The mathematician has often bemocked the metaphysician, who spins theories out of his brain, and he has pointed with pride to the solid fabric of his own science. Yet his science has no superior solidity; it may prove to be only a projection of his mind. It has not the dignity of a science of observation. The metaphysician is the sane man who wishes to satisfy himself whether this specious fabric may not be after all a vast cobweb of the mathematician’s brain. Given the idea of space, the science of geometry follows. But of the reality of space geometry has nothing to say. Such criticisms on the mathematical sciences have often been made before, and it would be less necessary to repeat them, were it not that such an axiom as “The whole is greater than its part” is pointed to as an instance of irresistible certainty. The axiom is necessary when once the idea of quantity is introduced, and if the universe is to be construed quantitatively, it is true of it that any whole in it is greater than its part. But is the universe to be construed quantitatively?
In regard to substance, Spinoza showed what may be, by a rigorous logic, deduced from the conception. But he simply assumed the conception. It never occurred to him that it might be simply a subjective idea, which advancing science and philosophy might dissipate. His ‘Substance’ was an idolum fori, if ever there was one.
That this is the nature of the necessities of our thinking, it is well to remember, in view of certain current refutations of scepticism. Professor Caird, in the passage already referred to, writes thus: “If I say that all I know is appearance, and that I do not, and cannot know the reality which is beyond appearance, I must have some positive reason for the distinction which I make between appearance and reality…. The last work of Scepticism is to disclose the basis of truth on which it must rest.” But doubt is scarcely to be exorcised by such an appeal. If the sceptic uses such a term as appearance, the correlative reality may be implied, and it is further possible that there is implied a power of distinguishing between the two. But what if the sceptic is doubtful of the claims of
