A mathematical and a metaphysical problem are not, then, problems of the same kind to be solved by the same method; nor is the conception of the mathematical Absolute reached in the same way as that of the metaphysical Absolute. We are even unable to say how far they correspond except in respect of their absoluteness.[14] But the contention of the mathematician to-day and of the epistemologist school of philosophy is not the identity of methods and results in the two sciences. It is the axiom of existence on which they both depend: the law of thought by which all methods are developed, and, above all, the _correlative value of each science to the other_, which allows us, in developing our knowledge from the standpoint of the two sciences, to recognise something of the greatness of the Absolute principle to which they both reach up, and in which their being consists. FOOTNOTES: [1] Of course, if we comprehend in our view only elementary geometrical and algebraical science, it is easy to show that they _do_ demand both axioms and intuitions. Take, _e.g._ Euclid I. I., where in the construction it is necessary to employ intuition for the assertion that the arcs really cut one another. There is no logical certainty that they do; in fact, in some other conditions, _e.g._ in those of other space dimensions, they might not. [2] This is, of course, not the space of experience. Logic and mathematics deal with implications of thought. See B. Russell (_Hibbert Journal_, 1904, pp. 809-12), who has shown that in all pure mathematics it is only the implications that are asserted, not the premiss or the consequence, as mathematicians used formerly to assume. [3] De Morgan, Peirce, Schröder, and B. Russell have worked out the logic of relations as well as the syllogism. [4] See Taylor, “Elements of Metaphysics,” p. 13. [5] See Dr. Caird, “Evolution of Theology in the Greek Philosophers.” [6] So Galileo, Newton, Huygens were philosophers in science. Descartes, Pascal, Leibniz were mathematicians as well as philosophers. [7] See S. Augustine, _De Civitate Dei_, Book XII. ch. xix.: “Ita vero suis quisque numerus proprietatibus terminatur, ut nullus eorum par esse cuicumque alteri possit. Ergo et dispares inter se atque diversi sunt, et singuli quique finiti sunt, et omnes infiniti sunt.” [8] See R. Dedekind, _Was sind und was sollen die Zahlen?_ 1893. [9] Two transfinite aggregates can have an ordinal correspondence with one another. [10] See G. Cantor, _Zur Lehre vom Transfiniten_. 1890. [11] _e.g._ Mr. P. Jourdain, _Philosophical Magazine_. 1904.