Abstract
In The Explanatory Dispensability of Idealizations, Sam Baron suggests a possible strategy enabling the indispensability argument to break the symmetry between mathematical claims and idealization assumptions in scientific models. Baron’s distinction between mathematical and non-mathematical idealization, I claim, is in need of a more compelling criterion, because in scientific models idealization assumptions are expressed through mathematical claims. In this paper I argue that this mutual dependence of idealization and mathematics cannot be read in terms of symmetry and that Baron’s non-causal notion of mathematical difference-making is not effective in justifying any symmetry-breaking between mathematics and idealization. The function of making a difference that Baron attributes to mathematics cannot be referred to physical facts, but to the features of quantities, such as step lengths or time intervals taken into account in the models. It appears, indeed, that it does not follow from Baron’s argument that idealizations do not help to carry the explanatory load at least for two reasons: (1) mathematics is not independent of idealizations in modelling and (2) idealizations help mathematics to carry the explanatory load of a model in different degrees.
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Notes
Baron admits that he should provide an account of mathematical difference-making. However, providing such an account, he claims, is beyond the scope of his paper (Baron 2014, p. 2). In what follows, I show that to provide such an account is fundamental in order to get the case of idealization right.
Even if idealization does not carry the explanatory load, yet Baron claims that it plays a crucial role in scientific models. It necessarily does so, by enabling a model to yield a particular explanandum, or by contributing to explanatory power by allowing us to build explanatory models that adequately yield the explananda we are interested in (role played in model building process), or, again, by avoiding the introduction of unwanted information into an explanation.
Baron believes that the memory independence idealization in the example of Lévy walk is a non-mathematical assumption about the memory capacity of sharks (Baron 2014, p. 7 footnote).
I follow here Leng (2010, pp. 111–122), who argues that idealization in scientific models are true mathematical claims of a certain kind that pose no trouble for mathematical realists.
See also Lange (2009) for the discussion on a whole class of arguments that fails to be explanatory, such as arguments by mathematical induction.
I shall present somewhere else the proposal of referring to mathematical explanation in terms of explication in order to distinguish its function and role from that of causal explanation in scientific models.
For a discussion of mathematical difference-making and counterfactuals, see Baker (2003).
For Baron, the difference between mathematics and idealization is that the latter does not carry the explanatory load, although both of them can unify similar instances under a common explanatory pattern. Nevertheless, it could be asked whether non-mathematical idealizations also share with mathematics-free formulations of the explanans a “weaker” degree or mode of explanatory power and see whether this can be portrayed in difference-making terms.
The role of idealization is to assert the explanatory irrelevance that is the failure to make a difference of a salient causal factor (Strevens 2008). However, since for Baron we cannot read the notion of difference-making in causal terms, then there is a lack of a criterion for identifying the asymmetry between idealization and mathematics with respect to what we are trying to explain. Furthermore, among the characteristics of an idealization the details are filled out in a certain way: relevant parameters are assigned a zero, an infinite or some other extreme or default value (Strevens 2008, p. 318). Infinite velocity of the group perfectly fits into Strevens’ picture of idealization assumption, whereas lack of memory capacity corresponds to what I identify with idealized consequences.
Raichlen et al. (2014) suggest that scale-invariant, superdiffusive movement profiles are a fundamental feature of human landscape use, regardless of the physical or cultural environment, and may have played an important role in the evolution of human mobility. The widespread use of this movement pattern among species with great cognitive variation suggests an important link between foraging/hunting patterns across different organisms, including humans and thus opens the search in evolutionary theory for a deep reason why these common patterns occur.
The idea that mathematics contributes to the explanatory power of current science (Colyvan 2001, 2012; Baker 2005) contributed to characterize mathematical entities as explanatory in scientific models. However, this approach does not appeal to a specific notion of mathematical explanation. As Colyvan (2012) admits: “No one, it seems, has a complete story of what is going on in the cases in question, so we are left with an unsatisfying standoff. We agree that further philosophical work is required to elucidate the notions of mathematical explanation under discussion”. In my view, even if we assume that in their practices scientists appeal to mathematical explanations, nevertheless this is not sufficient to elucidate the notion of mathematical explanation.
For an alternative account to Baker’s and Baron’s interpretation of the cicada case study, see Rizza (2011).
One of the fathers of the study on sharks population and Lévy flight, David William Sims, claimed: “I think we’re interested now in understanding how these patterns arise, not just the where and when, but also the how and the why. Those are the important questions” (Ornes 2013).
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De Bianchi, S. Which explanatory role for mathematics in scientific models? Reply to “The Explanatory Dispensability of Idealizations”. Synthese 193, 387–401 (2016). https://doi.org/10.1007/s11229-015-0795-0
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DOI: https://doi.org/10.1007/s11229-015-0795-0