3 / recursiveness extended

The trip from ABC across the spherical surface of the planet ij re-enacts the "moments" that are formalized in the bolagram. The diagram comparing transitive (ABC) with intransitive (ij) relationships explains how the values of the bolagram can be linked, further, with experiential and artistic concerns.

Translating Spencer-Brown's "cross" () into an "inverter switch" goes to the heart of the matter. Recursiveness, self-reference, self-similarity, and fractal forms involve connecting spaces that seem initially to be separated completely by boundaries that are reliably continent. The recognition of recursiveness typically occurs with a "flip" in a point of view, comparable to the phenomenon of anamorphosis, which involves a distorted image that, viewed from a particular angle, becomes recognizable. It is the speed of this transition from invisibility to visibility, unknown to known, that makes the flip so important in art. This is not a gradual, steady process but a revolutionary, catastrophic shift. In ritual, art, and cultural behavior, such changes have surprise value and, thus, ideological as well as esthetic power.

To understand the flip is also to understand the spatial and temporal conditions surrounding the creation and reception of works of art, which formalize this "journal" aspect of experience without oversimplifying into a rationalistic "map." In this extension of the planet ij diagram above, the main themes of motility, scale, and semblance will be cautiously attached to key elements. These translate mathematical conditions into themes encountered informally in non-mathematical contexts, and also afford critical vision its own anamorphic vision of structure concealed within works of art.

from the planet ij to the bolagram

Motility, scale, and semblance are not drawn from empirical experience but, rather, allowed to crystallize between the structures of self-reference and works of art, architecture, and landscape. The anchor point of these comparisons is the notion of the "recursive" cross or center, the point of the planet ij where the stone wall ceases to lengthen and begins to shrink as the kingdom expands. The ability of this cross to oscillate in value means that it is able both to separate spaces and times and to allow for an "illicit" re-entry into its own form. Any boundary drawn in self-referential conditions has this property, and it is useful to remember that all statements – philosophical, artistic, or otherwise – about the human condition are, because they are spoken and received by humans, inherently self-referential.

When works of art point explicitly to the recursiveness of art in general, the practice known as iconicity, they frequently create spaces or temporal situations that allow us to study the phenomenon in great detail. The window of Jefferies' apartment in Rear Window is, like the screen of the cinema, something that rises with the beginning of the film and lowers at the end. This synchronization makes every use of the window and its curtain between the beginning and end simultaneously symbolic of both events within the illusion of the story and the conditions of reception that affect the audience in general.

Motility can thus be a matter of the silence and immobility imposed on the audience during the process of reception; it can also be themes of travel, loss of freedom, delay, and disorientation with a story. Scale has to do with the realization of an "anamorphic" reality concealed within evident appearances, and also the shift in the point of view that this reality requires. In the audience's terms, it can be the balance of forces implied by the "willing suspension of disbelief." In terms of the work of art, it can be any conflict between alternatives such as rivalry, choice between alternative actions, levels of action (subplots), or the experience of contradiction or surprise.

Semblance has a special role in the dramatic work of art, where as "anagnorisis" (Aristotle's term) it stands for the concluding realizations made by the audience that bring the work to closure. Semblance can be literally expressed through themes of disguise, duplication, concealment, and escape. Or, semblance can relate to the large-scale patterns of relationships that affect the whole structure of the work. Semblance recognizes implicitly the recursive quality of the reality that has previously opposed one view to another. Like the ending of Oedipus Rex, the audience has the position of the gods who see that Oedipus was "destined" to murder his father and wed his mother through a rather ordinary sequence of causal events that insured, without any help from the gods, deus ex machina, that what would happen would happen.

When the diagram of travel on the planet ij is given characters, they initially take the form understood through action, reaction, and comprehension. The popular form of the murder mystery offers the handiest way of expanding this. Someone kills a victim, and someone solves the crime by identifying the killer. The killer is a part of a complex of causal conditions that dominate events with an invisible presence that can only gradually be mapped through the interpretation of clues.

The victim is a readably understood term. The detective or hero role has been given the coined term, "fictim," as a combination of the words fact, fiction, and victim. The hero takes the part, place, or cause of the victim, who is "immobilized." Fact and fiction are the ij formation, the anamorphic view produced by clues that are ignored by some (fiction) but seen as clues (facts) by the hero. Recursiveness of the boundary B has to do with the initial act that created a victim. The causes and agents are initially unknown, but clues are left behind. The clues conceal within them the factuality of the initial act. Thus, two devices for concealment, vent and poché, become important for translating B into . Vents are connections that, typically, involve no visual access or a twist of some kind. A cellar window is located at a high spot of a low space and, on the outside, is the low part of a high space. The figure of chiasmus (x), so common in Classical literature, comes into play, and the function of is to facilitate this double condition. A poché space borrows from the architectural term indicating concealed, interior spaces usually a part of the solid structure of buildings. But, the French word has affinities with the pocket (la poche) and physical poking (pocher) that relate directly to issues of anamorphy and discovery.

The "big other" is a structure of authority that "pulls strings" in the background, making characters into puppets without wills of their own. In mystery fiction, the big other can be the criminal who manipulates evidence, social relationships, and even the fictim (for example, Hitchcock's Vertigo, where the detective is hired by the killer). The big other is not always a villain, but he/she/it is always in the position of an authority which is unmasked or discovered through a process of anamorphic dis-covery. The authority's structure combines opposites in a way that is paradoxical and unknowable from the outside, but discoverable through the "tricky vent" that pokes its way past this inscrutability. The A position is labeled monstrum to connect to the ancient practices of divination that saw, in the anomalies of nature, signs from the gods. The connection between the highest order of meaning and misshapen or disfigured form fits into the anamorphic theme perfectly. And, the monstrum is often "monstrous" in the usual sense as well.

The reader is invited to go from one interpretation of the bolagram to others in order to discover particular shades of meaning that may be personally significant. The mathematical basis of recursiveness supplies a good experimental basis, a starting point for "testing out" various interpretive strategies. Examples of works of art, architecture, and place described in the boundary language web site extend this basis to areas where other issues are engaged. Cameo anecdotes, such as the Cretan Liar, the planet ij, and excerpts from texts such as Great Expectations and Chesterton's "The Queer Feet" offer laboratories for exploring nuances of boundary language.


see boundary language 'explain' Lewis Carroll's unique 'non-Booleian' logic