Comments on a Paper by Landauer and Bellman


A paper written by Christopher Landauer and Kirstie Bellman entitled “Theoretical Biology: Organisms and Mechanisms“, and published by the American Institute of Physics, critiques some aspects of the work of Robert Rosen in his book Life Itself. Landauer-Bellman come to the strong conclusion that “the Mathematics is incorrect, and the assertions remain unproven (and some of them are simply false).”

However, Landauer-Bellman are mistaken in their interpretation of Rosen’s work, to the extent that the arguments they present are not even relevant to Rosen’s claims. Because of the sweeping conclusions made by Landauer-Bellman, it is worth discussing the errors in their interpretation and arguments.






The Main Argument

Landauer-Bellman attempt to show that the relational modelling approach used by Rosen employs faulty mathematics and that the “published proof”, as they put it, is incorrect. They take as their point of departure the particular relational model created by Rosen and called an (M,R)-system model, shown here[1]:

This class of model is indeed a good choice for their argument, because it exemplifies for Rosen central characteristics of a living organism:

From this point of view, then, a cell is (at least) a material structure that realizes an (M,R)-system….Making a cell means constructing such a realization. Conversely, I see no grounds for refusing to call such a realization an autonomous life form, whatever its material basis may be.” [2]


In section 2 of their paper, Landauer-Bellman begin:

“In this section, we describe the basic Mathematical problem, and show that Rosen’s solution is simply wrong, by providing solutions to the distinguishing equations in several areas of Mathematics. In the next section, we continue with some other applicable Mathematics.

We start with the description from [Essays on Life Itself] of a model of living systems, his (M,R)-systems, since it gives the clearest description of what Rosen intended to do. Then we explain the equational distinction from [Life Itself], and discuss its implications. Finally, we describe our example solutions to the equations from three areas of Mathematics, to show that equational distinction does not work.” [3](bold added)


At this point, Landauer-Bellman have already erred. Landauer-Bellman have mistakenly interpreted the relational mappings in the (M,R)-system diagram as entirely equivalent to equations (see bolded in quote above). They spend p. 61 describing how the “equational formulation” is wrong. They conclude the details of their discussion:

“Since metabolism and repair change an organism, we should actually expect the function f that comes out of p(b) to be different that the one that goes into b(f) and transforms a to b, since the process is actually unfolded in time (this is the “helical” argument). This modeling approach, however, projects all of time into a single point.” [4]


That is, they argue that the specific f that is acting as the function of metabolism clearly cannot be the same f is being output by the repair process at a given moment in time. Hence, the logic behind the model must be faulty. However, relational models are precisely atemporal representations of relations. This is why chapter 5 of Life Itself is conspicuously entitled “Entailment Without States: Relational Biology”. Near the beginning of chapter 5, Rosen clearly states:

On the formal side, we shall see that the inferential structure characteristic of relational biology is much richer than, and at the same time very different from, the formalisms we have considered heretofore. Our systems are assigned no states, no environments, and there is no recursion.” [5]


This is later reiterated. After describing the basics of relational models, Rosen contrasts the relational approach with the Newtonian, state-based approach:

In the relational approach, on the other hand, the situation is quite different. As I have developed it so far, there is no time parameter, no states, no state transition sequences. There are only components (mappings), and the organizations, the abstract block diagrams, which can be built from them.” [6]


Thus, it is entirely erroneous for Landauer-Bellman to interpret relational models, such as the (M,R)-system, as being an “equational formulation”. They do not address their argument to the intended relational model, but instead to what they interpret as a set of equations. Therefore, their main argument and their main conclusion are entirely invalid.


On a more technical level, it has been pointed out [7] by Aloisius Louie, one of Rosen’s student’s, that the category used in the relational models of Rosen is the category Ens, whose objects are sets and whose morphisms are mappings among its objects. There are no additional mathematical requirements placed upon the objects in Ens. In particular, there is no requirement that all objects in Ens can be treated in an “equational formulation”. It is this very nature of Ens which facilitates its use in the representation of the state-less, recursion-less, atemporal approach of Rosen’s relational biology.

As Louie points out, Landauer-Bellman’s argument confuses necessity with sufficiency. Rosen’s argument is not that every interpretation of an (M,R) diagram must be a model of a cell; rather, only that there exists an (M,R)-system which is a model of a cell. Landauer-Bellman’s argument, which is directed against an interpretation of the (M,R) diagram as an “equational formulation” entirely misses this point.

Finally, it should be noted that although Rosen discusses the (M,R)-system model in Life Itself and Essays on Life Itself, perhaps the most complete paper on the (M,R)-system model is here[10]. This paper is not referenced by Landauer-Bellman.



Complexity and Computability of Models

Landauer-Bellman appear to also misunderstand Rosen in section 2.6 of their paper, after summarizing their main argument, when they say:

“More importantly, this [Rosen’s approach -TG] is probably not the right approach. After all, claims that complexity is non-computable contradict claims that formal systems are computable, since computability is only defined for formal systems, and cannot be proven or even properly defined for non-formal systems. There are many such confusions in the literature.” [8]


Again, by misinterpreting relational models as predicative equations, Landauer-Bellman mistakenly assert that these models are thus computable (i.e., Turing-computable). They further suggest that Rosen is claiming that noncomputability is a property of the organism, rather than of the formal model, and insinuate that Rosen must be confused, since it is (quite rightly) nonsensical to speak of computability of non-formal systems (such as organisms). On the contrary, it is Landauer-Bellman who appear to be confused. Rosen is quite clear that computability criteria apply to the models, not to the material system to which those models belong:

I call a material system with only computable models a simple system or mechanism. A system that is not simple in this sense, I call complex. A complex system must thus have noncomputable models.” [9]




Analytic vs. Synthetic Models

In section 3 of their paper, Landauer-Bellman spend 3 pages discussion analytic vs. synthetic models. They state at the beginning of the section that “we describe some other difficulties with the Theoretical Biology program as stated…”. [11] It is unclear to me what are the specific “difficulties” or why Landauer-Bellman perform this exercise since it appears they end up restating the same conclusion that Rosen makes in chapter 6 (“Analytic and Synthetic Models”) of Life Itself. Namely, that there are generically far more analytic models than synthetic models of a system:

“It is easy to show that a Synthetic Model is also an Analytic Model with trivial kernel, and that an Analytic Model with trivial kernel is also a Synthetic Model. However, there are Analytic Models that are not Synthetic Models (necessarily ones with non-trivial kernel).”[12]


As best I can discern, Landauer-Bellman seem to provide this argument because they mistakenly think Rosen is arguing that a “largest model” is the universal case: that analytic models are always only the inverse of synthetic models. Nothing could be more wrong. Rosen remarks near the outset of his chapter on analytic and synthetic models:

In a sense, it is the thrust of this entire work that this hypothesis of analysis = synthesis must be dropped. Above all, it must be dropped if we are to do biology, and hence a fortiori, it must be dropped if we are to do physics. By dropping it, we enter a new realm of system, which I call complex, and which in certain sense needs to have no synthetic models at all. The distinction between relational and Newtonian models of natural systems will become crucial here, because as we shall see, the former extend to the realm of complex systems, while the latter cannot.” [13]


Landauer-Bellman do not discuss the consequences of dropping the “analytic = synthesis” hypothesis, even though they (apparently) agree with Rosen that such a hypothesis is unwarranted by purely formal considerations. Rosen demonstrates in Life Itself that Turing-computability and synthetic models go hand-in-hand, and that it therefore follows that analytic models which are not also synthetic models are noncomputable (i.e., Turing-incomputable). Thus, if material systems exist which indeed possess analytic models that are not also synthetic models, then that system possesses noncomputable models and is what Rosen would call a complex system.

The (M,R)-system relational model is just such a noncomputable model: it is state-less, recursion-less, and embodies an impredicative entailment structure. As Landauer-Bellman found out, when one attempts to interpret such a model as if it were computable, in predicative terms (the “equational distinction”), they end with a infinite regress (their “helical argument”).




The paper by Landauer and Bellman appears to fail to correctly understand: 1) the fundamental concepts of Rosen’s relational modeling approach, 2) Rosen’s use of computability criteria of models, 3) the import of the consequences of Rosen’s discussion of analytic vs. synthetic models, as well as Rosen’s concept of complexity. Their paper misinterprets these concepts and their arguments are directed against those misinterpretations. As such, their paper inflicts no damage to Rosen’s actual concepts, approaches, or arguments. But by its gross misstatements, this paper does do harm to public perceptions about Rosen’s work.



References & Footnotes


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

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

FM: Rosen, R. 1978. Fundamentals of Measurement and Representation of Natural Systems. Elsevier Science

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

LB: Christopher Landauer and Kirstie L. Bellman. “Theoretical Biology: Organisms and Mechanisms”. AIP Conference Proceedings Vol 627(1) pp. 59-70. September 2, 2002

[1] LI ch. 10, EL. ch. 17, and Rosen, R. 1972. “Some Relational Cell Models: The Metabolism-Repair System”. Foundations of Math. Biology Vol. 2, 217-253. Academic Press.
[2] EL p. 263
[3] LB p. 60
[4] LB p. 61
[5] LI p. 109
[6] LI p. 134
[7] See post in list archives
[8] LB p. 65
[9] EL p. 325
[10] Rosen, R. 1972. “Some Relational Cell Models: The Metabolism-Repair System”. Foundations of Math. Biology Vol. 2, 217-253. Academic Press.
[11] LB p. 66
[12] LB p. 67
[13] LI p. 154

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