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
46.
is misled by a
form of expression are those in which we would say:
“he wouldn't talk as he does if he were
a
ware of this difference in the grammar of
such-and-such words, or if he were aware of this other
possibil
ity 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.