Relational Models and Measurement...

In the ongoing discussion about Rosennean Complexity, modeling, the modeling relation, "relational" models, etc... I thought it might be worthwhile to include some of Robert Rosen's own thoughts on the subjects at hand. The first 2 chapters of Life, Itself are very rich with discussions on these and other related topics. My father reveals a wry sense of humor as he discusses a few of his own surprises as he was starting out in science:

Robert Rosen wrote (Life, Itself, page xviii): I conclude these preliminaries with a personal word, regarding the genesis of these ideas reported below. I started from early childhood with a lust to do biology, which I retain. In pursuit of this lust, I acquired a great deal of mathematics; to me, this seemed natural, because as was said before ideas "fold"[similar to the way proteins fold and become active as they achieve the three-dimensional shape which brings active sites together which would otherwise be very far apart on a linear string or some other two-dimensional shape. He said, in Notes to the Reader, that "this particular volume is very heavily folded, indeed."] My main preoccupation in those days was to learn all about operator algebras, the language of quantum mechanics, which I thin believed would be sufficient. Almost by accident, I also absorbed a lot of Theory of Categories. It has turned out that the latter was much more important than the former.

My first steps in actually trying to do biology happened to be relational. I was early convinced that such considerations provided legitimate models, descriptions of natural systems that had as much right to be called models as any system of differential equations.

The trouble came when I tried to integrate relational and structural descriptions of the same biological systems. They did not seem to want to go together gracefully. Yet the MUST go together, being alternate descriptions of the same systems, the same material reality. Moreover, I needed both; biology seemed to require both.

They say that all science must start from experience. Mine was that relational models and mechanical models, drawn from physical analysis through reductionism, were not going together. That was a fact. My conclusion from that fact was that I was simply being stupid, or else there were some deep and essential things embodied in that fact. I was never able to rule out the first possibility, but the possibility of the second is what has led circuitously, in the course of time, to what is chronicled herein. In short, this is where I have been led by merely following the problem. It is the problem that imbues the path itself with whatever intrinsic logic is discernable in retrospect."

Relational descriptions are ones which would look, for example, at a function rather than at the structure from whence the functional behavior seems to be emanating. He asserts that relational descriptions retain meaning even when structural ones do not. Relational models allow science to model the entailment of a system, which then retains the aspects of organisms, such as "function" which are lost in mechanistic, purely structural models. As he observed; "Organisms sit at the other extreme of the entailment spectrum than mechanisms do; almost everything about them is entailed by something else about them." Admitting the existence of such a thing as "function" was and is considered "unscientific". My father argued that to exclude such notions from science based on artificial constraints of approach is what is unscientific:

(Page 4): "Let me begin with a few words about the relations existing between the mathematical universe and the perceptual one. It is a fact of experience, for instance, that 2 sticks plus 3 sticks equals 5 sticks. On its face, this is a proposition about sticks. But it is not the same kind of proposition as, say "sticks burn" or "sticks float". It differs from them in that it is also about something else besides sticks, and that "something else" takes us into the world of mathematics.

The mathematical world is embodied in percepts but exists independent of them. "Truth" in the mathematical world is likewise manifested in, but independent of, any material embodiment and is thus outside of conventional perceptual categories like space and time. These facts have indeed, from the time of Pythagoras on, spawned another profound dualism, a dualism between idealism (which at root is an attempt to extend the reality of number to the rest of the perceptual universe) and materialism (which is an attempt to include "mathematical reality" inside conventional perceptual realms).

Both science, the study of phenomena, and mathematics are in their different ways concerned with systems of entailment, causal entailment in the phenomenal world, inferential entailment in the mathematical. At root, the disagreement between "hard science" (quantitative) and "soft science" (qualitative) is precisely in their views about entailment, about what is entailed from a datum and about how that datum is itself entailed. Hence, at a sufficiently deep level, the controversy between them, and the dualism they represent, pertains to entailment itself, entailment in the abstract, free of any qualifying adjectives like "causal" or "inferential."

It is in this sense that I turn to the mathematical world in order to illuminate what it tells us about entailment. That is, I will be talking about entailment, rather than about mathematics, just as in the example above, I could talk about number while apparently talking about sticks."

He goes on to discuss the turbulence caused in the mathematical world by such things as the discovery of "Non-Euclidean Geometries" and Gödel's "Incompleteness Theorem" involving the basic fact that (p. 8 and 9): "syntactic truth in the formalization does not coincide with" (is narrower than) the set of truths about numbers, themselves. Or, put rather bluntly: "One cannot forget that Number Theory is about numbers."

"The fact that Number Theory is about numbers is essential, because there are percepts or qualities (theorems) pertaining to numbers that cannot be expressed in terms of a given, preassigned set of purely syntactic entailments. Stated contrapositively: no finite set of numerical qualities, taken as syntactical basis for Number Theory, exhausts the set of all numerical qualities. There is always a purely semantic residue, that cannot be accommodated by that syntactical scheme.

Gödel's Theorem thus shows that formalizations are part of mathematics, but not all of mathematics. Mathematics, like language itself, cannot be freed of all referents and remain mathematics. Any attempt to do this (i.e.; any attempt to capture every percept through a formalization of any finite set of percepts) must already fail in the Theory of Numbers.

On the other hand, Number Theory is still mathematics, still a system of inferential entailment in itself. It is only that it is not a purely syntactic system, not entirely a matter of word processing or symbol manipulation, independent of any external referent. In other words, Number Theory is not a closable, finite system of inferential entailment. These facts, as embodied in Gödel's Theorem, do not make us give up Number Theory as a part of mathematics nor even give up formalization as a strategy for studying certain kinds of mathematical systems. They express rather the limitations of formalization; it is not a universal strategy. If mathematics is a war against inconsistency, then that war is simply not as easily won as Hilbert believed.

The relation between Number Theory and any formalization of it concretely embodies certain features that bear essentially on the dualism with which we started (hard and soft science). The first thing to bear in mind is that both Number Theory and any formalization of it are both systems of entailment. It is the relation between them, or more specifically, the extent to which these schemes of entailment can be brought into congruence, that is of primary interest. The establishment of such congruences, through the positing of referents in one of them for elements of the other, is the essence of the modeling relation.

In a precise sense, Gödel's Theorem asserts that a formalization, in which all entailment is syntactic entailment, is too impoverished in entailment to be congruent to Number Theory, no matter how we try to establish such a congruence. There are thus qualities pertaining to numbers, and to Number Theory, that are missed by any such attempt; hence any entailments in Number Theory pertaining to these unencoded qualities are likewise inaccessible in the formalization. It would thus require, at best, an infinite number of distinct formalizations to capture all the qualities, and hence, all the entailments of Number Theory, in terms of syntax alone.

At this point, RR discusses how hard and soft science each take positions of extreme opposites and neither is correct. What they are really at odds over is the relation between them. Number Theory, as a "system of entailment" appears to be "soft" relative to its formalizations. So does that mean that Number Theory is therefore "immune" to mathematics? Hardly. It seems silly to suggest that Number Theory must be studied by other means simply because it is "more complex than" any set of formalizations of it. Furthermore, as he says on page 10:

"This is tantamount to abdicating to syntax alone the right to call itself mathematics and declaring thereby that nonsyntactical modes of entailment fall outside the scope of mathematics. The material counterpart to this line of reasoning is to exempt complex systems from the domain of science precisely because they are complex. This view, it seems to me, is equally mistaken.

As will be clear from the preceding discussion, it seems to me that the duality between "hard" or quantitative science and "soft" or qualitative science rests on an entirely fast presumption. It is not in fact a question of one versus the other, i.e.; a question of doing physics or not doing science at all. It is rather a relative question, of simplicity versus complexity.

There is, as yet, no comprehensive investigation of the ideas I have sketched in the course of the discussion above; they are too new. But it seems that such ideas, or ideas like them, are necessary in many ways I would in particular draw attention to the way such ideas ultimately rest on entailment alone, on systems of entailment in the material world (causal entailment) and in the world of formalisms or mathematics (inferential entailment), and on comparisons or congruences between such entailment systems. I have come to believe that the concept of entailment provides a reliable anchorage for the scientific enterprise itself, and I accordingly recommend it to your attention."