Pythagoras

Pythagoras was an ancient Greek philosopher and religious teacher whose community joined mathematical study with religious practices. Later traditions associated him with influential ideas about number, music, and the soul.

Pythagoreanism: A philosophical and religious tradition centered on the teachings and practices associated with Pythagoras and his followers. It names the community and tradition through which Pythagoras’s ideas persisted.
Ancient Greek mathematics: The mathematical practices and theories developed in the ancient Greek world. It provides the wider intellectual setting for mathematical traditions attributed to Pythagoras.
Music theory: The study of musical structures, including pitch, rhythm, harmony, and form. Pythagorean numerical accounts became a lasting way to explain musical intervals.
Pythagorean theorem in Babylonian mathematics: Evidence that Babylonian scribes knew numerical relationships equivalent to the Pythagorean theorem before Pythagoras lived. It cautions against treating the theorem’s later name as proof of his discovery.
Philolaus: A fifth-century BCE Greek philosopher who developed Pythagorean cosmological ideas. His surviving fragments offer rare evidence for early Pythagorean thought beyond later legend.
Pythagorean theorem: A geometric theorem stating that a right triangle’s hypotenuse squared equals the sum of its legs squared. Later tradition attached this theorem to Pythagoras, though its earlier history is uncertain.
Music of the spheres: An ancient and medieval idea that celestial motion produces a form of inaudible musical harmony. It extends the Pythagorean association between number, musical proportion, and cosmic order.
Pythagoras of Samos: The sixth-century BCE Greek philosopher and religious teacher traditionally identified as the founder of Pythagoreanism. The sparse contemporary evidence leaves his biography and specific teachings uncertain.
Archytas: A Greek philosopher, mathematician, and statesman associated with the Pythagorean tradition in the fourth century BCE. He shows how Pythagorean interests in mathematics and civic life continued after Pythagoras.
Ratios and proportions: Mathematical relationships comparing quantities through division and equality of ratios. Numerical ratios connect Pythagorean accounts of musical intervals with broader claims about order.

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