answer, questions about the truth of ideas that are just as vain as this.
When Mr. Spencer, according to the tradition of the long series of thinkers whom he in this respect follows, speaks, in a well-known passage, of “symbolic” ideas as essentially inferior in the conscious definiteness of their truth to ideas whose relation to their objects we can directly picture, he applies a criterion to the testing of ideas which is as crude as if one should argue that a razor is not as good a tool as a hammer, because, forsooth, the test of a tool shall be its weight, or the amount of noise that you can make when you use it. Many admirable ideas are, indeed, of the type of mental pictures. That is not only obvious, but worth remembering. There is no reason why such images should not be both valid and important. Sensuous experience may show you many sorts of truth that we cannot at present otherwise express. A man who sees a photograph sees truth, if he is intelligent enough to observe it. A man who sings a tune sings truth, if he is thoughtful enough to know what he is doing. And imageless abstractions, or algebraic symbols, are, indeed, not true by reason of their mere poverty of sensuous life. But, on the other hand, algebraic symbols are, for precisely the purposes of algebra, actually superior, as representations of objects, to any pictures of these objects. And this is not because by any chance we cannot picture the objects, but because, for this end, the symbols are truer than the pictures. The constructions of mathematics are oftener like razors, ideal tools that are all the better for their lack of bulk and grossness, and for the almost invisible fineness of their
