The result is lawfully fixed beforehand by the very essence of mathematical validity, i.e. by the very expression of your own final Will in its wholeness. Your calculation can only bring this result to light in your own private experience of numbers. It is an arithmetically true result quite apart from your instantaneous observation. Its triviality, as a mere matter for computation, is not now in question. Its eternal validity, however, interests us. Every number, then, speaking in terms of mathematical validity, already has its own thousandth power, whether you chance to have observed or to have computed that thousandth power or not. Yet, in any finite collection of whole numbers, those which are the thousandth powers of the whole numbers constitute at most an incomparably minute part of the whole collection. But, on the other hand, viewed with reference to the logically valid truth about all the numbers, these powers, as a mere part of the whole series of whole numbers, still occupy such a logically predetermined place that they are set, by their values, in a one-to-one relation to the members of the whole series; so that not a small portion, but absolutely all of the whole numbers, have their correspondents among the thousandth powers. Now, all these are facts of thought, just as valid as any conceptual constructions, however simple, and just as true as that 2 + 2 = 4. And by themselves these truths, trivial if you please, are, in all their wearisomeness, not "monstrous" at all, but simply the necessary consequences of an exact conception of the nature of number.
“Monstrous, however,” so one may reply, “would be the assertion that in any real world there could be determinate facts corresponding to all this merely ideal complexity.” On the contrary, as we might at once retort, it would be monstrous if all these truths were merely “valid,” in a purely formal way, without any correspondent facts whatever in the real world. Can mere validity hang in the void? Must it not possess a determinate basis?
The issue, then, is at once the issue about the Third Conception of Being in our list. Either the truth, the world of mere forms, can indeed hang in the void, valid, but nowhere
