Abstract
In an illuminating paper, Beisbart (Beisbart C, Good just isn’t good enough – Humean chances and Boltzmannian statistical physics. In: Galavotti MC, Dieks D (eds) New directions in the philosophy of science. Springer, Dordrecht, 2014) argues that the recently-popular thesis that the probabilities of statistical mechanics (SM) can function as Best System chances runs into a serious obstacle: there is no one axiomatization of SM that is robustly best, as judged by the theoretical virtues of simplicity, strength, and fit. Beisbart takes this “no clear winner” result to imply that the probabilities yielded by the competing axiomatizations simply fail to count as Best System chances. In this reply, we express sympathy for the “no clear winner” thesis, however we argue that an importantly different moral should be drawn from this. We contend that the implication for Humean chances of there being no uniquely best axiomatization of SM is not that there are no SM chances, but rather that SM chances fail to be sharp.
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Notes
- 1.
Or alternatively a Renyi-Popper measure Ch(p|q) that maps proposition pairs 〈p, q〉 onto the reals in the [0, 1] interval. (Plausibly, if one takes conditional chance as basic in this way, then it is redundant to include a “time” index to the chance function; see Hoefer 2007, pp. 562–565; Glynn 2010, pp. 78–79.)
- 2.
This notion of fit applies only if there are finitely many chance events. See Elga (2004) for an extension to infinite cases. In addition, if one wants to allow the possibility of statistical mechanical probabilities counting as chances, then one needs a notion of fit according to which one way a system may fit better is if its probability function assigns a relatively high probability to the macro-history of the world conditional upon a coarse-graining of its initial conditions (as well as assigning a relatively high probability to the micro-history conditional upon a fine graining of the initial conditions).
- 3.
In the quantum case, the uniform probability distribution is not over classical phase space, but over the set of quantum states compatible with the PH.
- 4.
- 5.
This proposal requires that initial conditions, such as PH, are potential axioms of the best system. The BSA has not always been construed as allowing for this. However, Lewis (1983, p. 367) himself seems sympathetic to the view that they may be.
- 6.
Callender (2011) calls attempts to derive the SM probabilities from a probability distribution over the initial conditions of the universe as a whole “Globalist” approaches to axiomatizing SM. The Mentaculus is one example of a Globalist approach. Beisbart points out that there are rivals. In contrast to Globalist approaches, “Localist” approaches (see Callender op cit.) attempt to derive the SM probabilities from probability distributions over the initial states of the various approximately isolated subsystems of the universe. Beisbart observes that there is a range of competing Localist approaches to axiomatizing SM. While, for reasons of space, we will here focus upon the competing Globalist approaches, many of our points will carry across to the competition between Localist axiomatizations, if one thinks that the Localist approaches are more promising (see Glynn unpublished).
- 7.
Indeed, as Beisbart points out, it is not clear precisely how low initial entropy is specified to be by the PH. Different precisifications of the PH yield different sized regions of phase space to which the uniform distribution is to be applied. So it seems that the number of competing systems may be larger still.
- 8.
At least this might be so if, with Frigg and Hoefer (2013), we exclude information about the precise micro-state of the world as inadmissable, since “chance rules operate at a specific level and evidence pertaining to more fundamental levels is inadmissible.” See Maudlin (2007) for a discussion of how one can derive typical thermodynamic behavior without committing to precise assumptions either about the initial probability distribution or the size of the initial phase space region.
- 9.
As we saw, Lewis claims that if there was not a unique best system, there would be nothing deserving of the name law, though he earlier (Lewis 1983, p. 367) said theorems entailed by all of the tied systems would count as laws. We’re sympathetic to his earlier position. If, for instance, the fundamental dynamics are the same in all the tied systems (this is not something that is disputed by Beisbart and others who have examined rivals to the Mentaculus), then they will come out as laws.
- 10.
Frigg and Hoefer (2013) themselves find it plausible that there may not be a unique best system for our world.
- 11.
- 12.
Also it’s not clear to us that an axiom system specifying that the universe was initially in a larger region of phase space than Γ0 is simpler than one specifying that the universe was in Γ0. As is well known (e.g. Lewis 1983, p. 367), simplicity is vocabulary-relative and, in order to avoid trivializing the desideratum of simplicity, we must take simplicity-when-formulated-with-unnatural-predicates to be less desirable than simplicity-when-formulated-with-reasonably-natural-predicates (and perhaps simplicity-when-formulated-with-perfectly-natural-predicates to be more desirable than both). In our not-too-unnatural macro-vocabulary, we may be able to formulate simple axioms that pick out moderately large regions of phase space like Γ0. Picking out smaller regions will often require employing complex microphysical predicates, thus increasing fit (and perhaps naturalness of predicates) at the expense of simplicity. But picking out larger regions than Γ0 may require disjunctions of reasonably natural macrophysical predicates. The resulting axiom systems will be both worse fitting and less simple (or worse fitting and formulated in less natural language). If so, they would be clearly inferior to an axiom system specifying that the universe was initially in Γ0.
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Acknowledgement
The authors would like to thank Claus Beisbart, Seamus Bradley and Leszek Wroński for helpful comments. We would also like to acknowledge the support of the Alexander von Humboldt-Foundation and the Munich Center for Mathematical Philosophy.
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Dardashti, R., Glynn, L., Thébault, K., Frisch, M. (2014). Unsharp Humean Chances in Statistical Physics: A Reply to Beisbart. In: Galavotti, M., Dieks, D., Gonzalez, W., Hartmann, S., Uebel, T., Weber, M. (eds) New Directions in the Philosophy of Science. The Philosophy of Science in a European Perspective, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-04382-1_37
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