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For while it is no doubt necessary in the highest degree, this quality springs directly out of what we may call an intuition of the equivalence of relations. We do not say that the exact quantitative equivalents of a third exact quantity must be equal to each other; we say that they are equal to each other, and this because the inference is one that rests directly on the equivalence of intuitions. The question whether any other species of certitude arises in the field of mathematical judgments is one the answer to which depends on the prior question as to whether the direct and mediate intuition of equivalence exhausts the possibilities of mathematical calculation. This we do not believe, for is there not a whole field of genuine mathemat- ical calculation in which results are reached by taking the terms of one kind of quantity as symbols of approximation for reaching judgments about another quantity of a differ- ent kind? This process will enter wherever the relation is one between a finite and an infinite quantity. Between the notion of the finite and that of the infinite there is a dif- ference of quality, since the finite is always greater than any of its parts and equal to the sum of its parts, whereas any of the parts of the infinite may be as infinite as the whole. prev next |