Page:Mind (New Series) Volume 8.djvu/533

KANT'S PROOF OF THE PROPOSITION, ETC. 519 could never be applied in Mathematics without leading at once to perfect absurdities. The circumference is ' a part ' of the circle if we use the term ' part ' in the general non- mathematical sense, and yet for geometry it is absolutely inane to say, that the circle is greater than the circumference, for a plane surface of two dimensions can never be super- posed upon a curved line of one and therefore can never be compared to it in magnitude. If we borrow Kant's language concerning Space generally we might say, that a geometrical whole contains all its parts in se, non sub se ; the parts of a line are lines and not points, a plane is divided by geometry into planes, a solid into solids. But the general or philosophic conception of a whole, from which is derived the analytic expression corresponding to Euclid's ninth axiom, contains its parts sub se and therefore the analysis of it can never give us a principle applicable as a geometrical axiom. The words 'whole' and 'part' have two quite distinct significa- tions in Philosophy and Mathematics, while Mathematics has simply nothing to do with the philosophic meaning of 'greater' and 'less'. Accordingly in his next line Kant continues, " What commonly causes us to believe here, that the predicate of such apodeictic judgments is already con- tained in our conception, and that the judgment is therefore analytic, is merely the equivocal nature of the expression ". Every word in Euclid's ninth axiom has a meaning peculiar to Mathematics and foreign to the ordinary usage, so that, unless we clearly distinguish the two applications of our one principle, we are apt to term it analytic in circumstances, under which its analytic meaning is at once false and absurd. And yet it is easy to see the danger of error, which must always confront the mathematician, when he expresses a geometrical axiom in words identical with those of a logical and analytical formula. "We must," says Kant, "join a certain predicate in thought t"o a certain given conception, and this necessity cleaves already to the conception. But the question is, not what we must join in thought to the given conception, but what we really think in conceptions though only obscurely. Then it becomes manifest that the predicate pertains to these conceptions, necessarily indeed, yet not immediately, but only by the mediation of an in- tuition, which must be added to the conceptions." In these axioms we use certain conceptions, which, to borrow a modern phrase, already have a certain connotation, but for purely mathematical purposes this connotation is quite valueless. Even though it be settled that a certain concep- tion is necessarily joined to certain predicates, this junction