Abstract
Much recent discussion in the philosophy of mathematics has concerned the indispensability argument—an argument which aims to establish the existence of abstract mathematical objects through appealing to the role that mathematics plays in empirical science. The indispensability argument is standardly attributed to W. V. Quine and Hilary Putnam. In this paper, I show that this attribution is mistaken. Quine’s argument for the existence of abstract mathematical objects differs from the argument which many philosophers of mathematics ascribe to him. Contrary to appearances, Putnam did not argue for the existence of abstract mathematical objects at all. I close by suggesting that attention to Quine and Putnam’s writings reveals some neglected arguments for platonism which may be superior to the indispensability argument.
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Notes
What it is to be abstract has also been a topic of discussion. Perhaps the most popular approach has been to say that an entity is abstract iff it lacks spatio–temporal location and is causally inactive. Nothing I will say here hinges on what abstractness is taken to be; see Hale (1987, chapter 3) and Rosen (2001) for more on the debate.
A few authors differ: Resnik (1997: 45 n. 3) is unsure whether Quine and Putnam intended the argument they are usually credited with; he produces a quotation from Putnam 1971 which he interprets as expressing a different, ‘pragmatic’ argument for platonism (Resnik 1997: 47 n. 7). And see footnote 5 below.
In this exposition of Quine’s ontological method, I have deliberately ignored his theses of ontological relativity and inscrutability of reference. For an explanation of why these doctrines do not undermine the project of ontology, see Hylton (2004), §V.
Not every author has agreed with this attribution. During a discussion of Colyvan, Pincock (2004: 67) asserts that Philosophy of Logic is an argument for Harvard realism, not platonism; and whilst surveying the varieties of nominalism found in print, Burgess and Rosen (1997: 201) spend a paragraph arguing for the same interpretation. Since these passages have met with a resounding silence, I judge that a more detailed discussion is required.
The references to Errett Bishop (a prominent intuitionist) and the scorn heaped on the view that numbers and functions are ‘mere fictions’ (74) make this clear.
‘This [if-thenist view of applied mathematics] is wrong. Cf. chapter 4 [i.e. Putnam 1975] in which it is argued that one cannot consistently be a realist in physics and an if-thenist in mathematics’ (33).
One ‘unconsidered complication’ of my own is that ‘mathematics as modal logic’ changed between 1967b and 1975. In the earlier paper, Putnam uses logical necessity in his modal translations; in the later one, he invokes ‘a strong and uniquely mathematical sense of “possible” and “impossible”’ (1975: 70). Field (1988: 270 n. 36) is baffled by the change, but I think that he himself provides an explanation (at 1984: 85 n. 7): if we use purely logical modality, every consistent theory will come out as equally ‘good’, and, since this is incompatible with bivalence (which the ‘objects picture’ is intended to support), the ‘equivalent descriptions’ thesis will be hard to sustain. Shifting to a less permissive sort of modality resolves the problem. I do not think that admitting Putnam’s change of heart threatens the argument in the text.
Indeed, there are passages where he seems to suggest that ‘mathematics as modal logic’ is superior to the standard picture: ‘The theory of mathematics as the study of special objects has a certain implausibility which, in my view, the theory of mathematics as the study of ordinary objects with the aid of a special concept does not…[P]uzzles… as to how one can refer to mathematical objects… can be clarified with the aid of modal notions’ (1975: 72).
Quine (1960: 242) argues that there is no syntactic criterion of ontological commitment that applies directly to natural language sentences: ‘[A]t best there is no simple correlation between the outward forms of ordinary affirmations and the existences implied’. To show this, Quine points out that the sentence ‘Agnes has fleas’ can be interpreted as ‘∃x (Fx & Gx)’ whereas other sentences of the same form, such as ‘Tabby eats mice’ and ‘Ernest hunts lions’, cannot. (see also Quine 1969a: 106.)
Maddy (1997: 133–143) attacks the conjunction of confirmational holism and Quine’s criterion of ontological commitment. Only recently has she chosen to lay the blame specifically with confirmational holism. For commentary on Sober, see Resnik (1997, chapter 7), Colyvan (2001: 126–134), and Leng (2002). For commentary on Maddy, see Colyvan (2001, chapter 5) and Leng 2002).
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Acknowledgments
This paper draws on chapter 3 of my Ph.D. thesis (Nominalist accounts of Mathematics, University of Sheffield, UK). I would like to give the warmest of thanks to Rosanna Keefe and Chris Hoookway for their painstaking supervision. I would also like to thank Hilary Putnam for discussing his work with me. For comments and advice, thanks to Chris Daly, Michael Scott, and an anonymous referee.
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Liggins, D. Quine, Putnam, and the ‘Quine–Putnam’ Indispensability Argument. Erkenn 68, 113–127 (2008). https://doi.org/10.1007/s10670-007-9081-y
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DOI: https://doi.org/10.1007/s10670-007-9081-y