Abstract
This paper argues that scientific realism commits us to a metaphysical determination relation between the mathematical entities that are indispensible to scientific explanation and the modal structure of the empirical phenomena those entities explain. The argument presupposes that scientific realism commits us to the indispensability argument. The view presented here is that the indispensability of mathematics commits us not only to the existence of mathematical structures and entities but to a metaphysical determination relation between those entities and the modal structure of the physical world. The no-miracles argument is the primary motivation for scientific realism. It is a presupposition of this argument that unobservable entities are explanatory only when they determine the empirical phenomena they explain. I argue that mathematical entities should also be seen as explanatory only when they determine the empirical facts they explain, namely, the modal structure of the physical world. Thus, scientific realism commits us to a metaphysical determination relation between mathematics and physical modality that has not been previously recognized. The requirement to account for the metaphysical dependence of modal physical structure on mathematics limits the class of acceptable solutions to the applicability problem that are available to the scientific realist.
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Notes
The conclusions of the indispensability argument remain disputed. For the purposes of this paper, it is assumed that the indispensability argument is sound. As the indispensability argument has scientific realists as its audience, scientific realism is also assumed here. For background on the connection between indispensability and scientific realism, see Colyvan (1999, 2001a, b, 2006). Broadly, the indispensability argument can be seen as an extension of the no-miracles argument to mathematical entities, as indispensability theorists take there to be no in-principle difference between the role that mathematical entities play in theoretical explanations of observable phenomena and the role that non-mathematical theoretical entities play in such explanations.
The version of scientific realism assumed here is a metaphysical one, though the conclusions of the paper are compatible with both standard entity realism and ontic structural realism.
See Berenstain and Ladyman (2012) for an argument that scientific realists must be committed to a non-Humean understanding of causation and laws of nature.
For what it’s worth, I take there to be numerous examples of both casual and non-causal relations holding between unobservable entities and the observable phenomena they explain. See Berenstain (2016) for further discussion.
The term ‘grounding’ as used here denotes the broad family of relations of metaphysical dependence. It is not meant to invoke the primitive relation that Wilson (2014) refers to as “Big-G Grounding,” nor does it imply anything about what specific relation of metaphysical dependence must hold between modal and mathematical structures.
This bears emphasizing, as not all scientific realists embrace the commitment to a robust conception of modality that scientific realism seems to require. See, for example, Psillos (2002).
These free parameters include the lepton charge and masses, the Fermi coupling constant of beta decay, the mixing angle \(\uptheta \), and the mass of the scalar particle.
Berenstain (2016) offers a view on how to understand the inherently modal nature of physical properties such as being a Z boson. The view differs from dispositional essentialism, though it shares many of its motivations.
It is helpful to distinguish between something’s being contingent on certain conditions holding and its being metaphysically contingent. The two are not the same. Consider, for instance, that the modal status of ceteris paribus law can be either necessary or contingent. That a law of nature is ceteris paribus does not thereby mean it is not necessary. A law’s being contingent upon certain conditions is not the same as the law’s being metaphysically contingent. For if the law holds in every world in which the conditions obtain then the law, though ceteris paribus, is metaphysically necessary. See Berenstain (2014) for examples and further discussion.
What Lyon (2012) did for the Bridges of Königsberg problem can also be easily done for the three-utilities problem.
The predator hypothesis covers parasitoids that maybe have attacked adults or eggs during the period of evolutionary development but that have since become extinct (Lloyd and Dybas 1966).
A framework of nested modality such as the one developed by Lange (2007) is useful here.
Of course identity is a two-way street. But when we talk about discovering that what we initially thought were two separate things, A and B, are actually one thing, this is frequently spelled out in terms of finding that A actually has all or most of the properties that we initially thought B had or vice versa.
Tegmark’s picture of the mathematical multiverse on which all mathematical structures are physically realized somewhere in the multiverse may contain the resources to respond to this concern.
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Acknowledgments
Thank you to the following people for their helpful and patient discussions of this paper: Derek Anderson, Jordan Baker, Anjan Chakravartty, Kathleen Connelly, Josh Dever, David Frank, Steven French, Joyce Havstad, Lina Jansson, Cory Juhl, Ashley Kennedy, Gal Kober, Rob Koons, James Ladyman, Bryan Pickel, Katherine Ritchie, John Roberts, Juha Saatsi, Marlin Sommers, and Audrey Yap
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Berenstain, N. The applicability of mathematics to physical modality. Synthese 194, 3361–3377 (2017). https://doi.org/10.1007/s11229-016-1067-3
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DOI: https://doi.org/10.1007/s11229-016-1067-3