And similarly outrageous seem to any calm considering the fictions of mathematics. This fact, indeed, was recently pointed out to me by my small son, in whom his governess was endeavoring to implant the conviction that two and two make four. But the child stayed sceptical. He was reservedly polite about a rational "Suppose you had two apples, and I gave you two more apples, how many apples would you have then?" He conceded with readiness, not unflavored with resignation to the obtuseness of grown-up persons, that in such circumstances he would have four apples, but could not eat that many without being real sick. Yet that two and two, in consequence, make four, he excluded as a logical inference: and he depreciated that inference by stating it did not mean anything. He was, of course, quite right. For that "two and two make four" becomes, the very instant that you play this familiar axiom the childish trick of thinking about it, at best an unprovable hypothesis. That two apples and two more apples compose four apples is, as my son admitted, plain enough. Or, you may change your unit to a penny, a match, a pencil, or to a bungalow, and still produce convincing evidence to prove your arithmetic. But the mathematician requests us to consider an abstract "two," to believe in two apples with the pomaceousness removed: his incorporal and incorporeal "two" has never existed and never can exist. His "two" is not merely a fiction, but an inconceivable fiction which the human mind can no more, really, imagine than it can his "four." You need only for one moment attempt to form some rational and clear-cut idea of this "two" to perceive that the governess in fact was (with all respect to her) talking about incredible fictions, just as my son affirmed.... And when the mathematician goes on from "two" and "four" into the higher branches of his romance weaving, and postulates as yet other realities his "lines" that have length but no breadth or thickness, or his "points" that have not even length, you face the choice between fleeing from his self-evident lunacy and accepting his insane but very useful fictions. § 13