[542] Said to be expressed by Grassman in the fundamental Axiom of Arithmetic (a + b) + 1 = a + (b + 1).
[543] Compare Helmholtz's more technically expressed Essay Zählen u. Messen, in the Philosophische Aufsätze, Ed. Zeller gewidmet (Leipzig, 1887), p. 17.
[544] For the original statements, cf. J. S. Mill's Logic, bk. ii. chap. vi. §§ 2, 3; and bk. iii. chap. xxiv. § 5.
[545] The subdivision itself consumes none of the space. In all practical experience our subdivisions do consume space. They consume it in our geometrical figures. But for simplicity's sake, in geometry we postulate subdivisions which violate experience and consume none of it.
[546] Cf. A. de Morgan: Syllabus of a proposed System of Logic (1860), pp. 46-56.
[547] Cf. Locke's Essay, bk. ii. chap. xvii. § 6.
[548] Some readers may expect me to plunge into the old debate as to whether the a priori truths are 'analytic' or 'synthetic.' It seems to me that the distinction is one of Kant's most unhappy legacies, for the reason that it is impossible to make it sharp. No one will say that such analytic judgments as "equidistant lines can nowhere meet" are pure tautologies. The predicate is a somewhat new way of conceiving as well as of naming the subject. There is something 'ampliative' in our greatest truisms, our state of mind is richer after than before we have uttered them. This being the case, the question "at what point does the new state of mind cease to be implicit in the old?" is too vague to be answered. The only sharp way of defining synthetic propositions would be to say that they express a relation between two data at least. But it is hard to find any proposition which cannot be construed as doing this. Even verbal definitions do it. Such painstaking attempts as that latest one by Mr. D. G. Thompson to prove all necessary judgments to be analytic (System of Psychology, ii. pp. 232 ff.) seem accordingly but nugæ difficiles, and little better than wastes of ink and paper. All philosophic interest vanishes from the question, the moment one ceases to ascribe to any a priori truths (whether analytic or synthetic) that "legislative character for all possible experience" which Kant believed in. We ourselves have denied such legislative character, and contended that it was for experience itself to prove whether its data can or cannot be assimilated to those ideal terms between which a priori relations obtain. The analytic-synthetic debate is thus for us devoid of all significance. On the whole, the best recent treatment of the question known to me is in one of A. Spir's works, his Denken und Wirklichkeit, I think, but I cannot now find the page.
[549] Book iv. chaps. ix. § 1; vii. 14.
[550] Chap. v. §§ 6, 8.
[551] Kant, by the way, made a strange tactical blunder in his way of showing that the forms of our necessary thought are underived from experience. He insisted on thought-forms with which experience largely agrees, forgetting that the only forms which could not by any possibility be the results of experience would be such as experience violated. The first thing a Kantian ought to do is to discover forms of judgment to which no order in 'things' runs parallel. These would indeed be features native to the mind. I owe this remark to Herr A. Spir, in whose 'Denken und Wirklichkeit' it is somewhere contained. I have myself already to some extent proceeded, and in the pages which follow shall proceed still farther, to show the originality of the mind's structure in this way.