Abstract
Lange (2013, 2016) argues that some natural phenomena can be explained by appeal to mathematical, rather than natural, facts. In these “distinctively mathematical” explanations (DMEs), the core explanatory facts are either modally stronger than facts about ordinary causal law or understood to be constitutive of the physical task or arrangement at issue. Craver and Povich (2017) argue that Lange’s account of DME fails to exclude certain “reversals”. Lange (2018) has replied that his account can avoid these directionality charges. Specifically, Lange argues that in legitimate DMEs, but not in their “reversals,” the empirical fact appealed to in the explanation is “understood to be constitutive of the physical task or arrangement at issue” in the explanandum. I argue that Lange’s reply is unsatisfactory because it leaves the crucial notion of being “understood to be constitutive of the physical task or arrangement” obscure in ways that fail to block “reversals” except by an apparent ad hoc stipulation or by abandoning the reliance on understanding and instead accepting a strong realism about essence.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Obviously, these are necessary, but not sufficient, conditions. Presumably, there should also be something like a relevance requirement, since if you take any of Lange’s examples and add the premise that 2 + 2 = 4, you still have an argument whose premises meet these conditions (see Lange 2016: 136). But note also that according to Lange (2013: 507), “there is no criterion that sharply distinguishes the distinctively mathematical explanations from among the non-causal explanations appealing to some mathematical facts. Rather, it is a matter of degree and of context. Insofar as mathematical facts alone are emphasized as doing the explaining, the explanation is properly characterized as distinctively mathematical”. In this paper, I am concerned with those explanations that Lange takes to be clearly distinctively mathematical (rather than in between), where the mathematical facts alone are emphasized. See Povich and Craver (2018) for a discussion of some of Lange’s other “explanations by constraint” and Craver and Povich (2017: 35 fn. 10) for a brief discussion of “really statistical explanations”. I thank an anonymous review for helping me be clear about this.
Or that Marta will always fail or that anyone will always fail; similarly, for the other examples. An anonymous reviewer has the intuition that when the explanandum is characterized this way, the case is not a DME. It might have been that Marta (or everyone on earth) failed to try all the possible paths and that this would make the mathematical fact irrelevant (see also Barrantes forthcoming). I see the force of that intuition, but I ultimately resist it because of the stronger, competing intuition that if something other than the mathematical fact explained a given failure to cross the bridges (or the fact that everyone has failed, or whatever) – not trying all paths, say - that would imply that trying all paths would have led to success, which is not the case. For the purposes of this paper, I will assume that the explanandum as characterized by Lange, Craver, and Povich does have a DME.
Prima facie, the fact that Terry does not have a trefoil knot in his shoelace is different from the fact that Terry does not have a trefoil knot in the shoelace he untied.
Lange’s (2016, especially Chapter 3) later account of “explanations by constraint” (of which DMEs are one variety) goes far beyond any traditional modal conception, but I do not have the space here to give his full account the attention it deserves. Here I am only concerned with the extent to which ontic, non-modal elements must be considered in an account of the explanatory power of DMEs. See Section 4 below for an argument that there are still “reversals” even if Lange is right that explanandum-constitution does not result in a DME.
There are similar statements in the book (Lange 2016: 30, 37, 38). I find these claims hard to square with Lange’s other claim that the explananda of DMEs need not be necessary (2016: 131).
X makes y impossible if and only if x makes ~y necessary.
If this seems like a definitional explanation, rather than an explanation of a natural fact, consider making the explanandum the fact that Patty’s pendulum does not have a flexible joint in it. The same explanation applies. Thanks to Mark Alford for this suggestion.
This is what Lange (2018) has in mind when he says, “Terry’s untying the knot does not help to make the knot trefoil (or non-trefoil)” (3). Obviously, it was possible from the beginning to argue that in Craver and Povich’s (2017) original “reversals” there was task-constitution as well; that they have stipulated a context in which the knot is considered as an individual such that Terry’s untying the knot helps to make it the kind of individual it is. However, this context (and this kind of individual) seems far more artificial than the one stipulated above, thus making the case far less plausible.
This is arguably false on the standard modal account of essential properties. See below.
K2,2 consists of two sets of two nodes, where each node of the first set is connected to each node of the second.
Perhaps you think that these arguments require additional premises – that the Thistlethwaite knot is isotopic to the unknot and that K2,2 permits an Eulerian walk, respectively. Even if that’s true, there is nothing in Lange’s account that implies DMEs can only have one mathematical premise. In fact, some of his other examples seem to have more than one mathematical premise, such as the pendulum case (Lange 2013).
References
Barrantes, M. Explanatory information in mathematical explanations of physical phenomena. Australasian Journal of Philosophy (Forthcoming).
Craver, C. F. (2016). The explanatory power of network models. Philosophy of Science, 83(5), 698–709.
Craver, C. F., & Povich, M. (2017). The directionality of distinctively mathematical explanations. Studies in History and Philosophy of Science Part A, 63, 31–38 https://doi.org/10.1016/j.shpsa.2017.04.005.
Lange, M. (2013). What makes a scientific explanation distinctively mathematical? The British Journal for the Philosophy of Science, 64(3), 485–511.
Lange, M. (2016). Because without cause: Non-causal explanations in science and mathematics. Oxford: Oxford University Press.
Lange, M. (2018). A reply to Craver and Povich on the directionality of distinctively mathematical explanations. Studies in History and Philosophy of Science Part A, 67, 85–88. https://doi.org/10.1016/j.shpsa.2018.01.002.
Povich, M., & Craver, C. F. (2018). Review of Marc Lange, Because without Cause: Non-Causal Explanations in Science and Mathematics. The Philosophical Review, 127(3), 422–426. https://doi.org/10.1215/00318108-6718870.
Povich, M. (2019). The narrow ontic counterfactual account of distinctively mathematical explanation. British Journal for the Philosophy of Science (Forthcoming). https://doi.org/10.1093/bjps/axz008.
Reutlinger, A. (2016). Is there a monist theory of causal and noncausal explanations? The counterfactual theory of scientific explanation. Philosophy of Science, 83, 733–745.
Salmon, W. (1989). Four decades of scientific explanation. Minneapolis: University of Minnesota Press.
Schaffer, J. (2010). The least discerning and most promiscuous truthmaker. The Philosophical Quarterly, 60(239), 307–324.
Sidelle, A. (1992). Rigidity, ontology, and semantic structure. The Journal of Philosophy, 89(8), 410–430.
Acknowledgements
I thank Carl Craver for helpful comments on earlier drafts and Marc Lange for invaluable discussion.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Povich, M. Modality and constitution in distinctively mathematical explanations. Euro Jnl Phil Sci 10, 28 (2020). https://doi.org/10.1007/s13194-020-00292-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13194-020-00292-y