When we say that by our method we try to counteract the misleading effect of certain analogies, it is important that you should understand that the idea of an analogy being misleading is nothing sharply defined. No sharp boundary can be drawn round the cases in which we should say that a man was misled by an analogy. The use of expressions constructed on analogical patterns stresses analogies between cases often far apart. And by doing this these expressions may be extremely useful. It is, in most cases, impossible to show an exact point where an analogy begins to mislead us. Every particular notation stresses some particular point of view. If, e.g., we call our investigations “philosophy”, this title, on the one hand, seems appropriate, on the other hand it certainly has misled people. (One might say that the subject we are dealing with is one of the heirs of the subject which we used to call “philosophy”.) The cases in which particularly, we wish to say that someone
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is misled by a form of expression are those in which we would say: “he wouldn't talk as he does if he were aware of this difference in the grammar of such-and-such words, or if he were aware of this other possibility of expression” and so on. Thus we may say of some philosophizing mathematicians that they are obviously not aware of the difference between the many different usages of the word “proof”; and that they are not clear about the difference between the uses of the word “kind”, when they talk of kinds of numbers, kinds of proofs, as thought the word “kind” here meant the same thing as in the context, “kinds of apples”. Or, we may say, they are not aware of the different meanings of the word “discovery”, when in one case we talk of the discovery of the construction of the pentagon and in the other case of the discovery of the South Pole.