Robert Rosen was a Columbia University Theoretical Biologist. He looked at biology from the depths of math and physics. We’ve apparently been looking through the wrong end of the telescope, the fundamental question is life.
I may misstate to a degree where he was going but the flavor included stuff like this:
The universe gets more complex when you go down not less. Complexity is the rule, not the exception.
Through Rosen’s eyes it seems physics becomes a subset of biology and not the other way around, life is the general case.
Now I will put words in his mouth but when he pointing out that genotype is not phenotype in a very serious way it was like… you don’t get away with thinking the bumps on a atom will dictate sequence and sequence will dictate form and form will dictate function. No there is a disconnect at each step. Where goes would seem to undermine the entire theory of pharmacology and notions of life based on chemistry and yet he is using math and complexity to point out some of the contradictions especially with reductive approaches.
Cut and past from the Panmere site follows
http://www.panmere.com/?page_id=16
Rosennean Complexity
The purpose of this page is to give an accurate, yet concise overview of Robert Rosen’s definition of complexity, its criteria, and a hint at its ramifications. The latter constitutes the bulk of Rosen’s investigations into complexity, and it cannot be easily surveyed here.
Rosen’s definition of “simple” and “complex” systems
Conceptual steps towards Rosennean complexity
Alternate phrasings of Rosennean complexity
Degrees of Rosennean complexity
A side note on Turing Machines
Characteristics of simple vs. complex systems
Simple systems as constrained complex systems
Ramification of Rosennean complexity for physics
References and footnotes
Rosen’s definition of “simple” and “complex” systems
“A system is simple if all its models are simulable. A system that is not simple, and that accordingly must have a nonsimulable model, is complex.”[1]
Conceptual steps toward Rosennean complexity :
The following provides a very brief sketch of one way of approaching Rosennean complexity. I chose this sequence since I believe the steps I have laid out also somewhat approximates the historical development of Rosen’s concept of complexity, as I perceive it through the sequence of his books. Thus, this layout serves two purposes at once. For the sake of brevity, I have relegated quotes and elaborations to the footnotes.
Our understanding of the world, particularly in science, is based in the activity of modeling. [3]
Modeling is the act of establishing congruence between the elements and entailment structures of two systems (namely, between an object system and a model of the system). [4]
The possibility of “complexity”, in an intuitive sense, arises when a system acts in unexpected ways; that is, in ways that do not match the predictions of our models. [5]
This notion of complexity is therefore rooted in a comparison of a system with its model(s).
Rosennean complexity is thus a relational property [6], not an intrinsic property, of a system. [7,8]
When a single dynamical description is capable of successfully modeling a system, then the behaviors of that system will, by definition, always be correctly predicted. [9] Hence, such a system will not have any “complexity” in the sense above, in that there will exist no unexpected or unanticipated behavior.
Conversely, systems that require multiple partial dynamical descriptions – no one of which, or combination of which, suffices to successfully describe the system – are complex, in the sense above. [10]
Another way of connoting this distinction is to say that simple systems will always have a single “largest model”. [11]
A set of models that can be combined (or “refined”[12]) into one largest model can be equivalently stated as a set of models that are each Turingcomputable. [13, 14]
Therefore, the precise relation that circumscribes the category of simple (noncomplex) systems is this: all the models of the system are what Rosen calls simulable [15] (i.e., Turingcomputable [16,17]).
In contradistinction, the category of complex systems is thus precisely the category of systems that have a model that is nonsimulable [18] (i.e., Turingincomputable). [19]
We have arrived at Rosen’s definition:
“A system is simple if all its models are simulable. A system that is not simple, and that accordingly must have a nonsimulable model, is complex.”[20]
So, we can see that Rosen’s definition provides a precise characterization of our intuitive notion of complexity. The property of simulability of models is one that not only delineates the realm of simple systems, it also carries with it a multitude of corollary characteristics – some of which are mentioned further below. These characteristics have profound consequences for the study of systems in general, and for science, specifically.
A subtle, yet important, item to note is that complex systems are defined essentially in terms of what they not (i.e., simple systems). This is done by specifying as minimal a criteria as possible (having at least one nonsimulable model) in order to provide a precise, yet unrestrictive, distinction between simple and complex. The realm of simple systems are wellunderstood to the extent that all their models restricted to Turingcomputability, and thus the characteristics of simple systems are likewise restricted. [20a] In contrast, complex systems are, in a sense, openended: there is no apparent upperlimit to how complex a system can be constructed, and therefore, no apparent limit to what characteristics such systems might have. Accordingly, complexity itself is defined by Rosen in an openended fashion. [20b]
Alternate phrasings of Rosennean Complexity
Taken out of its context, Rosen’s definition of simplicity/complexity can be misunderstood or misinterpreted. Here I have attempted to create other “standalone” definitions of Rosennean simplicity/complexity that I consider more selfexplanatory, while still remaining true to Rosen’s definitions.
A system is categorized as simple if all its models are Turingcomputable. A system that is not simple, and that accordingly must have a Turingincomputable model, is categorized as complex.
Rosennean simplicity is a relational property of a system if all the models of the system are Turingcomputable.
Rosennean complexity is a relational property of a system if at least one of the models of the system is Turingincomputable.
Degrees of Rosennean complexity
Just as a system is deemed simple or complex based on its set of models, so too can degrees of complexity be discussed. Within the category of complex formal systems, we can, at least in principle, compare two complex formal systems in the following way: if one system is inadequate to successfully model all the entailments in the second system, then we can say that the second system is more complex than the first system. [21] Note that this is a relative criterion, and not an absolute one.
A sidenote on Turing Machines
There sometimes exists a degree of confusion concerning what a Turing machine is or is not capable of computing. This often arises through the use of phrases such as “more powerful” when referring to certain kinds of computing devices, which seem to imply a larger scope of computability, but in actuality refer only to speed or efficiency of computation. As such, everything from Cray computers to cellular automata to neural nets to quantum computers are unable to compute anything that is outside the (theoretical) scope of a Turing machine. Also, adding “stochastic” devices, multiple read/write heads, etc. to a Turing machine does not allow the enhanced Turing machine to compute anything beyond what was already considered computable. [22,23,24]
Turingcomputability encompasses the realm of purely rote or algorithmic processes. Such processes are completely syntactic, as they constitute merely symbol manipulation. In the realm of mathematical systems, such systems that are bereft of semantic elements and entirely describable by rote processes are called formalizable.
Characteristics of simple vs. complex systems
Complex systems have remarkably different characteristics than simple systems. It is the investigations into these characteristics that Rosen pursued in order to better understand complex systems, particularly as they related to biological organisms. Here is a very brief summary of some of the differences between the two categories:
Simple System Complex System
Fully predicative Contains impredicativities
Fully fractionable Contains nonfractionable aspects
Has a single largest syntactic model Has no single largest syntactic model
Has no complex models Has complex and simple models
Has computable models Has noncomputable and computable models
Has no closed loops of entailment Has closed loops of entailment
Fully syntactic Has semantic aspects
Synthesis is the inverse of analysis Synthesis generally distinct from analysis
Epistemology coincides with ontology Epistemology generally distinct from ontology
Simple Systems as constrained complex systems
It is not unusual for people to imagine complex system as being, in some sense, more rare (or nongeneric), than simple systems. Often this notion is tied up with the erroneous idea that it somehow takes lots of simple pieces put together to make a complex system. [25] However, if we view simple vs. complex system in other terms, we will see that, in fact, it is simple systems that are nongeneric. We will do this by considering the way in which simple systems are complex systems that have been burdened with constraints. It is enlightening if we examine the comparative list of characteristics above, and reword them as follows:
Complex System Simple System
Contains impredicative and predicative structures Only predicative structures allowed
Contains at least some nonfractionable aspects Only fractionable structures allowed
Exceeds having a single largest syntactic model Confined to a single largest syntactic model
Has complex and simple models Only simple models allowed
Has noncomputable and computable models Only computable models allowed
Has closed loops, and can have linear chains, of entailment Only linear chains of entailment allowed
Has semantic and syntactic aspects Only syntactic elements allowed
Synthesis can vary from analysis Synthesis must be the inverse of analysis
Epistemology can vary from ontology Epistemology must coincide with ontology
Seen in this way, it becomes readily apparent that simple systems are a very restricted subset of systems, and that they are exceedingly constrained in comparison to complex systems. As such, simple systems require an inordinately rare (and rarified) set of circumstances to exist. It is thus quite evident that it is specious to presume that we can envision the world as composed of simple systems, or that we can understand the world via simple models alone.
Ramifications of Rosennean Complexity for Physics
Turingcomputability encompasses the class of recursive functions. As it turns out, the formalism of statebased Newtonian physics is just such a recursive formalism. [26] What this entails is that statebased Newtonian physics (and by extension, relativistic physics and quantum mechanics as well [27]) are all within the realm of Turingcomputability. The result is that these versions of physics are essentially adequate for modeling only simple systems; and conversely, are inadequate for modeling complex systems. [28]
The notion that Newtonianstyle physics is capable of describing physical reality is thus a tacit claim that the universe is simple, in Rosen’s sense. A more explicit version of this claim derives from Church’s Thesis. This (unprovable) thesis, based on Church’s lambda calculus, stated essentially that if a function is effectively calculable in an intuitive sense, then the function is Turingcomputable. This thesis has been widely exaggerated over the years to include not just “calculable” processes, but any type of effective processes – including anything in the physical world. Such “strong” or “material”[29] versions of Church’s Thesis are explicit restatements of what is built into the formalism of Newtonian mechanics. A corollary of this thesis is that reductionism is a valid universal program in science. [30]
However, the attempt to equate effectiveness with rote algorithm fails to include even most of mathematics, to the extent that most of mathematics is not formalizable. This is another way of understanding the consequences of Gödel’s Incompleteness theorems in Number Theory. [31] So, already Church’s Thesis fails if it is extended only slightly beyond a very limited notion of “effective process”. In physics, the failure of material versions of Church’s Thesis shows up, for example, in the inability to solve Nbody problems in closed form or by reductionistic methods. [32] At root, the idea that the physical universe adheres to some variant of Church’s Thesis is no more than an errant belief. Along with casting off that belief, so too is abandoned any hope for the program of reductionism as a universal strategy in physics, since reductionism relies upon a hypothesis of universal fractionability, a feature which is restricted to simple systems.A challenge in creating a physics for complex systems is in creating a formalism that is outside of the realm of simulable recursive functions, and therefore, is not statebased, and is not fully computable, in the Turing sense.[33] The methodology of relational biology [34,35,36] and its relational models created by Rosen’s mentor, Nicolas Rashevsky, provides an important avenue. In Essays on Life Itself, Rosen outlines some further approaches to creating a “new physics”. [37] The upshot is that we need not discard our Newtonian physics completely just because of its inability to fully capture complexity; but rather we need to enlarge physics. As Rosen says, “It does not say that we learn nothing about complex systems from simple models; it merely says we should widen our concept of what models are.” [38]
http://www.panmere.com/?page_id=18
The Rosen Modeling Relation
“I have been, and remain, entirely committed to the idea that modeling is the essence of science and the habitat of all epistemology.”

Robert Rosen, Essays on Life Itself
Overview
Natural and Formal Systems
Entailment Structures
Building the Modeling Relation
Summary
References and Footnotes
Overview
This commitment to the central role of modeling, as embodied in the quote above, began for Rosen during his early days as a student, and eventually manifested in his development of the Modeling Relation. [1] To better understand Rosen’s statement above, we need to understand the Modeling Relation.
The Modeling Relation is described primarily in Rosen’s books, Anticipatory Systems and Life Itself. The former, in particular, devotes many chapters to defining, describing, and illuminating the consequences of the Modeling Relation. Therefore, what I describe herein only scratches the surface of the detail and depth to which Rosen examined the Modeling Relation.
In its most general terms, the Modeling Relation is a way to compare synonymies between a system in one form and another system in another form. In other words, to establish a congruence relation between two systems.
Natural and Formal Systems
What is a system? For Rosen, systems are of one of two types: natural, or formal. Roughly speaking, the distinction is that natural systems are selected portions of the external world, while formal systems are based in symbols, syntax and rules of symbol manipulation – a formalism, for short.
In discussing natural systems, we must first make the distinction between our self and the world outside our self. This is the dualism between the inner world and the ambience. [1a] This is not something we can know with certainty, but it is what we experience and take to be the case. What is called the ambience we further consider to be an actual world outside of us, the external world, where objective reality and objective phenomena exist – where material objects exist. [1b]
A natural system is some portion of the external world that is, at root, based upon a collection of our sensory impressions, or percepts [2]. This portion of reality is something we actively choose: there are no a priori guidelines to tell us what aspects of the external world constitute the boundaries of a ‘system’. [3] Not surprisingly, the remaining portion of reality outside what has been chosen as the system is called the environment. We take these percepts to represent, at least to some extent, qualities of the external world. Indeed, unless we do so, science itself cannot exist. Not only do these percepts present various qualities of the external world to our mind, we also appear to comprehend various relations between percepts – relations that our organizing human mind actively builds, but which also appear to reflect in some way actual relations in the external world. [4] This brings us to the definition of a natural system:
“Specifically, we shall say that a natural system is a set of qualities, to which definite relations can be imputed. As such, then, a natural system from the outset embodies a mental construct (i.e., a relation established by the mind between percepts) which comprises a hypothesis or model pertaining to the organization of the external world.”
“In what follows, we shall refer to a perceptible quality of a natural system as an observable. We shall call a relation obtaining between two or more observables belonging to a natural system a linkage between them. We take the viewpoint that the study of natural systems is precisely the specification of the observables belonging to such a system, and a characterization of the manner in which they are linked. Indeed, for us observables are the fundamental units of natural systems, just as percepts are the fundamental units of experience.” [5]
An important point to emphasize is this: a natural system is itself a kind of “hypothesis or model” of the external world. At first glance, this might seem to invoke some kind of circularity in the Modeling Relation, in the sense of having the system under study be itself a model; however, the situation is more subtle than this. In particular, we must distinguish between a natural system and a material system.
The first subtlety is to recall Rosen’s philosophy (from the quote at top) that modeling is the “habitat of all epistemology”. That is, modeling is not merely the province of explicit Modeling Relations, but of all our ways of knowing, including innate or intuitive modes of knowing. As Rosen says: “…indeed the modelling relation is a ubiquitous characteristic of everyday life as well as science.” [5a] Therefore, when we subjectively define a natural system, we are engaging in a mental modeling act. As noted in the quote above, the relations between percepts are mental constructs. Further, it should be noted that even percepts are not to be regarded as some kind of direct knowing of the external world; but rather, percepts are themselves a kind of mental model (“sensory impressions”) of qualities we can only deem as plausible to actually have their source in the external world. [5b] Taken together, percepts plus relations comprise a mental model of the external world.
The second subtlety is that it would be impossible to engage in building a Modeling Relation without first carefully delineating the 1) observables and 2) linkage relations of the natural system. Otherwise we would have no welldefined basis from which to generate other models or to compare models. This act of delineation is itself an act of abstraction: a willful selection of certain finite number of both observables and relations among observables from out of the entirety of possible observables and relations. Rosen laments that contemporary science tends to be so prejudicial in its choices when defining natural systems that the result has been to isolate just those subsets of material systems in the external world that adhere to a mechanistic Newtonian notion of the world. [5c]
The third subtlety relates to realization. Realization is the process of working from a formal system to a natural system. [5d] It is essentially the notion of going from a blueprint to a working material version. When realizing a system, the process does not involve every intimate detail of its material structure. Instead, it is sufficient that the resultant material version embody the criteria from the formal model (the “blueprint”). So, the congruence relation established is only between the elements and relations specified in the formal model and a corresponding certain finite number of observables and relations in the material realization. As noted in the second subtlety, those identified observables and relations are abstractions. Whatever other additional material characteristics the realization might have, to the extent that they do not affect the congruence relation, such additional characteristics are not part of the natural system. Therefore, a natural system is not synonymous with a material system. Instead, a natural system is some subjectively defined subset, or abstraction, of an actual material system.
The definition Rosen gives of a formalism, or formal system, is as follows:
“We shall understand by a formalism any such “sublanguage” of a natural language, defined by syntactic qualities alone. That is, a formalism is a finite list of production rules, together with a generating family of propositions on which they can act, without any specification or consideration of extralinguistic referents. Thus, a formalism, as a fragment of natural language, could be “about” something (i.e., endowed with extralinguistic referents), but it need not be. A formalism, by its very nature, carries with it no “dictionary” associating its propositions with anything outside itself. It is propelled entirely by its own internal inferential structure, as embodied explicitly in its production rules. These and these alone determine the relations among the propositions of the formalism, which we have called inferential entailment.” [6]
The most typical type of formal systems encountered – particularly in modeling in science – are mathematical in nature. However, mathematical systems are only one class of formalisms. Other types of formalisms, meeting the conditions above, can and do exist.
Entailment Structures
The last sentence in the quote above refers to our next consideration: entailment structures. Entailment structures are certain types of relations between elements of a system. In the case of formal systems, entailment relations come in the form of implication or inference. That is, to say that “P entails Q” in a formal system, is to say that “P implies Q”, or that “Q is inferred from P”, via the production rules of the formalism. [7]
In a natural system, entailment takes the form of causality. [8] We understand intuitively, and experientially, that there does seem to exist entailment relations between phenomena in the external world. As such, we take causal entailment to be a real property of the external world, and hence, an entailment structure that is embodied in natural systems.
In both types of systems, entailment structures play a fundamental role in the organizational structure of the systems. Without inference rules, a formal system could generate no consistent set of propositions; without causal relations, the external world would have no discernable order at all. Therefore, any attempt at modeling must, of necessity, incorporate modeling of the entailment structure of the system under study.
Building the Modeling Relation
As stated at the outset, the idea of the Modeling Relation is to establish congruence between two systems; specifically, between the elements of each system and between the entailment structures of each system. By accomplishing both of these aspects, the orderly nature of one system can be be made to correspond to another system, to the extent that the two systems have a degree of correspondence.
In science, for example, the typical scenario involves creating formal models of aspects of the external world. Such formal models are often mathematical in nature, and the validity of these models is based upon the congruence of the system elements and the entailment structures, and also involves the processes of measurement and prediction. Rosen gives a generalized description of the way in which we relate a formal system (here called “F”) to a natural system (here called “N”):
“The essential step in establishing the relations we seek, and indeed the key to all that follows, lies in an exploitation of synonymy. We are going to force the name of a percept to be also the name of a formal entity; we are going to force the name of a linkage between percepts to also be the name of a relation between mathematical entities; and most particularly, we are going to force the various temporal relations characteristic of causality in the natural world to be synonymous with the inferential structure which allows us to draw conclusions from premises in the mathematical world. We are going to try to do this in a way which is consistent between the two worlds; i.e., in such a way that the synonymies we establish do not lead us into contradictions between the properties of the formal system and those of the natural system we have forced the formal system to name.”
“Another way to characterize what we are trying to do here is the following: we seek to encode natural systems into formal ones in a way which is consistent, in the above sense. Via such an encoding, if we are successful, the inferences or theorems we can elicit within the formal systems become predictions about the natural systems we have encoded into them; consistency then means that these predictions will be verified in the natural world when appropriately decoded into linkage relations in that world. And as we shall see, once such a relation between natural and formal systems has been established, a host of other important relations will follow of themselves; relations which will allow us to speak precisely about analogy, similarity, metaphor, complexity, and a spectrum of similar concepts.” [9]
This is displayed graphically in Figure 1 below:
The Rosen Modeling Relation
Figure 1
What we see is that the two systems are related via the encoding and decoding arrows. Encoding is the process of measurement: it is the assignment of a formal label (such as a number) to a natural phenomenon . Decoding is prediction: it is the taking of what we generate via the inferential machinery of the formal system into representations of expected phenomena . Additionally, the arrows for inference and causality represent the entailment structures of their respective systems .
As Rosen described in the above quote, the modeling relation provides us with a way of ascertaining congruence between the natural system, N, and the formal system, or model, F. What determines successful congruence is that the diagram, as a whole, commutes. That is, such that the numbered arrows meet the condition:
(1) = ( 2) + ( 3) + (4).
This means that our measurements (2), when run through the inferential machinery (3) of our model, will generate predictions (4), which will agree (when verified) with the actual phenomena (1) occurring in N. [10]
It bears mentioning that any encoding (measurement) from N to F is an abstraction . As such, the features of N that are represented in F will be condensed as a result of the particular encoding schema (and measuring instrumentation). There is nothing new in this; however, it is often overlooked that the process of doing measurement and experimentation involves a willful act of abstraction. [11]
Summary
The Modeling Relation thus provides us with a methodology for studying one system in terms of another system. This framework is quite remarkable and flexible. In the example above, we saw a Modeling Relation between a formal system and a natural system, something quite typical of activities in science; although the modeling process has rarely been so explicitly examined as Rosen has done.
Using the same concepts as in the above example, a Modeling Relation can just as readily be established between two formal systems. Or it can be established between multiple natural systems that encode into one same formal system, where both natural system are then analogs of each other. [12] Or a natural system might have Modeling Relations with multiple formal models, and one could then consider the prospect of Modeling Relations between those formal models. And so on.
Finally, it might be useful to note that the modeling relation diagram is essentially a categorytheoretic diagram. [13] The systems are essentially categories, the entailment structures are essentially morphisms, and the encoding and decoding arrows are essentially covariant and contravariant functors, respectively.
Now, certainly it will not be possible to force just any system to be congruent with another system in a Modeling Relation. It requires that we find systems that, together, will satisfy the conditions above, given the appropriate encoding and decoding “dictionaries” to translate back and forth between the two systems, consistently. Since these dictionaries do not exist prior to attempting to establish the Modeling Relation, nor is there any rote method for constructing such dictionaries, they must be created. In this way, modeling is as much an art, as it is a methodology. [14]
As a result, the Modeling Relation contains semantic elements [15] that cannot be replaced with syntactic elements alone. In other words, the Modeling Relation is itself complex. [16] This aspect of the Modeling Relation is discussed further in “The Modeling Relation as a Complex System“.