external reality. First, they must be simple, that is, their relations to one another must be easily handled; second, their relations must correspond more or less closely with the relations of some set of external things. They do not correspond absolutely. There are no external things which have the properties of mathematical straight lines except approximately. Take the annexed figure.
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One would not hesitate to call AB, BC, CB, straight lines, and to say that the triangle ABC has the sum of its angles equal to 180°. The error he would make (they are drawn with compasses) we always make in kind, though not in degree, in applying mathematics to realities. I wish to make clear that the relation between mathematical truths and external facts, is one of resemblance, not identity. What the essence of resemblance is I shall not discuss.
No external facts can do more than change the utility of the two geometries. At present, for simplicity and accuracy of resemblance to external facts, the Euclidean geometry need not fear being swallowed up. If, however, facts should be discovered which could be most simply correlated to transcendental truths, transcendental geometry might become important.
Let us recapitulate. We have tried to show that mathematics deals only with concepts, and that the two geometries are, therefore, also conceptual. Their apparent discrepancy we tried to account for by showing that they used different concepts. We showed that, although concepts might be originated by sensations, they were not, nor were affected by, external facts. The relation between mathematical truths and external facts is one of more or less resemblance, not of identity. Nor can the resemblance be ever proved to be perfect.
The Euclidean geometry has as great facility in accommodating itself to all known facts as the transcendental, and greater simplicity. It is therefore of greater practical utility. The mathematical truth of each is not affected by experience.
Thus transcendental geometry, with its egg-shells turned inside-out without cracking, its knots mysteriously untied, its worlds where the background of everything is a man's own head, is from its conceptual basis, as a creation of man's mind, true. It is a pretty mathematical diversion; it is, as yet, nothing more.
