The question of mathematical reasoning is more difficult. I think we may lay it down that, in mathematics, the conclusion always asserts merely the whole or part of the premisses, though usually in new language. The difficulty of mathematics consists in seeing that this is so in particular cases. In practice, the mathematician has a set of rules according to which his symbols can be manipulated, and he acquires technical skill in working according to the rules in the same sort of way as a billiard-player does. But there is a difference between mathematics and billiards: the rules of billiards are arbitrary, whereas in mathematics some at least are in some sense “true”. A man cannot be said to understand mathematics unless he has “seen” that these rules are right. Now what does this consist of? I think it is only a more complicated example of the process of understanding that “Napoleon” and “Bonaparte” refer to the same person. To explain this, however, we must revert to what was said, in the chapter on “Language”, about the understanding of form. Human beings possess the power of reacting to form. No doubt some of the higher animals also possess it, though to nothing like the same extent as men do; and all animals, except a few of the most intelligent species, appear to be nearly devoid of it. Among human beings, it differs greatly from one individual to another, and increases, as a rule, up to adolescence. I should take it as what chiefly characterises “intellect”. But let us see, first, in what the power consists. When a child is being taught to read, he learns to recognise a given letter, say H, whether it is large or small, black or white or red. However it may vary in these respects his reaction is the same: he says “H”. That is to say, the essential feature in the stimulus is its _form_. When my boy, at the age of just under three, was about to eat a three-cornered piece of bread and butter, I told him it was a triangle. (His slices were generally rectangular.) Next day, unprompted, he pointed to triangular bits in the pavement of the Albert Memorial, and called them “triangles”. Thus the form of the bread and butter, as opposed to its edibility, its softness, its colour, etc., was what had impressed him. This sort of thing constitutes the most elementary kind of reaction to form.