The (almost really) Complete Works of Lewis Carroll

Infinitesimal or Zero?

Source: Mathematical Questions and Solutions, from the “Educational Times”, XLIV, 1885; shorter version also in Educational Times, July 1885

The controversy above is treated on the example “A random point being taken on a given line, what is the chance of its coinciding with a previously assigned point?” Only the two sections by Carroll included here, one section is by Mr. Simmons, three by other writers.

It is surely too late, in a.d. 1885, to seriously discuss the question whether a converging series does or does not reach its limit—in other words, whether an infinitesimal is or is not equal to zero. If the ordinary text-books have not shown Mr. Simmons the difference between them, how can I hope to do it? I will, however, try a reductio ad absurdum. I present Mr. Simmons with a line AB in which I have selected a certain point C; and I ask him to take a point at random in AB, and to estimate its chance of coinciding with C. He will reply, ‘If its chance of falling on one side of C be k, its chance of falling on the other side is, with perfect accuracy, 1−k. Hence its chance of missing C is absolutely 1; and its chance of coinciding with it is absolutely zero.’ But the very same thing is true of any other point I might select in AB. Hence the new point has no chance of falling anywhere! If Mr. Simmons is partial to pitfalls, let me recommend this one to his notice; it is nice soft falling, and not very deep.

1. I re-affirm that the question whether a converging series does or does not reach its limit is, in other words, whether an infinitesimal is or is not equal to zero. E. g.—The converging series 2−1, 2−2, &c., 2−n, has for its limit, unity. Also its sum is 1−2−n. Hence, if when n is infinite, the series reaches its limit, the infinitesimal 2−n must be equal to zero.

2. I never assumed that ‘a line may be considered as wholly made up of points which can all previously be assigned,’ nor of points of any kind. A point, having no magnitude, can form no portion of a line.

3. I admit that, if the length of AC, one portion of a line AB, be k, the length of the other portion CB will with perfect accuracy be 1−k. And I am ‘prepared to deny that the two chances (of a point falling in the two portions) are represented quite accurately by k and 1−k.’ For this would omit the 3 chances of its falling at A, at B, and at C. Suppose that, when the point falls at C, it is reckoned as falling in AB, and not in BC. Then, to deal fairly with the two portions, we must exclude A, and make unity represent the chance of the point falling somewhere in the line AB, excluding A, but including B. Then k is the chance of its falling between A and C, or else at C; and 1−k the chance of its falling between C and B, or else at B.

4. I re-affirm, as absolutely axiomatic, that, when an event is possible, its chance of happening is not zero.