Mathematics is fundamental to the Pythagorean view. First, geometry originated in Egypt but was transformed in Greece. The Egyptians were flooded each year by the Nile, and so geometry (the study of land, as the name implies), was developed to help them retain their land boundaries. But when geometry was passed to the Greeks, who had no such physical or practical need, they pursued geometry and all mathematics as a “liberal” study—freed from practical needs. Not only was it a study of freemen as opposed to slaves, but it was a study capable of making a man free (Robinson, p. 67). Mathematics became a means of freeing the soul from practical, physical demands. To Pythagoras, mathematics is a way to free the soul and make it ready for the divine. So it could be said that Pythagoras invented the liberal arts.
Additionally, and more profoundly, Pythagoras comes to believe in number and ratio underlying all of nature. The monochord was a one-stringed instrument. By stopping the string at various intervals and plucking the string, you get the octave, the major fifth and the major fourth—the principles of Greek music. The ratios that create melody and harmony are what make the music on the string. This was Pythagoras’ chief insight: that number and ratio underly all of reality. As Aristotle describes the Pythagoreans:
They thought that the principles of mathematics were the principles of all things. . . Seeing that the properties and ratios of the musical consonances were expressible in numbers, and that indeed all other things seemed to be wholly modelled in their nature upon numbers, they took numbers to be the whole of reality, the elements of numbers to be the elements of all existing things, and the whole of heaven to be a musical scale and a number. (Metaphysics, I. 5. 985 b 23)
The Pythagorean view that number underlies all of nature, or that all things are number, may seem far-fetched. But two hundred years later, Archimides used it to create a science of mechanics, and later Galileo took it as foundational for his own work. Galileo writes:
Philosophy is written in the great book which is ever before our eyes—I mean the universe—but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles, and other geometrical figures without whose help it is impossible to comprehend a single word of it. (Galileo, Opere Complete di Galileo Galilei, Firenze, 1842, IV, 171)
To Pythagoras, not only is number the language of nature, but the celestial spheres make sounds (by virtue of their whirling) that are a vast, cosmic harmony. This idea, in addition to his view of the immortal soul in bondage within the body, influenced all subsequent Greek thought and through Plato, the whole of the Western tradition.