Intrinsic Explanation and Field’s Dispensabilist Strategy

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. Last, 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. Intrinsic Explanation and Field’s Dispensabilist Strategy, Page 1 §1: Overview Quine argued that we should believe that mathematical objects exist because of their indispensable uses in scientific theory. Hartry Field rejects Quine’s argument, arguing that we can reformulate science without referring to mathematical objects. Field provided a precedental reformulation of Newtonian Gravitational Theory (NGT) which has been refined, improved, and extended in the nearly thirty years since his original monograph. In this paper, I argue that Field’s impressive construction and the more recent developments based on it do not impugn Quine’s argument in the way that Field alleges that they do. I do not defend the indispensability argument, generally. I merely attempt to undermine this particular, influential line of criticism. By itself, Field’s reformulation is simply a formal construction. Field argues for its relevance in a defense of nominalism on the basis of a principle of intrinsic explanation. I argue that this principle is not sufficiently motived or defensible, and that it can not do the work for which Field uses it. I start with the relevant background for the Quine/Field disagreement, in §2, and a discussion of Field’s principle, in §3. In §4-§6, I present and reject Field’s arguments for that principle. Finally, in §7, I show that even accepting Field’s principle would not lead to his nominalist, or fictionalist, conclusion. Intrinsic Explanation and Field’s Dispensabilist Strategy, Page 2 See Quines 1939, 1948, 1951, 1955, 1958, 1960, 1978, and 1986. Other versions of the 1 indispensability argument are available. See Putnam 1971 (the success argument); Resnik 1997: §3.3 (the pragmatic indispensability argument); and Mancosu 2008: §3.2 (the explanatory indispensability argument) for a few examples. I focus on Quine’s argument, since Field’s response is directed at it. Carnap might be the instrumentalist toward which QI is aimed: “...[S]ome contemporary 2 nominalists label the admission of variables of abstract types as ‘Platonism’. This is, to say the least, an extremely misleading terminology. It leads to the absurd consequence, that the position of everybody who accepts the language of physics with its real number variables (as a language of communication, not merely as a calculus) would be called Platonistic, even if he is a strict empiricist who rejects Platonic metaphysics” (Carnap 1950: 215, emphasis added). See Melia 2000, Azzouni 2004, and Leng 2005 for more recent defenses of instrumentalism, mainly in response to QI. §2: Quine’s Indispensability Argument and the Dispensabilist Response Quine nowhere presents a detailed indispensability argument, though he alludes to one in many places. I interpret Quine’s argument as follows. QI QI1. We should believe the single, holistic theory which best accounts for our sense experience. QI2. If we believe a theory, we must believe in its ontological commitments. QI3. The ontological commitments of any theory are the objects over which that theory first-order quantifies. QI4. The theory which best accounts for our sense experience first-order quantifies over mathematical objects. QIC. We should believe that mathematical objects exist. An instrumentalist who believes that our uses of mathematics in science do not commit us to the existence of mathematical objects, may deny either QI1 or QI2, or both. Regarding QI1, there is some debate over whether we should believe our best theories. Regarding QI2, one might interpret some of a theory’s references fictionally. I shall return briefly to instrumentalist responses to QI in §7 of this paper. Quine’s procedure for determining the ontological commitments of theories, QI3, is even less controversial than QI1-2. Still, it is odd to talk of the commitments of a theory. People make commitments, or, better, believe things. It is more appropriate to talk about what a theory says, or means, though the connections among meaning, reference, and ontology may remain obscure. That the statements of a theory mean that there are mathematical objects need not entail that they refer to Intrinsic Explanation and Field’s Dispensabilist Strategy, Page 3 Contemporary discussions of the eleatic principle trace mainly to David Armstrong’s work. 3 Armstrong sometimes focuses on causation: “Against the suggestion that the world might contain...such things as possibilities, timeless propositions and “abstract” classes, I argued that these latter entities had no causal power; and that if they had no power there was no good reason to postulate them” (Armstrong 1978b: 46). At other times, Armstrong focuses on spatio-temporal location: “The world is nothing but a single spatio-temporal system” (Armstrong 1978a: 126). Other formulations are found in Oddie 1982: 286; Azzouni 2004: 150; and Field 1989: 68. The eleatic principle is difficult to formulate precisely, in part because necessary and sufficient conditions are always hard to come by, and in part because of its reliance on the concept of causation, which is itself notoriously unclear. See Shapiro 1983. Field says that he, “Came down more solidly in favor of the first order 4 formulations,” in Field 1985b. He continued to take the second-order version seriously in Field 1990. Burgess and Rosen 1997 elegantly collects the slew of reformulation strategies published in the 5 wake of Field’s monograph. See especially the construction at §IIA, based on Burgess 1984, in precisely the spirit of Field’s original work. Most reformulations replace mathematical references with modal ones. I focus on Field’s work since he explicitly defends the principle of intrinsic explanation. mathematical objects, let alone that by using the theory we must believe that there are mathematical objects. I believe that QI3 provides a misleading method for determining the ontological commitments of a theory, though I shall not argue directly against it. Still, instrumentalists who dismiss QI should be prepared to defend alternative criteria for determining their ontological commitments. One alternative to QI3 would be to adopt an eleatic principle on which the ontological commitments of any theory are, approximately, those objects in the causal realm. Debate over QI has focused mostly on QI4. To oppose QI4, Field provided two synthetic reformulations of NGT, replacing the standard analytic version of the theory, which relies on real numbers and their relations, with theories based on physical geometry. A second-order reformulation replaced quantification over mathematical objects with quantification over space-time points. A firstorder reformulation referred instead to space-time regions. There are technical questions about whether Field’s reformulations are adequate for NGT. Field mostly ceded the second-order reformulation, due to problems involving incompleteness. The first-order version, using Quine’s canonical language, is a more 4 appropriate response to QI anyway. There are also questions about whether analogous strategies are available for other current and future theories. I put these questions aside, for this paper, and suppose 5 Intrinsic Explanation and Field’s Dispensabilist Strategy, Page 4 See Field 1980, Chapter 7. See Field 1985a for his arguments for a substantivalist interpretation 6 of space-time. that reformulations in the spirit of Field’s construction are available for our best theories. My concern in this paper is whether such reformulations are better theories than standard ones, for the purposes of QI. The superiority of dispensabilist reformulations is important because the indispensability argument relies on the claim, at QI1-2, that we find our ontological commitments in our best theory. While Field defends his reformulation on the basis of a principle of intrinsic explanation, I argue that we do not really accept this principle, and that the standard theory is preferable to its dispensabilist counterpart. Thus, I reject Field’s claim that QI4 is false, not because reference to mathematical objects is ineliminable from science, but because the reformulated theories which eliminate quantification over mathematical objects are not our best theories. The value of dispensabilist reformulations has been questioned before. Pincock 2007 argues that the standard theory is better confirmed. To construct representation theorems which demonstrate that a reformulation is adequate, the dispensabilist adopts axioms about the physical world and its properties. For example, to measure mass or temperature, Field assumes the existence of spatio-temporal regions or points, and orderings among them, to do the work that connected sets of real numbers do in the standard theory. But, Pincock argues, those assumptions about the physical world are not as well confirmed as the 6 corresponding mathematical axioms and mappings between physical and mathematical structures. Against Pincock’s claim, even if the dispensabilist’s axioms are less well confirmed than the mathematical axioms they replace, they may derive some measure of confirmation from their adequacy. Furthermore, the dispensabilist reformulation, eschewing mathematical objects, makes fewer commitments. It is not clear how to balance the virtue of having fewer commitments with the benefit of having a greater degree of confirmation. Burgess and Rosen 1997 argue that a better theory should be publishable in scientific journals, and adopted by working scientists; since dispensabilist reformulations are not preferred by practicing Intrinsic Explanation and Field’s Dispensabilist Strategy, Page 5 On Burgess and Rosen’s suggestion: “While entertaining as rhetorical flourishes, such demands 7 leave a serious explanatory gap...” (Pincock 2007: 255). See Field 1980: viii, 8 and 41. 8 See Colyvan 1999 and Colyvan 2001: §4.3. 9 scientists, they are no better. This is a wrong way to measure the value of a theory. The practicing scientist wants a useful theory to produce and replicate empirical results. The scientist is mainly unconcerned with ontological commitments. Field’s defense of his reformulation correctly emphasizes concerns about ontological commitments. Dismissing such concerns, as Burgess and Rosen do, begs the questions raised by QI about whether scientific theories must quantify over mathematical objects. Much of the debate over whether dispensabilist reformulations are better than their standard counterparts has focused on their attractiveness. Field uses attractiveness as a criterion for acceptable reformulations in his original work. But, attractiveness is a vague and malleable criterion. One might 8 find a theory attractive based on its strength, simplicity, or explanatory power, for just a few examples. It is further unclear how to balance such considerations. “Of course, it is a deep and difficult question how the various attributes that contribute towards a theory’s attractiveness ought to be spelled out, and how these attributes are to be independently measured and weighed against each other” (Melia 2000: 472). Mark Colyvan, arguing that the standard theory is more attractive than Field’s reformulation, mentions the unification achieved by the standard theory, and its boldness, simplicity, and predictive powers. Unfortunately, Colyvan’s argument remains sketchy. “While I admit that I have remained 9 rather vague about the details of how to compare theories, nevertheless I have presented a case for accepting that mathematical entities directly contribute toward qualities such as boldness and unificatory power, which we see as properties of good theories” (Colyvan 2001: 87). This paper pursues and extends Colyvan’s argument, criticizing Field’s own criterion for attractiveness, his principle of intrinsic explanation. I present a specific explanation of why Field’s reformulation is not a better or more attractive theory than the standard one. Intrinsic Explanation and Field’s Dispensabilist Strategy, Page 6 Joseph Melia argues that space-time points are actually extrinsic to physical theories; see Melia 1