Fair warning: This post probably isn’t going to be a lot of fun unless you’re interested in George Spencer Brown’s Laws of Form, and then in adding to rather than taming its degree of abstraction by comparing it to Peirce. Ok? Ok.
Last week, Dirk Baecker traced an inspiration from a metaphor in George Spencer Brown’s explanations at AUM: Following Schönwälder et al. (2004:27) and then going beyond their description of the trope, he takes the — at first metaphorical — analogy of arithmetic and algebra as mirroring individual knowledge and sociological research seriously. While the arithmetician knows each number personally, and the arithmetic of form knows the two values of marked and unmarked by heart, variables allow the algebraist to focus on the “generality of numbers”, as GSB puts it: “He is more interested in the sociology of numbers that applies, whatever individual numbers come there.”
Note, and this is where my interest begins, that GSB calls the algebraist an algebraist, but does not name the arithmetician as I just did; instead, he refers to a number theorist as the algebraist’s opposite number, drawing on another analogy that sets up the arithmetic of form as the complement to the theory of numbers. While this second analogy is not metaphorical but firmly grounded in formal considerations, it still constitutes a ‘step of confusion’, that is, a simplified narrowing of equivalent but differently formed arrangements, albeit a productive one. This involves a contraction or expansion of reference, as described in the fifth canon of the LoF: For while the theory of numbers can be developed from the arithmetic of form, the plethora of numbers usually known to each number theorist goes well beyond the two values derived from the binary form. This is a necessary step for the prima facie plausibility of the metaphor: The number theorist, like the person among individuals, knows all these individuals and their differing values, whereas the algebraist deals with the more general forms that treat those values as unknowns. But crucially, a purely algebraic arrangement that is not interpreted as the starting point of a theory of numbers, but as a catject in sociology, only has two personal acquaintances, as taken from the form of the first distinction: That of the marked state, and that which is not. I think it is no coincidence that the Laws of Form deal with the introduction of this state under the headline of Knowledge (LoF ch. 2). Let’s assume, for the moment, that this is meant to indicate exactly the same sense of knowledge as when GSB describes his metaphor:
A number theorist knows each number in its individuality. He knows about the relationships it forms, and so on, as an individual, as a constant.
Dirk Baecker now continues along exactly those lines, and unfolds the narrowed reference like this:
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.
This touches on two problems that, to me, haunt sociological interpretations of the Laws of Form: One being the question of the observer, which GSB does deal with, but never exclusively for the arithmetic: He introduces a notion of reflexion in the algebra, and then uses it to discuss where “we imagine ourselves in” certain spaces of a formal arrangement when he re-unites arithmetic and algebra (ch. 8: Indicative space). This gives special credence to the idea that the concreteness of the observer can best be described by comparing arithmetic and algebra; but the evocative phrase that ‘imagines oneself’ within certain spaces of an arrangement at once completely visible to the operating individual leads to a number of questions.
The other problem is the concept of haecceitas. One way of thinking of the Laws of Form is as a continued genealogy reaching back to Charles Sanders Peirce’ Existential Graphs. The calculi appear immediately similar in notation, arranging spaces in successions of ‘cuts’, i.e. distinctive forms, that operate according to the game rules of a Boolean calculus and translate the envelopment of one expression within a cut or mark as the functional equivalent of formal negation (whatever that means in each interpretation). But that same similarity also immediately exposes the difference: Peirce, moving from Essential to Existential Graphs in a way not unlike (rhetorically, not formally) Spencer Brown’s move from arithmetic to algebra, soon introduces additional lines of identity or ‘tridentity’ (probably most clearly in the “Prolegomena To an Apology for Pragmaticism” in the Monist Series recently edited by Elize Bisanz). Tridentity operates on the question of a case defined purely by its formal aspects versus a case identified as being subjected to several formal descriptions: It adds Secondness, most easily recognized as indexicality, to the repeatable firstness and regular thirdness of the form. Not just a person born in Stagira and living in Athens and practicing philosophy, but this one.
Haecceitas, I believe, can be directly tied to this version of the problem of the observer: The observer reaches out, in addition to having defined the shape and rules of the form, and tries to point to the empirical thisness of what his form describes, sometimes over-extending his reach, sometimes attaining direct reference. An index, a pointed finger, that adds the indicated thisness to the formal whatness and howness. We can think of that Stagirean philosopher as a formal option; and then we can point him out, making that pretension of having found him that denotes the aesthetics of empiricity, here he is — Aristotle, by proper name, by index, by secondness.
So can we deal with both problems in this interpretation of catjects, separated along the lines of arithmetic and algebra as by their concern with empirical observers, and their stated generalities? I believe so, and I want to suggest that in doing so, we might better understand the semiotic dimension of that sense in which the algebraist, as opposed to the number theorist, knows individual values. Start by asking the calculus of form where its secondness has gone, thus re-introducing a pragmaticist view, and proceed from there.
The individual knowledge of an uncompromisingly concrete observation expressed in a catject by its arithmetic aspects cannot be that of the number theorist, drawing on a generative principle yielding infinite individual numbers to know. The principle is valid and known, as a principle, to the Laws of Form; but the LoF-algebraist’s focus is on the two individual values that precede the theory of numbers. Consider the difference in terms of managing contingency: The number theorist, in this sociological interpretation, is individually acquainted with a plethora of individuals constituting an enormous space of contingent actions that might take any of a great number of different forms. The algebraist of form knows of these as well, but his intimate knowledge concerns only the question of crossing: He is directly acquainted with being in a given form, or beyond it. What he recognizes as direct knowledge is the moment when a formal consideration is found, or refuses to be found, in the thisness of the individual pointed out.
This makes for two very different perspectives on contingency in terms of direct empirical knowledge on this side of sociology’s algebraic generality, both of which have a role to play: For the number theorist, contingency can be summed up as not knowing what any of the many, many individuals might do in any of the many, many circumstances, choosing from the many, many options they have at their command. Who knows, e.g., whether one of them might offer a certain communication to another, and whether that other accepts? By contrast, for the LoF-algebraist, the contingency appears as co-emergent with a specific form, which might or might not simplify to the marked state. That co-emergency concerns the form of a specific possible communication on the one hand, and the question of its realization on the other. For the algebra, an individual then does not pre-exist its contingent choice; what is individually known is the emergence of that individual and the form it fulfils at the same time. The communication that does take place produces its contingently involved individuals; their contingent choice is to be, and be in the form (and not, in terms of focus, to choose this form among many others).
Which then adds another angle on the role of variables in the algebraist/sociologist/generalist use of the Laws of Form: As the ‘particular observer states a particular generality’, or indeed generalisation, it is at those places where variables appear in his arrangement of forms that he reaches out to find instances of direct reference. He does not necessarily know, in his particular empiricity preceding generalisation, the individuals that might be involved in making its generality true. Rather, he has intimate knowledge of the moment in which particular individuals complement the form — and likewise intimate knowledge of that other moment, in which they fail to do so.
Rephrasing the idea of the individually acquainted number theorist as transferred to sociology’s particular precedent, then, we might consider in each case a pertinent question about the individual knowledge we might claim: Does it seem to describe an individually observed option among infinity? Or does it boast knowledge of fulfilled, as well as of failing, specific forms, general in their formality, and concrete in a quest for empirical reference? Which mask of contingency are we coping with by suggesting a given generalisation: That we know or don’t know what each of the contingent opposite numbers will do? Or that we know or don’t know whether a certain form and empiricity come together in a description, or act of finding, or reference?