Problem Solving & Research: Advice from Descartes

Research in mathematical sciences and computer science is in many ways a matter of problem solving. One tries to use whatever tools one has to get one’s way round a problem. I was recently impressed by someone from the seventeenth century who seemed to have exceptional insight into heuristics for problem solving and research in general. The following are some rules (for the direction of the mind) by none other than Descartes:

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In order to distinguish what is most simple from what is complex, and to deal with things in an orderly way, what we must do, whenever we have a series in which we have directly deduced a number of truths one from another, is to observe which one is most simple, and how far all the others are removed from this-whether more, or less, or equally. RULE VI

If in the series of subjects to be examined we come to a subject of which our intellect cannot gain a g6od enough intuition, we must stop there; and we must not examine the other matters that follow, but must refrain from futile toil. RULE VIII

We ought to turn our entire attention upon the smallest and easiest points, and dwell on them a long time, until we get accustomed to behold the truth by distinct and clear intuition. RULE IX

To gain sagacity, our mind must be trained on the very problems that other men have already solved, and it must methodically examine even the most trivial of human devices, but especially those which manifest or imply an orderly arrangement. RULE X

If, after gaining intuitive knowledge of several simple propositions, we are to draw some further inference from them, it is useful for us to run through them in a continuous and uninterrupted movement of thought, to reflect on their interrelations and to form, so far as we can, distinct conceptions of several at once. For this adds much to the certainty of our knowledge, and it greatly increases the scope of our mind. RULE XI

If we are to understand a problem perfectly, we must free it from any superfluous conceptions, reduce it to the simplest terms, and by a process of enumeration, split it up into the smallest possible parts. RULE XIII

The same rule must be applied to the real extension of bodies, and it must be set before the imagination by means of plain diagrams. For in this way it will be far more distinctly perceived by the understanding. RULE XIV

The same rule must be applied to the real extension of bodies ,and it must be set before the imagination by means of plain diagrams. For in this way it will be far more distinctly perceived by the understanding. RULE XIV

When we are dealing with a problem we must run over it in a direct course; in so doing, we must abstract from the fact that some of its terms are known, others unknown; and by valid processes, step by step, we must apprehend the interdependence of the terms. RULE XVII

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It is a pity that Descartes, who set out to write 36 rules, only finished a total of 21 rules. One thing that we can draw from the heuristics is that there is a lot in common between analytic philosophy and computer science.