Complementary Calculi: Distinction and Differentiation

George Spencer-Brown and Marin Mersenne

Headaches with complementary calculi

If two formal systems have a very close familiarity as a duality or even a complementarity, and are therefore to some degree nearly indistinguishable, but you nevertheless discovered in a strange situation of an insight a decisive difference between them. 

Then it might easily be possible, as in my case, that you get nightmares of endless oscillations and manifestations of something you don’t yet have access to, and what, as far as you guess, what it could be, you anyway wouldn’t like at all.

That’s what happens with the discovery of the complementary calculus of indication, a calculus I call a Mersenne calculus of differentiation and separation, in contrast to the Spencer-Brown calculus of indication and distinction. 

I have never been a friend of this calculus of The Laws of Form, therefore to get involved with its complementary calculus is no pleasure at all.

Obviously, to get rid of the headache with the CI and its ambitious and annoying celebrations, especially in German humanities, the best is to show, or even to prove, that there is a complementary calculus to the calculus of indication, too.

With that, the sectarian propaganda of the CI boils down to a strictly one-sided and utterly blind endeavour.

In-between I have written some papers dealing with the complementarity and applications of the concepts of the CI and the MC.

There might still be too much non-deliberated obfuscation involved, at least, some clear aspects of the new calculus of differentiation, CD, and its complementarity to the calculus of indication and distinction are now elaborated as far as it takes to get a primary understanding of the new situation.

Indication and differentiation in graphematics

Moshe Klein has given a simple introduction to George Spencer-Brown's calculus of indication (CI) as a special case of a bracket grammar. 

A context-free language with the grammar:
                                              S ⟶ SS|(S)| λ
is generating the proper paranthesis for formal languages. 



What was an act of a genius becomes an adhoc decision to restrict the grammar of bracket production.

Set the restriction of bracket rules to:
                                              (()) () = () (())
and you get the basic foundation of the famous CI as introduced by George Spencer-Brown. 

Nobody insists that this is an appropriate approach but it seems that it takes its legitimacy from the formal correctness of the approach.

Now, with the same decisionism, albeit not pre-thought by a genius, I opt for an alternative restriction,
                                             (( )) = ( ).
This decision is delivering the base system for a Mersenne calculus, interpreted as a calculus of differentiation, CD. 



I stipulate that both calculi, the CI and the CD, are complementary. And both calculi have additionally their own internal duality, delivering the dual calculi, i.e. the dual-CI and the dual-CD.

It will shown that, despite of its non-motivated adhocism, both calculi are well founded in graphematical systems, and are to be seen as interpretations of independent complementary graphematical calculi.

In fact, they belong, with the identity system for semiotics to the only two non-kenogrammatic graphematical systems of the general architectonics of graphematics. 

Abstract
The paper "Diamond Calculus of Formation of Forms. A calculus of dynamic complexions of distinctions as an interplay of worlds and distinctions” was mainly based on a deconstruction of the conditions of the calculus of indiction, i.e. the assumption of a “world” and “distinctions” in it. 

The present paper “Complementary Calculi: Distinction and Differentiation” opts for a graphematic turn in the understanding of calculi in general. This turn is exemplified with the George Spencer-Brown’s Calculus of Indication and the still to be discovered complementary Mersenne calculus of differentiations. 


First steps toward a graphematics had been presented with “Interplay of Elementary Graphematic Calculi. Graphematic Fourfoldness of semiotics, Indication, Differentiation and Kenogrammatics".

Graphematic calculi are not primarily related to a world or many worlds, like the CI and its diamondization. Graphematic calculi are studying the rules of the graphematic economy of kenomic inscriptions.


Graphematics was invented in the early 1970s as an interpretation of Gotthard Gunther’s keno- and morphogrammatics, inspired by Jaques Derrida’s grammatology and graphematics. 

Spencer-Brown’s calculus of indication has been extensively used to interpret human behavior in general (Niklas Luhmann). 

The proposed new complementary calculus to the indicational calculus, the Mersenne calculus, might not be applicable to human beings, but there is a great chance that it will be a success for the interaction and study of non-human beings, e.g. robots, aliens, and Others.

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