This Section develops the main framework of the paper. In Section 2.1, I give the necessary background about emergence, reduction, approximation, and reference: which I will use, in Section 2.2, to explicate ontological vs. epistemic emergence.

Emergence and related notions

In this Section, I give my preferred conceptions of theory, interpretation, and emergence. In Section 2.1.1, I discuss the first condition for emergence, namely dependence, or linkage. In Section 2.1.2, I recall the notion of novel reference: which I will use in Section 2.2 to define ontological novelty.

Talking about emergence in science of course forces us to talk about theories: and so I will take a conception of theory, from De Haro (2016: Section 1.1) and De Haro and Butterfield (2017: Section 2.2), that is appropriate for theories in the physical sciences.

The conception of a ‘theory’

The main task of applying a notion of theory to emergence is to make a conceptual distinction between the formalism and the interpretation of a theory. Thus we distinguish bare and interpreted theories:—

A bare theory is a triple \(T:={\langle \mathcal {S}},\mathcal {Q},\mathcal {D}\rangle \) comprising a structured state space, \(\mathcal {S}\), a structured set of quantities, \(\mathcal {Q}\), and a dynamics, \(\mathcal {D}\):Footnote 4 together with a set of rules for evaluating physical quantities on the states.Footnote 5

An interpreted theory adds, to a bare theory, an interpretation: construed as a set (a triple) of partial maps, preserving appropriate structure, from the theory to the world. The interpretation fixes the reference of the terms in the theory. A bit more precisely, an intepretation maps the theory T to a domain of application, D, within a (set of) possible world(s).Footnote 6 That is: there is a triple of maps, \(i:T\rightarrow D\). Using different interpretation maps, the same theory can describe different domains of the world. In this paper, we will restrict attention to interpretations that are empirically adequate, so that they describe some significant domain of the world in sufficient detail.

This general conception of interpretation is logically weak, because little is required for a structure-preserving partial map. But to discuss ontological emergence (cf. Section 2), we need interpretations that are “sufficiently good” within their domain of application. Thus, in addition to the interpretation being empirically adequate, I will impose the additional condition that every element in the codomain is described by at least some element of the theory. Though this looks like a strong condition, it is in fact innocuous. The idea is captured by the requirement that the map be surjective: and, as you might expect, this can always be achieved by restricting the codomain. We will return to this notion of interpretation, and develop it, in Section 2.1.2.

Discussions of emergence often work with models, i.e. specific solutions of the dynamical equations of the theory that are physically permitted, rather than with entire theories. (The possibility of seeing emergence this way will resurface in the example of Section 5.) In such cases, the interpretation maps are assigned to the states and quantities of a particular solution i.e. model, rather than to the whole theory: and the domain is then naturally embedded in a single possible world.

A further refinement of the notion of interpretation that will aid better understanding of the notion of emergence is the distinction between the intensions and extensions of terms.Footnote 7 The extension of a term is its worldly reference under a certain interpretation, i.e. the thing or entity being referred to, relative to a given possible world (with all of its contingent details). The intension is the linguistic meaning (cf. the Fregean sense) of a term, as described by the theory. Thus although ‘Evening star’ and ‘Morning star’ have different intensions (i.e. different linguistic meanings, and the stars they refer to are different in some possible worlds), they have the same extension in our world, viz. the planet Venus—thus it is a contingent fact about our world that, while their intensions differ, their extension is the same. I will also use ‘extension’ and ‘intension’ for theories, as the ranges of the corresponding interpretations of the collection of all the terms of a theory.

Scientific theories have both intensions and extensions,Footnote 8 which in the present framework can be modelled by two different kinds of maps, each with its own domain, depending on whether the interpretation is an intension or an extension.Footnote 9 Thus both intensions and extensions are structure-preserving partial maps from a bare theory or model to a domain relative to a possible world. The difference between the intension and the extension is in the kind of domain: explicitly, for states: an intension maps a state to a generic property (or physical arrangement) of a system mirroring the defining properties of the mapped state. Thus the image of the state and the domain abstract from contingencies such as how the system is spatially (and otherwise) related to other systems, and how the system is measured (so that the interpretation applies to all possible worlds that are described by the theory or model). By contrast, in the case of an extension, the image and domain are a fully concrete physical system: usually including also a specific context of experiment or description, and all the contingent details that are involved in applying a scientific theory to a concrete system.Footnote 10

I will construe the domain of application, D, introduced by the interpretation, as a set of entities, namely the elements of a set (fluids, particles, molecules, fields, charge properties, etc.), and relations between them (distances and correlation lengths, potentials and interaction strengths, etc.). Thus the criterion of identity of domains is the set-theoretic criterion: namely, the identity of the elements (and their relations).

To explain a bit more how this criterion will be applied in practice, consider that we normally use a language to talk about the world: we describe the domain of application linguistically (e.g. the theory may describe ‘this red ball I am kicking’), and express identity in terms of this language. This language is an aid (e.g. words used to mention people) to describe on paper, or in sound, the intended physical things and properties, and it is in general different from the mathematical language of the bare theory. It is ‘the language of experimental physicists (and lay people)’ used to describe the world, and it may contain extra-linguistic elements such as diagrams, images, ostension, etc. To give some examples of how this language picks out the objects in the world: (1) ‘A physicist’s description of a laser beam’ ↦ laser beam. (2) ‘A set of numbers on a computer screen’ ↦ the events in a particle detector. (3) The descriptions of experimental results (including the description of the working of the instruments) that we find in the pages of scientific journals.

Kuhn and Feyerabend said that the meanings of the terms of scientific theories cannot be compared, because theories are incommensurable. This cannot even be done empirically, because experimentation and observation are ‘theory laden’. Thus the Kuhn-Feyerabend critiques of meaning might lead one to think that the criterion of individuation of entities in the domain of application is problematic, if the language that we use to talk about them is theory-laden. I hope to address this issue in more detail elsewhere, but let me here make a comment of clarification in connection with the language which we use to talk about the domain of application.

For theories that, like those of the next Section, are related by linkage relations, the theory-ladenness of this language need not be an obstacle to comparing the domains of application, as my examples in Sections 5 and 6 will also illustrate: since the linkage relates the bare theories whose corresponding domains of application we compare. Therefore, combining linkage between bare theories, with the domain of application’s reflecting the bare theory (its being ‘theory-laden’), and with the empirical data about the domain of application, we can compare the domains of application after all. Thus we do not need a putative “theory-free” language to compare them. Also, incommensurability is usually only partial, and the domains of application are often partly overlapping and partly distinct.

Thus the practice of determining whether the elements of the domains of application of two theories are the same includes a combination of theoretical and empirical considerations. For examples of how to formalise and compare theories this way, see Sections 5 and 6.

I will expand on the criterion of identity of domains, in terms of intensions and extensions, in Section 2.2.2. Let me now introduce the conception of emergence.

The conception of emergence

Consider the opening words of the Stanford Encylopedia of Philosophy’s entry on ‘Emergent Properties’ (its only entry on ‘emergence’):

Emergence is a notorious philosophical term of art. A variety of theorists have appropriated it for their purposes ever since George Henry Lewes gave it a philosophical sense in his 1875 Problems of Life and Mind. We might roughly characterize the shared meaning thus: emergent entities (properties or substances) ‘arise’ out of more fundamental entities and yet are ‘novel’ or ‘irreducible’ with respect to them... Each of the quoted terms is slippery in its own right, and their specifications yield the varied notions of emergence.Footnote 11

On this general characterisation, emergence is a two-place relation between two sets of entities: the emergent entities, and the more fundamental ones (I will dub them the ‘top’ and ‘bottom’ entities, respectively). The top entities are ‘novel’ or ‘irreducible’ compared to the bottom ones.

‘Irreducibility’ here matches Broad’s ‘lack of deducibility’ quoted in the Introduction.Footnote 12 As the italicized phrase in the above quote suggests, irreducibility and novelty are related: which naturally leads us to ‘novelty’ as the weaker, more general alternative. For irreducibility is surely indicative of some type of novelty: if the top entities are irreducible compared to the bottom ones, then surely they have something novel. But the other way around is not the case: novelty need not be expressed as irreducibility. And so, I propose that ‘novelty’, in this general and—admittedly—still unspecified sense, is the more general notion that one should seek to define emergence. Part of our task in developing a theory of emergence, then, is to further clarify what we mean by ‘novelty’.

The above discussion is echoed by the recent philosophy of physics literature, which sees ‘emergence’ as a “delicate balance” between dependence, or rootedness, and independence, or autonomy (and the accounts also often compare theories rather than individual entities).Footnote 13 Roughly speaking, dependence means that there is a linkage between two theories (or between two items of a given theory): while in dependence means that there is novelty in one theory with respect to the other (or between the two items of the given theory). This matches the two aspects—the two-place relation and the characterisation of the top entities as ‘novel’—of the above quote.

In this Section, I shall make these notions more precise, within the framework for theories from Section 2.1. I will take these two aspects—linkage and novelty—to align with the two aspects comprising a theory: viz. the bare theory and its interpretation. Since these two aspects lie “along different conceptual axes”, they can be happily reconciled: which makes it unnecessary to define novelty somehow as a ‘lack of linkage’, ‘lack of deduction’ or ‘irreducibility’. In this subsection, I discuss linkage, and in the next, novelty.Footnote 14

Thus I take linkage to be a formal, i.e. uninterpreted, relation between bare theories (or between two items of a given bare theory), hence as an inter-theoretic relation between bare theories. More precisely, linkage is an asymmetric relation, link, among the bare theories in a given family—with the family containing at least two theories. For the case of just two theories, it is a surjective and non-injective map, denoted as: \(\textbf {link}: T_{\mathrm {b}}\rightarrow T_{\mathrm {t}}\). Here I adopt Butterfield’s mnemonic, whereby T b stands for ‘best, bottom or basic’, and T t stands for ‘tainted, top or tangible’, theory. Here, T b normally denotes a single theory, but it can be used to denote a family of theories. So, the idea is that the bare theory, T b , contains some variable(s) which provide(s) a more accurate description of a given situation than does T t .Footnote 15 (See the three cases of linkage, just below).

The idea of the linkage map, link, is that it exhibits the bottom theory, T b , as approximated by the top theory, T t . The map’s being surjective and non-injective embodies the idea of ‘coarse-graining to describe a physical situation’ (but linkage is not restricted to mere coarse-graining: see below). The linkage map (and so, the broad meaning of ‘linkage’, as an inter-theoretic relation, used here) can be any of three kinds which do not exclude one another, but are almost always found in combination:

(i) A limit in the mathematical sense. Some variable (e.g. \(V, c, \hbar \in \mathbb {R}_{\geq 0}\) or \(N\in \mathbb {N}\)) of the theory T b is taken to some special value (e.g. \(V\rightarrow \infty \), \(N\rightarrow \infty \), \(c\rightarrow \infty \), \(\hbar \rightarrow 0\)), i.e. there is a sequence in which a continuous or a discrete variable is taken to some value. T b then refers to the sequence of theories obtained when the variable is taken to the limit, while T t is the limit theory. In actual practice, the limit often comes with other operations, such as in (ii) immediately below—hence my warning above that these three kinds of linkage maps ‘are almost always found in combination’.Footnote 16 (ii) Comparing different states or quantities. Emergence often involves comparing a given physical situation, or system, or configuration of a system, to another that resembles it. This is often done by comparing some of the theory’s models, i.e. the solutions of the theory (for an example of this, see Section 5.3). Such physical comparisons have correlates in the formalism, which can be implemented prior to interpretation, by comparing different states or quantities or different models, here considered as uninterpreted solutions of the equations (cf. Section 2.1), between the bottom and top theories. In these cases, T b is the bare theory that describes the situation (or system, or configuration) one wishes to compare to, and T t is the bare theory that describes the situation (or system, or configuration) that one compares. The linkage map is then the formal implementation of this comparison, e.g. the comparison between the formal, i.e. uninterpreted, models. (iii) Mathematical approximations (whether good or poor). Here, one compares theories (or expressions within theories) mathematically: perhaps numerically, or in terms of some parameter(s) of approximation. For example, as when one cuts off an infinite sequence beyond a given member of the sequence (perhaps because the sequence does not converge, or perhaps because there is no physical motivation for keeping an infinite number of members of the sequence); or when, in a Taylor series, one drops the terms after a certain order of interest. This will implement the idea that “emergence need not require limits”.

Let me make three comments about how these three ways, (i)-(iii), of linking a theory to another, can define the map \(\textbf {link}:T_{\mathrm {b}}\rightarrow T_{\mathrm {t}}\). The first two comments are about how to apply (i)-(iii), while the third is about how linkage thus defined relates to emergence:

(1): Any of the three ways, (i)-(iii), of linking a theory to another, can define the map \(\textbf {link}:T_{\mathrm {b}}\rightarrow T_{\mathrm {t}}\). A given linkage map almost always involves a combination of several of the above approximations, as we will see in Sections 4 to 6. Also, the list is not meant to be exhaustive. (2): Bridge principles: before (i)-(iii) can successfully define a linkage map, one may need to “choose the right variables”, or make additional assumptions. For example, one may need to first recast the bottom theory in a more convenient or more general form (see an example of this in Section 5.1). This goes under the name of bridge principles or bridge laws, i.e. the map \(\textbf {link}:T_{\mathrm {b}}\rightarrow T_{\mathrm {t}}\) between the bottom and top theories may involve additional assumptions such as choices of initial or boundary conditions, or defining new variables to link the two theories’ vocabularies. However, since the comparison here is between bare theories, the bridge principles here meant are formal, i.e. uninterpreted, so that—recall the distinction at the end of Section 2.1—the ‘language’ just mentioned is the language of the bare theory, not the language we use to talk about the domain: thus it is mathematical rather than physical language.Footnote 17 (3): The ways of linking, (i)-(iii), as here defined are formal i.e. not interpretative, as I stressed in (ii): they link the bare theories, but they are physically motivated, and they often have a correlate in the theory’s interpretation (which serves as a constraining affordance on the kind of linkage map one is likely to adopt). Just how the theory’s interpretation correlates with the formal linkage map is in fact the very question of emergence itself, as I will discuss in the next few Sections:

Emergence : We have emergence iff two bare theories, T b and T t , are related by a linkage map, and if in addition the interpreted top theory has novel aspects relative to the interpreted bottom theory.Footnote 18

The linkage map thus specifies the dependence part of the emergence relation. To characterise emergence, we still need to specify the novelty.Footnote 19

Philosophers often distinguish between ontological and epistemic emergence. The intended contrast is, roughly, between emergence ‘in the world’ vs. emergence merely in our description of the world. Thus according to this distinction we can have two types of novelty: ontological or epistemic. Specifying novelty in the case of ontological emergence is the topic of the next Section.

Ontological emergence as novel reference

In this Section, I characterise the kind of novelty that is relevant to ontological emergence. First of all, let me point to an obvious question that comes to mind about the literature on ontological emergence. We saw, in Section 2.1.1, that it is reasonable to define emergence in terms of novelty: this being a more general notion than the stronger ‘irreducibility’—and, as I mentioned, this also seems to reflect a consensus in the recent philosophy of physics literature. Thus one would expect that, when defining ontological emergence, one would try to further characterise the so-far unspecified notion of ‘novelty’ in terms that are relevant to the ontology of the entities or theories involved in a relation of emergence.

But the literature usually takes some highly specific metaphysical relation instead, whose relation to novelty is not explicitly addressed, and uses that to characterise the emergence relation between the top and bottom entities or theories. I think this shift to specificity is understandable taking into account some of metaphysicians’ interests, but I will also argue that, lacking a general notion of ontological novelty, we run the risk of missing some more basic aspects of what we mean by ‘ontological emergence’.

For example, Wong (2010: p. 7) defines ontological emergence as ‘aggregativity generating emergent properties’: ‘Ontological emergence is the thesis that when aggregates of microphysical properties attain a requisite level of complexity, they generate and (perhaps) sustain emergent natural properties.’ And Wimsatt (1997, 2007: Chapter 12) takes the lack of aggregativity of a compound as the mark of emergence.Footnote 20

Further, in the Stanford Encyclopedia of Philosophy article quoted earlier, O’Connor and Wong (2002: Section 3.2) review various other uses of the phrase ‘ontological emergence’, none of which are obviously introduced as an explication of ‘novelty’: viz. ontological emergence as ‘supervenience’, as ‘non-synchronic and causal’, and as ‘fusion’.

I will not criticise these accounts here, since my claim is not that they are incorrect,Footnote 21 but rather that the accounts—interesting as they are—are very specific: in fact, too specific to be able deal with all the examples that physicists are interested in. For example, Wimsatt’s (1997, 2007) ‘failure of aggregativity’ does not seem applicable in cases where one compares systems that are neither spatial components of each other, nor aggregates of components. In other words, the relevant relation between the levels is not always one of spatial inclusion or material constitution. Thus already my examples from Sections 4 and 5 are not covered by Wimsatt’s criterion of failure of aggregativity. Also, it would seem that failure of aggregativity is most interesting if it leads to novelty. The same can be said about causation: it may play a role in important examples of emergence such as the mind, but it is hard to see how it plays a role in the kind of examples that I discuss in this paper, which are, I believe, “garden variety” for the theoretical physicist, and involve no causation from the bottom to the top entities.

Another reason to be sceptical about too specific accounts of ontological novelty is the requirement of consistency with the general description of emergence reviewed in Section 2.1.1. As I said above, having settled for novelty as the general mark of emergence, one naturally expects an explication of emergence to point to novelty in the top theory’s ontology. Thus I take it that the metaphysical accounts of emergence just discussed, in so far as they all capture aspects of emergence, aim to put forward a specific metaphysical expression of ‘novelty’. This is apparent from Wong’s mention of the ‘generation of emergent natural properties’: surely what makes these properties of the top theory ‘emergent’ (on pain of his own definition being circular) is that they are novel, relative to the properties of the bottom theory.

My account of ontological emergence is metaphysically pluralist in that, while it recognises that emergence is a matter of novelty in the world (as the general notion of emergence prompts us to say), it does not point to a single metaphysical relation (e.g. supervenience, causal influence or fusion) as constitutive of ontological emergence. And I believe that this is just right, for the reasons above: a single metaphysical relation does not seem to cover all the cases of interest. Indeed my main contention is that there is a more basic framework for ‘ontological novelty’ that needs to be developed before one fruitfully moves on to detailed metaphysical analyses. Furthermore, I take it as a virtue of the present framework that it allows for an explication of the phrase ‘ontological emergence’ without appealing to specific, and sometimes controversial, metaphysical notions. So it seems that an account satisfying these desiderata should exist (more on this in Section 3.2). In this sense, my account is “basic”, and can perhaps best be seen as an attempt to state the “almost obvious” in a more precise way.

To discuss ontological novelty, then, I take my cue from Norton (2012), which is a perceptive discussion of the difference between idealisation and—what I shall call—non-idealising approximation. Roughly speaking, he proposes the following contrast:

A [non-idealising] approximation is an inexact description of a target system.Footnote 22 An idealization is a real or fictitious, idealizing system, [possibly] distinct from the target system, whose properties provide an [exact or] inexact description of the target system’ (p. 209).Footnote 23

Norton summarises the main difference between the two as an answer to the question: ‘Do the terms involve novel reference?’ On Norton’s usage, only idealisations introduce reference to a novel system—notice that the word ‘system’ here may refer to a real or a fictitious system. In the case of idealisation, there is an ‘idealised system’ that realizes the ‘idealised properties’. In the case of a non-idealising approximation, an idealised system either does not exist, or is not accurate enough to describe the target system under study.Footnote 24

I propose to define ontological novelty in terms of novel reference thus construed. Thus, this gives an additional condition on the linkage map defined in Section 2.1.1: namely, the composition of the linkage map and the top theory’s interpretation map, i t ∘link, must have an idealisation in its range, in the sense that the top theory has a referent that is novel, relative to the bottom theory’s domain of application.Footnote 25

This is not merely the condition that the top theory must describe a physical system through its interpretation map, i t : for the top theory is linked to the bottom theory, and so this is an extra condition on the linkage map between the two theories.

Although this characterisation of ontological novelty as novelty of reference is, by itself, not a new construal of the notion (see the discussion in Section 3.3), it will give an interesting criterion for emergence: one formalised in terms of maps, in Section 2.2.

There is ‘novel reference’ when the terms of the bare theory refer (via the interpretation maps introduced in Section 2.1.1) to new things in the world, i.e. elements or relations that are (a) not in the domain of application of the bottom theory, T b , that T t is being compared to, but are (b) still in the world—they are in the domain of application of T t .

Novelty is here not meant primarily in the temporal sense, since it is conceptual noveltyFootnote 26 that counts (cf. Nagel (1961: p. 375)),Footnote 27 even allowing reference to other possible worlds (as Norton does when he refers to idealisations as referring to real or fictitious systems: p. 209; cf. also the definition of an interpreted theory in Section 2.1).

For example, Norton considers the eighteenth- and nineteenth-century theories of heat to be referentially successful, despite the fact that we now know that heat is not a fluid. According to Norton, the theory is nevertheless—

referentially successful in that the idealizing system is a part of the same system the successor theory describes. The “caloric” of caloric theory refers to the same thing as the “heat” of thermodynamics, but in the confines of situations in which there is no interchange of heat and work.

I propose to read Norton’s statement above as saying that the theory is referentially successful because it refers to an idealised worldFootnote 28—a possible world in which there is no interchange of heat and work—that approximates the target system of interest well, according to standard criteria of empirical adequacy: and that as such it is referentially successful. I develop this suggestion in the next Section.

The explication of ontological emergence as novel reference does not attempt to answer questions about scientific realism or referential stability across theory change.Footnote 29 In particular, T b should not strictly be seen as the successor of T t in a historical sense, nor need the continuity between their domains of applicability be assumed.

Ontological emergence, in more detail

In this Section, I propose an answer to the question of how to characterise ontological emergence for theories formulated as in Section 2.1. My proposal is based on a restatement of the notion of ‘novel reference’: namely as the failure of the interpretative and linkage map to mesh, i.e. to commute, in the usual mathematical sense of their diagram’s “not closing”. This idea is inspired by Butterfield (2011b: Section 3.3.2), though I here consider it for interpretations rather than for bare theories: a distinction which will turn out to be important (cf. Section 2.2.2).

I begin with two remarks which further specify the context to which these concepts apply:

(1): Emergence of theories: and of models, entities, properties, or behaviour. I will formulate the distinction in terms of theories, but the idea can be equally well formulated in terms of models (cf. Section 2.1), entities, properties, or behaviour (and I will adopt the latter perspective in the examples in Sections 4 and 5). (2): The space of theories. Properly defining a linkage map, especially if it involves a mathematical limit of a theory (so that one can then discuss the sequence of corresponding interpretations), often requires introducing the notion of a space of theories related by the linkage map. Sections 5 and 6 will take a simpler approach and consider sequences of states, quantities, and dynamics, without explicitly specifying the full details of the spaces in which the sequences are defined, so as to simplify the presentation.Footnote 30

Section 2.2.1 contains the core of this Section and the main proposal of this paper, i.e. the explication of ontological emergence. In Section 2.2.2, I discuss to what extent the proposal answers the questions in the Introduction.

Epistemic and ontological emergence as commutativity and non-commutativity of maps

In this Section, I reformulate the notion of novel reference in terms of the non-commutativity of the interpretative and linkage maps, or their “failure to mesh”: which I will use to give an explication of ontological emergence. I here mean non-commutativity of maps in the usual mathematical sense that, depending of the order in which the maps are applied, the image (value, output) that a given argument (input) is mapped to varies—the mismatch thus expressing the novelty of the reference.

Thus the starting point is a reformulation of the idea of novel reference, in terms of my notion of interpretation of a bare theory (Section 2.1). Recall that an interpretation is a surjective map, i, from the bare theory, T, to a domain of application, D, \(i:T\rightarrow D\), preserving appropriate structures. Norton’s idea can then be formulated as follows.

Consider two bare theories, a bottom theory, T b , and a top theory, T t . Their interpretations are corresponding surjective, structure-preserving maps (by our assumption, in the preamble of Section 2.1, that the interpretation is “sufficiently good”), from the theories to their respective domains of application. That is, there are maps \(i_{\mathrm {b}}:T_{\mathrm {b}}\rightarrow D_{\mathrm {b}}\) and \(i_{\mathrm {t}}:T_{\mathrm {t}}\rightarrow D_{\mathrm {t}}\).

Now, consider, as in Section 2.1.1, a linkage map, \(\textbf {link}:T_{\mathrm {b}}\rightarrow T_{\mathrm {t}}\), mapping the bottom theory to the top theory. There are two interpretative cases, as follows.

(1) Allowing epistemic emergence

If the two theories describe the same “sector of reality”, i.e. the same domain of application in the world, then the ranges of their interpretations must coincide. (Alternatively, the top range must be contained in the bottom range, i.e. \(D_{\mathrm {t}}\subseteq D_{\mathrm {b}}\): a condition that I will discuss separately, below). For the range of the interpretation is what the theory purports to represent: and these two theories represent the same sector of reality. This situation of identity is described by the commuting diagram in Fig. 1. As is clear from the diagram, there is no novel reference here, because the domains of application in the world are the same.

Fig. 1 Possibility of epistemic emergence. The two interpretations describe “the same sector of reality”, so that i b = i t ∘link Full size image

Notice that, since the domains of application are the same, the two maps map to the same elements within that domain of application (the same experiments, interactions, correlations, etc.): even though, as interpretations, they are of course different, because they map from different theories (and so, one theory could describe water in terms of molecular dynamics of the single molecules, or in terms of the dynamics of groups of molecules, even though both describe the same empirical data). In such a case, the two interpretations are related as follows: i b = i t ∘link.

Another type of epistemic emergence is if the bare theories, T b and T t , are equivalent bare theories. That is, we consider the commuting diagram in Fig. 1, but weaken the linkage map link, allowing it to be injective. There is no novelty in the world, but only in the bare theory and the interpretation map (but not in its range).Footnote 31 Dualities in physics give vivid examples of this type of emergence:Footnote 32 see Castellani and De Haro (2019) (see also Section 3.3.2).

Since in Fig. 1 there is no novel reference, there cannot be ontological emergence. However, there can be epistemic emergence: this will be the case when the two interpreted theories describe the domain of application differently. And so, the top theory T t can have properties or descriptive features that are indeed novel, and striking, relative to the bottom theory, T b . The novelty of these properties and descriptive features, however, is not a matter of reference: it does not rely on a difference in elements or relations between the bottom and top domains of application, since by definition we have a commuting diagram in Fig. 1. Rather, the novelty lies in the different relations between the bare theory and the domain that i t establishes, compared to i b , i.e. in the kind of statements that the interpreted theory makes about the domains of application.

I now discuss the special case of proper inclusion, i.e. D t ⊂ D b . The reason for allowing this case is that there are surely cases of epistemic emergence in which the diagram “does not close”. If i b is a “very fine-grained interpretation”, “respecting” the distinction of the many micro-variables of T b , i.e. an injective map, while link is of course non-injective, then we have a diagram which cannot possibly commute on all arguments. But this could be a case of mere coarse-graining with no ontological novelty, while the failure of the diagram to close would seem to suggest ontological emergence. In such cases, empirical adequacy (see Section 2.1) on the domain D b , so that it is a good range for i b , requires D b to “contain coarse-grained facts”, since i b needs to be surjective on D b . Thus in such cases, “good” interpretations of T b include a classification of microstates into macrostates.

To sum up: cases where the map link is mere coarse-graining, which results in the “dropping of certain fine-grained facts from the domain”, do not count as ontological emergence because then D t is a proper subset of D b .

Let me spell out the possibility of epistemic emergence a bit more. Though the domains and ranges of i b and i t ∘link are the same, the interpretative maps i b and i t are different and have different theories as their domain of definition. Thus the novelty lies in the existence of a new theory, T t , with its own novel interpretation map, i t ; and this novelty is relative to the bottom theory, T b , with its interpretation map i b . Therefore the emergence is epistemic, since it takes place at the level of the theory and its interpretation map (while the domains are the same): it is emergence in the theoretical description, rather than emergence in what the theory describes.Footnote 33

Epistemic emergence should not be dismissed as “merely” a matter of “words”, or even “taste”. For theories, and interpretations of the kind here discussed, are not the kinds of things that one can choose as one pleases—there may be virtually no freedom in choosing an appropriate linkage map giving rise to a consistent theory T t whose interpretation is as good as that of T b —i.e. the ranges of their interpretations must be identical (cf. Section 2.2.2).

The above is, as it should be, only a necessary condition of epistemic emergence: as a difference in theoretical description. To give a sufficient condition for epistemic emergence, one needs a more precise characterisation of epistemic novelty. Epistemic novelty is often characterised in terms of computational or algorithmical complexity, chaotic behaviour, possibility of derivation only through simulation, etc. Alternatively, it is seen as the failure of prediction or of explanation.Footnote 34 But since epistemic emergence will not be my main topic, I will leave such further characterisations aside (I will return to it in the example of Section 3.3.2; see also Castellani and De Haro (2019: Section 2.3)).

(2) Non-commutation: ontological emergence

But when there is novel reference, the two interpretations refer to different things—they may even refer to different domains of application. For example, hydrodynamics describes the motion of water between 0o C and 99o C at typical pressures, while the molecular theory of water potentially describes it in a different domain of application, since it can e.g. explain the boiling of water as the breaking of the inter-molecular hydrogen bonds due to molecular excitations. Here, one should not confuse the fact that the interpretations of the two theories attempt to explain the properties of the same target system (e.g. turbulence in a fluid) with the distinctiveness of their ranges: each of which is subject to its own conditions of empirical adequacy, within its relevant accuracy.

In all such cases of novel reference, the domains of application are different:

$$ \begin{array}{@{}rcl@{}} D_{\mathrm{t}}

subseteq D_{\mathrm{b}}~, \end{array} $$ (1)

and therefore so are the interpretation maps, that is:

$$ \begin{array}{@{}rcl@{}} i_{\mathrm{t}}\circ \text{link}&

ot=&i_{\mathrm{b}}\\ \text{ran}(i_{\mathrm{t}}) &~/ \subset&\text{ran}(i_{\mathrm{b}})~. \end{array} $$ (2)

Thus the range of the interpretation of the top theory, T t , is not the same as the range of the interpretation of the bottom theory, T b , nor is it a subset of it. This is what I call ontological emergence, and it is pictured in Fig. 2.

Fig. 2 The failure of interpretation and linkage to commute (i b ≠i t ∘link) gives rise to different interpretations, possibly with different domains of application, D b ≠D t Full size image

There are cases in which T t is not a good approximation to a single theory T b (cf. (i) and (iii) in Section 2.1.1), but only to one of the members of a family of theories T b . Let us label this family by x, so that the bottom theory becomes a function of x, which we write as T b (x). In other words, we take the emergence base (i.e. the class of bottom theories with respect to which the top theory emerges: cf. footnote 19) to be the entire family of theories, {T b (x)}, for any x in an appropriately chosen range. We will see this at work in the example of Section 5.

Since the emergence base is the entire set of theories, {T b (x)}, the bottom theories all have the same interpretation map, i b , but the range of i b now depends on, i.e. varies with, x through the bottom theory, i.e. we evaluate i b (T b (x)).

Now take x to be bounded from below by 0 and consider, as a special case, a linkage map that is taking the limit \(x\rightarrow 0\), i.e.:

$$ \begin{array}{@{}rcl@{}} T_{\mathrm{t}}:=\lim\limits_{x\rightarrow0}T_{\mathrm{b}}(x)~. \end{array} $$ (3)

In this case, the statement of ontological emergence, Eqs. 1–2, amounts to:

$$ \begin{array}{@{}rcl@{}} \text{ran}(i_{\mathrm{t}})

subseteq\text{ran} (i_{\mathrm{b}})|_{x=0} \end{array} $$ (4)

or, perhaps more intuitively:Footnote 35

$$ \begin{array}{@{}rcl@{}} D_{\mathrm{t}}:=i_{\mathrm{t}}(T_{\mathrm{t}})

subseteq D_{\mathrm{b}}|_{x=0}:=i_{\mathrm{b}}(T_{\mathrm{b}}(x))|_{x=0}~. \end{array} $$ (5)

In other words, the domains that we get by interpreting the top theory directly, or by first interpreting the bottom theory and then setting x = 0, are different, as we will see in the example below.Footnote 36

In view of the difference between Figs. 1 and 2, we can now reformulate novel reference as the linkage map’s failure to commute, or to mesh, with the interpretation. When linkage and interpretation do not commute, the two interpretation maps are different because the theories refer to different domains, and different systems—even if the underlying physical object, like the sample of water above, may of course be the same.

It is possible to formulate a robustness condition for the emergent behaviour—which I will assume in the rest of the paper—as the condition that the emergence base is a whole class of theories, i.e. {T b (x)} for a range of values of x, rather than a single theory for fixed x. Thus, emergence is robust if the interpretation does not change as we vary x over the base (see footnote 35).

To sum up: the meshing condition between the linkage and interpretation maps, formulated as the lack of commutativity or of closing of their joint diagram, Fig. 2, should be seen as a reformulation of novelty of reference—a reformulation that gives a straightforward formal criterion that can be used in examples.

Let me briefly discuss the relation between reduction and emergence. I will endorse the philosopher’s traditional account of reduction: namely, Nagel’s viewFootnote 37—as, essentially, deduction of one (here, bare) theory from another, almost always using additional definitions or bridge-principles linking the two theories’ vocabularies.Footnote 38 Not all linkage maps will be reductive. For the relations between physical theories are often much more varied than logical deduction with bridge principles allows (for example, the use of limits and related procedures, such as renormalization, often adds content to the theory). But when the linkage is reductive, the account explains at once why ontological emergence and reduction are independent of each other: for the former is a property of the theory’s interpretation (i.e. the novelty in the domain), while the latter, understood as a formal relation, is a property of the linkage map only, i.e. a relation between bare theories.Footnote 39

Is emergence ubiquitous? Regimenting the uses

In this Section, I point out two properties of the regimentation of ‘emergence’ I have proposed, that have the effect of reducing the number of putative cases of emergence; and I analyse the extent to which the framework gives a clear-cut criterion of emergence.

Understanding ontological emergence as novelty of reference, and epistemic emergence as novelty of theoretical description, gives a helpful regimentation of the uses of the word ‘emergence’. One problem which seems to have plagued the literature is the apparent ubiquitousness of emergence, which would seem to be sanctioned by my logically weak conception in Section 2.1 (cf. Chalmers (2006: Section 3)).

In addressing this problem, the first point to note is that, as I stressed: given the wealth of examples in which physicists justifiably talk about emergence (and which the philosophical literature has also endorsed), pervasiveness is something one may, to some extent, expect and accept.Footnote 40 But, more important: my regimentation has two implications which amount to strengthening the conception of ontological emergence:

(1): The requirement that T t , and a sufficiently good interpretation of it, must exist. My conception makes emergence less pervasive than it might at first sight appear, because of the requirement that T t must be a bare theory, presented in the same form as T b : usually as a triple, subject to the requirements in Section 2.1. Thus one cannot take an arbitrary bottom theory, apply to its elements \(\mathcal {S}\), \(\mathcal {Q}\), and \(\mathcal {D}\) some arbitrary map one calls ‘link’, and claim that one has emergence: for one is not guaranteed to get as the range of the map labelled ‘link’ a bare theory! Likewise for the interpretation, which must be sufficiently good, in the sense of the preamble of Section 2.1: it must be empirically adequate and the interpretation map must be surjective. Again, some arbitrary map one calls ‘link’ will in general not secure empirical adequacy of T t := ran(link), nor will the thus-obtained theory describe the whole domain. (2): The requirement of novel reference, for ontological emergence. Interpretations that are genuinely novel (because they refer to novelty in the physical system) and, at the same time, empirically adequate to within required accuracies, need not be abundant. The reason is as follows: Ontological emergence requires that we have a case of idealisation (see Section 2.1.2), so that there is a physical system to which the map refers. And this is, in fact, a considerable restriction on the linkage map: Norton (2016: p. 46) states that ‘merely declaring something an idealization produced by taking a limit is no guarantee that the result is well-behaved.’ In other words, ‘the limit operations generate limit properties only. They do not generate a single process that carries these properties, for these properties are mutually incompatible’ (p. 44).Footnote 41 Ontological emergence thus crucially depends on the choice of ranges: for \(D_{\mathbf {t}}

subseteq D_{\mathbf {b}}\) is the mark of novelty of reference. The ranges are to be determined by the theory’s best interpretation, which is subject to empirical adequacy and the other constraints mentioned in Section 2.1 (see also below).

To what extent do we get a clear-cut criterion of ontological emergence, as I promised in the Introduction? While I do not claim that formulating ontological emergence formally automatically dispels problems of ontology, which can be subtle (see Section 3.2): the criterion does give us a recipe for assessing whether ontological emergence obtains: it requires us to formulate theories, their interpretations, and the linkage relations, as formallyFootnote 42 as possible. Once that work is properly done, the decision about ontological emergence follows as the non-meshing of linkage with interpretation, provided both theories refer. The challenge lies in adequately and formally formulating interpretations and linkage maps. To illustrate, let us return to the example of water. Consider the questions:

Is liquid water “more than” the sum of molecules and their interactions? Should we consider liquid water as different from its molecular basis? What distinguishes the relation between the liquid state and its molecular basis from its cousin state, ice, and its molecular basis?

My proposal does not do away with the need to ask such questions, at one point or another. These questions will get implicitly answered once sufficiently precise interpretations have been developed. But, on my account, these are not the important questions for the philosophical account of emergence. The recipe is to develop as accurate as possible interpretations of the theories, subject to empirical adequacy and the other requirements from Section 2.1, and to then allow those interpretations to answer the question for us.

Interpretations allow for comparisons: one interpretation is better or worse than another because it covers more or fewer cases, and is more or less empirically adequate and precise, etc. So, one now asks: is there, by the interpreted bottom theory’s own lights, such a thing as ‘liquid water’? Or does the range, i b (T b ), only contain items such as ‘the collection of interacting molecules’? Here, one should not try to judge, independently of interpreting theories, whether ‘liquid water’ is the same as, or can be empirically distinguished from, ‘a collection of molecules’. The right question is whether the interpreted bottom theory, in its range, i b (T b ), has the hydrodynamic substance ‘liquid water’ as an element—for recall, from Section 2.2.1 (1), that we could have epistemic emergence if the range of the top theory was a subset of the range of the bottom theory. Now if our bottom theory is a theory of molecules and their interactions, it does not have ‘liquid water’ in its range: and so water is, indeed, ontologically emergent in the top theory, T t , with respect to its molecular basis in T b . For on an interpretation containing only collections of molecules, there is never a continuous state of matter, however numerous the number of molecules, and however small the size of the molecules, may be. This is because a discrete set of molecules and a continuous medium are referentially distinct (recall that interpretation makes reference to concrete objects: it is not just more theory!). Such an interpretation may not need to distinguish, for its limits of accuracy, between 1040 and 1040 + 1 molecules (the number of molecules may only be defined up to 5% accuracy, say). But it will distinguish a discrete from a continuous medium, because these are different kinds of objects, and so they constitute different domains.

The above is best understood when formulated using intensional semantics (see Section 2.1). The hydrodynamic and molecular theories have the same extension but different intensions (liquid water vs. a large collection of interacting molecules). The intensions differ because the theories are different and, although they may share many common terms (such as ‘volume’, ‘pressure’, ‘temperature’, etc.), ‘liquid water’ is not one of the terms that they share. Nevertheless, they both refer to the same experimental phenomenon in our world and, within certain margins of accuracy, their predictions are the same.

Many examples of emergence are of this type: the theories have the same extension, but different intensions. Of course, it is also possible that both the extensions and the intensions differ.Footnote 43

The distinction between extensions and intensions also further clarifies the criteria of identity of domains, from Section 2.1. For sameness of extension is chiefly (of course not solely!) determined by an empirical procedure, while sameness of intension is conceptual, and determined from within the theoryFootnote 44 (again, not without empirical input!).

Finally, I stress that there can be no “fiddling” with the interpretation in establishing whether there is emergence. This is the strategy I will adopt in my case studies in Sections 4–6: interpretations must be fixed (by other criteria) before one asks about emergence. This is especially true for intensions. In other words, ontological emergence can only be predicated relative to interpretations that are sufficiently good (on the criteria given in Section 2.1): and different interpreted theories must be assessed against each other employing the usual evaluation criteria given by the philosophy of science and in scientific practice. Thus my proposal is to decide for one’s best interpretation once and for all using independent methods, to take the interpretation literally, and to stick to it when enquiring about emergence. Changing one’s interpretation in the course of assessing emergence is liable to lead only to confusion.

Reduction and emergence. One should not confuse the question of emergence we are asking here with the question of reduction. We already agreed that, in this example, there is reduction between the bare theories: emergence and reduction are compatible! (see Section 2.2.1).Footnote 45 The point is precisely that reduction, when it obtains, is a relation between the bare theories,Footnote 46 while ontological emergence is in the range of the interpretation. Thus, the reduction of the bare theory cannot imply a putative reduction of the interpretation.Footnote 47