George Spencer-Brown invented a non-numerical mathematical system ("calculus") that extends the power of Boolian logic to include situations involving self-reference. The system, "laws of form," involves only one sign, , (called a "cross" or "call"), roughly analogous to the act of crossing from one space to another or naming a space. In Boolian logic, self-reference produces paradox and contradiction. In Spencer-Brown's calculus, self-reference "recursiveness" is a fundamental and formative act at the root of the formation of specific spaces and times.
Spencer-Brown's calculus is particularly good at characterizing situations where ambiguity serves a constructive purpose, where point of view becomes an active and unstable element in acts of perception, reception, or interpretation. Boundary Language uses Spencer-Brown's system in limited but hopefully consistent ways to focus on the phenomenon of self-reference as an economic expansion of terms into a diagram of relationships known as the bolagram.
the boundary as an inverter
A boundary is any distinction or division that creates, in addition to a partition such as a border between two countries, makes possible a condition of depth, where the status of one part directly affects and is affected by the status of the other. An example would be the boundary between "inside" and "outside," opposites that depend on each other for meaning. This might seem to be a special case. The majority of boundaries seem to be more like property lines, which sometimes divide two "equal" parcels of land, sometimes separate private from public land; but the binary depth condition pervades even these situations when polar oppositions such as public/private, city/country, occupied/unoccupied begin to contribute their separate meanings.
For the purposes of developing a boundary language, Spencer-Brown's "cross" will be treated as an inverter switch, using the analogy of electric current. A traveler crossing a boundary moves from a space conceived as, for example, "inside" to "outside." Because inside and outside are always "in the same place," the boundary is called "continent." The boundary "inverts" the traveler's journey from being inside to outside and back again.
self-reference / recursiveness
This is true as long as the traveler and his itinerary don't do anything that would constitute a "feedback loop" to connect an "inside" space with an "outside" space without having a boundary in between. How could this happen? Consider some statements that use self-reference:
"All Cretans are liars."
This sentance has three erors.
[the third error is that the sentence has two errors]
Ignore everything inside this box.
It is also possible to have ordinary-looking geometric figures that are self-referential. The most famous example is the "Golden Rectangle," made up of a series of squares combined with other squares whose size is in a consistent proportion to each other so that the largest rectangle and the smallest are all the same proportion.
These examples of self-reference cannot be explained or even adequately described using ordinary logic or standard Euclidean geometry. The problem with self-reference ("recursiveness") is that it threatens the "continence" of boundaries that separate and insulate one "space" from another. We might imagine the situation in terms of a circle whose "output" is connected to its "input." Using the middle term of the Golden Rectangle, B, we can see that, like all other lines, it is the short side of one rectangle and the long side of another. A's relation to B is the same as B's relation to C because B works to maintain a "growth rate" through its double role as the short side of one rectangle and the long side of another. The growth rate is related to a constant ratio, known as Ø. It is not clear how Ø is self-referential until we look at the equation that calculates the value of Ø.
The rectangle's "growth ratio" is A:B = B:C. By letting B = 1, an expression can be written entirely in terms of A. A : 1 = 1: A-1. Multiplying both sides by (A-1) and subtracting A sets up the equation for division by A, which, translated to Ø, we get an expression where Ø is involved in its own definition. Every time we substitute the expression 1 + 1/Ø for Ø, we extend the recursion process.
Can we model this process? Spencer-Brown's calculus permits an extremely effective way of symbolizing this. The boundary as "inverter switch" can be shown as a division of a looped flow. The double role of B is directly compared to the Cretan Liar's double role, as a liar and truth-teller.
When we say that B has "two sides to it," the inside is connected to the inside by a process that loops back onto itself. The Cretan Liar who says "All Cretans are liars" is referring to himself and his credibility in the process of making a claim. Inside becomes outside becomes inside . . .
The problem can be masked if there are two (or any even number of) inverter switches, so that the self-reference loop that changes inside to outside is "corrected" by a switch which fixes the problem by inverting the states back to their original.
Whether or not there is a masking effect, however, the feedback process of self-reference "flips" the relationship between inside and outside.
In these situations, B is a switch that both inverts a value and inverts itself. We use the shorthand symbol, , to indicate this process. This is not at all an abstract or rare situation, but one that can be found in abundance. B symbolizes any phenomenon which can be seen from two points of view, which serves two opposed but linked functions, such as the growth rate of the golden rectangle. In art, such elements pivot the audience between perceiving them as a representation part of the intended "message" or as an artifact, the material or background support of the work of art's structure. The opposition of representation and artifact extends to the audience's "suspension" between a "real" condition (sitting in an auditorium, holding a book, etc. and a "realistic" imaginary world created by the work of art. The famous "willing suspension of disbelief" is, itself, a double negation involving feedback that initiates the experience of art.
map and journal
Boundaries are used, thus, in two different senses. First, in an attempt to stabilize experience, they are spatial symbols that separate and contain spaces: a "map view" that appears to be stable and unchanging. Experience, especially experience of recursive or self-referential situations, is more accurately described by a "journal" which records events as they happen. Where a map shows boundaries solidly in place, a journal records that the names of the spaces on either side flip back and forth in the process of self-reference. Various ways of characterizing this flipping process include a space that "re-enters" itself, a statement of B's inside and outside nature, or a direct statement of B's "oscillation" between two possible states, 'i' and 'j', represented by a "square wave" ().