1. Introduction

The emergence of one or more properties of a system or state of affairs is commonly invoked in the physical sciences but rarely defined. Butterfield (Reference Butterfield2011a, Reference Butterfield2011b, Reference Butterfield2014) goes some way to illustrate this usage (and its relation with concepts of reduction and supervenience) in various examples from physics but, following tradition, does not provide a completely explicit or precise definition of emergence. The closest he comes to doing so is the statement that emergent properties are novel properties, where “‘novel’ means something like: ‘not definable from the comparison class’, and maybe ‘showing features (maybe striking ones) absent from the comparison class’” (Butterfield Reference Butterfield2011a, 921).Footnote ¹ But as Teller (Reference Teller, Beckermann, Flohr and Kim1992) has protested, mere invocations of some sort of novelty or other are too vague and weak to be of much help. Thus, the adjudication of Butterfield’s and others’ examples has continued to rely on informal intuitions and ostensive comparisons with purportedly intersubjectively agreed-on cases—hard to come by in the emergence literature.

Nevertheless, I believe Butterfield has set us in the right direction toward a precise and fruitful account of emergence for properties as described within scientific theories. My goal in the sequel is to take the next few steps. First, I follow Butterfield in focusing on emergence as it applies to scientific models, for example, of theories, rather than of systems or objects directly. I then propose understanding emergence as comparative dissimilarity: a property, as represented in a model, is emergent with respect to a comparison class just when is it relevantly dissimilar to the properties of the members of that class. This dissimilarity can come in different sorts, so in section 3 I distinguish four different types of emergence—weak, strong, nonreductive, and radical—partially ordered in strength. I also suggest several features that make examples of emergent properties more interesting than others.

The chief formal innovation I employ, introduced beforehand in section 2, is similarity structure, a structured set of similarity relations among those models under analysis—and perhaps their properties. Similarity structure is a generalization of topological structure.Footnote ² After explaining the interpretation and significance of these structures and using them to define in section 3 the aforementioned types of emergence, I apply the definitions in section 4 to a couple of examples from space-time and thermostatistical physics. Although these applications are both from physics, my definitions could extend to any sciences that use (mathematical) models to represent phenomena.

In section 5 I discuss some possible cases for such extension as well as three notable features of this account in general. First, some of its types are compatible with reduction, while others are not. This helps understand conflicting intuitions regarding this compatibility. Second, my account of emergence is relational and contextual. It is relational in that emergence is defined only relative to a comparison class, in agreement with Butterfield; it is contextual in that this comparison class, as well as the relevant notions of similarity, may vary according to the context of investigation. They are not determined from the formal structure of the models themselves. Third, the degree of precision with which one can identify which properties are emergent is limited in principle only by the precision with which one can specify the models of interest and the relevant ways those models are similar.

Before continuing in more detail, a few qualifications are in order in light of the apt remark of O’Connor and Wong (Reference O’Connor, Wong and Zalta2015) that “emergence is a notorious philosophical term of art,” one with a variety of conflicting usages and definitions. I have already specified that the notion of emergence with which I am concerned here pertains to properties as described in scientific or formal models (rather than, say, to entities or substances). Following the distinctions urged by Humphreys (Reference Humphreys and Humphreys2016), this sort of emergence is certainly epistemological, as I describe in section 3, in the sense that it is associated with a failure of deduction, of a sort, of the emergent property; sometimes it will count as conceptual emergence too when the emergent properties involve novel concepts. Whether a particular example is also a case of ontological emergence will depend on whether one’s (contextually appropriate) attitude toward the models in question (and in particular the putatively emergent property thereof) is more realist or antirealist, but that question can be set aside for current purposes, as its answer does not substantively affect the formal features of the analysis. Further, the notion of emergence here is synchronic: it will describe the relationships of models and novel properties thereof, not how properties of systems or states of affairs arise in time.Footnote ³

2. Similarity Structure

In order to define emergence as relative dissimilarity, first one must define similarity structure, whose constitutive unit is the qualitative, binary similarity relation.

The relation $y\sim x$ is interpreted as “y is similar to x.” Quasi-reflexivity requires that if any element is similar to another, then both the former and the latter are similar to themselves. Previous authors (Carnap Reference Carnap and George1967; Schreider Reference Schreider and Greendlinger1975; Mormann Reference Mormann1996; Konikowska Reference Konikowska1997) have commonly demanded a similarity relation to be not just quasi-reflexive but reflexive ($x\sim x$ for every $x\in X$ ) and symmetric (if $y\sim x$ , then $x\sim y$ ). However, while many useful similarity relations do satisfy these conditions, there are also some that do not. For example, one can consider the relation of (weak) observational indistinguishability among relativistic space-times (Malament Reference Malament, Earman, Glymour and Statchel1977; Manchak Reference Manchak2009) as a type of nonsymmetric similarity relation: one space-time (M, g) is similar in this sense to another (M′, g′) just when the past of each event in (M, g) is identical with the past of an event in (M′, g′). Observers in (M, g) cannot tell whether they are in (M, g) or (M′, g′), but some observers in the latter can tell that they are not in (M, g). Regarding reflexivity, a mundane example suffices: among all people, one can consider two to be similar in party affiliation just when they are members of the same political party. People registered with a political party are similar in party affiliation to themselves, but people who are unregistered are not, because they have no such affiliation.

In many contexts there are a variety of relevant ways in which the elements of a collection can be similar to one another. Considering multiple similarity relations on X allows comparisons of similarity in multiple respects.

This is fruitful precisely when there are multiple respects that could be relevant. For example, if there is a quantitative property relevant for comparison, then one could have a whole class of similarity relations, one for each positive ε, relating the models whose quantitative property values are within ε. (I discuss properties and their assignments more at the end of this section.)

Of particular interest will be the elements in a similarity space that are arbitrarily similar to others, that is, similar in each of the respects that a similarity structure represents. To define these, let ℘X be the power set of X, that is, the set of all subsets of X.

The value of the closeness operator acting on a set $A\subseteq X$ is the collection of all elements arbitrarily similar to some element in A, that is, those similar to an element of A in all respects determined to be relevant by 𝒮. The closeness operator of a similarity space is a generalization of the closure operator for a topological space: when restricted to sets of elements in the joint domain cl_𝒮(X) of the similarity structure, it satisfies the Kuratowski axioms for closure operators (Willard Reference Willard1970, sec. 3) except for idempotence, which states that for any $A\subseteq X$ , ${\mathrm{cl}}_{𝒮}\left({\mathrm{cl}}_{𝒮}\right(A\left)\right)={\mathrm{cl}}_{𝒮}\left(A\right)$ .

To understand the significance of these observations, it will be helpful to introduce some further definitions for both topological and similarity structures. One way to characterize a topology on a set X is to provide a neighborhood base for each of its elements (Willard Reference Willard1970, theorem 4.5).

A neighborhood base for a point is thus a set of subsets containing that point that are (up to taking supersets) closed under intersection and that are (allowing one to throw away some points therefrom) also basic neighborhoods of the other points they contain. Basic neighborhoods are also directly related to Kuratowski closure operators according to the following proposition (Willard Reference Willard1970, theorem 4.7c).

To see the parallels with similarity structures, consider one further definition.

The domain of ∼ at $x\in X$ is just the set of all the elements in X that are similar to x. Elementary logical manipulation of definitions 3 and 4 yields the following proposition.

The set of domains at $x\in X$ plays the same role in a similarity structure as a neighborhood base at x does for a topological structure. Indeed, every basic neighborhood B of a point x determines a certain minimal similarity relation ∼_B at that point: $y{~}_{B}x$ if and only if $B\in {ℬ}_x$ and $y\in B$ . Conversely, any similarity structure on X whose sets of nonempty domains at each x satisfy conditions 2 and 3 of proposition 1—condition 1 is always satisfied by definition—determines a topology for X. Hence:

When a similarity structure does not determine a topology, condition 2 or 3 must be violated. Because condition 2 can always be satisfied by closing a similarity structure under intersection, at issue is really condition 3. This corresponds with the general failure of idempotence for the closeness operator of a similarity space, which allows one repeatedly to enlarge a set by considering elements similar to it (see also n. 5).

For an example, consider the integers equipped with a single symmetric similarity relation that holds between any element and its successor (hence also its predecessor). The elements close to {0} are {−1, 0, 1}, the elements close to those are {−2, −1, 0, 1, 2}, and so on. This is not possible with a standard topology because topological basic neighborhoods must satisfy condition 3, leading to the discrete topology on the integers. Another example would be the real line equipped with similarity relations that hold between elements as long as their absolute difference is below a certain value ε. This might represent a kind of indistinguishability among sufficiently similar realizations of a quantitative property. But this also is not possible to represent with a standard topology for the same reasons as the previous example.

There are no restrictions on the kinds of entities represented by the elements of X. They could be objects, states of affairs, or even values of a property. So, a similarity structure can represent how a variety of objects, states of affairs, or values of a property are similar, including but not limited to when that structure is topological. The aforementioned elements, or models of these entities, are typically themselves highly structured, with subobjects, relations among them, and various other properties and pieces of architecture. For instance, a space-time model has properties (e.g., what temporal structure it supports; see sec. 4.1) but also subobjects such as smooth curves. These in turn have properties (e.g., being timelike or not) and subobjects (i.e., space-time points). In what follows in sections 3 and 4, I will be interested in the properties of the models themselves but also the properties of their subobjects, the values all these properties can take on, and so on. A bit more formally:

A property assignment represents a property intensionally, as a map from models to values. For properties of subobjects, instead of tediously and explicitly composing a property assignment with a map picking out a subobject of a model (or some more complex such composition), I often talk about and represent them simply as property assignments to the models themselves. One need just take care to ensure that the particular subobject is well defined.

Philosophers will already be familiar with (classical) propositional properties, those that either obtain or not for a given object or state of affairs. For example, consider a collection of paint color samples, whose various pigment properties we represent in a corresponding collection of models. There is a propositional property, “is red,” that applies to them and has the Boolean domain {⊤, ⊥} as its values.

Property assignments can also take on quantities as their values. For example, consider free Hamiltonian systems of a single particle. Although Hamiltonian systems are complex, having many subobjects and structures (e.g., symplectic structure), we can pick out the parts (e.g., the appropriate term in the Hamiltonian) from each one that describe the mass of the particle each represents. Thus, there is a property assignment representing “has mass m” to such models, whose space of values is the positive real interval $m\in (0,\infty )$ . There is also a property assignment representing “has velocity v (with respect to a certain reference frame F) at time t,” whose space of values is a three-dimensional real vector space, if the reference frame and time are taken as metavariables to be filled in with some constant and a more complex structure if not. It is important to accommodate quantitative and complex properties, as they are essential for evaluating the predictive success of many scientific models.

Property assignments to models can still pick out properties of subobjects and structures when there are many of them. Consider again a Hamiltonian system of countably many classical, distinguishable massive particles, and suppose (without loss of generality) that the particles have been labeled by a natural number. Then there are property assignments representing “particle 1 has mass m ₁,” “particle 1 has velocity v ₁ (with respect to a certain reference frame F ₁) at time t ₁,” “particle 2 has mass m₂,” “particle 2 has velocity v ₂ (with respect to a certain reference frame F ₂) at time t ₂,” and so on.

There are no restrictions on what kinds of systems and properties thereof the models and assignment maps respectively represent, including those represented by subobjects and properties thereof in the models. Any property assignment definable for the elements of X (and their subobjects), broadly understood, is a candidate for consideration.Footnote ⁴ This includes properties most naturally defined by functional composition. For instance, the disjunction of a collection of properties with Boolean values yields another property with Boolean values. And the kinetic energy of a simple particle in Newtonian physics, $K=(1/2)m|v{|}^2$ , is an algebraic function of its mass m and velocity v. Properties whose assignments are merely partial must have conditional definitions, of course, and expansions of such properties are also candidates for consideration. Whether such a given property is interesting is another matter. I discuss its relevance for emergence further in section 3.

3. Emergent Properties

Using the framework of similarity structures introduced in section 2, one can define the four types of emergence for properties mentioned in section 1. Each of them explicates the novelty of a property of a model, relative to some comparison class, in a slightly different way. I first state the formal definitions of these concepts and then discuss their interpretation, informal conditions of application, relations, and what makes particular examples of them interesting.

Before doing so, it will be helpful to note a point of terminology. The convention of common parlance is to talk of the potential emergence of properties, without reference to an assignment function representing which models have which properties. It will facilitate the technical aspects of my presentation to refer instead at times to emergent property values, as I have already occasionally done, rather than the properties simpliciter. But this should be understood as just a formal way of representing the common parlance: an assignment of a value to a model is merely a representation of the model’s properties (which is in turn perhaps a representation of some phenomenal or ontological properties). So, to say that some collection of property values is emergent is just to say that the particular properties those value assignments represent are emergent.

Suppose, then, that a collection of scientific models X has been equipped with a similarity structure 𝒮, as has the space of property values V with a similarity structure 𝒱 for some property assignment $\nu :X\nrightarrow V$ .

For each of these, it is said to obtain completely when both sides of its noninclusion are nonintersecting. Otherwise, it may said to obtain partially.

Note that while standard discussions of emergent properties restrict attention to a property of a single model, the above definitions as stated apply to the property values for sets of models. But, when they are so restricted, partial and complete emergence become equivalent.

The weakest of the above, weak emergence, requires only the mere nonidentity of the values of the property in question, ν[B], with any of those for the models in the comparison class, ν[A]. As Teller (Reference Teller, Beckermann, Flohr and Kim1992, 140) complains, however, an object’s property of having a mass of 10.74218 grams would be merely novel in this way with respect to the comparison class of objects without that mass, which seems uninteresting. Thus, one might demand not just mere nonidentity but a sort of unexpectedness or comparative inexplicability. This can be formalized along at least two dimensions, captured by strong and nonreductive emergence. The strong emergence of a property of a model requires that the property must also be not sufficiently similar to the properties of the models of the comparison class—it is unexpected because it is not even similar (in the relevant ways) to the properties available for consideration from the comparison class. This requires similarity structure on the space of values that a property can take on. The nonreductive emergence of a property of a model requires that the property must also be nonidentical with the corresponding properties of the models arbitrarily similar to those in the comparison class. This requires similarity structure on the joint collection of models. Finally, one can combine these criteria in radical emergence: the emergent property is not even sufficiently similar to the properties of the models arbitrarily similar to those in the comparison class. Accordingly, this requires similarity structure on both the space of values that a property can take on and the joint collection of models themselves.

These interpretations are of a set of formal definitions, but there is an important informal condition of application for them. Generally, models A whose putative emergent properties one is considering, as well as its comparison class, B, should have overlapping representational domains: that a property of avalanches, for example, is inexplicable from models of radioactive decay is hardly surprising or interesting and not even deserving of the name emergence.Footnote ⁶ Moreover, the only types of properties that are candidates for being emergent are those that the models aim to represent. This is not a formal requirement, as the structure of a model itself does not determine what the model represents. For instance, instrumentalist interpretations of the models of classical and quantum mechanics will permit only emergent empirical properties, while realist interpretation could permit more.

This suggests another condition of application: I require that the comparison class B should be at least as fundamental as the class A, at least with respect to properties that they aim to represent and that are considered as candidates for being emergent. This requirement distinguishes even weak emergence from mere nonidentity and captures the idea that emergent properties must “arise” from a more fundamental base. I do not here take a stand on any particular account of relative fundamentality, in part because the sort needed will depend on the aims of the models’ representations. Some realist-oriented possibilities include supervenience or composition between different scales or levels of description, reality, or explanation (List Reference List2019).Footnote ⁷ An anti-realist-oriented possibility could be empirical adequacy (van Fraassen Reference van Fraassen1980): one set of models is at least as fundamental as another when it is at least as empirically adequate for at least the phenomena under consideration. Although I have left this condition of application informal, these possible explications could likely be developed in formal detail, but this is not necessary for current purposes.

For models used in science, this last notion of relative fundamentality and the particularity of the properties represented seems to be especially important. To the extent that the candidate emergent property values are realized within a strict subset of the models of a theory, only those models will be empirically adequate representations of those property values. Even if some sort of empirical adequacy is an informal condition of application, one can still provide arguments about hypothetical emergence when such adequacy is unknown or counterfactual emergence when such adequacy is known to be false. This is important for capturing emergence claims in frontier science, such as quantum gravity, where a correct empirical description of the phenomena has yet to be established (Crowther Reference Crowther2016).

In any case, each of these types of emergence can be understood as epistemological, in the sense that, from the perspective of the comparison class, the emergent property in virtue of its novelty cannot be deduced. Whether any type is ontological or metaphysical will again depend on whether one takes a realist attitude toward the models and the properties they represent.

The different types of emergence bear various logical relations to one another. Radical emergence entails the other three types of emergence, while weak emergence is entailed by the other types. In general, neither strong emergence nor nonreductive emergence entails the other. And in each of these cases, emergence can obtain either because the property assignments are defined for the models of A but are different for the models of B or because they are not defined for the models of A. In the latter case, I suggest one could deem it a case of conceptual emergence, since the emergent property itself is novel, not just its values. Conceptually emergent properties are more plausibly interesting cases of emergence (Teller Reference Teller, Beckermann, Flohr and Kim1992, 140–41) since, for strongly emergent properties, at least, they are so independently of the similarity structure on the property’s space of values.

I cannot provide a list of necessary and jointly sufficient criteria for an emergent property to be interesting—indeed, I am skeptical that such a list could be feasibly produced. But all else being equal, the emergent property may be more interesting to the extent to which is it more robust, which “means something like: ‘the same for various choices of, or assumptions about, the comparison class’” (Butterfield Reference Butterfield2011a, 921). In other words, an emergent property of a model is interesting to the extent to which it is so for a relatively wide comparison class of other models. Many artificially defined properties, such as some “mere Cambridge properties” (Francescotti Reference Francescotti1999), will be uninteresting on this criterion because they do not hold when one varies assumptions about the comparison class that one has antecedent reason to believe are irrelevant.

Similar reasoning extends to conditionally defined properties whose domain of definition—the models on which the property assignment is defined—is needlessly restricted. An example would be the kinetic energy of a simple particle in Newtonian physics, restricted to particles of mass within a certain finite range. Expansions of conditionally defined properties—that is, of their assignment’s domain of definition—will also be uninteresting to the extent that they are deemed unprincipled. For instance, the stress and strain of a body modeled as a material continuum are the internal distribution of forces within the body and the deformation of the body from a reference (usually equilibrium) configuration. It would be unprincipled then to extend this definition to other materials (e.g., liquids and gases) by assigning them stresses and strains equal, for example, to their energy or temperature.Footnote ⁸ Here the requirement of overlapping representational domains provides additional constraint because the terms defined can have a meaning given in part externally, for example, through the means of empirical testing. Through these means one can often identify chimerical properties and judge whether their emergence is interesting.

One further positive criterion that makes an example of emergence interesting is whether it facilitates novel explanations (Knox Reference Knox2016). Each of the types of emergence defined involved a sort of comparative novelty in the values of a property found in a set of models with respect to a comparison class. Thus, models with emergent properties provide new resources for model-based explanations that cannot be found merely by abstracting away details from the models of the comparison class.

All such conditions on what make an example of emergence interesting are in a sense necessary concessions to what Humphreys (Reference Humphreys and Humphreys2016) calls the rarity heuristic for emergence: a satisfactory account of emergence should not make cases thereof too common. I have adapted the heuristic, though, not to emergence itself but to its interesting examples. So, although emergence may be fairly common on the above account, interesting examples of it are less so. While some may take this is as a conventional matter of shifting content from one definition to another, I find it useful to separate the relatively precise formal structure that allows us to make attributions to properties described by models and theories from how those attributions figure in broader intellectual and scientific projects. In other words, a property’s emergence, by itself as a partly formal property, does not portend significance, and we should recognize that such significance may be grounded in concerns external to those that determine how, say, various models are relevantly similar to one another. The context of investigation determines the similarity structure on a space of models, which in turn determines which properties are emergent with respect to others, but which of those emergent properties are interesting may depend on much more than this.Footnote ⁹

Before illustrating this account of emergence in the next section, let me acknowledge that, even constrained by the informal conditions of application, it may strike some readers as too weak to count as being of emergence rather than of mere dissimilarity—especially weak emergence. As I have stated myself, weak emergence requires very little, so some readers may wish to take it only as a precondition for more interesting forms of emergence. I invite them to consider even further conditions of application under which it would deserve the name, but I proceed in what follows without repeated qualifications.

4. Examples

In this section I illustrate two applications of the similarity structure approach to emergence. The first, in section 4.1, shows how properties related to absolute simultaneity are emergent in classical mechanics relative to special relativity—that is, in models of Galilean space-time relative to models of Minkowski space-time. The second, in section 4.2, concerns the emergence of phase transitions in certain thermostatistical systems, represented as nonanalytic points in the system’s free energy, under the thermodynamic limit—that is, when one considers infinite versions of the systems in question.

4.2. Phase Transitions

Phase transitions are sudden qualitative changes in the characteristics of a highly composite system, such as a drop of water freezing or an iron bar magnetizing. While there is a plethora of approaches to understanding phase transitions and philosophical debates surrounding them (Menon and Callender Reference Menon, Callender and Batterman2016), for simplicity of exposition I will focus on a crude toy model, in which a candidate (equilibrium) thermostatistical system is represented through, for example, a many-body Hamiltonian. The energy levels for the components determine energy levels for the total system, which in turn determine the (canonical) partition function for the system and its (Helmholtz) free energy function. A phase transition for the system occurs when the system enters into a state whose free energy is a nonanalytic point—that is, a point at which some order of its partial derivatives does not exist. Although having such a point (or not) is a property of a function, which is itself a derived property of a model of the system, for the reasons discussed in section 2 I refer to it as a property of the model itself.

Let us focus, without loss of generality, on a certain type of thermostatistical model (say, the two-dimensional Ising model, but the details will not concern us). Let T _F be the models with finitely many components and T _∞ be those with infinitely many, each element of which is represented by the (toy) free energy function of a single variable $F_n\left(x\right)=1/(1+e^{-\mathrm{nx}})$ , where n is the number of components and x is the thermodynamic variable.Footnote ¹¹ Meanwhile

$F_{\infty }\left(x\right)=\{\begin{array}{cc}0,& x<0,\\ 1/2,& x=0,\\ 1,& x>0,\end{array}$

the Heaviside step function. For real systems, the models with finitely many components are more fundamental because real systems are finite in just this way; the ones with infinitely many components are idealizations.

Further, let ${\nu }_a:T_F\cup T_{\infty }\to \{\top ,\perp \}$ be the property assignment whose value is ⊺ when evaluated on a model that is analytic and is ⊥ otherwise—the former evinces no phase transitions, while the latter does. Clearly ${\nu }_a\left[T_F\right]=\top$ while ${\nu }_a\left[T_{\infty }\right]=\perp$ . (We could also consider more specific properties, such as particular definite phase transition properties, which would not be defined for the models in T _F, but this will not matter for the broad point at hand.) Thus, phase transitions are weakly emergent for these crude toy models with infinitely many components with respect to the ones with finitely many. In fact, for the same reasoning as the simultaneity case described in section 4.1, they are also strongly emergent.

To answer whether they are also nonreductively or radically emergent, one must fix a similarity structure on the models $T_F\cup T_{\infty }$ . Two function space topologies that can indeed be interpreted as similarity structures are the topologies of pointwise and uniform convergence, respectively (Willard Reference Willard1970, sec. 42). The topology of pointwise convergence determines functions to be similar just when they are numerically close on a finite number of points. Here, the similarity relations vary over all such finite collections and all degrees of closeness within the positive real interval. The topology of uniform convergence, meanwhile, determines functions to be similar in much the same way but over the whole of real line, not just at finite points. (See the appendix for a more complete description.) It is elementary to show that $T_{\infty }\subseteq {\mathrm{cl}}_{\mathrm{𝒫𝒞}}\left(T_F\right)$ while $T_{\infty }\cap {\mathrm{cl}}_{\mathrm{𝒰𝒞}}\left(T_F\right)=\varnothing$ . So, by the same reasoning as in section 4.1, phase transitions are neither nonreductively nor radically emergent according to the topology of pointwise convergence but are radically emergent according to the topology of uniform convergence.

Which of these is the “right” verdict? Are phase transitions radically emergent or merely strongly emergent in infinite systems? Again, there is in general no context-free answer. If the goal is a description of the properties of infinite systems that are novel compared with those of finite systems only in light of the former’s success in actual empirical descriptions, then there is good reason to believe that the topology of pointwise convergence is a better choice than that of uniform convergence, since our actual data of the free energy functions of systems are confined to point measurements (Callender Reference Callender2001). But if the goal is such a description in light of a “global view” of the features of thermostatistical systems, regardless of whether they are reflected in what is observable, then the topology of uniform convergence may be better.

5. Features

Both of the examples of emergence in section 4 arise from a kind of infinite idealization: with respect to a more fundamental comparison class, the models with the emergent properties take some other property value to be infinite or infinitesimal, which allows for emergence. Although these are common in physics—see Fletcher et al. (Reference Fletcher, Palacios, Ruetsche and Shech2019) for more examples—they also appear in other scientific disciplines. For example, in biology, one might describe the emergence of deterministic genetic transmission in certain population genetics models compared with stochastic ones by introducing an infinite population (Abrams Reference Abrams2006; Strevens Reference Strevens2019); in economics, one might describe the emergence of unbounded rationality in models of agents with infinite patience, perfect homogeneity of choices, or undergoing infinite game-theoretic interactions (Brennan, Güth, and Kliemt Reference Brennan, Güth and Kliemt2008); and in linguistics, one might describe the emergence of languages with infinite expressions by abstracting away the material constraints on mental language representation (Nefdt Reference Nefdt2019).

In any case, one of the features of the current account of emergence concerns its relation with reduction. Both of the examples of emergence in section 4 could be compatible with reduction when the latter is understood as the models of one theory being arbitrarily similar to the models of another; that is, T reduces $T^{\prime }$ when $T^{\prime }\subseteq \mathrm{cl}\left(T\right)$ .Footnote ¹² In this case, both the weak and the strong emergence of a property in a model with respect to a comparison class are compatible with that model being arbitrarily similar to the members of the comparison class, but both nonreductive and radical emergence are not. So, some types of emergence represent a sort of relative dissimilarity that vindicates nonreductive intuitions concerning emergence, while some do not.

Which types of emergence (outside of the weak variety) pertain to a property of model, if at all, depends as well on how the model and its comparison class are relevantly similar to one another. I am skeptical that there will be a fixed similarity structure that accrues to a theory or collection of models once and for all, for similar high-level reasons as those Fletcher (Reference Fletcher2016) adduces in the case of general relativity. In that case, an explanatory question determines an investigative context. The properties of models that make a difference for answering the explanatory question are precisely those that are relevant to this context. Fletcher (Reference Fletcher2016) shows that different such questions—regarding the stability of certain temporal properties, on the one hand, and the deterministic nature of temporal evolution, on the otherFootnote ¹³—determine different enough such contexts that amalgamating them would yield absurd answers for at least one.

My suggestion for how to guide the selection of similarity structure in adjudicating cases of emergence is similarly contextual. In such, one is often interested in the explanatory resources one sort of model has with respect to another—the former with the putative emergent property with respect to the latter. Again, the properties of models that make a difference for answering the explanatory question are precisely those that are relevant to this context, so models are similar to the extent that they have similar such relevant properties. So, similarity in the space of models can be guided by similarity in the space of their relevant property values.Footnote ¹⁴ Often these values will represent empirical predictions, in which case the measurement theory for the property can often provide guidance as to the relevant similarity structure on the space of values. Boolean and real-valued property assignments, for example, would tend to have the same usual similarity structure as discussed earlier in this essay. Because the range and precision of these predictions can vary over time and between researchers according to what scientists can and have measured, and how and what they take the models in questions to represent, whether a property is emergent depends as well on these factors, which are not a part of the theory or its formalism. So, even though the definitions of emergence I provide are partly formal, they require input from outside the formalism itself.

That input, of course, can come in more or less precise forms. The extent to which the context of investigation can be made precise by determining the properties of models that make a difference to the questions being asked (and no more) is the extent to which the similarity structure can be made precise. And the extent to which the theory or models to which one is restricting attention can be made precise is the extent to which the resulting similarity space can be made precise. For physical theories, whose models and properties are already mathematized, this may be just a matter of technical ingenuity, but for other sorts of vaguer theories or models, this may require deliberate and determined investigation into how they can be made precise. I take this not as a vice of the similarity structure approach to emergence but a virtue: without offering false precision, it forces one to reveal, inasmuch as one can, one’s commitments to details hitherto obscured about just what questions one is asking of just which models or theories.