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Are non-accidental regularities a cosmic coincidence? Revisiting a central threat to Humean laws

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Abstract

If the laws of nature are as the Humean believes, it is an unexplained cosmic coincidence that the actual Humean mosaic is as extremely regular as it is. This is a strong and well-known objection to the Humean account of laws. Yet, as reasonable as this objection may seem, it is nowadays sometimes dismissed. The reason: its unjustified implicit assignment of equiprobability to each possible Humean mosaic; that is, its assumption of the principle of indifference, which has been attacked on many grounds ever since it was first proposed. In place of equiprobability, recent formal models represent the doxastic state of total ignorance as suspension of judgment. In this paper I revisit the cosmic coincidence objection to Humean laws by assessing which doxastic state we should endorse. By focusing on specific features of our scenario I conclude that suspending judgment results in an unnecessarily weak doxastic state. First, I point out that recent literature in epistemology has provided independent justifications of the principle of indifference. Second, given that the argument is framed within a Humean metaphysics, it turns out that we are warranted to appeal to these justifications and assign a uniform and additive credence distribution among Humean mosaics. This leads us to conclude that, contrary to widespread opinion, we should not dismiss the cosmic coincidence objection to the Humean account of laws.

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Notes

  1. Recent literature has discussed an objection in the vicinity according to which Humean laws cannot explain their instances, which is also sometimes phrased as the complaint that there is a circularity in the Humean account, as it takes the instances to explain the laws and vice versa (see e.g. Lange 2013; Marshall 2015). In this paper I reassess a stronger objection: one that concerns not just the general inability of Humean laws to explain simpliciter, but rather to their inability to explain a cosmic coincidence (i.e. the extremely regular arrangement of the instances). This stronger objection cannot be answered merely by claiming that the Humean mosaic is a brute fact that does not need to be explained, for its extremely specific arrangement calls for an explanation—or so the objection goes, as elaborated in Sect. 2.

  2. ‘Credence’ is to be understood for now in a neutral way, leaving open whether it refers to degrees of belief or full beliefs; only later we will conclude that the credences here can be represented as precise degrees of belief.

  3. Throughout the paper we use the terms ‘universe’, ‘world’, and ‘Humean mosaic’ interchangeably.

  4. Measure theory allows us to compare the sizes of uncountably infinite sets of the same cardinality, avoiding tricky orderings of these sets that make it difficult to compare them (as in Lewis 1986, Sect. 2.5). Still, we won’t follow this complex path, due to intractable problems with infinity (see Sects. 2.2.2 and 2.2.3); rather, we will aim for results on a finite but arbitrarily large space, which are approximate but nevertheless sufficiently significant and stand on firmer ground. More on this below.

  5. So understood, Hume’s scepticism did not amount to a metaphysical claim, but rather to a more modest emphasis of our epistemic limitations. (We cannot justify the rationality of inductive inference unless we justify our belief in the uniformity in Nature; yet we cannot rationally infer that Nature is uniform without making an inductive inference. But this is the epistemic problem of justifying the rationality of inductive inference, not the metaphysical problem of explaining the uniformity of Nature). Also, among the contemporary Humeans who we address in this paper, that is, those who do take the further step of drawing metaphysical conclusions, it is conceded that the non-Humean does explain the regularities of the actual world: see e.g. the influential [Beebee 2011, Sects. 2 and 6 (last paragraph)] and (Beebee 2006, p. 527). Here it is also worth adding that Beebee dismisses the cosmic coincidence objection by appealing to a posteriori evidence, which begs the question, as I argue in footnote 7.

  6. The metaphor of Fig. 1 of God arbitrarily choosing a mosaic is inadequate from the non-Humean’s point of view: the illustration was used by a non-Humean in reference only to the initial state of the universe! Even when restricted only to the initial state, the dialectics is different for the Humean and for the non-Humean. Whereas a Humean like Callender (2004) argues that there is no need to explain this (which I find unconvincing, but leave that aside), the non-Humean does not rule out that there might be some reason behind the very special initial state of the universe (Penrose himself proposes one). By leaving available the provision of some reason—a dialectical possibility unavailable to the Humean—the non-Humean is potentially able to avoid the scenario in which the IC has been arbitrarily actualized from a large possibility space.

  7. Finally, a temptation to avoid when reading Argument is to think that we now have a posteriori evidence about which mosaic is the actual world, and claim that given our current evidence it is no surprise that the world is as it is. In other words, P (the world turns out to be highly ordered | evidence ) = 1, given that evidence = ‘the world is highly ordered’. This trivially true statement is not what we are interested in. (This petitio principii is found in Beebee 2006, p. 527). We want to place ourselves before the actual world occurred, from a God’s eye viewpoint, in order to assess whether the actual world is a surprising event.

  8. I set aside other alleged problems related to updating and the impossibility of learning from experience, for they do not affect our scenario in which there is no updating.

  9. Minimax is the rule in decision theory that demands that an agent in the absence of evidence chooses the option that minimizes its maximum disutility (see Pettigrew 2016b, pp. 39–40 for details and justification). Minimax applies only to what David Lewis called ‘superbabies’: agents at the beginning of their credal life. We are such superbabies with respect to Argument: as pointed out in footnote 7, in assessing the plausibility of the regularity of the actual Humean mosaic we have to put ourselves before the actual evidence.

  10. When an agent’s credence function c is defined over finitely many outcomes \(E_i\), its ‘entropy’ or ‘uninformativeness’ is measured by the Shannon entropy, \(H(c)= - \Sigma _x c(E_i) \cdot log(c(E_i))\). When c is defined over uncountably many outcomes, its entropy is calculated as: \(h(c) = - \int _{\Pi '} c(E_i) log(c(E_i))dx\).

  11. More precisely, the best way to respect the condition is to follow a slight variation of this principle, which he calls the Maximum Sensitivity principle.

  12. The Principal Principle states that, if p is a certain proposition about the outcome of some chancy event and E is our background evidence at t, which must be admissible evidence, then: \(Cr(p|Ch(p) = x \wedge E) = x\). (Evidence E is admissible relative to p if it contains no information relevant to whether p will be true, except perhaps information bearing on the chance of p). See Lewis (1980, p. 86) and Hoefer (2019, Chap. 3).

  13. The point is that there is not a chance distribution related to the obtaining of this or that mosaic. Lewis (1994) tried to make sense of the whole mosaic having an objective chance, but in reductionist, Humean terms. He defined the fit of a system of laws to be equal to the chance that the system gives to the full mosaic that it supervenes on [cf. Loewer (2004); Albert (2012, Chap.1); Hoefer (2007, 2019)]. Yet, this is of course a different sense from the objective chance related to bringing about the mosaic. In any case, this attempt has been criticized in several ways, e.g. by Hoefer (2007), and more recently in Hoefer (2019, Chap. 4) where, for instance, it is shown that Humean chances must be restricted to small-scale phenomena within the mosaic in order for his proof of the Principal Principle to go through. Thus, we can talk of Humean chances within the mosaic-indeed we should, insofar as we are assuming the Humean point of view!-while accepting that no chance is assigned to the occurrence of the mosaic. It is, remember, just a brute fact.

  14. Optionally we could also add—although it is not necessary for our justification of additivity—that strict subsets are more probable than their constitutive elements. This has been argued in (Filomeno 2017). See Fig. 3.

  15. The book consists of a set of bets, each of which the agent views as fair, but which together guarantee that she will lose money come what may.

  16. Cf. the other conditions proposed in (Kraft et al. 1959; Luce 1967; Savage 1972; Suppes and Zanotti 1976); for discussion see Fine (1973, IIC and IIID), Krantz et al. (2006, Chap.5) and Suppes (1994).

  17. The use of infinitesimals has been disputed by Pruss (2012), Pruss (2013), Pruss (2014) and Williamson (2007). Benci et al. (2016) and Weintraub (2008) attempt to reply to such objections.

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Acknowledgements

I would like to express my deep gratitude to Carl Hoefer, John Norton, Alfonso Arroyo-Santos, and Sylvia Wenmackers for helpful comments and discussion. I would also like to thank Paul Bartha, Yann Benetreau-Dupin, Joan Bertran, Eddy Keming Chen, Juan Luís Gastaldi, John Horden, Pavel Janda, Ladislav Kvasz, Juergen Landes, Vera Matarese, Enrique Miranda, Richard Pettigrew, Carlos Romero, Miguel Ángel Sebastián, Teddy Seidenfeld, Alessandro Torza, Jon Williamson and audiences at the ALFAN 2015 conference, the ’Simefi’ seminar at UNAM, and the LOGOS seminar.

Funding

This work was supported by the Instituto de Investigaciones Filosóficas (Universidad Nacional Autónoma de México) through a fellowship from the postdoctoral fellowship program DGAPA-UNAM, and by the grant ‘Formal Epistemology - the Future Synthesis’, in the framework of the program Praemium Academicum of the Czech Academy of Sciences.

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Filomeno, A. Are non-accidental regularities a cosmic coincidence? Revisiting a central threat to Humean laws. Synthese 198, 5205–5227 (2021). https://doi.org/10.1007/s11229-019-02397-1

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