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 of numbers to enable him to deduce his theorems. We, since our object is philosophical, must grapple with the philosopher’s question. The answer to the question, “What is a number?” which we shall reach in this lecture, will be found to give also, by implication, the answer to the difficulties of infinity which we considered in the previous lecture.

The question “What is a number?” is one which, until quite recent times, was never considered in the kind of way that is capable of yielding a precise answer. Philosophers were content with some vague dictum such as, “Number is unity in plurality.” A typical definition of the kind that contented philosophers is the following from Sigwart’s Logic (§ 66, section 3): “Every number is not merely a plurality, but a plurality thought as held together and closed, and to that extent as a unity.” Now there is in such definitions a very elementary blunder, of the same kind that would be committed if we said “yellow is a flower” because some flowers are yellow. Take, for example, the number 3. A single collection of three things might conceivably be described as “a plurality thought as held together and closed, and to that extent as a unity”; but a collection of three things is not the number 3. The number 3 is something which all collections of three things have in common, but is not itself a collection of three things. The definition, therefore, apart from any other defects, has failed to reach the necessary degree of abstraction: the number 3 is something more abstract than any collection of three things.

Such vague philosophic definitions, however, remained inoperative because of their very vagueness. What most men who thought about numbers really had in mind was that numbers are the result of counting. “On the consciousness of the law of counting,” says Sigwart