The higher arithmetic presents us with an inexhaustible store of inter — Arithmetic

"Unfortunately for Pythagoras, his theorem led at once to the discovery of incommensurables, which appeared to disprove his whole philosophy. So long as no adequate arithmetical theory on incommensurables existed, the method of Euclid was the best that was possible in geometry. When Descartes introduced co-ordinate geometry, thereby again making arithmetic supreme, he [Descartes] assumed the possibility of a solution of the problem of incommensurables, though in his day no such solution had been found."

"Mystical doctrines as to the relation of time to eternity are also reinforced by pure mathematics, for the mathematical objects, such as numbers, if real at all, are eternal and not in time. Such eternal objects can be conceived as Gods thoughts. Hence Platos doctrine that God is a geometer, and Sir James Jeans belief that He is addicted to arithmetic."

"Not only philosophers were influenced by Plato. Why did the Puritans object to the music and painting and gorgeous ritual of the Catholic Church? You will find the answer in the tenth book of the Republic. Why are children compelled to learn arithmetic? The reasons are given in the seventh book."

"What mathematics, therefore are expected to do for the advanced student at the university, Arithmetic, if taught demonstratively, is capable of doing for the children even of the humblest school. It furnishes training in reasoning, and particularly in deductive reasoning. It is a discipline in closeness and continuity of thought. It reveals the nature of fallacies, and refuses to avail itself of unverified assumptions. It is the one department of school-study in which the sceptical and inquisitive spirit has the most legitimate scope; in which authority goes for nothing. In other departments of instruction you have a right to ask for the scholar’s confidence, and to expect many things to be received on your testimony with the understanding that they will be explained and verified afterwards. But here you are justified in saying to your pupil “Believe nothing which you cannot understand. Take nothing for granted.” In short, the proper office of arithmetic is to serve as elementary 268 training in logic. All through your work as teachers you will bear in mind the fundamental difference between knowing and thinking; and will feel how much more important relatively to the health of the intellectual life the habit of thinking is than the power of knowing, or even facility of achieving visible results. But here this principle has special significance. It is by Arithmetic more than by any other subject in the school course that the art of thinking—consecutively, closely, logically—can be effectually taught."

"Arithmetic and geometry are much more certain than the other sciences, because the objects of them are in themselves so simple and so clear that they need not suppose anything which experience can call in question, and both proceed by a chain of consequences which reason deduces one from another. They are also the easiest and clearest of all the sciences, and their object is such as we desire; for, except for want of attention, it is hardly supposable that a man should go astray in them. We must not be surprised, however, that many minds apply themselves by preference to other studies, or to philosophy. Indeed everyone allows himself more freely the right to make his guess if the matter be dark than if it be clear, and it is much easier to have on any question some vague ideas than to arrive at the truth itself on the simplest of all."

"Some writers maintain arithmetic to be only the only sure guide in political economy; for my part, I see so many detestable systems built upon arithmetical statements, that I am rather inclined to regard that science as the instrument of national calamity."