Iamblichus in the context of "Chaldean Oracles"

Nicomachus of Gerasa (Ancient Greek: Νικόμαχος; c. 60 – c. 120 AD) was an Ancient Greek Neopythagorean philosopher from Gerasa, in the Roman province of Syria (now Jerash, Jordan). Like many Pythagoreans, Nicomachus wrote about the mystical properties of numbers, best known for his works Introduction to Arithmetic and Manual of Harmonics, which are an important resource on Ancient Greek mathematics and Ancient Greek music in the Roman period. Nicomachus' work on arithmetic became a standard text for Neoplatonic education in Late antiquity, with philosophers such as Iamblichus and John Philoponus writing commentaries on it. A Latin paraphrase by Boethius of Nicomachus's works on arithmetic and music became standard textbooks in medieval education.

Neoplatonism is a version of Platonic philosophy that emerged in the 3rd century AD against the background of Hellenistic philosophy and religion. The term does not encapsulate a set of distinct ideas based on a form of Platonism as much as a series of Platonic thinkers coming primarily from a certain ancient historical period. Among the common ideas it maintains is monism, the doctrine that all of reality can be derived from a single principle, "the One".

Neoplatonism began with Ammonius Saccas and his student Plotinus (c. AD 205 – 271) and stretched to the sixth century. After Plotinus there were three distinct periods in the history of Neoplatonism: the work of his student Porphyry (third to early fourth century); that of Iamblichus (third to fourth century); and the period in the fifth and sixth centuries, when the academies in Alexandria and Athens flourished.

Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the ancient Greek colony of Kroton, in modern Calabria (Italy) circa 530 BC. Early Pythagorean communities spread throughout Magna Graecia.

Already during Pythagoras's life it is likely that the distinction between the akousmatikoi ("those who listen"), who is conventionally regarded as more concerned with religious, and ritual elements, and associated with the oral tradition, and the mathematikoi ("those who learn") existed. The ancient biographers of Pythagoras, Iamblichus (c. 245 – c. AD 325) and his master Porphyry (c. 234 – c. AD 305) seem to make the distinction of the two as that of 'beginner' and 'advanced'. As the Pythagorean cenobites practiced an esoteric path, like the mystery schools of antiquity, the adherents, akousmatikoi, following initiation became mathematikoi. It is wrong to say that the Pythagoreans were superseded by the Cynics in the 4th century BC, but it seems to be a distinction mark of the Cynics to disregard the hierarchy and protocol, ways of initiatory proceedings significant for the Pythagorean community; subsequently did the Greek philosophical traditions become more diverse. The Platonic Academy was arguably a Pythagorean cenobitic institution, outside the city walls of Athens in the 4th century BC. As a sacred grove dedicated to Athena, and Hecademos (Academos). The academy, the sacred grove of Academos, may have existed, as the contemporaries seem to have believed, since the Bronze Age, even pre-existing the Trojan War. Yet according to Plutarch it was the Athenian strategos (general) Kimon Milkiadou (c. 510 – c. 450 BC) who converted this, "waterless and arid spot into a well watered grove, which he provided with clear running-tracks and shady walks". Plato lived almost a hundred years later, circa 427 to 348 BC. On the other hand, it seems likely that this was a part of the re-building of Athens led by Kimon Milkiadou and Themistocles, following the Achaemenid destruction of Athens in 480–479 BC during the war with Persia. Kimon is at least associated with the building of the southern Wall of Themistocles, the city walls of ancient Athens. It seems likely that the Athenians saw this as a rejuvenation of the sacred grove of Academos.

Iccus of Taranto (Ancient Greek: Ἴκκος) (5th century BC) was a Magna Grecia Olympic athlete, a victor during the 84th Games (444 BC) or 70th Games (470 BC) according to older sources. He is considered the father of athletic dietology. He prepared himself physically before competing according to ethical-religious Pythagorean concepts by abstaining from sexual intercourse and a frugal diet specially prepared. He also taught these principles. Pausanias calls him the best gymnast of his age, and Plato also mentions him with great praise.

Iamblichus calls him a Pythagorean. According to Themistius, Plato reckoned him among the sophists. Specifically, in Plato's dialogue Protagoras, the sophist Protagoras lists Iccus alongside Homer, Hesiod, Simonides, Orpheus, Musaeus, Herodicus, Agathocles tutor of Damon, and Pythoclides [fr] as fellow sophists (that is, improvers of youth) who chose to present themselves as poets, athletes, or musicians for fear of public disapproval.

Cleinias of Tarentum (Ancient Greek: Κλεινίας; fl. 4th-century BCE), Magna Graecia, was a Pythagorean philosopher, and a contemporary and friend of Plato, as appears from the story (perhaps otherwise worthless) which Diogenes Laërtius gives on the authority of Aristoxenus, to the effect that Plato wished to burn all the writings of Democritus which he could collect, but was prevented by Cleinias and Amyclus of Heraclea. In his practice, Cleinias was a true Pythagorean. Thus, we hear that he used to assuage his anger by playing on his harp; and, when Prorus of Cyrene had lost all his fortune through a political revolution, Cleinias, who knew nothing of him except that he was a Pythagorean, took on himself the risk of a voyage to Cyrene, and supplied him with money to the full extent of his loss.

Two fragments of Pythagorean pseudepigrapha are attributed to Cleinias, one preserved by Stobaeus, the other in the Theology of Arithmetic attributed to Iamblichus.