linear

Linear and non-linear

March 2006 - G. Buccellati

Concepts and terminology
Sequentiality
Multi-layering
Reflection
The polyhedral argument
Linear and poly-segmental
Intuition
The digital dimension
Pre-digital non-linearity
Bibliography

Concepts and terminology

A central issue of digitally based analysis of data is the linearity vs. non-linearity of an argument. The very fact that a proper term is missing to refer to non-linearity suggests that the field remains wide open for a clarification of the issue. As a contribution in that direction, we should reflect on some key concepts, distinguishing between the form and the substance of the argument.
While I suggest alternative positive terms (multi-linear, polyhedral) in lieu of the negative term "non-linear," I do nevertheless retain the term "non-linear" simply because, by virtue of its oppositional value to "linear" and on account of its popularity, it provides a more immediate understanding of what is meant – as long as one attempts, as I do here, to explain what this meaning is.

Sequentiality

We must distinguish between the logic of an argument and the form in which it is presented. If we use the terms "linear" to refer to the modality, we may use the term "sequential" to refer to the substance, of the argument. We may thus say that the argument will always be sequential, regardless of whether the form it takes is linear or not.
In the schematic rendering below, the intermediate steps a-b-c-d must be in that sequence for the argument to hold. In this representation, these intermediate steps are all on the same plane, which results in an aligned set of arrows. But the steps may straddle across planes, resulting in a multi-linear or multi-layered arrangement – where sequentiality remains nevertheless the rule. Multi-layering

In most cases, an argument includes multiple layers or registers, interlocked with each other. In this respect an argument is not only linear, but multi-linear, with the various threads running parallel, and yet calling at the same time for cross-overs from one linear path to the other. Thus, in the schematic rendering below, A is the main register, which runs linearly from beginning to end, and B and C are secondary registers which overlap either wholly (C) or partly (only the central portion of B being relevant to A). The argument still flows sequentially, but with data and inferences drawn from multiple planes. Reflection

The mechanism described here helps to understand what is the nature of reflection. To read in the sense of studying means more than being led passively along the sequential line proposed by the author. Rather, the reader is expected to develop parallel lines of inquiry and to draw on parallel data sets while following the argument presented by the author. There is, in other words, a parallel set of layers that the reader juxtaposes to those already offered by the author. This multi-linear function can be greatly enhanced when articulated digitally – which is precisely the great promise of the medium. Access to these multiple layers is dramatically facilitated by the medium, because of the way in which they are structured: there are unlimited lines of inquiry that are built on equally unlimited bodies of data. The task of a digitally minded scholar is to capitalize on these possibilities of a non-linear approach understood as multi-linear, but developing a newly constructed digital text.
The non-linear, or multi-linear, perspective is interesting in that it shows both the positive and the negative aspects of the medium.
The positive side is that the bracketing of layers is practically unlimited, that a suggestion to explore a parallel layer can be elicited by explicit or implicit associative mechanisms (hyperlinks, search functions, etc.), and that within each layer one can pursue develop into
The negative side of things is that the ease with which one can dart from one topic to the next, and even more frequently from one detail to another  The polyhedral argument

The adjective "linear" refers to the geometric figure of a line, i. e. a point moving along a fixed direction. The adjective "polyhedral" refers to the geometric figure of a solid bounded by polygons, such as the cube represented as 1 in the figure below. A linear argument that proposes to link conceptually points A and B has to travel along points c and d (2 in the figure). A polyhedral argument, on the other hand, travels directly, across the solid, from A to B (3 in the figure).
The power and demonstrability of a polyhedral argument rely on a prior knowledge of the cube and of its properties. It is only in virtue of this knowledge that A can arguably be linked with B, since the whole structure of the cube is presupposed, hence the linear possibility of the link (as represented under 2) is virtually known, even if it is not followed. It is also as a result of the prior knowledge of the underly-ing structure (represented figuratively as a polyhedron) that the linkage takes place along the shortest line. Hence the power: greater prior knowledge allows the linkage. And hence the demonstrability: one can refer back to the nature of the solid and show how the link between the two is possible. Such a knowledge is "polyhedral" because it does not rely solely on points c and d, but rather on the whole solid figure (the cube or polyhedron), of which c and d are as much part as A and B. Schematic representation of linear vs. polyhedral arguments

Without a supporting structure such as the cube, points A and B are floating in space, and their linkage (as shown schematically under 4 in the figure) results from a hit or miss shot in the dark.

Linear and poly-segmental

It is further worth noting that, strictly speaking, even the linkage represented under 3 remains linear, since the linkage is indeed a line. To reflect properly the situation, the terms "poly-segmental" and "mono-segmental" are equivalent to "linear" and "non-linear." The argument's process represented under 2 is linear, but consists of many segments. The argument's process represented under 3, on the other hand, is also linear, but, as it cuts across the polyhedron in the most direct way, it consists of a single segment, and is therefore more effective. Obviously, the degree of effectiveness increases in proportion to the complexity of the structure.
The Urkesh Global Record is built in such a way as to allow precisely an extensive use of polyhedral arguments. In practice, this relies on systematic use of hyper-links, which are generated automatically and therefore in unlimited quantities. The arrow represented under 3 in Fig. 10-1 stands for such a hyperlink.

Intuition

It is worth noting that the situation represented under 4 describes properly the nature of intuition. A connection between A and B may well be perceived through a sort of logical short-circuit, one that bypasses the argument and cannot therefore be demonstrated – at least, not on the basis of the original intuition. But we all know that in most cases it is precisely such an intuition that initiates the process of discovery. A proper polyhedral argument is one that, building on such an intuition, shows how the linkage is possible, and therefore arguable.

The digital dimension

 The term “non-linear” has achieved nearly cultic status in contemporary parlance. It evokes a sense of mystery, which gains in awe and power the less we try to explain it. It is, however, no different than the case of Moličre’s bourgeois who felt he had reached a pinnacle at the discovery that he was able to speak in prose...      We have been conceptualizing our world in a non-linear fashion at least ever since writing was first invented, some 5000 years ago. The earliest ledgers, the earliest maps are based as much as today’s ledgers and maps on linkages that are not linear.      Consider this cuneiform tablet, from about 2000 B.C. It lists individual animals given to certain individuals (single circle), then it gives subtotals by types of animal (double circle), and finally gives the grand total (triple circle). It is so simple, anyone can “read” the numbers. Thus the grand total is 5 times 60 (the large vertical wedge), plus 10 (the oblique wedge head), plus 4 (the smaller vertical wedges). The connection is clear among all the various steps. It is non-linear, because it presupposes conceptual jumps, evinced by the sequence and general arrangement. 