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Heavy Duty Platonism

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Abstract

Heavy duty platonism (HDP) is of great dialectical importance in the philosophy of mathematics. It is the view that physical magnitudes, such as mass and temperature, are cases of physical objects being related to numbers. Many theorists have assumed HDP’s falsity in order to reach their own conclusions, but they are only justified in doing so if there are good arguments against HDP. In this paper, I present all five arguments against HDP alluded to in the literature and show that they all fail. In doing so, I establish two related truths: HDP has been unfairly ignored, and the arguments which take its falsity as a key premise should be re-assessed.

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Notes

  1. Melia assumes that taking mathematics to play an explanatory role in science is tantamount to endorsing HDP (see for example 1995: 228–229). If he is right, proponents of the indispensability argument (e.g. Colyvan 2001) are committed to HDP. Pincock’s (forthcoming) account of mathematical explanation in science sounds very much like HDP.

  2. Field perhaps took ‘not explainable in other terms’ and ‘fundamental’ to be synonymous; I don’t want to commit to this.

  3. See Stalnaker (1987: 9), Crane (1990: 227), Mumford (2006: 471–480), Molnar (2003: 131–137), and Ellis (2001: 114–115).

  4. The notion of an intrinsic relation is a generalisation of the notion of an intrinsic property, and is introduced in a similar manner:

    An n-place intrinsic relation is an n-place relation that n things stand in in virtue of how they are and how they are related to each other, as opposed to how they are related to things outside of them and how things outside of them are. (Weatherson and Marshall 2013: Sect. 1.3).

  5. See Mumford (2006: 471–480), Molnar (2003: 131–137), and Ellis (2001: 114–115).

  6. See Daly and Langford (2009: 643) for a similar argument.

  7. Wilczec (2001a, b, 2002) has provided more reasons for thinking Planck units will have fundamental significance in future scientific theories.

  8. Garay (1995) has also argued that a minimum length is a model-independent feature of all approaches to formulating a theory of quantum gravity.

  9. One problem with verificationism concerns the status of the verification principle itself: it appears unverifiable and so meaningless by its own lights. Another is that knowing the meaning of a sentence appears to be conceptually prior to knowing what would verify or falsify it. See Lycan (2000: 115–l28) for a good survey of the problems facing verificationism.

  10. See Churchland (1979: 105) and Stalnaker (1987: 8) for examples of this reasoning.

  11. See Rosen (2010) and Audi (2012) for more on metaphysical dependence.

  12. McKitrick (2005) has shown that, on the most plausible accounts of causality, dispositions are causal. See Mumford and Anjum (2011) for a theory of causation based on a metaphysics of dispositions.

  13. The apparent inconsistency of HDP was appealed to in order to undermine this commonplace view of intentional states. Having shown that HDP is not inconsistent, the heavy duty platonist is free to appeal to philosophy of mind to demonstrate that non-causal determination of effects by abstracta is widespread.

  14. Balaguer’s appeal to the non-causal nature of mathematical objects to argue that what ‘science says about the physical world… could be true even if there aren’t any mathematical objects’ (1998: 133) suggests another argument against HDP: Numbers are non-causal, so the number 10 does nothing to make it that a 10 kg brick is related to the number 10; therefore, there is something about the brick alone that does. I discuss this here because a reply can already be found in the literature: Baker rightly accuses Balaguer of sliding ‘from the claim that the physical world is not causally dependent on the existence of mathematical objects to the stronger claim that it is not dependent ‘in any way’ on their existence’ (2003: 250). The same accusation is warranted here.

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Acknowledgments

Most of all, I want to thank David Liggins and Chris Daly for their comments on drafts of this paper, and their sustained encouragement. Thanks also to Joel Smith, Catharine Abell and Leonid Tarasov for helpful discussion.

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  1. Department of Philosophy, University of Manchester, Oxford Road, Manchester, M13 9PL, UK

    Robert Knowles

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Knowles, R. Heavy Duty Platonism. Erkenn 80, 1255–1270 (2015). https://doi.org/10.1007/s10670-015-9723-4

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