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Optimal representations and the Enhanced Indispensability Argument

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Abstract

The Enhanced Indispensability Argument (EIA) appeals to the existence of Mathematical Explanations of Physical Phenomena (MEPPs) to justify mathematical Platonism, following the principle of Inference to the Best Explanation. In this paper, I examine one example of a MEPP—the explanation of the 13-year and 17-year life cycle of magicicadas—and argue that this case cannot be used defend the EIA. I then generalize my analysis of the cicada case to other MEPPs, and show that these explanations rely on what I will call ‘optimal representations’, which are representations that capture all that is relevant to explain a physical phenomenon at a specified level of description. In the end, because the role of mathematics in MEPPs is ultimately representational, they cannot be used to support mathematical Platonism. I finish the paper by addressing the claim, advanced by many EIA defendants, that quantification over mathematical objects results in explanations that have more theoretical virtues, especially that they are more general and modally stronger than alternative explanations. I will show that the EIA cannot be successfully defended by appealing to these notions.

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Notes

  1. A similar reconstruction of the original IA can be found in Colyvan (2001), p. 11.

  2. Although the number of species of cicadas is a matter of dispute, most biologists agree that there are seven species (cf. Cooley, magicidada.org 2016). Three of them (septendecim, cassini, and septendecula) have 17-year life cycles and each of them has at least one 13-year life cycles counterpart (thus: tredecim, neotredecim, tredecassini, and tredecula). Differences in morphology, behavior and calling signals are clear between species of the same life cycle. However—and here is where the dispute begins – these differences are not so evident between a species and its counterpart with the alternative life cycle. It is for this reason that some biologists claim that these counterparts may both belong to the same species (the only difference between subspecies would be the life cycle length). This is still an open question in the relevant scientific literature.

  3. The distinction between geometrical and mathematical properties can be found, among other places, in Melia (2002), p. 76.

  4. However, rather than rejecting mathematical Platonism, Bangu presents the ‘banana game’ as a case that requires a MEPP, and which explanandum is not committed to mathematical objects. I believe the account that I will develop below accommodates his example as well, but since it involves probabilities, addressing this example will require a larger discussion than the one I present here.

  5. There are many proposed nominalizations of the cicada case in the literature. What follows does not break new ground on this respect. However, I present my own nominalized version because it emphasizes how these explanatory facts depend on the extremely simple notions of combination and equality. The simplicity of these two notions will be crucial for defending my point below about these nominalizations providing generality and modal strength.

  6. m, n, p and q represent natural numbers; but, as we saw, using these representations is unproblematic.

  7. I believe that once we express these lemmas in empirical terms it becomes crucial, for the overall explanation to work, to provide an explanation of why these physical relations hold. Moreover, I believe that this empirical explanation can be tracked down by an appropriate proof of the mathematical lemmas, suitable interpreted. I will not argue for this point here, since it is not relevant for my thesis; I am planning to do this in further work.

  8. Three years ago the European Space Agency’s Planck mission found evidence that the age of the universe might be around that number.

  9. See also Bueno and Colyvan (2011), and Bueno and French (2012).

  10. See Weisberg (2007) and (2013) for a detailed discussion on minimal models and the sense in which they differ from other kinds of models. However, Weisberg does not include the notion of optimal representation in his analysis.

  11. Not to be confused with the notion of optimality models, which are models (mostly found in biology) that describe those traits that maximize fitness [see e.g., Orzack and Sober 2001; Rice 2015]. See also Baker (2016) for a description of the cicada case in terms of optimality models.

  12. The scheme below is adapted from Bueno, Colyvan and French’s presentation of the Inferential Conception (see Bueno and Colyvan 2011, p. 353; and Bueno and French 2012, p. 92.) However, here I emphasize that in the case of MEPPs the immersion step is done in terms of an optimal representation.

  13. This interpretation step is a crucial element of the Inferential Account (e.g., Bueno and Colyvan 2011, p. 357; Bueno and French 2012, p. 107 and ss).

  14. Cf. Baker (2016), p. 340.

  15. Thanks to an anonymous reviewer for raising this point

  16. See also Keas (forthcoming), Section 5.1.

  17. Lange (2013) also defends that MEPPs point to mathematical necessities, but he thinks this is independent from the IA.

  18. Perhaps geometrical relations are physical and yet stronger than mere nomologically necessary. I cannot go into more details about this at this point. However, I hope my overall strategy is clear: the existence of geometrical relations does not by itself support mathematical Platonism.

  19. I believe there are not, but I do not think this is relevant for my overall case against the EIA supporting Platonism.

  20. This point is perhaps clearer in the bridges of Königsberg case, which is another example of an alleged MEPP that is widely discussed in the EIA debate. The impossibility of performing a trip that crosses all the seven bridges of 18\(^{th}\) Königsberg without retracing one’s steps is explained by a graph-theoretical theorem. For me, what is crucial for understanding this case is that Leonard Euler’s proof of this theorem relied on the extremely simple fact that every time one crosses a bridge, two pieces of land are involved, the starting point and the ending point. It is often said that it is a matter of mathematical necessity that the trip over the bridges is impossible. This necessity would be both stronger than mere physical necessity, and about a physical system (cf. Lange 2013; Lyon 2011). As I said, I prefer not to enter this debate here. My point is that if the impossibility of performing an Eulerian trip over the bridges is described as a matter of mathematical necessity, then the fact that ‘every time that I cross a bridge two pieces of land are involved’ would also be mathematically necessary. But if this justifies mathematical Platonism, such argument does not need the complications of the EIA, or for that matter, of issues pertaining the applicability of mathematics in science. I will address this point in further work.

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Acknowledgements

I would like to thank Paul Humphreys for his comments on several versions of this paper. The final version also benefited from helpful discussions with Otávio Bueno, James Cargile, and Juan Durán, and the comments of two anonymous reviewers.

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  1. University of Virginia, 120 Cocke Hall, Charlottesville, VA, 22904–4780, USA

    Manuel Barrantes

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Barrantes, M. Optimal representations and the Enhanced Indispensability Argument. Synthese 196, 247–263 (2019). https://doi.org/10.1007/s11229-017-1470-4

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