## Abstract

This article elaborates the epistemic indispensability argument, which fully embraces the epistemic contribution of mathematics to science, but rejects the contention that such a contribution is a reason for granting reality to mathematicalia. Section 1 introduces the distinction between ontological and epistemic readings of the indispensability argument. Section 2 outlines some of the main flaws of the first premise of the ontological reading. Section 3 advances the epistemic indispensability argument in view of both applied and pure mathematics. And Sect. 4 makes a case for the epistemic approach, which firstly calls into question the appeal to inference to the best explanation in the defense of the indispensability claim; secondly, distinguishes between mathematical and physical posits; and thirdly, argues that even though some may think that inference to the best explanation works in the postulation of physical posits, no similar considerations are available for postulating mathematicalia.

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## Notes

- 1.
For an elaboration of unrestricted mathematical platonism, see Balaguer (1998).

- 2.
Note that my argument focuses on distinguishing those versions of the indispensability argument that draw ontological consequences concerning mathematicalia, and those that propose to read the indispensability claim in an epistemic key. I am aware of other versions of the indispensability argument. For a recent survey of the matter, see Panza and Sereni (2016).

- 3.
- 4.
Alternative constructions of the argument focus on the ontological reading as well. For instance, Resnik (1997) pays attention to the pragmatic component of the use of mathematics in scientific theorizing, whereas Baker (2009) endorses the reality of those mathematical entities that indispensably partake in scientific explanations. As in the standard OIA, however, both take their approaches to lead us to the conclusion that at least some mathematicalia exist.

- 5.
- 6.
Our argument may seem to fall prey of what Putnam calls the “intellectual dishonesty” (1971, 347) of not believing in the existence of what one daily presupposes. In our case, mathematics partakes in scientific theorizing routinely. Hence, why should we not believe in the reality of mathematicalia? (Or so the platonist would challenge us.) Against Putnam’s view, I am inclined to endorse the distinction between

*quantifier commitment*and*ontological commitment*advocated by Azzouni (2004) and Bueno (2005), i.e., scientific theories routinely quantify over mathematical posits without entailing ontological commitment. Beyond the mathematical setting, we likewise quantify over fictional entities in literature, folklore, or else, without entailing existence of any sort. - 7.
- 8.
For the sake of brevity, I shall look into these two criteria only in terms of experimental access and causal role. I am aware that other strategies are available in recent literature. Azzouni (1997) advances the criterion of thick epistemic access for separating the mathematical from the physical. Another criterion is discussed by Colyvan (1999) and Melia (2002), namely, unification in theory and in ontology. Concerning the latter, Melia (2002, 75) observes that the postulation of both mathematical and physical posits may contribute to the attractiveness and utility of theories, but points out that they do so differently. He suggests a case for demonstrating that mathematical unification does not entail physical unification. The use of complex numbers rather than real algebra helps us solve different equations in a unified way. This does not imply physical unification, he argues, nor does it prove that mathematical entities are on an ontological par with physical posits. By contrast, physical unification may give us a hint about the reality of new layers of phenomena, as in the case of Einstein’s postulation of a four-dimensional space-time, which unified the Newtonian gravitational theory and new phenomena related to gravitational lensing and redshift; or as in the case of Maxwell’s unification of electromagnetic phenomena and optical phenomena in his equations describing the behaviour of the electromagnetic field.

- 9.
This criterion has also been contested within the philosophy of mathematics. Bigelow’s (1988) physicalist Pythagoreanism and Franklin’s (2014) Aristotelian structural realism grant mathematical entities causal powers by placing them in the physical world. At present, for reasons of space, I leave these strategies aside. From a different perspective, Colyvan (2006) and Baker (2009) defend the view that mathematical entities play a genuine explanatory role in science, which, if granted, not only would justify their appearance as variables in current best scientific theories, but would also give us reason to believe that they are part and parcel of the fabric of reality. I briefly return to this issue in Sect. 5.

- 10.
We can also add now that the postulation of dark matter contributes to the unificatory power of explanations of physical phenomena, bringing together aspects as diverse as the total mass of the universe, the curvature observed in the orbits of galaxies, light lensing, and so forth.

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## Acknowledgements

This article is a result of the governmental funded research Grant FONDECYT de Iniciación, No. 11160324, “The Physico-Mathematical Structure of Scientific Laws: On the Roles of Mathematics, Models, Measurements, and Metaphysics in the Construction of Laws in Physics,” CONICYT, Chile.

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Soto, C. The Epistemic Indispensability Argument.
*J Gen Philos Sci* **50, **145–161 (2019). https://doi.org/10.1007/s10838-018-9437-9

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### Keywords

- Indispensability argument
- Epistemic approach
- Mathematics
- Science
- Inference to the best explanation
- Ontology