THE MATHEMATICAL ATTAINMENTS OF TRISTRAM SHANDY
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]
FOOTNOTES:
[82] Cf. a letter of De Morgan in Mrs. De Morgan's _Memoir of Augustus
De Morgan_, p. 324.
[83] Kant's tract was published in 1763, while _Tristram Shandy_ was
published in 1760.
[84] _P. M._, pp. 358-9 [Cf. _M._, vol. xxii., January 1912, p.
187.--ED.]
[85] Cf. _P. M._, pp. 350, 358-9; _M._, vol. xxii., 1912, p. 157.
[86] [Cf. for example, Cosmo Guastella, _Dell' infinito_, Palermo,
1912.--ED.]
[87] Cf. Russell, _P. M._, p. 360.