Living in Harmony: Nominalism and the Explanationist Argument for Realism

Abstract

According to the indispensability argument, scientific realists ought to believe in the existence of mathematical entities, due to their indispensable role in theorising. Arguably the crucial sense of indispensability can be understood in terms of the contribution that mathematics sometimes makes to the super‐empirical virtues of a theory. Moreover, the way in which the scientific realist values such virtues, in general, and draws on explanatory virtues, in particular, ought to make the realist ontologically committed to abstracta. This paper shows that this version of the indispensability argument glosses over crucial detail about how the scientific realist attempts to generate justificatory commitment to unobservables. The kind of role that the Platonist attributes to mathematics in scientific reasoning is compatible with nominalism, as far as scientific realist arguments are concerned.

Acknowledgements

I want to thank Jacob Busch, Angelo Cei, Mark Colyvan, Chris Daly, and Steven French for helpful correspondence, two anonymous referees for their comments, and Chris Daly also for his careful reading of an early version of the manuscript.

Notes

[1] There may be many ways of construing the indispensability argument. Perhaps it can be viewed as a package which includes all sorts of considerations that are not even mentioned here. The present discussion focuses only on this particular way of framing the argument as explicitly directed to the contemporary scientific realist. This version of the argument is very prominent in the literature cited.

[2] As Colyvan puts it, ‘How are we to understand the phrase “indispensable to our best scientific theory”? Much hangs on this question, and I'll need to treat it in some detail … In fact, whatever sense it is in which electrons, neutron stars, and viruses are indispensable to their respective theories will do’ (2001a, 12).

[3] These include, for example, simplicity, unificatory and explanatory power, and fecundity. See Colyvan (2001a, 78–9).

[4] For the purposes of this paper we can understand ‘scientific realism’ as optimism regarding the arguments rebutting such anti‐realist challenges, rather than as optimism about the approximate truth of our best theories, featuring mathematics and all. This statement of realism is more prudent than the familiar motto ‘our best theories are approximately true’, and it serves us better in the present context by more faithfully representing the contemporary realism debate.

[5] In so far as such qualified scepticism about the justification of theoretical beliefs about the unobservable is taken seriously, the realist is under pressure to go beyond Quinean scientific realism. The exact character of Quine's realism is difficult to pin down, and I cannot attempt to capture the position in the space available here. What matters for the issue at hand, however, are the idiosyncratic elements of Quine's philosophy, with respect to the general status of knowledge and his empiricism, that are not part and parcel of the contemporary realism debate. Although the present argument is not a wholesale repudiation of Quine's brand of holism and naturalism, I think it is safe to say that responding to my argument by appealing to idiosyncrasies of Quinean realism would transform the essence of contemporary scientific realism beyond recognition. For example, as is well known, Quine's ultimate response to underdetermination was to adopt a form of ontological relativism.

[6] Colyvan's claim that scientific realism tout court is incompatible with nominalism—that an insistent nominalist can be ‘forced to scientific instrumentalism’—is quite obviously overstated. Due to the grand difficulties in coming up with a credible global justificatory argument for scientific instances of IBE, many have opted for less. It is not clear why the proponents of the various entity realist arguments, say, would be at all moved by the kind of descriptive unity that the indispensability argument capitalises on. Ditto for a realist like Kitcher (2001), whose Galilean strategy provides another piecemeal justificatory device, turning on an assumption about the uniformity of the success‐producing conditions for theorising. But bypassing all these positions can perhaps be considered as a purely rhetorical move; after all, none of them depends on a wholesale espousal of IBE. (Note, though, that advocating a descriptive IBE‐model of the scientific method is compatible with each of these realist positions.)

Unqualified talk of the realist advocacy of IBE is also unwise. Some realists have put forward arguments that are not meant to apply across the board but only to suitably analysed scientific inferences of a particular form. Lipton (2004) and McMullin (1984), for example, argue that some scientific inferences—via causal and causal‐structural explanations, respectively—to the unobservable are of the same form as the kinds of inferences to the observable that the anti‐realist is happy with. Such restricted ‘bridge arguments’ for realism, even though they rely on the assumption that scientific reasoning is fundamentally abductive, do not seem to lend themselves to the Platonist analogy.

[7] Although Psillos in the above quote speaks of approximate truth of background theories without any anti‐Platonist qualifications, his more detailed discussion makes it clear that he means truth about concrete theoretical posits (1999, 79).

[8] Something like this was suggested by a referee.

[9] This point also functions as a partial response to Leng (2005), who argues that scientific realism, as formulated by Psillos, for example, is committed to the existence of mathematical abstracta purportedly referred to (or ‘posited’) in our best theories (1999, 74). Leng maintains that for this reason, and since the no‐miracles argument by itself fails as an argument for Platonism, scientific realists should be particularly worried about the Platonism/anti‐Platonism issue. Although I admit that there is room for increased precision in typical statements of scientific realism, in my view there is a natural, more charitable, reading of the realist commitments spoken for by the no‐miracles argument. Scientific realist talk of the ‘entities posited by a theory’, in particular, does not amount to mere quantification over these entities in some preferred logical formalisation of the theory, but takes into account the way these entities are introduced, by scientists, into our belief corpus.

[10] See also Melia (2002), Balaguer (1998, 137).

[11] Melia (2000) provides more detailed discussion of the way that mathematics can simplify theories, not the world. These details do not matter to us.

[12] Colyvan (2001a) discusses this point and also provides another response in a Quinean holistic spirit. The latter emphasises the difference in the stances that the nominalist and the Platonist hold with respect to gauging the super‐empirical virtues of a theory. The nominalist begins with the presupposition that the world just is the physical world, and hence the relevant theoretical utility (simplicity, or unification, say), assessed relative to the world, is measured relative to the physical content of the theory. The holist‐Platonist, in contrast, looks at the theory more open‐mindedly, as it were, and decides on questions of unification, and explanatory power, say, without such implicit physicalist bias. In effect, the nominalist is accused of question begging.

[13] I find Baker's reasoning seriously flawed at various points, but for the sake of the present argument we can take it showing that there is a sense in which mathematics can be considered genuinely explanatory.

[14] Some accounts of scientific explanation are more amicable to mathematics being explanatory than others. See, for example, de Regt and Dieks (2005).

Additional information

Notes on contributors

Juha T. Saatsi

Juha T. Saatsi is at the School of Philosophy, University of Leeds.