Neoplatonism was the last of the great schools of Greek philosophy; and the most mystical. It's founder Plotinus and his successors taught an elaborate emanationist cosmology.
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PORPHYRIAN SCHOOL |
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PERGAMENE SCHOOL |
Hypatia Plutarch of Athens
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Aeneas | |
(Christian) Hermiasas Proclus
GAZA SCHOOL | ___________________________|____________
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| Marinus Dionysius(Christian)
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John Philoponus Olympiodorus Simplicius Damascius
(Christian) | (Aristotlean)
school falls founds Imperial Academy
into Arab hands (Constantinople)
As the last great school of Greek philosophy and mysticism, Neoplatonism borrowed from all preceding schools. The influence of Aristotle himself can be found in the Neoplatonic philosophical method and propositions of logic. In its scepticism of empirical knowledge it draws from the Cynic and Pyrrhic line. In its dualistic emanationist metaphysics and aspiring to the Good in a transcendent spiritual sphere it is clearly a continuation of the Platonic school. Its derivation of all realities from a transcendent One is pure Neopythagoreanism. Its ethics have been adopted from Stoicism. And its conception of the action of the Divine in the world, and the essence and origin of matter, is clearly derived from the dynamic pantheism of Stoicism. Indeed, Neoplatonism could be seen as the culmination of Greek metaphysical thought, not as a mere eclecticism but a true synthesis.
Although Ammonius Saccus was reputedly the founder of the Neoplatonic school in Alexandria, none of his writings, and very little information about his life, has survived. In addition to Plotinus, his pupils include a number of others.
Plotinus (204-270 c.e.) is one of the giants of western spirituality - both mystic and philosopher - who in his Enneads describes reality in terms of a series of divine heirarchies or hypostases. He taught the rejection of worldly things and purification of the soul as the means for returning to the One.
Whilst Plotinus was the founder and dominant theorist of the Neoplatonic movement, his successors developed a more sophisticated cosmology and metaphysic within the general framework he laid out. And here we find two tendencies.
First is the reduction in the number of hypostases postulated (for example the identification of the One and the Nous), such as was taught by the Plotinian and Porphyrian schools. Plotinus and his immediate successors such as Porphyry and an anonymous commentator tended to "telescope" the Hypostases, to reduce them to a single pantheistic One  analogous to the immanent Absolute of Eastern Monism (Advaita Vedanta, Mahayana Buddhism, etc). Here we see the "mystical" side of Neoplatonism.
In contrast to this is the tendency to increase the number of hypostases, for example in th teachings of the Syrian Iamblichus (died c.326) and the Athenian Proclus (412-485). These later philosophers not only multiplied Plotinus' three hypostases, formulating a large number of metaphysical principles, but also introduced greater systematisation and complexity, and indeed also a strong element of rigidity, into Neoplatonic metaphysics.
Although Porphyry c.232-c.305) is best known for organizing and editing Plotinus's writings and lectures, he also made several original contributions regarding the nature of the hypostases, tending to adopt a more "monistic" and pantheistic position than Plotinus. His simplification of Plotinus' thought appealed to the practically-minded Romans, and influenced both pagans like Macrobius and Christians like Augustine[3b]
Porphyry’s Letter to Anebo & Iamblichus’ On the Mysteries - An analysis - on line text, with commentaries, by Stephen Ronan. Excellent.
Iamblichus was the chief representative of Syrian Neoplatonism. He modified the basic Plotinian metaphysic through a greater elaboration of the hypostases, a more systematic application of Pythagorean number-symbolism, and, under the influence of Oriental systems, less of an emphasis on the intellectual approach and more the occult-magical and mythical. By this latter he presented a philosophical interpretation of popular Hellenistic religion. His influence extends even to the Theosophists of the late 19th and early 20th century.
Aedesius was Iamblichus' chief student. He founded a school in Pergamum (now Albania). the school seems to have placed some emphasis on the Plotinian philoophical-mystical way of purification, rather than solely the theurgic.
Theodorus of Asine was another student; he broke away from the master and taught a synthesis of a number of different teachings, combining elements from Numenius, Plotinus, Amelius, Porphyry, and Iamblichus [3c]. He does not seem to have had any long-term influence.
Iamblichus and his successors rejected the pure philosophical speculation of
early Neoplatonists and increasingly emphasised theurgy ,
thus paralleling the greater emphasis on ritual magical procedures in later, Tantric, forms
of Hinduism and Buddhism. There was also a tendency towards increasing
scholasticism and textual fundamentalist adherence to Classical sources, along
with an increasing metaphysical complexity; in contrast to the simplicity of
Plotinus' hierarchy, the Iamblichus and the post-Iamblichean schools formulated
a proliferation of Hypostases, with ever more rigid divisions between
Until the beginning of the fifth century, Neoplatonic schools still flourished in the major cities of the Empire. But the murder of the female mathematician Hypartia by a mob of fanatical Christians showed that the Church would no longer endure the presence of "heathenism". Although the school in Alexandria maintained a lingering existence until the middle of the sixth century, it was elsewhere that Hellenistic philosophy found its last refuge. In Athens, which was by now a mere provincial town, a Neoplatonic centre still flourished.
In a time when classical civilisation was in decline, the Athenian academy returned to a stricter philosophical method and scholarship. It sought to interpret the entire Greek tradition, undertook in the light of Plotinus, to a comprehensive and tightly-knit system. Hence the earlier mysticism and magic was replaced by a drier and more intellectual approach; in other words scholasticism. For these Athenian Neoplatonists, the works of Plato, the Chaldean Oracles, the Orphic poems, and much more which was assigned to a great antiquity, were inspired divine writings, and formed the basic material, which was then elaborated through dialectic hermeneutics.
The first head of the school was Plutarch of Athens (d.432 c.e.) not to be confused with the great Roman biographer of the same name. The school flourished under his disciple Syrianus (d. c.437) an important commentator on Plato and Aristotle, and Proclus (412-85). the last great thinker of Greek thought, and the best known and most important of the later Neoplatonists.Thenceforth under Marinus, Isidorus, Hegias and Zenodotus a period of decline set in. Under the talented Damascius there was a revival of Platonism, and it could be speculated that this encouraged the emperor Justinian to close the school. 
The Athenian School - listing in the Internet Encyclopdia of Philosophy (scroll down)
The school of Alexandria is not the same as the vibrant acadamy under Ammonius. It seems to date back to the late fourth and early fifth centuries, represented by the mathematician Theon and his daughter Hypatia, who was martyred by a Christian mob under the instigation of the infamous church leader Cyril.
Persecution seems to have been common. Hierocles was flogged by the authorities in Constantinople, despite the fact that his teachings were more monotheistic than those of other pagan Neoplatonists. (Ironically, the school of Alexandria also included among its members a number of Christian philosophers, such as Aeneas and John Philoponus.)
It was only with Heimonius and his son Ammonius that a definite succession can be traced at Alexandria. Olympiodorus the Platonic commentator was the last pagan head of the school, after his death it passed into Christian hands under the Aristotlean commentators Elias and David. The school's last head, Stephanus, moved to and became head of an academy in Constantinople in 610. In 641 the Alexandrian school was captured by the Arabs. It thus played an important part in the transmission of Neoplatonic thought to both the Byzantine and the Islamic civilisations.
Whereas the Athenian school was strongly influenced by Iamblichus, and shared
his enthusiasm for metaphysics, ritual, and paganism, the Alexandrian school
concentrated instead on pure scholarship. But despite rivalry between
them, the relationships between the two schools were close and intermarriage
between their members common; and most representatives taught or studied in both
cities before settling down in one 
Proclus' works exerted a great influence on the next thousand years. They not only formed one of the bridges by which medieval thinkers rediscovered Plato and Aristotle, but also determined scientific method up until the sixteenth century, and through "Pseudo"-Dionysius gave rise to and nurtured the Christian mysticism of the middle ages.
In 529, Justinian closed the school of Athens. Damascius, the Aristotlean commentator Simplicius, and five other Neoplatonists set out for Persia, hoping they would be able to teach and continue there under Chosroes I. But conditions were unfavourable, and they were allowed to return to Athens.
Neoplatonism was the last of the great Hellenistic systems of thought to fall. Yet quite a lot of it did survive in Christian and Islamic form. In the West, Christian neoplatonism exerted a strong influence on philosophy and theology at least until the rise of scientific materialism in the seventeenth and eighteenth centuries. Neoplatonism had a profound effect on mediaeval Christian and Islamic mystical thought and on Jewish Kabbalah, Renaissance Hermeticism, the Cambridge Platonism of the 18th century, and 19th century Theosophy. In the more philosophical Islamic circles it is still going strong, appearing in the works of modern Islamic philosophers such as Fritjof Schuon and Sayyed Hossien Nasr.And through Theosophy its traces can be seen in the modern day "New Age" movements, and through Islam and Sufism (e.g. modern day writers like Fritjof Schuon) it made its way into the "New Paradigm" and transpersonal psychology arena.
Yet for all its influence, it is surprising how little it is in evidence on the electronic frontier of the Net itself (a few entries in on-line philosophy and theology encyclopaedias, and a digital version of Plotinus' Enneads.. It is hoped this project will go some small way to rectifying that shortcoming.
(Click on the refernce number to return to essay at that point)
 R. T. Wallis, Neoplatonism, pp.111f
 R. T. Wallis, Neoplatonism, p.1.
 Adolph Harnack and John Malcolm Mitchell, "Neoplatonism", in Encyclopaedia Brittanica, vol XIX, p.376, (Eleventh Edition, 1911); R. T. Wallis, Neoplatonism, p.94
[3b] R. T. Wallis, Neoplatonism, p.96
[3c] Ibid, p.95
 R. T. Wallis, Neoplatonism, p.93,
 Ibid, p.93, 123.
 Ibid, p.138