Abstract
Are there genuine mathematical explanations of physical phenomena, and if so, how can mathematical theories, which are typically thought to concern abstract mathematical objects, explain contingent empirical matters? The answer, I argue, is in seeing an important range of mathematical explanations as structural explanations, where structural explanations explain a phenomenon by showing it to have been an inevitable consequence of the structural features instantiated in the physical system under consideration. Such explanations are best cast as deductive arguments which, by virtue of their form, establish that, given the mathematical structure instantiated in the physical system under consideration, the explanandum had to occur. Against the claims of platonists such as Alan Baker and Mark Colyvan, I argue that formulating mathematical explanations as structural explanations in this way shows that we can accept that mathematics can play an indispensable explanatory role in empirical science without committing to the existence of any abstract mathematical objects.
Highlights
Are there genuine mathematical explanations of physical phenomena, and if so, how can mathematical theories, which are typically thought to concern abstract mathematical objects, explain contingent empirical matters? Lange (2016), for example, argues that mathematical explanations of physical phenomena are a species of noncausal explanations that he calls explanations by constraint
Discussion has been divided along platonist/anti-platonist lines, with most platonists agreeing that there are such explanations, and most anti-platonists disagreeing (notable exceptions are Brown (2012) on the platonist side, and Leng (2012) on the anti-platonist side). For those who reject the claim that mathematics does genuine explanatory work in our scientific theories, a standard strategy has been to point to the nominalistic content of putative mathematical explanations of physical phenomena, holding that while these explanations may be characterised mathematically, all the genuine explanatory work in these explanations is carried by their nominalistic content, with mathematics being used as a convenient—and perhaps indispensable—way of indexing the explanatorily relevant physical facts. (Examples of strategies along these lines include Brown, 2012; Daly & Langford, 2009; Melia, 2000; Saatsi, 2011) In Leng (2012) I side with platonists including Baker and Colyvan (2011) in suggesting that if we focus on the nominalistic content of mathematical explanations of physical phenomena, we lose explanatory power
Even many, mathematical explanations of physical phenomena best understood as structural explanations, explaining by showing that their target system is an instance of a mathematical structure? I have argued that the cicada explanation can be understood as a structural explanation, where the axioms of number theory are interpreted as truths about the system of years consisting of some initial years in which cicadas appear, with the ‘successor’ relation being the ‘the following year’ relation
Summary
Are there genuine mathematical explanations of physical phenomena, and if so, how can mathematical theories, which are typically thought to concern abstract mathematical objects, explain contingent empirical matters? Lange (2016), for example, argues that mathematical explanations of physical phenomena are a species of noncausal explanations that he calls explanations by constraint. My own view is there are features of these so-called ‘explanations’ that suggest that there is at least a case for including them as examples of genuine explanations They supply important modal information about their explananda: they tell us why they had to occur given the structural features of the physical situation. Even if supplying modal information about an observed phenomenon, and unifying disparate phenomena turn out to be not enough to count as providing an explanation in a strict sense, these still remain important theoretical roles played by mathematics in science that go beyond what would be available if we confined ourselves to purely nominalistically-stated alternatives This raises the question of whether, if what I am calling structural ‘explanations’ succeed where purely non-mathematical descriptions fail in enhancing our understanding of the physical world in these kinds of ways, this amounts to an indispensable theoretical role that supports platonism. I conclude, that viewing mathematical explanations of structural explanations provides an understanding of how mathematics can play a significant theoretical role in our understanding of physical phenomena that does not require us to adopt a platonist account of mathematical objects
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