Abstract
Enhanced indispensability arguments seek to establish realism about mathematics based on the explanatory role that mathematics plays in science. Idealizations pose a problem for such arguments. Idealizations, in a similar way to mathematics, boost the explanatory credentials of our best scientific theories. And yet, idealizations are not the sorts of things that are supposed to attract a realist attitude. I argue that the explanatory symmetry between idealizations and mathematics can potentially be broken as follows: although idealizations contribute to the explanatory power of our best theories, they do not carry the explanatory load. It is at least open however that mathematics is load-carrying. To give this idea substance, I offer an analysis of what it is to carry the explanatory load in terms of difference-making and counterfactuals.
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Notes
The predator–prey interaction is just one of two proposed explanations of the life-cycle length of North American cicadas. The other explanation involves the avoidance of cross-breeding with other cicada groups. As Baker (2005, pp. 230–232) notes, both explanations are underpinned by the same number-theoretic results. I have omitted the hybridization explanation, however, for ease of exposition.
I discuss this case as an example of extra-mathematical explanation in Baron (2014).
As a referee of this journal has pointed out, Batterman’s cases of asymptotic reasoning may provide clearer examples of explanatorily indispensable idealizations than the Lévy walk case described above, involving memory independence. If that’s right, then I would happily to defer to the cases at issue. The reason I don’t proceed in this way, however, is that I am attempting to sidestep a complication, namely that some idealizations are explicitly mathematical. This is particularly so for Batterman’s examples, in which the focus is on the application of limit-taking operations that admit of a precise mathematical formulation. This raises a difficult question as to whether we should treat these idealizations as false assumptions about the world, or—as Leng (2010, pp. 111–122) argues—simply as true mathematical claims of a certain kind, claims that pose no trouble for mathematical realism. To be clear, I don’t believe that all of Batterman’s cases involve explicitly mathematical idealizations and have included what I believe to be the least mathematical cases here. Still, someone might read the cases differently, and so it is important to have an idealization that cannot be so easily dealt with. I believe this to be true of the memory independence idealization in the Lévy walk example, which I take to be a non-mathematical assumption about the memory capacity of sharks.
The detour through explanation might seem unnecessary, quite apart from the detour through difference-making. If the goal is to argue for the truth of mathematical statements because of the modal properties of mathematical objects, then why not simply ‘go modal’ straight away as Hellman (1989) does? There are two reasons for not going this way. First, the case for realism about mathematics is supposed to ‘piggy-back’ on the case for scientific realism. The case for scientific realism, however, is made through explanation. So the case for mathematical realism should proceed via explanation as well. Going modal from the beginning without running the story through explanation threatens to separate the case for mathematical realism from the case for scientific realism in too great a manner. Second, and perhaps more importantly, Hellman’s account of mathematical theories is at odds with the ambitions of most mathematical realists. Hellman’s approach is to reformulate mathematical theories so as to avoid quantification over mathematical entities entirely, understanding those theories purely in terms of modality. If Hellman’s approach were successful it would obviate the need for mathematical realism.
Strevens (2008, Ch. 8) takes it to be definitive of idealizations that the features being idealized don’t make a difference to what we are trying to explain and, accordingly, that the purpose of an idealization is, in part, to convey this information.
A wrinkle: building specific details about memory capacity into the case may make the model more accurate. However, accuracy is not (necessarily) a good thing. For accuracy may trade-off against explanatory power as, for instance, when a more accurate model is also less general (i.e. explains fewer instances of a phenomenon). Strevens (2008, Ch. 3) provides some compelling examples of the trade-off between accuracy and explanatory power, and the important role that idealizations play in this trade-off.
I am grateful to Silvia De Bianchi for pressing me to consider Rizza’s response to Baker’s cicada case.
See e.g. Bangu’s (2013) banana game example.
If a case can be made for \({\textit{CF}}_2\) or \({\textit{CF}}_3\) this would, I believe, support realism about all integer-related facts. This is due to the integers forming a mutually supportive number-theoretic structure of facts such that realism about one part of the structure requires a realist attitude toward the whole. Idealizations, by contrast, are not situated in a structure in the same way. Because mathematical realism has this ‘one in, all in’ feature, arguments in favour of individual counterfactuals such as \({\textit{CF}}_2\) must be more persuasive than if only a limited realism about the number 13 were at stake. Some temperance regarding the conclusion of §5 is therefore required: while the case for mathematical difference-making remains open, it is perhaps also stacked against the realist.
I am grateful to a referee for raising this concern.
It may be too strong to say that we have no reason to accept that \({\textit{CF}}_2\) is trivially true and to therefore insist that the difference-making potential of mathematics remains open. The fact that we must appeal to impossible worlds at all might be seen as a cost, especially for those who are suspicious of counterpossibles. That’s a fair point, so I should temper my claim: one may still have some reason to believe that \({\textit{CF}}_2\) is trivially true if one has doubts about impossible worlds. Modulo such doubts, however, the issue remains open.
This issue is played out in a recent exchange between Lyon (2012), who offers an account of mathematical explanations in programming terms, and Saatsi (2012), who objects to Lyon’s view on the grounds that it requires relations between mathematical and physical entities of the kind just discussed. While Saatsi’s objection is aimed at Lyon’s account, I suspect it is indicative of a rather general concern with mathematical explanation. My point, then, is that if this concern can be addressed by making the relations Saatsi objects to respectable, then that may furnish us with the resources needed to elucidate mathematical difference-making.
The explanation concerning hybridization mentioned in fn. 2 is one possibility.
If, as in the shark case, we can gain evidence for \({\textit{CF}}_2\) by comparison with other species, then there is pressure to determine whether cicadas really were predated by predators with periodic life-cycles. Since if this can be substantiated, then there would be evidence that 13 is making a difference. Conversely, we may find that the evidence for \({\textit{CF}}_2\) is just not available in any terrestrial ecosystem and thus that there is no reason to think that 13 is a difference-maker. In short, further considerations coming from biology are likely to push the issue in one direction or another. Until the biology has been settled, then, some degree of circumspection is warranted regarding my conclusion that 13’s difference-making potential remains open.
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Acknowledgments
The author would like to thank Rachael Briggs, Mark Colyvan, Silvia De Bianchi, Arnon Levy, Raamy Majeed and Maureen O’Malley for discussion of and/or comments on earlier versions of this paper. The author would also like to thank two very helpful referees of this journal for their comments during the review process, as well as the editors Fabrice Pataut, Daniele Molini and Andrea Sereni, not least because they permitted me to offer an ad hoc submission to this special issue. Versions of this paper were presented at the University of Connecticut and Swarthmore College, and the author is grateful to members of the audience at both places for useful discussion of this paper, especially: Alan Baker, Richard Eldridge, Suzy Killmister, David Ripley and Lionel Shapiro. Research on this paper was partly supported under the Australian Research Council’s Discovery Projects funding scheme (Project Number DP120102871) and by a John Templeton Foundation grant held by Huw Price, Alex Holcombe, Kristie Miller, and Dean Rickles, entitled: New Agendas for the Study of Time: Connecting the Disciplines.
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Baron, S. The explanatory dispensability of idealizations. Synthese 193, 365–386 (2016). https://doi.org/10.1007/s11229-014-0517-z
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DOI: https://doi.org/10.1007/s11229-014-0517-z