Pythagoreanism

Pythagoreanism originated in the 6th century BCE, based on teachings, or beliefs held by Pythagoras and his followers, the Pythagoreans, who were considerably influenced by mathematics and mysticism. Later revivals of Pythagorean doctrines led to what is now called Neopythagoreanism or Neoplatonism. Pythagorean ideas exercised a marked influence on Aristotle, and Plato, and through them, all of Western philosophy.

Historians from the Stanford Encyclopedia of Philosophy pointed out

Two schools

According to tradition, pythagoreanism developed at some point into two separate schools of thought, the mathēmatikoi (μαθηματικοί, Greek for "Teachers") and the akousmatikoi (ἀκουσματικοί, Greek for "listeners").

John Burnet (1892) noted

According to Iamblichus (ca. 250-330 AD, 1918 translation) in The life of Pythagoras, by Thomas Taylor

Excerpt from a speech by the character ‘Aristotle’ in Protrepticus (Hutchinson and Johnson, 2015)

History

According to historians like Thomas Gale (based on Archytas account), Thomas Taler (based on the work of Iamblichus), or Cantor, Archytas (428 BC) became the head of the school, about a century after the murder of Pythagoras. According to August Böckh (1819), who cites Nicomachus, Philolaus was the successor of Pythagoras. And according to Cicero (de Orat. III 34.139), Philolaus was teacher of Archytas of Tarentum.