Abstract
This paper examines the implications of Bedau’s notion of weak emergence when combined with both a hierarchical ontological model and Wolfram’s principle of computational equivalence. I argue that if a system exhibits weak emergence, the very fact of weak emergence of some of its states must itself be a weakly emergent state. Interpreted through Wolfram’s principle of computational equivalence, this implies that a system with weakly emergent states would need to ‘decide’ whether its own dynamical trajectories are incompressible—which contradicts with the formal undecidability of the incompressibility predicate. I term this aporetic conclusion the paradox of weak emergence. After evaluating possible responses, I contend that Bedau’s approach exemplifies what I call the derivational account of metaphysical relations, a framework that I will show to be highly problematic insofar as it implies a mismatch between the intrinsic limitations of formal derivations and the explanatory ambitions of metaphysical notions.
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1 Introduction
The concept of emergence has been the focus of attention in both natural philosophy and the philosophy of science for the last two centuries. Its philosophical characterization is associated with the problem of understanding how macroscopic novel and autonomous features/entities may arise from their microscopic base. The most common examples of emergent phenomena are related to biological systems and psychological states, yet metaphysical explanations appealing to some notion of emergence are almost ubiquitous, from climate science to thermodynamics.
The “golden age” of emergence is traditionally identified with the rise of British emergentism in the XIX century, whose main exponents are John Stuart Mill, Samuel Alexander, Conwy Lloyd Morgan, and C.D. Broad. The common trait of these diverse views on emergence is the idea of the arising of new relations among given lower-level entities which may neither be characterized nor predicted in terms of lower-level compositions. The classical example of this sort of characterization is the emergence of chemical bonds and molecules from the atomic level: according to XIX century physics, mere combinations of intrinsic properties of atoms are insufficient to explain the formation of chemical bonds. Hence, even if the micro-physical base determines what happens at the molecular level, there is no scientific explanation of the formation of molecules in terms of the laws of atomic physics. The conclusion at that time was that chemical aspects of nature are genuinely “novel” and they “emerge” from their atomic base. Using the same explanatory schema, biological organization is an emergent feature of chemical systems, and psychological states emerge from biological organization. It is in virtue of this fundamental appeal to metaphysical novelty and epistemic unpredictability that nowadays we classify the account of British emergentists as a concept of strong emergence.
In a certain sense, the advent of quantum mechanics and the explanation of chemical bonds and atomic interactions in terms of atomic orbitals constituted a significant challenge to British emergentism. For now there were scientific explanations of the formation of chemical bonds in terms of notions dwelling in the province of particle physics.
The “fall” of British emergentism has induced a negative attitude of mainstream philosophical approaches toward the concept of emergence as a genuine metaphysical notion. The logical empiricist tradition tended to dissolve the so-called “doctrine of emergence” as a logical misunderstanding (Nagel, 1961, pp. 369–389); (Hempel & Oppenheim, 2008).
The notion of emergence resurfaced in the late 20th century, particularly in the context of complex systems theory, artificial life, and computational modeling. A major reason for this resurgence was the recognition that many natural phenomena exhibit emergent properties, where higher-level patterns arise from lower-level interactions without violating physical laws. Classic examples include the self-organizing behavior of tornadoes, the coordinated movement of a flock of birds, the emergence of life from non-living chemical processes, and the development of the mind from neural activity. In each case, the macroscopic phenomenon depends on its microphysical basis but also appears to exhibit a degree of autonomy, leading to a philosophical challenge: how can emergent phenomena be both dependent on and autonomous from their underlying substrates?
Metaphysical theories of emergence are often related to the attempt to reconcile this tension between dependence and autonomy. While emergent properties arise from lower-level processes, they are not always straightforwardly reducible to them, at least in terms of our explanatory practices. However, the renowned attention that the notion of emergence gained in the context of the sciences of complexity cannot be a simple rediscovery of the strong notion of emergence which in the past has proven to be in theoretical tension with scientific results. A new notion of emergence was required, possibly not framed in terms of ontological or causal novelty. It is within this theoretical context that the demand for a “weak” notion of emergence arises.
In a series of interesting papers (Bedau, 1997, 2008a, 2013), Mark Bedau offers a philosophically interesting and scientifically tractable account of the relation of weak emergence. To understand Bedau’s definition, his conceptual framework must be introduced. Bedau considers a system S whose states can be classified into “micro” and “macro” states. Clearly this classification may not be strictly related to the size of the parts of S which determine its micro or macro states. What is important here is that macro-states are determined by micro-states and that in some cases a particular macro-state is a sort of pattern which may be present in several distinct micro-states. Micro-states transitions are governed by S’s “microdynamics” D which models the causal interactions at the micro level. In simpler cases, state transitions in D may depend just on restricted spatial neighborhood of a each “micro-part” of S. The most common examples to understand these characterization may be taken from statistical mechanics or the computational science of cellular automtata.
Having clarified the relevant preliminary notions, it is possible to introduce Bedau’s definition of weak emergence:
“Macrostate P of S with microdynamic D is weakly emergent iff P can be derived from D and S’s external conditions but only by simulation.” Bedau (2008a)
Strictly speaking, weakly emergent features are not unpredictable from or theoretically irreducible to their microdynamics, i.e. their fundamental nomological basis. What makes a feature or a macrostate emergent in this case is precisely the peculiar way in which it may be predicted or derived. In particular, weakly emergent states cannot be derived by means of fixed general procedures describing the behavior of the system given the initial conditions, yet they are “irreductively historical”, for to derive them it is always necessary to go through the entire history of the system up to their occurrence.
Bedau’s notion of weak emergence may be understood from a different perspective. If we think of systems whose microdynamics can be conveniently described in terms of differential equations, then a case of weakly emergent macrostate is a case of absence of closed form solution, i.e. of a general formula to compute the macrostate of the system irrespective of its dynamcial history. In the case of systems modeled through computational approaches (e.g. cellular automata), a weakly emergent state is a state that cannot be derived by means of a recursively decidable procedure modeling the system’s microdynamics, but must be detected via simulation.
The diversity of contexts to which the notion of weak emergence may be applied lead Bedau to present a generalized formulation of his definition. In Bedau (2008b), Bedau define weakly emergent macrostates as states accountable in terms of the system’s microdynamics only by means of incompressible and generative explanations. According to Bedau, an explanation is generative “just in case it exactly and correctly explains how macro-events unfold over time, how they are generated dynamically.” (Bedau, 2008b, p. 445). In other words, generative explanations are derivations isomorphic to the web of causal interactions over time at the micro level. A generative explanation is incompressible if and only if “there is no short-cut generative explanation of that macro-property that is true, complete, accurate, and can avoid crawling the causal web.” (Bedau 2008b, p. 446). For instance, a certain macrostate of a linear system calculated by means of a closed form solution of the dynamical equations is not an emergent feature, for it may be derived by means of a compressed procedure (i.e. an analytic shortcut represented by a formula) that does not replicate the structure of causal interaction.
In this paper, I will show that placing Bedau’s notion of weak emergence within the broader contexts of a hierarchical ontological model and Wolfram’s principle of computational equivalence leads to paradoxical consequences. In particular, I will argue that the fact that a particular state of a system weakly emerges must be itself a weakly emergent state of the system. When this consequence is interpreted in the light of Worlfram’s principle of computational equivalence, it follows that a system having weakly emergent states must be a system capable of “deciding” whether its own dynamical trajectories are incompressible which contradicts the fact that incompressibility is recursively undecidable. I call this aporetic conclusion the paradox of weak emergence. In Sect. 2 I will present the hierarchical ontological model to be adopted, with reference to Kim’s critical view of the mulit-layered model. In Sect. 3 I will present an argument to the effect that facts about weakly emergent states of a system S are themselves weakly emergent states of S, i.e. there is a self-referentiality of the application of the relation of weak emergence. In Sect. 4 I will derive the paradox by applying Wolfram’s principe of computational equivalence, which roughly says that natural processes may be seen as computations and natural systems as Turing machines. In Sect. 5, after presenting some possible ways out, I will argue that they are unacceptable from an emergentist perspective and I will present general conclusive remarks regarding a critical aspect of Bedau’s approach to emergence.
2 Weak Emergence and Ontological Hierarchy
What is the impact of introducing a relation of weak emergence—and a relation of emergence in general—on our metaphysical picture of reality? How our ontological models are to be influenced by adding this sort of relation? Answering these questions is a crucial point if we aim at a bold metaphysical account of emergence, a task which goes far beyond the formulation of a mere definition. If Bedau in his extended series of works on weak emergence has attempted the definition and the preliminary application of this concept, here we are concerned with the subsequent goal of harmonizing Bedau’s characterization of emergence with already available ontological models.
When the issue is emergence, it is natural to think about a particular model of reality, i.e. the idea of a multilayered ordered ontology where the ordering is marked by non-reciprocal supervenience. This view dates back to British emergentism and it is explicitly endorsed in (Morgan, 1923, pp. 9–14). According to this model, the natural world is conceived as a pyramidal hierarchical structure where more fundamental entities occupy “lower” layers and emergent or derivative entities stand on the “higher” layers. Clearly, in such a characterization there is a big deal of metaphorical language which may induce several misunderstandings associated either with the attribution of some sort of “values” to the higher positions or with the confusion between distinctness of layers and irreducibility. I will try to clarify these possible confusions as long as they appear in the discussion. However, the combination of the multilayered ontological model with the general idea of emergence has been traditionally considered as a natural blend, insofar as the problems associated with this view are related more to the concepts of emergence and ontological layer taken separately rather than to their combination in one metaphysical picture.
There are two main difficulties associated with the attempt to fit a relation of weak emergence in the multilayered model. Firstly, the idea of emergence traditionally fitted into this model is a concept of strong emergence which, insofar as involves metaphysical novelty and theoretical unpredictability, seems to be compatible with the idea of distinctness of ontological layer. In other words, if a state \(\phi \) emerges on a base B in the strong sense of British emergentists, it is reasonable to suppose that \(\phi \) belongs to an ontological layer higher than that of B, for \(\phi \)’s main characteristics are neither “present” nor derivable from B. On the other hand, weak emergent states are not required to be “underivable”. On the contrary, Bedau’s account of weak emergence implies reduction and derivability from the basal system, even if such relations are taken to be incompressible and realizable only by simulation. On this point Bedau seems clear and explicit:
“Many scientists and philosophers conflate emergence and anti-reductionism. Some forms of emergence are inconsistent with some forms of reductionism. But weak emergent properties are not. Weak emergent properties apply specifically to complex bottom-up wholes that are nothing more than organized combinations of parts. Weak emergent properties are generated precisely by causal webs that are so complex that their future states are impossible to derive except by crawling step-by-step through all of the contingencies in the web.” (Bedau, 2013, p. 336)
Hence weakly emergent properties are still derivable from their basal theory/model, even if in a very peculiar way. This seems to suggest the idea that ontological layers are not “flat” but have an internal structure, where merely resultant features are to be kept distinct from weakly emergent features. We are thus taking the relation of weak emergence as an intra-layer relation. This because weakly emergent features are still explainable in terms of the fundamental laws of the ontological layer from which they emerge and it seems reasonable to consider part of a certain ontological layer L whatever may be explained (even only through incompressible explanations) by the set of laws identifying L. In other words, we are accepting the following principle:
Layer Identification (LI)
A property/state of affairs \(\phi \) may be derived from and explained by the fundamental laws of a layer L iff \(\phi \) belongs to L.
The second main difficulty in including weak emergence in the multi-layered model is related to the model itself. As pointed out by Kim (2002), the theoretical implementation of the doctrine of emergence in the context of the multi-layered model is quite poorly understood and it is unclear how to make a consistent ascription of layer to complex entities which belong to different natural kinds. Take for instance human beings and (hypothetical?) intelligent robots; the former seem to emerge from the biological layer, whereas the latter may be reasonably taken to emerge from the physical layer. Given that the biological layer emerges from the physical one, the awkward consequence is that human beings must emerge from robots, at least in terms of type-type emergence. For this reason Kim proposes a “pardigm shift” in the ontological model we should adopt:
‘If we want a hierarchical representation, the right picture seems to be a tree-like structure with multiple branches, not a single ladder like system, with levels belonging to different branches not being comparable in respect of higher and lower. Even this may be too neat and orderly: branches may merge as we ascend further on the tree (we will see an example that will illustrate this possibility). If a comprehensive levels ontology is wanted, a tree-like structure is what we should look for” (Kim, 2002, p. 17)
The example to which Kim is referring is precisely that of humans and robots. The physical layer branches at some level of complexity: on one side, we have entities such as electro-mechanical artifacts, on another side of the branch “biological entities”. Clearly, the domain of artifacts and that of biological entities are now “segregated”, for none ontologically depends on the other yet both supervene on the physical layer. Kim imagine that robots may achieve a level of complexity such that they share some fundamental functional features with human beings to the extent that the two segregated branches may merge.
The new ontological model Kim proposes does not have layers but “domains of entities” which represent the nodes of the tree-like structure. Given that the right metaphysical picture to bear in mind seems to be more the tree-like structure rather than the ladder-like structure, how does such a paradigm shift impact our contextualization of weak emergence in the preferred ontological model? I don’t see a dramatic conceptual revision, at least insofar as the present discussion is concerned. The only impact we shall evaluate is related to the possible ontological relations between distinct entities. If in the classical multi-layered model either two entities belong to the same layer or one supervenes on the other, in the tree-like structure there is also the possibility for two entities to lay on segregate branches and thus to be neither ontologically equipollent nor supervenient. More specifically, given two entities \(E_1\) and \(E_2\) belonging respectively to domains \(D_1\) and \(D_2\), there are three possibilities:
-
1.
\(E_1\) and \(E_2\) belong to the same domain, i.e. \(D_1 = D_2\);
-
2.
\(E_2\) supervenes on \(E_1\) yet \(E_1\) does not supervene on \(E_2\), i.e. \(D_2\) occupies a position higher than \(D_1\) on the same branch;
-
3.
\(E_1\) and \(E_2\) are ontologically independent, i.e. \(D_1\) and \(D_2\) occupies distinct and segregated branches.
Another important change in our conceptual framework is related to the principle of layer identification (LI), which must be suitably adapted:
Domain Identification (DI)
A property/state of affairs \(\phi \) may be derived from and explained by the fundamental laws of a domain D iff \(\phi \) belongs to D.
It is by virtue of (DI) that we endorse the fact that weakly emerging features belong to the same domain from which they emerge, i.e that the relation of weak emergence is an “intra-domain” relation and thus not imply ascension on a given branch.
The general ontological model has thus been sketched.
3 Emerging Facts and Facts About Emergence
In this section I will be concerned with the ontological status of “facts about weak emergence”, e.g. the fact F that a certain property/state \(\phi \) of a system S weakly emerges on a set of natural laws L and initial/boundary conditions x. For instance, L may be a system of differential equations modeling the behavior of a natural system S, x may be the set of initial conditions and parametric assignments, and \(\phi \) may be a certain pattern (e.g. an attractor) in the state space of S. Another interesting example is the case of cellular automata, where L represents the set of transition rules, x contains the parameters of the cell grid (e.g.dimensions, size, etc...) and the initial conditions, and \(\phi \) is a certain pattern of cells (e.g. a glider in Conway’s Game of Life).
Before disputing the ontological status of facts about emergence, it is worth clarifying the structure of this sort of facts in terms of the relevant ontological categories. In general, emergence may be seen as a relation between a property (i.e. the emerging feature), a certain set of natural laws identifying an ontological domain, and certain specific conditions identifying both the particular system having the emerging property and its initial configuration. For instance, a certain fluid dynamical pattern (e.g. a tornado-shape) is said to emerge from the laws of statistical mechanics in certain conditions which identify a certain system (e.g. a particular stream of air in a particular environment) in a given initial conditions. Hence, emergence (weak or strong) ends up being a three-place relation between a property, a set of natural laws, and a set of conditions defining and situating the system at issue. For this reason we may say that in a fact about emergence there is the emerging component (i.e. the emerging property), a nomological component (i.e. the natural laws from which the property emerges), and a situational component (i.e. specific parameters of the system plus boundary or initial conditions). There seems to be no particular reason to take emergence as a non-homogeneous relation, i.e as a relation holding of entities of different ontological categories. Hence, we may consider emergence as a three-place relation between facts or state of affairs, being the first argument-place of the relation occupied by the fact that a certain natural system presents a certain feature, the second argument-place by a conjunction of “nomic facts”, i.e. the occurrence of certain universal regularities, and the third argument-place by particular facts identified the natural system at issue and the initial/boundary conditions. According to these remarks, we represent a fact about emergence by means of the following symbolism:
which stands for ‘the fact that the system S has a property \(\phi \) weakly emerges on natural laws L under specific conditions x’.
An additional remark on the nomological component is needed. Notice that the proposed account is completely neutral with respect to a particular metaphysical conception of natural laws, for I have not specified the nature and the internal structure of the nomological component L. For instance, according to the universalist view (Armstrong, 1983), L should be interpreted as a state of affairs composed of a necessitation relation between Aristotelian universals; alternatively, L may be taken to be a conjunction of Humean regularities (Lewis, 1994) or a set of primitive dispositions (Mumford, 1998). What matters to me is that the proposed account of the emergence relation does not explicitly rely on any particular view on the ontological status of laws of nature and it is compatible with the main existing proposals.
Having set the general form and categories of the constituents of facts about emergence we are ready to discuss the ontological status of such facts.
According to the previous reasoning, the fact \(\phi \) should be considered as belonging to the ontological domain picked out by natural laws L, which we denote by ‘\(D_L\)’. How should we classify \({\mathbf {E}}_w(\phi , L, x)\) in our ontological model? In which domain and on which branch of the tree-like structure should it be located? According to previous remarks, there are three possibilities:
-
(i)
\({\mathbf {E}}_w(\phi , L, x)\) belong to the domain \(D_L\);
-
(ii)
\({\mathbf {E}}_w(\phi , L, x)\) supervenes on L but L does not supervene on \({\mathbf {E}}_w(\phi , L, x)\), i.e. \( {\mathbf {E}}_w(\phi , L, x)\) occupies a position higher than \(D_L\) on the same branch;
-
(iii)
\( {\mathbf {E}}_w(\phi , L, x)\) is ontologically independent from L, i.e. the domain of \( {\mathbf {E}}_w(\phi , L, x)\) and \(D_L\) are segregated in the tree-like structure.
Case (iii) may be ruled out by considering that weak emergence—as any sort of emergence—is commonly taken to entail nomological supervenienceFootnote 1 (Wilson, 2021, p. 41), (Mclaughlin, 2008); (Kim 2006). More precisely, if \(\phi \) weakly emerges on L under conditions x, then \(\phi \) is a metaphysical consequence of L and x; in other words, in all possible worlds in which the natural laws L hold, \( {\mathbf {E}}_w(\phi , L, x)\) is the case (provided that the conditions x are satisfied by the system S). Yet ontological independence requires that there is at least one possible world in which the natural laws L are true, conditions x are satisfied, and \({\mathbf {E}}_w(\phi , L, x)\) does not occur, i.e. the system S does not reach the state \(\phi \) under the conditions specified in x; clearly this is incompatible with the nomological supervenience of \(\phi \) on L plus x, and thus with the hypothesis that \(\phi \) weakly emerges on L and x.
Case (ii) needs a wider discussion, for it calls into question additional metaphysical notions. If \({\mathbf {E}}_w(\phi , L, x)\) supervenes on L (i.e. if the property of being a weakly emergent state of S supervenes on all the nomological properties of S) without belonging to \(D_L\), then \({\mathbf {E}}_w(\phi , L, x)\) is neither explainable in terms of nor deducible from L and x. In other words, case (ii) represents the view that facts about weak emergence represent cases of mere supervenience on their nomological basis, i.e. although they are determined by such nomological basis, they are not reducible to it.
Is mere supervenience (i.e. supervenience without reduction) of facts about weak emergence plausible? It seems that some metaphysical considerations on the nature of the relation of weak emergence forces us upon a negative answer. Indeed, it seems that the relation of weak emergence must be taken to be an internal relation between facts, i.e. a relation having application conditions completely determined by the intrinsic properties of its relata. In other words, once facts X, Y, Z are fixed, it is thereby determined whether the higher-order fact \({\mathbf {E}}_w(X,Y,Z)\) occurs or not, i.e. given a state/property of a natural system plus laws and conditions “governing” its behavior, the weak emergence of the given state is already decided. Thus there seems to be compelling reasons to conceive the ternary relation of weak emergence as an internal relation.
Internal relations are traditionally taken to be reducible to the same ontological domain of their relata (Armstrong, 1978, p. 86), (Campbell, 1990, p. 90): indeed, they are considered as a sort of “ontological free lunch”. Moreover, the relata of the relation of weak emergence constituting the fact \({\mathbf {E}}_w(\phi , L, x)\) are all members of the domain \(D_L\) (in virtue of the fact that weak emergence is an “intra-domain” relation). As a consequence, the fact \({\mathbf {E}}_w(\phi , L, x)\) cannot be a case of mere supervenience on the laws L, for, being a case of an internal relation holding of members of \(D_L\), it must be a fact belonging to the domain \(D_L\). Therefore, case (ii) cannot be possible and the fact \({\mathbf {E}}_w(\phi , L, x)\) must belong to the domain \(D_L\). By (DI), \({\mathbf {E}}_w(\phi , L, x)\) must be explainable by and deducible from the fundamental laws of the domain \(D_L\), i.e. the set of natural laws L.
If the presented argument is sound, then of the three possibilities listed at the beginning of this section only (i) actually occurs. Now there are two “subcases” which shall be discussed: either (i.1) \({\mathbf {E}}_w(\phi , L, x)\) is merely resultant from the natural laws L under the conditions x, or (i.2) \({\mathbf {E}}_w(\phi , L, x)\) is itself a weakly emergent fact from the laws L under the conditions x.
Assume (i.1). If \({\mathbf {E}}_w(\phi , L, x)\) is merely resultant form L and x, then there is a compressible explanation of the occurrence of \({\mathbf {E}}_w(\phi , L, x)\) under conditions x in terms of the natural laws L. As a consequence, there is a “derivational shortcut”, i.e. a deduction computationally shorter than the simulation of the effective dynamics of the system, that allows to derive \({\mathbf {E}}_w(\phi , L, x)\) from L and x. Notice that the following principle seems to be hardly doubtable:
Factivity of Weak Emergence (FWE)
If \(\phi \) weakly emerges from natural laws L under conditions x, then in every nomologically possible world, if conditions x obtain, then S reaches the state \(\phi \).
In other words, if a state \(\phi \) occurs by weakly emerging on L and x, then, given L and x, \(\phi \) occurs; this seems to hold out of analytic necessity. Hence, if we have a compressible deduction of \({\mathbf {E}}_w(\phi , L, x)\) from L and x, we may have a compressible deduction of \(\phi \) from L and x, for it suffices to add an application of Modus Ponens between \({\mathbf {E}}_w(\phi , L, x)\) and (FWE). Thus we have shown that if \({\mathbf {E}}_w(\phi , L, x)\) is merely resultant from natural laws L under conditions x, then also \(\phi \) is merely resultant from the same nomological base. Yet this contradicts our hypothesis that \(\phi \) is weakly emergent from L and x, i.e. that there is no compressible explanation of \(\phi \)’s occurrence in terms of L and x. Therefore, if only one between (i.1) and (i.2) must be the case and (i.1) is inconsistent with our hypotheses, it follows that (i.2) must be the case. We have thus come the first important conclusion of our argument: facts about weak emergence are weakly emergent facts, i.e. if a certain fact X weakly emerges, under suitable conditions Z, from a given set of natural laws Y, then the fact that X emerges from Y under Z is in turn a weakly emergent fact from the same set of natural laws and under the same conditions.
In the next section I will show how to interpret this conclusion and discuss some of its consequences.
4 Weak Emergence and Computational Equivalence
We have seen that if a certain metaphysical picture of reality is true, facts about weak emergence are themselves weakly emergent facts. In this section I will try to understand in details some consequences of this conclusion and develop further remarks on the computational aspects of natural systems.
Firstly, we have to recall that a fact \(\phi \) weakly emerges on natural laws L under conditions x iff
-
(i)
\(\phi \) is “generatively derivable” from L and x (Generative theoretical reduction);
-
(ii)
Every derivation of \(\phi \) from L and x is of computational complexity greater than or equal to the computational complexity of a simulation of the dynamics of the system S as defined by natural laws L and conditions x, i.e. the derivation of \(\phi \) from L and x is “incompressible”. (Incompressibility)
Clause (i) has to do with the fact that a derivation of a weakly emergent state from its nomological basis must be based on the web of causal interactions among S’s micro-constituents. As mentioned in the Introduction, a perspicuous way to understand clause (ii) of this definition is to consider the case in which a certain fact \(\psi \) is not emergent, i.e. is a merely resultant state of the system S. As we have seen, in this case there is a compressible derivation of \(\psi \), i.e. a computational shortcut that allows to determine that the system S reaches the state \(\psi \) under conditions x in a number of steps lesser than the length of the simulation of S’s dynamics. A classical example is that in which we have a recursively decidable procedure—e.g. a general algebraic formula—to determine whether a possible state \(\psi \) of S is reached under conditions x after some interval of time (or a some number of time-steps). Hence, the problem of determining the occurrence of a state \(\psi \) is just a matter of applying the same recursively decidable procedure, irrespective of the “generative dynamical history” through which the system S must go to reach \(\psi \). On the contrary, if \(\phi \) is a weakly emergent state, then there is no recursively decidable procedure at hand and we must go through the entire generative dynamical history of the system S until, possibly, S reaches the state \(\phi \) (and if there’s no trajectory that lead to \(\phi \) under conditions x, then our simulation will never halt). In this case, there is no way to compress the dynamical history of S which must always be “inspected” step by step, “crawling the causal web”.
With these ideas in mind let’s try to understand the meaning of the statement that \({\mathbf {E}}_w(\phi , L, x)\) weakly emerges from L and x, i.e.
By the previously given definition, (2) is the case iff:
-
(iii)
\({\mathbf {E}}_w(\phi , L, x)\) is generatively derivable from L and x (Generative theoretical reduction);
-
(iv)
Every derivation of \({\mathbf {E}}_w(\phi , L, x)\) from L and x is of computational complexity greater than or equal to the computational complexity of a simulation of the dynamics of the system S as defined by natural laws L and conditions x. (Incompressibility)
Combining clauses (i) and (ii) with (iii) and (iv):
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(iii.1)
The fact that \(\phi \) is generatively derivable from L and x is in turn generatively derivable from L and x (Generative theoretical reduction of generative theoretical reducibility);
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(iii.2)
It is generatively derivable from L and x that the generative derivation of \(\phi \) from L and x is incompressible. (Generative theoretical reduction of incompressibility)
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(iv*)
Both the derivation in (iii.1) and that in (iii.2) must be incompressible (Incompressibility)
Let’s examine these new clauses. Firstly, it is important to point out that the clauses are a bit obscure, for I have not presented a detailed explication of the meaning of ‘generative derivation’. As a preliminary approach, I propose to understand these terms by means of examples neglecting for the moment Bedau’s idea of generative explanation and considering just a formal notion of derivation. Clause (iii.1) may be considered in the context of the formal system of Peano arithmetic (henceforth PA). It is well known that it is possible to define in the formal language of PA a proof predicate and thus the predicate ‘being a theorem of PA’. Thus in formal systems containing a certain significant “amount” of arithmetic, it is possible to prove that certain formulas are theorems. This is roughly what clause (iii.1) requires when the natural laws L and the conditions x are taken to be axioms of a formal system L and L is taken to be “strong enough” to admit a codifying system for its formulas and a proof predicate as in the case of PA. Being ‘\(\text {Proof}_{\mathbf {L}}\)’ the proof predicate of L and ‘\(\langle ... \rangle \)’ a schematic representation of the codifying operation for L’s formulas, clause (iii.1) may be represented as follows:
Therefore, if the behavior of the system S may be modeled by means of an axiomatic system L, by (iii.1) L must be “strong enough” to have a definable proof predicate. Clearly, as a consequence of (iv*), the set of elements satisfying the formula ‘\(\exists {z} \text {Proof}_{\mathbf {L}}(z,y)\)’ will not be recursively decidable. Indeed, if there were a recursively decidable procedure to determine whether a certain formula describing a certain state of the system S is a theorem of L or not, then the fact that S reaches such state under conditions x would not be derivable exclusively by simulation, for there would be a general algorithm to solve this problem irrespective of the particular “trajectory” S have taken to reach the state at issue.
How should we understand clause (iii.1) in more concrete cases of generative derivations within “real systems” and not formal axiomatizations? We may consider a range of systems whose behavior is not conveniently described by axiomatizable mathematical models (e.g. sets of differential equations) (Wolfram, 2002, pp. 8–9) but rather in terms of rules for state transition; consider, for instance, systems that can be interestingly modeled by means of cellular automata (or cellular automata themselves as systems under study). As already mentioned, according to Bedau (2008a) the case of cellular automata as theoretical descriptions of natural systems seems to be an eminent case study for the notion of weak emergence.Footnote 2 For this sort of systems the notion of ‘generative derivability’ mentioned in clause (iii.1) may be interpreted in the context of Stephen Wolfram’s “new kind of science”. Wolforam’s approach may be seen as an alternative to the traditional modeling of dynamic system theory; the fundamental idea is that natural processes may be seen as computations and dynamic systems as computational devices. Wolfram presents this approach by formulating the so-called Principle of Computational Equivalence:
“But on the basis of many discoveries I have been led to a still more sweeping conclusion, summarized in what I call the Principle of Computational Equivalence: that whenever one sees behavior that is not obviously simple—in essentially any system—it can be thought of as corresponding to a computation of equivalent sophistication.” (Wolfram, 2002, p. 5)
“The key unifying idea that has allowed me to formulate the Principle of Computational Equivalence is a simple but immensely powerful one: that all processes, whether they are produced by human effort or occur spontaneously in nature, can be viewed as computations.” (Wolfram, 2002, p. 715)
The idea is essentially simple and yet has significant philosophical consequences. Firstly, if natural systems are to be conceived as computational devices and if the concept of computability is delimited by the paradigm of Turing machines, then the highly complex behavior of certain natural systems may be often explained by a set of few simple rules, as well as a Turing machine defined by few lines of code may perform incredibly complex operations and calculations. In other words, the principle of computational equivalence may provide an interesting explanation of how complexity arises from simplicity. Secondly, given that physical systems are viewed as “natural Turing machines”, the complexity of their behavior seems to have an upper bound, which is represented by the computational powers of a universal Turing machine, i.e. by what in the tecnical jargon is called “computational universality”. In the third place, systems characterized by remarkable complexity—i.e. systems endowed with computational universality—cannot be compactly described and predicted, for there is no system (e.g. in the form of an intelligent being or of an electronic computer) capable of “compressing” or “outsmarting” the computational complexity of the system under study (Wolfram, 1994), (Wolfram 2002, pp. 737–750).
However, the applicaion of Wolfram’s Principle of Computational Equivalence is still in need of further clarifications. Computational devices may be described in terms of an input, a program describing the particular function to be applied, and the output. The input and the program may be conceived as encoded in the initial conditions of the system, whereas the output is represented by a certain state reached during the dynamical evolution of the system; the computational specifics—such as fundamental description of the particular Turing machine the system realizes and its physical limitations—are defined by both the natural laws governing the system and its specific parameters.
Before using the principle of computational equivalence to spell out clauses (iii.1) and (iii.2), it is important to introduce an adequate conceptual apparatus supported by a useful notation. Given that systems under study are to be conceived as Turing machines, I will henceforth denote by ‘S’ the characteristic function of the system S viewed as a Turing machine whose general rules and computational powers are encoded in the set of natural laws L (instantiated with the specific values of S’s characteristic dynamical parameters). Since the inputs and outputs of the machine are states of the system S, S’s micro-configurations are to be conceived as strings of our “physical” symbolism, i.e. the way the system S may encode information.
Since a natural system may be capable of computing the values of a plurality of functions (possibly being a universal Turing machine, capable of computing every partial recursive function modulo its physical limitations), the fact that it may be suitably set to compute a specific one must be explicitly expressed. In order to do this, we must express the fact that specific initial conditions encode both the function that we are willing to compute (i.e. the program) and the input being the argument of such function. The main states of the machine may also be rendered as particular states/state transitions of the physical system S. We do not need to require the system to effectively halt when it reaches the state encoding the corresponding output.Footnote 3 All these features will be expressed using the following notational conventions:
-
1.
As previously mentioned—with a little abuse of notation—the expression ‘S’ will denote the characteristic function of the system S viewed as a Turing machine, where \(S(x, y)=1\) if the system S reaches the state y from initial conditions x, otherwise \(S(x, y)=0\);
-
2.
The expression ‘\(\langle S; x, y \rangle \)’ denotes the string—in the “physical code” of states of the system S—representing the entire step-by-step derivation of y from x (viz. the step-by-step simulation of S’s own dynamics from x to y);
-
3.
‘\(\overline{\alpha }\)’ denotes the codification of the expression \(\alpha \) of a certain symbolic language into a state of the system S;
-
4.
‘|x|’ denotes the length of the string x;
-
5.
‘C(x)’ and ‘C(x|y)’ denote respectively plain Kolmogorov complexity of string y and plain conditional Kolmogorov complexity of x given yFootnote 4;
A last piece of notation is needed. As previously mentioned, we will consider the case in which our system S may be viewed as a Turing machine able to compute a certain function, say h. In this case, we need to encode in the input of S both the initial state—which codifies the argument of the function h to be computed by S—and the sequence of instruction which corresponds to the computation of h. Being u the argument of the function h we are willing to compute through S, the complex input S receives will be expressed as follows:
Hence, the fact that the system S computes the value v a certain function h in correspondence of the argument u will be expressed as follows:
(4) says that whenever the system S is set to initial conditions identified as ‘\(\overline{[h; u]}\)’ which stands for a certain array –i.e. a configuration of fundamental elements of S—encoding both the function to be computed h and the input u, after a certain interval of time (or after a certain amount of discrete steps), S reaches a certain state which encodes the result \(v= h(u)\). The notation used in (4) may be generalized by considering h as an n-place function of arguments \(u_1,..., u_n\):
The last concept I have to introduce to discuss clauses (iii.1), (iii.2), and (iv*) is that of an incompressible derivation of a state of the system S. To this end, I define the characteristic function of the incompressibility predicate for the system S as follows:
(6) says that the derivation within S of a state y from initial conditions x is incompressible iff the length of the sequence of steps of S’s dynamics that takes from x to y is less than or equal to the plain conditional Kolmogorovian complexity of y given x. Recall that the plain conditional Kolmogorovian complexity of y given x represents the additively optimal length of a program that describe y having x as input. As a consequence, incompressibility defined in (6) means that the process by means of which S “computes” a certain state y from initial conditions x cannot be optimized or shortened in a significant way, for it is already shorter than or identical in size to the most optimized computational effort to describe y given x. Roughly speaking, two states x and y of S are connected by an incompressible dynamical trajectory in S’s phase space iff when this trajectory is seen as a computational process, it already represents the most efficient way of computing y given s.
We are now in position to provide an interesting interpretation of (iii.1), (iii.2), and (iv*) using Wolfram’s Principle of Computational Equivalence. I will start by re-formulating these clauses in the light of the introduced notions/notations:
- (iii.1*):
-
The system S is capable of computing its own dynamics from x to \(\phi \), i.e.
$$\begin{aligned} S(\, \, \overline{[S; x \phi ]} \,, \, \overline{S(x, \phi )}\, \,) = 1 \quad \text {iff} \quad S(x, \phi )=1 \qquad (7) \end{aligned}$$ - (iii.2*):
-
The system S is capable of computing the incompressibility of its trajectory from x to \(\phi \), i.e. is capable of computing the value of the characteristic function of the incompressiblity predicate relative to S:
$$\begin{aligned} S(\, \, \overline{[I_S; x \phi ]} \,, \, \overline{I_S(x, \phi )}\, \,) = 1 \quad \text {iff} \quad I_S(x, \phi )=1 \qquad (8) \end{aligned}$$ - (iv**):
-
Both computations descibed in (7) and (8) are incompressible;
Clause (iii.1*) represents the re-formulation of clause (iii.1) in the light of the interpretation of natural processes as computations via the principle of computational equivalence. The expression (7) basically says that the system S is capable of self-simulation, i.e. is capable of encoding its own states and perform operations on them in such a way that given two states x and \(\phi \), S returns an output encoding an affirmative answer to the problem of the existence of a trajectory in its phase space that goes from x to \(\phi \).Footnote 5 When S is conceived as a Turing machine, S’s characteristic function is the counterpart of the “theoremhood” predicate occurring in (3). As usual, we may think of S as a cellular automata whose initial grid/string partly encodes the states \(x, \phi \) and partly encodes an operative description of the function S. The idea of “self-simulation”, though, shall not induce misunderstandings: the machine S cannot compute its own characteristic function, i.e. there is no recursively decidable procedure that can be implemented in S that tells us whether the system S starting from x reaches the state \(\phi \). The only way in which (7) may be true is through simulation, i.e. the only way in which the system S may “prove” that there is a trajectory from x and \(\phi \) is by a “wait-and-see” procedure by means of which S simulates its own dynamical trajectories of encoded states. We may interpret this fact also in terms of incompressibility, thus showing how clauses (iii.1*) and (iv*) are jointly met: even if the system S may be capable of simulating itself, it cannot predict its own behavior through a recursively decidable shortcut and every attempt of determining if a certain state \(\phi \) is reachable under given initial conditions x cannot be computationally more economic than the actual dynamical history of the system between x and \(\phi \). In other words, the incompressibility of self-simulations is a consequence of the logical limits of Turing machines, i.e. of the impossibility for a Turing machine of solving its own version of the halting problem.
Consider now clause (iii.2*). (8) says that S must be able to determine the incompressibiilty defined in (6), i.e. it must be able of “proving” that trajectories leading to weakly emergent states are incompressible. This may be done by performing a search on all possible descriptions of the state \(\phi \) given the state x, then comparing each description with the simulated dynamics \(\langle S; x, \phi \rangle \), and finally returning a positive answer if the simulated dynamics has already the minimum length. Nevertheless, there are two main difficulties regarding the possibility for S of satisfying this clause. Firstly, we want S to correctly determine the incompressibility of a given dynamical trajectory and it is easy to imagine a way in which it may fail. Suppose that the trajectory \(\langle S; x, \phi \rangle \) is compressible, i.e. there is a method of computing \(\phi \) given x of complexity lower than \(\langle S; x, \phi \rangle \); assume also that any possible way of optimizing the computation of \(\phi \) given x is not computable by S and that any method computable by S has complexity greater than or equal to that of \(\langle S; x, \phi \rangle \). Under these assumptions, S will falsely output a positive answer to the problem of deciding whether \(\langle S; x, \phi \rangle \), simply because is not capable of compressing this trajectory, i.e. is computationally blind regarding any possible optimization. This difficulty may be overcome assuming that S is a universal Turing machine, i.e. is capable of computing every partial recursive function. This fact will ensure that if there is a computational method to describe \(\phi \) given x more “efficient” then the effective dynamical trajectory of S, then S will be able to perform it.
However there is a second much harder difficulty according to which clause (iii.2*) is inconsistent. As a matter of fact, the incompressibility predicate is undecidable, thus the set of all incompressible strings is not recursively enumerable and the incompressibility of a computational process cannot be decided even by the “wait-and-see” approach of simulations. It is not hard to understand the reason of such limitation: the search for possible optimizations of \(\langle S; x, \phi \rangle \) may correspond to an infinite task, for only when S has effectively “examined” the absolute totality of possible ways of computing \(\phi \) given x, S may make a sound decision about the incompressibility of \(\langle S; x, \phi \rangle \); and yet there seems to be infinite computable functions having \(\phi \) as value in correspondence of the argument x.
It seems that we have come to a dead end. In the next section I will examine possible ways out of the presented difficulty and I will argue that none is satisfactory, concluding thus that the attempt of cotextualizing Bedau’s notion of weak emergence in a more general ontological model leads unavoidably to an aporetic situation.
5 The Paradox of Weak Emergence
Time to take a stock. We started from Kim’s general metaphysical picture of reality as a tree-like structure of ontological domains and we have shown that this general model combined with the idea that weak emergence is an internal relation implies that facts about weak emergence are weakly emergent facts. We have interpreted this latter conclusion in the light of Wolfram’s Principle of Computational Equivalence and thus we have derived a series of computational features of systems having weakly emergent states. In particular, the fact that systems instantiating the phenomenon of weak emergence should be capable of simulating their own dynamics (clause (iii.1*)), of showing the incompressibility of the trajectories that lead to weakly emergent states (clause (iii.2*)), and that both results are achieved by means of incompressible derivations (clause (iv**)). At this point we have found ourselves in an apparently contradictory situation: the capability of a system of computing the incompressibility of its trajectories seems to contradict the fact that the characteristic function of the incompressibility predicate is not computable, i.e. the set of incompressible strings (including encoded dynamical trajectories) is not even recursively enumerable. I call this the problem of undecidability of incompressibility.
It is reasonable to consider the train of thought that lead to an inconsistent conclusion unsound, either in its metaphysical part—i.e. the argument showing that facts about weak emergence are weakly emergent facts—or in its computational part—i.e. the interpretation of (iii.1), (iii.2), and (iv*) in the light of the principle of computational equivalence. Since the presented arguments seem to be valid, there must be a problem with at least one of their premises. For this reason, I will list the main assumptions introduced and discussed in this paper in order to investigate possible grounds to reject at least one of them:
- (P\(_1\)):
-
Ontological model: reality is a tree-like structure of ontological domains partially ordered by the relation of non-mutual supervenience;
- (P\(_2\)):
-
Domain identification: a property/state of affairs \(\phi \) belongs to the ontological domain D iff \(\phi \) may be derived from and explained by the fundamental laws of D;
- (P\(_3\)):
-
Factivity: weak emergence is factive, i.e. if a state \(\phi \) weakly emerges from natural laws L under conditions x, then given conditions x, \(\phi \) occurs (in all nomologically possible worlds);
- (P\(_4\)):
-
Internality: weak emergence is an internal relation between facts;
- (P\(_5\)):
-
Reductionism about internal relations: If intrinsic facts about the relata \(x_1,...,x_k\) of a certain k-ary internal relation R belong to the same ontological domain D, then the fact that R holds of \(x_1,...,x_k\) also belongs to the domain D;
- (P\(_6\)):
-
Computational equivalence: dynamical systems can be viewed as Turing machines where their initial conditions encode the input and the program, their evolution represent the computational process, and the reached state after a given time represent the output.
Let me do a bit of classification. The premises (P\(_1\)), (P\(_2\)), and (P\(_5\)) represent the metaphysical background of the proposed argument, i.e. they shape the ontological model, the big picture representing the context of the discussion; (P\(_3\)) and (P\(_4\)) are specifically about the notion of weak emergence, and (P\(_6\)) represents the computational background. Clearly, as it is common in philosophical discussions, each assumption of this list is controversial to a certain extent, and someone may have reasons to reject it or, at least, see it as suspicious. To my mind, the most rational attitude here seems to adopt a principle of minimal mutilation, in the sense that it would not seem plainly rational to reject some general metaphysical or computational principles due to undesirable bearings on a specific notion of emergence. Therefore, a detailed discussion of the reasons one may have to reject (P\(_1\)) or (P\(_5\)) will not be my priority in this paper. Moreover, I have serious concerns regarding the comprehensibility of the concept—or the family of concepts—of metaphysical emergence outside the context of a multi-layered or multi-branched picture of reality: in fact, authors that reject a multi-level ontology tend to reject the meaningfulness or the fruitfulness of the concept of emergence altogether (consider (Ladyman & Ross, 2007, pp. 45–57) as a paradigmatic case)
(P\(_2\)) seems to have a more direct impact on the discussion of the concept of weak emergence, thus it is worth examining its possible consequences and interpretations in more detail. Notice that (P\(_2\)) has been formulated as biconditional between membership in ontological domain and theoretical reduction. In other words, (P\(_2\)) states that theoretical reduction to certain natural laws is necessary and sufficient for ontological reduction to the ontological domain “governed” by those laws. The main problem with this assumption is that it seems to rule out non-reductive physicalism. Non-reductive physicalism—roughly speaking—corresponds to the view according to which all facts are physical and yet not all of them are reducible to the physical base; as a consequence, there are physical facts which cannot be fully explained or predicted by physical laws. At first glance, non-reductive physicalism denies that ontological reduction—e.g. the membership to the physical realm—implies theoretical reduction—e.g. explanation and prediction by virtue of physical laws. We may use non-reductive physicalism to block our inference from the idea that facts about the weak emergence of a property \(\phi \) from natural laws L belong to the domain identified by L to the idea that facts about weak emergence must be theoretically reducible to the nomological basis L. For if there are facts that belong to a certain ontological domain and are not derivable from the laws of that domain, then it is possible for facts about weak emergence to be merely supervenient on their nomological basis.
It is not clear what to do with non-reductive physicalism in the context of Kim’s tree-like ontological model.Intuitively, non-reductive physicalism—insofar as it is a form of physicalism—should be understood as a thesis about a singled-domain reality; in other words, non-reducible physical facts do not constitute a distinct ontological domain. Clearly we may jettison an ontological model consisting of multiple ontological domains, embrace non-reductive physicalism, and solve the contradiction in (iii.2*). However, there are two main difficulties related to the effectiveness of this strategy: the acceptance of non-reductive physicalism itself and its relation with Bedau’s notion of weak emergence. Firstly, non-reductive physicalism may be rejected on independent grounds—e.g. Kim’s exclusion argument (Kim, 1989). Yet one may object that the concept of emergence has its significance precisely within the framework of non-reductive physicalism and thus that if we accept a certain idea of emergence as a fruitful tool to understand nature, then there is no point in rejecting non-reductive physicalism. I have nothing to object to this remark as a general qualification about the concept of strong emergence; however, Bedau’s notion of weak emergence has been introduced and developed quite independently of —and sometimes in opposition to—the big metaphysical picture of non-reductive physicalism. As previously mentioned (see Bedau’s quotation in section 2) Bedau’s notion of weak emergence stems from the idea that emergence and reduction may be not only be consistent, but logically connected. Therefore, the notion of weak emergence under discussion is expected to be compatible also with physicalism in its reductive version. However, if the only way to avoid inconsistencies (as in (iii.2*)) in Bedau’s concept of weak emergence is to spouse non-reductive physicalism (thus rejecting the left-to-right implication in (P\(_2\))), then we should conclude that Bedau’s notion of weak emergence is incompatible with reductive physicalism....which is quite undesirable and unexpected from a philosophical view whose main virtue is the conciliation between emergence and reduction! For these reasons, in spite of the fact that non-reductive physicalism may be acceptable on different grounds, I do not believe that a rejection of (P\(_2\)) is a valid strategy to avoid the contradiction.
Discussing the possibility of dropping (P\(_3\)) is out question due to its “analytic flavor”. Assumption (P\(_4\)) is directly related to the metaphysical interpretation of the concept of weak emergence and thus is extremely relevant to the present discussion. The provided justification for endorsing (P\(_4\)) is that given a certain state \(\phi \) of the system S, a nomological basis L, and conditions x, the truth value of the sentence ‘State \(\phi \) weakly emerges on the nomological basis L under conditions x’ seems to be thereby determined. I do not see any reason to put in doubt (P\(_4\)), yet given its crucial role in the argument and its strict relevance, it will be interesting to attempt the philosophical exercise of imagining possible circumstances in which it fails. An interesting question would thus be: what else would be needed to determine the weak emergence of a certain state besides its nomological basis and the given conditions? To answer this question we must conceive a possible world in which laws L hold, conditions x obtain, and yet \(\phi \) does not weakly emerge. Given that \(\phi \) is a metaphysical consequence of L and x, the case in which \(\phi \) simply does not occur must be discarded. Therefore, we must imagine a situation in which \(\phi \) occurs, however either it is undecided whether its occurrence must be seen as a weak emergence or it simply does not emerge, i.e. it is a merely resultant state. How this could be possible? Recall that weak emergence is defined as underivability from certain laws and conditions except by simulation; perhaps laws and conditions might not determine the underivability except by simulation when the way in which a certain state is derived from the given nomological basis depends upon the performances of another system, i.e. an observer system or a simulator. More specifically, the fact that a certain state of a system S weakly emerges on a certain nomological basis is relative to another system \(S'\) which, at least in principles, defines the derivability conditions of the state at issue, i.e. the perspective from which it is said that there is a computational shortcut or not. In other words, according to this view, if a state \(\phi \) of the simulated system S weakly emerges on L or not is something that can be settled only with respect to the computational powers of a simulating system \(S'\); as a consequence, the relation of weak emergence is not internal for its holding also depends on “external conditions”, i.e. on the system that defines the deliverability conditions. I call this view perspectivism about weak emergence (henceforth just ‘perspectivism’).
Perspectivism is an extremely uncomfortable position if we endorse Bedau’s definition. This because it opens the way to the idea that weak emergence is a relative and epistemic notion, thus unable to carve the structure of an ontological domain at its joints. Indeed, a logical consequence of perspectivism is that underivability except by simulation is a condition which may be determined only relatively to the system which “performs or defines the specifics of the simulation”. In particular, if we think of us humans with our culture and technology as main observer systems, then the notion of weak emergence will irremediably turn out to be anthropocentric and epistemic. However, the main problem with perspectivism is not its being unpalatable to the emergentist, yet the fact that it seems to contradict some uncontroversial facts about the computational profile of natural processes. More specifically, incompressibility is commonly taken to be an intrinsic property of computational processes. Clearly the conditions under which a certain process may effectively be computationally compressed depend upon the physical system that realizes the data compression; nevertheless, Bedau’s notion of underivability except by simulation is to be considered in principles and not as a concrete implementation. The fact that the dynamics of a certain system is incompressible seems to be independent on the fact that a certain system is effectively able to determine it. Hence I deem the rejection of (P\(_4\)) based on perspectivism as unconvincing.
However, there seems to be a more radical way of questioning (P\(_4\)), namely the denial of the existence of a relation of weak emergence as an extra ontological ingredient over and above the system’s microelements. More precisely, one may object that the weak emergence of a certain state \(\phi \) is simply a property of the way \(\phi \) has been derived within a description of the system S; as a consequence, there is neither a “real relation” of weak emergence nor an extra fact about this relation holding between the state \(\phi \), conditions x, and the nomic component L. In absence of a real emergence relation and real facts about emergence there is no possibility of self-reference, and thus no paradox of weak emergence.
Should we be anti realists regarding Bedau’s relation of weak emergence and facts about this relation? The correct answer depends upon the role we are open to ascribe to explanations framed in terms of the Bedauvian notion of weak emergence. If these explanations are purely epistemological, i.e. if the only work they do is to make us understand the complexity of certain natural systems, then we may be open to reject any specific ontological import associated with them. In this case, Bedau’s definition is not interpreted as a way of tracking a real relation through a feature of the system’s description. Yet if, on the other hand, we attribute explanatory value to Bedau’s notion of weak emergence in virtue of the fact that it also describes the structure of reality in itself, then it seems that we should take Bedau’s definition with ontological seriousness. Bedau himself seems to suggest this latter view:
“Weak emergent phenomena might also have ontological, non-epistemological aspects. In particular, the distinctively incompressible micro-causal explanations of weak emergence presumably are due to a distinctively incompressible form of micro-causal structure in reality. It is presumably not an accident that one sort of micro-causal structure is incompressible and another sort is compressible.” (Bedau, 2008b, p. 451)
It is precisely in virtue of the existence of a real dynamic structure of micro-causal interactions of enough complexity that we say that a certain state weakly emerges from a given initial micro-configuration. As a consequence, eliminativism about the relation of weak emergence and states of affairs involving such relations does not seem to be consistent with Bedau’s account.
Consider now (P\(_5\)).This assumption plays the role of defining the metaphysical background of my argument, like (P\(_1\)) and (P\(_2\)) and is concerned with the ontological status of facts involving internal relations. More specifically, I take facts about internal relations to belong to the same domain of their relata (assuming that the relation at issue is homogeneous, i.e. all of its relata belong to the same ontological domain). Recall that the main debate about the ontological status of internal relations is commonly represented as a contraposition between eliminativism and reductive realism. Hence, the idea that facts about internal relations belong to an ontological layer distinct from that of intrinsic facts about their relata does not seem to be a theoretical option.
Still, one may be open to the fact that certain special internal relations may not be reducible to the domain of their relata. For instance, one may consider weak emergence as a general metaphysical relation shaping the fundamental structure of reality as a whole; as a consequence, the relation of weak emergence should be considered as having relata of all possible ontological domains and facts about weak emergence as meta facts which merely supervene upon all facts of the natural world.
Surely Bedau’s relation of weak emergence has a metaphysical value and contributes to our understanding of the structure of facts. However, there is another important aspect of Bedau’s definition that deserves attention: a strong form of naturalism. According to Bedau, weak emergence is a property of natural systems which may be detected within and characterized in terms of the system’s physical behavior, i.e. according to the level of complexity of micro-causal interactions. For this reason, Bedau does not need any metaphysical conceptual apparatus to define and apply his notion of weak emergence. Moreover, in Bedau (2013), while considering interesting examples from synthetic biology, it is argued that weak emergence is even engineered to produce systems with desired physical properties; for instance, gene refactoring is presented as an example of a case in which a natural system is manipulated in such a way that certain configurations of interest—which were originally weakly emergent—are turned into resultant properties in order to be controllable and programmable. Here the emergence of certain features is something on which we can intervene and the fact that certain configurations are emergent is itself object of physical interaction. Notice the strength in the kind of naturalism of Bedau’s approach: the notion of weak emergence is not just scientifically informed or abstractly continuous with natural science; it may be generically characterized within the scientific language, roughly in the same way as many other concepts of non-linear system theory. Moreover, it is detectable as a property of the behavior of empirical models, even if only by incompressible means. And, more importantly, the identification of weak emergence with certain aspects of the web of micro-interactions is not a merely stipulative move, for it vindicates a series of commonly accepted philosophical views on emergence (e.g. non straightforward predictability, robustness of macroscopic patterns, or even a moderate form of downward causation). Hence, I take these facts to imply that the Bedauvian notion of weak emergence is not referring to a sort of “meta-relation” framing the metaphysical structure of reality as a whole, yet it is tracking a physical relation inhabiting the domain of the physical system at issue.Footnote 6 It seems, then, that facts about Bedauvian weak emergence are not meta-facts that merely supervene upon the totality of physical facts, yet physical facts themselves, part of the fabric of the natural history of complex systems. Clearly, the fact that the relation of weak emergence does not belong to the domain of “meta facts” about the whole world does not mean that it carries no information at all about ontological structures; within a given ontological domain D, relations of weak emergence are still important to define the internal structure of D and facts about weak emergence are still constitutive facts of the relevant layer of reality.
Regarding (P\(_6\)) I also adopt a conservative stance: clearly, the idea of conceiving processes as computations may turn out to be inadequate—at least in some specific cases– yet that should be on independent grounds and not with the aim of solving a paradox about weak emergence. Moreover, the very notion of weak emergence Bedau is proposing seems to be framed in terms of the idea of “real computations”, i.e. simulations of the dynamics of the system where nothing prevents to identify the “simulator” with the system itself (as the examples involving cellular automata seem to suggest).
However, it is important to make clear and explicit the actual role that (P\(_6\)) plays in the derivation of the paradox, thus avoiding possible misinterpretations of this principle. The principle of computational equivalence ensures that every dynamic evolution of the system S—i.e. every evolution governed by laws L under conditions x—may be identified with a computation performed by the system S viewed as a natural Turing machine. The principle of computational equivalence has thus the role of bridging between physical and computational facts. However, in order to derive the paradox of weak emergence, we need to show that all facts belonging to the ontological domain \(D_L\) and supervening on conditions x within \(D_L\) are computable by S viewed as a Turing machine. In other words, what is needed is a “longer bridge”, i.e. a principle connecting metaphysical facts—i.e. membership in the domain grounded in natural laws L-to computational facts-computability by S. Clearly, the principle of computational equivalence alone cannot do all the work here. Yet the transition from computational to metaphysical discourse is supported by a further assumption, i.e. (P\(_2\)). In other words, it is being argued that physical occurrence in the dynamics of S is necessary and sufficient for computability by the system S, computability by the system S is necessary and sufficient for theoretical reducibility to the nomic base of S, and theoretical reducibility to the nomic base of S is necessary and sufficient to ontological reducibility to the domain defined by such nomic base. Only the first step of this chain is granted by the principle of computational equivalence. As a consequence, there is no implicit strengthening of this principle, which has not been used to make a wild conceptual leap from computation to metaphysics.
Having discussed the assumptions that have lead to the contradictory conclusion of the partial decidability of incompressibility and not having found clear and compelling reasons to reject any of them, I can draw my main conclusion: Bedau’s concept of weak emergence is metaphysically inadequate for it is inconsistent with the big metaphysical picture implicitly suggested in the context of its formulation. In other words, it seems reasonable to expect the notion of weak emergence to be at least compatible with a structured ontology, with the sufficiency and necessity of theoretical reduction for ontological reduction, with the nature of internal relations, and with the principle of computational equivalence. However, I hope to have shown that this set of assumptions is inconsistent with Bedau’s concept of weak emergence. The inconsistency of weak emergence with the presented assumptions and the fact that there are no clear reasons to reject at least one of them, puts the advocate of Bedau’s concept of emergence in a paradoxical situation, at least in the context of a general metaphysical theory of emergence. For from her perspective neither the definition of weak emergence nor the underlying assumptions deserve to be dropped. Such an aporetic condition represents a problem also for whom in spite of not considering Bedau’s definition as a satisfactory metaphysical account of weak emergence, still thinks that Bedau’s approach is on the right track to pick out weakly emergent behavior in natural systems.
Once a paradoxical circunstance is detected, it is desirable to go beyond a mere logical analysis of what explicitly engendered the contradiction and aim at a deep diagnosis even of the implicit ideas which in the majority of cases, although seem plainly acceptable, are part of the problem. To my mind, one of the leading ideas of Bedau’s concept of emergence which may be partly blamed for the inconsistency is what I call the derivational account of metaphysical concepts. The idea consists in trying to capture relations constituting the ontological structure of a certain domain of reality (or of reality as a whole) by appealing to the way in which the relata of such relations may be “derived” or “operationally obtained” from one another. A paradigmatic example of this approach is represent by the ontological interpretation of Nagelian reduction: the fact that a certain state of affairs \(\phi \) may be deduced from a certain theory T plus relevant bridge laws is interpreted as the fact that \(\phi \) is ontologically reduced to the level of reality described by the theory T. In general, the derivational account of metaphysical concepts has the virtue of being clearly applicable to elucidating examples and neatly detectable in the scientific practice; however, when it comes to providing a metaphysical theory—and thus to the attempt of finding an explanatory conceptual characterization—the approach may fall short of this goal. What is wrong, then, with the derivational account of metaphysical concepts and why is it responsible for the paradoxical situation about weak emergence? Arguably, there may be a theoretical conflict within the core idea of the derivational account of metaphysical relations, namely the mismatch between the intrinsic limitations of the notion of derivability in all of its forms (e.g. formal deduction, recursive decidability, simulation, etc...) and the highly demanding explanatory role of the metaphysical notions allegedly captured in terms of derivability. Let me try to present such a mismatch by way of example. The incompleteness of formal arithmetic is clearly a limitation of the notion of arithmetical derivability; such a limitation is intrinsic to the axiomatic method in the sense that it does not depend on our inferential capabilities as finite beings. Now consider the metaphysical problem of the status and the extension of concepts such as that of ‘mathematical universe’ and ‘mathematical truth’; it seems unwise to define these concepts in terms of the notion of “derivability in formal arithmetic”. For instance, it may not be reasonable to define the mathematical universe as the set of all facts derivable in formal arithmetic and the set of mathematical truths as the set of all propositions derivable in formal arithmetic; this because there is no clear reason why the limitations applying to the notion of derivation in formal arithmetic must apply to metaphysical concepts. On the contrary, many semantic interpretations of the First Incompleteness Theorem rely on the idea that there are mathematical truths and mathematical objects inaccessible to the formal means of arithmetical derivation; and this account may be acceptable on independent grounds, e.g. in order to explain the existence of non-constructive mathematical objects or the legitimacy of non-constructive mathematical axioms. Metaphysical theories are explanatory ambitious and if they are to be understood in terms of derivability, constructibility, or other germane notions, they are doomed to crash with the formal limits imposed to these latter concepts. The example of incompleteness of formal arithmetic shows that, in spite of the fact that a derivational account may be helpful in understanding metaphysical notions from a pragmatic and operational perspective, there may be no clear reason why an ontological concept must obey the same limitations as a certain notion of derivability.
In the specific case of the relation of weak emergence, the contradiction seems to be engendered precisely by the sort of conflict I have briefly described. On the one hand, we require the notion of weak emergence to harmonize with a broadly articulated metaphysical view, i.e. the tree-like structure ontological model and reductionism about internal relations. A consequence of this attempt of metaphysical contextualization is the self-referentiality of the relation of weak emergence, i.e. the conclusion that facts about weak emergence must be weakly emergent facts; this may represent the extent to which our metaphysical ambitions allegedly go. On the other hand, if we are to understand weak emergence in terms of computational incompressibility, we are forced to interpret the self-referentiality of weak emergence in terms of decidability of incompressibility, which clearly breaks the limits imposed by foundational results in computability theory such as the undecidability of the halting problem and Rice’s theorem. Verdict: the derivational account of metaphysical relations implies the risk of a theoretical conflict between the explanatory ambitions of our metaphysical theory and the logico-mathematical barriers imposed by the particular notion of derivability at issue.
One last remark. Since the discovery of the incompleteness theorems or the undecidability of the halting problem, we have not stopped doing formal arithmetic or implementing Turing machines; on the contrary, those limitative results have proven themselves fruitful in making us understand profound aspects of arithmetic derivability or computability and their applications. In a similar spirit, I do not claim that Bedau’s notion of weak emergence must be abandoned for it cannot be useful in our understanding of complex systems; on the contrary, Bedau’s examples from syntetic biology (Bedau, 2013) represent a remarkable case of the elucidating role of the notion of weak emergence as underivability except by simulation. What I do claim instead, is that Bedau’s characterization cannot be used as a definition of a metaphysical concept, for it presents difficulties in being harmonized with broader metaphysical views. Future works may interestingly try to extrapolate a metaphysical characterization of the notion of weak emergence from Bedau’s underivability except by simulation, progressively making this notion less dependent from the derivational account and richer by conciliating its limitation with the overarching scope of metaphysical investigations.
Notes
Butterfield (2011) and Howard (2007) present examples of cases of emergence without supervenience. However, the examples are allegedly cases of strong emergence, for they involve metaphysical novelty and failure of reduction. The fact that weak emergence entails supervenience may also be shown by considering that Bedau’s notion of weak emergence entails reduction and the entailment between reduction and supervenience seems to be out of question.
Notice that the point here is not axiomatizability in itself, for most of the commonly discussed types of cellular automata may be—at least in principles—axiomatized; the point here is that a system of axioms is not the most perspicuous way of presenting the behavior of a cellular automaton, for the steps of a formal deduction may not correspond to the steps of a generative dynamical history.
For interesting and detailed examples of descriptions of cellular automata as Turing machines see (Wolfram, 2002, pp. 637–714).
For a complete survey of these notions see (Vitany & Li, 2019).
Cellular automata capable of self-simulation are not uncommon; the most famous example is ‘Life within life’ representing the simulation of Conway’s game of life within Conway’s gama of life.
One may even think that Bedau’s definition is tracking a family of natural relations at different levels (e.g. chemical, biological, etc...) characterized by structural similarities in the causal web of micro-interactions.
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Ciccarelli, V. Weak Emergence, Ontological Hierarchies, and Computational Equivalence. Minds & Machines 36, 41 (2026). https://doi.org/10.1007/s11023-026-09794-9
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DOI: https://doi.org/10.1007/s11023-026-09794-9