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The holistic presumptions of the indispensability argument

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Abstract

The indispensability argument is sometimes seen as weakened by its reliance on a controversial premise of confirmation holism. Recently, some philosophers working on the indispensability argument have developed versions of the argument which, they claim, do not rely on holism. Some of these writers even claim to have strengthened the argument by eliminating the controversial premise. I argue that the apparent removal of holism leaves a lacuna in the argument. Without the holistic premise, or some other premise which facilitates the transfer of evidence to mathematical portions of scientific theories, the argument is implausible.

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Notes

  1. Mark Colyvan credits Frege and Gödel; see Colyvan (2001, §1.2.1). Garavaso (2005) properly disputes the attribution to Frege. Gödel’s indispensability argument is at most an intra-theoretic argument, rather than a standard inter-theoretic argument. (For the difference, see Marcus 2010, § 8.)

  2. See Maddy (1997): §III.4.

  3. See Morrison (2010) for recent relevant discussion.

  4. “Let us call a claim brute if its obtaining or not obtaining doesn’t depend on anything else; barren if no phenomena from a different domain depend on it; and absolutely insular if it is both brute and barren: (Field 1993, p. 296). Alternatively, see Mark Balaguer’s principle of causal isolation which denies causal interactions between mathematical and physical objects: (Balaguer 1998, p. 110).

  5. Resnik’s defense of RP precedes Robert Batterman’s recent work on asymptotic reasoning (see  Batterman 2003) but can also be seen as a way of attempting to avoid concerns which arise from considering the idealizations that Batterman convincingly argues are central to scientific discourse.

  6. Similarly, see Sober (1999, pp. 52–53).

  7. See Balaguer (1998, Chap. 7).

  8. Baker has other purposes, including appealing to the explanatory role of mathematics in addition to its representational role. Some critics of the indispensability argument (e.g. Melia 2000; Leng 2010) claim that the merely representational role of mathematics in science is insufficient to justify our mathematical beliefs.

  9. See Lyon and Colyvan (2007) on the honeycomb conjecture; Mancosu (2011) on the twisting tennis racket theorem (among other explanations); and Bangu (2012) on uses of mathematics in economics.

  10. Other accounts of explanation fare worse. As I argue in Marcus (2014), the version of ‘explanation’ on which Baker must rely in order to distinguish EI from the standard indispensability argument is too weak to support Baker’s first premise, whether or not holism is presumed.

  11. See Bangu (2012, Chap. 8) and Morrison (2012, p. 276). Morrison finds a lacuna similar to the one I describe in this paper.

  12. See Azzouni (2004), Chap. 6.

  13. Morrison argues that some defenses of naturalism in indispensability arguments depend on holism; see Morrison (2012): § 2.

  14. Liggins’ Harvard realism is what I earlier called sentence realism. The Harvard realist takes existentially quantified mathematical sentences to be true without accepting the existence of abstract mathematical objects. Alternative semantics for such sentences include taking mathematical terms to refer to modal properties or to arrangements of physical objects (or to possible such arrangements).

  15. An exegetical aside: Busch and Sereni support their claim that holism is inessential by referring to Putnam’s elision of holism: “So far I have been developing an argument for realism roughly along the following lines: quantification over mathematical entities is indispensable for science, both formal and physical, therefore we should accept such quantification; but this commits us to accepting the existence of the mathematical entities in question” (Putnam 1971, p. 347; cited at Busch and Sereni 2012, p. 350). Busch and Sereni go on to say: “No mention is made here of naturalism nor holism.”

    Busch and Sereni are correct that Putnam hints at a non-holistic version of the indispensability argument, one like some of the non-holistic versions here. But the fact that Putnam does not mention holism is no evidence that it is not lurking in the background. Putnam’s argument is really a forebear of Resnik’s RP and similarly presumes transfer of evidence from science to mathematics.

    Similarly, Busch and Sereni find support for their non-holistic version from a thin reading of Quine’s work. First, they quote Quine: “Ordinary interpreted scientific discourse is as irredeemably committed to abstract objects – to nations, species, numbers, functions, sets – as it is to apples and other bodies. All these things figure as values of the variables in our overall system of the world. The numbers and functions contribute just as genuinely to physical theory as do hypothetical particles” (Quine 1981, pp. 149–150). Then they claim that holism is absent from the argument: “Even though [holism] might have been a working hypothesis of Quine’s throughout his works, there is no explicit mention of it in the quotation above” (Busch and Sereni 2012, p. 351).

  16. They ascribe the appeals to theory contribution to Colyvan (2001) and Baker (2009). I see no evidence for the ascription in Colyvan’s work, which Busch and Sereni misquote. (Compare Colyvan 2001, p. 37 with Busch and Sereni 2012, p. 348, fn 6).

  17. The goal of their paper is to clarify the Quinean roots of the indispensability argument and to argue that what we ordinarily ascribe to Quine is not really his argument. The exegetical point is beyond the range of this paper.

  18. For examples, see Quine (1948, p. 13) and Putnam (1971, p. 356). Relatedly, see Field (1980, p. 2).

  19. See especially Melia (2000), Colyvan (2002), Melia (2002), Leng (2002), Daly and Langford (2010), Melia (2010); and Colyvan (2010).

  20. Dieveney puzzlingly cites Resnik (1997, pp. 101–110) in support. Resnik is working within a holistic framework, so his arguments are inapplicable to a non-holistic response to the weasel.

  21. QI is my version of Quine’s argument, one which I think best captures Quine’s central intent. See Quine (1939, 1948, 1951, 1955, 1958, 1960, 1978, 1986); also see Marcus (2010) and Marcus forthcoming.

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Acknowledgments

My thanks to David Bordeaux and other members of the audience at a meeting of the Albritton Society, UCLA, at which a portion of this work was presented, as well as to three anonymous reviewers, for helpful comments and criticisms.

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    Russell Marcus

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Marcus, R. The holistic presumptions of the indispensability argument. Synthese 191, 3575–3594 (2014). https://doi.org/10.1007/s11229-014-0481-7

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