from the unknown. Look at the data! Could you derive something useful from the data? This suggests starting the work from the data." (George Pólya, "How to solve it", 1945) 

"We should give some consideration to the order in which we work out the details of our plan, especially if our problem is complex. We should not omit any detail, we should understand the relation of the detail before us to the whole problem, we should not lose sight of the connection of the major steps. Therefore, we should proceed in proper order." (George Pólya, "How to solve it", 1945)

"I believe, that the decisive idea which brings the solution of a problem is rather often connected with a well-turned word or sentence. The word or the sentence enlightens the situation, gives things, as you say, a physiognomy. It can precede by little the decisive idea or follow on it immediately; perhaps, it arises at the same time as the decisive idea. […]  The right word, the subtly appropriate word, helps us to recall the mathematical idea, perhaps less completely and less objectively than a diagram or a mathematical notation, but in an analogous way. […] It may contribute to fix it in the mind." (George Pólya [in a letter to Jaque Hadamard, "The Psychology of Invention in the Mathematical Field", 1949])

"The problems are solved, not by giving new information, but by arranging what we have known since long." (Ludwig Wittgenstein, "Philosophical Investigations", 1953)

"In picking that problem be sure to analyze it carefully to see that it is worth the effort. It takes just as much effort to solve a useless problem as a useful one." (Charles F Kettering, 1955)

"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)

"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)

"An important symptom of an emerging understanding is the capacity to represent a problem in a number of different ways and to approach its solution from varied vantage points; a single, rigid representation is unlikely to suffice." (Howard Gardner, "The Unschooled Mind", 1991)

"Alternative models are neither right nor wrong, just more or less useful in allowing us to operate in the world and discover more and better options for solving problems." (Andrew Weil," The Natural Mind: A Revolutionary Approach to the Drug Problem", 2004)

"Mostly we rely on stories to put our ideas into context and give them meaning. It should be no surprise, then, that the human capacity for storytelling plays an important role in the intrinsically human-centered approach to problem solving, design thinking." (Tim Brown, "Change by Design: How Design Thinking Transforms Organizations and Inspires Innovation", 2009)

"A problem thoroughly understood is always fairly simple. Found your opinions on facts, not prejudices. We know too many things that are not true." (Charles F Kettering)

"Divide each problem that you examine into as many parts as you can and as you need to solve them more easily." (Descartes OEuvres, vol. VI)

"Each problem that I solved became a rule which served afterwards to solve other problems." (Descartes, Oeuvres, vol. VI)

"When a problem arises, we should be able to see soon whether it will be profitable to examine some other problems first, and which others, and in which order." (Descartes, OEuvres, vol. X)

"I do believe in simplicity. It is astonishing as well as sad, how many trivial affairs even the wisest thinks he must attend to in a day; how singular an affair he thinks he must omit. When the mathematician would solve a difficult problem, he first frees the equation of all encumbrances, and reduces it to its simplest terms. So simplify the problem of life, distinguish the necessary and the real. Probe the earth to see where your main roots run." (Henry D Thoreau)

Previous <<||>> Next