Xenocrates research paper due and don’t know how to start it? How about like this?
Xenocrates (396-314 B.C.E.) was a philosopher who succeeded Speusippus, who had himself succeeded Plato, as head of Plato’s Academy. We know of him and his work chiefly through a biography written by Diogenes Laertius. Dancy notes that Diogenes listed some seventy titles of Xenocrates’ works, but that none of these survive and that we have not even a single direct quotation from the mouth or pen of Xenocrates. Our knowledge of his philosophy is therefore largely inferential and based on the work of Aristotle who does not mention him even once by name. As head of the Academy, however, Xenocrates was prominent in the intellectual life of Athens and, as the general lines of what was occurring in the Academy are known, we do know something about this philosopher.
As a mathematician Xenocrates moved in a very different realm than what we usually associate with mathematics. His approach to numbers was metaphysical rather than computational. At the root of his conception of number lay the Platonic theory of forms, the idea that there are abstract entities, objects of the intellect, that lie behind and “generate” all the concrete objects that are objects of perception in the world. The former are more perfect than the latter. The idea of the “formal” circle is superior to any earthly circle that we could draw or see. A debate had arisen in Athens as to whether “formal numbers” were the same as “mathematical (computational) numbers.” Aristotle claimed that they were not. Apparently, Xenocrates claimed that they were and that the distinction between the two was therefore meaningless.
At the heart of this debate lay something akin to Pythagorean mysticism, which was fashionable at the Academy during the time in which Xenocrates was there, a mysticism which sought to explain metaphysical and cosmological phenomena in terms of the concept of number. This kind of “math” is very different from our own.