framed from making any mention of the special problems about continuity, or of the conceptions of the Calculus. And it has also been noted that Cantor, who has done so much to make specific the positive concept of das Eigentlich-Unendliche, and who has also given us one of the very first of the exact definitions of continuous quantity ever discovered, himself rejects the actually infinitesimal quantities as quite impossible; and does so quite as vigorously as he accepts and defends the actually infinite quantities; so that he fully agrees that the infinitesimal must remain where the Calculus leaves it, namely, simply the variable small at will.[1] It must therefore be distinctly understood that, in the discussion of the reality of the infinite quantities and multitudes, appeal need no longer be made to the conceptions of quantity peculiar to the Calculus; while, in general, the majority of those concerned in this inquiry expressly admit that the logic of the Calculus is quite independent of the present issue, and that the infinite of the Calculus is simply the variable large at will, which therefore need not be at any moment, even notably large at all.[2]
And now, finally, there is also urged against any conception of the actually Infinite the well-known consideration that the conception of such infinity involves an empty and worthless repetition of the same, over and over, — a mere “counting when there is nothing to count,” or, in the realm of explicit reflection, a vain observation that I am I, and that I am I, again, even in saying that I am I, — or an equally inane insistence
- ↑ See Cantor’s statement in the Zeitschr f. Philos., Bd. 88, p. 230; and in the same journal, Bd. 91, p. 112, in a passage there quoted from a letter addressed by Cantor to Weierstrass. I am unable to understand how Mr. Charles Peirce, in his paper in the Monist (1892, p. 537 of Vol. 2) is led to attribute to Cantor his own opinion as to the infinitesimals.
- ↑ Mr. Charles Peirce, as I understand his statements in the Monist (loc. cit.), appears to stand almost alone amongst recent mathematical logicians outside of Italy, in still regarding the Calculus as properly to be founded upon the conception of the actually infinite and infinitesimal. In Italy, Veronese has used in his Geometry the concept of the actually infinitesimal.
