Sunday, 25 January 2009

Plato's Indeterminate Dyad and the division of the unlimited

"We assume that the 'Indeterminate Dyad' is a straight line or distance, not to be interpreted as a unit distance, or as having yet been measured at all. We assume that a point (limit, monas, 'One') is placed successively in such positions that it divides the Dyad according to the ratio 1 : n, for any natural number n. Then we can describe the 'generation' of the numbers that follows. For n = 1, the Dyad is divided into two parts whose ratio is 1 : 1. This may interpreted as the 'generation' of Twoness out of Oneness (1 : 1 = 1) and the Dyad, since we have divided the Dyad into two equal parts. Having thus 'generated' the number 2, we can divide the Dyad according to the ratio 1 : 2 (and the larger of the ensuing sections, as before, according to the ratio 1 : 1), thus generating three equal parts and the number 3; generally, the 'generation' of a number n gives rise to a division of the Dyad in the ration 1 : n, and with this, to the 'generation' of the number n + 1. (And in each stage the 'One' intervenes afresh as he point which introduces a limit or form or measure into the otherwise 'indeterminate' Dyad to create a new number...

Karl Popper, Conjectures and Refutations, (Routledge, p. 122, n. 35)