Abstract
One of the debates concerning so-called mathematical explanations of physical phenomena (MEPPs) revolves around the question of whether mathematics plays a genuinely explanatory role (e.g., Baker 2005) or a mere representational one (e.g., Leng 2021). In a recent paper, Baker concedes that at one level these explanations use representational models, but this does not disqualify them from being MEPPs. He argues that, once these models are implemented, a purely mathematical reasoning occurs at a higher level, which is what really explains the explanandum, thus rendering the whole explanation a genuine MEPP (Baker 2021, 15) (Huneman 2018 holds a similar view). Baker introduced this distinction with the bipedal gait explanation, which consists in explaining the 3:1 ratio between the energy consumed at the stance phase and the swing phase of bipedal animal walking. For Baker, straightforward representational models capture the energy costs of the different phases (level 1). But the ratio itself is calculated by a minimum value theorem that not only proves it, but also explains it (Baker 2021, 7), which is what makes this case a MEPP (Baker 2021, 16). I think, however, that this explanation can also be understood from a representationalist perspective. The minimum value theorem is not used as a black box that provides results detached from physical reality. Rather, it is understood as tracking down the relevant physical relationships that explain the invariant ratio. The higher-level reasoning occurs precisely because the level 1 use of mathematics successfully captures the relevant stages of walking. But there is no guarantee that the higher-level reasoning will continue to be successful in the same way, because there is always the possibility that it will bring extraneous results that do not necessarily make sense for the physical system. This is acknowledged by the author of the study cited by Baker, who pointed out that his model also predicted a third optimal gait that, he thinks, may be a mere artifact (Srinivasan 2006, 108). Since the higher-level mathematical results are explicitly interpreted in empirical terms (Srinivasan 2006, 106), this supports the view that they play a representational role as well. Against this, it may be pointed out that Srinivasan’s study includes a mathematical explanation of the minimization theorem (Baker 2021, 11-12; Srinivasan 2006, 104), and that this would show that the overall scientific explanation depends on a mathematical explanation, rendering this case a genuine MEPP. However, this objection is incompatible with Baker’s stance in a related debate, namely, the view that MEPPs only need to cite, but not explain, the mathematical theorems they use (Baker 2012). The objection is compatible with an alternative view though, that Baker rejects, which holds that MEPPs must include purely mathematical explanations (Steiner 1978; Colyvan 2018). However, as Barrantes (2020) has pointed out, and scientific practice seem to suggest (Baker 2012), the merits of these derivations (e.g., whether they qualify as purely mathematical explanations) are unimportant for empirical purposes