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CHAPTER XIII.[1]
ON THE METHODS OF MATHEMATICS.
Euclid's method of demonstration has brought forth from its own womb its most striking parody and caricature in the famous controversy on the theory of parallels, and the attempts, which are repeated every year, to prove the eleventh axiom. This axiom asserts, and indeed supports its assertion by the indirect evidence of a third intersecting line, that two lines inclining towards each other (for that is just the meaning of "less than two right angles") if produced far enough must meet – a truth which is supposed to be too complicated to pass as self-evident, and therefore requires a demonstration. Such a demonstration, however, cannot be produced, just because there is nothing that is not immediate. This scruple of conscience reminds me of Schiller's question of law: –
"For years I have used my nose for smelling. Have I, then, actually a right to it that can be proved?" Indeed it seems to me that the logical method is hereby reduced to absurdity. Yet it is just through the controversies about this, together with the vain attempts to prove what is directly certain as merely indirectly certain, that the self-sufficingness and clearness of intuitive evidence appears in contrast with the uselessness and difficulty of logical proof – a contrast which is no less instructive than amusing. The direct certainty is not allowed to be valid here, because it is no mere logical certainty following from the conceptions, thus resting only upon the relation of the
1 This chapter is connected with § 15 of the first volume.
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