Cutting up Space, Part 2: The Laws of Form

G. Spencer Brown's fabulous book on the calculus of indications, The Laws of Form, begins thus:

The theme of this book is that a universe comes into being when a space is severed or taken apart. The skin of a living organism cuts off an inside from an outside. So does the circumference of a circle in a plane. By tracing the way we represent such a severance, we can begin to reconstruct, with an accuracy and coverage that appear almost uncanny, the basic forms underlying linguistic, mathematical, physical, and biological science, and can begin to see how the familiar laws of our own experience follow inexorably from the original act of severance.

Who would not want to continue reading this book after that enticingly fabulous introduction? Especially someone like myself who is interested in both language and urban space (specifically psychogeography and poststructuralist theory).

While the above paragraph sounds very philosophical, the book is nevertheless a mathematics-based book also, which includes Boolean algebra, the algebra of logic. This mostly appears in the form of symbols and simple formula, most of which is beyond me. However, I do understand the philosophical material, al lot of which is redolent of non-dual eastern philosophical discussion - and indeed G. Spencer Brown was a student of R. D. Laing (many of the terms Brown uses are translatable into psychoanalysis: for example, condensation and compensation). The concepts can also be applied to theories of self and other, of which Laing has written.

The book was described in its day as a mathematics of consciousness and became useful as a springboard into the theory of autopoiesis (Maturana and Varela) and in second-order observation.

The basic symbol of the theory is the cross: the 'mark' or cross'.

It is a mark that forms a boundary, separating one space from another by creating a distinction. An inside and an outside, or a 'this' and everything else that isn't 'this, if you will. It also assumes an observer of the differentiation. Various actions can then be taken that involve crossing the boundary - once or twice, crossing then returning, etc. - and the result of carrying out these transactions. This mark, the cross, when it exists creates a marked state. When it does not it is an unmarked state, or the void, or nothingness (hence it's relationship to non-dual Eastern religion).

Already you may be able to see the potential relationship with deconstruction and its interest in the binary oppositions inherent in language. Not as much as you might expect has been written about the relationship between the two, although Niklas Luhman has done so, since he is a proponent of second-order observation theory too. There is also some parergonal logic in the Brown text. As discusses by Jacques Derrida in The Truth in Painting (in very simple terms, where the framing of the artwork creates both a division but also a bridge, this concept also being applied to theory itself).

My potential interest in the book in relationship to its uses in psychogeography are around the ideas of inclusion/exclusion, directions of observation, urban planning and zoning, crossing boundaries, self and 'the other', etc. I'd be interested in any mathematical psychogeographers who might have some more thoughts about it.

I'll sign off with this brilliant passage from the notes at the end of the book:

[In order for the universe to have the function of seeing itself] evidently it must first cut itself up into at least one state which sees, and at least one other state which can be seen. In this severed and mutilated condition, whatever it sees is only partially itself. We may take it that the world undoubtedly is itself (i.e. is indistinct form itself), but, in any attempt to see itself as an object, it must, equally undoubtedly, act so as to make itself distinct from, and therefore false too, itself. In this condition it will always partly elude itself.

NB: I have a beautiful second edition copy of the book. I had wanted a copy for a long time so treated myself to one for completing my Masters. It has a great 1970s-esque cover, shown below:

Please click here for Part 1 of this blog: Cutting up Space, Part 1: L = S - [l + c + i = e + p]