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Should scientific realists be platonists?

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Abstract

Enhanced indispensability arguments (EIA) claim that Scientific Realists are committed to the existence of mathematical entities due to their reliance on Inference to the best explanation (IBE). Our central question concerns this purported parity of reasoning: do people who defend the EIA make an appropriate use of the resources of Scientific Realism (in particular, IBE) to achieve platonism? (§2) We argue that just because a variety of different inferential strategies can be employed by Scientific Realists does not mean that ontological conclusions concerning which things we should be Scientific Realists about are arrived at by any inferential route which eschews causes (§3), and nor is there any direct pressure for Scientific Realists to change their inferential methods (§4). We suggest that in order to maintain inferential parity with Scientific Realism, proponents of EIA need to give details about how and in what way the presence of mathematical entities directly contribute to explanations (§5).

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Notes

  1. We will proceed on the assumption that EIA is employed to argue in favour of platonism: the claim that mathematical objects exist and that mathematical objects are abstracta, as this makes EIA more interesting than if it is used to argue for mathematical realism construed non-platonistically. If EIA is supposed to be used to argue for a non-abstract ontology for mathematics, it is unclear exactly why we would need it, since presumably there are alternative epistemic routes for establishing this conclusion.

  2. We have in mind the EIA that appear in the work of, among others, Alan Baker (Baker 2001, 2005, 2009), Mark Colyvan (2001, 2002, 2006), Lyon and Colyvan (2008), and Bangu (2013).

  3. These two populations may overlap.

  4. This suggestion is similar to formulations found in (Lycan 1988; Josephson and Josephson 1994; Psillos 2002, 2007; Mackonis 2013). We will not evaluate what the correct formulation of IBE should be. The qualification that ‘H would, if true, explain E’ reflects Lipton’s (2004) qualification based on the observation that we only take hypotheses or theories to be actual explanations in so far as they are true. Any hypothesis that acts as an explanation for some E is a potential explanation but only true hypotheses can be actual explanations. Sorin Bangu (2008, 2012) and Mary Leng (2005, 2010) suggest that construing explanation factively is question-begging in this context.

  5. Further GME cases are discussed in Batterman (2010).

  6. See the exchange between Colyvan (2002) and Joseph Melia (2002). Despite their differences, Melia (nominalist) and Colyvan (platonist) agree that if mathematics can be shown to be contributory “in the appropriate way,” then EIA should be considered persuasive (Colvyan 2002, p. 70).

  7. Colyvan relies upon the idea that this is a genuine dilemma for Scientific Realists in order to set up a subsequent challenge for prospective nominalists. The subsequent challenge is that nominalists must either take an ‘easy road’ to nominalism, and deny that there are GMEs, or else take a ‘hard road’ and successfully carry out an ambitious nominalisation project (similar to Field’s 1980 proposal), and he believes that there is no ‘easy road’ (Colyvan 2010, 2012). We argue below that Scientific Realists can avoid the initial dilemma, and so it follows that the ‘easy’ and ‘hard’ roads do not exhaust their options for being nominalists about mathematical entities.

  8. The horizontal/vertical inference distinction is discussed by Peter Lipton (Lipton 2000: 185–6).

  9. John Norton’s ‘material theory’ of induction makes a case for this claim quite neatly – whether a particular ampliative inference is rationally defensible depends upon a ‘material postulate’ which is local to the domain within which the inference is being made. See Norton (2003).

  10. See Busch (2011) for a comparison between the role of IBE in defending Scientific Realism and its role in arguing for Scientific Realism.

  11. For discussion of a related point, see Saatsi (2009).

  12. Grover Maxwell (1970: 17) “Causal connection must be counted among these structural properties, for it is by virtue of them that the unobservables interact with one another and with observables and, thus, that Ramsey sentences have observable consequences.” For discussion of this issue, and the general claim that the ontology of Scientific Realism is wedded to causal features, see Chakravartty (2007, esp. 27–45).

  13. Unification is often championed as one of the virtues that mathematics contributes to the bestness of an explanation or a theory (see Friedman 1974, Kitcher 1981). Note that Salmon’s treatment of these cases does not explicitly appeal to IBE, but to the Common Cause principle.

  14. Part of the debate here might turn on determining what is a permissible or acceptable set of operative norms for Scientific Realism: can there be a blanket ban on the involvement of any abstract objects? Bangu (2013: 257) makes this point: if our starting point is some form of philosophical naturalism that eschews any sort of involvement with abstracta then it is clear that the Indispensability Argument will not work. But both he and Baker seem to be at home with another version of naturalism, one that takes actual scientific practices seriously. They think that since practising scientists seem to be at home with invoking abstracta in their explanations of physical phenomena – not just mathematical objects, but such abstract objects as ‘species’ – then our Scientific Realist should have no a priori restriction against the involvement of abstracta. The methodology of the philosophy of science is not something we will explore further here.

  15. Medeleev’s inference to the existence of germanium was certainly more complicated than this, since it included attending to the gappiness of his table. But its gaps could only be identified as being gaps in virtue of the assumption of a periodic law.

  16. An anonymous referee suggested that we need to give some sort of justification as to why scientific realists are allowed to be selective, otherwise they may appear to be unprincipled or engaging in some sort of ad hoc practice. We shall attempt no defence here (although we note that the selectiveness is not motivated by or embarked upon in an attempt to avoid being committed to abstracta), the interested reader can consult Chakravartty (1998; 2007: 48–51) and Saatsi (2005) for examples of how to be selective. Rather, our point is a broader one: that the Mathematical Realist should not assume that the Scientific Realist has a general purpose strategy for inferring the existence of entities that can also be exploited to generate Platonism.

  17. We note that Baker (2012) is a recent attempt to pin down the nature of mathematical explanation in science, and in this sense, Baker may be coming round to accepting a burden of proof that he has previously repudiated. In that paper, Baker suggests that, given the existence of GMEs, the nature of mathematical explanation in science poses a significant challenge to general accounts of scientific explanation. The point of our argument here is show that GMEs can only be used in EIA if parity with Scientific Realists accounts of explanation is maintained. Baker is right to try to pin down the nature of how mathematical entities can contribute to scientific explanations; our argument here is that, independent of anything else he might reveal about mathematical explanation in science, unless explanation functions in way which maintains parity, Scientific Realists have no reason to accept EIA. Thanks to Josh Hunt for encouraging this point.

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Acknowledgments

Earlier versions of this paper were presented at the Indispensability and Explanation conference hosted by Institut d’histoire et de philosophie des sciences et des techniques, Université Paris Sorbonne, and the Irish Philosophical Club in February 2013, the University of Sheffield’s Departmental Seminar in December 2013. We wish to thank the participants in those discussions for their valuable contributions. We benefitted immensely from discussions and suggestions from Josh Hunt. We are also grateful for help, discussion and contributions from Sorin Bangu, Darragh Byrne, Gerry Hough, David Liggins, Christopher Pincock, Darrell Rowbottom, Juha Saatsi, Naomi Thompson, Richard Woodward and several anonymous reviewers for Synthese. Our thanks to the editors for their efforts in putting together this special edition.

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  1. Department of Culture and Society: Philosophy and History of Ideas, Aarhus University, Aarhus, Denmark

    Jacob Busch

  2. School of Politics, International Studies and Philosophy, Queen’s University Belfast, Belfast, UK

    Joe Morrison

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Correspondence to Jacob Busch.

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Busch, J., Morrison, J. Should scientific realists be platonists?. Synthese 193, 435–449 (2016). https://doi.org/10.1007/s11229-015-0676-6

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