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The Indispensability Argument for Mathematical Realism and Scientific Realism

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An Erratum to this article was published on 01 December 2012

Abstract

Confirmational holism is central to a traditional formulation of the indispensability argument for mathematical realism (IA). I argue that recent strategies for defending scientific realism are incompatible with confirmational holism. Thus a traditional formulation of IA is incompatible with recent strategies for defending scientific realism. As a consequence a traditional formulation of IA will only have limited appeal.

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Notes

  1. Quine (1948, 12–13) proposed that the ontological commitments of a discourse are determined by first regimenting the discourse into first order logic and then looking at what falls under the existential quantifier. So in the case of theories, if a theory implies that a first order statement of the form ∃(Rx) is true then the Rs exist (according to that theory). There is an extensive debate over the status of Quine’s criterion of ontological commitment. Yablo (2000) suggests that we can sometimes read the existential quantifier metaphorically, and so as non-committing in the ontological sense. Azzouni (2004) argues that even when understood non-metaphorically the existential quantifier can sometimes be understood as non-committing (this will be determined in the meta-language) and explores the option that commitment to an object is determined by whether that object falls under a particular existence predicate as well as falling within the range of the quantifier. These issues deserve further discussion but fall outside the scope of this paper.

  2. It is not my intention to beg the question against the Quinean who believes that mathematical entities are in fact theoretical entities. I am aiming at showing that there will be evidential difference between mathematical entities and entities like electrons molecules etc. and reserve the term ‘theoretical entities’ for the latter kinds of objects, to allow myself a vocabulary for arguing this. Also, Resnik (1997) has interestingly argued that a Bohmian interpretation of quantum mechanics lends credibility to viewing quantum objects as quasi-mathematical, thereby putting pressure on the idea that there is a clear-cut distinction between mathematical entities and what I call theoretical entities. The line of argument motivating this claim takes off from the observation that classic entity metaphysics is violated at the quantum level, but accepting this, the conclusion that quantum objects are quasi-mathematical doesn’t follow. A recent interpretation of quantum physics, ontic structuralism, accommodates the violation of classic metaphysics while upholding a division between mathematics and the entities (structures) described at the quantum level. As such, whether quantum objects are quasi-mathematical or not is at best underdetermined. I have to set a further discussion of this issue aside as it constitutes a separate topic in and by itself. I refer the reader to Ladyman and Ross (2007) for a full elaboration of the structuralist position.

  3. Sober (1993) has challenged IA by arguing that theories are only ever confirmed (or disconfirmed) relative to one another, thereby denying that confirmational holism is central to scientific practice.

  4. One can agree that mathematics is indispensable to science but deny that this entails that mathematical entities are indispensable to scientific theories. Field (1980) has argued in this way, proposing a nominalist theory of mathematics (though most agree unsuccessfully) and Hellman (1989) has suggested a version of mathematical structuralism according to which structures are modal structures, but I do not intend to enter into a discussion of these views here.

  5. By formulating the structure of the argument in the way that I do below I do not intend to indicate that the conditionals are meant to express logical entailment. For starters, implication is between propositions and what I say here is something about how, for example (in the conclusion) the adoption or non-adoption of a particular theory of confirmation plays a role for whether we may employ a particular variety of the indispensability argument. In premise 1. I suggest that if confirmational holism as an approach to confirmation is incorrect (or is not employed) then we cannot establish the parity clause. As I mentioned above, this is only correct on the assumption that confirmational holism were the only motivation for the parity clause. Furthermore, one should not read this as strict implication and take premise 1. to assert that if in fact the parity clause was established, that if mathematical entities and theoretical entities were found to be epistemically on a par, we could infer that confirmational holism was correct. The premises are meant to express the structure of the argument, and the argument is only to be understood in the context of assuming that only confirmational holism can be used for motivating the parity clause. (I wish to thank a referee of this journal for prompting me to make this explicit).

  6. There are different formulations of the PMI argument in the literature. The following formulation from Putnam is sufficient for our purposes here: “What if all the theoretical entities postulated by one generation (…) invariably don’t exist from the standpoint of later science? (…). One reason this is a serious worry is that eventually the following meta-induction becomes overwhelmingly compelling: Just as no term used in the science of more than fifty (or whatever) years ago referred, so it will turn out that no term used now (except maybe observational terms, if there are such) refers” (Putnam 1978, 25).

  7. Psillos (1999) draws on Kitcher (1993) in order to formulate his own version of the strategy which he calls the divide et impera strategy. There is a multitude of current realist positions that opt for the same strategy in answer to PMI. Worrall’s (1989) epistemic structuralist position, Chakravartty’s (1998) semi-realist position, and recently Saatsi’s (2005) eclectic realist position are all instances of the compartmentalization strategy.

  8. By ‘practice’, Kitcher (1993, 142, fn. 21) could be understood to mean ‘theory’ (he talks about success spread over sentences), so we can think of these two kinds of posits as introduced within scientific theory.

  9. Resnik (1997) defends a variety of IA that argues for the existence of mathematical entities by appeal to pragmatic considerations and Baker (2009) argue for a variety of IA that is based on inference to the best explanation. Colyvan (2001) also provides a version of IA based on inference to the best explanation reasoning in addition to the version that proceeds by appeal to confirmational holism.

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Acknowledgments

I wish to thank Ioannis Votsis for his encouraging feedback on an early version of this article. I would also like to thank Asbjørn Steglich-Petersen, James McAllister and two anonymous referees for this journal for their useful comments.

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Busch, J. The Indispensability Argument for Mathematical Realism and Scientific Realism. J Gen Philos Sci 43, 3–9 (2012). https://doi.org/10.1007/s10838-012-9184-2

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