To Solve a Problem

To Solve a Problem - Part II

Thanks for all the superb comments.

Indeed, it is noted that the heavyweights have been mentioned, notably Poincare, Hadamard, Archimedes, and others. The great Albert Einstein, who used thought experiments to both pose and then solve problems should be included. These were the true problem creators and solvers in history. History is dotted with great problems solvers in almost every genre, though their methods were not always known. When the "prince" of mathematicians, Johann Carl Friedrich Gauss (1777 – 1855) , was asked why it was unclear how he constructed his marvelous solutions and theories, he responded that he wished not to leave the scaffolding apparent.

Henry James (1843 – 1916), an American philosopher, anticipated some of this with his pragmatic philosophy by which the solution (i.e. recourse) that works is the solution to accept. Let me sketch a bit on problems, pre-problems, solutions, sub-solutions, pre-solutions, and problem determination. From these, a suggested checklist might evolve. Yet, forming a checklist combines both a problem and a solution in the meta-sense.

On Problems

Some problems come to us delivered, wrapped up as it were and clear, seeking a solution. Rules are unmitigated.
Some problems come to us as situations, not with a clear problem defined, but needing clarity. Here the task is problem formulation, and with that an accepted formulation.
Some problems come as questions, wherein the solver is relegated to the task of giving an answer. Many of these have no solution.
Some problems are fundamentally semantic, and devolve into issues of our highly flawed language. (Logicians work this end of the street. Demagogues uses these for political points.)

On Solutions.

There is but one solution, as the problem is clearly defined.
Some solutions offer only partial resolution and then only to those with this particular view. Often there are contradictory alternative solutions.
Some problems have only incomplete solutions, in some cases a solution to a substantial but only partial, or sub-problem. Such really difficult problem solutions are developed in this manner. A normal pattern, this is. E.g. Fermat's theorem, and the Poincare conjecture - now theorem. This engenders the idea of solution evolution.
Some problems have multiple solutions, each admirable in and of itself, but each incomplete, possible due to incomplete or very low information.
Some solutions require true innovation. Such require the inclusion of new and different problem solutions, and new ideas.
Some solutions involve the psychological, the logical, the pragmatic, the philosophical, the intuitive, the programmatic, and even the sociological aspects of the multi-faceted manner(s) in which we think. See http://used-ideas.blogspot.com/2013/02/problem-solving-your-marvelous-brain.html

I know these lists are incomplete but am hoping they illustrate the complexity. It is difficult for me to see a seminal work arising from this contradictory, and conflicting melee of possibilities.

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