Classes in logic are not only represented by circles with a hard firm
outline, whereas in fact they have no such definite limits, but also
there is a constant disposition to think of all names as if they
represented positive classes. With words just as with numbers and
abstract forms there have been definite phases of human development.
There was with regard to number, the phase when man could barely count
at all, or counted in perfect good faith and sanity upon his fingers.
Then there was the phase when he struggled with the development of
number, when he began to elaborate all sorts of ideas about numbers,
until at last he developed complex superstitions about perfect numbers
and imperfect numbers, about threes and sevens and the like. The same
was the case with abstract forms; and even to-day we are scarcely more
than heads out of the vast subtle muddle of thinking about spheres and
ideally perfect forms and so on, that was the price of this little
necessary step to clear thinking. How large a part numerical and
geometrical magic, numerical and geometrical philosophy have played in
the history of the mind! And the whole apparatus of language and mental
communication is beset with like dangers.
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