Abstract
Relationships of counterfactual dependence have played a major role in recent debates of explanation and understanding in the philosophy of science. Usually, counterfactual dependencies have been viewed as the explanantia of explanation, i.e., the things providing explanation and understanding. Sometimes, however, counterfactual dependencies are themselves the targets of explanations in science. These kinds of explanations are the focus of this paper. I argue that “micro-level model explanations” explain the particular form of the empirical regularity underlying a counterfactual dependency by representing it as a physical necessity on the basis of postulated microscopic entities. By doing so, micro-level models rule out possible forms the regularity (and the associated counterfactual) could have taken. Micro-model explanations, in other words, constrain empirical regularities and their associated counterfactual dependencies. I introduce and illustrate micro-level model explanations in detail, contrast them to other accounts of explanation, and consider potential problems.
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Notes
Woodward lists a number of conditions that need to be satisfied for a permissible intervention on the cause variable. For more details see Woodward (2003, 98).
Another example of his is the macro-explanation of the rise in the price of oranges in terms of a shortage of supply, as opposed to the (fictional) micro-explanation of the same phenomenon in terms of the fundamental physics of “human behavior involved in oranges selling at price P” (233). See Bradley (2020) for a discussion of the claim that “higher level” explanations are generally better because they omit details.
Explanandum regularities of MLM explanations can also be probability distributions. See Section 5.3
Another form could be a quadratic relation so that low or high values in X result in low values of Y, but middle range value X results in high values of Y.
I speak of physical necessities because the necessities are meant to hold about the physical world. They are different from metaphysical necessities or mathematical necessities (such as the necessity involved when Mother fails to distribute 23 strawberries among her three children; see Lange (2017)). See also Section 5 for more on necessities.
In standard usage, the attribute “contingent” applies to objects (such as IGL) which depend on how our actual world is like / happens to be like. That is how I intend to use this attribute also in this paper. Contingency should not be confused with arbitrariness, though. For example, the famous “all gold spheres are less than 1 mile in diameter” is contingent, but also arbitrary and is therefore generally considered not to be a law of nature. Quite in contrast to “all uranium spheres are less than 1 mile in diameter”, which supports counterfactuals (such as “had there been spheres of 1 mile in diameter, they would just have blown up immediately”). It is of course those latter kinds of regularities that I’m interested in here as targets for micro model explanations.
See Strevens (2008) for a detailed discussion of the idealizing assumptions of KT. KT is called a theory, rather than a model, only for historical reasons.
In my derivation I follow mostly Holton and Brush (2001). For very similar derivations see the textbooks by Levine (1995), McQuarrie and Simon (1997), which are both cited by Strevens (2008). In contrast, Doyle et al. (2019) use a partition function to derive the IGL. A partition function gives the sum over all energy states of the system, and as such is quite different from a derivation in KT that is based on the (classical) dynamics of the individual molecules (although information about molecules of the gas, such as their number, also enters the calculation of the partition function).
In the Rydberg formula, n1 determines the kind of spectral line series, e.g., the Balmer series for n1 = 2.
For example, Bohr assumed – quite boldly – that the frequency of emitted light (which is inversely proportional to its wavelength) is equal to the average of the frequencies of the electron on its orbit before and after the “jump”.
One of my own points of criticism of Bokulich’s account draws attention to a tension between Bokulich’s demand that model fictions be justified by true theories in order for them to count as explanatory on the one hand, and her view that model fictions can provide deeper explanations than true theories on the other hand. If explanatory model fictions must be justified by true theories, however, then the explanations provided by such models ultimately will counterfactually depend on the explanans identified by those true theories (and the model explanation can therefore not be deeper than the explanations provided by those true theories) (Schindler, 2014, 1747). This point of criticism strikes me as being rather similar to the second horn of Nguyen’s dilemma (Nguyen, 2021, 3240) – contrary to what Nguyen’s remarks in footnote 32 of his paper may suggest. (The first horn of Nguyen’s dilemma is the point mentioned above, namely that the target cannot counterfactually depend on fictions.).
In view of the second point of our definition in Section 3.1, we can say more precisely that Dalton’s atomism represents each chemical element (the macroscopic variable) with its own kind of atoms (the microscopic variable), all of which are indivisible, and which therefore combine with atoms of other elements only in multiple integers.
Again, we may say more precisely that Mendel’s model represents phenotypic traits (the macroscopic variable) in terms of alleles (the microscopic variable).
Strevens (2008) seeks to solve the puzzle by requiring that explanatory principles describe the aggregation of real causal influences and by ruling out cases as the Kepler-Boyle example as causally irrelevant to the explanation (pp. 275–6). Other attempts are discussed in Salmon (1989). See also Kitcher (1976).
An anonymous referee referred me to the account by Doyle et al. (2019). Doyle et al. introduce the distinction between an object and a base of understanding, whereby they define the former as the “thing to be understood” and the latter as the “thing to provide understanding”. As one of their examples, they consider the explanation of the IGL (an object of understanding) by the so-called partition function in statistical mechanics. However, Doyle et al. do not provide any detail as to how a philosophical account of regularity “understanding”, let alone explanation, might look like. Instead, their primary goal is to defend non-factivism about understanding, that is, the view that “radical departures from the truth are not always barriers to understanding” (345) (to which I myself would subscribe to).
Woodward (2003) introduces the notion of “change-relating relationships” between explanans and explanandum which he defines as relating “changes in variables in the explanans to changes in the explanandum variable” (202). Again, for Woodward these change-relating relationships mostly consist of generalizations such as IGL (rather than relationships between a model and a regularity).
In virtue of their reductive nature, MLM explanations contrast quite sharply with so-called “minimal model” explanations, as identified by Batterman and Rice (2014). Minimal “minimal models” in science explain their targets (some macro-behavior) on the basis of the model and the target being members of the same universality class. Many of the microscopic details of the systems considered does not matter to the explanations provided by minimal models. For criticisms of Batterman and Rice’s account see e.g., Lange (2015). Incidentally, I agree with Lange that there does seem to be a sense in which minimal models in their own examples explain their targets by sharing some (abstract) features (contra Batterman and Rice).
Nagel’s original account seems to come with all kinds of positivistic presumptions (e.g., a distinction between observational and theoretical language), but it has been argued more recently that the basic idea of Nagelian reduction does not require such baggage (Dizadji-Bahmani et al., 2010).
A reason for that belief is the idea that “often, it is in fact not possible to derive the exact laws [of the reduced theory]”, as for example in the second law of thermodynamics (Dizadji-Bahmani et al., 2010, 398).
See the literature on reduction of Mendelian genetics to molecular biology (see Waters, 2013).
I am not in principle opposed to using the label of “mechanistic explanation” for MLM explanations: so long as the basic elements of the features identified by my account are respected (see Section 3.1), this would then merely be a terminological matter.
Logical possibility is believed to be the most encompassing modality there is. Metaphysical modality is believed to more restricted, but the details are disputed. See Vaidya (2017).
The same point holds for antireductionist views such as Maudlin’s, which stipulate the necessity of the laws of nature as a primitive in their metaphysics (Maudlin, 2007).
There is a sense in which the explanation of regularities automatically entails the explanation of single events, since regularities are based on (or even grounded in) events. I take this sense of explaining events when one explains regularities to be trivially true. Salmon certainly had something else in mind when raising his challenge to the modal account.
It bears some irony that Mendel himself has been suspected of bias because he presented numbers from his experiments which seemed too good to be true (Franklin et al., 2008).
Note that the Duhem thesis seems also relevant to this point: theories are never applied to the world without a number of auxiliary assumptions about the experimental setup, the experimental apparatus, background effects, etc. Cartwright (1983) also speaks of a “prepared (i.e. idealized) description” of the phenomena which theories explain (rather than the nitty gritty of the actual world).
The analogous considerations – in principle – apply to quantum mechanics, Salmon’s other alleged counterexample to the modal account: quantum mechanics predicts probabilities of possible measurement outcomes with necessity, but it doesn’t predict any particular measurement outcomes. I would be prepared to argue that it is (also) by virtue of necessitating these probabilities that quantum mechanics possesses explanatory power. Of course, quantum mechanics is a special case, since the veritable measurement problem poses its very own difficulties for any account of explanation. See e.g., Salmon (1998, 325). It’s not clear how even Woodward’s more liberal account of causation could accommodate quantum mechanics, and in particular, the Einstein-Podolsky-Rosen paradox (de Regt, 2004).
There are other solutions to this dilemma. Strevens (2008) suggests that idealizations in reality just flag those causal factors that don’t make a difference to the explanandum. Bokulich (2011, 2012) argues that model explanations are justified through “translation keys” with true theories. See Schindler (2014) and Nguyen (2021) for criticisms of the role of these translation keys in Bokulich’s account. See also Section 3.3.1.
Thanks to Ioannis Votsis for suggesting this to me. See also Pincock (2022) for a ‘veritist’ proposal along those lines.
The structure retained in this example seems a purely empirical structure we know by virtue of IGL. But structural realism requires structural continuity at the theoretical level. See Worrall (1989).
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Acknowledgments
I thank the two anonymous referees for this journal for their detailed critical remarks, which helped to improve the paper tremendously. I’m also grateful to Kareem Khalifa, Daniel Kostić, and Insa Lawler, who kindly read distant relatives of this paper and made me rethink the framing of my paper. Previous versions of this paper were presented at the Workshop on Understanding from Models at the IHPST in Paris (2017), the EPSA 2017 in Exeter, the research colloquium at Simon Fraser University (2019), and the Cognitive Science Research Group seminar at the New College of the Humanities, London (2019). I thank the audiences at each of these talks for their valuable feedback.
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Schindler, S. Micro-level model explanation and counterfactual constraint. Euro Jnl Phil Sci 12, 40 (2022). https://doi.org/10.1007/s13194-022-00465-x
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DOI: https://doi.org/10.1007/s13194-022-00465-x