Mathematical explanation

world. The theory of explanation used for computer simulation needs to take this time-dependency of theories and models into account.

Having abandoned the deductive nomological model of scientific explanation, I now continue my analysis of theories of explanation by discussing mathematical expla-nation of physical phenomena.

Suppose that a bunch of sticks are thrown into the air with a lot of spin so that they twirl and tumble as they fall. We freeze the scene as the sticks are in free fall and find that appreciably more of them are near the horizontal than near the vertical orientation. Why is this? The reason is that there are more ways for a stick to be near the horizontal than near the vertical. To see this, consider a single stick with a fixed midpoint position. There are many ways this stick could be horizontal (spin it around in the horizontal plane), but only two ways it could be vertical (up or down). This asymmetry remains for positions near horizontal and vertical, as you can see if you think about the full shell traced out by the stick as it takes all possible orientations.

This is a beautiful explanation for the physical distribution of the sticks, but what is doing the explaining are broadly geometrical facts that cannot be causes. (Lipton, 2004, 31-32)

As Peter Lipton points out, this is an explanation of a physical phenomenon us-ing geometrical, non-causal facts about properties of the sticks in free fall. Such an explanation makes use of mathematical procedures, objects, attributes of these objects, relations among them, and the like; it is an explanation that ignores (and is required to ignore) various physical details about the system of interest. With this in mind, we can now answer the fundamental question about mathematical ex-planation of physical phenomena; namely: “how is mathematics applied in scientific explanations and descriptions?” (Shapiro, 2000, 36).

In 1978, Mark Steiner published his first work on mathematical explanation of physical phenomena. In that work he states that every scientific explanation requires mathematical, as well as physical truths. For instance, the physical space is a three-dimensional Euclidean space, and the rotation of a rigid body around a point generates an orthogonal transformation. In Steiner’s mind, then, mathematical explanations of physical phenomena consist in a typical mathematical proof once all the physical concepts have been removed. In other words, in order to explain p, one must first remove all the terms in p referring to physical properties and then carry out a typical mathematical proof.⁵² Steiner, then, endorses the idea that once the terms referring to physical properties have been removed, all that remains is the mathematical explanation of a mathematical truth.

There is an important metaphysical assumption here consisting in deleting the boundaries between explanation of physical phenomena and explanation in mathe-matics by ‘removing the physics.’ In addition, the continuity between the natural sciences and the mathematical sciences is grounded in methodological alikeness:

both describe an objective world of entities, both make use of methods that are remarkably similar, and one can be subsumed under the other by ‘mathematical analogies’ that allows for the removal of the physical concepts. Basically, a ‘math-ematical analogy’ is a way to bring the realm of mathematics into the realm of physics, since both share equivalent properties (e.g., the linearity of a second or-der equation is ‘analogous’ to the applicability of the principle of superposition in

physics⁵³).

Thus understood, mathematical explanation has few things to offer to computer simulations. First, it is very unlikely that one can perform a mathematical proof of a simulated phenomenon. And even if such a proof is possible, it would most likely take an incredible amount of time to make it humanly feasible. After all, com-puter simulations are powerful tools for systems otherwise cognitively inaccessible.

Second, mathematical properties of physical entities cannot always be accurately represented in computer simulations; therefore, it is not true that one can remove the physical terms in order to carry out an explanation. Steiner imposes unaccept-able conditions for a theory of mathematical explanation of physical phenomena.

A more suitable account seems to be Robert Batterman’s asymptotic explana-tions. According to this author, explanations here “illuminate structurally stable aspects of a phenomenon and its governing equations” (Batterman, 2002, 59) by means of highly sophisticated mathematical manipulations. Similarly to Steiner, Batterman considers that we have reasons for ignoring many details of a physical system that we cannot control. In an asymptotic explanation, then, the explanatory power is given by systematically ‘throwing away’ the various causal and physical de-tails. However, unlike Steiner, this is not done by ‘removing the physics,’ but rather by abstracting and idealizing the stable aspects of the phenomenon.

The way asymptotic reasoning explains, therefore, is by yielding understanding of the transformations of one mathematical formula into another. These transfor-mations are carried out by means of taking the limit of some quantity, often 0 or infinity. Batterman illustrates his account with a couple of examples: one is flow, and the other is heat.⁵⁴ Allow me to omit the details of these examples and get directly into the objections to Batterman’s account.

Christopher Pincock objects that taking the limit of a function can have positive as well as negative effects. The positive effect is that we eliminate from the rep-resentation the irrelevant details, while the negative effect is that in the transition from one representation to another, we also lose relevant representational aspects of the phenomenon.⁵⁵ Batterman’s account also suffers from a serious drawback that makes it uninteresting for computer simulations: it leaves unexplained cases where asymptotic explanation cannot be applied. Consider applying asymptotic expla-nation to Lipton’s example: there are no limits to take in the function describing the position of the sticks. It follows that no explanation is possible. Just like my objection to Steiner, Batterman suggests a kind of explanation that works only for specific cases, and is therefore unappealing for computer simulations.

There are many other interesting accounts of mathematical explanation, such as the mapping account championed by Pincock,⁵⁶ or its ‘refined’ version, the inferen-tial account defended by Otávio Bueno and Mark Colyvan.⁵⁷ Unfortunately, these accounts share similar assumptions that make them inapplicable or unappealing for explaining simulated phenomena. Take for instance the usual metaphysical assump-tion that explanaassump-tion depends solely on the mathematical machinery that governs the phenomenon. In this sense, it assumes that the phenomenon under study is governed by certain mathematical structures, and that an explanation is possible because there is a suitable language in place. However, such an assumption limits explanation to a pure mathematical language, forcing the conclusion that expla-nation in computer simulations is only possible in mathematical terms. It follows that a non-mathematical explanation could not be used for explaining simulated phenomena, a clearly undesirable consequence.

Restricting explanation to a purely mathematical domain seems to be too high a price for the epistemology of computer simulations. The kind of computer sim-ulations we are interested in cannot always yield mathematical insight into the phenomenon that is being simulated (albeit that simulation yields insight into such phenomenon). Moreover, the computer simulation would not be responsible for carrying out the explanation since the explanation itself is located somewhere else:

whether on the notion of asymptotic reasoning (in Batterman) or in the notion of proof (in Steiner). Hence, we cannot properly grant computer simulations explana-tory capacity.

Despite these objections and drawbacks, we still need to consider mathematical explanation as important for computer simulations. This is especially important because many models simulated are mathematical models, and an account of expla-nation in computer simulation needs to be powerful enough to include these models as well.

Now suppose for a moment that instead of acomplete mathematical explanation of phenomena, the explanation is onlypartial, that is, only a part of the explanation relies on mathematics whereas other equally relevant aspects rely on some other domains. Consider the example of the hexagonal structure of hive-bee honeycombs in evolutionary biology. The nature of the problem is contrastive, that is, why hive-bees chose to create honeycombs with a hexagonal structure rather than any other polygonal figure or combination. Part of the explanation depends on mathematical facts, part on evolutionary facts. Bees that use less wax and thus spend less energy have a better chance at evolving via natural selection. The explanation is completed

by pointing out that “any partition of the plane into regions of equal area has a perimeter at least that of the regular hexagonal honeycomb tiling” (Hales, 2001, 1), or “the existence of a honeycomb tiling depends on the shape and size of the lattice”

(Hales, 2001, 7). Thus, the hexagonal tiling is optimal with respect to dividing the plane into equal areas and minimizing the perimeter.

This fact, known as the ‘honeycomb conjecture,’ shows how mathematical ex-planation of physical phenomena and classic scientific exex-planation depend on each other. It also motivates the claim that a successful theory of explanation for com-puter simulation should account for both of these kinds of explanation, given the very nature of computer simulations themselves. Here lies another reason why I be-lieve that the unificationist account is the most suitable candidate for simulations.

According to this latter account, scientific explanation is treated in the same logical way as mathematical explanation of physical phenomena, merging into one account two allegedly different kinds of explanation (recall my first objection to Hempel’s de-ductive nomological model). In this vein, simulated phenomena could be explained regardless of their mathematical nature or their physical characteristics. I discuss the unificationist account in more detail in Chapter 5.

Let me turn to another interesting account of explanation: model explanation.

Given the model-like nature of computer simulations, it is also natural to envisage it as a possibility.