But the element in philosophical thought which, employing the psychological method, tends to the discussion of a theory of being rather than that of knowledge, and thus to the realisation of an ethical system rather than to metaphysical discovery, is averse from accepting these conclusions. It remains, therefore, for us to examine the criticism offered by the psychological school on what they call the mathematising of philosophy; and it will be found that the attack deals both with the ground of the alliance and its results.
A typical exponent of this school is Moisant, who, in the Revue Philosophique for January 1905, attacked what he considered to be the characteristic of modern philosophy and also its vice. It will be observed that at the outset he reverses the rôles of philosophy and mathematics as we have apprehended them. Philosophy, he says, should expect to be inspired by mathematics, but should avoid its method. Next, he connects the modern movement with the theories of Leibniz, who aimed at substituting general formulæ for elementary forms of reason and calculation. These short cuts, which seem to the mathematician to liberate the mind from a burden which prevents it from employing its full activity, seem to the psychologist to tend to a mechanical method, in which the thinker is only aware of premises and results, and in which the mathematical concept tends to replace the real idea. Then he attacks the new definition of mathematics as the science of relations, asserting that it still contains notions of space.[13]
Finally, he comes to the real question at issue, and enters into the comparison of a metaphysical and a mathematical problem. He takes as his subject the argument from the known to the unknown. Descartes had said that argument should lead from the known to the unknown, simple to complex, and had defined the first as that which could be known without the help of the second. This logical order of reasoning has been attributed to mathematics, but has been considered to be inapplicable to philosophy. Mathematics, in its recent development, by the argument from the Finite to the Infinite and back again, starts from two propositions, neither of which can be said to be axiomatic, because each in turn can be proved from the other, but in the course of argument from either mathematics makes use of the logical process. The real axiom, as has been shown, is that of existence or being. A metaphysical argument has the same root—that of existence—but a metaphysical problem deals with paradoxes, with questions which are sometimes defined as having two answers, each equally correct, and sometimes as yielding no answer at all. The method of thesis, antithesis, and synthesis is in the Hegelian logic applied to their solution.
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.”