The World as Will and Idea, Vol. I — Arthur Schopenhauer, trans. Haldane & Kemp

That which is most certain, and yet always inexplicable, is what is involved in the principle of sufficient reason, for this principle, in its different aspects, expresses the universal form of all our ideas and knowledge. All explanation consists of reduction to it, exemplification in the particular case of the connection of ideas expressed generally through it. It is thus the principle of all explanation, and therefore it is neither susceptible of an explanation itself, nor does it stand in need of it; for

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every explanation presupposes it, and only obtains meaning through it. Now, none of its forms are superior to the rest; it is equally certain and incapable of demonstration as the principle of the ground of being, or of change, or of action, or of knowing. The relation of reason and consequent is a necessity in all its forms, and indeed it is, in general, the source of the concept of necessity, for necessity has no other meaning.

If the reason is given there is no other necessity than that of the consequent, and there is no reason that does not involve the necessity of the consequent.

Just as surely then as the consequent expressed in the conclusion follows from the ground of knowledge given in the premises, does the ground of being in space determine its consequent in space: if I know through perception the relation of these two, this certainty is just as great as any logical certainty.

But every geometrical proposition is just as good an expression of such a relation as one of the twelve axioms ; it is a metaphysical truth, and as such, just as certain as the principle of contradiction itself, which is a metalogical truth, and the common foundation of all logical demonstration.

Whoever denies the necessity, exhibited for intuition or perception, of the space-relations expressed in any proposition, may just as w'ell deny the axioms, or that the conclusion follows from the premises, or, indeed, he may as well deny the principle of contradiction itself, for all these relations are equally undemonstrable, immediately evident and known a priori.

For any one to wish to derive the necessity of space-relations, known in intuition or perception, from the principle of contradiction by means of a logical demonstration is just the same as for the feudal superior of an estate to wish to hold it as the vassal of another. Yet this is what Euclid has done.

His axioms only, he is compelled to leave resting upon immediate evidence; all the geometrical truths which follow are demonstrated logically, that is to say, from

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the agreement of the assumptions made in the pro* position with the axioms which are presupposed, or with some earlier proposition; or from the contradiction between the opposite of the proposition and the assumptions made in it, or the axioms, or earlier propositions, or even itself. But the axioms themselves have no more immediate evidence than any other geometrical problem, but only more simplicity on account of their smaller content.