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Abstract
Hartry Field defended the importance of his nominalist reformulation of Newtonian Gravitational Theory, as a response to the indispensability argument, on the basis of a general principle of intrinsic explanation. In this paper, I argue that this principle is not sufficiently defensible, and can not do the work for which Field uses it. I argue first that the model for Field’s reformulation, Hilbert’s axiomatization of Euclidean geometry, can be understood without appealing to the principle. Second, I argue that our desires to unify our theories and explanations undermines Field’s principle. Third, the claim that extrinsic theories seem like magic is, in this case, really just a demand for an account of the applications of mathematics in science. Finally, even if we were to accept the principle, it would not favor the fictionalism that motivates Field’s argument, since the indispensabilist’s mathematical objects are actually intrinsic to scientific theory.
Notes
1 See Quine Citation1939, Citation1955, Citation1958, Citation1960, Citation1978, Citation1980a, Citation1980b, and 1986. For other versions of the indispensability argument, see Putnam Citation1975 (the success argument); Resnik Citation1997: §3.3 (the pragmatic indispensability argument); and Mancosu Citation2008: §3.2 (the explanatory indispensability argument). I focus on Quine's argument because Field's response is directed at it.
2 See Carnap 1950. For more recent defenses of instrumentalism in response to QI, see Melia 2000; Azzouni 2004; and Leng 2005a.
3 Contemporary discussions of the eleatic principle trace mainly to David Armstrong's work. Armstrong sometimes focuses on causation (see Armstrong Citation1978b: p. 46), at other times on spatio-temporal location (see Armstrong Citation1978a: p. 126). Other formulations are found in Oddie Citation1982: 286; Azzouni Citation2004: p. 150; and Field Citation1989a: p. 68.
4 See Shapiro Citation1983; Field Citation1989c, Citation1990.
5 Burgess and Rosen (Citation1997) elegantly collects the slew of reformulation strategies published in the wake of Field's monograph. See especially the construction at §IIA in the spirit of Field's original work. Most reformulations replace mathematical references with modal ones.
6 See Field Citation1980, Chapter 7. See Field Citation1989b for his arguments for a substantivalist interpretation of space-time.
7 On Burgess and Rosen's suggestion: ‘While entertaining as rhetorical flourishes, such demands leave a serious explanatory gap...' (Pincock Citation2007: p. 255).
8 See Field Citation1980: pp. viii, 8 and 41.
9 See Colyvan Citation1999 and Colyvan Citation2001: §4.3.
10 Joseph Melia argues that space-time points are actually extrinsic to physical theories; see Melia 1998: pp. 65–7.
11 See Mancosu Citation2008, §3.2, for a formulation of the explanatory argument, and Baker Citation2005; Colyvan Citation2001; Colyvan Citation2010; and Lyon and Colyvan Citation2007 for defenses of the argument.
12 If a mathematical theory is conservative over a nominalist physical theory, then we can use the mathematics to facilitate derivations in the physical theory with assurance that we will not derive any unacceptable empirical consequences. The conservativeness of mathematics would assure us that Field's reformulation need have no consequences for working scientists.
13 See Hilbert 1971. On Hilbert's influence, see Field Citation1980, Chapter 3.
14 Hilbert mentions both in a letter to Frege on his motivation for axiomatizing geometry (Frege Citation1980: Letter IV/4).
15 See Kitcher Citation1981: p. 508.
16 The nominalist theory may also be used in an account of the applicability of mathematics in empirical science. But, though the platonist can use any nominalist account just as well, so this does not serve to distinguish the nominalist from the platonist.
17 ‘On the one hand, the indispensability argument sides with nominalists in avoiding any presupposition that mathematical statements are intrinsically privileged. On the other hand, the argument sides with Platonists in taking mathematical statements at face value, as making genuine ontological claims … This evenhandedness is an important strength of the indispensability argument …' (Baker Citation2003: p. 50).
18 See Leng Citation2005b and Yablo Citation2005.
19 As an anonymous reviewer notes, Field could object that there is a difference between the kinds of bridge principles involved when T1 and T2 are both scientific theories and when one is scientific and one is mathematical: such principles are causal in the former case and acausal in the latter. In such a response, PIE is doing no work. The difference would be based on an eleatic principle. Such a move, then, would be consistent with the claim of this paper that PIE does not support Field's reformulation of NGT.