Mathematical Explanations of Physical Phenomena

The most natural way of understanding the role of mathematics in scientific explanations is as representing physical explanatory facts and helping to draw inferences about those facts. According to Joseph Melia, for example, because mathematics offers good representations of the physical world, some scientific explanations require the use of mathematics, but this does not mean that mathematics is in itself explanatory. If we say, for example: ‘F occurs because P is 2 meters long’, despite the fact that we are mentioning the number 2 in the explanation, it is the actual physical length of object P, not the real number 2 by which we represent it, that does the real explanatory work (cf. Melia 2002, 76). The main idea is that mathematical statements feature in scientific explanations because of these representational and inferential roles. By performing derivations over these mathematical statements, we can learn how the relevant physical facts explain the explanandum.

But it has recently been argued that mathematics may be able to do more than this, that there can be mathematical explanations of physical phenomena (MEPP). In recent years, there has been much discussion about the nature of these MEPPs. Authors wonder what exactly it means for mathematics to explain a physical phenomenon; whether MEPPs are genuinely different from ordinary scientific explanations that use mathematics; and, if MEPPs are indeed different, which ontological consequences follow from the fact that there are MEPPs in science.

Many purported cases have been advanced in recent literature, and there have been many attempts to determine what exactly the distinctive feature of each of these cases is, and whether they all belong to the same category; and although there is no current consensus, most authors agree that in these explanations mathematics is involved in a special way. For example, almost all accounts agree that these are scientific explanations that depend on the mathematical model they use in the explanans in a way such that without the mathematical model these explanations would not stand. In other words, the mathematical part of these explanations is indispensable for the explanation to work as an explanation.

However, many authors have gone further, and claim that in these cases mathematics itself is playing an explanatory role in science. In fact, some even suggest that MEPPs ontologically commit us to the existence of mathematical entities, following a new version of the Indispensability Argument. If mathematical statements feature indispensably in some scientific explanations, then, if we are scientific realists, we ought to be committed to the existence of the mathematical entities that make those mathematical statements true (just as, say, explanations of quantum phenomena commit us to the existence of subatomic particles).

I think, however, that we should not go that far. In my dissertation, I offer an account of MEPPs that emphasizes the representational role of mathematics. That is, I argue that in MEPPs the role of mathematics is (merely) to represent physical facts, but I also maintain that these explanations are special. I combine elements of James Woodward’s counterfactual account of scientific explanation, and Otávio Bueno, Marc Colyvan, and Stephen French’s Inferential Conception of the Applicability of Mathematics. The main aspects of my account are the notions of optimal representation and explanatory mathematical derivations. My goal is to advance an account of MEPPs that is counterfactual, noncausal, and where the role of mathematics is representational.

The dissertation is divided in five chapters. I give some background to the topic of scientific explanation and introduce Woodward’s account in chapter 1. In chapter 2 I discuss the role of mathematics in scientific explanation. In chapter 3 I introduce my own account of MEPPs. In chapter 4 I discuss other accounts of MEPPs; and in chapter 5 I show that the existence of MEPPs does not justify mathematical realism.