The whole argument against the abuse of statistics is summed up in the story of the man who explained what an average was. "If you were struck dead by lightning at my side and I remained safe and sound, we should both be on the average half dead."
But there is a deeper final cause for the suspicion of numbers on which one must touch very carefully, because though it has led to the wildest extravagance in philosophy, and has weakened the use of the human reason in modern times, it none the less has a basis of truth.
Numbers, absolute though they be, and the expression of the human mind's divine capacity for measure, escape us in their ultimate use. The science of numbers, when you have pursued it far enough, lands you where every human analysis lands you—in a contradiction or a mystery. On this account it is that men have been tempted of late to a monstrous denial of mathematical truth, and that as a part of their general revolt against reason. But it is true that you approach regions beyond which what had hitherto been the necessary laws of number cease to stand. You have it wherever there is a sudden passage from an indefinitely increasing number to an indefinitely small one; a sudden passage through what we call "the infinite" to "Zero." And you get it in the "limit" of any mathematical process.
For instance, as you stretch out an ellipse it ought to get more and more like a parabola, at any rate at the business end, if I may so express myself; and it ought gradually at the end of the process to merge into a parabola. Then, as you go on tilting your plane, you ought logically to be able to follow the process of the parabola turning into an hyperbola. Well, the thing does not happen that way. The ellipse remains an ellipse though its foci get farther and farther apart. It remains an ellipse right up to the critical moment, when the distant focus vanishes into infinity, and, then, suddenly, the curve becomes a parabola, and then in a flash, it turns inside out, splits into two, and is an hyperbola. And no one will ever be able to tell you what happened out in the infinities, where the transmogrification took place.
There is a lot more.... Every operation of subtraction, or of the addition of opposite signs, is a mystery; for the conception of the negative in mathematics is a mystery. As for imaginaries, and the dear old root of minus one, you may call it a convention if you like, but it cannot be altogether a convention since it is a necessary basis for arriving at a truth; nor is it a lie, though it certainly dresses up like one. Its name is iota and I loved it long ago. One might quote of this formidable being what I think St. Augustine said of some Bible-story:
"Non est mendacium sed mysterium."
You may tell me that by far the greater part of men are not concerned with these ultimate far goals of bewilderment and of despair to which the science of numbers will lead them out of their happy homes. I am not so sure. Nowadays when people are so fond of the word "sub-conscious" (and I have myself worked the animal as hard as was possible on meagre rations) we may drag it in here without extravagance. There is, I think, an instinct in men the least acquainted with mathematics that it is possible to push them too far; and though men begin to tremble long before the limits of human reason are reached—in fact, as a rule, they get the wind up before they reach the Calculi—yet a god tells them in what direction they are going, and warns them that too much curiosity is ill for man.