Turning now to their philosophical teaching, the first thing that we have to understand is that we cannot speak of the philosophy of Pythagoras, but only of the philosophy of the Pythagoreans. For it is not known what share Pythagoras had in this philosophy or what share was contributed by his successors. Now we recognize objects in the universe by means of their qualities. But the majority of these qualities are not universal in their scope; some things possess some qualities; others possess others. A leaf, for example, is green, but not all things are green. Some things have no colour at all. The same is true of tastes and smells. Some things are sweet; some bitter. But there is one quality in things which is absolutely universal in its scope, which applies to everything in the universe--corporeal or incorporeal. All things are _numerable_, and can be counted. Moreover, it is impossible to conceive a universe in which number is not to be found. You could easily imagine a universe in which there is no colour, or no sweet taste, or a universe in which nothing possesses weight. But you cannot imagine a universe in which there is no number. This is an inconceivable thought. Upon these grounds we should be justified in concluding that number is an extremely important aspect of things, and forms a fundamental pad of the framework of the world. And it is upon this aspect of things that the Pythagoreans laid emphasis. They drew attention to proportion, order, and harmony as the dominant notes of the universe. Now when we examine the ideas of proportion, order, and harmony, we shall see that they are closely connected with number. Proportion, for example, must necessarily {35} be expressible by the relation of one number to another. Similarly order is measurable by numbers. When we say that the ranks of a regiment exhibit order, we mean that they are arranged in such a way that the soldiers stand at certain regular distances from each other, and these distances are measurable by numbers of feet or inches. Lastly, consider the idea of harmony. If, in modern times, we were to say that the universe is a harmonious whole, we should understand that we are merely using a metaphor from music. But the Pythagoreans lived in an age when men were not practised in thought, and they confused cosmical harmony with musical harmony. They thought that the two things were the same. Now musical harmony is founded upon numbers, and the Pythagoreans were the first to discover this. The difference of notes is due to the different numbers of vibrations of the sounding instrument. The musical intervals are likewise based upon numerical proportions. So that since, for the Pythagoreans, the universe is a musical harmony, it follows that the essential character of the universe is number. The study of mathematics confirmed the Pythagoreans in this idea. Arithmetic is the science of numbers, and all other mathematical sciences are ultimately reducible to numbers.