Thus we see that mathematics may be defined as the science of the relations of concepts. Its vocabulary, too, must be one of fixed connotation. That is why symbols are so useful; their connotation does not vary unconsciously.
Benjamin Peirce defines mathematics as the science that draws necessary conclusions. Mill says, "The problem is—given a function, what function is it of some other function?" It is obvious that necessary conclusions can be drawn only so far as there are relations fixed whence to draw them; the function must be given before we find its relations with other functions.
Now, I wish to insist, as strongly as I can, that any set of concepts become fit for mathematical handling as soon as their relations are unfolded, and this is what I have so far proved. If you ask, "Whence these concepts?" my answer is, "From experience." From it comes the "element of intuition" that Stallo says is an element in every geometrical axiom. Space itself is but a product of experience. If a man could only hear or taste, would he have our concept of space? I trow not.
Let us now, after this long digression, return to our transcendentalists. Euclidean geometry and non-Euclidean alike are mathematical. Verbally they come to different conclusions, but neither conclusion affects facts. The difference is here, it seems to me. Transcendental geometry is the offspring of analytic, though some have tried to treat it otherwise. The relations that it handles are at first algebraic relations that may apply to anything. Then applying the geometric nomenclature to algebraic expression, calling expressions of the first degree linear, etc., it interprets these results geometrically. Its definitions, thus, are different from those of Euclid; the ideas connoted by its vocabulary are different; its concepts are not the same. It is not wonderful, then, that it gets a broader field of relations.
We decide, then, that from their respective definitions the Euclidean and the transcendental geometry are true. And this is, perhaps, the most important point to settle, for the transcendentalists have said that, although the geometrical definitions were true, the axioms need not be. We, however, say that the axioms, or what you will, of parallelism, etc., are part of the connotation of the words defined, and are simultaneously given. Of course, some experience is necessary to make us form any concepts.
The question now to be answered is, then, Which are the best definitions? But it must be remembered that, as long as we are dealing with mathematics, we are never dealing with real things. Thus Helmholtz is wrong in saying that by adding any mechanical axioms or principles we can obtain an empirical science out of geometry, if the science thus obtained is purely mathematical.
Mathematical concepts can have two virtues in varying degrees, namely, simplicity and resemblance to, or rather correspondence with,