Perseity and the coincidence of opposites

While it’s more common to argue for the coincidence of opposites by limit arguments, it is more interesting to argue for them by the perseity of predicates.

It’s clear that speaking per se about things is part of any development of thought. Bulls might attack red flags, but they don’t attack red per se but fluttering things, or, to use Aristotle’s example, it’s not isocoles triangles per se that have angles equal to 180 degrees but just triangles. Truth consists in rightly saying one thing of another, and “rightly” requires speaking per se.

But while Aristotle took geometry as his paradigm case of learning, persity unifies what is opposite in it: the straight and curved, discrete and continuous, the rational and the irrational. What exactly is π or (to list a quantity dearer to Aristotle) the golden ratio (φ)? For the ancients, these were, respectively, the ratio of the circumference of a circle to its diameter and a line cut such that the whole: larger part :: larger part : smaller part. But it seems pretty clear that these are not what these quantities are per se but just the first instances of the quantity that we learn. Both quantities pop up in all sorts of places that have nothing to do with ratios in circles or lines. But if there is anything that is π or φ per se – and this seems to be a condition for speaking of them scientifically at all, which we certainly can – then there has to be something transcending the discrete and continuous, and so a fortiori the straight and curved. Again, if we allow for a real number system then we have homogenized the rational and irrational.

Dekoninck tries to make a more or less sharp distinction between a science and a dialectical extension of that science, but I find this hard to believe. The progress of geometry and arithmetic prove that the thing they are about is not given from the outset, but only in a form that is closest to our imagination.