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 ity a little more rather than a little less. The neat little circles the logician uses to convey his idea of pink or not pink to the student are just pictures of boundaries in his mind, exaggerations of a natural mental tendency. They are required for the purposes of his science, but they are departures from the nature of fact.

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 geometric magic, numerical and