square. But when the relation of the angles of the triangle to the length of the sides has to be determined, the law of non-contradiction is comparatively irrelevant. Thought has a new method for its new subject. Different from all these, and each possessing a law of its own, are the conceptions of substance, and causality, and spirit.
In each of these cases the laws of thought are on the same footing, and possess the same necessity, as the primary ‘laws of thought.’ The axioms of mathematics, and all the deductions from them, are as necessary to thought as the laws of Identity and Contradiction. In at least part of the system of Spinoza may be seen the necessity laid upon us when we touch the conception of Substance. Thus each conception has a law of its own. The laws of identity and non-contradiction are the logical methods we follow when we use the conception of Being. Each other conception that we employ has a method of its own. So that we have other certainties than the so-called ‘laws of thought.’ These are simply the mind’s employment of one conception. It has other certainties, as it has other conceptions.
Further, the relation of these necessities of thought to the actual world is, in all cases, of the same kind. The laws of identity and non-contradiction obtain, so long as we hold to the conception of Being. And it is a necessary corollary to this to say that if there is a reality answering to the conception, it is governed by these laws of thought. And it may also be said, that a world to which none of our conceptions apply, is for us a nonentity. But this is not to say that Being is a true category, that is, a counterpart of the actual universe. In fact, as we have already seen, it is unable to represent the reality, and as usually employed, has in it an element of false hood.
Mathematics also is an ideal construction. It starts from an assumption. There may be tridimensional space; and if there is, it is a space governed by the principles of geometry. But the existence of geometry does not prove the objectivity of space.
