(Sektion 143: Philosophische Untersuchungen von Ludwig Wittgenstein).
This is the section that begins, according to many readers, the famous rule-following discussions in the Philosophical Investigations. Wittgenstein here begins to consider what is meant by following a rule, in what does following a rule consist, and how are we to determine (and specifically the extent of a logical determination) the correct way to follow a given rule.
(The third paragraph in the original German) ... Aber merke: Es gibt keine scharfe Grenze
zweischen einem regellosen
und einem systematischen
Fehler. D.h., zwischen dem, was du einen "regellosen", und dem, was du einen "systematischen Fehler" zu nennen geneigt bist. ...
143. Now we examine the following kind of language-game: B should at the order of A write down series of signs according to a specific rule of formation.
The first of these series should be that of the natural numbers in decimal notation.--How does he understand this notation?--First of all number-series will be written down before him and he will be made to copy them. (Do not balk at the word "number-series"; it is not being used incorrectly here!). And already here there exists a normal and an abnormal reaction of the learner.--We guide him at first in writing down the series 0 to 9; but then the possibility of understanding will hang on his going on to write it down himself.--And here we can imagine for example that he does copy the figures down by himself, but not in order, rather randomly this one or that one. And then communication stops at that point.--Or again, he makes "mistakes" in following the series. The difference between this and the first case is of course the frequency.--Or: he makes a systematic mistake when he for example always copies only every second number; or he copeis the series 0, 1, 2, 3, 4, 5, . . . . . as: 1, 0, 3, 2, 5, 4, . . . . . Here we shall almost try to say that he has understood us wrong.
But notice: There are is not a sharp line between a random and a systematic mistake. That is, between what you are inclined to call "random" and what you are inclined to call "systematic".
Perhaps one can help him away from the systematic misake (as from a bad habit). Or perhaps one can accept his way of copying and try to teach him the normal, kind of as an offshot, or a variation, of his.--And here also the pupil's capacity to learn could come to an end.