Hegel on Arithmetic

Suppose we begin with a free group generated by a single symbol: \(M = \langle s, + \rangle\). We can get to this point by a variety of means, but we'll suppose it as our starting point. I think it is then interesting to ask, from where does multiplication emerge if we start from this vantage point? The answer is quite simple, many probably already know– all that is required is consideration of the mappings \(f : M \rightarrow M\): the endomorphisms. For any two endomorphism \(\phi\) and \(\psi\), we have a third endomorphism \(\phi \circ \psi\) as well as \(\phi + \psi\), and these operations behave in a manner so as to produce a ring.

Actually, more generally, any left \(R\)-module \(M\) can be viewed as a ring homomorphism from \(R\) into the ring of endomorphisms for \(M\). The point of all this today is that it suggests a relatively natural road to consideration of multiplication. In our simple free cyclic group, any homomorphism is determined by its image on the generator. Therefore \(s\) will be sent to \(\Sigma^n s = ns\) which is a natural analogue for multiplication by \(n\). Composition of these homomorphisms corresponds to multiplication of the numbers they correspond to, and likewise addition of them corresponds to the sum in an obvious way. So we've reconstructed the ring of integers.

Hegel had an alternate picture as to the construction of arithmetic, starting from his notion of quantity:

This definition warrants a bit of elaboration: what exactly does this mean? It's easier to read in the full context of the doctrine of being, but more or less the idea is that the individual entities in quantity, or the "being" underlying them, are unimportant. What is important is their limit, the division betwen them, but since there is no real difference between them when considering them from the point of view of quantity, it is not entirely proper to think of quantity as a "delimiting." Hegel singles out the nullification of difference in this form of determination (where things are separated and made what they are) when he says "a limit which is just as much no limit." In other words, quantity is a category of division, determination, wherein which the particular divisions between constituents are unimportant.

When we consider things as either being-for-self or being-for-other, we consider them as separate from the other, or distinct from the other, or the self, whatever way we have it. In quantity, the difference between this and that has been suppressed. It is essentially the "repulsion" or divisiveness of the beings in question which constitute the coherence of their quantity. It's important to note that we are considering some-things as "a quantity," not to say that we are considering quantity as merely a property of something else, to the extent that you find this distinction meaningful. When we consider a quantity as a quantity, the characteristics described above are evidenced.

When quantity itself is determined, for Hegel, an indifferent multitude is given structure by recognizing the continuity within the unities (singles) which comprise the multitude as well as the discreteness present in their separation, yielding a definite number. Continuity in this sense is self attraction, the lack of repulsion, self-sameness. All of this is to say that when we consider a multitude of discretely divided objects, each with continuity rendering them indistinguishable or undivided on their own, we find a definite number of objects. Hegel likes to proceed from here to construct arithmetic. Hegel identifies unity as the part of number which corresponds to continuity, and sum as the portion which corresponds to discreteness, the separation of otherwise indifferent continuities into sums.

So we have two aspects present: summation or combination into a discretely separated union, and unity or the presence of a continuous, self-same-ness. Unity in this sense corresponds not to additive unities in a group, but instead to the single free generator of the kind of group we mentioned in the introduction. Let's follow Hegel's exposition:

This is, to my reading, more or less an emphasis of beginning with a free structure: all that really matters is our choice of unity and our method of agglomeration.

Here we are introduced to the cyclic free monoid on our choice of unity. When Hegel says that "it is necessary that what we count up should be numbers already," he is referring to the capacity to consider sums as numbers, or as "no longer a mere unit." Addition is the inductive process by which the many inequal numbers are constructed, by discrete combination with a unity, something is incremented.

What Hegel posits here is uncannily similar to what was written in the introduction. Multiplication corresponds to equating a number to unity, and then considering the effect of this equation on numbers. This is more or less equal to taking the image of a homomorphism wherein which the generator (unity) is sent to some other member of the group. Hegel further comments on commutativity.

When we evaluate a number under its own multiplication map, or when we consider it as the discrete structure or sum, as well as treating it as the identity, we get the first nonlinear operation: squaring. This for Hegel is a natural concluding point, since it involves continuity and discreteness equated in the structure of number. I find it interesting to arrive at an entirely similar view of arithmetic in this way, despite the technological differences in mathematical machinery. I find this treatment amenable to my own natural views on numbers, of course. I think often of nonlinearity, since it involves such an insidious and serious difficulty in mathematics today.

Sorry, I'm still getting my blogging setup oriented for citations. I'm more used to a pure latex situation for this sort of thing! The citations were a little wonky, but in this informal article, I figured it'd be fine.

[1] George Frideric Hegel, A.V. Miller, 1969, Allen & Unwin, Science of Logic, https://www.marxists.org/reference/archive/hegel/works/hl/hlbeing.htm
[2] George Frideric Hegel, William Wallace, Shorter Logic, 1975, Claredon Press, https://www.marxists.org/reference/archive/hegel/works/sl/slquant.htm