Wednesday, August 16, 2017
Cut loose your thinking and refine it; examine the problem piece by piece, correctly sorting and investigating ... and if you hit a dead end with one of your ideas, toss it aside and abandon it, then later try putting it in play again with your mind and weigh it up.W.J. Verdenius, "Notes on Aristophanes' Clouds," Mnemosyne 6.3 (1953) 178-180 (at 179):
σχάσας τὴν φροντίδα 740
λεπτὴν κατὰ μικρὸν περιφρόνει τὰ πράγματα
ὀρθῶς διαιρῶν καὶ σκοπῶν ...
... κἂν ἀπορῇς τι τῶν νοημάτων,
ἀφεὶς ἄπελθε, κᾆτα τῇ γνώμῃ πάλιν
κίνησον αὖθις αὐτὸ καὶ ζυγώθρισον. 745
744 τῇ γνώμῃ Reiske: τὴν γνώμην codd.
740-1 σχάσας τὴν φροντίδα / λεπτὴν κατὰ μικρὸν περιφρόνει τὰ πράγματα: κατὰ μικρόν should not be connected with περιφρόνει, but with σχάσας. Socrates adds κατὰ μικρόν as an explanation of λεπτήν (which could be misunderstood as an apposition): "into small pieces". Cp. Xen. An. VII 3, 22 ἄρτους διέκλα κατὰ μικρόν.Liddell-Scott-Jones, s.v. ζυγωθρίζω:
weigh, examine, Ar. Nu. 745, acc. to Sch.: but acc. to Poll. 10.26 from ζύγωθρον (the bar of a door), lock up.On Socrates' suggestion to cut the problem into small pieces, cf. G. Polya, How to Solve It: A New Aspect of Mathematical Method, 2nd ed. (1957; rpt. Princeton: Princeton University Press, 1973), pp. 35-36:
If your problem is very complex you may distinguish "great" steps and "small" steps, each great step being composed of several small ones. Check first the great steps, and get down to the smaller ones afterwards....Consider the details of the solution and try to make them as simple as you can; survey more extensive parts of the solution and try to make them shorter; try to see the whole solution at a glance. Try to modify to their advantage smaller or larger parts of the solution, try to improve the whole solution, to make it intuitive, to fit it into your formerly acquired knowledge as naturally as possible. Scrutinize the method that led you to the solution, try to see its point, and try to make use of it for other problems. Scrutinize the result and try to make use of it for other problems.Cf. also id., pp. 75-85, on "Decomposing and Recombining."
On Socrates' suggestion to lay aside the problem until a later time, cf. Polya, op. cit., pp. 197-198:
Subconscious work. One evening I wished to discuss with a friend a certain author but I could not remember the author's name. I was annoyed, because I remembered fairly well one of his stories. I remembered also some story about the author himself which I wanted to tell; I remembered, in fact, everything except the name. Repeatedly, I tried to recollect that name but all in vain. The next morning, as soon as I thought of the annoyance of the evening before, the name occurred to me without any effort.
The reader, very likely, remembers some similar experience of his own. And, if he is a passionate problem-solver, he has probably had some similar experience with problems. It often happens that you have no success at all with a problem; you work very hard yet without finding anything. But when you come back to the problem after a night's rest, or a few days' interruption, a bright idea appears and you solve the problem easily. The nature of the problem matters little; a forgotten word, a difficult word from a crossword-puzzle, the beginning of an annoying letter, or the solution of a mathematical problem may occur in this way.
Such happenings give the impression of subconscious work. The fact is that a problem, after prolonged absence, may return into consciousness essentially clarified, much nearer to its solution than it was when it dropped out of consciousness. Who clarified it, who brought it nearer to the solution? Obviously, oneself, working at it subconsciously. It is difficult to give any other answer; although psychologists have discovered the beginnings of another answer which may turn out some day to be more satisfactory.
Whatever may or may not be the merits of the theory of subconscious work, it is certain that there is a limit beyond which we should not force the conscious reflection. There are certain moments in which it is better to leave the problem alone for a while. "Take counsel of your pillow" is an old piece of advice. Allowing an interval of rest to the problem and to ourselves, we may obtain more tomorrow with less effort. "If today will not, tomorrow may" is another old saying. But it is desirable not to set aside a problem to which we wish to come back later without the impression of some achievement; at least some littIe point should be settled, some aspect of the question somewhat elucidated when we quit working.
Only such problems come back improved whose solution we passionately desire, or for which we have worked with great tension; conscious effort and tension seem to be necessary to set the subconscious work going. At any rate, it would be too easy if it were not so; we could solve difficult problems just by sleeping and waiting for a bright idea.