The Calculus Interpreted for Sociology
In session two of the 1973 AUM conference with George Spencer-Brown on his Laws of Form he talked about his distinction between the primary arithmetic and the primary algebra emphasizing that the first is dealing with constants and the latter with variables. See here for the transcript. Constants are written as crosses, variables as letters a, b, …
Tatjana Schönwälder, Katrin Wille, and Thomas Hölscher in their deserving introduction into the Laws of Form draw the attention to Spencer-Brown’s illustration of this distinction in terms of a number theorist in arithmetics being interested in the individuality of numbers, whereas an algebraist would be interested in the sociology of numbers (Schönwälder/Wille/Hölscher 2004, p. 27).
If we take this illustration seriously, we end up with crosses indicating individual observers drawing their distinctions and variables indicating a knowledge they refer to to organize their relations of order and exchange.
As far as I can see this would define an altogether new methodology for constructing catjects. First of all, there can only be empirical examples of catjects. Any catject indicates the observation of observers drawing specific distinctions. There are no general catjects which should not be referred back to a particular observer stating a particular generality.
Note that “there are two constants in the calculus, a mark or operator, and a blank or void” (Laws of Form, 2008, p. 76). As any second order equation featuring a re-entry is unsolvable with respect to mark or blank, the oscillation between mark and blank, operator and void, might give the solution to Fichte’s and Hegel’s attempts in their Wissenschaftslehre and Philosophie des Rechts, respectively, to come up with a notion of radical, i.e., absolute, subjectivity. See Hegel 1821, §§ 4-7.
Any re-entry, however, assumes second-order observations. Thus we need to introduce a multiplicity, diversity, and heterogeneity of observers observing each other. They do not have to be human. This takes us beyond German idealism and dialectic materialism. It may even take us beyond sociology perhaps toward a new and ecological kind of culture theory.