Page:Carroll - Euclid and His Modern Rivals.djvu/94

56 Nie. That may be the natural process: but surely you will allow our Definition to be a legitimate one?

Min. I think not: and I have the great authority of Kant to support me. In his 'Critique of Pure Reason,' he says (I quote from Meiklejohn's translation, in Bohn's Philosophical Library, pp. 9, 10), 'Mathematical judgments are always synthetical … "A straight Line between two points is the shortest" is a synthetical Proposition. For my conception of straight contains no notion of quantity, but is merely qualitative. The conception of the shortest is therefore wholly an addition, and by no analysis can it be extracted from our conception of a straight Line.'

This may fairly be taken as a denial of the fitness of the Axiom to stand as a Definition. For all Definitions ought to be the expressions of analytical, not of synthetical, judgments: their predicates ought not to introduce anything which is not already included in the idea corresponding to the subject. Thus, if the idea of 'shortest distance' cannot be obtained by a mere analysis of the conception represented by 'straight Line,' the Axiom ought not to be used as a Definition.

Nie. We are not particular as to whether it be taken as a Definition or Axiom: either will answer our purpose.

Min. Let us then at least banish it from the Definitions. And now for its claim to be regarded as an Axiom. It involves the assertion that a straight Line is shorter than any curved Line between the two points. Now the length of a curved Line is altogether too difficult a subject for a beginner to have to consider: it is moreover unnecessary that he should consider it at all, at least in the earlier