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Naturalistic metaphysics sometimes engages in exploring the ontological commitments of science, as in the case of the so-called indispensability arguments. I will now examine the connection between these arguments and scientific explanationism.

The best-known indispensability argument views mathematical realism as a corollary of scientific realism (Colyvan 2006). Briefly and roughly put, in the Quinean holistic picture of science, the empirical justification for theoretical assumptions “bleeds over” to (applied) mathematics. As Psillos (2012, 53) puts it,

[I]ndispensability arguments capitalise on the strengths of scientific realism, and in particular of the no-miracles argument (NMA), in order to suggest that a) the reality of mathematical entities (in their full abstractness) follows from the truth of (literally understood) scientific theories; and b) there are good reasons to take certain theories to be true.

An explanationist can also add that empirical justification is “mediated” via inference to the best explanation, as this is how we can arguably construe the relation between our best theories and their empirical support. Furthermore, the argument for scientific realism can itself be an instance of inference to the best explanation, as in the case of the no-miracles (NMA) argument:

The epistemic optimism characteristic of scientific realism is based on NMA. The argument, roughly put, is that empirical success (suitably regimented so as to include novel predictions and the like) offers good reasons to believe in the truth of theories, since it is best explained by the claim that theories are true. Thus conceived, NMA is blind to a distinction between abstract entities and concrete ones insofar as commitment to both types is implied by the truth of (literally understood) scientific theories. (Psillos 2012, 53)

The original (Quine-Putnam) indispensability argument, recited by Psillos, concerns ontological commitment of literally true theories. What should we make of the fact that in light of the above discussion we are not justified in taking explanatory inferences in science as delivering literal truths ? If mathematical realism is but a corollary of scientific realism, presumably it matters what kind of scientific realism it is rational for us to maintain? Taking on board our complete understanding of science and its explanatory endeavors, we must admit that the extent of our ability to answer ontological questions with science, naturalistically, depends on whether or not, and in what sense, scientific explanationism is reliable. The history of science indicates that the reliability of explanationism is curtailed: explanationism allows us to track the unobservable reality only in limited respects (cf. section 4.1). Such curtailed reliability can still generate considerable empirical successes (including novel predictions and the like), if the outputs of explanatory inferences selectively latch onto appropriate features of the world—namely those features that entail the right predictions, and so forth.[1] Hence, the sort of no-miracles argument alluded to by Psillos above is overoptimistic, since the best realist explanations of the empirical success of science need not be in terms of the literal truth of theories. In the history of science, the realist must repeatedly explain the empirical success of theories that are only partially veridical, in a way that is compatible with the curtailed reliability of the scientific method.

We have reason to worry about indispensability arguments, since we have reason to worry that realist explanations of empirical success fail to support mathematical realism. It is a considerable, hitherto unsettled question whether there is any


support for mathematical realism (or other paradigmatically metaphysical views, such as presentism) to be found in the best scientific realist arguments. I am skeptical, but I will not argue for this here. My present point is that our overarching understanding of scientific explanation and explanationism can challenge the strategy of confirmational holism according to which the scientific realist should by her own lights be committed to paradigmatically metaphysical views.

The recent literature on the indispensability argument displays a broad consensus according to which scientific realist commitments cannot be read off from our best theories construed as true simpliciter. Philosophers are nowadays much less concerned with mathematics’ unqualified indispensability in science—the unvarnished Quinean notion that our best scientific theories simply quantify over mathematics. The focus has rather shifted to mathematics’ indispensability for scientific explanation.s.[2] This shift presents a further explanationist twist on the strategy of confirma- tional holism.[3] The champions of the new “explanatory indispensability argument” envisage a more direct route from scientific realism to mathematical realism, based on the notion that (for a scientific realist) ontological commitment and scientific explanation are directly connected. Realist commitment to mathematics allegedly follows from the admission that mathematics plays a genuine “explanatory role” in science.

Although the focus on explanation admittedly enhances the indispensability argument, our best understanding of scientific explanation yet again considerably complicates the key issue at stake. The (burgeoning) literature on the explanatory indispensability argument has by and large taken for granted the connection between explanatory indispensability and ontological commitment, without any reference to a particular conception of explanation to underwrite this connection. Prima facie innocent assumptions concerning explanations’ ontological commitments turn out to harbor various complications, however, when examined in relation to specific accounts of explanation. In particular, the literature on the indispensability argument is rife with references to “mathematics’ (indispensable) explanatory role,” but the key notion of “explanatory role" has been mostly left unanalyzed. This is a major shortcoming: the notion of “mathematics’ explanatory role” must be examined in relation to different analyses and conceptions of explanation to properly judge whether mathematics plays the kind of ontologically committing explanatory role that matters for the indispensability argument. This is critical because in the context of different accounts of explanation we can delineate different kinds of explanatory roles. The admission that mathematics is in a sense “genuinely explanatory,” or is “playing an explanatory role,” does not by itself at all imply that it is playing the right kind of explanatory role. Saatsi (2016a) draws some critical distinctions between different types of explanatory roles—in connection with some leading “ontic” accounts of explanation in particular—pointing to various hitherto unappreciated challenges faced by the strategy of confirmational holism. A properly informed analysis of scientific explanation is again required in order to see whether scientific practice can underwrite a naturalistic argument to a metaphysical conclusion (viz., mathematical realism).

  • [1] For example, although Fresnel’s ether theories of optics or Newtonian gravity are far from being literally true,and their explanatory posits do not exist, these theories arguably selectively latch onto reality so as to generatehugely impressive empirical successes.
  • [2] See, e.g., Colyvan 2012.
  • [3] Baker (2005, 225) puts it thus: A crucial plank of the scientific realist position involves inference to the best explanation (IBE) to justify the postulation in particular cases of unobservable theoretical entities... . [T]he indispensabilitydebate only gets off the ground if both sides take IBE seriously, which suggests that explanation is ofkey importance in this debate.
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