Pythagoreanism is the name given to the system of philosophy and science developed by Pythagoras, which influenced nearly all the systems of Hellenistic philosophy that followed. Two schools of Pythagorean thought eventually developed; one based largely on mathematics and continuing his line of scientific work, while the other focused on his metaphysical teachings, though each shared a part of the other.
In Ancient Greece, the sophists were a category of teachers who specialized in using the tools of philosophy and rhetoric for the purpose of teaching aretê—excellence, or virtue—predominantly to young statesmen and nobility.
The Cynics were an ascetic sect of philosophers beginning with Antisthenes in the 4th century BC and continuing until the 5th century AD. They believed that one should live a life of Virtue in agreement with Nature. This meant rejecting all conventional desires for wealth, power, health, or fame, and living a life free from possessions.
The Cyrenaics were a hedonist school of philosophy founded in the 4th century BCE by Aristippus of Cyrene, who was a student of Socrates. They held that pleasure was the supreme good, especially immediate gratifications; and that people could only know their own experiences, beyond that truth was unknowable.
Platonism is the name given to the philosophy of Plato, which was maintained and developed by his followers. The central concept was the theory of Forms: the transcendent, perfect archetypes, of which objects in the everyday world are imperfect copies. The highest form was the Form of the Good, the source of being, which could be known by reason. In the 3rd century BC, Arcesilaus adopted skepticism, which became a central tenet of the school until 90 BC when Antiochus added Stoic elements, rejecting skepticism. With the adoption of oriental mysticism in the 3rd century AD, Platonism evolved into Neoplatonism.