Some time ago, I had two posts (here and here) on infinite series, in which Hegel was also quoted saying that what is expressed in a ratio, can be only deficiently represented as aggregation (as infinite series).
While I tried to explain how that is so through an example, I didn’t try to give account of why it is so, which is of course more important that pointing to a fact. I will try to do this now, though in somewhat superficial way.
It has to do with the nature of the number…
Take for example some quantum… for example a distance in space. By itself that quantum is not determined as a number. It is not neither 1, nor 2, nor any other number by itself. It becomes number only in its relation to other quantum, to some other distance. So, it is their ratio which is 1:1, 2:1, 3:1, or any other. By, and within themselves both the distances aren’t any specific number. Only in their synthesis there is a number. Here we should be careful, and note that the fact that 1:1, 2:1 and 3:1, can be also presented by number as aggregate (i.e. 1=1, 2=1+1 or 3=1+1+1) in which the notion of ratio is left out, doesn’t mean that they are merely some kind of representations of that aggregate. On contrary, what is argued here, is that the number in its proper concept is always a form of ratio. So we can say that 1 is ratio of two qualities, 2 is also ratio of two other qualities and so on. Or said differently, the form of number as aggregate is abstraction from the richer concept of number as ratio. It shouldn’t come as surprise, then, that there will be a problem of expressing in a form of aggregate what is expressed in a form of ratio.
Further, concept of number as a ratio, also allows for natural connection to the rest of metaphysics in which it needs to relate to other “non-mathematical” concepts. And that is visible in the concept of measurement – there one quantum of some specific quality is compared with an another. The distance in space is quantum only as long it is compared with another distance, the amount of time is quantum only if it is compared with other amount of time. Abstracting from the specific quality of the quanta compared (which is implicit in the act of measurement) we are left with the concept of number as ratio. What should be kept on mind in the measuring then, is that it is not a quantum as something in itself, but it is properly understood as quantum as ratio. Such understanding is implicit in all numbers which appear for example in natural science, and there too every measurement needs to be seen as a ratio (except in the counting of discrete things, where the unit presents itself as ontologically basic, i.e. – one thing, and where we can’t change it to something smaller or larger without changing the nature of the thing. For example, there is not much sense in counting in half-planets, or double-planets).
I should notice here that Hegel’s account of numbers isn’t left in this kind of “outside” connection to measure, which as I said at the start is somewhat superficial. In Science of Logic, the movement from quantity to measure, and further to essence is seen as a necessary resolution of contradictions that appear in those concepts when taken as separate and abstract. In the movement in Hegel’s Logic, number doesn’t stop at its proper understanding as ratio also, but is sublated further (brought into) in the concept of inverse ratio, and ratio of powers, before it “develops” into measure, and further into essence. But I don’t think I understand enough this particular development to write about it. If you are interested, you can check David Gray Carlson’s paper Hegel and the Becoming of Essence. (I had problems opening it directly from the web, but it opened fine after downloading it).