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
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.)
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’.
See (Rosen (2010), p. 118) on the relation between grounding and necessitation.
Yablo cannot be straightforwardly classed as a nominalist, but his view is close enough to abstract expressionism to be worth mentioning here.
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.
Melia’s point echoes Mundy’s criticism of appeal to representation theorems, mentioned in Sect. 2 above.
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.
See Baker (1993) for a related point.
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.
The example is Colyvan’s but strongly echoes Melia (2000, p. 469).
Explaining how the content is conveyed is also a challenge for abstract expressionism: see Liggins (forthcoming) for discussion.
Lyon and Colyvan (2008) discuss such examples.
See Balaguer (1998), pp. 13–14 for related remarks.
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.
References
Baker, A. (2003). Does the existence of mathematical objects make a difference? Australasian Journal of Philosophy, 81, 246–64.
Baker, A. (2005). Are there genuine mathematical explanations of physical phenomena? Mind, 114, 223–238.
Baker, L. R. (1993). Metaphysics and mental causation. In J. Heil & A. Mele (Eds.), Mental causation (pp. 75–95). Oxford: Clarendon Press.
Balaguer, M. (1998). Platonism and anti-platonism in mathematics. New York: Oxford University Press.
Balaguer, M. (2011). Fictionalism in the philosophy of mathematics. In Edward N. Zalta (ed.) Stanford Encyclopedia of Philosophy (Fall 2011 edition). http://plato.stanford.edu/archives/fall2011/entries/fictionalism-mathematics/.
Clark, M. J., & Liggins, D. (2012). Recent work on grounding. Analysis, 72, 812–823.
Colyvan, M. (2001). The indispensability of mathematics. New York: Oxford University Press.
Colyvan, M. (2010). There is no easy road to nominalism. Mind, 119, 285–306.
Colyvan, M. (2012). Road work ahead: heavy machinery on the easy road. Mind, 121, 1031–1046.
Correia, F., & Schnieder, B. (2012). Grounding: An opinionated introduction. In F. Correia and B. Schnieder (Eds.), Metaphysical grounding: Understanding the structure of reality. Cambridge: Cambridge University Press.
Crane, T. (1990). An alleged analogy between numbers and propositions. Analysis, 50, 224–230.
Creath, R. (1980). Nominalism by theft. American Philosophical Quarterly, 17, 311–318.
Daly, C., & Langford, S. (2009). Mathematical explanation and indispensability arguments. Philosophical Quarterly, 59, 641–658.
Dasgupta, S. (2013). Absolutism vs comparativism about quantities. Oxford Studies in Metaphysics, 8, 105–150.
Davidson, D. (1965). Theories of meaning and learnable languages. In Y. Bar-Hillel (Ed.), Logic, Methodology and Philosophy of Science: Proceedings of the 1964 International Congress (pp. 383–394). Amsterdam: North-Holland.
Field, H. (1980). Science without numbers: A defence of nominalism. Oxford: Blackwell.
Field, H. (1984). Review of Dale Gottlieb’s Ontological economy: Substitutional quantification and mathematics. Noûs, 18, 160–165.
Field, H. (1989). Realism, mathematics and modality. Oxford: Blackwell.
Fine, K. (1995). Ontological dependence. Proceedings of the Aristotelian Society, 95, 269–90.
Hawthorne, J. (2006). Quantity in Lewisian metaphysics. Metaphysical essays (pp. 229–243). Oxford: Clarendon Press.
Horgan, T. (1984). Science nominalized. Philosophy of Science, 51, 529–549.
Horgan, T. (1989). Attitudinatives. Linguistics and Philosophy, 12, 133–165.
Künne, W. (2006). Properties in abundance. In P. F. Strawson & A. Chakrabarti (Eds.), Universals, concepts, and qualities: New essays on the meaning of predicates (pp. 249–300). Aldershot: Ashgate.
Leng, M. (2010). Mathematics and reality. Oxford: Oxford University Press.
Liggins, D. (2003). On being twice as heavy. Philosophia Mathematica, 11, 203–207.
Liggins, D. (2012). Weaseling and the content of science. Mind, 121, 997–1005.
Liggins, D. (forthcoming). Abstract expressionism and the communication problem. British Journal for the Philosophy of Science, axt012.
Liggins, D. (MS). Grounding, explanation, and multiple realization in mathematics and ethics.
Lyon, A., & Colyvan, M. (2008). The explanatory power of phase spaces. Philosophia Mathematica, 16, 227–243.
Lyon, A. (2012). Mathematical explanations of empirical facts, and mathematical realism. Australasian Journal of Philosophy, 90, 559–578.
Malament, D. (1982). Review of Hartry Field’s Science without numbers. Journal of Philosophy, 79, 523–534.
Melia, J. (1995). On what there’s not. Analysis, 55, 223–229.
Melia, J. (1998). Field’s programme: Some interference. Analysis, 58, 63–71.
Melia, J. (2000). Weaseling away the indispensability argument. Mind, 109, 455–479.
Melia, J. (2003). Defending the indispensible. Metascience, 12, 55–58.
Melia, J. (2006). The conservativeness of mathematics. Analysis, 66, 202–208.
Meyer, U. (2002). Is science first-order? Analysis, 62, 305–308.
Miller, K. (2010). Three routes to contingentism in metaphysics. Philosophy Compass, 5, 965–977.
Morrison, J. (2012). Evidential holism and indispensability arguments. Erkenntnis, 76, 263–278.
Mundy, B. (1987). The metaphysics of quantity. Philosophical Studies, 51, 29–54.
Nagel, E. (1931). Measurement. Erkenntnis, 2, 313–333.
Nolan, D. (1997). Impossible worlds: A modest approach. Notre Dame Journal of Formal Logic, 38, 535–572.
Pincock, C. (2007). A role for mathematics in the physical sciences. Noûs, 41, 253–275.
Pincock, C. (2012). Mathematics and scientific representation. New York: Oxford University Press.
Plebani, M. (2014). Nominalistic content, grounding, and covering generalizations: Reply to ‘Grounding and the indispensability argument’. Synthese (this volume).
Putnam, H. (1971). Philosophy of logic. New York: Harper. Reprinted in his 1979 Mathematics, matter and method: Philosophical papers, (2nd edn., Vol. 1, pp. 323–357). Cambridge: Cambridge University Press.
Railton, P. (1998). Moral explanation and moral objectivity. Philosophy and Phenomenological Research, 58, 175–182.
Resnik, M. D. (1985). How nominalist is Hartry Field’s nominalism? Philosophical Studies, 47, 163–181.
Resnik, M. D. (1997). Mathematics as a science of patterns. Oxford: Clarendon Press.
Rosefeldt, T. (2008). ‘That’-clauses and non-nominal quantification. Philosophical Studies, 137, 301–333.
Rosen, G. (2010). Metaphysical dependence: Grounding and reduction. In B. Hale & A. Hoffmann (Eds.), Modality: Metaphysics, logic, and epistemology (pp. 109–135). Oxford: Oxford University Press.
Sayre-McCord, G. (1988). Moral theory and explanatory impotence. Midwest Studies in Philosophy, 12, 433–57.
Schiffer, S. (2003). The things we mean. Oxford: Clarendon Press.
Shapiro, S. (1983). Conservativeness and incompleteness. Journal of Philosophy, 80, 521–531.
Trogdon, K. (2013). An introduction to grounding. In M. Hoeltje, B. Schnieder, & A. Steinberg (Eds.), Varieties of dependence: Ontological dependence, grounding, supervenience, response-dependence. Munich: Philosophia.
Turner, J. (2011). Review of Yablo’s Things: Philosophical papers, (Vol. 2). http://ndpr.nd.edu/news/25473-things-philosophical-papers-volume-2/.
Williamson, T. (1999). Truthmakers and the converse Barcan formula. Dialectica, 53, 253–270.
Williamson, T. (2011a). What is naturalism? New York Times (online), 4 September 2011. http://opinionator.blogs.nytimes.com/category/the-stone/.
Williamson, T. (2011b). On ducking challenges to naturalism. New York Times (online), 28 September 2011. http://opinionator.blogs.nytimes.com/category/the-stone/.
Williamson, T. (2014). Logic, metalogic and neutrality. Erkenntnis, 79, 211–231.
Yablo, S. (2001). Go figure: A path through fictionalism. Midwest Studies in Philosophy, 25, 72–102.
Yablo, S. (2002). Abstract objects: A case study. Philosophical Issues, 12, 220–240.
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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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DOI: https://doi.org/10.1007/s11229-014-0478-2