1.1 A Brief History

The philosophical literature on scientific explanation has experienced notable shifts since foundational work in the late 1940s, with the rise and fall of different accounts of explanation (Woodward and Ross Reference Woodward and Ross2021). A helpful starting point is the influential deductive-nomological (DN) model, articulated by Hempel and Oppenheim in a “first wave” of recent philosophical work on explanation (Hempel and Oppenheim Reference Hempel and Oppenheim1948; Baker Reference Baker2012, 243).Footnote ¹ This model suggests that scientific explanations are deductive arguments, in which (1) initial conditions and (2) laws of nature are used to (3) deduce and “explain” a target of interest (Hempel and Oppenheim Reference Hempel and Oppenheim1948).Footnote ² In order to see the structure of the DN account, consider a mercury thermometer that is placed in boiling water (Hempel Reference Hempel1965). When this thermometer is placed in boiling water, why does the mercury column first dip down in the glass thermometer case and then quickly rise? The explanation for this involves the initial expansion of the glass thermometer casing, which increases its inner volume such that the mercury level drops. However, once the temperature reaches the mercury inside, its level rises as mercury has a larger coefficient of expansion than glass. According to Hempel, this explanation is provided by antecedent conditions and general laws (Hempel Reference Hempel1965). The antecedent conditions include the constituents of the thermometer (the glass casing surrounding the mercury) and the fact that the thermometer was placed in boiling water. The general laws include laws about the thermic expansion of glass and mercury and the thermic conductivity of the glass. In this manner, the antecedent conditions and general laws are said to explain and “entail the consequence” that the mercury will drop and rise. On this account, the explanatory target is a “logical consequence” of the relevant antecedent conditions and general laws, with a structure shown in Figure 1.

Figure 1 The Deductive-nomological model of scientific explanation, rewritten from (Hempel Reference Hempel1965, 336). In this model, the initial conditions are represented by $C_1$, $C_2,\dots ,C_k$, while the general laws are $L_1,L_2\dots ,L_k$. The initial conditions and general laws make up the explanans (what does the explanatory work), which deductively lead to the explanandum (or outcome to be explained).

While the DN model has its strengths, it also suffers from significant problems. First, this model fails to accommodate explanations in sciences that often lack strict, universal laws of nature, such as biology and the life sciences. Genes and environmental factors are cited in explaining traits in organisms, but these explanations rarely (if ever) appeal to general laws from which these traits deductively follow. A second problem is that the DN model fails to capture the directionality of explanation. This is seen in the well-known “flagpole problem,” in which a flagpole’s height, the sun’s location, and laws of optics explain the length of the flagpole’s shadow (Hempel Reference Hempel, Peigel and Maxwell1962; Bromberger Reference Bromberger1992).Footnote ³ The issue for the DN model is that it incorrectly counts the reverse direction as also explanatory – it implies that the shadow’s length (with the sun’s location and same general laws) explains the flagpole’s height, as this also meets the DN criteria. These and other limitations began to receive attention in the 1960s, which led to increased interest in alternative models of scientific explanation (Woodward and Ross Reference Woodward and Ross2021).

One celebrated solution to these problems was found in accounts of causal explanation. On accounts of causal explanation, an outcome is explained by citing the causes that produce it, as causes explain their effects. This solves directionality because causation is asymmetric and contains a clear direction, from cause to effect. This allows one to count the first flagpole scenario as the true explanation, because the explanatory direction is captured by causality – the sun’s rays cause, and thus explain, the shadow and not vice versa (Salmon Reference Salmon1989). Another advantage is that the causal framework accommodates explanations in sciences that lack strict laws of nature. These explanations only require causal regularities, which should hold in some set of conditions but need not hold universally. If gene X causes brown coat color in guinea pigs in some background conditions, this explains the trait even if such a regularity does not hold for other animals or in other conditions.

For these and other reasons, causal accounts superseded the DN model as the new reigning view of scientific explanation. This change was supported by increased attention to examples of explanation outside of the physical sciences – such as biology, neuroscience, ecology, and sociology – which were better captured with causal frameworks than the DN model. This encouraged a growing literature on causal explanation and work examining various definitions of causality.Footnote ⁴ This literature would explore a multitude of topics such as causal selection, absence causation, explanatory reduction, causal complexity, causal pluralism, and mechanistic explanation. In this work, causal selection has to do with how to “select” the most explanatory causes (Lewis Reference Lewis1986; Waters Reference Waters2007; Ross Reference Ross2018), absence causation concerns whether absences are genuinely causal or not (Beebee Reference Beebee, Hall and Paul2004), and explanatory reduction pertains to whether lower-level causes are more explanatory than higher-level causes (Sober Reference Sober1999; Bickle Reference Bickle2006). Causal complexity can refer to situations in which causes are numerous, where causes are related in intricate ways, and where causes are difficult to identify in the world (Wimsatt Reference Wimsatt1972; Lewis Reference Lewis1986; Mitchell Reference Mitchell2009; Ross Reference Ross2023b). Causal pluralism refers to different projects, including views that there are different legitimate definitions of causation, that there are distinct methods for identifying causality, and that there are different types of causal explanation in science (Cartwright Reference Cartwright2004; Hitchcock Reference Hitchcock and Machamer2007; Godfrey-Smith Reference Godfrey-Smith, Beebee, Hitchcock and Menzies2013). Other work has examined causal mechanisms and mechanistic explanation, in which outcomes are explained by citing the causal mechanisms that produce them. The mechanistic research program has become a mainstream philosophical account of explanatory practice in the life sciences (Machamer et al. Reference Machamer, Darden and Craver2000; Bechtel and Richardson Reference Bechtel and Richardson2010; Glennan Reference Glennan2017) and what some consider the “dominant view of explanation in the philosophy of science” (Kaplan and Craver Reference Kaplan and Craver2011). Some work in this area implicitly assumes that all scientific explanations are causal, while other projects directly acknowledge support of this view (Skow Reference Skow2014).

While the philosophical literature experienced an increase in fruitful work on causation, the view that all scientific explanations are causal would be short-lived. An appreciation for other, non-causal forms of explanation has been supported by examples from many scientific domains. The life sciences have played a central role in these debates as many purported examples of non-causal explanation come from biology, neuroscience, ecology, and so on. Examples of these non-causal explanations include explanations of robust and fragile ecosystems (Huneman Reference Huneman2010), of neural coding and firing behaviors (Ross Reference Ross2015; Chirimuuta Reference Chirimuuta2018), of the prime number life cycle of cicadas (Baker Reference Baker2005; Batterman Reference Batterman2010), the hexagonal-shaped honeycombs of bees (Lyon and Colyvan Reference Lyon and Colyvan2008; Lange Reference Lange2013), and why animal size is limited by the square-cube law (Ross Reference Ross2023c). In these examples, it is suggested that an important part of the explanatory power comes from mathematics, which is not the case in causal explanations. There are extensive, ongoing debates about how exactly these non-causal, mathematical explanations work – what their features, details, and nature are. As these debates continue, a growing consensus in the field views non-causal, mathematical explanations as legitimate and accepts that scientific explanation is far more diverse than earlier work assumed. While earlier theories of scientific explanation searched for the single, universal account that captures all examples of scientific explanation,Footnote ⁵ more recent work views explanation as a diverse enterprise (Woodward Reference Woodward, Reutlinger and Saatsi2019; Woodward and Ross Reference Woodward and Ross2021). This brings us to a relatively new position in the field – an appreciation for various types of scientific explanation, without expecting that they all fit a single model, but still requiring that they specify rigorous standards for what counts as explanatory.

This Element focuses on two main categories of scientific explanation – causal and non-causal explanation. These categories of explanation are examined in the context of biology and the life sciences. The discussion of causal explanation (Section 2) will consider foundational topics in this area and three main types of causal explanation – these include explanations that appeal to mechanisms, pathways, and cascades (Ross Reference Ross2021a, Reference RossIn Press). In providing an analysis of non-causal explanation (Section 3), three main classes of non-causal, mathematical explanation will be considered, including topological and constraint-based explanations, optimality and efficiency explanations, and minimal model explanations (Batterman Reference Batterman2001; Baker Reference Baker2005; Lange Reference Lange2018). This work will examine similarities and differences across these explanatory patterns – their shared structural features, different guiding principles, and paradigmatic examples of each type. This work balances an appreciation for distinct explanatory patterns, with the view that explanations should meet rigorous standards.

While this work focuses on causal and non-causal explanation, other categories of scientific explanation have been studied. Examples of these other accounts include unificationist explanations (Friedman Reference Friedman1974; Kitcher Reference Kitcher1989), narrative and historical explanations (Roth Reference Roth1988), functional explanations (Wright Reference Wright1976), and structural explanations (Garfinkel Reference Garfinkel1981), among others. While causal and non-causal explanation have received significant attention in recent work, questions remain about how they both relate to the other explanation classes mentioned earlier.

This Element is organized in the following manner. The next subsection (1.2) introduces the basics of scientific explanation that will help in understanding the distinct accounts of explanation to come. The second section (2) examines topics in causal explanation including various foundational questions and forms of causal explanation, including mechanism, pathway, and cascade explanations. The third section will discuss types of non-causal, mathematical explanation, including topological and constraint-based explanation, optimality and efficiency explanations, and minimal model explanations. The fourth section will provide concluding remarks on the diversity of scientific explanation in biology and the life sciences.

1.2 The Basics of Explanation

Scientific explanations often involve three main components – the explanandum, explanans, and dependency relation, shown in Figure 2. The explanandum is the explanatory target or phenomenon to be explained. The explanans is what does the explanatory “work” or provides the explanation. A final component is the dependency relation between the explanandum and explanans, which captures how they are related to each other. These elements are found in most explanations, but their features vary across different types of explanation (as will be discussed more soon). Many accounts of scientific explanation elaborate on versions of this three-part model in order to capture the standards that scientific explanations need to meet (Reutlinger Reference Reutlinger2016; Jansson and Saatsi Reference Jansson and Saatsi2017; Woodward Reference Woodward, Reutlinger and Saatsi2019). Identifying these standards is important because it would allow us to distinguish genuine explanations from non-explanations and to determine when explanations are better or worse. The notion of explanation that is examined in this work does not include all broader uses of “explanation” in everyday life, such as explaining the meaning of an expression, how to bake a cake, or why a decision was made (which can involve a detailed description or justification) (Woodward Reference Woodward2003, 4). Instead, it focuses on explanations of natural phenomena, often why they have occurred or why they have particular features.

Figure 2 The three-part model of scientific explanation contains: the explanandum (phenomenon to be explained), the explanans (what does the explanatory work), and the dependency relation that connects both. Most philosophical accounts of scientific explanation fit this three-part model, but they differ with respect to what counts as the explanans and how to understand the dependency relation.

Discussions of scientific explanation frequently start with the explanatory target. A scientific explanation cannot be provided until the explanatory target is sufficiently clear.Footnote ⁶ In the philosophical literature, the explanatory target is presented as an outcome that scientists are interested in, that they “stumble across by accident” and find surprising, and that calls out for explanation (Hempel Reference Hempel1965; Baker Reference Baker2005). The target of interest is couched in terms of a why-question, which frames the inquiry and ensuing explanatory quest.

Providing an explanation requires precisely specifying the explanatory target. This precision involves identifying the contrastive focus of interest and meaning of terms in the why-question. Consider the explanatory why-question: “Why does guinea pig A have spotted-coloring on its trunk?” Without further clarification, this question is ambiguous and does not sufficiently specify the explanatory target. For example, this question might be asking why guinea pig A has spotted-coloring on its trunk versus solid coloring; or why guinea pig A has spotted-coloring on its trunk as opposed to its legs; or why guinea pig A has this spotted-coloring as opposed to guinea pig B. Each of these contrasts refers to a different explanatory why-question that may have different answers. Specifying the explanatory target with precision is essential for ensuring that scientists are discussing the same explanatory question and not talking past each other. Clarity here helps avoid situations in which scientists mistakenly compare explanations without realizing that the explanations have different targets of interest and, thus, aim to explain different things. This is especially important because natural phenomena can be represented in different ways and have distinct features, which emphasize different contrasts that we might want to explain.

Designating a clear explanatory target also requires defining relevant terms in the explanatory why-question and explanandum. If scientists aim to explain a phenomenon that lacks clear characterization, is ill-defined, or invokes multiple definitions, there are likely to be challenges in providing an explanation. Consider the question, “Why do human beings have lungs?” which we might seek an answer to. As Nagel states, this “question as it stands is ambiguous” because it can be interpreted in distinct ways (Nagel Reference Nagel1961, 19). The principle outlined here is that, if it is not clear what we want to explain, then we are not equipped to provide or assess an explanation. The explanatory target can change through scientific work, but if explanatory potential is to be assessed then some target must be fixed and specified. As a further example, if “consciousness” is defined in myriad or ambiguous ways, then determining how to explain consciousness and whether it has been explained are likely to be contentious and unsettled. If there is little agreement on how to characterize Y, there will likely be little agreement on whether some account explains Y or not.Footnote ⁷ In short, providing an explanation requires that one is very clear about the phenomenon to be explained.

A second main component of scientific explanations is the explanans or what does the explanatory work. This is associated with the “answer” that is provided to the explanatory why-question. If we want to explain why guinea pig A’s trunk is spotted as opposed to solid-colored, we might appeal to a gene that is responsible for this phenotype. Perhaps other phenotypes (behaviors, disease states, etc.) are caused by environmental factors, such as stress, temperature, and contagions. Factors that are cited in explaining the outcome are said to have “explanatory power” with respect to the target of interest. It is common to view these explanatory factors as “difference-makers” for the target of interest, such that changes in the explanans “make a difference” to the states of the explanandum. Relatedly, the explanandum is said to “depend” on the explanatory factors specified in the explanans.

Most philosophical work on scientific explanation is focused on the explanans and criteria that it needs to meet to ensure that a legitimate explanation is provided. Explanations provide deep understanding in a way that is distinct from other scientific projects such as mere description, prediction, and classification. While it is important to describe, predict, and classify distinct guinea pig coat coloring patterns, these are different tasks than explaining why one of these colorings is produced. Accounts of scientific explanation should exclude cases in which a purported explanans simply redescribes or predicts the explanandum. A classic example of the latter is a common cause scenario, shown in Figure 3, in which (A) air pressure causes two distinct outcomes, namely, (1) changes to a barometer reading and (2) changes in storm occurrence (Woodward Reference Woodward2003, 14–15). Notice that the (1) barometer accurately predicts (2) storm occurrence, but we would not say that it explains this outcome. The correlation between (1) and (2) allows this relationship to be useful and reliable, but we are more likely to view both (1) and (2) as caused and explained by the (A) air pressure. This type of example has led to claims that explanatory factors need to do more than simply predict outcomes or make them more probable. Accounts of scientific explanation aim to specify exactly what standards the explanans and explanatory factors need to meet, to ensure that a genuine scientific explanation has been provided.

Figure 3 Air pressure is a common cause of both the barometer reading and storm occurrence – air pressure causes and explains both of these outcomes. While the barometer reading accurately predicts (and is correlated with) the storm, it does not cause or explain it.

The third feature of scientific explanations is the dependency relation that connects the explanans and explanandum. It has been suggested that many diverse forms of scientific explanation share this dependency relation feature – that they involve some specification of how the explanandum is dependent on factors in the explanans (Woodward Reference Woodward2003; Reutlinger Reference Reutlinger2016; Jansson and Saatsi Reference Jansson and Saatsi2017). In some cases, this dependency relation helps distinguish types of explanations, including causal explanations from non-causal, mathematical explanations (Woodward Reference Woodward, Reutlinger and Saatsi2019). For causal explanations, this dependency relation is causal as the factors in the explanans and explanandum are connected causally. Alternatively, for some accounts of non-causal explanation, the dependency relation is said to be mathematical and non-empirical in character (Woodward Reference Woodward, Reutlinger and Saatsi2019). While causal dependencies are revealed through an empirical, a posteriori study of the natural world, mathematical dependencies are identified through mathematical and a priori considerations. This difference between causal and mathematical dependencies bears similarity to Hume’s distinction between “matters of fact” and “relations of ideas,” referred to as “Hume’s fork” (Hume Reference Hume1985). While “matters of fact” are a type of knowledge revealed by experiences of the world, “relations of ideas” are “discoverable independently of experience” as they are logical and mathematical propositions, that are true by definition and known through reason alone (Morris and Brown Reference Morris and Brown2019). Other unique features of these mathematical dependencies are that they can exhibit a stronger form of necessity than standard causal relations (Lange Reference Lange2018) and they can contain other types of non-causal information (Chirimuuta Reference Chirimuuta2018). This is discussed in more detail in Section 3, which covers different types of non-causal, mathematical explanation.

The focus of this work will be to identify the standards and “formal pattern” of explanation in biology and the life sciences (Nagel Reference Nagel1961). This work aims to strike a balance between capturing standards that scientific explanations should meet, while appreciating that there are different patterns of explanatory practice. Accounts of scientific explanation should capture actual scientific work, but they should also rule out non-explanations. Scientific explanation is not an “anything goes” type of project – correct explanations guide medical treatments, public policy interventions, regulations on our environmental impact, and accounts of how to control and change outcomes in the world. An important role of philosophy involves clarifying the features, characteristics, and structure of scientific explanations and how they give us a principled understanding of the natural world.

2.1 Foundational Questions

In the context of causal explanation in biology and the life sciences, three foundational topics include defining causation, causal selection, and reductive explanation. With respect to the first, accounts of causal explanation require defining causation, which involves specifying the characteristic or hallmark features of causality. Once these features are specified, they can be used to distinguish true, genuine causal relationships from everything else, including non-causal relationships and mere correlations. The ability to define causality and reliably make these distinctions is very useful and it has occupied significant attention in philosophy and in science. As causal explanations cite causal factors, a definition of causality is required to identify which factors, relationships, and systems are legitimately causal and capable of providing these explanations. While many different definitions of causation have received attention, four of the most commonly discussed include regularity accounts (Hume Reference Hume1985; Mackie Reference Mackie1965), connected process accounts (Salmon Reference Salmon1984; Dowe Reference Dowe2018), probabilistic accounts (Suppes Reference Suppes1970; Cartwright Reference Cartwright1979), and interventionist accounts (Woodward Reference Woodward2003).Footnote ⁸

While all of these accounts of causation have proponents, one of the most influential and widely used accounts of causation is Woodward’s interventionist framework (Woodward Reference Woodward2003).Footnote ⁹ This account is motivated by causal reasoning in scientific contexts, empirical studies of human cognition, and formal methods of causal analysis (Woodward Reference Woodward2021). On the interventionist account of causation, the relata of causal relationships are variables (X, Y, Z, etc.), which represent properties of interest, such as phenotypes, genes, and environmental factors. These variables take on different values ($0,1,2$, etc.) that represent the states of these properties, such as coat color that is brown, gray, and black, or a gene variant that is present or absent. On the interventionist account of causation, to say that X is a cause of Y means that if X were intervened upon and changed, in background conditions B, this would lead to changes in Y. For example, to say that gene G causes a guinea pig’s coat to be brown, as opposed to gray or black, means that changes to this gene G (in early development) would have produced a different coat color. In this manner, causes are factors with “control” over their effects, as manipulating them would produce changes to the effect of interest.

This helps reveal the connection between causation and causal explanation. Explaining why, for example, a guinea pig has coat color that is brown, as opposed to other colors, requires citing the cause or causes that “make-a-difference-to” this outcome (Woodward Reference Woodward2003; Waters Reference Waters2007). In this case, it involves citing a genetic cause because if the genetic cause had been different, in the same set of background conditions, an alternative coat color would have been produced. This reveals the importance of a contrastive focus in specifying an explanatory target and how explanatorily relevant causes should relate to this target. If an environmental factor (such as changes in temperature) met these criteria, then the environmental factor would be viewed as the cause or main cause of coat color. However, true background conditions – such as the presence of oxygen, diet, and so on – should be excluded as main causes and explanations of this phenotype. While background conditions are present and supportive, they are not causally responsible for this phenotype because manipulating them does not control whether the phenotype is in one state or another. If oxygen were manipulated (from present to absent) this would control whether the guinea pig is alive or dead, but this is not the contrastive focus of interest. We are interested in what explains one coat color versus other colors in a living guinea pig. When background conditions are required for life, they are necessary conditions for phenotypes to manifest, which prevents them from explaining, causing and controlling the presence of one phenotype versus another. This is why many background conditions do not explain the outcome of interest (even if they seem relevant to it) – they are not explanatory if they do not “make a difference” to the contrastive focus in question.

The interventionist account does not require that causes are actually manipulable with current technology or according to ethical standards. Interventionism just requires that we can consider hypothetical manipulations, which involve what would happen if the intervention were to occur (Woodward Reference Woodward2016). This is consistent with causal reasoning in scientific and everyday life contexts, in which we accept causal claims even when we cannot perform interventions on the relevant causal factor(s). Examples of legitimate causes that we cannot intervene on include: historical or past causes (“yesterday the medicine cured my headache”), causes that are not manipulable with current technology (“the location of the moon causes the tides to ebb and flow”), and causes that we cannot manipulate for ethical reasons (“the patient’s genes have caused their disease”). The interventionist account includes these as genuine causes because they meet the hypothetical manipulation requirement – in all of these cases, we can consider hypothetical manipulations of the cause, and our evidence and theory supports the claim that this hypothetical manipulation would change the effect of interest.Footnote ¹⁰ If a purported causal claim cannot be associated with a hypothetical manipulation – perhaps because the suggested changes to the candidate cause are inconsistent with scientific theory or because the purported causes and effects are not sufficiently defined (as mentioned in subsection 1.2) – then this gives good reason to be skeptical that a causal claim has been provided (Woodward Reference Woodward2003, Reference Woodward2016).

Significant amounts of philosophical work has focused on the question of how to define causality. While interventionism provides one answer to this question, there remains the further topic of how many definitions of causation we need to capture causal relationships in the world. This relates to “causal pluralism,” which is associated with different views in the literature (Cartwright Reference Cartwright2004; Hitchcock Reference Hitchcock and Machamer2007; Godfrey-Smith Reference Godfrey-Smith, Beebee, Hitchcock and Menzies2013). Three forms of causal pluralism to keep distinct are definitional pluralism, methodological pluralism, and (what I call) structural pluralism. Definitional pluralism about causation claims that we need two or more definitions of causation to capture causality in the world. Definitional monism about causation claims that we only need one definition of causation to capture causality in the world. Alternatively, methodological pluralism refers to the number of methods that can identify (or provide evidence of) causality in the world (Shadish et al. Reference Shadish, Cook and Campbell2002; Reiss Reference Reiss2009). A methodological pluralist claims that many methods can be used to identify causal relationships, while a methodological monist claims that only one method can provide this (with a main contender being randomized control trials).Footnote ¹¹ A third type of causal pluralism–structural pluralism about causation – captures distinct causes and causal systems that can meet the same definition of causation, but differ with respect to other features. Examples discussed later in this section are mechanisms, pathways, and cascades, which are distinct causal systems (Ross Reference Ross2021a, Reference RossIn Press). Further examples are differences between causes that are proximal, distal, predisposing, exciting, and organized in different manners (such as linear chains, feedback loops, etc.). While these causal systems can all meet the same definition of causation, they differ with respect to other features. While most scholars accept some type of structural pluralism, versions of structural monism are seen in claims that all causal systems are best understood as “mechanisms” no matter what their differences are (Craver Reference Craver2007; Levy and Bechtel Reference Levy and Bechtel2013; Craver and Tabery Reference Craver and Tabery2015).Footnote ¹²

A second foundational topic in the area of causality has to do with causal selection, which refers to selecting or identifying causes that are the most explanatory for an outcome. Consider that the number of causally relevant factors for any target of interest can be seemingly “infinite” (Lewis Reference Lewis1986, 214). For any explanatory target, there appear to be an enormous set of factors far “back” in its causal history (extending back to the Big bang) and far “down” at lower scales (such as fundamental physics). While this list of factors is nearly endless, we do not view all of these causes as equally explanatory for the outcome – only a few are selected as the main, explanatory causal factors. A main question for any account of explanation is how to determine which factors are most explanatory and why. What guidelines or criteria distinguish causal factors that are more explanatory from those that are less explanatory? In philosophy of science, there are longstanding debates about whether a cogent rationale for causal selection exists or if it is hopelessly unprincipled. For example, Mill and Lewis have claimed that causal selection is guided by “capricious” and “invidious” sentiments, respectively (Mill Reference Mill1874; Lewis Reference Lewis1986). Lewis suggests that when we select the main causes of an outcome, that this is guided merely by our interest in these causes, the fact that we find these causes to be good or bad, or simply because they are causes under our control. These claims suggest that causal selection is guided by arbitrary, subjective, and non-scientific reasons.Footnote ¹³ Such a position is problematic for the view that scientific explanation has a principled nature – if causal explanation and causal selection are guided by principled considerations, these principles and the role they play need to be specified.

In the biological sciences, a central example of causal selection is genetic causation, in which genes are viewed as the main cause for various traits (sickle cell disease, Huntington’s disease, etc.). Significant literature has focused on what rationale justifies privileging genes as the main causes for these outcomes. An important tool that can be used to address these questions involves clarifying various “distinctions among causation,” which capture objective differences across types of causal factors Woodward (Reference Woodward2010, Reference Woodward2021). For example, some causes are more specific, stable, strong, or fast than others – these differences can figure in why some are more or less explanatory for an outcome. It has been suggested that genes are prioritized in scientific explanations, because they are more specific and stable than other causal factors (Waters Reference Waters2007; Woodward Reference Woodward2010; Weber Reference Weber2017, Reference Weber2022). A specific cause (in the sense used in these discussions) is a cause that produces fine-grained control over an outcome – this is seen when specific changes in DNA result in fine-grained changes in the protein product created.Footnote ¹⁴ This type of specificity can be understood as a cause variable with many values, in which each values corresponds to a unique value in the effect variable. (This is similar to a dimmer switch that has fine-grained control over the brightness of a light.) This is contrasted with a non-specific cause, in which binary values of the cause variable control a binary outcome of the effect (as seen in an on/off switch for the on/off of a light bulb). Additionally, a stable cause is a factor that exerts its causal influence across a wide range of changes in background conditions. Consider that various gene mutations are stable in the sense that they cause disease in patients even when there are differences in diet, childhood upbringing, and other environmental factors (Kendler Reference Kendler2005; Woodward Reference Woodward2010). Research in empirical psychology shows that humans view stable causes as more paradigmatically causal than less stable causes, and stable causes clearly have advantages when it comes to understanding and control (Cheng Reference Cheng1997; Lombrozo Reference Lombrozo2010; Vasilyeva et al. Reference Vasilyeva, Blanchard and Lombrozo2018).

Another important causal distinction, especially in the context of causal selection and explanatory causes, is causal strength. The strength of a causal relationship refers to the degree to which a cause increases the probability of its effect or produces a large magnitude of change in its effect.Footnote ^15, Footnote ¹⁶ A classic example are Mendelian genes, which are frequently described as having causal action that is “deterministic” in the sense that the presence of the gene ensures occurrence of the trait (Kendler Reference Kendler2005). Other causes are said to be “probabilistic” as they increase the probability of their effects, without guaranteeing or determining them (Parascandola and Weed Reference Parascandola and Weed2001; Kendler Reference Kendler2005).Footnote ¹⁷ Similar to specificity and stability, causal strength is a feature that comes in degrees, as causes can differ continuously with respect to how probability boosting their causal influence is. Many paradigmatic causes in biology and the life sciences are not only highly stable (according to the aforementioned description), but also rank high in terms of strength. This is seen for Mendelian traits and diseases that fit the monocausal model – most of these cases cite single causes that produce their effects with a high probability and in a wide range of contexts.Footnote ¹⁸ While stability is similar to the “generalizability” of a causal relationship, strength is similar to scientific discussions of “deterministic” and “probabilistic” causes, which concerns whether a cause guarantees its effect or not (or how probability boosting it is for its effect). Factors that are often referred to as “sufficient causes” often rank high in terms of both stability and strength (and notions of sufficient causation can problematically conflate these distinct causal varieties).

Many other distinctions among causation have been studied in recent work. These include whether causes produce their effects in ways that are fast or slow, irreversible or reversible, in ways that transfer material to the effect or not, and in ways that are proportional (Woodward Reference Woodward2010; Ross Reference Ross2018; Ross and Woodward Reference Ross and Woodward2022; Ross Reference Ross2023a).Footnote ¹⁹ Research into these and other causal distinctions is a helpful approach, in efforts to capture how main, explanatory causes can (and should be) distinguished from less explanatory causes. The fact that scientific communities often reach consensus on the main causes of outcomes – and that these causes serve specific goals, such as control – supports the view that causal selection in science is often guided by principled considerations, as opposed to ever-changing subjective preferences.

A third foundational topic in the causation literature is explanatory reduction, which refers to how far “down” an explanation should go when citing causal factors (Mayr Reference Mayr1989).Footnote ²⁰ These debates commonly reference a “layer cake” picture of the world, in which phenomena differ with respect to the “scale” or “level” they reside on (Batterman Reference Batterman2001; Waters Reference Waters and Ruse2009; Potochnik Reference Potochnik, Brooks, DiFrisco and Wimsatt2021). This view suggests a hierarchical picture in which distinct sciences focus on different scales or levels – the lower levels are studied by physics, the next higher levels by chemistry, and further increasing levels are the focus of biology, psychology, and various social sciences (Wimsatt Reference Wimsatt, Globus, Maxwell and Savodnik1976; Sober Reference Sober1999).Footnote ²¹ Suppose an explanatory target is specified in the context of biology, such as an organism’s phenotype. With respect to this phenotype, are explanations that cite factors from lower biological scales – such as molecular biology – better than explanations that cite factors from higher-scales such as cellular networks, neural circuits, pathways, and system-level topologies? Will the best explanations of these outcomes cite causes from the biological scale or should it ultimately include causes from chemistry and physics? Many who argue for explanatory reduction claim that the best explanations – even of biological outcomes – will ultimately cite causal factors from lower-levels (Sober Reference Sober1999; Strevens Reference Strevens2008; Butterfield Reference Butterfield2011).

These questions about explanatory reduction can be illustrated with Sober’s discussion of explanations of lung cancer (Sober Reference Sober1999). In this case, the higher-level causal regularity is “smoking cigarettes causes lung cancer,” which is represented by “P $\to$ Q” in Figure 4. In this example, causal variable P “smoking cigarettes” is realized by different lower-level carcinogens, namely, carcinogens $A_1$, $A_2,\dots A_n$. Similarly, effect variable Q “lung cancer” is realized by distinct cellular cancer types, namely, $B_1$, $B_2,\dots B_n$. A main question with respect to this example is which causal factors best explain the higher-level lung cancer phenotype Q? In particular, are causes at lower or higher levels more explanatory?

Figure 4 Based on Fodor (Reference Fodor1974) and Sober (Reference Sober1999).

Sober claims that lower-level causal details provide objectively superior explanations when compared to higher-level causes. He claims that, while higher-level causal detail might be preferred by some scientists and might give explanatory breadth, this higher-level causal detail never provides an objectively superior explanation compared to lower-level causal details (Sober Reference Sober1999; Ross Reference Ross2020). Sober argues for this position with three main points. First, he claims that lower-level causes can always be included in an explanation, without reducing explanatory power, while the same cannot be said for higher-level causes. He suggests that lower-level causal detail is never unexplanatory – it might be more than a scientist wants to hear, but at worst it (simply) “explains too much” (Sober Reference Sober1999). In other words, just because we “may not want to hear the gory details $\dots$ does not mean that the details are not explanatory” (Sober Reference Sober1999, 549). Second, Sober claims that lower-level causes are what do the real “work” in producing the explanatory target of interest and that this justifies their explanatory priority over higher-level causes (Sober Reference Sober1999, 548). A final point he makes is that lower-level causal detail – such as detail from fundamental physics – has the advantage of providing “causal completeness,” which cannot be supplied by higher-level causal detail. As he states, “if singular occurrences can be explained by citing their causes, then the causal completeness of physics ensures that physics has a variety of explanatory completeness that other sciences do not possess” (Sober Reference Sober1999). In other arguments in the literature, explanatory reduction is supported by views that higher-level detail provides “shallow explanations” compared to “deeper accounts” provided by lower-level causal information (Waters Reference Waters1990).

In contrast to explanatory reduction, I suggest that a level-agnostic view of causal explanation is more compelling. On a level-agnostic view of causal explanation the most explanatorily relevant causes for an outcome can be at any scale or level (not just at lower-levels). Indeed, in some cases, higher-level causes are more explanatory than lower-level causal details. The guiding principles of this framework are that (i) the level of causal detail that is explanatory depends on the explanatory target of interest and (ii) the explanatorily relevant causes should “make a difference to” and provide control over this explanatory target. In some cases, higher-level causes better meet these standards than lower-level ones – this captures the principled reasons for why the higher-level causes are more explanatory. The pithy way to capture this level-agnostic view is to say—causal explanation is not a game of how low can you go, but what gives you control. In some cases, the causes with control (over the effect) are at higher scales.

The strength of this level-agnostic view of causal explanation can be illustrated with Sober’s lung cancer example and discussions of multiple realizability (Putnam Reference Putnam, Capitan and Merrill1967; Fodor Reference Fodor1974; Ross Reference Ross2020). Multiple realizability refers to situations in which some higher-level phenomenon x (such as smoking cigarettes) is multiply realized by distinct lower-level details, $z_1$, $z_2,\dots z_n$ (such as distinct carcinogens that vary across cigarette types). In the context of Sober’s smoking example, the multiple realization of x relates to a type of causal complexity – called causal heterogeneity – in which distinct causes or combinations of causes ($z_1$, $z_2,\dots z_n$) are all individually sufficient to produce the same effect (y). This is seen in Figure 5 that captures how distinct lower-level carcinogens ($z_1$, $z_2,\dots z_n$) are all able to cause the same higher-level lung cancer outcome (y).

Figure 5 This diagram is based on Fodor (Reference Fodor1974) and Sober (Reference Sober1999). The property x is multiply realized by $z_1$, $z_2,\dots z_n$, and each of these realizers is a cause of y, making y causally heterogeneous.

Suppose that our explanatory why-question is: “What is the cause of lung cancer in this population?” If we want to explain this type or population level target y – all cases of lung cancer – there is a problem with appealing to a lower-level carcinogen, such as z₁. The problem is that this lower-level cause z₁ only “makes a difference” to and explains a small fraction of lung cancer cases and we want to explain all of them. Intervening on z₁ does not control or account for all lung cancer outcomes, as the other lung cancer outcomes are produced by other causes ($z_2$, $z_3\dots$). Alternatively, a clear advantage of the higher-level causal factor “smoking cigarettes” (x) is that it does “make a difference to” and provide control over all (or most) instances of this disease. Preventing most or all lung cancer outcomes is possible with smoking cessation, unlike interventions on specific carcinogens, which differ across cigarette types. This is why public health campaigns purposefully target smoking habits (and not individual carcinogens) in their efforts to reduce lung cancer. However, if the explanatory target is a token or single instance of lung cancer, the lower-level carcinogen can easily surpass the higher-level cause in explanatory power. In this case, the lower-level cause “makes-a-difference” to the outcome and provides control over it. An issue with Sober’s analysis is that it focuses on the token-outcome explanation, while multiple realizability arguments are concerned with type-level explanatory targets (Ross Reference Ross2020). As life scientists often want to understand and explain reoccurring, type-level outcomes, the challenge of multiple realizability (and citing lower level-causes) is significant. A central message in this analysis is that the causal factors that are explanatory and the “level” that they reside at can change with the explanatory target of interest.Footnote ²²

A main suggestion of this analysis is that different explanatory targets can require different levels of causal detail – there is not a particular level of causal detail that is always privileged or objectively better in providing scientific explanations. Once an explanatory target is specified, explanatory causes need to be factors that provide control over the target. In some cases, causal factors at higher-levels provide better control and explanation of the explanatory target of interest, when compared to causal factors at lower-levels.

Finally, it is worth addressing claims about whether biology can be – in principle or in some future science – completely reduced to or explained away with fundamental physics. Some challenges for such claims are that they are often guesses or desires about future science, without reflecting actual scientific work. These guesses differ from person to person, and they are difficult to support with evidence or argument. In other cases, such claims are motivated by metaphysical or philosophical assumptions about the nature of the world (perhaps about how all biological phenomena supervene on fundamental physics). The work provided in this Element takes more of a methodological orientation in using science and scientific methods to settle such questions. Attention to actual work in the life sciences encourages skepticism of strongly reductive claims about explanation. Researchers in biology and the life sciences have a strong track record of successfully identifying causes and providing causal explanations. Medical researchers have successfully identified causes of diseases, neuroscientists identify axonal, circuit, and network causes of behavior, and ecologists discover the causes of ecosystem perturbations. That scientists are successful is supported by their use of these causal relationships to change the world – they have eliminated diseases from the planet (such as smallpox), developed effective treatments for neurological conditions (such as epilepsy), and restored ecosystem stability (as in the reintroduction of the gray wolf population to Yellowstone National Park). These successes are not attributable to fundamental physics – they are the fruits of the biological and life sciences. Capturing scientific explanation and scientific successes in biology, neuroscience, and ecology requires appreciating the actual scientific methods, assumptions, and reasoning that are present in these scientific domains.

2.2 Mechanistic Explanation

So far, this discussion has focused on definitions of causation and the rationale behind viewing particular causes as explanatory. Much of this work focuses on single (or a few) main causal factors, but most outcomes in biology and the life sciences are multicausal, as they are the result of many interacting causes (L. N. Ross Reference Ross2023b). Mechanistic explanations accommodate this multicausal perspective as they cite causal mechanisms – which contain multiple causes – that produce the explanatory target of interest. Accounts of mechanistic explanation have been extremely influential, leading to views that mechanistic explanation is the “dominant view of explanation in the philosophy of science at present” (Kaplan and Craver Reference Kaplan and Craver2011).

Modern accounts of mechanistic explanation have three main influences. A first influence is the seventeenth-century mechanical philosophy views of Descartes, Newton, Boyle, and others (Henry Reference Henry2001). This mechanist work opposed vitalist conceptions of living systems, as vitalist conceptions explained natural phenomena by appealing to occult powers, vital forces, and magical properties. Alternatively, the mechanist framework explained living systems with the “mathematical discipline of mechanics,” in which systems were divisible into lower-level entities that interact through “contact action” to produce behaviors of the system (Clatterbaugh Reference Clatterbaugh1999; Henry Reference Henry2001). This framework was reductive in appealing to causes at lower-scales and it defined causation in terms of action, physical forces, and matter in motion. This mechanistic perspective relied heavily on the analogy of living systems to machines – both were said to contain lower-level parts that mechanically interact, similar to the levers, pulleys, and pipes in simple machines (Henry Reference Henry2001). Modern accounts of mechanistic explanation are referred to as “new” mechanist accounts in order to acknowledge their origin in this earlier work, but also to distinguish them from it. These modern accounts retain a focus on mechanisms as constitutive (in containing lower-level parts) and as providing reductive explanations, but they sometimes distance themselves from direct analogy of mechanisms to machines (Craver and Tabery Reference Craver and Tabery2015).

A second main influence on new mechanist accounts of explanation is Salmon’s “causal mechanical” model of explanation, which relies on a connected process view of causation (Salmon Reference Salmon1984). Connected process accounts of causation maintain that genuine causal processes are capable of transmitting marks through their steps, while non-causal, “pseudo-processes” are incapable of reliably transmitting marks. Examples of causal processes and the marks they transmit are: scuffs on a fly baseball, snow on the roof of a railcar, carved initials on a flying arrow, and chalk marks on a sequence of colliding billiard balls (Reichenbach Reference Reichenbach1971; Salmon Reference Salmon1984; Woodward Reference Woodward2016). In all of these cases, some physical mark moves through the causal process in question. A shadow that moves alongside an object in motion is a pseudo-process – the shadow is non-causal because it cannot reliably transmit a mark (while the traveling object can). Although there are different conceptions of what counts as a “mark,” in much of this work various types of properties have counted, such as “constituent material, bonding forces $\dots$ geometrical shape” (Dowe Reference Dowe2018) and “momentum, energy, or electric charge” (Salmon Reference Salmon1997). A significant feature of Salmon’s causal mechanical model is his suggestion that the causal structure of the world is mechanistic. He views explanation as a project of fitting an explanandum into the “causal nexus” of the world, and this causal nexus is exclusively mechanistic. This supports an expansive notion of mechanism as mechanism captures the entire causal structure of the world. While modern accounts of mechanistic explanation seldom rely on Salmon’s connected process account of causation, they share many other features. For example, contemporary accounts of mechanistic explanation adopt Salmon’s view that the causal structure of the world is mechanistic, they often view all causal explanations as mechanistic, and they often agree that mechanisms come in constitutive and etiological varieties (Salmon Reference Salmon1984). With respect to defining causation, many mechanistic analyses either rely on an interventionist account or they are silent on what definition of causation should be used to understand their notion of mechanism (Craver Reference Craver2007; Bechtel and Richardson Reference Bechtel and Richardson2010; Craver and Tabery Reference Craver and Tabery2015).Footnote ²³

Finally, a third main influence on – and motivation for – mechanistic accounts is the fact that scientists commonly use the mechanism concept in biology and the life sciences. Biologists and life science researchers frequently use the term “mechanism” when they describe causal systems and provide explanations. This is seen in appeals to the mechanisms of gene expression, mechanisms of neuron signal propagation, a drug’s mechanism of action, reference to various circuit and network mechanisms of the brain, and so on (Pickrell et al. Reference Pickrell, Marioni and Pai2010; Cole et al. Reference Cole, Ito, Bassett and Schultz2016; Masse et al. Reference Masse, Yang, Song, Wang and Freedman2019). While scientists commonly appeal to mechanisms in explaining outcomes, philosophers are interested in specifying what type of causal structure mechanism refers to, such that the standards, features, and nature of these explanations can be clarified and understood.

The philosophical literature on mechanisms and mechanistic explanation is vast. There are many different accounts of mechanism and mechanistic explanation and they are applied to various questions in philosophy (Andersen Reference Andersen2014a, Reference Andersen2014b). While there is no consensus on how to understand mechanisms, some common claims emerge in the literature. These include claims about the features of mechanisms, their association with the machine analogy, and claims that they are studied with the investigative strategies of decomposition and localization.

In terms of what causal structures count as mechanisms, a first common claim is that mechanisms are hierarchical, in the sense that they contain lower-level causal parts that interact together to produce a higher-level outcome of the system.Footnote ²⁴ This feature of mechanisms is also referred to as “part-whole,” “constitutive,” “componential,” or as having a “nested character,” as these terms capture the difference in level or scale of causal parts to their effect (Craver Reference Craver2007, 108). This feature is often represented with an illustration similar to the one shown in Figure 6, in which lower-level causal parts all interact to produce a higher-level outcome of the system. Emphasis on this hierarchical feature reveals how these new mechanist accounts are similar to seventeenth-century notions of mechanism and to Salmon’s constitutive notion of mechanism. It also shows how the new mechanist framework supports a reductive picture of scientific explanation, in which outcomes are explained by lower-level causes.

Figure 6 Mechanism. This figure captures the hierarchical feature of mechanisms, in which they are systems with lower-level causes that produce higher-level outcomes of the system. Similar representations are found in Craver (Reference Craver2007) and Craver and Tabery (Reference Craver and Tabery2015).

Second, it is commonly claimed that mechanisms contain significant amounts of fine-grained detail. In this manner, simple monocausal models (that specify one cause for an effect) do not count as mechanisms and highly sparse network models do not count as mechanisms either. This point is argued by Craver who captures the completeness of mechanism representations on a “continuum” of more or less detail (Craver Reference Craver2007). On this picture, less-detailed causal representations are mere “sketches” or “schema” that are not yet complete mechanisms. These “incomplete” mechanism representations leave gaps and use filler terms that “veil failures of understanding” or provide an “illusion of understanding” (Craver and Darden Reference Craver and Darden2013, 113–114). On this view, making “progress” in providing causal explanations involves moving along this continuum to gain more detail about the mechanism (Craver and Darden Reference Craver and Darden2013, 113–114). These points are also seen in claims that sparse causal structures are “incomplete” and reflect a “shallowness” of understanding and that monocausal models are “[t]he sketchiest of mechanism sketches” (Craver and Darden Reference Craver and Darden2013; Glennan Reference Glennan2017).Footnote ²⁵ On these accounts, in order for a causal structure to count as a mechanism and to be sufficiently explanatory, it must contain some significant amount of causal detail.

A third common claim about mechanisms is that the causal relationships they contain are described in terms of mechanical interactions, such as force, action, and motion. In a mechanism, it is not enough to say that X causes Y – one needs to say how X causes Y, often in terms of mechanical interactions and detail. For example, these mechanistic interactions can involve specifying that X splices, activates, triggers, opens or binds to Y.Footnote ²⁶ This third feature of mechanisms is related to the second feature in the sense that when you give additional information about how a cause produces its effect you also serve the second feature by providing more detail about the system.

In addition to these three features, mechanisms in the life sciences are often analogized to machines and they are studied with causal investigative strategies that include decomposition and localization (Bechtel and Richardson Reference Bechtel and Richardson2010). With respect to the former, causal structures that scientists refer to as mechanisms are often claimed to be machine-like. Examples from molecular biology include referring to enzymes as “motors,” “biological ratchets,” and “molecular machines” (Mahadevan and Matsudaira Reference Mahadevan and Matsudaira2000; Endow Reference Endow2003). The fact that mechanisms in the life sciences are analogized to machines is unsurprising, because many everyday life machines share the three main mechanism features listed earlier.Footnote ²⁷ Finally, it is often suggested that mechanisms are studied with the investigative strategies of decomposition and localization (Bechtel and Richardson Reference Bechtel and Richardson2010). These strategies involve fixing an explanatory outcome and then drilling down to identify the lower-level causal parts that produce this outcome. These strategies are related to the hierarchical organization of mechanisms and the fact that they are defined by and relative to single explanatory targets (Ross Reference Ross2021a). The explanatory target of interest is what fixes the particular “parts” and circumscribes the bounds of the causal mechanical system.

Mechanistic accounts of explanation have received enormous attention in recent philosophical literature on scientific explanation. Central questions in this literature concern the exact character of mechanistic explanation and its prevalence in biology and the life sciences. With respect to the prevalence of mechanistic explanation, consider two questions that capture different views on this topic:

1. 1. Are all explanations in the life sciences mechanistic or is mechanistic explanation one type of explanation in this domain?

2. 2. Are all causal explanations in the life sciences mechanistic or is mechanistic explanation one type of causal explanation in this domain?

In early work, many new mechanists supported an explanatory monist position with respect to (1) and argued that all explanations in biology and the life sciences are mechanistic (Machamer et al. Reference Machamer, Darden and Craver2000; Craver Reference Craver2007; Bechtel and Richardson Reference Bechtel and Richardson2010; Kaplan and Craver Reference Kaplan and Craver2011). On this view “explanations are said to be adequate to the extent, and only to the extent, that they describe the causal mechanisms that maintain, produce, or underlie the phenomenon to be explained” (Kaplan and Craver Reference Kaplan and Craver2011). These claims were viewed by many as overly bold and “imperialist” in claiming that mechanistic explanation is the exclusive form of explanation in the life sciences (Kaplan Reference Kaplan2017). These explanatory monist claims were strongly countered by various discussions of the limits of mechanistic explanation (Weber Reference Weber2008; Dupré Reference Dupré2013; Woodward Reference Woodward2013; Skillings Reference Skillings2015; Halina Reference Halina2018; Ross Reference Ross2021a) and myriad examples of non-causal, mathematical explanations, which explain without appealing exclusively to causal information (Silberstein and Chemero Reference Silberstein and Chemero2013; Batterman and Rice Reference Batterman and Rice2014; Ross Reference Ross2015; Chirimuuta Reference Chirimuuta2018). These non-causal explanation types have been overwhelmingly accepted in the philosophical literature on scientific explanation (and they are discussed more in the next subsection). Many new mechanists have been convinced by positions on non-causal (and, thus, non-mechanistic) explanation and they have revised their views to acknowledge the legitimacy of these explanation types.

However, in revising their views, many new mechanists have adopted a second monist position, captured by the second (2) question earlier. These new mechanists have adopted monism about causal explanation, which claims that all causal explanations in the life sciences are mechanistic.Footnote ²⁸ The tenability of this position depends on how “mechanism” is defined and understood. Notice that if mechanism is defined narrowly by the three features earlier (constitutive organization, fine-grained detail, and mechanical interactions), or by any particular features, this conflicts with the monist position – this is because there are a great variety of distinct causal structures that are cited in biological and life science explanations, and there is no specific set of features that are shared across all of these causal structures. For example, some explanations cite higher-level causes that produce lower-level outcomes, other explanations appeal to causal systems that are sparse and abstract away from detail (such as topological and network models), and yet other explanations refer to causal systems with linear, causal chain organizations. As these do not share the previously outlined mechanism features, they qualify as non-mechanistic causal systems. These are just three alternative causal structures – many other diverse causal systems exist and are cited in life science explanations. Thus, an important question to ask is, given the variety of causal systems that figure in life science explanations, which of these causal systems count as mechanisms?

While many new mechanist accounts began with specific, narrower conceptions of mechanism (such as the three-feature account earlier), recent accounts have broadened the notion of mechanism to the point of equating it with nearly any type of causal system. This broad notion of mechanism is seen in accounts that refrain from specifying any defining or characteristic features of mechanisms. For example, some of these accounts claim that mechanisms can be highly detailed or not, machine-like or not, reductive or not, driven by push-pull dynamics or not, and so on (Craver and Tabery Reference Craver and Tabery2015). Whichever feature it is suggested that some mechanisms have, it is quickly stated that other mechanisms need not have this feature. Similarly, when these new mechanists state what does not count as a mechanism, they tend to only provide examples of non-causal systems (these are obviously not mechanistic because they are not causal). Insofar as helpfully articulating the mechanism concept requires specifying what does and does not count as a mechanism, many of these broad views have been found to be overly expansive and wanting by many. As Dupré states, “if the concept of a mechanism is to do any work, we must surely have some sense of what isn’t a mechanism” (Dupré Reference Dupré2013). I will add that we must also have a sense of what causal systems are not mechanisms; otherwise, the term is only as meaningful as “causal system,” despite being advertised as much more.

It is clear that many distinct types of causal systems figure in explanations in biology and the life sciences. The question is whether to view some of these causal systems as mechanisms (narrow mechanism view) or to view all of these causal systems as mechanisms (broad mechanism view).Footnote ²⁹ Challenges for the broad mechanism view are that it threatens to over-expand and trivialize the notion of mechanism by equating it with generic notions of “causal system” or “causation.” In many scientific and everyday life contexts, mechanism implies much more than “causal system” or “causality” – “mechanism” is often a high-status causal term that communicates knowledge of significant detail and information (Hutchinson Reference Hutchinson2007; Ankley et al. Reference Ankley, Bennett and Erickson2010). Furthermore, if “mechanism” just means “causal system,” then accounts of mechanistic explanation are limited in what they add to our understanding of causal explanation in science. Since the fall of the DN model, accounts of causal explanation have received significant attention. These accounts claim that causes explain their effects and they define causation in order to capture how these explanations work. Accounts of mechanistic explanation have entered this scene and their main contribution is that “causal explanations cite mechanisms.” However, if “mechanism” is synonymous with “causal system,” then these accounts are merely stating that “causal explanations cite causal systems.” This is obvious, tautological, and something we already knew – it is implied by the causal explanation framework and it fails to add to our understanding of how these explanations work. Thus, it is worth questioning whether the broad notion of mechanism advances our understanding of explanation in the life sciences and whether it captures the use of mechanism in these domains.

Whether one adopts a narrow or broad notion of mechanism, most agree that accounts of causal explanation should capture the diversity of causal systems that are explanatory. This Element adopts a narrow conception of mechanism – captured by the three features earlier – and examines other types of causal systems and causal explanation in the life sciences. Two other types of causal systems are examined – these include the pathway and cascade concepts. The next section explores these two causal concepts and how they have unique features, are associated with distinct analogies, and are studied with particular causal investigative strategies. However, the analysis in the next section does not require that one commit to the narrow mechanism notion. If the broad notion of mechanism is preferred, then these three causal systems (which are called “mechanisms,” “pathways,” and “cascades” in this Element) can all be considered distinct types of causal mechanisms. The important point is not what we call these causal systems, per se, but that we have an account that captures the distinct features of these causal systems and different types of causal explanations they provide.

2.3 Pathways

When biologists and other life scientists provide causal explanations, what types of causal systems do they refer to? What causal language is important for capturing the causal systems they find explanatory? While “mechanism” is a common causal term and concept, many other causal concepts are found in these scientific fields. Examples of causal concepts in the life sciences include pathways, cascades, circuits, triggers, causal cycles, and structuring causes. This subsection examines the pathway and cascade concepts. I argue that pathways and cascades differ from mechanisms in terms of their characteristic or hallmark features, the analogies they are associated with, and the causal investigative strategies used to study them. This analysis is used to suggest that explanations that appeal to pathways, cascades, and mechanisms are distinct types of explanation because they cite different types of causal systems.

A first causal concept to consider is the notion of a pathway, which is commonly found in biology, neuroscience, physiology, ecology, and numerous other life sciences (Boniolo and Campaner Reference Boniolo and Campaner2018; Ross Reference Ross2018). Examples of the pathway concept include gene expression pathways, metabolic pathways, neural pathways, vascular pathways, developmental pathways, and ecological pathways, as seen in Figure 7. In these contexts, pathways are causal systems that have four main features. These features include: a sequence of causal steps, the flow of information or material, abstraction from causal detail, and causal relationships that emphasize causal connection (as opposed to mechanical interactions) (Ross Reference Ross2021a). Additionally, while mechanisms are often analogized to machines, pathways have their own unique analogy – pathways are often analogized to roadways, highways, and city streets. The reasons for this analogy will become clear as the main features of the pathway concept are discussed in detail.

Figure 7 Pathway examples in the life sciences, including (a) metabolic pathways, (b) developmental pathways, (c) vascular pathways (or blood vessels), and (d) ecological pathways (which capture prey–predator relationships).

A first feature of pathways is that they involve a sequence of causal steps – these steps outline a causal route from upstream, to intermediate, to downstream factors.Footnote ³⁰ In their simplest form, pathways are linear but they can be organized in more complex branching configurations. This sequential feature of pathways is seen in the butterfly developmental pathway in Figure 7, in which a sequence of causal steps – from egg to larvae, to pupa, to adult – is represented.

Second, pathways involve the flow of some entity along their sequence of causal steps. In many cases, pathways involve the flow of some material or information that is relevant to the system of interest. For example, metabolic pathways involve the flow of metabolites, neural pathways guide the flow of information, vascular pathways dictate the flow of blood, ecological pathways channel the flow of energy through ecosystems, and developmental pathways trace the flow of cells, tissues, or organisms, as they mature. In this manner, pathways involve both an entity that changes and a sequence of steps that limits, guides, and channels how the entity changes over time.Footnote ³¹ These scientific pathways are analogized to roadways, because roadways outline a route that guides, channels, and constrains the flow of traffic.

A third feature of pathways, and one that clearly distinguishes them from mechanisms, is that they abstract from large amounts of causal detail. One way that pathways abstract from detail is by representing complex processes in an economy of causal steps. We see this in the developmental pathway of a butterfly, which represents this entire developmental process (the entire life cycle) in just three main steps. While any one of these steps could be divided into further causal links, this is not represented with the pathway concept, which abstracts from this information. A second way that pathways abstract from detail is that they omit lower-level, mechanistic information about how an entity moves from one causal step to the next. When vascular, developmental, and neural pathways capture a sequence of causal steps, they outline and emphasize the different steps that the entity moves through, but not the lower-level causal details that are involved in this process. Again, similar to the roadway analogy, roadways capture macro-level information about where an object can flow in a system, but not how it flows (or moves) through this space. A roadway reveals where cars can travel, but not the mechanistic details of how the car moves from one location to another. Similarly, metabolic pathways capture causal routes along which metabolites travel, but not the lower-level enzymatic details that drive this travel. For example, the glycolytic pathway captures a ten-step metabolic process that is shared and “conserved” across nearly all species on the planet. However, it is only the higher-level, causal steps outlined in the glycolytic pathway that are shared – this pathway is instantiated by different causal-mechanical details in different systems (such as different enzymes at a given step of the pathway). By abstracting from detail, the pathway can capture what all these systems share and it can explain the glycolytic process (conversion of glucose into pyruvate) generally, across all systems. Including lower-level mechanistic detail would prevent this, because such details are not shared across all systems.

A fourth feature of pathways is that their causal relationships emphasize causal connection, but not lower-level detail or mechanical detail. In other words, pathways capture that X is a cause of Y, but they do not reveal mechanistic details about how X causes Y. Their role is to highlight the set of causal connections in some domain – namely, who is causally connected to who, but not fine-grained information about these causal connections. For example, neural tracts can outline the flow of neural signals from one location to another, but they do not convey how these signals flow, travel, or move. This emphasis on causal connection with the pathway concept is seen when scientists organize many causal pathways together into network diagrams or what they sometimes call “roadmaps” of causal connections. This is seen in reference to “roadmaps” of metabolic pathways, “roadmaps” of stem cell developmental pathways, and “roadmaps” of ecological pathways (Marina et al. Reference Marina, Salinas and Cordone2018; Ly et al. Reference Ly, Lynch and Ryall2020; Zheng et al. Reference Zheng, Tan and Li2021).

If pathways are unique causal systems, with the four features mentioned earlier, how do they provide explanations? How do pathways provide causal explanations that differ from mechanistic explanation? Consistent with the three-part model of explanation (shown in Figure 2), pathway explanations are cases in which pathways serve as the explanans and explain the outcome of interest. As pathways refer to a causal structure that differs from mechanisms, citing them provides a distinct type of causal explanation. On this view, one way to capture different types of causal explanations is by appreciating the distinct causal systems they cite that perform the explanatory “work.” As mentioned earlier, consider explanations of glycolysis, which is a process by which organisms breakdown glucose into energy. When scientists explain glycolysis across a large number of species (or in living systems in general) they cite the “conserved” glycolytic pathway, with ten chemical conversion steps (from glucose to pyruvate). This pathway abstracts from lower-level mechanistic details, such as the enzymes that catalyze each step of the process. The reason for this abstraction is that the ten-step glycolytic pathway is shared across all systems in the explanatory target, while the lower-level enzymes are not. As the lower-level mechanisms in these systems are not shared, they cannot be cited to explain their identical behavior of glucose production. Similar reasoning is found in many explanations in biology (Sober Reference Sober1999; Ross Reference Ross2020) and in explanations of shared or universal behaviors across systems that differ in terms of their lower-level details (Batterman Reference Batterman2001; Ross Reference Ross2015; Woodward Reference Woodward and Kaplan2017).

Another way to see the uniqueness of pathway explanations is with the network and “roadmap” diagrams of pathways. Consider the map of ecological pathways, shown in Figure 7. These pathway maps allow scientists to answer unique explanatory why-questions that mechanisms do not address. These explanatory why-questions include: (1) Given a starting point on the map, which downstream locations can (and cannot) be reached? (2) Given a downstream location, which upstream positions can (and cannot) arrive at this location? (3) What is the most (and least) direct way to get from any two points in the map? While there are other questions that this pathway map can answer, the important point is showing how pathway information helps scientists answer distinct types of explanatory why-questions. The questions here concern a possibility-space of outcomes and how an entity moves through this space. This matters for determining the flow of toxins as they damage species in an ecosystem, how blood clots move through blood vessels to damage some organs in the body (and not others), and how to manufacture particular chemical species from available chemical reactions (Ross Reference Ross2021a, Reference Ross2023b). Examining these pathway maps makes it clear that they do not represent mechanistic information – these maps do not represent a mechanism, in the sense of lower-level causal parts (or gears) that interact together to produce a single outcome of the system. Instead, these pathway maps capture a possibility space of routes in the system that capture where some entity can and cannot flow through this space. While pathways provide dynamic information about possible outcomes within a complex space, mechanisms explain why a particular, specific outcome presents.

Finally, in addition to their four main features, pathways are studied with unique causal investigative strategies. Scientists often study these systems by exploiting their “flow” feature. In particular, they often use tags and tracers to mark some entity, and then watch it as it flows along the steps of the pathway. This is seen in radioactive tracer and dye experiments, in which these materials are used to illuminate the steps of many different types of pathways, including metabolic, vascular, neural, and ecological (Ross Reference Ross2021b). For example, radioactive tracers are attached to carbon atoms that flow through metabolic pathways, radioactive tracers are introduced into prey and followed through prey–predator relationships in ecosystems, and dyes are attached to material that flows through neural pathways and tracts (Ross Reference Ross2021b). These are not techniques that can always or easily be used with mechanisms, because many of them lack the reliable flow of some entity through their causal steps. This is similar to the gears of a watch mechanism – while these gears are involved in the causal process, there is no material that reliably moves along them (from the first gear, through all intervening gears, to the final time keeping behavior of interest). However, for metabolic, vascular, and ecological pathways, there are materials that reliably flow through the causal steps and that are targeted by tracers in the study of these systems. This reveals how these systems have distinct features that matter for how they are studied. Thus, while mechanisms are often studied by “drilling down,” pathways are studied by dropping tags or tracers into a system and “expanding out” along causal routes. These tracers reveal the causal sequences of interest in some domain, but not the fine-grained, mechanical details of the causal routes of interest.

Before moving on, it might help to consider some objections that mechanists have voiced toward viewing pathways as an explanatory causal structure. Does the sparse detail of pathways make them less explanatory than mechanisms? Are pathways mere mechanism sketches that should be suffused with more detail in order to explain? These points are supported by Craver and Darden (Reference Craver and Darden2013) who consider pathways to be explanatorily deficient due to their lack of detail. They associate pathways with the “vice of chainology,” in which “one becomes fascinated by nodes in a causal chain but loses sight of how the nodes work to produce, underlie, or maintain the phenomenon” (Craver and Darden Reference Craver and Darden2013, 91). Along these lines, they claim that pathways are causal structures that are “incomplete” and that they reflect a “shallowness” of understanding (Craver and Darden Reference Craver and Darden2013, 91–92).

These statements misunderstand scientists’ use of the pathway concept and its explanatory role. The sparse nature of pathways is central to their explanatory power because it captures shared higher-level causal connections across systems with different lower-level details (Ross Reference Ross2020). This is seen in explanations of glycolysis and other cases in which shared behaviors present across systems that differ in terms of their mechanistic details. In addition to capturing shared, macro-level details, pathways also capture distinct types of causal details, as seen in the “roadmaps” of causal connections. Mechanisms are equipped to capture lower-level causes that produce a particular behavior of interest, but they are not suited to capture a map of possible causal routes through a system. For this type of causal information, the pathway concept is much better-suited, as is the analogy of these pathways to roadways, highways, and city streets. Here we see that mechanisms are limited, not just because they involve lower-level details that we do not need, but because mechanisms do not capture information about flow and possible routes, which is required for some explanatory why-questions. Further varieties of causal (and non-causal) explanation are considered in the next subsection and section.

2.4 Cascades

Another causal concept that commonly figures in explanations in biology and many other scientific domains is the notion of a cascade (Ross Reference RossIn Press). Examples of cascades are plentiful in science – they include cell signaling cascades in physiology, trophic cascades in ecology, ischemic cascades in neuroscience, cascading reactions in physics and chemistry, and cascading disasters (or failures) in the social sciences, as seen in Figure 8 (Macfarlane Reference Macfarlane1966; Ripple et al. Reference Ripple, Estes and Schmitz2016; Smolyak et al. Reference Smolyak, Levy, Vodenska, Buldyrev and Havlin2020). Similar to mechanisms and pathways, cascades are causal structures with unique features, analogies, and causal investigative methods. Cascades have three main features: they involve an initial trigger, sequential amplification, and stable progression. In examining cascades in science it will help to introduce them with the analogies they are associated with. Cascades are often analogized to systems such as the snowball effect, ripple effect, and waterfalls (synonymous with “cascades,” their namesake). We see why this is the case, by exploring their features more next.

Figure 8 Cascade examples, including (a) cascading reactions in physics and chemistry, (b) cascading disease spread in epidemiology, (c) ecological cascades, and (d) cell-signaling cascades.

First, cascades are initiated by a trigger, which is often conceptualized as a binary switch (on/off) that sets the process off. In order to see this, consider one of the first causal systems to receive the “cascade” label in modern biology, namely, the blood coagulation cascade (Davie and Ratnoff Reference Davie and Ratnoff1964; Macfarlane Reference Macfarlane1966). Blood coagulation is a process that functions to stop massive bleeding after blood vessel injury – this injury triggers an enzyme cascade, which culminates in a large clot to stop the bleeding. The trigger for blood coagulation is this vessel injury, which initiates the cascade process. Other examples of triggers in cascades are seen in the causal systems they are analogized to. The snowball effect is triggered by a small amount of snow, the ripple effect is triggered by a single drop of water, and waterfalls (or natural cascades) begin with a small amount of upstream water, which progressively disperses.

Second, the central feature of cascades is sequential amplification, which involves a small cause that produces an increasingly large amount of some effect.Footnote ³² At the scale of a single causal step, there are at least two types of causal amplification: a small amount of cause can produce a large amount of a single effect, called single-product amplification, or a small cause can produce many different effect types, called multi-product amplification (Ross Reference RossIn Press). An example of single product amplification is seen in enzyme cascades, such as the blood coagulation cascade, as these involve a small amount of some enzyme “a” that produces a large amount of downstream enzyme “b”. In this enzyme case, there is amplification of a single product, as there is a large amount of enzyme “b.” An example of multiproduct amplification is a disaster cascade, such as when an earthquake causes many different downstream outcomes, such as fires, collapsing bridges, and water damage from broken water pipes. In this disaster cascade, there is amplification in the sense of the production of many different types of effects.

While these two types of amplification help capture the notion of causal amplification at a single step, cascades involve sequential amplification, in which there are a sequence of amplifying steps. This is important because when amplifiers are arranged in series, this drastically increases the overall amplification of the system – the overall amplification of the system is the product of the gain at each step. For example, if one unit of enzyme “a” produces ten units of enzyme “b” and one unit of enzyme “b” produces ten units of enzyme “c,” the overall process has a gain of one hundred (producing one hundred units of “c” for every single unit of “a”). This allows blood coagulation and other physiological cascades to produce “explosive” outcomes in which colossal amounts of a product are produced. In some cases, this amplification serves a physiological function, such as clot formation in the blood coagulation cascade and signal amplification in hormonal cascades. These cascades are also exploited in various technologies, such as polymerase-chain reaction (which amplifies trace amounts of some substance). In other cases, the amplification is non-functional and produces significant damage, as seen in ischemic cascades, disaster cascades, and cascading failures (Smolyak et al. Reference Smolyak, Levy, Vodenska, Buldyrev and Havlin2020).

The sequential amplification feature of cascades is seen in the analogies they are associated with and the diagrams used to represent them. Notice that the cascade analogies all involve a small cause that amplifies a downstream effect. For example, the snowball effect refers to a small amount of snow that causes a huge avalanche, the ripple effect involves a small drop that produces outward ripples that increase in size, and waterfalls involve a narrow stream of water that progressively amplifies in speed and distribution of spread. In analogizing scientific cascades to these systems, a main goal is to highlight and communicate the amplification in these causal systems. This amplification feature is also emphasized in the diagrams used to represent scientific cascades. These diagrams often depict one-to-many causal systems, in which one cause gives rise to many effects in succession (this is seen in the illustrations in Figure 8). Interestingly, as the degree of amplification (or gain) at any given step of the cascade is larger than can be easily drawn, scientists find other ways to represent this significant amplification (sometimes by representing the amplification numerically, with a 10³ at a given step).

A third feature of cascades is that once initiated they involve stable progression to their final effect. In other words, cascades gain in momentum and are propelled forward as they move through their sequence of causal steps (Dodge et al. Reference Dodge, Malone and Lansford2009). One implication of this is that, once initiated, cascades can be very difficult to stop. They have the potential to “run-away” or “become uncontrolled” (Bloomfield and Stephens Reference Bloomfield and Stephens1996, 168, 171). This is similar to descriptions of the snowball effect, in which a small snow fall triggers a growing, unavoidable avalanche. As Stein states, “cascade refers to a process that once started, proceeds stepwise to its full, seemingly inexorable, conclusion [$\dots$] the danger of a cascade is that it can be inappropriately triggered [$\dots$] once triggered, it is virtually impossible to stop” (Stein Reference Stein1990). This stable progression feature is seen in examples such as the initiation of a chemical chain reaction, the spread of COVID through the population, and cascading failures triggered by natural disasters. Due to this feature, scientists often recommend intervening early on in a causal cascade as a way to modify or stop it, as it becomes harder to control as its steps unfold.

If the narrow notion of mechanism is adopted (see Subsection 2.2), the cascade concept is an additional alternative to mechanism. In order to see this, consider that cascades do not always have the hierarchical organization of mechanisms (in which lower-level causes produce a higher-level effect). Cascades are level-agnostic with respect to their causes and effects. Cascades can have causes and effects at the same level, they can have higher-level causes that produce lower-level effects, and they can have lower-level causes that produce higher-level effects. Enzymatic cascades are an example of the first type (causes and effects at the same level) as the cause and effect are both enzymes. An example of higher-level causes producing lower-level effects are energy cascades involved in turbulence, which involve the transfer of energy from “large scales of motion to the small scales” ((Richardson Reference Richardson1922, 66); see also Wilson (Reference Wilson2021)). Yet another example are traumatic experiences that alter gene expression, referred to as “downward cascades” (Masten and Cicchetti Reference Masten and Cicchetti2010, 492). Finally, a cascade example with a lower-level cause that produces a higher-level effect is a hormone cascade, in which a hormone trigger produces some system-level behavior. Another example is a pharmacological intervention that alters behavior, sometimes called an “upward cascade” (Masten and Cicchetti 2010 Reference Masten and Cicchetti2010, 492).

Yet another difference between cascades and mechanisms is that unlike mechanisms, cascades do not always have a single, main effect of interest. Recall that the decomposition and localization methods used to study mechanisms require first fixing a single explanatory target before “drilling down” to identify the causal parts of the system. Cascades cannot be reliably studied in this way because they often have many distinct effects, as seen in multiproduct amplification cases. Furthermore, even when cascades do have a single main explanatory target, their causes cannot always be identified by “drilling down” because, as just mentioned, their causes are not always at lower-levels. Instead, cascades are often studied with tools that intervene on their initial trigger, using this as a way to study what their downstream effects are.

Finally, even if a broad notion of mechanism is adopted – such that the aforementioned cascade and pathway examples are said to count as mechanisms – it will still be important to distinguish across kinds of mechanisms. As suggested in this analysis, these varieties of causal systems (whatever they are called) have different features that set them apart from each other, require different causal investigative strategies, provide different types of control over effects, and figure in distinct types of explanations. Capturing the nuanced picture of causal varieties in the world and types of causal explanations in science requires a framework that appreciates these distinct causal systems.

2.5 Causal Explanation Conclusions

Providing an account of causal explanation in biology and the life sciences requires specifying the types of causal structures that scientists cite in their explanations. These causal structures serve as the explanans in the three-part model of explanation, and they capture what does the explanatory “work” in these cases. The analysis in this section is compatible with the claim that “causes explain their effects.” What is highlighted is that there are many distinct types of causes and causal systems that play this explanatory role. Referring to all of these explanatory causal systems as “mechanisms” involves adopting a broad, expansive notion of mechanism, on which mechanism is synonymous with “causal system.” As others have voiced, while it may be possible to “shoehorn descriptions of biological systems into talk of mechanisms,” the “attempt to do so risks reducing the idea of a mechanism to vacuity” (Dupré Reference Dupré2013).Footnote ³³ Alternatively, referring to particular causal systems as mechanisms adopts a narrow notion of mechanism – a notion in which “mechanism” refers to causal systems with specific features. Whichever view of “mechanism” one adopts, the important point is that different types of causes, causal relationships, and causal systems are explanatory. Identifying and distinguishing these causal types is an important part of providing an account of scientific explanation that captures work in biology and the life sciences.

3.1 Topological and Constraint-Based Explanations

A first type of non-causal, mathematical explanation are topological explanations. These explanations are commonly found in network and systems science areas, in which life science cases are studied with network models, graph theory, and pathway analysis. In these examples, systems are represented with network models containing nodes and edges – nodes capture properties in the system (metabolic compounds, areas of the brain, species in an ecosystem, etc.), while edges capture relations between these properties (chemical conversions, anatomical connections, prey–predator relations, etc.). In the context of network neuroscience, some of these models capture the brain’s structural and anatomical connections, which are referred to as the brains “network architecture,” “structural topology” and “hard-wired connection topology” (Zöller et al. Reference Zöller, Sandini and Schaer2021). Similar methods are used to represent networks in other domains, such as enzyme interactions in molecular biology, cellular interactions in immunology, prey–predator relationships in ecology, and many others (Montoya and Solé Reference Montoya and Solé2002; Kitano and Oda Reference Kitano and Oda2006; Taylor et al. Reference Taylor, Wang and Rosa2013). These network diagrams reveal the “global shape” of connections in some domain and, because of this, are also referred to as “circuit diagrams” and “wiring diagrams” of connections in the system (Karuza et al. Reference Karuza, Thompson-Schill and Bassett2016).

Representing systems with network and graphical formats can reveal important topological properties of the system. These topological properties capture how relations between entities in the system are arranged and organized – in this manner, the “topology of a graph defines how the links between system elements are organized” (Fornito et al. Reference Fornito, Zalesky and Bullmore2016). Examples of topological properties include: connection or linkage density, link distance number, small-worldness, nested relations, power-law scale-free distributions and scaling properties (Bassett and Bullmore Reference Bassett and Bullmore2017). In many cases, these topological properties are cited in explaining unique behaviors of the system. While these topological properties are sometimes characterized as “mathematical” themselves, they are also claimed to bear a mathematical relationship to the system-level behaviors that they produce. In this manner, it is suggested that there is a mathematical dependency relation between the topology of a system and the behavior produced.

As an introductory example, consider a topological explanation that is commonly discussed in the philosophical and mathematics literatures (Euler Reference Euler1956; Pincock Reference Pincock2012; Lange Reference Lange2018). While this example is not drawn from the life sciences, it motivates and captures an explanatory pattern that is identified in many life science contexts. This is the well-known case of the Königsberg bridges. In this example, a set of rivers runs through the eighteenth-century city of Königsberg, shown in Figure 9 (Adams and Franzosa Reference Adams and Franzosa2008). In this layout, there are seven bridges that span the river, such that the two central islands are connected to surrounding land. According to this story, there was interest in determining whether it was possible to walk a path across each bridge exactly and only once. After much deliberation an answer was provided by Euler in the form of a mathematical proof. Euler represented the bridge system graphically, as shown in Figure 9, in which landmasses are represented as nodes and the bridges as edges.Footnote ³⁶ He used this graphical depiction to demonstrate that in order for there to be a walking path of this nature – what we now call an Eulerian path – two conditions needed to be met: (3.1a) all nodes should be connected and (3.1b) the amount of nodes with an odd number of connections (or edges) should be zero or two (Euler Reference Euler1956; Ross Reference Ross2021). In this example, the explanatory why-question is “In the Königsberg bridge case, is it possible to walk across each bridge exactly and only once?”Footnote ³⁷ Importantly, the answer to this question is “no,” and the explanation of this is provided by mathematics, in particular the mathematical proof and failure to meet both conditions mentioned earlier (3.1a, 3.1b).

Figure 9 The Königsberg bridges. The figure on the left represents the seven Königsberg bridges, which cross different sections of river. This structure of bridges is represented graphically in the figure on the right, in which the landmasses are represented as nodes and the bridges as edges (Adams and Franzosa Reference Adams and Franzosa2008).

This example is viewed as a non-causal explanation for a number of reasons. First, the causal properties and constituents of the bridges are irrelevant to the explanation – all the matters is the macroscale topology of the system. If the same arrangement of bridges is present, but you change the lower-level materials they are made of (steel, iron, wood, etc.) the explanation still holds (Pincock Reference Pincock2012). The topological structure lacks lower-level physical-causal details, but it also lacks causal details at macro-scales. As Huneman states, this type of “topological explanation” is one that “abstracts away from causal relations and interactions in a system, in order to pick up some sort of ‘topological’ properties of that system and draw from those properties mathematical consequences that explain the features of the system they target” (Huneman Reference Huneman2010, 214). In this example, math is used to capture abstract features of the system that are explanatory and independent of causal details. Second, not only does the explanans (the topological structure) have a mathematical character in this case, but the dependency relation in this explanation is mathematical as well. This explanation contains a dependency relation between topological properties (meeting the two criteria mentioned earlier) and presence of an Eulerian path. Whether an Eulerian path is present or not depends on the topology of the system. However, in the Königsberg case, this dependency relation is mathematical and not causal (Woodward Reference Woodward, Reutlinger and Saatsi2019). Causal dependencies have an empirical character that mathematical dependencies lack. Causal dependencies require empirical study to be identified and studied in the world. Identifying that a gene, drug, or vitamin deficiency causes some outcome requires empirical study in the world and cannot be revealed by mathematics alone (proofs, derivations, etc.). However, for the Königsberg case and others, once the topological features of the system are identified, mathematical understanding alone can reveal consequences of the system and answer explanatory why-questions. In these examples, topological properties have “mathematical consequences” that power the explanation, as opposed to causal consequences that require empirical discovery (Huneman Reference Huneman2010).

Consider similar explanations in the context of ecology and neuroscience. A first example involves explanations of ecosystem robustness in the face of species extinctions. An ecosystem’s prey–predator relationships can be represented with graphical models, in which nodes represent different species and edges capture the relations between species. With this graphical model, the extinction of species from the ecosystem can be represented with the deletion of nodes. One way to explain the robustness of an ecosystem to species extinction is to determine how the system responds to random deletions to nodes in the network. In this case, topological properties of the network can explain whether the ecosystem will be stable (or robust) in the face of a random node deletion in the network. In particular, if the graph has a scale-free or small world structure, in which there are few highly connected nodes, the system is more stable as a less-connected node is more likely to be deleted (Huneman Reference Huneman2010; Ross Reference Ross2021). Removing a less-connected node is less disturbing to the network than removing a highly connected node. Alternatively, if a system lacks this scale-free structure (and has many highly connected nodes) a random deletion is more likely to hit a highly connected node, which would lead to more disruption and instability. Similar to the Königsberg case, explaining why a system is more or less robust has to do with abstract, topological features of the system and not with the system’s physical constituents, realizers, or higher-scale causal connections. This allows the same explanation to be provided for other systems that share the same topological structure, even when they differ in terms of lower-level and causal details.

Other examples of topological explanation are found in neuroscience, such as those involving human brain networks and neural networks of model organisms, such as C. elegans. Studies of the C. elegans neural network reveals a “small-world” network topology characterized by high clustering and a short path length (Watts and Strogatz Reference Watts and Strogatz1998). This small-world “connection topology” – found in C. elegans, other living organisms, and various non-living systems – has been cited in explaining particular behaviors, such as signal-propagation speed, computational power, and synchronizability (Watts and Strogatz Reference Watts and Strogatz1998, Bassett and Bullmore Reference Bassett and Bullmore2017). In other cases, the small-world structure of neural networks is cited in explaining the economical character of the system, in the sense of minimizing biological costs, while maximizing topological integration (Bassett and Bullmore Reference Bassett and Bullmore2017).Footnote ³⁸ These explanations fit the non-causal category because the explanatory target of interest – unique behaviors of the system – are a mathematical consequence of the system’s topology, as opposed to an empirically identified cause-effect relationship. In these explanations, the “abstract, dimensionless” topological features are unique in that they “tell us nothing about the physical layout” of the nervous system or how it is “embedded in anatomical space” (Bassett and Bullmore Reference Bassett and Bullmore2017).

All three of these examples – the Königsberg, ecological, and neural network cases – share explanatory features. Each of these cases specifies how the topology of the system (the explanans) “makes a difference’ to the system’s behavior (the explanandum), in which this difference-making relation (dependency relation) is specified mathematically. In these cases, the explanatory work is supplied by topological properties, which abstract from lower-level and causal details. The fact that these explanations abstract from lower-level details, gives them the ability to generalize across microstructurally distinct systems. A system’s scale-free structure explains its robustness across contexts – this explanation holds in the contexts of enzyme networks, cellular networks, brain networks, ecological networks, social networks, and so on. This captures the global and domain-general nature of these explanations, as the same topological properties can be instantiated by different lower-level details.

While these topological cases are examples of non-causal explanation, this does not mean that topology only explains in a non-causal manner. In some situations, topology can capture higher-level causal connections in a system, which provide causal explanations (Ross Reference Ross2021). Examples of these causal topologies include the bowtie (fan-in, fan-out) network structure of human immune cell interactions, the chokepoint topology in enzyme networks, and unique connection topologies in ecosystem networks (Ross Reference Ross2021). The abstraction involved in both the non-causal and causal topological explanations captures their autonomy from lower-level detail and their domain general character. After significant emphasis on reductive explanation in the philosophical literature, these cases have proved helpful in demonstrating the rationale behind non-reductive explanations and cases in which macro-scale structures have explanatory power.

These non-causal topological explanations are related to another account of non-causal explanation that Lange refers to as “explanations by constraint” (Lange Reference Lange2018, 3). Lange provides an account of non-causal explanation, in which factors are explanatory in virtue of the fact that they constrain the system in some way. Notice that in the Königsberg bridge case, the topology is explanatory in virtue of its constraining influence on potential walking paths. Lange elaborates on this to identify two important features of these explanations. First, these constraint-based explanations are non-causal because (i) the constraining influence of mathematics is stronger than causal influence. The strength of the constraining influence in these mathematical explanations, sets them apart from the weaker causal influence found in causal explanations. In this manner, mathematical explanations have a stronger form of necessity than causal relationships – this allows mathematical explanations to show why an outcome “was inevitable to a stronger degree than could result from the action of causal powers” (Lange Reference Lange2018, 6). Second, (ii) the strength of the constraining influence of the explanatory mathematics in these cases allows them to provide a unique type of “impossibility” explanation. These explanations do not just show why some outcomes are not realized, but why it is strictly impossible for them to occur. This is intended to capture how the Königsberg topology has explanatory impact that is stronger than causality and how it explains why it is impossible to take an Eulerian path in the system.

Lange uses these two features (i, ii) to identify other non-causal explanations, which need not involve topology. Consider a non-scientific explanation that illustrates this: suppose a mother has twenty-three strawberries and she wants to divide them evenly among her three children. What explains why this is impossible? According to Lange, this impossibility is explained by mathematical facts and it does not have to do with causal relationships in the world. While causal relationships have influence over which outcome of a set of possible outcomes manifests, they lack the strong form of necessity implied by the explanatorily relevant mathematics in these cases. The strong type of necessity in these mathematical explanations is what allows them to explain impossibilities – not just why something did not happen, but why it could not possibly happen at all. As Lange states, “[t]he Königsberg bridges as so arranged were never crossed because they couldn’t be crossed. Mother’s strawberries were not distributed evenly among her children because they couldn’t be” (Lange Reference Lange2018, 9).

Consider a similar non-causal explanation from the contexts of human genetics. In humans, genes contain four different types of nucleotides (A,C,T,G), which are arranged linearly in DNA. When genes are expressed to produce proteins, these nucleotides are “read” sequentially, three at a time. Each three nucleotides in a gene – referred to as a “codon” – codes for a particular amino acid, which are small units that make-up proteins. Scientific research has revealed which three-nucleotide sequences (codons) code for particular amino acids and it has become clear that there are only 64 possible codons in humans. What explains this limit on codon types? If our explanatory-why question is, “Why are there 64 possible codons in humans as opposed to more or less?” the answer to this is provided by mathematics. The fact that there are only four nucleotide options, which are organized in sequences of three, mathematically entails that there are 64 possibilities (as $4^3$ or $4\times 4\times 4=64$).Footnote ³⁹ In this case, the explanatory power is not derived from causality, but from mathematics. The mathematics explains why there are exactly 64 possibilities and why it is impossible for there to be more. In other words, the math constrains possible codon number and it explains why it is impossible to have more than 64 codon types in humans. Similar to the Königsberg, ecological, and neuroscience examples, there are some empirical facts that are accepted when formulating the explanatory why-question. However, the answer to this question is provided from mathematical derivation alone and not empirical-causal assessment. There are surely other types of these mathematical, impossibility explanations in the life sciences. In exploring these types, it will be fruitful to focus on cases that matter to scientists, that they view as furthering their understanding of the natural world, and that capture their explanatory practices, aims, and goals.

This analysis reveals overlap and dissimilarities between two accounts of non-causal explanations, namely, topological and constraint-based explanations. Both explanation types involve a mathematical explanans and a mathematical dependency relation. Differences include the fact that sometimes the explanatory math is topological, while other times it is not (such as in the strawberry and genetics examples). Furthermore, both types contain some impossibility explanations (as seen in the Königsberg and genetics examples), but in topological cases there is sometimes more interest in explaining unique behaviors of the system (such as fragility, robustness, energy costs, etc.) as opposed to whether an outcome is impossible or possible for the system.

3.2 Optimality and Efficiency Explanations

A second main category of non-causal, mathematical explanation are optimality or efficiency explanations, which often arise in the context of evolutionary biology. Modern interest in these cases is motivated by indispensability arguments and attempts to capture the diversity of explanatory practice in science (Baker Reference Baker2005; Lyon and Colyvan Reference Lyon and Colyvan2008; Lyon Reference Lyon2012; Pincock Reference Pincock2012; Lange Reference Lange2013). Many of these cases involve surprising traits in organisms, which are explained by appealing to mathematical considerations. Examples of these surprising traits include explaining why cicadas emerge on prime-number-year life cycles, why honeybees have hexagonal-shaped honeycombs, and why sunflower seeds are arranged according to a Fibonacci-sequence (or Golden ratio) on the heads of sunflowers (Baker Reference Baker2005; Lyon and Colyvan Reference Lyon and Colyvan2008; Ross Reference Ross2023c). For many of these cases, the explanation involves two main components: (i) a background assumption that evolutionary processes select traits that are more efficient, optimal, or advantageous for the individual or species, and (ii) an argument – often mathematical – for how the trait in question meets this standard of being more efficient, optimal, or advantageous. The point is not that these explanations lack causal information – they clearly include this by appealing to evolutionary processes. The claim is that the explanatory work is not provided by causality alone, as it also requires mathematics.

Optimality and efficiency explanations build on the structure of the previous topological and constraint-based explanations in an important way. While cases in the previous subsection explained the system’s behavior by appealing to its mathematical features, the explanations in this subsection explain the presence of mathematical features in the system by appealing to the evolutionary advantage that these mathematical features convey.Footnote ⁴⁰ The similarity is that explanations in this and the last subsection capture how mathematical features have particular consequences for the system – the difference is that explanations in this subsection use these consequences to explain why the mathematical features are present in the system. As will be discussed more soon, optimality and efficiency explanations share a similar structure to “existence” or functional explanations, which explain the existence of some property in a system on the basis of the consequence it produces or entails (Wright Reference Wright1976).

A first example of optimality and efficiency explanations is the case of cicada life cycles. Cicadas are insects that spend part of their life cycle underground as nymphs, after which they emerge on predictable intervals. One species of cicada (the Magicicada genus) reliably emerges on intervals of prime number years – in particular 13 and 17 years – and there is interest in explaining why this is the case (Baker Reference Baker2005, 230). To be clear, there is interest in explaining why these cicadas emerge on prime number years as opposed to more common non-prime number year intervals. Explanations of this prime number year emergence are provided by the fact that prime number life cycles reduce the cicadas overlap with predator species. It is suggested that the evolutionary advantage of this reduced overlap has led to the selection and permanence of this prime number year life cycle.

The mathematical component of this explanation requires explaining why prime numbers reduce predator overlap. This explanation is provided by the notion of the lowest common multiple (lcm) within number theory in mathematics, as this shows how the frequency of intersection is minimized with prime numbers (Baker Reference Baker2005, 230–232). Given that cicadas can have life cycles between thirteen and 18 years (due to ecological constraints), this is the age range that is considered. In considering this range, the advantage of primes is evident by the mathematical fact that “12 (a non-prime) intersects with 1, 2, 3, 4, and 6; while 13 (a prime) only intersects with 1” (Lyon Reference Lyon2012, 561). For these reasons, mathematics, and number theory in particular, are said to play “a genuinely explanatory role in accounting for the cycle lengths of periodical cicadas” (Baker Reference Baker2005, 237). The mathematics in this case helps explain why “primes are optimal,” which is necessary to explain why they have been selected through evolutionary processes (Baker Reference Baker2005, 232).

In considering the structure of this explanation, it helps to examine the role of the mathematics involved. The explanatory target in this case is presented as the prime number year life cycle of cicadas. In traditional accounts of scientific explanation, one selects the explanatory target and then identifies factors “upstream” of this target that account for, produce, or cause the target, and that the target “depends” on. This captures the directionality of the explanation from explanans to explanandum. For this prime number year life cycle there are “upstream” evolutionary considerations that “selected” for this trait. However, these explanations also cite the prime number year as an “upstream” factor that has mathematical consequences for the “downstream” outcome of species intersection frequency. This is natural because changing whether life cycles are prime or not “makes-a-difference” to and explains the degree of intersection in the species. And, relatedly, species intersection “depends” on presence of prime number year life cycle. However, this is somewhat nonstandard because it places the original explanatory target (prime number), in the upstream “explanans” position and the consequences of this math (the species intersection) in the downstream explanandum position. Interestingly, explaining primeness does not involve merely citing the upstream factors that selected it, but also appealing to its forward influence on the cicada species. The need to appeal to what produces and is produced by the prime-year life cycle makes this explanation unique.

This cicada example bears similarity to types of “functional explanation,” in which one explains the existence of a biological trait on the basis of the trait’s function in an organism or species (Wright Reference Wright1973). In standard functional explanations, the link between the trait and the goal it serves is causal – for example, the heart exists in mammals because it causes the blood to circulate (and serves the function of blood circulation). As circulation is a subgoal that serves main goals of the organism (survival and reproduction), this explains the existence of the heart. The cicada example differs from this, in that the dependency relation between the prime-year life cycle and predator overlap is mathematical, as opposed to causal.Footnote ⁴¹ This mathematical dependency relation captures what is importantly non-causal and mathematical about this example, which distinguishes it from causal explanation.

A main question raised by this cicada example is how to integrate the mathematical and evolutionary components in the explanation. This explanation requires both components and they play different roles. One natural interpretation is to separate the explanation into two explanatory why-questions, which both need to be addressed. A first explanatory why-question is: Why do cicadas have the feature of prime-number year life cycles? Answer: Because this feature is evolutionarily efficient, optimal, or advantageous. A second explanatory why-question is: Why are prime-number year life cycles evolutionarily efficient, optimal, or advantageous? Answer: Because they reduce predator overlap, as explained by mathematical features of the least common multiple principle. The full explanation is not provided until both questions are answered, and they require causal and mathematical responses, respectively. While many philosophical accounts of evolutionary explanations assume that existing traits have been selected because they are optimal, efficient, or advantageous, this piece often requires careful justification (Wakil and Justus Reference Wakil and Justus2017).

Consider two further optimality explanations that fit a similar pattern. A second optimality explanation involves the explanatory why-question, “Why do bees have hexagonal-shaped honeycombs as opposed to honeycombs of other shapes?” The explanation of this geometrical trait involves a mathematical component revealing that hexagonal honeycombs are more efficient than other shapes because they use less honey and wax. The explanation of this optimality is provided by the honeycomb theorem, which “explains why a hexagonal grid is the optimal way to divide a surface up into regions of equal area” (Lyon and Colyvan Reference Lyon and Colyvan2008, Lyon Reference Lyon2012). The second piece is the evolutionary explanation, which specifies that this optimality makes these bees “fitter” and explains why the trait is selected for in the population. A third example is the Golden ratio spiral configuration of seeds on sunflower heads. Why do sunflowers pack their seeds in this configuration and not another? The answer here involves the fact that with this Golden ratio structure “the optimal packing of sunflower seeds is achieved” and that such optimality, due to the benefit it provides the species, is selected for through evolutionary processes. In both cases the mathematical trait makes a difference to some optimal outcome (less use of materials and higher packing density, respectively), and this difference is explained in virtue of mathematical relationships, dependencies, and information.

Other candidates for optimality and efficiency explanations are Sober’s “equilibrium explanations” (Sober Reference Sober1983), Chirimuuta’s efficient coding explanations (Chirimuuta Reference Chirimuuta2014), and Rice’s discussion of various optimality explanations in biology (Rice Reference Rice2015). Each of these cases explains an outcome by appealing to some evolutionary advantage – whether it is the “reproductive advantage” of a 1:1 sex ratio in a population, “evolutionary principles” that explain current behaviors such as the ability of neurons to “transmit more information,” and other cases in which optimality models are used to “explain why a system has evolved the optimal strategy” (Sober Reference Sober1983, Chirimuuta Reference Chirimuuta2014, Rice Reference Rice2015). While the optimality and efficiency explanations discussed in this section provide one class of non-causal, mathematical explanation, this does not imply that all evolutionary explanations or optimality explanations are non-causal. Some evolutionary explanations are causal in character and this framework is open to optimality and efficiency explanations outside of evolutionary biology.Footnote ⁴²

3.3 Minimal Model Explanations

A third class of non-causal mathematical explanations are “minimal model” explanations (Batterman Reference Batterman2002, Reference Batterman2021). These involve “minimal models,” which abstract from various types of information, including causal detail, in order to have explanatory power. Batterman introduces this form of explanation by distinguishing two types of explanatory-why questions. The first is a type (i) why-question, which “asks for an explanation of why a given instance of some pattern obtained” (Batterman Reference Batterman2001, 23). The second is a type (ii) why-question, which “asks why, in general, patterns of a given type can be expected to obtain” (Batterman Reference Batterman2001, 23). In this manner, type (ii) why-questions ask why the same pattern is exhibited generally or universally across a large group of different systems.

Minimal model explanations provide answers to type (ii) why-questions – they aim to explain why behaviors generally obtain and why they are shared across distinct systems. In scientific contexts, these shared behaviors are captured with the notion of “universality” as they are universal behaviors that are shared across physically distinct systems (Kadanoff Reference Kadanoff1990; Batterman Reference Batterman2021). Examples of universal behaviors in the physical sciences include how the Navier-Stokes equations capture the behavior of many microstructurally distinct fluids, that physically distinct pendulums exhibit periods that are proportional to the square root of their length, and that systems as diverse as fluids and magnets exhibit identical behavior at their critical points (Batterman Reference Batterman2001; Batterman and Rice Reference Batterman and Rice2014). While Batterman’s account of minimal model explanations was first examined in the context of physical science examples, later work applied his account to biological and life science explanations. Examples of universal behaviors in the life sciences include firing behaviors that are shared across physically distinct neurons (such as class I excitability), cases in which the same phenotype has different genetic and environmental causes (such as Parkinson’s disease), and processes that are “conserved” across nearly all living systems (such as glycolysis). The central explanatory why-question in these contexts is why systems with different physical details all exhibit the same universal behavior.

Batterman suggests that universal behaviors are often explained with principled techniques that isolate a “minimal model,” which is a model that “most economically caricatures the essential” features of the system (Goldenfeld Reference Goldenfeld1992; Batterman Reference Batterman2001). In many of these explanations, mathematical techniques are used to remove irrelevant detail from models of distinct systems until the reduced models collapse into the same “minimal model” or the same “universality” class of models. Given that these techniques allow for the principled removal of detail – which preserves the qualitative behavior of systems – the fact that distinct systems are all reduced to the same model or class explains why they exhibit shared behavior. A main feature of these cases is the use of mathematical techniques in providing the explanation. While other mathematical explanations involve “mathematical entities” (a number, a graph, a geometrical feature, etc.), minimal model explanations involve an “essential operation” that is mathematical and does the explanatory work (Batterman Reference Batterman2010).Footnote ⁴³

Consider an example of the minimal model approach in neuroscience. In the mid-twentieth century, Hodgkin used voltage clamp studies of single crab neurons to identify three different types of neural excitability, referred to as class I, class II, and class III excitability (Hodgkin Reference Hodgkin1948). For class I neurons the frequency–current relationship increases continuously from zero, for class II neurons it is discontinuous, and for class III neurons it is undefined (Ross Reference Ross2015, 40). Hodgkin identified that class I neurons are found in many different animals and it was later discovered that these neurons differ greatly in terms of their microstructural details. In order to appreciate this, consider mammalian pyramidal neurons, many of which exhibit class 1 behavior. These neurons have three main types of voltage-gated ion channels (selective for Na⁺, K⁺, and Ca²⁺), and each of these can have hundreds of molecularly distinct subtypes. From this variety, each neuron can express over a dozen different types of voltage-gated ion channels and these vary in density along the membrane producing unique voltage-dependent conductances. This helps capture the vast molecular diversity of mammalian pyramidal neurons with this behavior – the diversity across all neurons with this behavior (beyond mammalian pyramidal neurons) is, of course, much greater.

Neuroscientists have been interested in explaining why neurons that differ in terms of their lower-level details all exhibit the same excitability behavior. A central component of this explanation was provided by Ermentrout and Kopell, who derived a canonical model for class I excitability (Hoppensteadt and Izhikevich Reference Hoppensteadt and Izhikevich1997). This work involved using mathematical abstraction procedures to condense models of molecularly diverse neural systems to a simple model, referred to as a canonical model. In this case, “when mathematical abstraction techniques are used to abstract away from details of mathematical representations of neural systems, all representations converge onto the same canonical model” (Ross Reference Ross2015, 41). It is this convergence of the distinct models onto a shared single model that explains why they exhibit the same behavior despite their lower-level differences. In other words, because these mathematical reduction techniques eliminate detail while preserving qualitative behavior, the fact that all reduced models converge on the same model or class of models confirms and explains their qualitative similarity. This mathematical operation captures features of these systems that are “stable under perturbation of their microscopic details” (Batterman Reference Batterman2001).Footnote ⁴⁴

The minimal model account captures a non-reductive form of explanation that is supported by actual methods that scientists use in their work. Among the advantages of this explanatory framework is that it clarifies exactly why reductive forms of explanation are limited for particular explanatory targets. This limitation is that the shared, universal behavior of interest – that scientists want to explain – does not have a particular lower-level story, account, or explanation. In a line of reasoning that extends back to the work of Putnam (Reference Putnam, Capitan and Merrill1967) and Fodor (Reference Fodor1974) these shared behaviors are multiply-realized by different lower-level details, in a way that makes citing such lower-level details insufficient (Batterman Reference Batterman2001; Ross Reference Ross2020). This is seen in cases in which the same metabolic pathway is instantiated by different lower-level enzymes, where the same bridge topology can have different lower-level material constituents, and where the same neural firing behavior is made-up of different ion channel details. This clarifies a strong problem with assuming that these explanatory targets should fit a reductive model of explanation. The next step in these cases, involves providing the principled rationale that guide which factors do (and do not) explain and what justifies this. Distinct strategies for finding and representing abstract features propose different frameworks to address this second question, from those that emphasize causal relevance and control, topological properties with pertinent consequences, and mathematical abstraction techniques that expose qualitative similarity.

3.4 Non-Causal Explanation Conclusions

The three types of non-causal explanation presented in this section have unique features, they share similarities, and sometimes the border between them is less distinct. Similar to minimal model explanations, topological explanations abstract from large amounts of lower-level detail. In fact, it is a main feature of topological representations of systems that these representations are indifferent to lower-level, physical instantiations. This is what allows the same topology and explanation to capture neural networks in C. elegans, immune cell networks in humans, and social networks in societies. This is comparable to how the physical material of the Königsberg bridges are irrelevant to whether they have an Eulerian path or not. And a similar autonomy from lower-level details are found in some optimality and efficiency explanations. This is seen in Sober’s equilibrium explanations for the 1:1 sex ratio in many species at reproductive age (Sober Reference Sober1983). In this case, the 1:1 sex ratio is explained by showing “how the event would have occurred regardless of which of a variety of causal scenarios actually transpired” (Sober, Reference Sober1983, 202). It is not factors in the actual causal history of any outcome that matter for these explanations – instead a variety of disjuncts captures possible outcomes, without the need to specify which actually occurs. Similarly, the mathematics underpinning the optimalities and efficiencies in the cicada, honeybee, and sunflower cases do not depend on or require particular lower-level details. The lowest common multiple principle, honeycomb proof, and Golden rule hold regardless of the details of the system. All accounts provide strong, principled reasons for why particular lower-level details are irrelevant to the explanation.

However, these types of non-causal explanation also differ in notable ways. Topological and constraint-based explanations tend to lean on coarse mathematical relationships, with little (to no) information about a system’s goals or purposes. Unlike other forms of non-causal explanation, these include the unique impossibility explanations, which explain why some outcomes are impossible or off-limits for a system (Ross Reference Ross2023c). Optimality and efficiency explanations, on the other hand, have the unique feature of including assumptions about a system’s goals and how these are supported by the system’s features. These are often found in evolutionary or design cases, in which there is interest in explaining why a system has a particular feature. Despite their differences, in many of these cases, insights into novel forms of explanation in biology and the life sciences have been provided by revealing unique types of explanatory targets and explanatory why-questions.