On the (un)reasonable effectiveness of mathematics in ecology

An article appeared last week in Ecological Modeling that has the intention to be thought-provoking; it looks at effectiveness of mathematics in ecological theory [1], but it just as well can be applied to bioinformatics, computational biology, and bio-ontologies. In short: mathematical models are useful only if they are not too general to be trivially true and not too specific to be applicable to one data set only. But how to go about finding the middle way? Ginzburg et al fail to clearly answer this question, but there are some pointers worth mentioning. In the words of the authors (bold face my emphasis):

A good theory is focused without being blurred by extraneous detail or overgenerality. Yet ecological theories frequently fail to achieve this desirable middle ground. Here, we review the reasons for the mismatch between what theorists seek to achieve and what they actually accomplish. In doing so, we argue on pragmatic grounds against mathematical literalism as an appropriate constraint to mathematical constructions: such literalism would allow mathematics to constrain biology when the biology ought to be constraining mathematics. We also suggest a method for differentiating theories with the potential to be “unreasonably effective” from those that are simply overgeneral. Simple axiomatic assumptions about an ecological system should lead to theoretical predictions that can then be compared with existing data. If the theory is so general that data cannot be used to test it, the theory must be made more specific.

What then about this pragmatism and mathematical literalism? The pragmatism sums up as a “theories never work perfectly” anyway and, well, reality is surpassing us given that “we face an ever-increasing number of ecological crises, social demand will be for crude, imperfect descriptions of ecological phenomena now rather than more detailed, complex understanding later” (as aside and to rub it in: the latter is a different argumentation for pragmatism than the ‘I need a program from you today in order to analyse my lab data so that I can submit the article tomorrow and beat the competition’). The former I concur with, the latter on preferring imperfection over more thought-through theories is a judgment call and I leave that for what it is.
Mathematical literalism roughly means strict adherence to some limited mathematical model for its mathematical characteristics and limitations. For instance, in several ecological models (and elsewhere) processes are interpreted as strictly instantaneous—the “mechanistic” models—whereas those models that do not are mocked as “phenomenological”. But, so the authors argue, we should not fit nature to match the maths, but use mathematics to describe nature. Now this likely does ring a bell or two with developers of formal (logic-based) bio-ontologies: describe your bio stuff with the constructs that OWL gives you! And not—but probably should be—which formal language (i.e, which constructs) do I actually need to describe my subject domain? (Some follow-up questions on the latter are: if you can’t represent it, what exactly is it that you can’t represent? Do you really need it for the task at hand? Can you represent it in another [logical/conceptual modeling] language?)

It is not this black-and-white, however. As Ginzburg et al mention a couple of times in the article (kicking in an open door), trying to make a mathematical model of the biological theory greatly helps to be more precise about the underlying assumptions and to make those explicit. This, in turn aids making predictions based on those assumptions & theory, which subsequently should be tested against real data; if you can’t test it against data, then the theory is no good. This is a bit harsh because it may be that for some practical reasons something cannot be tested, but on the other hand, if that is the case, one may want to think twice about the usefulness of the theory.
Last, “The most useful theories emphasize explanation over description and incorporate a “limit myth” (i.e., they describe a pure situation without extraneous factors, as with the assumption in physics that surfaces are frictionless).” While it is true that one seeks for explanations, this conveniently brushes over the fact that first one has to have a way to describe things in order to incorporate them in an explanatory theory! If the theory fails, then thanks to a structured approach for the descriptions—say, some formal language or [annotated] mathematical equations—it will be easier to fiddle with the theory and reuse parts of it to come up with a new one. If the theory succeeds, it will be easier to link it up to another properly described and annotated theory to make more complex explanatory models.

Overall, the contents of the article is a bit premature and would have benefited from a thorough analyses of the too-general and too-specific theories other than anecdotal evidence with a couple of examples. Also, the “method for differentiating theories” advertised in the abstract is buried somewhere in the text, so, some sort of a bullet-pointed checklist for assessing one’s own pet theory on too-general/specific would have been useful. Despite this, it is good material to feed a confirmation bias for being against too much and too strict adherence to mathematics… as well as against no mathematics.

[1] Lev R. Ginzburg, Christopher X.J. Jensen and Jeffrey V. Yule. (2007). Aiming the “unreasonable effectiveness of mathematics” at ecological theory. Ecological Modeling, 207(2-4): 356-362. doi:10.1016/j.ecolmodel.2007.05.015

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