1.2. An account of reduction: Ontology-first or theory-first?

To clarify, neither the ontology-first approach nor the theory-first approach is meant to provide an account of reduction, which concerns what reduction is. Usually, such an account forthrightly specifies what kind of reduction it is an account of (for instance, whether its relata are objects or theories). It then identifies necessary and sufficient conditions that a successful reduction satisfies. Nagel’s (Reference Nagel1961) account, one of the most prominent accounts of reduction, takes the relata of reduction to be scientific theories. According to this account, one theory is reduced to another theory if (roughly speaking) the former can be derived from the latter. Different accounts of reduction may identify different kinds of relata of reduction. Smart (Reference Smart1959, 143), to consider another example, offers a tentative account that takes entities to be the relata of reduction.

The two approaches to reduction, in contrast, concern the priority relation between ontological and inter-theoretic reduction and can be conceived of as a way of classifying various accounts of reduction. By specifying what the relata of reduction are—whether they are objects or theories, a particular account of reduction takes reduction to be either primarily or exclusively an ontological relation (or an inter-theoretic relation). We thus can ask: For any given specific account of reduction, does it follow the ontology-first approach or the theory-first approach?

To answer this question, we need to identify whether that account takes reduction to be primarily (or exclusively) a relation between objects or primarily (or exclusively) a relation between theories. For example, Smart’s account takes reduction to be primarily about entities; hence, it is classified as following the ontology-first approach. Nagel’s account, prima facie, may be seen as following the theory-first approach, since it takes reduction primarily to be a relation between theories. ^{Footnote 7} If an account admits more than one correct way to understand reduction, we need to identify whether that account takes ontological reduction to be prior (or primary) and inter-theoretic reduction to be derivative (or secondary), or the other way around.

What does it mean that ontological reduction is prior to inter-theoretic reduction (or the other way around)? Various senses of priority are adequate to flesh out the relation between these two kinds of reduction (and accordingly, the distinction between the two approaches to reduction). For instance, x is prior to y if y is dependent on, derived from, a consequence of, grounded by, justified by, or explained by x. These different senses of priority are not mutually exclusive but could be complementary.

An account of reduction follows the ontology-first approach if, for instance, what it is to be inter-theoretic reduction relies on ontological reduction, or understanding inter-theoretic reduction requires understanding ontological reduction to begin with. For example, Oppenheim and Putnam’s account of micro-reduction, which concerns reducing one theory to another (or reducing a branch of science by another branch), requires that the ontology of a branch of science “possess a decomposition into proper parts” of the ontology of another branch (Reference Oppenheim and Putnam1958, 6). Since they take this ontological reduction to be “the essential feature of a micro-reduction,” their account follows the ontology-first approach. In contrast, an account of reduction follows the theory-first approach if ontological reduction is only a consequence of, and depends on, inter-theoretic reduction. New Wave Reduction is an account that most explicitly commits to the theory-first approach: it takes reduction to be primarily a relation between theories; more importantly, it is essential to this account that “the ontological consequences of a given reduction [that is, ontological reduction relations] are secondary to and dependent upon the nature of the theory reduction relation” (Bickle Reference Bickle1996, 65, 74).

An account of reduction can be classified as ontology-first or theory-first, but committing to an account of reduction is not the only way for one to follow the ontology-first or the theory-first approach.

1.3. The ontology-first versus theory-first approach to reduction: Further explication

The ontology-first approach takes it as given that there is a reduction relation between higher-level objects ${O_H}$ and lower-level objects ${O_L}$ , and if there is a reduction relation between the theory of ${O_H}$ and the theory of ${O_L}$ , then this inter-theoretic reduction relation is a consequence of the ontological reduction relation. In short, the ontology-first approach takes ontological reduction to be prior to inter-theoretic reduction. This direction of priority is illustrated by the arrow in the middle in Figure 1.

Figure 1. The ontology-first approach starts with the ontological reduction between O _H and O_L , illustrated by the arrow on the right. Once O_H and O_L are specified, scientific theories are then meant to describe, explain, and make predictions about O_H and O_L . The arrow in the middle depicts the core of the ontology-first approach: inter-theoretic reduction (illustrated by the arrow on the left) follows from ontological reduction as a consequence.

In contrast, the theory-first approach takes it as given that there is a reduction relation between two scientific theories ${T_H}$ and ${T_L}$ . Once ${T_H}$ and ${T_L}$ are each interpreted with an ontology, the approach states: if there exists a reduction relation between the ontology of ${T_H}$ and the ontology of ${T_L}$ , then this ontological reduction relation is a consequence of the inter-theoretic reduction relation. In short, the theory-first approach takes inter-theoretic reduction to be prior to ontological reduction. This direction of priority is illustrated by the arrow in the middle in Figure 2.

Figure 2. The theory-first approach starts with the inter-theoretic reduction between T_H and T_L , illustrated by the arrow on the left. Given a theory, we can then interpret it with an ontology. The arrow in the middle depicts the core of the theory-first approach: ontological reduction (illustrated by the arrow on the right) follows from inter-theoretic reduction as a consequence.

Stating these two approaches precisely requires specifying what ontological reduction and inter-theoretic reduction are, which requires specifying an account of reduction that takes objects as relata and another that takes theories as relata. However, neither the ontology-first approach nor the theory-first approach relies on any particular account of ontological reduction or inter-theoretic reduction. For our purposes, it suffices to get an intuitive idea of ontological reduction by thinking of, say, a mereological relation. An example of such a relation is the composition relation between chlorine gas and chlorine molecules. Ontological reduction can also be understood in terms of supervenience, identity, realization, or elimination (van Gulick Reference van Gulick2001, 4–9). For instance, the states of chlorine gas supervene on the states of chlorine molecules. The general idea of ontological reduction is, as Schaffer (Reference Schaffer, Sider, Hawthorne and Dean2008, 83) puts it, “[w]hat reduces is grounded in, based on, existent in virtue of, and nothing over and above, what it reduces to” and, metaphorically, “to create what reduces, God would only need to create what it reduces to.”

Crudely and tentatively, one can take inter-theoretic reduction to mean something like the following: ${T_H}$ can be reduced to ${T_L}$ if and only if ${T_H}$ can be fully explained by, or derived from, ${T_L}$ . Consider, as a simplified example, reduction as derivation. In this case, what the theory-first approach takes as given are ${T_H}$ , ${T_L}$ , and a derivation of ${T_H}$ from ${T_L}$ .

The main motivation behind the ontology-first approach is that ontological reduction is about what the world is like, and what the world is like is independent of, and prior to, how we theorize about the world. In other words, “[o]ntological reduction is independent of how we conceptualize entities, or theorize about them. Ontological reduction is a thesis about mind-and-theory-independent reality” (Schaffer Reference Schaffer, Sider, Hawthorne and Dean2008, 83). Meanwhile, our scientific theories and any relations between them, including inter-theoretic reduction, depend on what the world is like. Because ontological reduction means that objects ${O_H}$ are “nothing over and above” ${O_L}$ , the theories of ${O_H}$ and ${O_L}$ should also bear some kind of reduction relation—it would be deeply puzzling if they didn’t. Altogether, it suggests that inter-theoretic reduction depends on and follows from ontological reduction, not the other way around. This relation between ontological and inter-theoretic reduction is explicitly characterized by, for example, Fodor (Reference Fodor1974, 97): “the assumption that the subject-matter of psychology is part of the subject-matter of physics is taken to imply that psychological theories must reduce to physical theories.”

One (but not the only) way for ontological reduction to be prior to inter-theoretic reduction, or more generally, how we theorize about objects, is if an ontology of a certain domain is taken to be prior to its theory. This way suggests two sufficient but not necessary conditions for an ontology-first approach.

First, scientific theories are primarily about objects. That is, given the objects from a certain domain, a scientific theory is meant to provide descriptions, predictions, and explanations of these objects. Hence, a theory, especially a physical theory, should forthrightly specify or postulate its ontology. Once what the ontology is has been made clear, only then does the theory say what the ontology does, how it behaves, or what its dynamics is. ^{Footnote 8} A physical theory is thus necessarily attributed with an ontology. An uninterpreted mathematical formalism, even if it is successful at making novel predictions, does not count as a physical theory unless it is interpreted with an appropriate ontology.

Second, we can have some kind of grasp of what an ontology is like prior to its theory. We may not know exactly what the ontology consists of or what specific properties it possesses. Rather, what we can grasp are pre-theoretical or metaphysical constraints on what the ontology is like. That is to say, what the ontology of a theory is like is not only constrained by what is said by the theory but also by pre-theoretical or metaphysical considerations. In particular, ontological reduction can be one of these considerations.

For example, Poidevin (Reference Poidevin2005) argues for the principle of recombination as a constraint on what chemical elements are physically possible, that is, on what the ontology of a chemical theory could be. Elements in the periodic table (such as potassium [with atomic number 19] and calcium [20]), which form a discrete series, are physically possible. In contrast, anything with atomic number between 19 and 20 (say, $19.356$ ), which forms a continuous series, is merely logically possible. According to the principle of recombination, the physical possibility of being an element is constituted by a recombination of actual instances of electron distributions (Poidevin Reference Poidevin2005, 129–130). Because there isn’t any intermediate position between, say, having two electrons in one orbit around the nucleus and having only one, anything with atomic number between 19 and 20 cannot be the result of a recombination of actual electron distributions and thus is ruled out as a physical possibility by the principle of recombination. This principle identifies a reduction relation between the higher-level objects, elements, and the lower-level objects, electrons, and it is this ontological reduction that determines what elements are physically possible and what are merely logically possible. Particularly, Poidevin (Reference Poidevin2005, 131) emphasizes that the property of being a chemical element is theory neutral. Hence, the reduction relation involved is indeed ontological rather than inter-theoretic.

Consider another example in which ontological reduction acts as a constraint on what the ontology of a theory could be and, consequently, on the theory itself. The primitive-ontology version of Bohmian mechanics has been defended by arguing that a fundamental physical theory (such as quantum mechanics) without a primitive ontology should be avoided. Primitive ontology was introduced as “the basic kinds of entities that are to be the building blocks of everything else” (Dürr et al. Reference Dürr, Goldstein and Zanghì1992, 850). Its role, as Allori (Reference Allori2015, 110) puts it, is to “ground a scheme of explanation” in which the behavior of the primitive ontology determines the properties of macroscopic physical objects. That means: introducing the primitive ontology secures an ontological reduction relation between the fundamental ontology and familiar higher-level objects (like tables, chairs, and measurement pointers). Given this relation between the fundamental ontology and familiar macroscopic objects and the latter being local and three-dimensional, the former needs to have these properties as well. It is in this way that ontological reduction imposes a constraint on what the ontology of the fundamental theory could be like; consequently, whatever the fundamental quantum theory turns out to be like, its ontology needs to contain the primitive ontology, or else it would not be the right theory. ^{Footnote 9} (This argument would not work under the theory-first approach, for neither the higher-level theory that describes macroscopic objects like tables and pointers nor its inter-theoretic reduction relation [with quantum mechanics] even comes up. Thus, inter-theoretic reduction is not primary in this argument.) ^{Footnote 10}

In contrast to the ontology-first approach, the theory-first approach does not require that each theory be attributed with an ontology. In other words, it is not necessary for a theory to forthrightly specify or postulate an appropriate ontology in order to be physical or carry any physical significance (instead of merely being a mathematical tool). How does a theory establish its status as a physical theory, then? Via its usefulness or efficiency at describing patterns, making predictions, and providing explanations and practical applications. In physics, this is usually achieved by offering a new robust and autonomous dynamics (e.g., Maxwell’s equations offered such a dynamics for electromagnetic phenomena). Accordingly, a physical theory can be a mathematical formalism that is only partially interpreted, as long as it can be tested empirically, make novel predictions, and provide explanations. ^{Footnote 11}

Nevertheless, the fact that a theory is not necessarily attributed with an ontology does not imply that we cannot subsequently interpret the theory with an ontology. It’s just that such an ontology does not play a primitive role in the theory. Anything that can be known about the ontology is given by the theory and how it’s used. Whether or not a theory is physical, or what its ontology is like, is not constrained by any metaphysical preconceptions about the ontology or the ontological reduction relation. Rather, the ontology is only taken to be secondary or derivative to its theory, especially to its dynamics. (This is easier to see with physical theories like quantum theories, less so with, say, biological theories.) A supplement on how we can attribute an ontology to a theory might be needed—for instance, something along the lines of functionalism or Dennett’s (Reference Dennett1991) pattern theory. The theory-first approach may demand a metaphysical picture that is radically different from what we are accustomed to: one no longer centered around objects with intrinsic properties moving in spacetime.

The ontology-first and theory-first approaches are not meant to exhaust all possible views on the relation between ontological and inter-theoretic reduction. They are better thought of as representing families of views by two ends of a spectrum. Another view on this spectrum: ontological and inter-theoretic reduction are interdependent—there is no ground to prioritize one over the other, and they are on par. This view assumes that ontological and inter-theoretic reduction always come together, which is contentious. For instance, inter-theoretic reduction may not follow from ontological reduction because it is computationally intractable to derive ${T_H}$ from ${T_L}$ , even though the ontology of ${T_H}$ is reduced to the ontology of ${T_L}$ . ^{Footnote 12}

The ontology-first and theory-first approaches are not necessarily subject to such challenges. Recall: the ontology-first approach has a conditional, which leaves open the possibility that there isn’t any inter-theoretic reduction following from ontological reduction. Thus, the view that ontological reduction is the only correct way to understand reduction or the only kind of reduction that holds in certain cases still counts as ontology-first. Nonreductive physicalism is one such example: it is reductionist only about ontological reduction but not inter-theoretic reduction. Similarly, the view that there is only inter-theoretic reduction and no ontological reduction still counts as theory-first.

Moreover, drawing the distinction between the ontology-first and the theory-first approach is not necessarily incompatible with the view that ontological and inter-theoretic reduction are interdependent. Because this view might not specify exactly how, or in what sense, they are interdependent on each other, the ontology-first and theory-first approaches together can be seen as a way to further explicate this view.

The two approaches are competing if they are both taken to be metaphysical (see section 1.3, especially the end). But that’s not the only way to understand these two approaches. Instead of metaphysical priority, one can understand the two approaches, for instance, in terms of explanatory priority. The ontology-first approach then states: ontological reduction explains inter-theoretic reduction; that is, the fact that objects ${O_H}$ reduce to ${O_L}$ (say, chlorine molecules are composed of chlorine atoms, etc.) explains the fact that the theory of ${O_H}$ reduces to the theory of ${O_L}$ (say, a chemical theory reduces to atomic physics). Similarly, the theory-first approach states that inter-theoretic reduction explains ontological reduction. Alternatively, the ontology-first approach can be understood in terms of metaphysical priority and the theory-first approach in terms of epistemic priority. In either case, the ontology-first and the theory-first approach do not oppose each other but can be seen as complementary: each spells out a particular aspect of the relation between ontological and inter-theoretic reduction.

2.1. The Boltzmannian framework

BSM uses the concept of macrostate to describe the system at the statistical-mechanical level. A macrostate is characterized by macroscopic parameters, such as local pressure and local density of regions that are large enough to contain many molecules but small compared to the size of the box. It is related to the micro-description of the system, namely, a microstate, as follows: the system in a particular macrostate could be in one of many different microstates, whereas the system in a particular microstate is in a unique macrostate. This is because what the macrostate of a system is is fully determined by the state of its microscopic constituents, but not vice versa. If we slightly change the location or velocity of just one particle in the system, it would no longer be in the same microstate, whereas its macrostate would not be affected. Mathematically, the phase space can be partitioned into regions such that the microstates in each region correspond to the same macrostate—a macrostate is identified with one of those regions. Regardless of whether it is the macrostate or the microstate that is under consideration, what is taken to be the object of study for BSM is clearly, its advocates emphasize, an individual system (e.g., Goldstein Reference Goldstein2019).

Given the concept of macrostate, entropy and equilibrium are defined: the Boltzmann entropy of a system with macrostate M is

(1) $${S_B} \equiv {k_B}{\rm{ln}}{\mu _M},$$

where ${k_B}$ is the Boltzmann constant, and ${\mu _M}$ is the phase-space volume of M. For any given energy, there will be some macrostate that has the maximal Boltzmann entropy among all the macrostates with that energy. This state is designated as the equilibrium state in BSM. As it turns out, the phase-space volume of the equilibrium state of a system at a given energy is overwhelmingly larger than any other macrostates with the same energy. This feature of equilibrium is key to the Boltzmannian characterization of how systems approach equilibrium (e.g., how gas that is initially confined in a corner of a box will uniformly spread out to the whole box later) and their explanation of the prima facie inconsistency between the time-irreversibility of thermodynamics and the time-reversibility of its underlying micro-dynamics (i.e., classical mechanics). Many Boltzmannian advocates take the primary task of SM to be to provide a microphysical description and a justification of thermodynamics and, in particular, to explain the time-irreversibility of thermodynamics. ^{Footnote 16}

2.2. The criticized Gibbsian framework

Compared to BSM, recent philosophy of physics has paid less attention to developing GSM into a systematic framework, despite the fact that it is the standard tool in practical applications of SM (Wallace Reference Wallace and Allori2020) and widely used among working physicists (Frigg and Werndl Reference Frigg and Werndl2020). Consequently, it is not clear what exactly GSM is (Frigg and Werndl Reference Frigg and Werndl2020). For this reason and to demonstrate the disputes between Boltzmannians and Gibbsians more sharply, I first present the version of GSM that has been criticized by Boltzmannians. In sections 4 and 5, I discuss possible conceptual modifications that can be made to GSM to respond to those criticisms.

In contrast to the object of study being individual systems in BSM, the core object of study for GSM is commonly taken to be ensembles (e.g., Frigg Reference Frigg and Rickles2008; Pathria and Beale Reference Pathria and Beale2011, xxiii) or probability distributions (Wallace Reference Wallace and Allori2020). An ensemble is usually understood as an infinite collection of systems of the same kind, which only differ in their configuration and velocities at a time point. ^{Footnote 17} The state of an ensemble at time t is represented by a probability density function $\rho \left( {q,p;t} \right)$ over the phase space. The time evolution of $\rho $ is given by Liouville’s equation:

(2) $${{\partial \rho } \over {\partial t}} = - \left\{ {\rho, H} \right\},$$

where H is the Hamiltonian, and { , } is the Poisson bracket.

The Gibbs fine-grained entropy is defined as

(3) $${S_G}\left( \rho \right) \equiv - {k_B}\mathop \int \nolimits_{\rm{\Gamma }} \rho {\rm{ln}}\left( \rho \right)d{\rm{\Gamma }},$$

where ${\rm{\Gamma }}$ is the phase space, and $d{\rm{\Gamma }}$ is the standard Lebesgue measure. It is invariant over time, as a consequence of Liouville’s equation. Since thermodynamic entropy increases when the system evolves from a nonequilibrium state toward an equilibrium state, the Gibbs fine-grained entropy is inadequate to be the microphysical counterpart of thermodynamic entropy (this is almost universally recognized).

The Gibbs coarse-grained entropy, in contrast, is not invariant in time. Abstractly, coarse-graining is a procedure of averaging over details of the system that are irrelevant to its description at a higher level. We can represent such a procedure by a projection operator J, which is a map on the space of probability distributions such that ${J^2} = J$ (i.e., the result of coarse-graining twice is the same as coarse-graining once). J acts on the original probability density $\rho $ , yielding the coarse-grained density: ^{Footnote 18}

(4) $$\bar \rho = J\rho .$$

One particularly important way, at least conceptually, to think of coarse-graining is as partitioning the phase space into small cells. We define $\bar \rho $ such that it is uniform over each cell and assigns the same probability to a cell as $\rho $ . $\bar \rho $ is coarse-grained in the sense that the details of $\rho $ within each cell are disregarded. The Gibbs coarse-grained entropy ${\bar S_G}$ has the same form as equation (3) but substitutes $\rho $ with $\bar \rho $ :

(5) $${\bar S_G}\left( \rho \right) \equiv {S_G}\left( {\bar \rho } \right) = - {k_B}\mathop \int \nolimits_{\rm{\Gamma }} \bar \rho {\rm{ln}}\left( {\bar \rho } \right)d{\rm{\Gamma }}.$$

In GSM, the microphysical counterpart of thermodynamic entropy is ${\bar S_G}$ .

Accordingly, equilibrium is defined as a state for which $\bar \rho $ is invariant in time. ^{Footnote 19} GSM characterizes how systems approach equilibrium in terms of the increase of ${\bar S_G}$ . To describe and make quantitative predictions about thermodynamic systems at equilibrium, GSM associates each macroscopic parameter with a phase function $f:{\rm{\Gamma }} \to \mathbb{R}$ . The phase average $\langle f\rangle $ of f,

(6) $$\langle f\rangle = \mathop \int \nolimits_{\rm{\Gamma }} f\left( {q,p} \right)\rho \left( {q,p;t} \right)d{\rm{\Gamma }},$$

gives the values of these macroscopic parameters. ^{Footnote 20} Precisely because the macroscopic parameters are insensitive to coarse-graining, we in fact attain the same value for $\langle f\rangle $ whether we use $\rho $ or $\bar \rho $ .

To summarize, BSM and GSM offer different descriptions of the same physical system at the statistical-mechanical level; in particular, they differ in whether such descriptions should involve probability. For GSM, probability or ensemble is indispensable to describe the system and define key notions like entropy and equilibrium. For BSM, it’s not.

4.1. Problems of ensemble and probability

First of all, Boltzmannians criticize GSM for taking “ensembles of infinitely many systems” as its core object of study, in particular, for using ensembles to represent actual individual systems (e.g., Callender Reference Callender1999; Goldstein et al. Reference Goldstein, Lebowitz, Tumulka, Zanghi and Allori2020). In fact, some Boltzmannians describe GSM as “ensemblist,” in contrast to their own framework being “individualist” (Goldstein Reference Goldstein2019). The criticism goes as follows. SM should be about actual individual physical systems. An ensemble, which is a collection of infinitely many systems, is neither actual nor individual. More specifically, equilibrium and entropy are supposed to be properties of an individual system. But if they are defined in terms of probability distributions over ensembles, an individual system can no longer be said to be in equilibrium or have certain entropy. Moreover, we cannot infer the behavior of an individual system from the behavior of an ensemble (Frigg Reference Frigg and Rickles2008, 174). Thus, actual individual systems cannot be represented by ensembles.

Gibbsians have an immediate response to this criticism: an ensemble is only a fictitious set of all possible microstates of the system. It is introduced merely for convenience or as a heuristic. GSM can be presented without mentioning ‘ensemble’: statistical-mechanical systems are represented directly by probability distributions over phase space (Wallace Reference Wallace and Allori2020).

But this is criticized by Boltzmannians as well. They argue that actual, individual statistical-mechanical systems cannot be represented by probability distributions. For example:

For Callender, any function that is not “a function of the dynamical variables of real individual systems” is inadequate to be a part of the explanation for the thermal behaviors of actual individual systems, and probability is one such function.

Why can’t probability distributions represent “individual real systems”? A probability distribution describes how likely it is for a system to be at one of the many possible microstates. But at any given time, there is only one definite microstate at which the actual system can be. Goldstein (Reference Goldstein2019, 443) thus asks:

By pointing out that a nontrivial probability distribution over many possible microstates does not refer to anything actual, Goldstein is effectively arguing that it is problematic to use probability to represent an actual system.

(One may argue that an actual individual system can be represented by probability, if probability is interpreted as subjective in the sense that it measures how much we know about the system. In that case, Gibbs entropy, which is defined in terms of probability, would be subjective as well. However, the reason why thermodynamic entropy of an isolated system does not decrease cannot be subjective, because that fact holds regardless of how much we know about the system. Thus, Gibbs entropy as a subjective notion is not adequate to capture this objective fact about the thermodynamic entropy (e.g., Albert Reference Albert2000). Accordingly, interpreting probability as subjective is not a viable solution to the problem that probability can’t represent actual individual systems.)

The key to the Boltzmannian criticisms of the Gibbsian use of ensemble and probability lies in their claim that SM should be about actual individual systems. If a framework of SM is not about individual systems for whatever reason, it is plainly a drawback of that framework. Here’s Callender (Reference Callender1999, 357; emphasis added):

For Albert (Reference Albert2000, 70), it is just “sheer madness” that entropy, equilibrium, and the laws of thermodynamics are associated not with individual systems but with ensembles or probability.

Why is SM supposed to be about actual individual systems? We can answer this if we take scientific theories to be primarily about objects (section 1.3). Then it makes sense to think that appealing to objects and their actual states is the only admissible way to characterize a given physical system, and appealing to “some abstract entity,” such as probability (Maudlin Reference Maudlin1995, 147), is not. Accordingly, the core object of study of a physical theory cannot be a fictitious collection of many possible states. Because taking theories to be primarily about objects is a sufficient condition for the ontology-first approach (section 1.3), this approach needs to be assumed as a consequence. In sum, to justify their claim that SM is supposed to be about actual individual systems in this way, Boltzmannians assume the ontology-first approach.

Maudlin (Reference Maudlin1995, 147) gives a slightly different explanation, which arguably appeals to ontological reduction:

That is, because thermodynamics is about individual systems, SM is supposed to be about individual systems as well; making SM be about probability distributions, even though it preserves thermodynamics, has the cost of making SM no longer be about individual systems; this cost is so devastating that it is tantamount to defeat. But why is SM supposed to be about individual systems? Simply because thermodynamics is about individual systems? If the ontology-first approach is assumed, then ontological reduction acts as a constraint on what the ontology of a theory could be (section 1.3). Given that there is an ontological relation of reduction between thermodynamic and statistical-mechanical systems and that thermodynamic systems are individual systems, statistical-mechanical systems should be individual systems as well.

4.2. Problems of coarse-graining

Moreover, GSM is criticized by Boltzmannians for its use of coarse-graining without proper justification. For example, Callender (Reference Callender1999, 360) argues that the sole purpose of coarse-graining is to get a notion of Gibbs entropy that monotonically increases in time (i.e., the Gibbs coarse-grained entropy; see eq. 5). That is to say, the coarse-graining projection operator J is chosen opportunistically, and thus the Gibbs coarse-grained entropy is introduced without any justification apart from it matching the increase of thermodynamic entropy in time.

BSM, however, also employs coarse-graining without providing a justification. Although Boltzmannians do not always use the word ‘coarse-graining’, the idea of partitioning the phase space into finite regions is employed. For example, the definition of Boltzmann entropy appeals to coarse-graining:

Hence, Boltzmannians apply a double standard in criticizing the Gibbsian use of coarse-graining. ^{Footnote 22} What justifies this double standard? A plausible answer is that BSM tacitly assumes the ontology-first approach. This assumption licenses BSM to take ontological reduction as given, which justifies its choice of coarse-graining, more specifically, its choice of partitioning the phase space into “macroscopically small but microscopically large cells” (Goldstein Reference Goldstein, Bricmont, Ghirardi, Dürr, Petruccione, Galavotti and Zanghi2001, 42). This particular choice of the size of the cells can be justified, as Frigg (Reference Frigg and Rickles2008, 135) points out, if there exists an objective separation of the relevant macroscopic and microscopic scales. A reduction relation between objects at a microscopic and a macroscopic scale provides just such a natural and objective micro–macro separation. It’s simply “carving nature at its joints” that there are such objects at such and such scales. The two scales involved in coarse-graining are thus not chosen arbitrarily, but are picked out because they are ontologically significant (i.e., they are associated with the relevant ontologies in the relevant reduction relation). Put another way, the choice of coarse-graining is justified because it gives rise to the right higher-level objects, which are marked off by what is macroscopically indistinguishable and what can be meaningfully measured. Without the assumption of ontological reduction, it would be puzzling why being macroscopically indistinguishable matters to any individual microstate.

GSM cannot appeal to the same kind of justification for coarse-grained probability density. It is unclear in what sense probability distributions are ontological and stand in an ontological reduction relation with individual microstates. Indeed, this is exactly what GSM is criticized for—under the assumption of the ontology-first approach (section 4.1). Therefore, Boltzmannians would argue, there is no relevant ontological reduction relation for GSM to employ to justify their choice of coarse-graining. ^{Footnote 23}