«Abstract In many of the special sciences, mathematical models are used to provide information about speciﬁed target systems. For instance, ...»
The Undeniable Eﬀectiveness of Mathematics
in the Special Sciences
In many of the special sciences, mathematical models are used
to provide information about speciﬁed target systems. For instance,
population models are used in ecology to make predictions about the
abundance of real populations of particular organisms. The status of
mathematical models, though, is unclear and their use is hotly contested by some practitioners. A common objection levelled against the use of these models is that they ignore all the known, causally-relevant details of the often complex target systems. Indeed, the objection continues, mathematical models, by their very nature,
away from what matters and thus cannot be relied upon to provide any useful information about the systems they are supposed to represent. In this paper, I will examine the role of some typical mathematical models in population ecology and elsewhere. I argue that while, in a sense, these models do ignore the causal details, this move can not only be justiﬁed, it is necessary. I will argue that idealising away from complicating causal details often gives a clearer view of what really matters.
And often what really matters is not the push and shove of base-level causal processes, but higher-level predictions and (non-causal) explanations.
1 The Philosophical Problems of Applied Mathematics The applications of mathematics to empirical science raise a number of interesting philosophical issues. Perhaps the most well known of ∗ Department of Philosophy, University of Sydney, Sydney, NSW, 2006, Australia and Stellenbosch Institute for Advanced Study (STIAS), Wallenberg Research Centre at Stellenbosch University, Stellenbosch, South Africa. Email: firstname.lastname@example.org these issues is the so-called unreasonable eﬀectiveness of mathematics.
The issue here is to account for the success of mathematics in helping empirical science achieve its goals. It is hard to say precisely what the crux of the issue is supposed to be, let alone what an adequate explanation would look like. The problem is usually attributed to Eugene Wigner  in his well known essay on the topic,1 where he suggests that “[t]he miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.” [34, p. 14] I take it that the problem, in its most general form, is to account for the applicability of mathematics in empirical science. Put this way, though, there are a number, of interrelated problems. There’s the unreasonable eﬀectiveness of arithmetic, of calculus, of diﬀerential geometry, of algebraic topology, and so on.2 There’s the way different philosophies of mathematics draw diﬀerent conclusions to help explain the applications of mathematics.3 There’s the issue of the diﬀerent roles mathematics can play in science—the diﬀerent ways mathematics might be thought to be unreasonably eﬀective. And, of course, physics is not the only scientiﬁc consumer of mathematics.
Mathematics might also be thought to be unreasonably eﬀectiveness in economics, in biology, in chemistry, in psychology, and elsewhere.
Finally, there’s the problem of understanding the nature of the modelling process itself and why mathematical modelling is so often an eﬀective way of advancing our knowledge.4 Many of these issues are interrelated but, still, a great deal of confusion has resulted from running some of the issues together and failing to state exactly what is supposed to be unreasonable about the eﬀectiveness in question. Having been guilty myself of such carelessness in the past [8, p. 15], my aim here is a modest one. I intend to look at the use of mathematical models in the special sciences. As my primary example I’ll consider the use of mathematics in population ecology.
The issue here is that the mathematical models in question seem to leave out the relevant causal detail, yet still manage to both predict and (arguably) explain population-level phenomena. The task, then, There has been a great deal of subsequent discussion on the issue, for example [1, 10, 20, 22, 33, 35] and this discussion has helped clarify the problem and its solution.
Not to mention the much less appreciated problem of the unreasonable eﬀectiveness of inconsistent mathematics .
See, for example, [9, 17, 33].
See, for example, [4, 6].
is to give an account of how mathematical models can succeed in such tasks.
dP = βV P − qP dt Here V is the population of the prey, P is the population of the predator, r is the intrinsic rate of increase in prey population, q is the per capita death rate of the predator population, and α and β are parameters: the capture eﬃciency and the conversion eﬃciency, respectively.
These equations can give rise to complex dynamics, but the dual outof-phases, population oscillations of predator and prey are the best known.
Of course both these mathematical models are overly simple and are rarely used beyond introductory texts in population ecology. For example, the logistic equation treats the carrying capacity as constant, and the Lotka-Volterra equations treats the predators as specialists, incapable of eating anything other than the prey in question. Both these assumptions are typically false. These models do, however, serve as the basis for many of the more realistic models used in population ecology. The more serious models add complications such as age structure, variable growth rates and the like. These complications do not matter for my purposes in this paper, though. Even in these more complicated models, biological detail is deliberately omitted and yet the models are adequate for the purposes at hand. The issues I am interested to explore can be raised with the more complicated models, but it’s easier to see the issues in the simpler models. We will not be losing any generality by focussing our attention on the simpler text-book population models.
We are now in a position to state the philosophical problem posed by mathematics in population ecology. Population abundance is completely determined by biological facts at the organism level—births, deaths, immigration and emigration—but the (standard) mathematical models leave out all the biological detail of which individuals are dying (and why), which are immigrating (and why), and so on.
That is, the mathematical models ignore the only things that matter, namely, the biological facts. The mathematical models here—the relevant diﬀerential equations—seem to ignore the biology, and yet it is the biology that fully determines population abundances. How can ignoring that which is most important ever be a good strategy?
We might put the point in terms of explanation: the mathematical models are not explanatory because they ignore the causal detail. The model may tell us that the abundance of some population at time t is N, but without knowing anything about the organism-level biology, we will not know why the population at time t is N and will have little conﬁdence in such predictions. A full account of the relevant biology, on the other hand, would include all the causal detail and would provide the required explanations. Let’s focus on this explanatory version of the puzzle because I think it is what underwrites the less-speciﬁc worries expressed in the previous paragraph.
Before I go any further, it will be useful to say a few words about explanation and philosophical theories of explanation. First, I take it that we simply cannot deny that there are population-level explanations in ecology. To deny this would, in eﬀect, amount to giving up on explanation in the special sciences. Unlike physics, in the special sciences we do not have the option of reserving all genuine explanation for the fundamental level (or the fundamental laws). So the issue we are meant to be addressing is not that there can be no explanation in the special sciences. Rather, we take it for granted that there are explanations in the special sciences but that the mathematical models used in special sciences such as population ecology can not deliver explanations.
Next we might reasonably ask for a philosophical account of explanation, so that we are all on the same page. But that turns out to be diﬃcult for a number of reasons, not least of which is that there is no generally-accepted philosophical account of scientiﬁc explanation.
So, for present purposes, I shall be rather liberal about what counts as an explanation. I suggest that an intuitive understanding of an explanation as an answer to a “why questions” will do.6 It is important to keep in mind that explanation should not be confused with a more limited class of explanation known as causal explanation. There is no denying that causal explanation—tracing the relevant causal history of an event of interest—is one kind of explanation. I deny, however, that this is the only kind of explanation.7 Explanations must be enlightening, and that’s about all we really need to assume here.
3 The Role of Mathematics Now I turn to the task of investigating what makes these mathematical models in ecology tick. I will argue that there is no reason to suggest that mathematical models in ecology are not explanatory. I will suggest three diﬀerent ways in which the models in question can I also take an explanation to be that which is accepted as such in the relevant scientiﬁc community. This, of course, is not a philosophical account of explanation; it’s just a constraint that I take very seriously. I think philosophical accounts of explanation need to (largely) agree with scientiﬁc uses of the notion of explanation. A philosophical account of explanation that does violence to scientiﬁc practice is of little interest to we naturalistic philosophers.
It would take us too far aﬁeld to argue for this here, but see, [9, chap. 3], , , and .
explain. First, the mathematical models do not ignore the biological detail—at least sometimes the models in question are oﬀering biological explanations, albeit explanations couched in mathematical terms.
Second, understanding a system often does involve ignoring, or rather, abstracting away from, causal detail in order to get the right perspective on it. Finally, I’ll suggest that mathematics can oﬀer explanation for empirical phenomena.
Recall that we started out with the charge that mathematical models leave out all the relevant biological detail. But this is not quite right. Often the mathematical model is just representing the biology in a mathematical form. For example, in the logistic equation, all the information about births, deaths, immigration and emigration is packed into r and all the information about the resources is packed into the constant K. The information about the predators’ impact on the per capita growth rate of the prey is summarised in the LotkaVolterra equation in α—the capture eﬃciency parameter—and the information about the predators’ ability to turn prey into per capita growth of the predator population is summarised by β—the conversion eﬃciency parameter. You might have misgivings about the representation of this information8, but this is a diﬀerent objection. It’s now a concern about the simplicity of the model. As I mentioned before, we can provide more complex models that relinquish some of the more unrealistic idealisations. These more complex models also have their idealisations, though. Indeed, it is part of the very enterprise of modelling that some details are ignored. So the basic concern about biological detail not being represented in the mathematical models under consideration is misplaced. Of course not all the biological detail is present in the model, but the fact remains that many of the key terms of the mathematical models have natural biological interpretations, or at least are representing or summarising the biological information in mathematical form. The mathematical models have a lot more biology represented in them than is typically appreciated.
In cases where the biology is represented in mathematical form, the model is indeed capable of oﬀering perfectly legitimate biological explanations. For instance, think of the standard story of how population cycles arise as a result of predator–prey interactions. The cycles in question are solutions to the coupled diﬀerential equations in question [5, chap. 9] but there is also a very natural biological explanation that can be extracted from the mathematical model: when the predator population is high the predators catch many of the prey so that the latter’s population falls, but then there is less food for You might, for example, object that r and K are represented in the logistic model as constants.
the predators, so after a time the predator population also falls; but now there is less pressure on the prey population, so it recovers and this, in turn, supports an increase in the predator population (after a similar time lag). This cyclic behaviour falls out of the mathematics, but the explanation, once suitably interpreted, is in fact a perfectly respectable ecological explanation.
Next, notice that ignoring some detail can lead to insights via analogy.9 Sometimes similarities between systems will not be apparent until certain details are ignored. Mathematics is a particularly useful tool for drawing out such similarities because mathematics allows one—indeed forces one—to abstract away from the causal detail and notice abstract similarities. For example, Newton’s law of cooling/heating is just the logistic equation with abundance replaced with temperature of the body in question, and carrying capacity replaced with ambient room temperature.10 Why are such connections between systems important? One reason is that it saves work: one can import results already at hand from work done elsewhere. Once the connection between the logistic equation and the cooling/heating equation are recognised, results from either area can be used by the other area (suitably interpreted, of course). Moreover, these rather abstract connections—often only apparent via the mathematics—can lead to new developments and, as we’ll see shortly, even help with explanations.