The difference so far is merely that the mathematician makes his empirical objects, and does not wait to see if ordinary natural processes will furnish them to him. His world, therefore, seems at first quite plastic. It is, as we have said, his fairyland. He plays with it. Yet none the less, as he plays, he observes the empirical results of his play. And while he does this, he is as much a student of given facts as is a chemist or a business man. The results of this observation are often unexpected. And once seen (just here lies the mystery of the realm of validity), — once seen, they are also seen to stand for unalterable truth. How this can be, is precisely our present problem. The mathematician, in his own exact way, is thus like Browning’s lover. His instant is an eternity. He sees in a transient moment. Every one of his glimpses of fact is like the flash of the moonlight on the water. Yet what he sees outlasts the ages of ages. But nothing in all this eternal validity of his outcome makes him less empirical in his actual scrutiny. The validity is to be eternal. But his form of his experience is precisely that of any other human creature of the instant’s flight. In examining his diagram, he is as faithful a watcher as the astronomer alone with his star. The mathematician has made his diagram, but he cannot wilfully alter its consequences. And they must first be seen. Then alone can they be believed. Here is the strange antithesis between the empirical form and the eternal content of the realm of mathematical validity.
The valid, then, even the eternally valid, enters our human consciousness through the narrow portals of the
