Amicable numbers were known to the Pythagoreans, who accredited them with many mystical properties.
Though this narrative is a mixture of truth and fiction, it may be said with certainty that a thorough study of the philosophy of Peripatetics and Pythagoreans, Stoics and Platonists, brought home to Justin the conviction that true knowledge was not to be found in them.
He was taught first by his father Spintharus, a pupil of Socrates, and later by the Pythagoreans, Lamprus of Erythrae and Xenophilus, from whom he learned the theory of music. Finally he studied under Aristotle at Athens, and was deeply annoyed, it is said, when Theophrastus was appointed head of the school on Aristotle's death.
P. 43) In music he held that the notes of the scale are to be judged, not as the Pythagoreans held, by mathematical ratio, but by the ear.
It had been the dream of this man's whole life to supersede both forms of Christianity by a semi-pagan theosophy deduced from the writings of the later Pythagoreans and Platonists.
He was the author of notes on Cluvier's Italia antiqua (1624); an edition of portions of Porphyrius (1630), with a dissertation on his life and writings, described as a model of its kind; notes on Eusebius Against Hierocles (1628), on the Sayings of the later Pythagoreans (1638), and the De diis et mundo of the neo-Platonist Sallustius (1638); Notae et castigationes in Stephani Byzantini ethnica (first published in 1684); and Codex regularum, Collection of the Early Rules of the Monastic Orders (1661).
He owed something to Lucretius, something to the Stoic nature-pantheism, something to Anaxagoras, to Heraclitus, to the Pythagoreans, and to the Neoplatonists, who were partially known to him; above all, he was a profound student of Nicolas of Cusa, who was indeed a speculative Copernicus.
A famous problem concerning the cube, namely, to construct a cube of twice the volume of a given cube, was attacked with great vigour by the Pythagoreans, Sophists and Platonists.
C. 330) declares: " this was admirable " amongst the Neo-Pythagoreans " that they ascribed everything to Pythagoras; but few of them acknowledge their own works as their own " (de Pythag.
3), he was an adherent of the Heraclitean firedoctrine, whereas the Pythagoreans maintained the theory that number is the principle of everything.
This strange, exotic, ascetic view was adopted by some philosophers, and especially by the Pythagoreans, and so transmitted to Plato.
He gradually became a logician out of his previous studies: out of metaphysics, for with him being is always the basis of thinking, and common principles, such as that of contradiction, are axioms of things before axioms of thought, while categories are primarily things signified by names; out of the mathematics of the Pythagoreans and the Platonists, which taught him the nature of demonstration; out of the physics, of which he imbibed the first draughts from his father, which taught him induction from sense and the modification of strict demonstration to suit facts; out of the dialectic between man and man which provided him with beautiful examples of inference in the Socratic dialogues of Xenophon and Plato; out of the rhetoric addressed to large audiences, which with dialectic called his attention to probable inferences; out of the grammar taught with rhetoric and poetics which led him to the logic of the proposition.
32 was first proved in a general way by the Pythagoreans; but, on the other hand, we learn from Geminus that the ancient geometers observed the equality to two right angles in each kind of triangle - in the equilateral first, then in the isosceles, and lastly in the scalene (Apoll.
Halleius, p. 9), and it is plain that the geometers older than the Pythagoreans can be no other than Thales and his school.
He wrote a great work on the doctrines of the Pythagoreans, and tried to show that the successors of Pythagoras had made no additions to the views of their founder, but had merely borrowed and altered the phraseology.
It is an advance on this when Heraclitus 2 opposes to the eyes and ears which are bad witnesses " for such as understand not their language " a common something which we would do well to follow; or again when in the incommensurability of the diagonal and side of a square the Pythagoreans stumbled upon what was clearly neither thing nor image of sense, but yet was endowed with meaning, and henceforth were increasingly at home with symbol and formula.
In this they differed from the Eleatics and the Pythagoreans who thought in the abstract, and explained knowledge and existence in metaphysical terminology.
That such dietary restrictions were merely ceremonial and superstitious, and not intended to prevent the consumption of meats which would revolt modern tastes, is certain from the fact that the Levitical law freely allowed the eating of locusts, grasshoppers, crickets and cockroaches, while forbidding the consumption of rabbits, hares, storks, swine, &c. The Pythagoreans were forbidden to eat beans.
The avoidance of wine, therefore, by Rechabites, Nazirites, Arab dervishes and Pythagoreans, and also of leaven in bread, is parallel to and explicable in the same way as abstention from flesh.
The Pythagoreans and Orphic mystae so abstained all their life long, and Porphyry eloquently insists on such a discipline for all who "are not content merely to talk about Reason, but are really intent on casting aside the body and living through Reason with Truth.
As he designedly wrote nothing, and, with the aid of his pupils, kept his views secret, after the manner of the Pythagoreans, his philosophy must be inferred mainly from the writings of Plotinus.
Another feature of his works was the large number of excellent sentiments expressed in a brief proverbial form; the Pythagoreans claimed him as a member of their school, who had forsaken the study of philosophy for the writing of comedy.
These solids played an important part in the geometry of the Pythagoreans, and in their cosmology symbolized the five elements: fire (tetrahedron), air (octahedron), water (icosahedron), earth (cube), universe or ether (dodecahedron).
The doctrine of the Pythagoreans that the essence of justice (conceived as equal retribution) was a square number, indicates a serious attempt to extend to the region of conduct their mathematical view of the universe; and the same may be said of their classification of good with unity, straightness and the like, and of evil with the opposite qualities.
It was important, no doubt, to express the need of observing due measure and proportion, in order to attain good results in human life no less than in artistic products; but the observation of this need was no new thing in Greek literature; indeed, it had already led the Pythagoreans and Plato to find the ultimate essence of the ordered universe in number.
Thus, whereas the Ionians, confounding the unity and the plurality of the universe, had neglected plurality, and the Pythagoreans, contenting themselves with the reduction of the variety of nature to a duality or a series of dualities, had neglected unity, Parmenides, taking a hint from Xenophanes, made the antagonistic doctrines supply one another's deficiencies; for, as Xenophanes in his theological system had recognized at once the unity of God and the plurality of things, so Parmenides in his system of nature recognized at once the rational unity of the Ent and the phenomenal plurality of the Nonent.
In thus reverting to the crudities of certain Pythagoreans, he laid himself open to the criticisms of Aristotle, who, in his Metaphysics, recognizing amongst contemporary Platonists three principal groups - (1) those who, like Plato, distinguished mathematical and ideal numbers; (2) those who, like Xenocrates, identified them; and (3) those who, like Speusippus, postulated mathematical numbers only - has much to say against the Xenocratean interpretation of the theory, and in particular points out that, if the ideas are numbers made up of arithmetical units, they not only cease to be principles, but also become subject to arithmetical operations.
Of the Pythagoreans) that the human soul does not die with the body but is " born again " in new incarnations.
Zeno commenced, then, as a Cynic; and in the developed system we can point to a kernel of Cynic doctrine to which various philosophemes of other thinkers (more especially Heraclitus and Aristotle, but also Diogenes of Apollonia, the Pythagoreans, and the medical school of Hippocrates in a lesser degree) were added.