The Modeling Relation as a Complex System


Note: It is recommended that the reader first understand Rosennean Complexity, and the Rosen Modeling Relation prior to reading this page.


 

Decoding and Encoding Dictionaries

One of the most interesting aspects of the Rosen Modeling Relation is the fact that the Modeling Relation is itself complex, in Rosen’s sense of the word.

As we recall from the modeling diagram, a notable part of establishing congruence between two systems in a Modeling Relation requires that we find the appropriate encoding and decoding “dictionaries” to translate back and forth between the two systems, consistently. Without the proper encoding and decoding, there can be no comparing of the two systems, and no way to establish congruence between them.

Whence these dictionaries? They do not exist prior to attempting to establish the Modeling Relation. What is the procedure (if any) for generating the necessary encoding and decoding? Rosen explains, using an example of a Modeling Relation between two formal systems, F1 and F2:

“The first matter of importance is to note that, from the standpoint of the formalisms being compared, the encoding and decoding arrows…are unentailed. In fact, they belong to neither formalism, and hence, cannot be entailed by anything in the formalisms. The comparison of the two inferential structures, like F1 and F2, thus inherently involves something outside the formalisms, in effect, a creative act, resulting in a new kind of formal object, namely, the modeling relation itself. It involves art.” [1]

In other words, the Modeling Relation contains semantic elements [2] that cannot be replaced with syntactic elements alone. Another way to say this is that there are no rote or algorithmic methods for constructing such dictionaries. As we know, systems that embody such characteristics belong to a certain category; namely, the category of complex systems. In other words, the Modeling Relation is itself a complex system. [3]

 

 

Complexity and Objectivity

There is a longstanding obsession in science to equate objectivity with context independence and fractionability. This derives from a belief in a universe that is a vast mechanism, or simple system. [4] Such a view posits that 1) there is an objective reality “out there”, and 2) that it is possible to maintain a clean boundary between the subjective and the objective. However, this cannot be the case, as shown below.

Any attempt to divide the world into distinct subjective and objective portions founders almost immediately. The problem arises because the placement of such a division – intended as a separator between two distinct portions – is itself a subjective choice, and therefore the dividing line paradoxically also must be inside the subjective portion. This issue is not merely a matter of clever wording, but is just another version of a well-known set of paradoxes, or antinomies, that have plagued mathematics and logic since ancient times. Henri Poincaré identified the source of the problem as resting in impredicative definitions. The mathematician Bertrand Russell attempted to ban such definitions as follows: No set S is allowed to contain members m definable only in terms of S, or members m involving or presupposing S. [5] Unfortunately, this effort and similar efforts to banish impredicative definitions only resulted in tortuous (and ultimately unsatisfactory) constructions, such as Russell’s Theory of Types [6]; or, it resulted in enfeebled systems of mathematics. Ultimately, in order to be able to do real mathematics, impredicative definitions had to be allowed. As a result, most of mathematics cannot be formalized into a fully syntactic (rote) system; instead, real mathematics contains irremovable semantic aspects and impredicativities. [7] The consequence for objectivity is clear: the placement of such a dividing line between subjective and objective is an impredicative definition. [8] As a result, the mechanistic notion of objectivity fails.

 

 

The Observer and the Modeling Relation

The impredicativity in the above discussion, resulting from attempting to unequivocally divide the subjective from objective, does not mean that we must abandon science. Nor does it mean that we must abandon objectivity. However, it does mean that objectivity can no longer be equated with a simple, mechanistic view of the world. Science, in fact, has been facing this prospect for nearly a century, in the form of the “measurement problem” in quantum mechanics. [9] What makes it a “problem” is not so much the actual measurement process: it is instead a problem to the extent that the interactions challenge the belief in a mechanistic objectivity and its concomitant methodology of physics.

In terms of the Modeling Relation, the inherent creative, or semantic, nature of the encoding/decoding process is simply an example of the complex (non-mechanistic) interaction between the subjective self and the external world. Just as with the measurement problem, it is futile to attempt to minimize or ignore this inevitable situation. Instead, we must incorporate it into an larger framework – in other words, we must objectify the impredicativities. [10]

We therefore find that the construction of the Modeling Relation varies not only with the system and the model under comparison, but that it also varies with the role played by the observer. The exact formulation of the dictionaries will be, at least in part, dependent upon the creativity of the observer. Moreover, when modeling natural systems, to the degree that the observer (the modeler) can increase the ways in which he can interact with the natural system, and thus the number and variety of non-equivalent models that he can create, and generate encoding/decoding dictionaries that establish congruence, the modeler will be, in a sense, increasing the degree of complexity. [11] So, not only does the Modeling Relation involve creative acts, but those creative acts can significantly impact the nature of the modeling results.

The point of all this is not to conclude that we ought to give up on modeling as a task, any more than we conclude from the measurement problem that we ought to give up on physics. Quite the contrary. Instead, the point is that by acknowledging the presence of the semantic elements and the role they play in the process of modeling, we are objectifying the complex system as a whole. [12] In doing so, we maintain an objectivity; but one that is unlike the mechanistic version: namely, this objectivity allows a context-dependency, in the sense that we are acknowledging that each instance of a Modeling Relation will carry with it unique semantic aspects that cannot be fractionated from that specific Modeling Relation without destroying it. This may run counter to a typical notion of objectivity that can only abide context-independence, however it is no less rigorous.

This way of dealing with complexity in the Modeling Relation also hints at a way of dealing with complexity in other situations, even outside of formal systems, such as the measurement problem in physics. At root, a notion of objectivity tied to complete independence from an observer is symptomatic of a science that will inevitably run into paradoxes, such as the measurement problem. [13] But, as Frege and Russell found out, much to their dismay, no amount of predicative machinations could make the inherent impredicativities go away, and ultimately the impredicativities were grudgingly accepted and further attempts to strip them from mathematics largely ceased. The science of physics, on the other hand, has remained largely vexed, in this regard, since the prospect of admitting impredicative structures requires relinquishing some deeply held beliefs about the physical world and the kinds of models necessary to describe that world.

 


 

 

References & Footnotes

 

EL: Rosen, R. 1998. Essays on Life Itself. Columbia University Press

LI: Rosen, R. 1991. Life Itself. Columbia University Press

AS: Rosen, R. 1985. Anticipatory Systems. Pergamon Press

FM: Rosen, R. 1978. Fundamentals of Measurement. Elsevier Science

Eves, H. 1997. Foundations and Fundamental Concepts of Mathematics. Dover.

[1] LI p. 54
[2] EL p. 159: “They [encoding and decoding] introduce an obvious further semantic element into the model, over and above what semantic (e.g., nonformalizable) features may already be present in the model.”
[3] EL p. 138
[4] EL p. 83-86
[5] Eves, H. 1997. p. 264-265
[6] EL p. 91
[7] EL p. 92-93
[8] EL p. 139
[9] EL p. 139
[10] EL p. 93
[11] AS p. 83, 322
[12] EL p. 94
[13] EL p. 89

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