METHODOLOGY \ PRINCIPLES \ Digital Thought \312r
1: G. Buccellati, February 2008
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.
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SequentialityWe 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.
Schematic representation of sequentiality within argument
Multi-layeringIn 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.
Schematic representation of a multi-layered argument
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 concomitant searches with the greatest of ease.
The negative side of things is that the very ease with which one can dart from one topic to the next, and within a topic from one detail to another, may deflect our attention and blind us to the reality of the initial goal. Instead of reflection, we then have distraction.
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The polyhedral argumentThe 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.