of equally ideal objects which are not present to him, and which never will be present to any human being.
To illustrate, — suppose that a mathematician wants to prove something about the value of π, or about the universal laws of Arithmetic, or about the properties of a continuous function, or about the sum of an infinite series, or about the mathematical relationships of two infinite collections of ideal objects. What he is concerned to demonstrate, lies in the realm of the infinite, and of the eternally valid. And our direct experience gives us only the passing data and the fragmentary ideas of the moment. Does the mathematician then, like the rationalistic metaphysician of old, hereupon merely appeal to so-called first and fundamental principles? Does he write down axioms, and merely defy you to deny them? Does he assert a priori that this or that cannot or shall not be questioned? No, the modern mathematician has no dogmas. He waits for his facts. He asks you to construct, and then to observe these facts with him. What he does is to build up before your eyes something, as Mr. Peirce well says, that either is a diagram or else resembles one, — a collection of observable symbols, or of figures in space, arranged in a certain deliberately planned way. In brief, he shows you empirically present inner constructions. He builds up these artificial objects before your eyes, and then he experiments upon them, and asks you to watch the result of the experiment. This result he first reads off, with as much the sense that he is recording present facts of observation, as one would have who should observe, on the street, that yonder horse is in front of yonder cart.
