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An Observers’ Mathematics

March 8, 2012

As systems come and go, network is sure to stay. That is why systems is a functional term, network, a structural. Systems focus on differentiation, reproduction, and functional equivalence (Niklas Luhmann), network emphasizes identity, control, and structural equivalence (Harrison C. White). As functions mobilize via delimiting comparison, structures ensure that alternatives are right at hand.

Thus, it is systems and network together that describe how action selects among “means” (Bruno Latour) enacted by experience. We propose George Spencer-Brown’s (Laws of Form, 1969) notion of form and his calculus of indications as a way to keep systems and network apart within one approach which uses both of them to describe and explain how action and experience come about. Form combines actuality (marked state), potentiality (unmarked state), and negativity (in two versions: cross and re-entry, i.e., operation and indeterminacy). It needs systems to bring about recursive eigen-values (Heinz von Foerster). It needs network to risk synthesis (Louis H. Kauffman). And it needs culture to build up a memory which never stops to oscillate with respect to contingency.

This project is about the mathematics of form. Spencer-Brown’s calculus is still to be discovered and tested within sociological theory and empirical research. Within mathematics’ axiomatic stance and interest in multiple operation (Alain Badiou) we may search for an understanding of how sets of indications are numbered, ordered, and exchanged by observers associated among each other who more or less inclusively exist by counting themselves among the indications they are producing. Positive and negative languages rely on each other to code events and structure situations via both cognition and volition (Gotthard Günther).

Spencer-Brown’s mathematics is the first to pay explicit attention not only to operation, self-reference, and time, but also to the observer. That turns his calculus of indications into an analytical instrument of utmost interest to sociology. Sociologists are observers observing other observer who at their turn observe observers as well. Reality is the eigen-value which emerges as that kind of concatenated second-order observing begins to observe and mistrust itself. That is when distinctinctions are re-entered into their form to be able to watch that form and to reflect on its performance.

Note that Spencer Brown’s notion of form is akin to Claude E. Shannon’s notion of information as the latter one defines information as the selection of a message out of a set of possible messages. Dropping the assumption of that set of possible messages being technically given, we move from a mathematical theory of communication to a self-referential theory of communication. The next step is to flesh out both the mathematics and the sociology of such a self-referential theory of the social. Doing without the assumption of technically given sets of possible messages means to turn from probabilistics to modalities and self-reference. Stochastics still reign, yet they are invested by systems’ functions, by network’s structure, and by observers’ distinctions. What kind of mathematics is up to this view of the problem of the social?


Let us try the following axiom: There is no consciousness (William James). There may even be no communication (Niklas Luhmann). There are observers observing observers (Heinz von Foerster). They number, order, and exchange states of world inside a universe of modalities (George Spencer-Brown and Gotthard Günther). Accounting for preferences, optimization (or, satisfaction), and equilibrium, as in economics, is a special case thereof.

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