Tristram Shandy[82] said that his father was sometimes a gainer by misfortune; for if the pleasure of haranguing about it was as ten, and the misfortune itself only as five, he gained “half in half,” and was well off again as if the misfortune had never happened.

Suppose that the unit (arbitrary) of pleasure is denoted by A, Tristram Shandy, by neglecting, in this ethical discussion, to introduce negative quantities (Kant’s pamphlet advocating this introduction into philosophy was made subsequently[83]), apparently made 15A to result, and this can hardly be maintained to be the half of 10A. It is possible, however, that Tristram Shandy succeeded in proving the apparently paradoxical equation

15A = 5A

by remarking that the axiom “the whole is greater than the part” does not always hold. This remark follows at once from what Mr. Russell[84] has called “The Paradox of Tristram Shandy.” This paradox is described by Mr. Russell as follows:

“Tristram Shandy, as we know, took two years writing the history of the first two days of his life, and lamented that, at this rate, material would accumulate faster than he could deal with it, so that he could never come to an end. Now I maintain that, if he had lived for ever, and not wearied of his task, then, even if his life had continued as eventfully as it began, no part of his biography would have remained unwritten.”

This paradox is strictly correlative to the well-known paradox of Zeno about Achilles and the Tortoise.[85] “The Achilles proves that two variables in a continuous series, which approach equality from the same side, cannot ever have a common limit: the Tristram Shandy proves that two variables which start from a common term, and proceed in the same direction, but diverge more and more, may yet determine the same limiting class (which, however, is not necessarily a segment, because segments were defined as having terms beyond them). The Achilles assumes that whole and part cannot be similar, and deduces a paradox; the other, starting from a platitude, deduces that whole and part may be similar. For common-sense, it must be confessed that it is a most unfortunate state of things.” And Mr. Russell considers that, in the face of proofs, it ought to commit suicide in despair.

Now, I suggest the extremely unlikely possibility that Tristram Shandy, by reflection on his own life and literary labours, was led to the correct course of accepting the paradox which resulted from this reflection and rejecting the Achilles. Thus, he concluded that an infinite whole may be similar (or, in Cantor’s terminology, “equivalent”) to a proper part of itself, and hence, by a confusion of similarity with identity (or equivalence with equality) which he shares with some subsequent philosophers,[86] that a whole may be equal to a proper part of itself. If A is an infinite class, it is not difficult to see that we can have

10A = 5A.

In this way many have avoided an opinion which rests on no better foundation than that formerly entertained by the inductive philosophers of Central Africa, that all men are black.[87]

CHAPTER XXXII