from without, to refer to them, so that if such knowledge vanished from the universe, or if no external knowledge of them had ever come to be, the real beings would remain just what they are. On the other hand, however, the realities of the present type exist explicitly as Objects of Possible Knowledge. Their whole defined Being is exhausted by their validity when regarded from the point of view of such possible knowledge. If nobody had ever recognized the British Constitution, or the prices, credits, debts, marks, and ranks aforesaid, these objects could not be said to be able to retain any being, although now that they are recognized, such objects appear to have a genuine being, and to be relatively independent of this or that individual judgment.
The case of the eternal truths, such as the ethical, or still more obviously the mathematical truths, is more like the case of the atoms or monads of a thoroughgoing realism, since the eternal verities are said to have been valid before any human mathematician or moralist conceived them, and to remain true even if men forget them, or, as in case of the value of π, are physically unable to verify them in concrete circles. Yet, on the other hand, their case has its own peculiar puzzle, in that, when the mathematician himself first conceives of his equations and of his functions, he seems, as we have said, to be engaged in an act of perfectly free construction, as if he were building in fairyland. Yet the familiar miracle of this mathematical realm is that, after one has built, he discovers that the form of his edifice is somehow eternal, and that there are existences which this form has preëstablished, so that he himself looks with wonder to find whether this
