Mathematics and Explanatory Generality: Nothing but Cognitive Salience

Erkenntnis https://doi.org/10.1007/s10670-019-00146-x ORIGINAL RESEARCH Mathematics and Explanatory Generality: Nothing but Cognitive Salience Robert Knowles1 · Juha Saatsi1 Received: 25 November 2018 / Accepted: 28 June 2019 © The Author(s) 2019 Abstract We demonstrate how real progress can be made in the debate surrounding the enhanced indispensability argument. Drawing on a counterfactual theory of explanation, wellmotivated independently of the debate, we provide a novel analysis of ‘explanatory generality’ and how mathematics is involved in its procurement. On our analysis, mathematics’ sole explanatory contribution to the procurement of explanatory generality is to make counterfactual information about physical dependencies easier to grasp and reason with for creatures like us. This gives precise content to key intuitions traded in the debate, regarding mathematics’ procurement of explanatory generality, and adjudicates unambiguously in favour of the nominalist, at least as far as explanatory generality is concerned. 1 Introduction The debate surrounding the enhanced indispensability argument for mathematical platonism (EIA) has reached an impasse, descending into intuition-trading regarding purported examples of mathematical explanations of physical phenomena. Progress demands an independently motivated understanding of these explanations that favours either nominalism or platonism. Indeed, ‘[i]f there is a point of agreement in this debate, it is that we could do with a better understanding of mathematical explanation’ (Colyvan 2013: 1044). In this paper, we make significant progress towards such an understanding. Drawing on a counterfactual theory of explanation, well-motivated independently of the debate, we provide a novel analysis of ‘explanatory generality’, highlighted by advocates of EIA as a key virtue of certain mathematical explanations, and how mathematics is involved in its procurement. On our analysis, mathematics’ B Robert Knowles Juha Saatsi 1 School of Philosophy, Religion, and History of Science, University of Leeds, Leeds, UK 123 R. Knowles, J. Saatsi sole explanatory contribution to the procurement of explanatory generality is to make counterfactual information about physical dependencies cognitively salient: easier to grasp and reason with for creatures like us. This gives precise content to the conflicting intuitions about explanatory generality traded in the debate, and adjudicates unambiguously in favour of the nominalist. Since it is independently motivated, this verdict is not question-begging, demonstrating our methodological conclusion: working with an independently well-motivated theory of explanation can push past conflicting intuitions in the EIA debate. To be clear, we do not aim to vindicate nominalism once and for all. Our aims are more modest than that. First, we aim to provide non-question-begging reasons for thinking that the explanatory generality of certain mathematical explanations is compatible with nominalism. Second, in achieving this, we aim demonstrate the methodological conclusion mentioned above. This only addresses one virtue claimed for certain mathematical explanations, though one that has thus far been central to the platonist’s case (see below). Although we regard the counterfactual theory in question capable of capturing a very broad range of explanations, it is not our aim here to argue for this account’s universal superiority, nor do we commit to explanatory monism. Nevertheless, this is real progress. In light of it, if it is claimed that a purported mathematical explanation or a virtue thereof escapes our analysis, this claim had better not be supported by intuition alone. Our opponent must provide reasons, similarly independent of the EIA debate, to suppose our analysis fails to capture something of genuine explanatory worth. Before presenting our analysis, we provide some background on the EIA debate (below), a presentation of our chosen counterfactual theory of explanation (Sect. 2), and a toy example of the kind of mathematical explanation our analysis is supposed to capture (Sect. 3). The impasse regarding EIA stands as follows. According to EIA, scientific realists should be mathematical platonists, in light of examples of scientific explanations which seem to turn on mathematical facts. Even if alternative, nominalistic explanations can be offered, arguably such alternatives are worse explanations. Thus, a realist who infers to the best explanation cannot but accept commitment to whatever the mathematical facts involve. So runs the most prominent naturalistic argument for platonism (e.g. Baker 2005; Colyvan 2002, 2013; Lyon 2012). In response, nominalists can either deny that the mathematically presented explanations are better, or deny that mathematics’ contribution to them is ontologically committing. There is a nearconsensus that the first horn is untenable. The second horn has been popular, but here the impasse looms. While platonists take mathematics’ explanatory indispensability to evidence the existence of explanatory mathematical features of reality, nominalists take mathematics to merely increase our expressive capacity, allowing us to represent the physical features of reality that are doing the real explanatory work. On the one hand, the distinction between ‘really explanatory’ and ‘merely expressive’ can only be drawn fairly on some principled, non-question-begging grounds, and nominalists’ appeal to this distinction has hitherto not convinced platonists.1 On the 1 For example, Saatsi (2016) draws an interesting distinction between ‘thin’ and ‘thick’ explanatory roles, arguing that a nominalist can make sense of mathematics’ explanatory indispensability by maintaining 123 Mathematics and Explanatory Generality: Nothing but… other hand, the indispensability of mathematics for providing an explanation is not enough in and of itself to convince the nominalist that its explanatoriness springs from correctly representing mathematical features of reality. Thus, the debate has reached a serious impasse (e.g. Baker 2017: 2; Knowles and Liggins 2015: 3403–3407), and the result is predictable: intuition trading, subtle dialectical manoeuvring, and charges of question-begging. The claim that explanatory generality is what makes certain mathematical explanations better than any nominalistic alternative is popular among platonists. Indeed, it has been widely expressed for more than 10 years. For example, Colyvan (2002) defends EIA against Melia (2000) by noting that mathematics is indispensable for a ‘unified approach’ to presenting and solving disparate equations, and hence ‘genuinely explanatory’, since ‘unification is linked to explanatory power’ (p. 72). Alan Baker and Colyvan (2011) defend EIA by noting that any nominalised explanation of cicada periods is ‘both less general and less robust’ (p. 331). Colyvan (2013) defends EIA against Yablo (2013) by arguing that Yablo’s ap (...truncated)