A 2006 paper attempting to criticize some of Rosen’s arguments, entitled “In Defense of Mechanism” by A.J. Wells , was recently brought to my attention. The paper makes a lengthy series of erroneous arguments based on misinterpretations of Rosen’s arguments. Wells begins with a list of statements drawn from Rosen’s Life Itself . He then proceeds to critique the validity of these statements. As this goes on for many pages, I shall not bother to refute each minor point, only some major ones, for the sake of some degree of brevity.
Defining Mechanisms and Machines
Here are Wells’ Point 1 and 4:
1. A system, natural or formal, is a machine if it simulates, or can simulate, something else. Simulation is what machines do. (7B, 7D, 7E)
4. A natural system, N, is a machine if and only if it is a mechanism such that at least one of its models is already a mathematical machine. (8B)
Wells then goes on to remark (p.47):
Turning now to the status of the argument itself, Point 1, which takes simulation to be the key criterion for distinguishing machines from other systems, is simply wrong. Most machines do not simulate. Think of the functions of the everyday machines that surround us: refrigerators cool things, cars transport us from place to place, scissors cut things, pianos produce music, and so forth. None of these machines is a simulator. Machines that do simulate, of which computers are the most obvious examples, are a special, highly organized class. Rosen’s (1991) definition of a machine is, therefore, far too narrow.
However, Wells fails to understand that Rosen is constructing a specific and unequivocal definition of “machine” – Rosen is not attempting to utilize some vague colloquial definition of “machine”. As Rosen states, “In a nutshell, the present section is devoted to establishing a characterization of machine in terms of the notion of simulation.” [LI 7B]) Further, it is difficult to see how Wells misunderstand this, since Wells’ point 4 is actually Rosen’s explicit definition of “machine” [LI 8B]:
Let us give a name to this class. We shall say that a natural system N is a mechanism if and only if all of its models are simulable.
We shall further say that a natural system N is a machine if and only if it is a mechanism, such that at least one of its models is already a mathematical machine.
On the face of it, these seem peculiar ways of characterizing mechanisms and machines from among the class of natural systems. But this peculiarity stems only from my expression of these concepts in terms of the models of N, rather than try to talk about N itself. This is all that Natural Law entitles us to do. We have so far in this volume nothing that transcends those entitlements, and I shall do nothing in what follows that transcends them; on the other hand, it is my aim to utilize precisely these entitlements to the full extent. It is my main contention, in fact, that contemporary science, as a whole, does not do this; by the time we are done, this fact and its consequences will be quite apparent.
In fact, my characterization of mechanism will be seen to be nothing but Church’s Thesis, explicitly and fully manifested in its true material garb. As I noted earlier, the intent of Church’s Thesis was initially to characterize vague notions of “effective” calculability and algorithm, and hence programmability in Turing machines. But because it is so easy to equivocate on the word “machine” and because everything that happens in the material world must be considered effective, Church’s Thesis has always been tacitly supposed to have a physical content as well….My definition of mechanism above merely characterizes the world in which this equivocation is legitimate (i.e., within which Church’s Thesis becomes elevated to a Law of Nature) and investigates the consequences.
As usual, Rosen is very explicit and precise about the terminology he uses. Indeed, in doing so, he is also precisely characterizing the mechanist view, and the consequences of the mechanist view.
Largest Model of a Mechanism
Wells Point 5:
5. If N is a mechanism, it has a unique largest model. (8C)
Wells contends (p. 50):
The more specific points to be made concern the status of the argument that purports to establish Point 5, the claim that every machine has a unique largest model. The argument in section 8C of Life Itself is faulty. It takes the form of a reductio ad absurdum. Starting from the assumption that N is a mechanism, but that the category of all its models C(N) contains no largest model, Rosen claims that we can find an infinite sequence of increasingly refined models. Because N is a mechanism, all its models must be simulable and each of them must have a program of finite length.We can then form the intersection of all the models and, by hypothesis, this is also a model. Unless the sequence of models terminates after a finite number of iterations, the model formed from the intersection is larger than any of the other models. Because it is simulable, it must also have a program. The conclusion of the argument, according to Rosen, is the following: “We thus end up with a countable family of distinct programs, each of which is a distinct word of finite length on a finite alphabet. This is clearly impossible” (p. 205).
In fact it is perfectly possible. Thus even if the premises of the argument were correct, which is questionable, the conclusion would not follow. The decimal representations of the integers, for example, are the elements of a countably infinite set of distinct words of finite length on a finite alphabet and integers are commonly used to represent programs. It is also always possible to extend a program without altering its functionality by adding new instructions that do nothing (cf. Rogers, 1967, p. 22, Theorem III). This possibility contradicts the assertion that a machine must have a unique largest model.
First, let me point out the odd fact that here Wells criticism is actually arguing against computability – against a thesis of mechanism. By arguing that there is no largest model of a mechanism, he is in fact arguing that any given natural system can be successfully and completely modeled only by an infinity of computer programs, which contradicts Church’s Thesis requirement of effective calculability. Thus, if Wells actually holds this view, then he must reject the mechanist thesis as the two are fundamentally incompatible.
However, Wells argument that any program can be extended arbitrarily misses the point. Note that the programs in question are the simulations of the models of N, not the models themselves. Arbitrarily extending a simulation of a model is irrelevant to Rosen’s argument – Rosen’s criteria in this regard is the property of simulability itself, which is satisfied by the program absent any arbitrary extension. Of course, simulability also requires that the programs in question be finite, which is why Rosen notes that each program would be a “distinct word of finite length on a finite alphabet”.
Wells analogy with the integers is also incorrect. Note Rosen’s statement “the model formed from the intersection [of the countable infinity of models] is larger than any of the other models”. However, this would mean that there must be a distinct model which is larger than any member of, yet also not already among, the countable infinity of models, and which must accordingly possess a finite-length simulation program. But clearly, if such a model existed, it would already be a member of the countable infinity of models, unless it had an infinitely long program. This is what is impossible given finite length strings on a finite alphabet. A correct version of Wells analogy would be: find a distinct integer (which of course means it must have a finite decimal representation) which is strictly larger than any member of, and also not already a member of, the countable infinity of integers, which is clearly impossible.
Simulation and Entailment
Wells p. 45:
Part of his [Rosen’s] reason for focusing on simulation as the key activity of machines is his belief that there is a fundamental divide between simulation and modeling. He claims that whereas a model lays bare the entailment structure of the system it models, a simulation hides it. “In causal terms, simulation involves the conversion of efficient cause, the hardware of that being simulated, into material cause in the simulator. In essence, this means that one can learn nothing about entailment by looking at a simulation” (Rosen, 1991, p. 193).
This is an extraordinary claim that, once again, is simply wrong. If it were true, there would be absolutely no point in the many computer simulations, for example, of airflows over aircraft wings and the development of weather patterns, which are used to study causal relations in complex systems. Far from hiding the details of causal interactions in the systems that are modeled, simulations enable them to be studied in great detail and at a variety of time scales. A simulation is rather like a high speed film of the impact of a bullet on a particular material in which time can be slowed down on playback precisely to enable the investigator to understand better the extremely rapid succession of causal interactions between the bullet and the material on which it impacts. A simulation should, if anything, be described as a kind of “supermodel.”
This indicates a misunderstanding of Rosen’s remark (and which is but one remark out of an entire chapter devoted to the topic, entitled On Simulation in Life Itself). Briefly, in the Rosennean epistemology, models are models primarily because their inferential entailment structure can be shown to be congruent with causal entailment structure in the natural system being modeled. This is a necessary condition of a model. Models allow us to ask “why?” of a model in order to learn about the “why” (the entailments) of the natural system under study. When a model is converted into software for a Turing machine, then the simulation consists of the Turing machine plus the software. So there is no longer a congruence of entailment, since the relationship is between the natural system and [TM+software]. To restate it, suppose a natural system N has a model M. For M to be a model it must have a congruence of inferential entailment to causal entailment in N. To produce a simulation, M is converted to software s, to run on some Turing machine T, such that we end up with the simulation T(s). Plainly, there is no necessary congruence of entailment between M and T(s), since the extent of the entailment in T(s) is the entailment of next_state from current_state of a Turing machine, while the entailment structure that was in M, and is now the software s, has become simply the sequence of values that T operates on. Likewise, no necessary relationship exists between entailment in N and entailment in T(s). In short, we cannot ask “why?” questions of T(s) as a means to learn more about entailment in N.
From this, however, it does not follow that simulations are useless or irrelevant, nor does Rosen make such a claim. When a model is carefully converted into a simulation, then it is of course useful. Rosen’s point is that simulations are not of the same epistemological status as models: models are what exhibit congruence with natural systems, simulations (when they are derived from such models) provide mechanical means to grind out the calculations which arise from those models. Science does not proceed by creating simulations; it proceeds by creating models, from which simulations can then be constructed, if we choose, and if the models are simulable.
Simulations are certainly not “supermodels” in any sense; indeed, they have no virtue or validity independent of the models from which they are derived. Suppose they did have virtue or validity independent of the models. Since there can be no necessary congruence of entailments between a simulation and a natural system it simulates, the simulations can only be judged by their ability (or not) to mimic the behavior of the natural system. If we suppose that the virtue of a simulation T(s) rests on the notion its behavior (i.e., the output it produces) successfully mimics the behavior of a system N under study, then we must also accept that another simulation T(s’) which produces the same output as T(s) must have the same virtue with respect to N, even if T(s’) is simply based on some arbitrary mathematical function that has no relationship whatsoever to the causal relations actually occurring in N. But this of course is not what we would call science. In short, mimicry of behavior is what simulations do, but mimicry of behavior is not science.
Too often, the idea of model and simulation are conflated, which leads to epistemological confusion. Rosen’s work clarifies the epistemological differences and relationships between these ideas.
Infinite Regress and Complexity
Wells (p. 53) attempts to discern a contradiction in Rosen’s Essays in Life itself:
At this point the crux of the argument has been reached. Rosen claims that the price to be paid for escaping the infinite regress is that the systems thus arrived at are complex, noncomputable, and contain closed loops.
Breaking off such an infinite regress does not come for free. For it to happen, the graphs to which we have drawn attention, and which arise in successively more complicated forms at each step of the process, must fold back on each other in unprecedented ways. In the process, we create (among other things) closed loops of efficient causation. Systems of this type cannot be simulated by finite-state machines (e.g., Turing machines); hence they themselves are not machines or mechanisms. In formal terms, they manifest impredicative loops. I call these systems complex. (p. 24)
The particular point to notice is the claim that it is the process of breaking off the infinite regress of system expansions that generates complex systems.
The argument in chapter 1 (Rosen, 2000) contradicts and is contradicted by another of Rosen’s arguments in chapter 20 of Essays on Life Itself.
Rosen notes that the example of the thermostatically controlled room is characterized by a single state variable but says that the analysis can be generalized to deal with any finite number of state variables. “The analysis, of course, grows increasingly complicated, but, in effect, we now have a much larger family of cascading control loops, each of which creates the potentiality for infinite regress” (p. 303).
The crux of the argument has again been reached. In the earlier example Rosen claimed that the price to be paid for breaking off the infinite regress was that the resulting systems were complex. In the latter example, however, he reaches the opposite conclusion:
If every such cascade breaks off after a finite number of steps, then the system itself, and its environment, must both be simple. Conversely, if a system (or its environment) is not simple, then there must be at least one cascade of simple controls that does not break off…A system that is not simple in this sense (i.e., is not a mechanism) I call complex. (p. 304)
To drive home the point, a little later in the chapter Rosen argues that “there is a sense in which complex systems are infinitely open” (p. 307) and it is for this reason, he claims, that side effects are the norm rather than the exception in medical interventions.
The contradictory nature of the two arguments cited is perfectly clear. In one case Rosen argues that the existence of a break point that prevents an infinite regress of system openings leads to complex systems, in the other he argues that the break point leads to simple systems.
However, there is no contradiction because Wells draws the wrong conclusion by selectively focusing on the first sentence of the quote on p. 24 and ignoring the rest of the paragraph. Specifically, he ignores that Rosen explicitly says that in order to break off the infinite regress “the graphs to which we have drawn attention, and which arise in successively more complicated forms at each step of the process, must fold back on each other in unprecedented ways”, which refers to the impredicative loops, as Rosen plainly states in the last two sentences of the quote from p.24. (Consider, for example, the Liars Paradox which contains an impredicative loop, and which results in an infinite chain if one attempts to open the loop and cast it in a predicative form.)
Impredicative Loops and Turing machines
Wells makes a further remark on impredicativity (p. 55):
An argument that is largely implicit in Life Itself but which is given greater prominence in Essays on Life Itself claims that machines cannot contain closed causal loops because such loops are “impredicative” and are forbidden in formal systems such as Turing’s. One instance of this argument, from Rosen’s autobiographical sketch, was mentioned in the introduction. Here is another:
Impredicativity … was identified as the culprit in the paradoxes springing up in Set Theory. Something was impredicative…if it could be defined only in terms of a totality to which it itself had to belong.…Formalizations are simple systems (in my sense) and, in particular, cannot manifest impredicativities or self-references or “vicious circles.” This is precisely why such a simple world seemed to provide a mathematical Eden, inherently free from paradox and inconsistency. Alas, as Gödel showed, it was also free of most of mathematics. We cannot dispense with impredicativity without simultaneously losing most of what we want to preserve. (Rosen, 2000, pp.293–294)
There are several flaws in this argument. One is the implicit claim that all closed loops are impredicative. They are not: The closed loops in the finite state control automata of Turing machines are not necessarily defined impredicatively, although they may be. Thus, closed causal loops could be found in machines even if impredicative loops were forbidden.
Wells assertion: “One is the implicit claim that all closed loops are impredicative” is incorrect. Instead, definitionally, Rosen specifically uses to the phrase “closed causal loop” to refer to natural system counterparts of formal system impredicative loops. Wells may wish to define the term “closed loop” to refer to other possibilities, but he cannot thereby simply assign that usage to Rosen’s writings and so distort Rosen’s meanings.
There are no impredicative loops of entailment in Turing machines: two states in a Turing machine cannot entail each other simultaneously; instead, there can only be a predicative chain or sequence of entailments, where current_state entails next_state. This is the full extent of entailment in a Turing machine. Of course, it is quite possible that a chain of these states can form a loop of states, although these are more properly identified as iterative loops to clearly distinguish them from impredicative loops. On p. 60 Wells describes a Turing machine (TM1) with just such a property:
This function is a map from Q to Q and is implemented as a closed loop of entailments. The starting state of the machine is q1: q1 entails q2, q2 entails q3, q3 entails q4, and q4 entails q1. Once started, the machine cycles endlessly through this processing loop. It is precisely because the four internal states of its control automaton are structured as a closed loop that the fixed, finite machine TM1 can output the infinitely long sequence 010101 … . TM1 provides a definitive and conclusive rebuttal of Rosen’s lengthy and complex argument in Life Itself. TM1 is a machine and it has a closed causal loop of the kind that Rosen says machines cannot have.
Again, this is an iterative looping sequence of states and not an impredicative loop, and thus the argument is irrelevant to Rosen’s discussions of impredicative loops. But I also quote this passage since in it Wells interestingly considers that an infinitely iterative looping TM is somehow a “conclusive rebuttal of Rosen’s lengthy and complex argument in Life Itself” because it shows “that infinite sequences can be produced by finite means” (p.60). Yet, infinite computations do not meet the mechanist criteria of “effective calculability”, which, as Rogers  (p.5) and Davis  (p. 10) point out, requires that the program halt after a finite number of steps. As such, TM1 is antithetical to the mechanist thesis since it violates its basic tenet.
Complexity swallows mechanism
Wells’ use of the term “antimechanist” when referring to Rosen sets up the dichotomy that either one is a “mechanist” or one is “antimechanist”, as though the two concepts constitute entirely conflicting, non-overlapping sets of ideas. However, the view of Rosennean complexity is that mechanism constitutes a subset of complexity, and that, accordingly, mechanistic mathematical scientific models constitute a subset of the universe of mathematical scientific models. Thus, computable models are entirely valid and it is plainly obvious that they are very effective and worthwhile. What Rosennean complexity instead argues is that there is no a priori basis to restrict modeling of the natural world to only computable models, and that indeed, the inherent limitations of computability make such a restriction both unscientific and foolhardy. Since Rosennean complexity loosens these a priori restrictions, it generalizes to a larger universe of mathematical models, of which mathematical computable models are but a portion: Rosennean complexity swallows mechanism. Likewise, for any given natural system, the set of models for that system is likely comprised of both computable and noncomputable models.
Wells states in his conclusion:
The antimechanist arguments of Robert Rosen have been used by some ecological psychologists to support an attack on computational methods and mechanist thinking generally. Close examination of Rosen’s views shows that his epistemology assumes a strong form of indirect realism and his arguments, if valid, would constitute a denial of some of the fundamental principles on which ecological psychology is based. However, his arguments are not valid and do not show that organisms have properties that cannot be captured by machine models. For me this is important because I believe that antimechanist thinking in contemporary ecological psychology is based on a restricted view of the possible types of computational models and is acting as a barrier to the development of a genuinely ecological form of computational psychology.
If the first sentence is indeed true, then it is a mistake for some to have apparently concluded from Rosen’s arguments “to support an attack on computational methods”, for mechanistic models indeed constitute part of the universe of mathematical models. (It is sometimes overlooked that Rosen’s first two books, Optimality Principles in Biology  and Dynamical Systems Theory , are replete with discussions and models that would hardly raise a mechanist’s eyebrow.)
The question which Rosennean complexity raises – and let us take the case of psychology specifically – is whether there is a priori justification for restricting models to only computable models. (Further, in what way would such an a priori restriction be scientific?) If one holds a strict computationalist view of mind, then one could certainly claim there is justification for such a restriction. However, this simply moves the question of computability from the models of the system to the system itself, and does not really strengthen the stance. One could instead argue that Church’s Thesis is true of the universe, and a fortiori must be true of the human mind. But this of course simply makes the a priori claim of computability that much more grandiose and tenuous. One could argue that the historical success of computable models is itself empirical evidence for such a claim. However, such evidence cannot logically entail any particular outer boundary of model types, it can only entail that the total universe of models is at least as large as the universe of computable models.
From an empirical perspective, there is good evidence that a science of the mind cannot be accomplished with computable models alone. In particular, neither natural language nor mathematics can be rendered from an algorithm. That is, while we can computationally model fragments of such systems, we do not thereby gain a model of the whole. Also, we are capable of conceptualizing impredicative loops (indeed, we would have no such word if we could not!), which are not computable. In all these cases (and there are obviously many more) the evidence raises a fundamental question of how computable models alone suffice for the study of mind.
Additionally, as long as we consider mind to be brain-based, then all the inferential entailments of mind must be realized physically by corresponding underlying causal entailments: there can be no less entailment causally than there is inferentially. Therefore, to physically realize an inferential structure in the mind, such as an impredicativity, requires an underlying closed causal loop of entailment which will thus necessarily possess a noncomputable model. Thus, as long as we consider human beings (and so too therefore the activities in their minds) as physical systems subject to physical laws, then obviously those laws of physics must be capable of encompassing these entailment structures. As such, adherence to only strictly mechanist laws (i.e., laws comporting with Church’s Thesis) of physics would be too restrictive.
Acknowledgements: I am grateful to Dr. Aloisius Louie for discussion on Rosen’s Largest Model of a Mechanism [LI 8C].
 Wells, A.J. “In Defense of Mechanism”. Ecological Psychology. 18(1):39-65
 Rosen, R. 1991. Life Itself. Columbia Univ. Press
 Rosen, R. 2000. Essays on Life Itself. Columbia Univ. Press
 Rogers, H. 1967. Theory of Recursive Functions and Effective Computability. McGraw-Hill
 Davis, M. 1982. Computability and Unsolvability. Dover
 Rosen, R. 1967. Optimality Principle in Biology. Plenum Press
 Rosen, R. 1970. Dynamical Systems Theory. John Wiley & Sons