Abstract
This paper argues that in some explanations mathematics are playing an explanatory rather than a representational role, and that this feature unifies many types of non-causal or non-mechanistic explanations that some philosophers of science have been recently exploring under various names (mathematical, topological, etc.). After showing how mathematics can play either a representational or an explanatory role by considering two alternative explanations of a same biological pattern—“Bergmann’s rule”—I offer an example of an explanation where the bulk of the explanatory job is done by a mathematical theorem, and where mechanisms involved in the target systems are not explanatorily relevant. Then I account for the way mathematical properties may function in an explanatory way within an explanation by arguing that some mathematical propositions involving variables non directly referring to the target system features constitute constraints to which a whole class of systems should comply, provided they are describable by a mathematical object concerned by those propositions. According to such “constraint account”, those mathematical facts are directly entailing the explanandum (often a limit regime, a robustness property or a steady state), as a consequence of such constraints. I call those explanations “structural”, because here properties of mathematical structures are accounting for the explanandum; various kinds of mathematical structures (algebraic, graph-theoretical, etc.) thereby define various types of structural explanations.
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Notes
Some mechanicists like Bechtel or Glennan are reluctant regarding an ontic conception because they think that the model of the mechanism itself explains, while the mechanism stands in the world. Hence they view themselves as favoring an epistemic view of explanation—but this difference is not relevant for my present purposes.
See also Rice (2012) for a non-causal understanding of optimality explanations.
Provisionally, this term names all the explanations cited here, to the extent that they refer to formal properties. A full justification of this label is given in Sect. 4.
Mechanisms may be very abstract—the issue of distinguishing abstract mechanisms, or very abstract types of representation of mechanisms (this distinction is not crucial here) from the types of explanations considered here will be addressed in Sect. 5—but see also Felline (2015) and Huneman (2015): the upshot here is that if a mechanism is too abstract, it trivializes the very idea of a mechanism.
I use “generality” about the fact that propositions holds for a wide variety of cases, abstracting away for their specificities. By contrast “genericity” points to the difference of ontological domains in which a property holds. “Generic” behaviors are for instance features of phase transition, that are identical in epidemiology, condensed matter physics or financial markets, considered as networks. Genericity is often defined in terms of limit behaviors that are manifest through a passage to an infinite limit, like in the paradigmatic case of phase transitions.
Given that this paper intends to be as neutral as possible regarding the ontology of mathematics, I won’t assume more than that. I clearly touch upon issues that are not unrelated with problems familiar to philosophers of mathematics: ontology of mathematical entities, indispensability of mathematics, abstraction, explanations within mathematics. However, since I aim at outlining a theory of explanation for the empirical sciences I try to minimally assume theses from the philosophy of mathematics.
See Huneman (2015) for a critique of the argument that topological explanations are indeed the description of very abstract mechanisms.
See Gaucherel et al. (2011) for this issue in ecology.
To prevent any objection based on the preceding discussion, notice that in the case of optimality reasoning, the function involving fitness did not describe a process taking place anywhere—in the sense that no single entity is converting each phenotypic value unit into a fitness value, but the ‘converter’ is each time a distinct organism.
Notice that simplifying animal shapes by treating them as spheres is a rather conservative estimate since this figure has the smallest surface area to volume ratio. For all other shapes, the rule is even more justified.
This is approximated here as a sphere, but the only important point is that forms are comparable, which obtains since we consider a given clade or genera.
One could see those facts as physical facts represented in mathematics; however, they are still different from physical facts such as the melting temperature of iron—since they are properties that space necessarily should have. Mathematics is the set of those necessary facts. Physical space has the features it has—for instance, that any triangle build with a diameter of a circle and a point on this circle will be a rectangular triangle—in virtue of those mathematical facts, that are by themselves necessary. The question of the relation of these facts to physical facts commits one to an interpretation of the status of mathematical truths, which I don’t want to address here. The account of mathematical properties as standing in scientific explanations should be as independent as possible from such kinds of issues proper to the philosophy of mathematics.
The distribution need not be normal, indeed.
I focused here on the case of normal distributions because it was the easiest one and the most familiar.
Later sections will answer the objection that this could be only a very abstract mechanism—yet the above mentioned papers by Felline and Huneman suggest answers to this objection, yet not in the present context of trying to define mathematical explanatoriness.
Unlike for instance Bergmann’s rule, where the assumption that a specific “mechanism” causing dominant phenotypes is at work.
They can also be physically represented, for sure, but this representation can in turn often be mathematically captured.
Actually there is even one journal in mathematics devoted to fixed-point theorems: Fixed-point theory.
Actually Nash uses also Brouwer’s theorem in another demonstration in his dissertation, but of course the logics of the argument is the same (Hofbauer 2000).
When brought about by migration they are by definition rare at the beginning.
I considered here functions and analysis, but an analogous reasoning easily holds for other mathematical objects such as matrices etc.
It is also the topic of the more general “category theory” but this paper does not intend to commit to category theory.
Of course a function can be seen as a set of n-tuples and therefore the sets of sets and the functions of functions are isomorphic. This plays a crucial role when addressing the issue of (non-)equivalence between different kinds of structural explanations distinguished by the nature of the mathematical properties involved—see below.
The question of the interpretation of those probabilities can be left aside. There is a way of interpreting them causally that would turn such independence into causal independence and open the way for an argument where this condition is in the end a causal condition—but the point made here is supposed to hold independently, or prior to any interpretation of probabilities. Moreover Strevens (2015) shows how causal and probabilistic independence are not in principle equivalent.
Reutlinger (ms) develops a strong critique of the argument that derives non-causality from the abstractness of explanations.
Note that all this should hold whatever the correct position regarding mathematical entities (fictionalism, Platonism, etc.).
A second issue untouched here is the relation of this account to category theory, which is in mathematics a very recent and general theory precisely concerned with the structures proper to various mathematical theories and their relations. This should be done but the main point of this paper was to introduce and describe this kind of explanations in general. Thus, this specific mathematical theory, whose use is often controversial, needs not be brought into the picture for now—though it will be when the question of the relations between those “subfamilies” will come to the fore.
Another issue is the relation between those explanations and structural realism. It’s left for another investigation. From their notion of structural explanations, Dorato and Felline (2011) suggest an account of that.
Lange has just published a book about non-causal explanations (Lange 2016), which develops the views presented in (2013a, b). Given that the present paper has been written before the publication of this book, I won’t compare its theory to the ones he elaborates there. This would anyway be much too long here and will be left for a coming work.
Humphreys’ paper also includes a consideration of the mathematical hierarchy, connected with the generality and depth of explanations—which concurs with the ideas of reference and hierarchies used here in order to make sense of explanatoriness.
Actually this distinction is exactly the one made in terms of distinct why-questions when Huneman (2010) about topological explanations distinguishes the explanation of the stability properties of a ecosystem having given topological properties from the explanation of the fact that a given ecosystem indeed has a network of interaction that realizes those properties.
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Acknowledgements
I am grateful to Carl Craver, Stuart Glennan, Paul Humphreys and Anya Plutynski for insightful comments and constructive criticism on drafts of this paper, and to Robert Batterman, Laura Felline, Tom Polger, Larry Shapiro and Rasmus Winther for discussions on the arguments. Thanks to Andrew McFarland for a thorough language checking of the text. I also warmly thank an anonymous reviewer for critiques and suggestions that substantially improved the paper. This work is supported by the Grant ANR 13 BSH3 0007 “Explabio”.
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Huneman, P. Outlines of a theory of structural explanations. Philos Stud 175, 665–702 (2018). https://doi.org/10.1007/s11098-017-0887-4
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DOI: https://doi.org/10.1007/s11098-017-0887-4