From my perspective as a scientist, the assessment of a claim as to how scientific models are formulated and constrained comes down to answering two questions:

• • Is the claim conceptually coherent and empirically adequate?

• • Does the claim lead to normative procedures and recommendations that improve the quality of models and thereby lead to better understanding of nature?

In our Reference Orzack and Sober1993 paper, “A Critical Look,” Elliott Sober and I concluded that the famous claim about model formulation and constraints made by Richard Levins in his influential Reference Levins1966 article on model building in population biology is neither true nor normative. I now comment upon the claim of Odenbaugh (Reference Odenbaugh2003) that the conclusions of “A Critical Look” are incorrect.

In “A Critical Look,” Sober and I discussed Levins’ two-part claim (Reference Levins1996, 422): that (1) there are three kinds of models, each of which result from a necessary ‘sacrifice’ of one of three desired model qualities—generality, precision, and realism—so as to increase the other two, and (2) models with enhanced generality and realism, achieved at the expense of precision, are preferable. We demonstrated by a formal analysis of model properties that it is possible to simultaneously increase (or decrease) generality, precision, and realism. Accordingly, Sober and I disagreed with Levins’ claim that there are only three types of models; instead, many different kinds of models are possible.

Odenbaugh (Reference Odenbaugh2003, 1497) disagreed with our claim in that he believes both that formal analysis of model properties is not sufficient by itself to understand the possible constraints on models in practice and that Levins’ first claim was not only about such properties; instead, it was about the pragmatic features of model building. Odenbaugh (Reference Odenbaugh2003, 1505) argued that the ‘cognitive limitations’ of scientists are also important in regard to assessing the constraints on models. In particular, he claimed that present models describe biology that is so complex that the models themselves are situated at the limit of our cognitive ability to understand them. In such a situation, given additional assumptions, it is correct that a further elaboration of such a model in order to increase, say, realism, must result in a decrease of either generality or precision (see Odenbaugh Reference Odenbaugh2003, 1504–1507). Only models that are ‘cognitively inaccessible’ would have increased generality and precision as realism increases.

There are two points to be discussed in regard to this claim about the content and meaning of Levins’ claim. The first is whether the substance of Odenbaugh's claim follows from Levins’ claim. The second point is whether the claim is true. I discuss these in turn.

Levins did allude in his article to the role played by human cognitive abilities in determining what models are used and are useful in science. He wrote,

Here, we see Levins making separate claims about human cognitive abilities and about the antagonistic relationship among model characteristics. Note that the latter claim does not reference human cognitive abilities. In my opinion, this separation does not support Odenbaugh's apparent assertion that Levins inherently tied his claim about the relationship among model attributes to the way the human mind works.

Nonetheless, suppose that such a link is plausible, and indeed, integral to Levins’ claim. This brings us to the question of whether Levins’ and Odenbaugh's claim is true. To be so, one must believe that the complexity of present models is as great as it can be without the models being incomprehensible to us. Otherwise, as Odenbaugh (Reference Odenbaugh2003, 1505) stated, they could be elaborated so as to have increased generality, precision, and realism. Levins (Reference Levins1993) also agreed that this is possible. For example, consider how one could elaborate the stochastic model of reproductive effort evolution of Orzack and Tuljapurkar (Reference Orzack and Tuljapurkar2001) so as to make it more general, more precise, and more realistic. Suppose that we regard the uninstantiated model as ‘realistic’ because it accounts for temporal variation in reproductive biology, which an otherwise identical deterministic model does not. Now we modify the stochastic model so that it accounts for autocorrelation of the temporal variation. As compared to the original stochastic model, the modified model is more general (as it applies to species inhabiting more types of environments), it is more realistic (it accounts for autocorrelation), and it will be no less precise (in that it generates a point prediction).

It is simply an unsupported assertion that present models are complex enough that they are at our cognitive ‘limit’. Perhaps this is so, but it is unclear as to how one can meaningfully conclude this about any scientific model if only because the history of science provides so many examples of ‘incomprehensible’ models becoming the commonplace understanding of subsequent generations. (The history of models in quantum mechanics comes to mind in this regard. See Bell Reference Bell1987.) For what it is worth, many of the models that have been classified as being one of Levins’ three types are not obviously just shy of being ‘cognitively inaccessible’ (e.g., the Hardy-Weinberg Law, which was classified by Clark and Terrell Reference Clark and Terrell1978 as being general and precise, but not realistic).

In this context, it is also worth commenting upon biological complexity, which is supposed to drive model complexity to our cognitive limit. The claim that the world is ‘complex’ has a checkered history in biology. In many instances, claims of complexity have often vanished in the face of findings that explain broad patterns in nature. So, for example, nineteenth century claims about genetic mechanisms often defaulted to claims about the complex variety of mechanisms that ‘surely’ must be present among animal and plant taxa. Yet the discovery of Mendel's laws, and later the discovery of the molecular structure of DNA, revealed that much (but not all) of this complexity was more apparent than real.

Similarly, until the 1980s there were many claims about the complex variety of developmental mechanisms that ‘surely’ must be present among these taxa. Yet, the discovery of homologous molecular mechanisms of development in very disparate groups of animals (and plants) suggests that much (but not all) of this complexity can be understood as something much simpler than was originally suspected (e.g., see Panganiban et al. Reference Panganiban1997).

So, too, has complexity observed in the study of ecology and evolution or in the social sciences often putatively been resolved into something simpler (e.g., see Romney et al. Reference Romney, Moore and Rusch1997).

The point is not whether these (or any) claims that ‘remove’ complexity prove true in the long run. The minimal point is that the complexity, as invoked by Levins and Odenbaugh, is not a given; it may or may not be relevant to a model. Yes, the world is complicated. Yet, it is a separate issue of whether this complexity inheres within the biological domain that motivates any given model. Surely, sometimes such complexity is present … but not always. To this extent, even if biological and model complexity match one another, as is claimed by Odenbaugh, it need not mean that the model is at the limit of our comprehension because the extent of the relevant complexity may be limited.

However, for the sake of argument, let us agree that ‘full’ biological complexity is always expressed ‘locally’. In such a circumstance, what is not obvious is that model complexity must match the biological complexity. It may be that present models are still at a stage of development that allows for increases in generality, precision, and realism. For that matter, in some instances, models of biology phenomena may sometimes be more complex than the biology they were devised to explain; discrete-time models that predict chaotic dynamics (which appear to occur rarely in nature) come to mind in this regard (see Zhou et al. Reference Zhou1997).

To this biologist, Levins’ claim have distracted many investigators from the hard work of assessing the meaning of and relationship among present models; they have instead found convenient quasiphilosophical cover for deeming some types of models acceptable and others not so (see also Palladino Reference Palladino1991). This has often allowed for model selection without the real work of comparing and empirically testing models. This is a temptation to be avoided if we are to make real progress at understanding nature.

Perhaps Levins’ trichotomy of models would be useful if it were possible to unambiguously assign a given model to a category. Yet, as Sober and I noted, the assignment of a model can depend on whether it is instantiated or not, i.e., whether the parameters in the model are assigned numerical values so that the model makes numerical predictions (Reference Levins1993, 536). For this and other reasons, it is not obvious that his classification scheme can be applied in any kind of objective manner; Levins (Reference Levins1993, 549) agreed with this point, although he regarded this ambiguity as a virtue. The history of apparently arbitrary classification of models is a testimony to this ambiguity (see Orzack and Sober Reference Orzack and Sober1993, 541–542). Indeed, I am unable to identify even one uninstantiated model that cannot be classified as each of Levins’ three types, depending upon the set of models to which it is compared. The slippery nature of classification is readily appreciated once one examines any given model. Consider the model of Orzack and Tuljapurkar (Reference Orzack and Tuljapurkar2001) mentioned above. This stochastic model is more realistic than related deterministic models, since it accounts for temporal variability of the environment. Yet, it is less realistic than other models that account for temporal autocorrelation. So, can we identify this as a realistic model in a reasonably unambiguous way? My answer is no. Similar comparisons resulting in ambiguous assessments of the qualities of generality and precision can be readily constructed. This model is typical in regard to generating such ambiguity.

Even if Levins’ specific claim about model qualities and about types of models is not credible, it is still possible that it somehow underlies a special normative framework for the process of modeling, as claimed by Levins (Reference Levins1993) and Taylor (Reference Taylor2000). Perhaps this is true. However, the general guidelines putatively stemming from Levins’ claim (at least those described by Taylor Reference Taylor2000, 205) appear to be no more than an expression of a common understanding of typical scientific practices; indeed, these guidelines were also described by Holling (Reference Holling and Watt1966). Such tenure may or may not be a virtue. Of course, regardless of whether or not Levins’ claim was common understanding in 1966, perhaps the research strategy he promoted, with its emphasis on general and realistic models, was useful at the time. The important issue going forward is whether his claim is still useful. For what it is worth, the research practices of population biologists (and of many other kinds of scientists) have changed dramatically since 1966 because of the availability of cheap, fast computers. Both this shift and the resulting resurgence of simulation models in population biology as a supposed means to understanding biological complexity (Peck Reference Peck2004) conflicts with Levins’ prescription that such models are less appropriate for this task as compared to other types of models. Nonetheless, it is reasonable to believe that substantial progress has been made in our understanding of nature. For example, we owe to the numerical simulation analysis of very complex climate models the inference that global warming is, at least partially, the result of human activity (e.g., see Thorne et al. Reference Thorne2003). It is not clear to me how qualitative analysis of a general and realistic (but ‘imprecise’) model would lead us to the same conclusion. Of course, one could claim that even more progress would have been made if Levins’ prescription had been followed; at present, this claim is unsubstantiated.

Because Levins’ claim about the tradeoff between model properties lacks logical coherence, because of the arbitrariness of the model classification it inspires, and because of its lack of normative utility, I view this claim as providing no useful insight into the nature of models and model building in population biology. To me, continued citation of Levins’ paper by scientists who classify a model as being one of his three types is best understood as something whose meaning inheres in the act itself and not in its content; this is especially true for the many scientists who have labeled a model as being of Levins’ preferred Type III—a model that maximizes generality and realism at the expense of precision. I would guess that many such scientists wish to emphasize the importance of a model by situating it in the preferred part of the larger scientific enterprise implied by Levins’ trichotomy of models. This is an innocent (and plausibly appropriate) practice, but it reflects only upon the social context in which science is done, rather than indicating that Levins’ claim provides insight into the nature of models or into the nature of the world around us.