In one form or another the problem known as that of the One (Hen ) and the Many (Polla ) pervades the whole history of Greek philosophy. According to Aristotle (Physics I, 2–3), it arose first in the pre-Socratic inquiry into whether there is one first principle or source—for example, water alone, or air alone—for things, or whether there is more than one first principle. If we are to avoid "coming into being out of nothing," we must either deny with Parmenides that there is any multiplicity arising from the first principle, or else we must suppose that somehow or other multiplicity is already present within the unitary first principle. If we choose the second supposition, we are faced with a problem that is no longer purely physical, namely, how one thing can also be many.
In the period after Parmenides it became clear that this problem arose at a number of different levels of thought, though it usually seemed natural to suppose that a solution at one level would solve the problem at other levels as well. It arose in relation to the phenomenal world in three ways: How one thing can possess a number of different characteristics, how one thing can change into another, and how one thing can have many parts. It arose, above all for Plato, as a problem concerning metaphysical entities such as forms—how unitary forms can be split up among many particulars, and how one form can possess a number of attributes. It arose in the theory of predication as the question of how a number of predicates can be applied to a singular subject. It arose in logic especially as the problem of how classification of many things under one head can be justified. It was also discussed (by Aristotle in the Physics ) in terms of the number of first principles, even when the question of their number was no longer seen from within a purely physical framework.
Only some of the more important treatments can be mentioned here. The earlier Pythagoreans subordinated One and Many to Limit and Unlimited in their table of opposites. It remained a standard problem for the Eleatics after Parmenides, and Zeno leveled some forty arguments against plurality, of which one or two survive. The Sophist Protagoras tackled the problem at the level of perception, and Gorgias's pupil Lycophron dealt with it as a problem of predication by banishing the word is in statements such as "Socrates is white"; Antisthenes, the Megarians, and the Eretrians, according to some, followed the same path. Plato discussed the question repeatedly, most explicitly in Part Two of the Parmenides and in the Philebus, but of vital importance is his approach to predication in the Sophist. His discussion was continued by Speusippus, and by Aristotle in the Metaphysics (I, 6) and in the Physics, where he propounded a solution in terms of "substrate," "privation," and "form." The Neoplatonists saw it as the problem of how the multiplicity of the world order can proceed from the ultimate absolute unity.
Virtually the whole of F. M. Cornford's Plato and Parmenides (London: K. Paul, Trench, Trubner, 1939) is concerned with the Hen/Polla problem in one way or another.
G. B. Kerferd (1967)