Pythagoras of Samos, most infamously known for his Pythagorean theorem, lived from 560 B. C. to 480 B. C. as a Greek philosopher and religious leader who was responsible for important developments in the history of astronomy, theory of music, and of course, mathematics. He moved to Croton and founded a religious and philosophical school; unfortunately, because the school practiced both secrecy and communalism and no reliable contemporary records survived, the contributions of Pythagoras himself as well as those of his followers could not be distinguished.

Pythagoreans believed ultimately that all relations could be reduced to number relations, in other words, all things are numbers. Focusing on music, Pythagoras and his followers discovered that vibrating strings produce harmonious tones when the ratios of the lengths of the strings are whole numbers.

They then related these different theories to the so-called Pythagorean theorem that any triangle whose sides were in the ratio 3:4:5 was a right-angled triangle. In astronomy, the Pythagoreans believed that the periodic numerical relations of these heavenly bodies, or celestial spheres of the planets, were thought to produce a harmony called the music of the spheres.

One of the most important discoveries of Pythagoras was his belief that whole numbers and their ratios could account for geometrical properties; as a result, the world was introduced to the existence of irrational numbers. Whereas much of the Pythagorean doctrine that has survived consists of numerology and number mysticism, the influence of the idea that the world can be understood through mathematics can solely be credited to Pythagoras himself.