Multilevel Ensemble Explanations: a Case from Theoretical Biology

Multilevel Ensemble Explanations: A Case from Theoretical Biology

Luca Rivelli University of Padua, Philosophy Department (FISPPA)

I analyze a well-known argument by Stuart Kauffman about complex systems and evolution to show it contains a hierarchy of non-mechanistic, non- causal explanations—which I would call, following Kauffman, “ensemble explanations”—quite closely resembling the explanations of the structural kind proposed in Huneman (2017), but lacking their absolute mathematical certainty, being based on results of non-exhaustive computer simulations. In Kauffman’s core argument ensemble explanations form an explanatory chain along a hierarchy of levels, where each explanans at one level gets itself recursively explained at the lower level. Explanations at adjacent levels turn out to be related not by mereological containment as in a multilevel mechanistic explanation, but by an analog to the relationship between two specifications at different levels of a specification/implementation hierarchy as understood by computer science. A mechanistic explanation grounds the whole hierarchy enabling the explanatory chain. Interestingly, the preliminary production of ensemble explanations enables the multilevel mechanistic explanations of systems manifesting what Bedau (1997) defines as weak emergence.

1. Introduction In this paper I will reconstruct and analyze a famous argument by Stuart Kauffman about complex systems and evolution, in order to highlight the use in theoretical biology of a kind of non-mechanistic and non-causal explanation which I propose to call, following Kauffman, ensemble explanation. The aim is to contribute to the ongoing philosophical debate about non-causal explanations in the special sciences, kinds of explanation apparently extraneous to the received causal-mechanistic view. Ensemble

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc_a_00301 by guest on 26 September 2021 Perspectives on Science 89

explanations resemble quite closely the explanations of the structural kind proposed by Philippe Huneman (2017), which—unlike mechanistic explanations—do not explain by exposing the causal structure of a phenomenon, but by virtue of structural-mathematical “laws” (theorems) holding for all abstract models of a certain class, a class to which a valid abstract model of the phenomenon to explain turns out to belong. Simi- larly, ensemble explanations explain by virtue of a law holding for a class of abstract models, with the difference that the law is not ascertained deduc- tively, but is inductively derived via non-exhaustive sets of computer-based simulations on random samples taken from the class under consideration. This circumstance renders such law only probably true, setting ensemble explanations apart from structural explanations. I will show how in Kauffman’s work ensemble explanations form an explanatory chain structured along a hierarchy of levels, in which each explanans at one level, becoming in turn an explanandum, gets recursively explained at a progressively lower level, until the hierarchy bottoms out into a classic mechanistic explanation. I argue that the nature of this hierarchy is not part-whole composition as in mechanistic multilevel explanations, but an analog to the specification/implementation hierarchy of computer science: along it, ensemble explanations fulfill the rather peculiar role of specifications explaining other specifications, while the same hierarchy, if needed, would allow for the production of a corresponding full functional explanation making explicit the details of the implementation at each level. Interestingly, it turns out that, far from being hard compet- itors, ensemble explanations coordinate with mechanistic explanations fruitfully: while a mechanistic explanation grounds the whole hierarchy at its lowest-level step, the production of ensemble explanations constitute in turn an enabling preliminary step for a further multilevel mechanistic explanation of those complex systems manifesting what Bedau (1997) calls a weakly emergent dynamics. It is safe to say that any attempted generalization, based on the single case studied in this work, is risky. I do not claim to have highlighted a universal articulation between ensemble, functional, and mechanistic explanations, but, at least, a possible way in which these explanation kinds can be related.

2. Topological and Structural Explanations Since the late 1990s, the received view on explanations in the special sciences has been the so-called new mechanical philosophy, introduced by seminal works such as Bechtel and Richardson (1993) and Machamer, Darden, and Craver (2000): for mechanists, to explain a phenomenon consists in presenting the mechanism producing it, that is, showing in detail how a coordinated ensemble of parts and activities engaging in causal

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc_a_00301 by guest on 26 September 2021 90 Multilevel Ensemble Explanations

interactions brings about the phenomenon.1 More recently, a variety of non-causal and non-mechanistic explanations in the special sciences have caught the attention of philosophers. Huneman (2010) and Huneman (2015) are among the first philosophical works to identify in the biological literature cases of recourse to a specific type of non-mechanistic scientificexpla- nation based on topological properties of abstract representations of a system, that is, properties invariant under a class of possible continuous deformations. The idea comes from mathematical topology, a discipline studying, intuitively, invariant properties of the “form” of an object, or the structure of connectivity of networks. To clarify, a network is a mathematical structure2 constituted by a set of items, the nodes, variously connected with one another by links, or edges. In a directed network links have a direction, so they can be plausibly seen as inputs going into nodes or outputs coming out of them. As a dynamics can occur on the network when nodes represent active elements capable of changing state and of reciprocal influence through input and output signals, networks have been more and more employed as theoretical dynamical models of a multitude of empirical phenomena. Of course, the connectivity structure of the network, its topology, constrains the possible dynamics. Topology changes only by connecting or disconnecting nodes, not by deforming the network. Certain networks show a form of modularity called community structure, a topological property consisting in the fact that the network can be partitioned into subnetworks, the modules or communities, with sparse connections between communities, and a higher density of connections among nodes internal to communities. An example is in fig. 1. Mechanistic explanations explain by citing how the coordinated causal interactions between the relevant physical parts that make up the overall mechanism produce the mechanism’s behavior to be explained. When possible, these explanations proceed by individuating a multi-level structure of nested mechanisms, where each of the parts composing the encompass- ing mechanism is itself a whole mechanism, possibly in turn composed of parts at the lower level and so on, until the hierarchy bottoms-out in a level—determined by pragmatic considerations—of what are considered elementary parts. While mechanistic explanations explain by citing causal chains of events involving physical parts embedded in a multilevel containment constitutive structure, in topological explanations the properly explanatory role is fulfilled by topological facts, which are mathematical facts. Huneman (2017) expands this pioneering topological conception outlining a more general theory of non-mechanistic structural explanations

1. For reasons of space, I take for granted that the reader is well acquainted with this established theoretical position. An outstanding overview is Glennan 2017. 2. Properly, a graph.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc_a_00301 by guest on 26 September 2021 Perspectives on Science 91

Figure 1. A network with modularity by community structure. Colored discs surround the modules.

where what is explanatory is the reference to formal or structural mathematical properties of a model of the system. Here is an outline of Huneman’s view: i. Structural explanations omit speciﬁc causal spatio-temporal trajectories of entities—that is, they omit the core of any mechanistic explanation—and explain instead by means of some mathematical feature of a representation of the system. ii. As mathematical properties—and not simply abstract representations of physical features—these explanatory features endow structural explanations with the modal force of a mathematical statement, whose truth traverses possible worlds, as opposed to empirical laws, which concern only the actual world. Speciﬁcally, this mathematical modal force out- classes the force of causal explanations, marking the different nature of structural explanations with respect to causal-mechanistic ones.3

3. This does not mean that mathematically certain explanations are to be preferred to causal explanations: it is a fact in certain cases of scientiﬁc explanation of empirical phenomena reported in the literature, that what does the explaining is not a causal connection, but some mathematical truth. Some examples are in Huneman 2017 and Lange 2013.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc_a_00301 by guest on 26 September 2021 92 Multilevel Ensemble Explanations

iii. The generality and abstractness of the representation employed in such explanations allows for a substitution of any mechanism with the right topology—that is, belonging to the right class—as refer- ent of the abstract representation. iv. As the explicative property characterizes a class of mathematical structures, often the explanation proceeds by showing that a theoretical model of the observed system belongs to this class: a theorem states that a certain class of mathematical structures has a certain property P, and it is shown that a theoretical model of the system belongs to this class. So, by virtue of the theorem, this model will undoubtedly sport the property P. And, if the model validly represents the system, the system’s actual behavior will itself inevitably show, with mathematical modal force, the same property P. Such an explanation is evidently non-causal and non-mechanistic, because it neither involves causal facts or considerations, nor it mentions any speciﬁc spatio-temporal path inside a mechanism. A simple example of structural explanation4 is an economic explanation of the persistence of casinos in the world: it is the law of large numbers together with the unfair ratio of winning numbers (1–36) to possible numbers (0–36) of the roulette, a mathematical fact ensuring casinos realize net gains in the long run, what explains why they do not go bankrupt, and thus explains their persistence.

3. Kauffman on Genetic Regulatory Networks and Their Evolvability In this and the following four sections I will analyze and reconstruct a classic proposal in theoretical biology by Stuart Kauffman, in order to show that it contains explanations of a peculiar non-mechanistic kind—that I will propose to call ensemble explanations—somewhat resembling Philippe Huneman’s structural explanations. Stuart Kauffman’scoreargumentiscen- tered around two basic concepts: fitness landscapes and genetic regulatory networks. A fitness landscape (an idea originally introduced by Sewall Wright, one of the fathers of the modern synthesis) is a 3D representation comprising a plane where an aggregating function positions each genotype according to its overall allelic configuration, and a metric making the distance between two points proportional to how much the corresponding genotypes differ in terms of number of loci with different alleles. A third dimension represents the fitness each genotype manifests. Darwinian phylogenetic evolution of a population can then be viewed as a walk of a group of points in the “landscape” toward a “peak” of high fitness.

4. From Huneman 2017.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc_a_00301 by guest on 26 September 2021 Perspectives on Science 93

The idea of genetic regulatory network views the genome as a parallel processing network, where each node—each gene—is activated or inhibited according to a function of its inputs, a set of links coming from other genes. The resulting dynamics of gene activation constitutes the functioning of the genome during ontogeny or during the normal activity of the cell. As abstract models of genetic networks Kauffman adopts boolean networks, dynamical networks in which each gene can be only completely on or off and time flow is discretized into separate consecutive time steps. The binary state each node assumes at the next time step is determined by a boolean function, which can vary from node to node, of its inputs, that is of the states, at the current time step, of the nodes connected to it. At each successive time step the states of all nodes are updated synchronously, and the corresponding change in the overall configuration of states gives rise to the dynamic evolution of the system. Kauffman defends the validity of this discretized model of a natural continuous system. In Kauffman’s major works fitness landscapes and genetic networks are involved in a web of interconnected themes:

• An organism is the result of a process of ontogenetic development controlled by the unfolding of the dynamics of a parallel processing system, the genetic regulatory network. • To produce the degree of complexity we see in actual living beings, such a dynamical process must proceed in an ordered complex and coordinated manner. • In this view, phylogenetic evolution consists in the evolution of the ontogenetic dynamics, by means of the selection over evolutionary times of different genomes encoding different developmental programs. • Evolvability—that is, a feasible evolution—needs a reasonably smooth (a correlated) fitness landscape with sufficiently high fitness areas: a too rugged and multi-peaked landscape, where even small variations of the genotype make fitness vary wildly and unpredict- ably, would hinder the ability of natural selection to progressively shape organisms toward the optimal fitness. • The point is: the fitness landscape of an organism depends on the structural features of its genetic regulatory network. A question arises from these premises: given that organisms have indeed evolved, and so their genomes’ fitness landscapes are certainly correlated, why do genomes have correlated fitness landscapes? So, I take the following statement to be the main explanandum of Kauffman’s research: E1: Genomes have a correlated fitness landscape. The same premises suggest another question: what relation subsists between order exhibited by ontogenetic processes and features of the genetic

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc_a_00301 by guest on 26 September 2021 94 Multilevel Ensemble Explanations

network? Or, by virtue of what do genetic networks manifest an ordered dynamics? This leads to a second explanandum present in Kauffman’s work: E2: Ontogenetic developmental processes are organized and ordered. To explain E1 and E2, Kauffman follows a more general route: he sets out to study not how real-world genetic networks, but boolean networks, that is, very simplified theoretical models of genetic networks, are able of such a kind of ordered dynamics and evolvability. To generalize, Kauffman sets out to investigate if even random boolean networks of a certain type can give rise to a sufficiently ordered dynamical behavior. The apparently surprising expectation that even random systems can give rise to order stems from Kauffman’s cultural background. Since the early ’70s, complex systems theory was showing through computer simulations how ordered dynamics can spontaneously emerge in certain classes of systems composed of myriads of interacting elements. Around the same era, doubts had started to spread about the absolute power of freely shaping genomes, and thus organisms, attributed by neo-darwinism to natural selection. In his most prominent work (Kauffman 1993), Kauffman claims that, far from being a product of natural selection, spon- taneous order manifesting in dynamical complex systems like the genomes is what actually enables natural selection, by making fitness landscapes sufficiently correlated.

4. Kauffman’s Findings: Classes of NK Networks and Network Dynamics To show that an intrinsic source of order is the emergent dynamics of certain complex systems, Kauffman focuses on random boolean networks,in which both the connections between nodes and the specific boolean function each node implements are chosen at random. Specifically, Kauffman studies random boolean networks with a fixed number N of nodes and a fixed number K of inputs for each node, the so-called NK networks, as K, N and the distribution of the boolean function vary (Kauffman 1971, p.150; Kauffman 1990, p.140). He proceeds by running computer simulations on random samples of the space of possible NK networks: for each random choice of N, K and of a particular probability distribution D of the type of boolean functions assigned to each node, randomly wired networks with such parameters are simulated to evaluate their dynamics. In a set of tests the D distribution is biased towards assigning canalyzing functions5 to a majority of nodes, or, in other tests, towards a prevalence of functions with

5. Explained below.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc_a_00301 by guest on 26 September 2021 Perspectives on Science 95

ahighP—the percentage of times a function assumes the same logical value on the set of all possible combinations of its inputs. The canalyzing functions are boolean functions for which, when one input is in a certain speciﬁc state, the output remains in a certain “forced” state, regardless of the possible changes of state of the other inputs (Kauffman 1990, p.143). For instance, the OR function is canalyzing: the presence of 1 on any input will make the output be 1 whatever the combination of states of the other inputs. This logical feature makes subnetworks whose nodes predominantly implement canalyzing functions progressively assume a dynamical state that after a ﬁnite number of time

Figure 2. A forcing structure: a subnetwork with a prevalence of canalyzing functions. Thick arrows are forcing inputs, which if in a certain state will make the output remain in a certain forced state. Due to the presence of loops, after a certain number of time steps, forced states become self-sustaining and the whole structure freezes. (After Kauffman 1971, p. 154).

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc_a_00301 by guest on 26 September 2021 96 Multilevel Ensemble Explanations

steps becomes frozen: after a certain time, the state of each node of the subnetwork does not change anymore, regardless of the state of its other input links. These subnetworks are called forcing structures (see fig. 2). Subnetworks where high-P functions prevail act similarly. As we will see, frozen structures are of great importance in explaining the dynamical behavior of the networks in which they appear. Kauffman argues that, for physico-chemical reasons concerning the structure and functions of the macromolecules involved in activation and repression of genes, the obtainment in nature of non-canalyzing functions appears difficult (Kauffman 1993, p. 454). Thus, he reasons, most functions regulating genes in real-life are probably canalyzing, and cites empirical research confirming their prevalence (Kauffman 1993, pp. 444–54). Interestingly, here the logical feature of being canalyzing is explained in a mechanistic way, by appeal to physical causal facts. Based on computer simulations of random boolean networks, Kauffman classifies NK networks into three subclasses—called here C1, C2 and C3— differing for certain properties concerning either the dynamical behavior of the network (an ontogenetic view) or its fitness landscape (a phylogenetic view). These are findings of a probabilistic nature, due to their being sup- ported by partial simulations; the use of random sampling for the simulations is absolutely necessary because even the space of finite boolean networks with a limited number N ofnodesisenormous6 and cannot be explored exhaustively. Here are Kauffman’s results:

• Class C1: networks with K<2. There is maximum independence between nodes, because they are sparsely connected with one another (the equivalent of a completely Mendelian genome for K ¼ 0). The ﬁtness landscape is maximally smooth and correlated, with only a very few peaks. This would ensure maximum evolvability, but in this class dynamical behavior is still too simple and uncoordinated to guide the ontogenetic process of complex and structured organisms. In K=0 networks, as N increases, a so-called error catastrophe curtails the height of ﬁtness peaks. • Class C2: networks with 2

ÀÁ K N 6. Cardinality 2ðÞ2 .

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc_a_00301 by guest on 26 September 2021 Perspectives on Science 97

the current state gets amplified and spreads without bounds, producing unpredictable outcomes. Fitness landscapes are maximally rugged, with many peaks of different height, for in such highly connected networks any mutation in any gene will influence most of the others, making the fitness value—an overall function of the dynamics of every gene—vary in an uncontrolled and non-linear manner even for single-gene mutations. Consequently, evolvability is very low: natural selection would have a hard time in shifting a population gradually and smoothly toward the global maximum fitness peak through such a rugged landscape, so populations would tend to remain stuck at nearby peaks. Moreover, as N grows, a so-called complexity catastrophe lowers the height of reachable peaks. • Class C3: networks with K=2 or K not much higher than 2 and with a majority of canalyzing or high-P functions. Dynamical behavior is structured and well ordered, while remaining a complex and flexible behavior, seemingly capable of computation (Kauffman 1990, p. 151).7 These are dynamics able to guide developmental processes in complex organisms. Fitness landscape is moderately smooth and walkable by evolution, with reachable local peaks of reasonably high fitness. This allows evolvability of this class of systems by natural selection. Please note that, for the sake of simplicity, henceforth I will use the following terminology:

• GOODMICROSTRUCT: the condition, typical of class C3 networks, of having K=2 or a majority of canalyzing functions or of functions with high P. Being these functions features of single nodes, this condition represents a micro—that is, a low-level—feature of the connectivity of the network.8 • COMPBEHAVIOR: a structured, well ordered and at the same time complex and flexible network dynamics resulting in a “computational” behavior, able to guide an ontogenetic process. This is an overall high-level feature of the system. • GOODLANDSCAPE: the condition of having moderately smooth fitness landscape with reasonably high-fitness reachable local peaks. Again, a high-level, global feature. • MECHAGOODMICROSTRUCT: will label all the physico-chemical mechanistic explanations and the observational data Kauffman

7. I omit discussing this attribution of “computation.” 8. See section 8 for a discussion.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc_a_00301 by guest on 26 September 2021 98 Multilevel Ensemble Explanations

employs to claim that real genomes tend to have a numerical prevalence of canalyzing or high-P functions and average number of inputs per node not much higher than 2.

5. An Interpretation of Kauffman’s Findings in Terms of Modularity From a dynamical point of view, seemingly the only class of NK networks capable of COMPBEHAVIOR is class C3: too simple (class C1) or too cha- otic (class C2) dynamics are not capable of guiding the developmental process of actual organisms. C3 networks also show some robustness: temporary moderate perturbations of their dynamical state during the developmental process would not disrupt ontogenesis, being possibly com- pensated by homeostatic or homeorhesic trajectories toward the original path. Interestingly, C3 networks also show stability regarding their phylogenetic evolution: their fitness will not vary wildly after a limited structural mutation. This is the reason why the fitness landscapes of C3’s networks sport GOODLANDSCAPE, allowing their evolvability. A specific property explains these two features of class C3 networks: their temporal dynamics manifests, due to GOODMICROSTRUCT, a form of emergent self-organization, consisting in the internal formation and spreading of frozen components,9 subnetworks that become frozen in their dynamical state. Frozen components act as insulating “walls” through which no dynamical perturbation can spread, coming to constitute variously shaped boundaries partially partitioning the whole network into more or less separated unfrozen “liquid” islands, whose nodes can instead continue to change of state. Inside these islands, processing can take place, and the thin remaining liquid “channels” connecting different liquid areas can act as communication channels for the passing of information. The sparsity of these channels explains why dynamical perturbations in such a network remain localized, allowing for an ordered but flexible behavior. From a phylogenetic standpoint, mutations—changes of the network structure in terms of the distribution of links and/or of the boolean functions nodes perform—when affecting nodes (genes) embedded in a frozen part, will not affect the dynamics of any other part of the system, because the boundaries of the frozen part remain unchanged. And, mutation affecting nodes in a liquid island will mostly affect the intra-island dynamics, for the limited inter-island communication channels will prevent a drastic change of the system’s overall dynamics. For this reason, the fitness landscape of C3 networks remains reasonably smooth.

9. See section 4.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc_a_00301 by guest on 26 September 2021 Perspectives on Science 99

The above can be interpreted in terms of community structure network modularity.10 When frozen components consolidate, they start acting as stable, permanent impenetrable walls structurally isolating parts of the network (the modules): from that moment on, the network’s dynamics will occur as if occurring on a structurally modular network with a topological structure different from the original topology of the network. This new structural modularity induces a correspondent form of dynamical modularity, constituted by a limited passing of signals between the dynamical activities of different modules (the liquid islands), and temporal decou- pling between intra-module and inter-module dynamics: intra-island connections are denser and thus intra-island interactions happen, on average, more frequently than the inter-module exchange of signals. It is this form of dynamical modularity that enables the observer to view the system as “computational,” capable of ﬂexible and complex structured dynamic behavior, where each module performs a kind of input/output function. What makes this high-level modularity (high-level because modules are groups of nodes)11 dynamically emerge is the low-level (regarding single nodes) structure of the network, with its peculiar feature of connectivity GOODMICROSTRUCT: This low-level topological condition explains the rise of a high-level topological condition. We see that topological features can be ideally situated at different levels of description, as will be discussed in section 8, where a full hierarchy of topological explanations is taken into account.

6. A Reconstruction of Kauffman’sArgument I will try to show that in Kauffman’s line of reasoning a series of non- mechanistic explanations of a peculiar type can be recognized. Rational reconstruction and restructuring of Kauffman’s core argument are necessary ﬁrst, for his argument is made up of a constellation of intertwined questions and explanations sometimes spread across whole volumes, not easy to disentangle. I would summarize Kauffman’s argument as follows: we want to explain E1, that is, to answer the question “Why do genomes have a correlated ﬁtness landscape?” Answer: a. Since both real genomes and NK networks are dynamical directed networks, we take NK networks as plausible candidate models of real genetic regulatory networks.

10. See section 2. 11. Low-level modularity trivially consists in seeing each node as a module.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc_a_00301 by guest on 26 September 2021 100 Multilevel Ensemble Explanations

b. By randomly sampled simulations on the space of possible NK networks, we identify 3 classes: C1, C2 and C3. c. C3 is the class of NK networks with GOODMICROSTRUCT. d. C3 appears to be the sole class with GOODLANDSCAPE property. e. Now, by MECHAGOODMICROSTRUCT, it seems likely that real- life genomes have GOODMICROSTRUCT too. f. So, it follows that it is likely that real genomes are validly modeled only by class C3’s NK networks, with which they are most similar in terms of common features of their low-level network structure. g. So, for points d and f, real genomes will have the same GOOD- LANDSCAPE property of C3 networks. h. Putting it very simply, real genomes have good ﬁtness landscape because they12 belong to the only class of networks with the property of having that kind of landscape.

While, admittedly, this may seem a vacuous explanation, let’s examine for the moment what kind of explanation it is. In some passages, Kauffman hints at it as an anomalous type of “theory,” which he calls ensemble theory.13 In his words: The theory is not concerned with particular dynamic systems, but with the behavior of entire classes of dynamic systems. The strength of such a theory is that it offers hope of explaining some of the ubiquitous global properties of cellular gene control systems in terms of membership of all those control systems in the same “good” class of dynamic systems. However, the theory makes no statements at all about the detailed pattern of behavior of any hypothetical member of the class of systems studied, so it cannot be directly tested by attempting to describe in detail, for example, the modes of biochemical oscillations of any given cell type. (Kauffman 1971, p. 174) Indeed, this quotation recalls quite closely the features of Huneman’s structural kind of explanation, summarized in point iv of section 2. Kauff- man’s explanation, however, is not, apparently, an actual instance of structural explanation, for it is not based on mathematically proven facts, but only on a probabilistic, inductive proof that the class with GOODMICRO- STRUCT is characterized by GOODLANDSCAPE. Other similar explanations can be found in Kauffman’swork,the more prominent being the explanation of E2,thatis,theanswerto

12. Actually, their models. 13. In Kauffman 1993, chapter 11.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc_a_00301 by guest on 26 September 2021 Perspectives on Science 101

the question: “Why are ontogenetic developmental processes organized and ordered?” The answer is completely analogous to the answer to E1, provided that we substitute points d, g,andh with: d. C3 appears to be the sole class with the COMPBEHAVIOR property. g. So, for points d and f, real genomes will have the same COMP- BEHAVIOR property of C3 networks. h. Putting it very simply, real genomes have computational behavior— a behavior able to produce an organized and ordered ontogenesis— because they14 belong to the only class of networks with the property of having that kind of dynamical behavior. While the above explanations manage to explain in a very generic way why real genomes have such evolutionary and dynamical features, they seem vacuous, immediately suggesting new explananda:

• E3: Why do class C3 networks have GOODLANDSCAPE? • E4: Why do class C3 networks show COMPBEHAVIOR? Or, in general:

• E5: Why do class C3 networks have GOODLANDSCAPE and COMPBEHAVIOR? This scarce ability to satisfy the enquirer is not necessarily present in properly structural explanations: for example (Huneman 2017, p. 10), an explanation of why the average size of birds tends to augment in northern populations based on the geometrical fact that a body with bigger volume has a smaller surface/volume ratio, and that this reduces heath loss, seems a quite satisfying explanation.15 The likely reason is that there is a mathematical proofofthisratio,while,inKauffman’s case, deductive proofs are missing: through random sampling, he found that networks of a class have a certain feature, but that amounts to a phenomenological recap of the limited sample of observed networks, an observational statement lacking the deductive cogency of a mathematical theorem. The enquirer’s dissatisfaction is then probably due to a lack, in Kauffman’s case, of a cogent rule, expressible in a concise way as a mathematical theorem would be, which in turn explains why membership in C3—that is, the fact of having GOODMICROSTRUCT—is tied to GOOD- LANDSCAPE or COMPBEHAVIOR: actually, a phenomenological recap of observed data seems like the statement of a brute fact, and that apparently is not explanatory enough. Hence, the need to explain E5 itself persists.

14. Actually, their models. 15. Provided, of course, that natural selection has been at work.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc_a_00301 by guest on 26 September 2021 102 Multilevel Ensemble Explanations

7. A Hierarchy of Ensemble Explanations in Kauffman’sArgument Fortunately, at least in Kauffman’s case, there is an explanation of E5: actually, two whole chains of explanations (CHAIN1 and CHAIN2 henceforth), explaining E3 and E4 respectively. See them schematically represented in fig. 3 and fig. 4. In these figures the thick vertical dotted red arrows stand for “explains,” symbolizing the relation between an explanans and its explanandum: the vertical stack of properties linked by these arrows constitutes the chain of explanations. In each chain, the top property—the explanans—gets explained by the property placed immediately below it, the explanandum. Interestingly, along the chain, each explanans becomes in turn an explanandum explained by an immediately

Figure 3. CHAIN1–the explanatory chain explaining GOODLANDSCAPE.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc_a_00301 by guest on 26 September 2021 Perspectives on Science 103

Figure 4. CHAIN2–the explanatory chain explaining COMPBEHAVIOR.

lower explanans, and so on. Features of the single explanations composing the chain are listed in the vertical stack of boxes (EXPLANATION 0, EXPLANATION 1…) to the right of the explanatory chain. For instance, the explanans in EXPLANATION 2 (frozen components) becomes the explanandum in EXPLANATION 1. This reproduces, recursively, the reiterated demand for explanation, highlighted above, after an explanation at a

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc_a_00301 by guest on 26 September 2021 104 Multilevel Ensemble Explanations

certain “level” has already been provided. Seemingly, this reiterated demand does not arise in structural explanations, as in the example of the birds’ size. Possibly, this conﬁrms that Kauffman’s explanations are not proper structural explanations: they are, in a way, explanations of a structural kind, although probabilistic.16 The explanations above EXPLANATION 0 concern exclusively the abstract realm of NK networks, which are theoretical models of real genomes. Following Kauffman’s expression “ensemble theory,” I will call explana- tionsofthiskindensemble explanations. In ensemble explanations the explanatory device is the recourse to a law stating that abstract systems belonging to the class deﬁned by the explanans property also have the explanandum property. Ensemble explanations differ from proper structural or topological explanations in that this law, which is what does the explaining, is not endowed with absolute mathematical certainty, but holds only probabilistically, being based on the results of sets of non-exhaustive simulations. Both chains of proper ensemble explanations bottom-out with EXPLA- NATION 1: the explanans GOODMICROSTRUCT, concerning boolean functions, is a logical, low-level17 property seemingly not calling itself for explanation. One of its possible instances, the property of being canalyzing, is a logical feature we cannot possibly decompose into constituents. Explanation could end here, and its nature would be akin to mathematical explanation: ultimately, what actually explains, by sustaining from the bottom the chain of ensemble explanations above it, is a logical property. But on what grounds does such a property hold in the system? This question concerns not the logical level, but its possible physical realization: a mechanistic explanation seems the only possible answer. Indeed, EXPLA- NATION 0 is such a mechanistic explanation. At the same time, EXPLA- NATION 0 grounds the validity of class C3 networks as models of real genomes: this mechanistic step is what guarantees the applicability of the upper chain to real-world systems, in addition to its already immediate applicability to the formal model. Step 0 has a representational role,18 war- ranting the existence of a mapping between the real phenomenon (the genome) and its theoretical model (the boolean network)—a necessary grounding. But this mechanistic step is not able to explain the whole explanandum: it does not allow us to say that, by virtue of it, E3/E4 are explained: here, there is no mechanistic direct reconstruction of the causal processes bringing from level 0 to the top-level explananda. Instead, all the

16. But see discussion in section 9. 17. Concerning single nodes. 18. See Huneman 2017.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc_a_00301 by guest on 26 September 2021 Perspectives on Science 105

explanatory burden seems to lie on the shoulders of steps 1–5, which are the non-causal steps. Nevertheless, the mechanistic EXPLANATION 0 does explain the empirical numerical prevalence of a certain logical property of the networks’ nodes, a logical property which in turn grounds, non- causally, all the chain toward the top: ensemble explanations are enabled by a low-level mechanistic “starter.” So why, after having explained mechanistically the lowest level doesn’t Kauffman produce a complete mechanistic explanation of the top explananda? I think there are at least two plausible reasons:

• First, he follows an ensemble approach, studying why a whole class of systems, the genomes, have in general certain global properties, so he must not focus on specific mechanisms, which are hard to generalize: from the mechanistic explanation of a specific genetic network, we wouldn’t obtain an explanation of common features of the whole class. Moreover, Kauffman does not have the complete schematic of any real genomic network, and resorts to seek general properties of the class. For these and other reasons19 he focuses on random networks: the basic statistical idea is that a sufficiently ample sample of random networks will represent the typical features of any dynamical network of the set from which the sample is taken. • Second, a complete mechanistic explanation of E3/E4 is probably an arduous task: we are dealing here with systems of thousand or more interacting elements connected by at least the double of links. How to describe the particular causal trajectories of all the parts’ interactions? Such complex systems can be feasibly explained mechanistically only recurring to a hierarchical modular decomposition of the system into levels, where at each level the mechanism is recursively represented as composed of submechanisms: that is, mechanistic explanation of complex enough systems is unfeasible unless the system is modular. In absence of modularity, mechanistic explanation is seemingly precluded. Not so with topological explanation: we could still recur to global high-level-only ensemble explanations, such as sequence a–h of section 6, even for non-modular complex systems. Fortunately, in Kauffman’s case, class C3 networks are modular, so, we could potentially explain any of them mechanistically. Now, how can we know beforehand these networks are modular? We surely know it now, after Kauffman’s simulations. As the so-called science of complexity teaches us, large complex systems, composed by myriads of interacting elements, often defy intuition and manifest an emergent behavior surprising

19. Section 3.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc_a_00301 by guest on 26 September 2021 106 Multilevel Ensemble Explanations

the observer. In its minimal conception, weak emergence (Bedau 1997),20 emergence consists in the unpredictability—defying expectations—of the behavior of completely deterministic complex systems, at least, unpredictability via a closed-form law applied to the system’s initial conditions. Emergent behavior is thus explorable only by letting the system naturally express its dynamical functioning or by actually running a simulation of it. During these runs, in certain systems some surprising conﬁgurations emerge. The appearance of frozen components is an example of a weakly emergent behavior: it wouldn’t have been easy for Kauffman, before ever running the simulations, to tell that frozen components would have appeared in certain classes of networks. He actually discovered this by simulation. The explanatory chain explains that it is the emergent dynamics—the emergence of frozen components—that which restructures the original topology of the system into a modular one: the crucial explanation is the sequence EXPLANATION 1–EXPLANATION 2, explaining the presence of structural modularity with the emergence, during the dynamical functioning of the system, of frozen components partitioning the network into dynamically isolated subnetworks (the modules). In turn, frozen components are explained by GOODMICROSTRUCT, the numerical prevalence in the network of nodes computing logical functions of a certain type. The emergence of this kind of modularity is what allows for multilevel mechanistic explanations: a higher-level mechanism would have as its single parts not single nodes of the network, but whole dynamical modules. These are plausibly relevant parts of a high-level mechanistic explanation because they are the natural high-level active parts of the system restruc- tured by its own emergent dynamics: the modules—the liquid islands naturally circumscribed by the emerged frozen components—are the only areas still capable of change of state and of interacting reciprocally through narrow communication channels inside a completely static landscape. This makes them the only remaining sources of any form of dynamical behavior of the system, and as such naturally relevant for most mechanistic explanations. Each module naturally appears as one elementary part of a high- level system, thus a system with a much more limited number of parts with respect to the low-level network,21 andassuchmuchmoreeasily explainable mechanistically.22

20. I do not endorse here any “stronger” emergence of an ontological kind. 21. In which each node is an elementary part. 22. Of course, an exception is that a purely intra-module phenomenon has to be mechanistically explained: such an explanation will be at a lower level.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc_a_00301 by guest on 26 September 2021 Perspectives on Science 107

Interestingly, the identification of the relevant high level of mechanistic explanation, the level of modules, is a result of the preliminary production, by means of simulation and observation of the simulation’s results, of the chain of ensemble explanations, which is what guarantees, by explaining its emergence, the existence of a modular hierarchy potentially relevant for mechanistic explanation. So, when dealing with complex weakly emergent systems, it seems ensemble and mechanistic explanation are indeed complementary. That preliminary step is not sufficient, of course: mechanistic explanation, being based on specific spatio-temporal causal stories, would still require that the specific structure of the modularity of the system—the specific wiring of the dynamical connections between the modules—be determined. All that the chain of ensemble explanations has provided us is a general fact: that there is modularity in the class of system considered, and what hierarchy a subsequent multilevel mechanistic explanation could plausibly exploit.

8. Multilevel Ensemble Explanations A remarkable feature of the chain of ensemble explanations is that at each explanatory step different classes of active parts are involved in the explanation.23 These are clearly not classes of physical parts, because explanations above EXPLANATION 0 concern not physical systems but abstract models: the boolean networks. Nevertheless, a network can be decomposed into subnetworks, and a subnetwork into its nodes, showing the possibility to form an abstract whole-part hierarchy. In the chain, between explanandum and explanans, though, there is not always a part-whole relation; dynamical modularity, a global property, is not engendered by features of its spatial components, but by structural modularity, another global property. So, this is not a uniform whole-part containment hierarchy as a hierarchy of mechanisms.24 However, it appears clearly that there is some sort of hierarchical order of some kind here, because each higher-level explanandum explanatorily depends on the lower-level explanans and not vice versa, even if they are both apparently global features. Moreover, by going top-down it appears we are in general dealing with progressively ﬁner-grained parts involved in the explanation, so in some way a micro/macro scale appears. How can we reconcile these apparently opposite features of the chain? The risk here is to confuse two kinds of hierarchical levels: mechanistic levels, based on physical part-whole relations (Craver 2015),25 and some other

23. See involved parts in ﬁg. 3 and 4. 24. See Craver (2015). 25. A capacity of the whole system is mechanistically explained by the interactions between its causally relevant parts.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc_a_00301 by guest on 26 September 2021 108 Multilevel Ensemble Explanations

kind of levels based on a sort of “logical” whole-part or even means-end relation. A possible answer could come by considering the hierarchies of functions resulting from functional analysis as proposed in Cummins (1975), where the production of a high-level capacity is explained in terms of how a set of sub-capacities, the functions, work in a coordinated manner. This is the very same relationship that holds between the specification of a computer program and its sub-functions;26 actually functional analysis constitutes a form of computational explanation. Loosely speaking, a specification states what a program should do (e.g., manage e-mail), while the implementation, usually expressed in the form of a program—a set of commands acting on a set of variables according to a structure of condi- tional rules—describes how the specification is brought about. Thus, the specification—that is, the overall high-level phenomenon we want to produce—is in a sense produced by a set of lower-level interactions and entities, lower-level in the sense that they provide more fine-grained information about how to bring about the phenomenon than the higher-level specification. This is a conception of levels of abstraction—abstraction understood as a process by which some information is omitted—in which the lower-level entities do not always correspond to proper constituent parts of the specification as a whole,27 but to means: detailed methods to produce it. As in a mereological hierarchy, the notions of specification and implementation of computer science are relative: a specification is a global property that does not mention in detail the way and means of its realization, delegating this task to the implementation. But implementation too can sometimes lack specific details on how to be brought forth, and will in turn require an even lower-level implementation of itself. Actually, a full hierarchy of specifications and implementations can be imagined; this is the craft of software engineering. The physical realization of a computation is of course obtained by taking as a specification the lowest software level and physically—mechanistically—implementing it in the hardware, similarly to what happens at EXPLANATION 0 of Kauffman’s chain. Ensemble explanations in the chain constitute a hierarchy of specifications and implementations, not based on containment of parts into wholes. But, this same hierarchy—or at least some segment of it—coincides with the hierarchy of functional explanation, which explains the overall capacity

26. See Galton 1993. 27. In some cases they do: single machine language instructions, at the lowest level of a software stack, are proper parts of a higher-level language single instructions, which are sequences of machine language instructions.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc_a_00301 by guest on 26 September 2021 Perspectives on Science 109

characteristic of a certain level by showing how it is realized by the dynamics of the immediately lower-level functional parts, and this by spec- ifying in detail how these parts are structured and how specifically they are related and interact. On the contrary, in a chain of ensemble explanations, even the lower-level explanation omits any implementation details. So, how are functional and ensemble explanation related? To clarify, let’s look at fig. 5. The two red diagonal arrows indicate the “way” of functional analysis, going from “what” to “how,” namely from specification to implementation: starting from a “what”—the specification of an overall capacity of the higher level (such as the presence of frozen structures)— functional explanation explains by producing a detailed and specific description of how the lower level brings about (implements) the higher- level capacity. In functional analysis the higher level/lower level relation is a complete analog of the specification/implementation relation. The blue diagonal arrows go instead from “what” to “what”:froma specification to another specification. I would call this the way of ensemble explanation: ensemble explanations, in a peculiar fashion, explain a “what”—an overall feature of a given level—by mentioning another, lower-level what. In other words, ensemble explanations explain specifications with lower-level specifications, without ever touching upon the implementation details. While functional or computational explanations explain “how,” ensemble explanations answer instead exclusively to “why” questions, making them a very peculiar kind of computational explanation. Despite omitting “how” details, chains of ensemble explanations are nevertheless able to give the questioner some sense of epistemic satisfaction, because they provide a description of progressively finer- grained explaining units, albeit in the form of the description of a common, distributed and thus global, typical feature of these units, without ever scrutinizing other details of the units and of the organization of their interactions, as instead a functional (or mechanistic, on the ontic side) explanation would require to do. It seems ensemble explanations along the chain are explanatorily exhaustive because, despite mentioning at each level a very general and generic fact as explanans, they mention as explanandum precisely the fact that confers to a possible functional explanation at the correspondingly lower level its explanatory power in relation to the higher level’s “why” question. For example, in answering the question “Why are there frozen structures at level n?,” an ensemble explanation could answer “Because at level n-1 a majority of the network’s nodes compute canalyzing boolean functions.” This generic fact concerning level n-1 is precisely the feature that would make any level n-1 functional explanation explain how the phenomenon of frozen structures at level n is produced. Since it is precisely by

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc_a_00301 by guest on 26 September 2021 110 Multilevel Ensemble Explanations

Figure 5. A fragment of the explanatory hierarchy.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc_a_00301 by guest on 26 September 2021 Perspectives on Science 111

virtue of this feature of being mostly composed of nodes computing canalyzing functions28 that subnetworks become frozen, largely independently of the specific structure of their internal connections, any functional explanation would proceed by citing how, in detail, the canalyzing function at each node, in interaction with the other connected nodes, progressively tends to maintain the fixed state of the node (and this for each node). In other words, this is the explanatory feature common to all good functional explanations of level n in terms of functions at level n-1. The sensation of “epistemic satisfaction” chains of ensemble explanations provide is also attributable to the fact that the very same hierarchy they in- habit can, if needed, be exhaustively investigated by functional or mechanistic analysis (depending on the level and on the nature of the system), obtaining this way the fullest amount of explanatory information about the system. Another important point is that ensemble explanations explain by mentioning only global properties29 typical of the entire class of objects at each level, so hierarchies of ensemble explanations allow for a higher economy of description in comparison to hierarchies of functional or mechanistic explanations, which must provide implementation details. This clearly distinguishes them from mechanistic explanations, which explain specificbehaviorofspe- cific systems. The economy of description allowed by ensemble explanations turns out to be really useful in case the system being explained is too large and complex to allow for a non-multilevel mechanistic explanation describing the detailed structure of the innumerable causal interactions engendering the overall functioning of the system. A purely non-multilevel functional explanation describing in full detail the network of functions implementing the global capacity to explain would encounter the same difficulty. In these cases, ensemble explanation is still possible, and its production by way of simulation seems, prima facie, to constitute the only feasible way of explanation, and certainly the most economical. As we have seen,30 this same preliminary step is able to identify, in systems manifesting an emergent modularity, a chain of ensemble explanations, and with it the hierarchy that could subsequently support a mechanistic or functional multilevel explanation. It seems then ensemble explanations are an enabling or at least a facilitating precondi- tion for other kinds of explanation of weakly emergent systems.

9. Discussion I argued above that in complex systems showing weak emergence like those considered by Kauffman, a direct mechanistic explanation of the

28. Or high-P functions. 29. A feature of topological explanations proper, as Kostic 2018 highlights. 30. Section 7 and this section.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc_a_00301 by guest on 26 September 2021 112 Multilevel Ensemble Explanations

emergent phenomenon is not humanly feasible, but an ensemble explanation is still possible and paves the way for the production of multilevel mechanistic explanations. A possible objection is that at least in Kauffman’s cases a direct mechanistic explanation is still possible because they do not show real weak emergence: in some of his works31 Kauffman actually manages to explain in detail,32 by means of some examples, how forcing structures make frozen components emerge. If these explanations were produced before running any simulation, Kauffman could have predicted—or at least suspected beforehand, that in networks where canalyzing functions prevail, the formation of frozen components is likely. So, there would be no unpredictable “emergent” phenomenon here. A reply: Kauffman’s detailed explanation is itself an ensemble explanation, expressed by a general statement, such as: “systems with a prevalence of canalyzing functions tend to have frozen components.” This statement could not result from the functional study of a specific single structure taken as example: Kauffman must have made multiple attempts, sketching and studying several example networks, before coming up with such a generalization. In a way, this is a sort of “paper simulation” on “random” samples. So, even if this were a case of bland weak emergence, without this paper simulation and the consequent ensemble explanation, we would be unaware of the modularity of the system, and remain unable to produce any high-level mechanistic explanation describing the overall behavior in terms of specific causal chains of interactions between dynamical modules. Even if Kauffman’s case actually lacked real emergence, at least a class of elementary cellular automata—a subclass of K ¼ 2 boolean networks with regular network structure and the same rule for all nodes—actually shows more evident cases of emergent behavior, unpredictable before running simulations (see Bedau 1997). The point is: a chain of ensemble explanations similar to Kauffman’s chain could certainly be produced even for these automata, and in this case the chain would have to precede a possible mechanistic explanation, in order to highlight the relevant explanatory levels in terms of which the latter can explain. Ensemble explanations somewhat resemble abstract mechanistic explanations as conceptualized in Arnon Levy and William Bechtel (2013), explanations omitting specific details of a mechanism in order to highlight its causal connectivity, an abstract schematization, usually in network form, of how the different parts of the mechanism are connected. Appar- ently, GOODMICROSTRUCT is a feature of the connectivity of the network. However, GOODMICROSTRUCT is not a specification of any

31. E.g., Kauffman 1971, p. 154 and Kauffman 1990, p. 143. 32. Space precludes a clariﬁcation here, but refer to ﬁg. 2 and its caption.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc_a_00301 by guest on 26 September 2021 Perspectives on Science 113

particular scheme of connectivity. In fact, it is a generic feature that applies to random networks. But, as examples in Levy and Bechtel (2013) show, abstract mechanistic explanations should refer in any case to certain specific recurrent wiring schematics between nodes, for example the so-called network motifs, usually realizing specific dynamical control circuits, such as specific types of feedback loops. GOODMICROSTRUCT instead generi- cally claims only that nodes with canalyzing or high-P functions numeri- cally prevail, not how these nodes are specifically connected in the network; the connection schemata considered are chosen randomly. Another crucial difference: mechanistic, even abstract mechanistic explanations, explain a specific behavior of a certain system (such as the arabinose regulation in bacteria, as in Levy and Bechtel 2013), while ensemble explanations target another kind of explanandum, namely, generic, global properties of all systems of a class. Just look again at Kauffman’s quotation in section 6. It states that an “ensemble theory”33:

• is not concerned with particular dynamical systems, but with classes of them; • it explains some of the ubiquitous global properties of cellular gene control systems; • it makes no statement whatsoever about detailed patterns of behavior of any speciﬁc system of the class. This probably shows ensemble explanations differ essentially from abstract mechanistic explanations. It can be observed that ensemble explanations resemble Robert Batterman’s minimal model explanations (Batterman and Rice 2014), which explain by citing general features of the class of abstract models to which a minimal model of the phenomenon belongs. However, Batterman downplays the importance of the representational role of the minimal model with respect to its explaining power, while the mapping provided by the mechanistic explanation at the bottom is essential for the justiﬁca- tion of the chain of ensemble explanations. Without this grounding, these explanations would not be guaranteed to apply to valid models of empirical phenomena. Another crucial feature differentiating ensemble explanations from minimal model explanations and, for that matter, from the so-called distinctively mathematical explanations described in Lange (2013) and from Philippe Huneman’s structural explanations, is that these explanations are based on mathematically certain facts, while ensemble explanations rest on only probable laws obtained by partial simulation on classes of abstract systems.

33. That is, an ensemble explanation.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc_a_00301 by guest on 26 September 2021 114 Multilevel Ensemble Explanations

A brief discussion of the probabilistic nature of the laws employed by ensemble explanations is needed. Properties of boolean networks, of their topology and dynamics, are of an abstract, logico-mathematical nature, since boolean networks are abstracts mathematical models, of the same nature as, say, the Turing machine. Besides, GOODMICROSTRUCT, grounding the chain, is a purely logical property. Thus, even if probabilistic, the “law” employed in ensemble explanations, connecting GOODMICRO- STRUCT with higher-level properties, is apparently not empirical, as would instead be a law derived from the observation of empirical facts. Even the probabilistic aspect of such laws seem different in nature from the probabilistic aspect of statistical empirical laws: the “law” results from a non-exhaustive investigation of the space of certain completely deterministic abstract mathematical structures—the boolean networks—an investigation comparable, say, to a computer-based investigation of the distribution of prime numbers. Besides, Kauffman actually proves analytically (Kauffman 1972) certain theorems about the probability distribution of forcing structures as the number of nodes in the network vary, showing that forcing structures arise with maximum probability for K ¼ 2. While not proving that any C3 network will show frozen components, this proves that this outcome is most likely. Here, probabilities derive not from observation of stochastic phenomena, but from a proof of the objective distribution of formal properties of abstract mathematical objects; statements about these probabilities have mathematical force. In principle, by exhaustively simulating a limited subspace of the boolean networks, instead of probabilistic generalizations, proper theorems proving dynamical features of the networks of each class of the partition C1–C3 of this subspace could be obtained, similarly as to how certain theorems (famously, the four-color theorem) are obtained computationally by exhaustively exploring spaces of possibilities, an approach actually attempted by Kauffman for small networks (Kauffman et al. 2004). So, the probabilistic aspect here is, in principle, possibly only apparent and provisional, even if, of course, in actuality a vast exhaustive exploration of significant portions of the space of possible networks is forever precluded by excessive computational complexity. It seems then we are not dealing here with proper statistical generalizations on partially stochastic phenomena, as for example a law regarding social norms; there is a lack of empirical content here. If the above arguments hold, ensemble explanations appear endowed with a quasi-mathematical modal force and with a lack of empirical content in their specifically explanatory aspect. This makes them definitely closer to Huneman’s structural explanations, and, together with the omission of details about chains of specific events, makes ensemble explanations clearly a case of non-causal explanations.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc_a_00301 by guest on 26 September 2021 Perspectives on Science 115

Although comprising a mechanistic explanation, Kauffman’s chain of ensemble explanations appears non-mechanistic, because the mechanistic lowest-level explanation serves as a representational mapping between the real genome and its theoretical model, while the properly explanatory device is a property of the class to which this abstract model belongs.

10. Conclusions I analyzed a famous argument by Stuart Kauffman and showed it contains a set of non-mechanistic, non-causal explanations of a kind that I propose to call, following Kauffman, ensemble explanation. This kind closely re- sembles Philippe Huneman’s structural explanation, but is different in that it lacks absolute mathematical certainty. Kauffman’sexplanations form a top-down chain in which each explanans is in turn recursively explained as an explanandum at a lower step, until the chain bottoms-out. This constitutes a hierarchy of explanations positioned at different levels along a scale corresponding to the classic specification/implementation hierarchy involved in functional analysis. Along this scale, ensemble explanations explain higher-level specifications by means of lower-level specifications. This constitutes a very economical and still epistemically satisfying style of explanation, allowing for the production of more classic functional explanations along the same hierarchy, if more detail on the specific processes of the system is needed. Interestingly, the whole chain of ensemble explanations requires a mechanistic micro-explanation at its bottom as a grounding condition guaranteeing that the theoretical model, employed by the whole chain of non-mechanistic ensemble explanations above it, represents the physical system. Nevertheless, when dealing with complex systems showing weakly emergent behavior, ensemble explanations themselves allow a preliminary identification of the relevant mechanistic hierarchy necessary for a subsequent multilevel mechanistic or functional explanation of features of the system. It seems that ensemble explanations do not compete directly with mechanistic explanations, but can fruitfully integrate with them. It is understood that the kind of explanation proposed here is still in need of further investigation, and is especially in need of more good examples taken from actual science to enable good generalizations.

References Batterman, Robert W., and Collin C. Rice. 2014. “Minimal Model Expla- nations.” Philosophy of Science 81 (3): 349–76. Bechtel, William, and Robert C. Richardson. 1993. Discovering Complexity: Decomposition and Localization as Strategies in Scientiﬁc Research. Princeton, NJ: Princeton University Press.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc_a_00301 by guest on 26 September 2021 116 Multilevel Ensemble Explanations

Bedau, Mark A. 1997. “Weak Emergence.” Noûs 31: 375–399. Craver, Carl F. 2015. “Levels.” Pp. 1–28 in Open MIND. Edited by Thomas K. Metzinger and Jennifer M. Windt. Frankfurt am Main: MIND Group. doi: 10.15502/9783958570498 Cummins, Robert. 1975. “Functional Analysis.” The Journal of Philosophy 72 (20): 741–65. Galton, Antony. 1993. “On the Notions of Speciﬁcation and Implementa- tion.” Royal Institute of Philosophy Supplements 34: 111–36. Glennan, Stuart. 2017. The New Mechanical Philosophy. Oxford: Oxford University Press. Huneman, Philippe. 2010. “Topological Explanations and Robustness in Biological Sciences.” Synthese 177 (2): 213–45. Huneman, Philippe. 2015. “Diversifying the Picture of Explanations in Biological Sciences: Ways of Combining Topology with Mechanisms.” Synthese 195 (1): 115–146. Huneman, Philippe. 2017. “Outlines of a Theory of Structural Explana- tions.” Philosophical Studies 175 (3): 665–702. Kauffman, Stuart A. 1971. “Gene Regulation Networks: A Theory for Their Global Structure and Behaviors.” Current Topics in Developmental Biology 6: 145–182. https://doi.org/10.1016/S0070-2153(08)60640-7 Kauffman, Stuart A. 1972. The Organization of Cellular Genetic Control Systems. Pp. 63–116 in Some Mathematical Questions in Biology. Edited by Jack D. Cowan. Providence, RI: American Mathematical Society. Kauffman, Stuart A. 1990. “Requirements for Evolvability in Complex Systems: Orderly Dynamics and Frozen Components.” Physica D: Non- linear Phenomena 42 (1–3): 135–52. Kauffman, Stuart A. 1993. The Origins of Order: Self-Organization and Selec- tion in Evolution. Oxford: Oxford University Press. Kauffman, Stuart A., Carsten Peterson, Björn Samuelsson, and Carl Troein. 2004. “Genetic Networks with Canalyzing Boolean Rules Are Always Stable.” Proceedings of the National Academy of Sciences 101 (49): 17102–17107. Kostic, Daniel. 2018. “The Topological Realization.” Synthese 195 (1): 79–98. Lange, Marc. 2013. “What Makes a Scientiﬁc Explanation Distinctively Mathematical?” The British Journal for the Philosophy of Science 64 (3): 485–511. Levy, Arnon, and William Bechtel. 2013. “Abstraction and the Organiza- tion of Mechanisms.” Philosophy of Science 80 (2): 241–61. Machamer, Peter K., Lindley Darden, and Carl F. Craver. 2000. “Thinking About Mechanisms.” Philosophy of Science 67 (1): 1–25.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc_a_00301 by guest on 26 September 2021