Some tweets on arithmos ("series")

One can only appreciate the significance of arithmetic for Greek thinkers if one understands that arithmos is fundamentally "series", and not "number" in the way we think of it, blinded by our numerals. When Greek thinkers strove for an arithmetic of things, it was a question of relations, of all the kinds of relations possible for things, all the ways in which things can enter into series, series that are not abstract, but appropriate (oikeios) to those things. What we think of as "mathematics" is just the most abstract form of this, the bare seriality of quantity. This bare seriality allows us to say of any three things, "three"; but the other one was never fully realized, perhaps. This truly philosophical "arithmetic" would have been a comprehensive science of relations. (Jacob Klein is on the trail of this, it seems, with what he terms "theoretical logistic".)

One starts from the simple game of what things "belong together", and how. At the furthest limits, one arrives at what I termed "bare seriality": the threeness of three things with no other relationship. That's the basis of what we think of as arithmetic, which is purely quantitative. Plato, however, proposes a qualitative arithmetic. To us, the notion of a qualitative arithmetic is almost meaningless. And yet just as quantitative mathematics branches off from this wider "arithmetic", so too does ontology. But we might say that ontology goes all the way in a qualitative direction, just as mathematics goes all the way in the quantitative. The philosophical arithmetic I am interested in recovering operates prior to this split, as a pure science of relation.