Abstract
Alan Baker’s enhanced indispensability argument (EIA) supports mathematical platonism through the explanatory role of mathematics in science. Busch and Morrison defend nominalism by denying that scientific realists use inference to the best explanation (IBE) to directly establish ontological claims. In response to Busch and Morrison, I argue that nominalists can rebut the EIA while still accepting Baker’s form of IBE. Nominalists can plausibly require that defenders of the EIA establish the indispensability of a particular mathematical entity. Next, I argue that IBE cannot establish that any particular mathematical entity is indispensable. Mathematical entities do not compete with each other in the way physical unobservables do. This lack of competition enables alternative formulations of scientific explanations that use different, but compatible, mathematical entities. The compatibility of these explanations prevents IBE from establishing platonism.
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Notes
This pertains to versions of the Quine-Putnam indispensability argument. These arguments include non-explanatory indispensability arguments as well, such as Colyvan’s (2001, cf. p. 8).
I thank Busch and Morrison for suggesting this name for their position.
For a careful catalog of these styles of indispensability arguments, confer Panza and Sereni (2013).
In the next section, I consider Busch and Morrison’s reservations about this statement.
Daly and Langford extend Melia’s (2000) nominalist strategy, which they refer to as “Melia’s indexing argument” (2009, p. 642). By saying that mathematics indexes concrete physical relations, Daly and Langford intend to generalize the idea that the number “1” can be used to stand for the length of a meter-bar: the number indexes the physical length of a concrete object (2009, p. 644).
By “the 2-Step,” I refer to Busch and Morrison’s formulation. Alternative formulations could modify their second step to allow alternative criteria for ontological commitment.
I am indebted to Busch’s conference presentation at the 2012 IHPST Indispensability and Explanation workshop; responding to this presentation was important for developing my argument.
Citing this same objection, Rizza has extended Field’s nominalization program to rebut the EIA (2011, pp. 111–112).
I thank two anonymous reviewers for asking me to clarify this point.
Baker’s discussion of “science driven mathematical explanations” in his (2012) also does not meet this burden of proof.
Baker does, however, acknowledge a similar burden of proof: “Does the platonist need to give a positive argument for why the mathematics in [Baker’s example] is explanatory in its own right, or does the nominalist need to give a positive argument to the contrary?” (2009, p. 624). However, Baker claims that “it is reasonable to place the burden of proof here on the nominalist” (2009, p. 625). I attempt to meet this burden of proof as well.
An anonymous reviewer has pointed out that this requirement is an instance of the fallacy of division. Nevertheless, in the context of genuinely explanatory mathematical objects, it is plausible that this fallacy does not apply. Since my primary goal is to motivate a demand for the stronger \(\exists \forall \) claim, I do not address this further.
I consider a version of this refinement in Sect. 4.2. I argue that it goes outside the dialectical bounds of the enhanced indispensability argument by forcing scientific realists to assume a specific form of structuralism.
Since indispensability arguments are directed at scientific realists—most of whom accept IBE—it is dialectically impermissible to reject inference to the best explanation in order to defend nominalism.
I thank an anonymous reviewer for asking me to clarify this point.
I thank an anonymous reviewer for asking me to clarify this point.
Baker has also formulated the indispensability argument in terms of mathematical theories, contending that platonists are “not arguing that individual numbers are indispensable for science or play an explanatory role in science, but rather that certain mathematical theories are indispensable and explanatory” (2005, p. 227). I return to this point below.
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Acknowledgments
I thank Jacob Busch, Joe Morrison, and the anonymous reviewers for their advice and criticism. For guidance on earlier versions of this essay, I thank Kyle Stanford, Alan Baker, Giovanni Valente, Julia Bursten, Thomas Ricketts, and the participants of the Indispensability and Explanation conference. I especially thank Kenneth Manders for help in the early stages of this project.
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Hunt, J. Indispensability and the problem of compatible explanations. Synthese 193, 451–467 (2016). https://doi.org/10.1007/s11229-015-0667-7
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DOI: https://doi.org/10.1007/s11229-015-0667-7