The Mysteries of Plato's Parmenides Dispersed: From musical intervals to contacts/"hapseis" to "logoi"

Abstract

Plato's Parmenides has long been considered the most difficult and enigmatic of Platonic dialogues, marked by seemingly contradictory arguments. We present an interpretation showing that the One of the Second Hypothesis (142b-155e) is an intelligible Being whose structure constitutes the philosophical analogue of a dyad in periodic anthyphairesis. First, by linguistic analysis and Proclus' Platonic Theology, we show that the One and the Being form an indefinite dyad satisfying the philosophical analogue of infinite anthyphairesis. We analyze Plato's initial introduction and eventual rejection of eristic numbers formed with unequal units from the infinite sequence of remainders (143c-144c). Second, we reveal that the presence of the One in the Being (144c-d, 138a) is achieved via contacts ("hapseis", 148d-149d). Based on the Pythagorean musical rule relating terms and intervals, a contact is identified with the ratio between consecutive anthyphairetic remainders. The circularity of contacts (138a) expresses Theaetetus' Logos Criterion for anthyphairetic periodicity, defining dialectical numbers with equalized units and establishing the finiteness of the dialectical number of parts. This framework resolves key Platonic enigmas: Plato's Indivisible Line is identified with the intelligible Being; the Sophist's assertion that "not-Being is a Being" corresponds to the equalization of the dyad; and the compresence of opposite properties separates intelligibles from sensibles. Finally, the self-similar Oneness of an intelligible Being refutes Vlastos' Non-Identity principle, resolving the Third Man Argument.