How, again, would Hegel have rejoiced in a definition of thought and existence which would bridge over the logical gulf in his system! Hegel asserted that thought and existence were one. He is objected to by many philosophers, who ask where is the _tertium quid_ which makes it possible to reach from one to the other, or predicate their essential unity? But the mathematician defines existence as something which is not self-contradictory. Thought, then, to him is a form of existence, for thought is not self-contradictory; but the two, thought and existence, are not necessarily conterminous.[12] Hence, to say that non-contradiction is a fundamental condition of true thinking is as much as to say that it is a fundamental characteristic of real existence, and he identifies thought with reality. Dr. Caird remarks that the secular conscience conceives of the Infinite as opposed to the Finite; the religious conscience treats the Infinite as real, presupposed by the illusory Finite. Where does the truth lie? Mathematics does not admit the necessity of adopting either view at the expense of the other. Metaphysics standing alone produces results that may be disproved, but cannot be proved. Mathematics standing alone produces results that are susceptible of proof. Both are based on logic, and rest on the prerequisites of thought. Together they are a field for the best powers of human reason: metaphysics supplies insight, intuition, imagination; mathematics offers the indubitable proof and translates the ideal into the actual. 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.