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The Calculus Interpreted for Sociology

June 3, 2012

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.

From → Catjects, Culture, Form

  1. Perfect! I think that’s exactly the point of difference between Luhmann & GSB on the one hand, and Peirce on the other: For Peirce’ pragmaticism, you’re not dealing with anything that can be true if it doesn’t include (the possibility of an ever realisable) direct reference, i.e. secondness. There is a version of something very like idealism in Luhmann/Brown there that Peirce’ fundamentally Aristotelian monism is opposed to — which, perhaps, also allows unknown individuals referring to a distinction so fundamental as a form that it can be referenced even without any pre-established joint concrete frame for such reference.

    • I see. So Peirce’s secondness is tantamount to Fichte’s and Hegel’s radical subjectivity of the Ich being necessary and empty (empty of firstness and thirdness) at the same time? Luhmann and Spencer-Brown do acknowledge this phenomenon of secondness — and regard it as being withdrawn within the elusive sphere of the self-reference of the observer. Their basic tact demands to leave it there and not to call it by any names.

      • I know far too little about Fichte to follow here; though from Peirce’ point of view, I’m not sure how we took the step from the necessity of reference to self-reference/subjectivity, which is *not* necessary for his idea of signification. I’m beginning to think I’m missing a central point here: Isn’t the withdrawal into self-reference precisely typical of a lack of direct or empirical reference (indicating a lack of secondness)?

      • One may call this the idealist insight, advanced by Pascal and Montaigne, avoided by Kant, formulated by Fichte and Hegel, dealt with by Marx, Kierkegaard, Nietzsche, Husserl, and Heidegger (did I forget somebody?😉 ): the self-reference of any individual being empirically (!) empty. See with respect to Heidegger who is almost explicit about this my German paper “Kalkuel des Seins“.

      • Thank you, I’m looking forward to reading that. 🙂

  2. Remember good old Kant? Never refer to names and places if you want to be sure of an attempt of a scientific explanation? Luhmann liked to quote this. I never checked where it actually is to be found. But this would mean that Peirce’s secondness is useful for a semiotics of the everyday world but misleading for a semiotics of sciences. Indeed, it would be secondness which protects everyday life from science, or at least from scientific explanations. For a sociology of the form of distinction this means that the pragmatist dimension of meaning is to be located elsewhere.
    My reading of algebra and arithmetic, moreover, is a little different. Arithmetic is dealing with unknown individuals, taken as constants and producing their indications as, for them, identical to the distinction itself, and so by no means contingent, but as variable for the second-order observer whose most prestigious gift in all matters of affair is the contingency of distinction, indication, and first-order observer along with them. Knowledge, thus, would be the knowledge of a constant being a variable, while, of course, paying all due respect to the individual observer going about her/his/its business.

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  1. Secondness and Individuality in a Sociological Calculus of Form « Signifying Media

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