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Grounding and the indispensability argument

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Abstract

There has been much discussion of the indispensability argument for the existence of mathematical objects. In this paper I reconsider the debate by using the notion of grounding, or non-causal dependence. First of all, I investigate what proponents of the indispensability argument should say about the grounding of relations between physical objects and mathematical ones. This reveals some resources which nominalists are entitled to use. Making use of these resources, I present a neglected but promising response to the indispensability argument—a liberalized version of Field’s response—and I discuss its significance. I argue that if it succeeds, it provides a new refutation of the indispensability argument; and that, even if it fails, its failure may bolster some of the fictionalist responses to the indispensability argument already under discussion. In addition, I use grounding to reply to a recent challenge to these responses.

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Notes

  1. For introductions to grounding, see (Correia and Schnieder 2012; Clark and Liggins 2012; Trogdon 2013).

  2. A footnote of Colyvan’s (2001, p. 12 fn. 19) suggests he would have no quarrel with my formulation. On the appeal to confirmational holism, see (Morrison 2012). On naturalism, see Williamson (2011a, b).

  3. Some philosophers will object to this definition by claiming that (i) if platonism is true, nominalism is metaphysically impossible; and (ii) it is a trivial matter how things would be were some metaphysical impossibility the case. But neither of these claims should command our confidence; still less should their conjunction. In particular, (i) lacks a secure motivation and there are serious arguments against it; and the Lewis–Stalnaker semantics for counterfactuals—the usual motivation for (ii)—is not beyond question. I cannot cite all the relevant literature here, but regarding (i), see Miller (2010); regarding (ii), see Nolan (1997), Sect. 2. (Philosophers who object the definition given in the text, whether on these grounds or others, may think of nominalistic properties and relations as those that do not ‘involve’ mathematical objects. Thanks to an anonymous referee for this alternative definition.)

  4. This resembles the view typically attributed to Pythagoras. It is similar to the position Field (1989, pp. 186–193) calls ‘heavy duty platonism’—but not quite the same. Field (1989, p. 181 n. 16) insists he is talking about predicates, not properties, and he envisages the alternative to ‘heavy duty platonism’ to involve an appeal to representation theorems (187), whereas that does not follow from the view under discussion here. So my option (a) is not exactly Field’s ‘heavy duty platonism’.

  5. See (Rosen (2010), p. 118) on the relation between grounding and necessitation.

  6. Yablo cannot be straightforwardly classed as a nominalist, but his view is close enough to abstract expressionism to be worth mentioning here.

  7. Note that I do not claim that the fast lane philosopher should hold that ‘S-g’ and suchlike are semantically primitive predicates. Neither do I claim that platonists should hold this either. Thanks to an anonymous referee for pressing me to be clearer on this.

  8. Field (1980, pp. ix, 6, 43, 45–46, 1989, pp. 18–19) claims these two virtues for his own reformulations of science. But see Melia (1998) for criticism.

  9. Melia’s point echoes Mundy’s criticism of appeal to representation theorems, mentioned in Sect. 2 above.

  10. See Liggins (2012), Sect. 3 for related discussion.

  11. Instead of ‘\(\forall x \exists y \hbox {M-}yx\)’, the fast lane philosopher might try to express the claim that each of the objects has a mass using ‘\(\forall x \exists F (\Phi (F) \wedge Fx)\)’, where ‘\(\Phi (F)\)’ holds just in case ‘\(F\)’ ascribes a mass-property. This option also goes beyond the resources of predicate calculus—even second-order predicate calculus. It raises similar issues to the formulation in the text. Thanks to an anonymous referee for this suggestion.

  12. We should not assume that the explanation of how the new language functions has to take place in predicate calculus, or that it must avoid the use of the new resources: see Williamson (1999, 2014).

  13. See Baker (1993) for a related point.

  14. One may well worry that step (ii) fails in the mathematical case: the graph-theoretic explanation of why no-one has ever walked across Königsberg crossing each bridge exactly once seems more explanatory than the explanation appealing to the physical property in virtue of which the graph-theoretic property is instantiated; the nominalistic explanation is messier, less elegant. I am grateful both to an anonymous referee and to Matteo Plebani for raising this worry (see Plebani 2014). It echoes earlier criticisms of inference to the best explanation arguments for moral realism (see e.g. Sayre-McCord 1988, p. 449; Railton 1998, pp. 179–180). In my ‘Grounding, explanation, and multiple realization in mathematics and ethics’ I discuss this worry and the earlier metaethical arguments in detail. In brief: I suggest that these ‘multiple realizability’ worries can both be laid to rest if opponents of realism can provide explanations of the phenomena which—while remaining non-moral or non-mathematical—are not too specific to forfeit explanatory insight. I argue that realists should think their alleged moral or mathematical facts have grounds which have the right degree of specificity; and that their opponents can and should appeal to these facts in their explanations. So I think that the fast-lane philosopher can deal with the worry provided they are careful to select the appropriate alleged grounds.

  15. The example is Colyvan’s but strongly echoes Melia (2000, p. 469).

  16. Explaining how the content is conveyed is also a challenge for abstract expressionism: see Liggins (forthcoming) for discussion.

  17. Lyon and Colyvan (2008) discuss such examples.

  18. See Balaguer (1998), pp. 13–14 for related remarks.

  19. Thanks to Chris Daly and to participants in the Paris conference, especially Mark Colyvan. Thanks also to the editors of this volume and the anonymous referees for this paper. Particular thanks to Matteo Plebani for many illuminating and enjoyable discussions.

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    David Liggins

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Liggins, D. Grounding and the indispensability argument. Synthese 193, 531–548 (2016). https://doi.org/10.1007/s11229-014-0478-2

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