Pythagoras placed great value in mathematics, and his mathematical achievements were prompted by a discovery in the field of music. Working with a monochord (comparable to a guitar with one string), Pythagoras discovered that the note produced by plucking a string of a certain length could be ‘reproduced’ one octave higher by plucking a string exactly half as long, or one octave lower by plucking a string exactly twice as long. Continuing on, he discovered the beautiful symmetry of strings and vibrations: simple mathematical proportions that account for the pleasing harmonies we hear when the right notes of the musical scale are played together as chords. Pythagoras found that these harmonic proportions can be expressed quite simply as ratios of the numbers 1, 2, 3 and 4. Other ratios of string length produce irritating disharmony.
Thus Pythagoras discovered that ‘Number’ somehow underlies the phenomena and experience of music. Like other Greek philosophers of his era, he was looking for a fundamental ‘something’ that could unify and explain life, mind, and nature. But rather than a primary physical substance, he came up with a primary idea. All things, he concluded, in their essence, are ‘Number’. He found that the motions of the heavenly bodies follow regular patterns which are understandable and predictable in terms of the same numerical principles of harmony and proportion − and thus was born the ancient notion of ‘the harmony of the spheres’; he saw that the various surfaces of tangible objects can be viewed as examples and illustrations of the perfect figures of geometry; he saw that beauty and physical health are dependent upon a harmony of material elements, and he saw that psychological health was also to be achieved through temperance or moderation, again requiring a proper harmonious balance. The Pythagoreans would eventually conclude that the perfection of a soul requires the restoration of inner harmony, and that achieving this is not dissimilar to achieving the perfect attunement of a musical string.
For Pythagoras, then, as for Plato who followed him and for many ancient people, the domain of mathematics was sacred. It could also be practical of course, but this did not detract from its divine essential nature.