Now “matter” and “form” can be placed, as in the Aristotelian philosophy, in a hierarchy. From a triangle we can advance to a polygon, thence to a figure, thence to a manifold of points. Then we can go on and turn “point” into a formal concept, meaning “something that has relations which resemble spatial relations in certain formal respects”. Each of these is a step away from “matter” and further into the region of “form”. At each stage the difficulty increases. The difficulty consists in having a uniform reaction (other than boredom) to a stimulus of this kind. When we “understand” a mathematical expression, that means that we can react to it in an appropriate manner, in fact, that it has “meaning” for us. This is also what we mean by “understanding” the word “cat”. But it is easier to understand the word “cat”, because the resemblances between different cats are of a sort which causes even animals to have a uniform reaction to all cats. When we come to algebra, and have to operate with _x_ and _y_, there is a natural desire to know what _x_ and _y_ really are. That, at least, was my feeling: I always thought the teacher knew what they really were, but would not tell me. To “understand” even the simplest formula in algebra, say (x + y)² = x² + 2xy + y², is to be able to react to two sets of symbols in virtue of the form which they express, and to perceive that the form is the same in both cases. This is a very elaborate business, and it is no wonder that boys and girls find algebra a bugbear. But there is no novelty _in principle_ after the first elementary perceptions of form. And perception of form consists merely in reacting alike to two stimuli which are alike in form but very different in other respects. For, when we can do that, we can say, on the appropriate occasion, “that is a triangle”; and this is enough to satisfy the examiner that we know what a triangle is, unless he is so old-fashioned as to expect us to reproduce the verbal definition, which is of course a far easier matter, in which, with patience, we might teach even a parrot to succeed. The meanings of complex mathematical symbols are always fixed by rules in relation to the meaning of simpler symbols; thus their meanings are analogous to those of sentences, not to those of single words. What was said earlier about the understanding of sentences applies, therefore, to any group of symbols which, in mathematics, will be declared to have the same meaning as another group, or part of that meaning.