Plato's Theory of Number

Abstract

In a well-known passage Aristotle ascribes to Plato, or as some think to his followers, the dictum, ⋯ γ⋯ρ ⋯ριθμóς ⋯στιν ⋯κ ⋯νòς κα⋯ τ⋯ς ⋯ορίστον (Met. 1081a, 14), ‘Number is (formed) from the one (unit, monad) and the undetermined (indefinite, unbounded) dyad (duality)’, but what this apparently simple statement means has remained a mystery until modern times. In other passages Aristotle expands it to explain that the indefinite duality is a duality of the great and small, e.g., ὡς μ⋯ν οὔν ὕλην τò μ⋯γα κα⋯ τò μικρòν εἶναι ⋯ρχ⋯ς, ὡς δ' οὐσίαν τò ἔν. ⋯ξ ⋯κείνων γ⋯ρ κατ⋯ μ⋯θεξιν το⋯ ⋯νòς εἶναι τοὺς ⋯ριθμο⋯ς (Met. 987b20–22). ‘As the matter (of number) he posits the great and small for principles, as substance the one; for by the mixture of the one with them he says numbers (arise).’ This identification of the dyad with the great and small, elsewhere called τò ἄνισον (‘the unequal’) and τò ἄπειρον (‘the unbounded’), gives a first clue to its nature. In a notable article in Mind 35 (1926), 419–40, continued in vol. 36, (1927), 12–33, and amplified by D' Arcy Wentworth Thompson in vol. 38 (1929), 43–55, A. E. Taylor first suggested a connexion between the indefinite duality and the modern theory of continued fractions. In the light of subsequent research in the history of Greek mathematics it may now be asserted with a high degree of confidence that his conjecture was almost certainly correct; but it was then no more than a conjecture, and when he looked for confirmation he looked in the wrong direction.