Hume and Berkeley on the Proofs of Infinite Divisibility

Abstract

idea, but instead a sign that can be successfully used to refer to indefinitely many particulars. Berkeley puts it just this way in Query #6 in the Analyst: the diagrams in geometrical demonstrations are not to be considered as signs, of all possible finite figures, of all sensible and imaginable extensions or magnitudes of the same kind? The expected answer to this question is yes. But those who are under the influence of the theory of abstract ideas are liable to misunderstand the manner in which diagrams are employed.44 Indeed, the tendency toward misunderstanding seems entirely natural, for: a line in the scheme, but an inch long, must be spoken of, as though it contained ten thousand parts, since it is regarded not in it self, but as it is universal; and it is universal only in its signification, whereby it represents innumerable lines greater than it self, in which may be distinguished ten thousand parts or more, though there may not be above an inch in it. Then the key passage: After this manner the properties of the lines signified are (by a very usual figure) transferred to the sign, and thence through mistake thought to appertain to it considered in its own nature.45 Berkeley's point here is remarkably subtle. He is not simply saying, as Wittgenstein would,46 that a misunderstanding of the 44See Query 7 in the Analyst. 45P, #126. Query 17 in the Analyst makes the same point this way: Whether the considering geometrical diagrams absolutely or in themselves, rather than as representatives of all assignable magnitudes or figures of the same kind, be not a principal cause of the supposing finite extension infinitely divisible; and of all the difficulties and absurdities consequent thereupon? 46The remarkable similarity between Berkeley's and Wittgenstein's treatment of mathematics is obvious. ## 1 19122 of the Principles, which concerns arithmetic, is similarly very close to Wittgenstein.