Abstract
In a recent paper Alan Baker has argued for the thesis that the use of a stronger mathematical apparatus in optimization explanations can reduce our concrete ontological commitment, and this results in an increase of explanatory power. The import of this thesis in the context of the Enhanced Indispensability Argument is significant because it sheds light on how the Inference to the Best Explanation principle, on which the Enhanced Indispensability Argument crucially depends, may work at the level of concrete and mathematical posits in scientific explanations. In this paper I examine Baker’s position and I argue that, although the employment of additional mathematical resources in some explanations can enhance explanatory power, it is highly controversial that Baker’s example of cicadas can have a strong import in the platonism vs nominalism debate. I conclude with a general discussion of the way in which a stronger mathematical apparatus may sometimes lead to an increase of explanatory power.
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Notes
- 1.
Baker points out that his analysis pushes the platonist case because the nominalist who accepts the more parsimonious explanation must also embrace the stronger mathematics, and it seems too difficult for him to offer a nominalized version of the mathematics used in such an explanation: “For if I am right then there is a general push in such cases towards more sophisticated mathematics, driven by considerations of concrete ontological, ideological, and structural parsimony. This more sophisticated mathematics tends to be more difficult (perhaps impossible) for the nominalist to paraphrase. And this gives more ammunition to the platonist cause, even when it rests on examples of MES [mathematical explanations in science] that look deceptively simple” (Baker 2016, p. 350). Although the possibility of offering a nominalistic reconstruction of Baker’s more parsimonious explanation is left open, I do not tackle this issue here.
- 2.
Baker also considers that Expl2 is more explanatory because it permits us to answer various counterfactual questions. More on this point in the next section.
- 3.
With his example, Baker also claims in favor of quantitative parsimony: “strengthening of the mathematical results involved can lead to a reduction in the number [emphasis added] and variety of concrete posits” (Baker 2016, p. 334). Although he offers no argument in defense of the claim that the use of additional mathematical resources leads to quantitative parsimony, the idea seems to be that by reducing the number of predator-types Pi to P2 and P3 (using Lemma2), it also follows a reduction of the total number of individual predators postulated. This is obviously true for the case considered by Baker. Nevertheless, it is worth noting that such an argument is not conclusive because, in general, qualitative parsimony does not entail quantitative parsimony.
- 4.
In a previous work, Baker has defended the idea that quantitative parsimony yields explanatory power: “Quantitative parsimony tends to bring with it greater explanatory power. Less quantitatively parsimonious hypotheses can match this power only by adding auxiliary claims which decrease their syntactic simplicity. Thus the preference for quantitatively parsimonious hypotheses emerges as one facet of a more general preference for hypotheses with more explanatory power” (Baker 2003, p. 258).
- 5.
Clearly, we may deny that there exists a link between parsimony and explanatoriness, however maintaining that parsimony is a theoretical virtue. In this case, we could still prefer Expl2 over Expl1 (because we consider the first as based on a more theoretically virtuous hypothesis). Although this standpoint is plausible, by adopting it we lose a crucial ingredient of Baker’s point, namely the connection between parsimony and explanatory power.
- 6.
Obviously, evidence for the existence of present, or past predators (for which periodicity cannot be observed from fossil evidence), may be found. On the other hand, lack of direct evidence of periodical predators is certainly a weakness of Baker’s explanations, as acknowledged by Baker himself in his last paper on the subject (Baker 2017b, p. 3).
- 7.
This explanation may be based on features that do not involve the existence of periodical predators, as for instance those discussed later in this section. After all, from a biological point of view the hypothesis that cicadas have no (or had not, in their evolutionary past) predators is perfectly plausible. In biology, a species with no predator is called an ‘aphex predator’.
- 8.
This is relevant to the present discussion because, as observed by Baker, Webb’s model assumes only the existence of periodical predators with periods of 2 years or 3 years, which are the periodical predators assumed in Expl2 (Webb 2001, p. 389). Furthermore, Webb’s assumption that the predator dynamics is independent of the cicada dynamics is also present in Baker’s explanation.
- 9.
Shift in the periodicities of magicicadas having either 13 or 17 years life cycles has been also considered theoretically by other biologists. For instance, the evolutionary biologist Peter Grant suggests that “these life cycles evolved earlier than the Pleistocene and involved an abrupt transition from a nine-year to a 13-year life cycle, driven, in part, by interspecific competition” (Grant 2005, p. 169). On the fixation of the cicadas’ periodicity see also Ito et al. (2015).
- 10.
In his more recent paper on the subject, Baker explicitly addresses the phenomenon of period length shifts and proposes a renewed version of Expl1 (Baker 2017b). This version of Expl1 includes the ‘genetic’ premise “Periodical cicadas are limited by genetic constraints to periods of the form 4n + 1” and is capable of accounting for shifts of +4 or −4 years.
- 11.
Nariai et al., for instance, believe that genetic factors have an important role in the life switch of a 17-year cycle population to a 13-year cycle (Nariai et al. 2011). More precisely, they show how life cycle switching by gene introduction appears to be possible under fitness reductions at low population densities of mating individuals (‘Allee effect’). But Nariai et al. are not alone in thinking that genetic mechanisms play a decisive role in the cicadas’ emergence phenomenon. Lehmann-Ziebarth et al. propose that “the explanation for prime-numbered periods, rather than just fixed periods, may reside in physiological or genetic mechanisms or constraints” (Lehmann-Ziebarth et al. 2005, p. 3200), and yet the sort of explanation that these biologists have in mind seems to considerably diverge from the kind of explanation proposed by Baker.
- 12.
For the unconvinced reader: if \(1=\sin {2\theta }\), then \(\sin ^{-1}{(1)}=2\theta \); therefore \(\frac {\pi }{2}=2\theta \), which means that \(\theta =\frac {\pi }{4}\).
- 13.
En passant, let me note that my argument in this section does not necessitate that the optimization explanation considered be distinctively mathematical. On the other hand, it seems quite natural to consider it as such because it essentially depends on a mathematical fact which cannot be expressed in causal terms. Moreover, my impression is that this particular explanation can be captured in terms of a particular account of mathematical explanation in science, as for instance the account advocated by Marc Lange in his (2013). However interesting, I shall not pursue this issue here and I leave it for future work.
- 14.
Note that λ = dM∕dk and therefore λ approximates the change in M resulting in a one unit increase in k.
- 15.
The mathematics of Lagrange multipliers is more general in the sense that it embeds the mathematics used in the first explanation, which indeed can be recovered using the Lagrange method, and moreover it includes the larger mathematical resources of multivariable calculus.
- 16.
The explanation also permits to answer counterfactual questions, as for instance ‘What if the mass of the cannonball were the same but the kinetic energy were doubled?’. From \(R(v_{x}^{*},v_{y}^{*})=(2/mg)E\) we have that the maximum range will be doubled too. Or ‘What if the maximum range remain the same but the mass of the cannonball were doubled?’. In this case we should expect the energy E to be twice its initial value.
- 17.
Presumably, Baker would be sympathetic to this point. For instance, in another work he explores the issue of how mathematics can provide topic generality when used to account for a scientific phenomenon, and how such form of generality is a source of explanatory value (Baker 2017a). He maintains that the cicada explanation exhibits topic generality. Nevertheless, he does not link his analysis to the issues addressed in Baker (2016), namely ontological parsimony and the import of additional mathematical resources.
- 18.
It may be observed that in the explanation using the Lagrange multipliers we need to consider the mass and the kinetic energy of the cannonball as fixed, and that this assumption alters our concrete ontology. Consider, however, that the mass and the kinetic energy are properties of the object cannonball. And therefore an assumption about these properties does not influence the number of types of concrete objects or the number of individual objects that we should consider in our explanation (although it affects the relations between some properties of the concrete objects that are involved in our explanation).
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Acknowledgements
I wish to thank the organizers of the Second FilMat Conference and the members of the audience for useful discussion of this paper. I benefitted immensely from suggestions from Marco Panza, Matteo Morganti, Mauro Dorato, Marc Lange, Davide Vecchi, Stathis Psillos, Andrea Sereni, Michèle Friend, Pierluigi Graziani, Achille Varzi, Francesca Poggiolesi, Mary Leng, Luca Incurvati, Andrew Arana, Josephine Salverda, Claudio Ternullo and Giorgio Venturi. I would also like to thank an anonymous referee for his helpful comments.
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Molinini, D. (2018). Parsimony, Ontological Commitment and the Import of Mathematics. In: Piazza, M., Pulcini, G. (eds) Truth, Existence and Explanation. Boston Studies in the Philosophy and History of Science, vol 334. Springer, Cham. https://doi.org/10.1007/978-3-319-93342-9_11
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