1. Introduction

In recent work on the foundations of statistical mechanics and the arrow of time, Barry Loewer (Reference Loewer1996, Reference Loewer, Price and Corry2007) and David Albert (Reference Albert2001, Reference Albert2009) have developed a view that defends both a Humean account of laws and a physicalist fundamentalism. A proper understanding of the foundations of statistical physics, they argue, provides the missing piece for a successful completion of David Lewis's project of a unified account of laws, causes, counterfactuals, and probabilities that is at once Humean and foundationalist. In this article, I will critically examine their account and argue that there exists a tension between their account of laws and their reductivism or foundationalism. Albert and Loewer's version of a best system account of laws departs from Lewis's own account by further emphasizing the pragmatic dimension that arguably was already implicit in Lewis's account. Here I find myself in broad agreement with them. But, as I will argue here, this pragmatism is in tension with the foundationalism to which they also subscribe.

I will proceed as follows. In section 2, I briefly summarize Albert's and Loewer's account of the foundations of statistical physics and its place in a Lewisian conception of the world. In sections 3 and 4, I discuss worries that arise from the fact that their account, unlike Lewis's original account, allows for initial conditions to be laws and for terms referring to nonfundamental properties to be part of the language in which the laws are formulated. In section 5, I argue that pragmatic considerations similar to the ones that Albert and Loewer themselves invoke suggest that the laws of the special sciences should be added as additional axioms to the best system. In section 6, I critically examine an argument by Albert aimed at defending his fundamentalism against an objection due to Philip Kitcher. I side with Kitcher and argue that the special sciences can offer explanations of the phenomena in their domain that are explanatorily independent from—and, at least in many contexts, explanatorily superior to—putative derivations of these phenomena from fundamental physics.

2. From Boltzmann to Lewis

According to the Boltzmannian account defended in Albert (Reference Albert2001), the thermodynamic asymmetry that the entropy of a closed macroscopic system never decreases can be explained by appealing to a time-symmetric micro-dynamics and an asymmetric constraint on initial conditions. If we assume an equiprobability distribution of microstates compatible with a given macrostate of nonmaximal entropy, then it can be made plausible that, intuitively, ‘most’ microstates will evolve into states corresponding to macrostates of higher entropy. However, if the micro-dynamics governing the system is time-symmetric, then the same kind of considerations also appear to show that, with overwhelming probability, the system evolved from a state of higher entropy. This undesirable retrodiction can be blocked if we conditionalize the distribution of microstates not only on the present macrostate but also on a low-entropy initial state of the system. Since the same reversibility objection can be raised for any time in the past as well, Albert and others argue that we are ultimately led to postulate an extremely low-entropy state at or near the beginning of the universe. Albert calls this postulate “the past hypothesis” (PH). Albert and Loewer maintain that the statistical mechanical account (SM), consisting of the past hypothesis; a probability postulate (PROB), which postulates a uniform probability distribution over the physically possible initial microstates of the universe; and the dynamical microlaws (which are assumed to be deterministic) provide us with a complete physics—a fundamental theory of everything.

According to Albert and Loewer, both PH and PROB should be thought of as nomic constraints, even though they are statements about the initial conditions of the universe, since then the Boltzmannian account provides the missing piece in a broadly Lewisian account of counterfactuals and causation. Lewis himself attempted to derive the temporal asymmetry of counterfactuals and causation from the thesis of an asymmetry of overdetermination. But this thesis is false (see, e.g., Frisch Reference Frisch2005, chap. 7). In its stead, Albert and Loewer appeal to the SM account as the source of the asymmetry. The idea is roughly that we evaluate counterfactuals by calculating the probability of the consequent, conditional on the laws of the actual world, the counterfactual antecedent event c at some time t, and the actual macrostate at t outside of the region where c occurs. The asymmetry is, then, meant to be a consequence of the fact that the laws include a time-asymmetric constraint, PH. And the reason for why PH is a nomic constraint rather than merely a de facto initial condition is that it is an axiom of the Lewisian best system—the theory of the world “couched in a vocabulary that includes a conditional probability function, mathematical notions, and terms that represent fundamental properties” that “best combines simplicity, informativeness and fit” (Loewer Reference Loewer, Price and Corry2007, 305).

3. Initial Conditions and Simple Systems

Loewer argues that PH and PROB are part of the best system, since adding the two conditions to the micro-dynamical laws “results in a system that is only a little less simple but is vastly more informative than is the system consisting only of the dynamical laws” (Loewer Reference Loewer, Price and Corry2007, 305). That is, on Loewer's view, all axioms of the best deductive system qualify as laws, in contrast with Lewis's original proposal, according to which only genuine regularities but not initial conditions could qualify as laws. On Lewis's account, the strength of different deductive systems is evaluated by including a statement of the relevant initial conditions, which themselves, however, are not candidates for being laws.

Allowing initial conditions to be laws is problematic, however. Consider a relatively simple ‘world’ that consists of nothing but a simple gravitational system satisfying Newton's gravitational theory. If we compare a deductive system for this world consisting of Newton's laws alone with one that, in addition, includes a statement of the world's instantaneous state at some time, then the relative ranking of the two systems seems obvious. The initial conditions, specifying the instantaneous positions and momenta of the planets, are not overly complex, while the gain in informativeness over a system without initial conditions is quite dramatic: Newton's laws on their own do not determine the positions or momenta of any planet, but adding the initial conditions determines the entire history of the world. Thus, according to Loewer's account, the statement of the initial conditions ought to be a law of that world. But this has the consequence that the entire evolution of this world is nomologically necessary, since the state of the world at any time is derived from nomic constraints alone. Thus, the intuitive distinction between laws and nomologically contingent initial conditions cannot be drawn in such a world.

This strikes me as problematic. One condition of adequacy of any account of the notion of scientific laws is that it allow us to recover core features of the usage of the term ‘law’ by scientists, and one such feature is the distinction between laws and contingent boundary and initial conditions applied even to relatively simple systems. Moreover, Loewer's account has the seemingly absurd conclusion that for each different initial state of a physical system, that system could not have possibly been in any other state, since it is a law of the system that it started out in that state.

4. Comparing Different Systems

Lewis's best system account requires that the axioms of the best system relate only fundamental properties. Albert and Loewer's account, by contrast, allows for PROB and PH, which contain terms referring to nonfundamental macro-properties, to be laws as well. Yet, once we allow axioms to be formulated in different vocabularies and do not restrict our language to that of fundamental physics, the question arises how to compare deductive systems formulated in terms of different languages. As Cohen and Callender (Reference Cohen and Callender2009) argue, both simplicity and strength are immanent notions that are relative to an inventory of basic kinds or predicates, and there exists no language-transcendent metric of comparison. Thus, PH is a very simple hypothesis only from the perspective of a language that includes terms referring to macro-properties, but it is extremely complicated from the perspective of the language of micro-physics. As Cohen and Callender (Reference Cohen and Callender2009) say, “if we translate the claim that entropy was low into a more fundamental microlanguage we end up with a long gruesome mess. … The translated claim would still be strong, but hardly simple. For this reason, written in the language of the (ex hypothesi) fundamental kinds, the Past Hypothesis cannot be a[n] MRL law” (10).

Moreover, even within a single language, allowing initial conditions to be laws poses problems for the comparison of different deductive system. In relatively simple worlds, as we have seen, adding a statement concerning initial conditions leads, on balance, to a better system: since the dynamical laws on their own do not imply any particular matter of fact, adding relatively simple initial conditions vastly increases the strength of the system with only a minimal loss in simplicity. But how do we balance simplicity and informativeness if we are trying to adjudicate between the system that contains a statement of the actual dynamical microlaws of our universe and the system that contains, in addition, a statement of the precise microscopic initial state of the universe? In this case, it is no longer obvious how to rank the two systems. On the one hand, a statement of the complete initial conditions of the universe at one time would be extraordinarily complicated; on the other hand, the gain in informativeness is enormous: the system consisting only of the dynamical laws implies not a single particular fact about the instantiation of any properties at any space-time point, while the system that includes a statement of the initial conditions implies the complete history of the universe. It is unclear, at least to me, how one might possibly adjudicate between the two systems and how to balance the dramatic increase in complexity against the (equally?) dramatic increase in informativeness. (Note that this problem does not arise for Lewis's account, according to which the two systems—with and without initial conditions, respectively—have exactly the same laws.)

One might reply that there need not be a complete ordering of all candidates for best system. It might be enough if there were a partial ordering with a clear ‘winner’—a winner that Albert and Loewer would insist is the system consisting of the true micro-dynamics in conjunction with PH and PROB. Yet, if we compare this system with the system that contains a statement of the precise initial state of the universe, it is still not clear which system maximizes simplicity and strength and how we should balance the dramatic increase in informativeness in the system with complete microscopic initial conditions against the vast increase in complexity.

Now, Albert and Loewer's account has the resources to answer both of these problems of comparison. First, Loewer proposes that the simplicity of a system be measured not only in terms of the number or syntactic complexity of its axioms but also in terms of the lengths of the proofs required for deriving different theorems. Second, Albert and Loewer introduce a strongly pragmatic element into the metric for comparing different systems, which is brought out particularly vividly in their imaginary tale of how the best system is revealed to us during an audience with God. Here is how Albert describes the scenario. Imagine that you have an audience with God, who provides you with as much information about the particular facts of the world as you could possible want to have. One way to provide this information is to recite long lists of particular facts concerning which properties are instantiated at which spatiotemporal locations. Yet, as God begins to recite particular matters of fact,

The meaty and pithy information with which God provides you, Albert claims, will consist of the micro-dynamical laws together with PH and PROB.

Once we emphasize that it is our standard of simplicity and usefulness that is the ultimate criterion by which we adjudicate between systems, both Cohen and Callender's worry and the problem of comparing systems with or without initial conditions can be answered. The yardstick for simplicity and informativeness is how practically useful a system is for us—how well it allows us to make our way about in the world. For beings like us, the deductive system that includes the PH is clearly simpler—even though, from the perspective of the language of the fundamental micro-theory, stating the PH would be a ‘gruesome mess.’ And arguably both the system consisting only of the dynamical laws and that consisting of the laws and the exact initial conditions fail dramatically as far as their usefulness for us in making our way about in the world is concerned: the former system does not allow us to derive any particular goings-on in the world, not even in principle, while the latter is much too complicated to be of any practical use. Indeed, specifying the exact initial conditions of the universe, Albert says, would “violate” the stipulation of providing a simple summary. Thus, practical usefulness, for him, provides not only a criterion of relative goodness for a system but also a necessary condition for being minimally acceptable: a system that includes axioms that are too complex is practically useless and hence could not be the best system, no matter how informative it might be in principle or how much more informative it might be than any of its competitors.Footnote ¹ The lesson to draw from the discussion so far is that the strong pragmatic element is a necessary component of Albert and Loewer's view: without construing simplicity in terms of what is useful for beings like us, their account has no answer to the problems of comparison.

5. Why Stop Here?

Allowing predicates referring to thermodynamic macro-properties as part of the best system constitutes an expansion of Lewis's original proposal, which only permits axioms containing predicates referring to fundamental properties and kinds. And as we have seen, this expansion can be both motivated and defended by appealing to a strongly pragmatic construal of what it is for a system to be the best: the best system is the one that summarizes as much information about the world as possible in a way that is useful for us. Now, Albert and Loewer want to stop there. They insist that once we have expanded the system consisting of the micro-dynamical laws to include PROB and PH, we have found the best system and have arrived at a complete, unified, and fundamental theory of the universe. Yet, the very same considerations to which Albert and Loewer themselves appeal suggest that we ought to expand our system further and include the laws of the special sciences as well.

According to Albert and Loewer, adding PH and PROB to the micro-dynamical laws results in a better system than adding a statement of the exact microscopic initial state, since the latter statement would be absurdly complex—too complex to be even minimally acceptable as part of any contender as overall best system. But consider the derivation of any ‘law’ or regularity of the special sciences, such as Fisher's principle, which explains why the sex ratio of most species is . Any derivation of this phenomenon from the fundamental micro-dynamics, PH and PROB would be absurdly complicated—in fact, too complicated to be of any help to us in making our way about in the world. Thus, it seems that the SM account, as well, violates the minimal condition of adequacy proposed by Albert of providing “something that will serve you well, or reasonably well, or as well as possible, in making your way about in the world.” By contrast, a system that contains, in addition to the laws of physics, also the relevant laws of evolutionary biology would only be minimally more complex—it would contain only a few more simple axioms—yet, the gain in deductive simplicity would be enormous.

Now, Albert and Loewer entertain the hypothesis that all the laws of the special sciences follow from the axioms of the SM account as theorems, presumably once we also introduce an appropriate translation manual. This hypothesis is open to challenge, but even if we, for the present purposes, grant this claim, we can insist that the laws of the special sciences be added as axioms to the deductive system. The expanded deductive system will not be deductively stronger, and it will be less simple along one of the dimensions of simplicity identified by Loewer: the system will have more axioms. But, first, the same holds for the SM account vis-à-vis the system consisting of micro-dynamics, an initial microstate, and a translation manual: the former is not deductively stronger than the latter. And, second, the loss of simplicity along one dimension will be more than made up for by a gain in simplicity along another dimension that Loewer has identified—the length of the proofs required to derive different theorems. Thus, contrary to their own intentions, Albert and Loewer's pragmatism supports a view similar to the “relativized” and antifoundationalist best system account defended in Cohen and Callender.Footnote ² Once we take the best system to be the one that best allows us to make our way about in the world, not only the postulates of thermodynamics but the regularities of the special sciences end up as part of our overall system, as well.

6. Against Explanatory Fundamentalism

Explanatory considerations also count in favor of including higher-level laws in the best system. Albert (Reference Albert2009) defends the explanatory universality and completeness of physics against an objection by Philip Kitcher. Kitcher contrasts R. A. Fisher's evolutionary explanation of a regularity discovered by Arbuthnot—that in each of the 82 years from 1623 onward the preponderance of children born in London were boys—with a microphysical derivation of the sex of every child and maintains that even if per impossibile we possessed a microphysical derivation of the sex of each child, starting with the microscopic initial state of the world at each copulation, this derivation would not advance our understanding of the sex ratio and would not show that Arbuthnot's regularity is anything but a gigantic coincidence. Albert argues against Kitcher that if we assume the statistical mechanical account consisting of PROB, PH, and the micro-dynamics—that is, if we start with an equiprobability distribution over all initial microstates of the universe compatible with PH, evolve that distribution forward until the beginning of Arbuthnot's observation period in accord with the (assumed to be) deterministic microlaws, and conditionalize this on the existence of the solar system and on everything else taken for granted in the discussion of relative birth rates in London—then the resulting conditionalized probability distribution should be exactly in accord with the predictions based on Fisher's evolutionary explanation. If Fisher's explanation is correct, then, Albert maintains, the probabilities to which this explanation appeals must follow from the true and complete physics. Thus, Fisher's explanation provides us with some information about the “true and complete and universal physical theory,” but it could not possibly add to that theory.

I here want to side with Kitcher. I will, first, point to a sense in which Fisher's account does add to the SM account and, second, argue that Kitcher's claim that, according to the microphysical account, the regularity is a coincidence is correct. Let us grant that Albert is right and that in principle a probability for each male birth in London is derivable from the SM account. But what Fisher's account provides us is a probability distribution conditional on certain facts about the state of the earth and its vicinity. The problem for Albert's putative SM derivation is that there are myriad such distributions, each conditionalizing on different macro-facts. Moreover, Fisher's account, like any macro-theory, involves idealizations and abstractions, and what is taken for granted in the account is left to some extent vague. Thus, arguably none of the conditional probability distributions that might be in principle recoverable from the SM account will match exactly the probability distribution derived by Fisher. Finally, in order to derive the sex ratios of different species at different times, the putative SM account would have to conditionalize on different features of the macrostate at different times—macrostates that will be similar in countless ways and will differ from one another in countless others.

All this raises the worry that even if we were to grant Albert that the SM account allowed in principle derivations of probability distributions for sex ratios, the simple account given by Fisher relating birth ratios to sex ratios at maturity (and unifying the probability distributions for different times and species) would be deeply buried within that account. Fisher's account does not merely give us an observation about a particular probability distribution that follows from the SM account, as Albert suggests, but it gives us an argument why it is this conditional probability distribution, rather than one of the many others implied by the SM account, that is useful and interesting for us. Only from the perspective of evolutionary biology and its way of carving up the world, it seems, can we readily detect those conditional probability distributions that are relatively simple and informative about sex ratios at birth. What Fisher's account adds, then, is a reason for focusing on certain conditional probability distributions rather than others—it provides us with reasons to expect that certain specific conditional probability distributions among the very large number of such distributions in principle implied by the SM account are particularly simple and useful for beings like us to make our way about in the world.

Kitcher's claim that a putative micro-account could not explain why Arbuthnot's regularity is more than a gigantic coincidence can be defended, too, against Albert's criticism. Consider a gerrymandered set (S) of events all of which have probability P = .53—the probability of having a boy, according to Fisher's account—conditional on some relatively localized macrostate of the world, such as the state of London in 1623. One would expect there to be a very large number of such events, including, for example, the probability that a certain not entirely symmetrical coin comes up heads, say. Now compare the SM account of the events in S with an SM account of the events in the set B of male births. The events in both S and B have negligible probability given just PH and PROB, but all have the same probability P conditional on certain later macrostates. Indeed, from the perspective of the microphysical theory, the two accounts are equivalent: both proceed by evolving the initial probability distribution forward in time and then conditionalizing on later macrostates and showing that all events in the two sets have conditional probability P. Nevertheless, it seems intuitive that the fact that all the events in the gerrymandered set S have probability P is a mere coincidence, while the fact that all male births have probability P is not: in some intuitive sense, the probabilities of the events in S, unlike those of the events in B, have nothing to do with one another.

This intuition, I want to suggest, can be cashed out in terms of the existence of unifying theories: the events in B can be unified at the macro-level in terms of the evolutionary account, while the events in S cannot be similarly unified. The evolutionary account provides an explanation of the probabilities for the events in B that is both simple and unified. By contrast, by assumption no such simple unified explanation of the events in S is possible. Of course, the SM account unifies the phenomena as well—if successful, it provides the ultimate unified account of the world—but it only provides us with conditional probabilities of events on the complete microphysical goings-on in entire ‘slices’ of the world, rather than singling out certain higher-level structural features that may account for the phenomena we are interested in. Note also that the events in S are not spatiotemporally more unified: it is easy to pick a set of events B such that the macrostate on which we have to conditionalize is spatiotemporally much more localized than the respective macrostate in the case of male births. From the perspective of the micro-level, both sets of events simply follow from facts about the to-ing and fro-ing of the microscopic particles. By contrast, Fisher's explanation points to certain higher-level structural features of evolutionary systems that are alone relevant for a derivation of the sex ratio, abstracting from additional microscopic details irrelevant to that phenomenon.

Thus, if we assume, as seems plausible, that it contributes to the goodness of an explanation if it does not add irrelevant details and explicitly singles out the factors explanatorily relevant in a given context, then Fisher's explanation is explanatorily superior to the putative explanation posited by Albert. The evolutionary account shows that Arbuthnot's regularity holds no matter what the microphysical details as long as they allow for the existence of evolutionary systems. The point here is not merely that these details would be absurdly complex and, hence, would violate Albert's minimal condition of adequacy but that it is a virtue of an explanation if it accounts for its explanandum only by appealing to genuinely relevant features of the system at issue.

One might nevertheless have the intuition that the microphysical account is in some sense deeper since any higher-order regularities are ultimately ‘due to’ the fundamental laws. The fundamental laws, one might want to say, are what are ultimately responsible for the existence of the higher-level laws, and it is in this sense that Fisher's account does not add anything. But this reply is not open to Humeans such as Albert and Loewer. All there is, ultimately, for a Humean is the mosaic of particular matters of fact. The role of the laws, at all levels, is to provide us with particularly useful ways of summarizing features of that mosaic, and there is no genuine sense for the Humean in which higher-level laws are the product of lower-level laws.

7. Conclusion

I find Albert and Loewer's emphasis of the pragmatic dimension in a Lewisian best system account very attractive. But, as I have argued here, this pragmatism sits ill with their insistence on a physical foundationalism. If the role of laws is, indeed, to provide us with a meaty and pithy and useful summary about what happens in the world in a manner “that will serve [us] well, or reasonably well, or as well as possible, in making [our] way about in the world,” then the laws of the special sciences ought to be part of the best system no less than the past hypothesis or the probability postulate of statistical mechanics.