Xenocrates was the third headmaster of the Academy (339-314 B.C.), after Plato and Speusippus. He was a prolific writer of works on every field of philosophy; unfortunately, none has survived. He tried to achieve the organic integration of Platonism and Pythagorism.
The information we draw from(Lives of Eminent Philosophers IV, 14) add up to Xenocrates' birth in 396 B.C., in Chalcedon of Propontis. He moved to Athens at an early age, and became a member of the platonic circle. He accompanied , probably in his second expedition in 367. Xenocrates succeeded as head of the in 339; a post that he held until his death in 314 - upon which took office.
The anecdotal reports for his life focus on his exemplar temperance, and his parsimonious conduct in life. Athenians must have held him in high esteem, since they included him -a metic- in the official delegation to the regent of Macedonia Antipater, upon Athens' defeat in the Lamian war in 322 B.C.
According to the 76 titles that Diogenes registers inbibliography, we assume that he must had been a prolific writer. The titles alone (for the books are not extant) imply a broad range of interests - comparable with that of . His work consists mainly of treatises. He must have avoided dialogues due to lack of narrative traits. This assumption gains support by the alleged advice Plato gave to Xenocrates: "sacrifice to the Graces" (Diogenes, Lives IV, 6).
Sextus Empiricus reports (Adversus mathematicos I 16) that Xenocrates was the first to divide philosophy into three topics: Physics, Ethics, and Logic. At any rate, it is expected that Xenocrates, in being a devoted Platonist, should have started with the theory of being - which, in his tripartite division, falls under the Physics.
In the same vein as Speusippus, Xenocrates predicates his ontology on the two platonic principles of the "": the One, and the Indefinite Dyad. The name he used for the second principle, that of plurality, has not come down to us; it is presumable that he might have had kept the platonic term. Of greater importance is that he also designates the first principle -that is, the One- as Mind. This designation runs against Speusippus, but along the lines of Aristotl'es designation of the Unmoved Mover as "thought thinking itself", and God. The combination of the One and the Indefinite Dyad generates the superior among being: the Forms, and the numbers. To be sure, Xenocrates identified Forms with numbers - as Plato of the "unwritten doctrines" had probably done. But what is the nature of these numbers? Are they the platonic formal numbers, or the mathematical numbers? The general assumption, that we draw on various sources, has it that Plato asserted the distinction between Forms and mathematical numbers; that Speusippus ruled out the Forms, and affirmed the existence of numbers; and that Xenocrates identified Forms with the numbers. It is possible that Xenocrates tried to preserve the platonic Forms even though he reduced their broadness by rejecting them for artefacts, negative terms, and maybe particular entities. He only affirmed Forms for "the things that are always constituted according to nature". On top of that, Xenocrates, an advocate of , exalted numbers onto the highest ontological level, thus identifying them with the Forms.
According to Theophrastus, Xenocrates succeeded in developing a complete ontological system, for, within the entire Pythagorean and Platonic tradition, he alone "places all things somehow around the world-order, alike perceptibles and intelligibles, i.e. mathematicals, and again even the divine [things]" (Metaphysics 6b6-9, trans. Ross & Fobes). Aristotle is on the same page when he says that "some again hold that the Forms and numbers have the same nature, and that other things—lines and planes—are dependent upon them; and soon back to the substance of the visible universe and sensible things" (Metaphysics VII 1028B24-27). The exact logic behind this generation is unknown. However, it is more than certain that the first four numbers must have had something to do with it: number 2 was at work in the generation of lines; 3 in planes; and 4 in solids.
Xenocrates must have conjured up a peculiar atomism of mathematical nature, for many sources ascribe to him the existence of "indivisible" or "uncuttable" lines. He might have formed this idea based on the assumption that lines, which are by nature extended, cannot consist of points without extension. By the same token, he must have assumed that there are also plane "atoms", and solid "atoms". This sort of atomism is not incompatible with the platonic tradition: in, Plato explained the creation of material beings by reducing them to elemental triangles. We are aware that Xenocrates was drawing a lot on the interpretation of the Timaeus. This could have also been the source for his definition of the soul as "number that moves itself" (Aristotle, On the Soul 404b27-28). The cosmic soul has, according to Plato, two distinctive characteristics: first, it knows, and, second, it moves. Therefore, knowledge of the superior beings (that is, numbers) requires that the soul is of the same nature as them, i.e., a number. Second, if the soul is to induce motion, it has to be able to move itself.
Given that Xenocrates attempted a thorough systematization of Platonism, he could not leave out ethics (something that is also attested by the titles of his lost works). Nevertheless, the little evidence we have cannot account for an original moral theory. Along the platonic lines, Xenocrates opposes hedonism, and supports the life of theory and the cultivation of the soul (which he designated as the "demon" that each person carries).
The designation of the soul as a "demon" suggests that Xenocrates' Pythagorism was not confined to the significance of the numbers, but also extended to matters of religion. I have already pointed out that he called the first principle "Mind". What is more, Aëtius reports that he considered the One to be a male god, father of all, king of the heavens; and gave it the names "Zeus", "Odd", and "Mind". On the other hand, the Dyad was a female goddess, and a mother. He thought of the heaven and the stars as gods, and believed that demons lurk behind the material forces in the sublunary sphere. He assigned the symbolic name of "Athena" to the thought of the divine Mind; and the names of "Hades", "Poseidon", and "Demeter" to the elements of air, water, and earth respectively. It seems, therefore, that he developed a complicated theology, or demonology; and also that he considered a cryptic system of symbolic names to be a measure for the apprehension of reality. With regards to this point, Xenocrates' impact onand must have been significant.
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