Pythagoreanism

Pythagoreanism refers to the philosophical doctrine of the early Pythagorean school represented by Pythagoras in Greece and formed in the ancient Greek colonial city-state of Croton in the southern part of the Italian peninsula. Like the Elia school, it was committed to the search for abstract principles, which promoted the formation and development of the ancient Greek idealist philosophy, preaching the idea of the immortality of the soul and reincarnation, believing that the soul is immortal, and the human soul can migrate to other organisms, and can appear repeatedly, so that all living things are related by blood. Given that there are abstract things in the world besides material beings, and find that there is some abstract quantitative relationship in everything.

The Pythagorean Diet
Scholars debate whether Pythagoras advocated an early form of strict vegetarianism, but as it was widely believed he did, those who later followed a strict vegetarian diet (more similar to the modern vegan diet than lacto-ovo vegetarianism) were called "Pythagoreans" or adherents to the "Pythagorean Diet".

The word "vegetarian" wouldn't be coined for several centuries to come, around 1839, and included those that followed varying forms of vegetarianism. The word "vegan" was coined in 1944 in order to distinguish vegetarians who ate no animal products from vegetarians who ate dairy and/or eggs, etc. and very soon after to define those who explicitly rejected the use of animals for any purpose.

Eudoxus of Cnidus, a student of Archytas and Plato, wrote that "Pythagoras was distinguished by such purity and so avoided killing and killers that he not only abstained from animal foods, but even kept his distance from cooks and hunters".