Abstract
This paper proposes a methodology to investigate how the conceptual structure of a model may have borrowed from an already existing model in a different field. From a reverse perspective, it amounts to hypothesizing that an already existing model was somehow transferred to a new field and incorporated into a newly constructed model with the purpose of explaining some additional phenomena. We use two well-known theoretic models, Robert Malthus’ model of human population and Darwin’s model of organism population included in the theory of evolution by natural selection, to exemplify this “incorporation of a conceptual structure” methodology. By highlighting how the conceptual structure of Malthus’ model actually sits within that of Darwin’s model, we show how Darwin’s model may have been constructed, on the one hand, and how Malthus’ model may have been transferred, on the other hand. In order to show how the construction and transfer occur, we outline five epistemic strategies that could be underlying incorporation. Overall, this methodology can be viewed as a diadic and bijective perspective on models that can be applied to other similar cases, using “construction” and “transfer” as complementary handles to probe the available conceptual, structural, and historiographical material.
1. Introduction
The topic of knowledge transfer has become a hot issue in the philosophy of science. “Knowledge transfer” means that knowledge originating from a discipline is transferred to another discipline whose subject matters and domain differ from the original discipline. There are at least two kinds of approaches to the issue of knowledge transfer.
The first kind of approaches focuses on evidential knowledge transfer: pieces of knowledge from a discipline are transferred to engage with various practices in another discipline. For examples, Chapman and Wylie (2016) explore how different types of isotope analysis, such as radiocarbon dating and lead isotope analysis, originating in nuclear physics, were transferred to support archaeological practices and offer a new type of evidence.1 In a similar undertaking, and following Chapman and Wylie’s framework, Downes (2019) investigated how genome sequencing of ancient human DNA, originating in molecular biology, was transferred to contribute to archaeology.
The second kind of approaches focuses on model and theory transfer, because models and theories are regarded as the major vehicles for providing descriptions, explanations, and predictions of phenomena of interest. The question of knowledge transfer can thus be usefully narrowed down to that of models and theories transfer. Model and theory transfer raises a problem regarding the specific relationship between the source and target models. Until recently, philosophers of science tended to view such a transfer as a mere application or extension of an exemplar model, describing relations between source and target models in terms of analogousness (Kuhn 1970) or similarity (Giere 1990). Since then, other philosophers of science have contributed to this topic by investigating a variety of cases, revealing specific modes of model transfer, and describing relationships between two models beyond the frameworks of analogy and similarity (Herfeld and Lisciandra 2019; Humphreys 2002, 2019; Morgan 2014).
The approach we adopt in this paper belongs to the second kind, led by Humphreys’ template-based approach, which has been followed by a few philosophers and applied to analyze the cross-disciplinary transfer of knowledge (Knuuttila and Loettgers 2012, 2014, 2016; Houkes and Zwart 2019, Knuuttila and Deister 2019, Price 2019). However, our approach is still somewhat different from the template-based one, because philosophers focus on mathematical/computational models and their computational aspect, but pay little attention to non-computational models and other aspects.2 In this paper, we aim to investigate non-computational models and the conceptual aspects of models in general, and to introduce a new mode of model transfer that involves the successful incorporation of a source model’s conceptual structure. Because a new model can arise via incorporation of an already existing model, it is also a question of how a new model is constructed from an existing one. Indeed, the topic of model transfer usually goes hand in hand with the topic of model construction (Humphreys 2002, 2004; Knuuttila and Loettgers 2012, 2014, 2016).
Scientific models and theories do not arise from thin air. Presumably, there are many modes of model and theory construction. Some theories or theoretic models might have been conjectured (Popper 1959), or guided by heuristic criteria (Post 1971), or mediated by middle-level models from raw empirical data (Morgan and Morrison 1999), while others might have been constructed based on some antecedents through analogy (Hesse 1966; Holyoak and Thagard 1994; Chen 2022). Most likely, models and theories have been constructed and are still being constructed in different ways such as, for example, Weisberg’s abstract direct representation (Weisberg 2007, 2013). In this paper, we present a specific methodology to help understand how a theoretical model may have been built by incorporating a source model into a novel structure, even across different fields or disciplines. As we focus on the evolution of science, regardless of the period or the time frame considered, the methodology introduced here allows adopting the complementary perspectives of model construction and of model transfer, alone or in combination, drawing from the available conceptual, structural, and historiographical material.
In the framework we are introducing, incorporation is to be understood as the transfer of a kernel of conceptual components from one theoretic model to another, broader, one. This, however, does not require that all components from the source model be kept unchanged. On the contrary, in the process of being incorporated or embedded, some conceptual components of the source model may be modified to fit a new context. Regardless of the replacement, removal or modification of some conceptual components, the conceptual structure from the source model incorporated in the new model should be found as a whole, unchanged. Hence, this form of incorporation can be dubbed “structural incorporation.” Structural incorporation implies that the broader model has been constructed from, or is based upon, the source model, and that a specific conceptual structure is involved.
To demonstrate the meaningfulness of structural incorporation in addressing concrete study cases, we present a historical case involving two very well-known scientific works, Robert Malthus’ theory of human population and a part of Charles Darwin’s theory of evolution (i.e., population of organisms). Although this case has been repeatedly analysed and investigated in the past decades (Flew [1957] 1986; Vorzimmer 1969; Herbert 1971; Kohn 1980; Young 1985; Bowler 1990; Ruse 1999), introducing a new perspective on the relation between theories to revisit the Malthus-Darwin case is still valuable. Like most philosophers of science today, we take a model-based view of theories to understand Malthus’ theory of population as a theoretic model and Darwin’s theory of evolution as a combination of several sub-theoretic models.
When addressing the construction of Darwin’s theoretic models, it is important to specify that we are not immediately concerned with the actual process through which Darwin constructed his entire theory of evolution (cf. Hodge 1983, 1989; Okasha 2000). Consequently, we will not attempt to trace the development of Darwin’s theory via his notes, correspondences, early works, and the likes. Rather, we intend to offer a methodological framework that explains (in a sense of modelling) the being-embedded relation between Malthus’ and Darwin’s models of population. This specific being-embedded relation shows that Darwin could have constructed his model of organism populations by incorporating Malthus’ model. More specifically, it shows that the modelling of socio-economic changes in human populations has been transferred into the modelling of changes in other populations within a broader biological domain.
The next problem we will consider is the ways according to which structural incorporation may occur. We will describe five epistemic strategies: conceptual inheritance, conceptual modification, structural abstraction, structural projection, and expansion from conceptual modification and structural projection.3 We do not claim that Darwin actually applied any of these strategies while developing his model; instead, we argue that, starting from Darwin’s premises, the use of any of these strategies could have led to the result Darwin painstakingly came up with. Our position is that if any of these five strategies can be used to outline the incorporating/being-incorporated relation between two theoretic models, then such a strategy can help elucidate the constructive procedure of Darwin’s model and the transferring procedure of Malthus’ model via incorporation, even if Darwin did not actually move along that way. In a sense, our methodology is itself a form of conceptual modelling, as it is well aligned with Weisberg’s definition of modelling as “the indirect theoretical investigation of a real world phenomenon using a model” (Weisberg 2007, p. 209).4 In our methodology, Darwin’s construction of evolutionary theory by natural selection is a real world phenomenon, while our methodology of structural incorporation is an explanatory model for part of the phenomenon. We take a meta-modelling approach to attempt filling a blank in history—Darwin himself did not reveal what and how he drew something from Malthus’ theory of human population to construct his theory of organism population.
This paper is divided into seven sections. Following the introductory section, we fully expound our methodology of structural incorporation and characterize the conceptual aspect of scientific models. This section also introduces the five epistemic strategies possibly used for structural incorporation. The third and fourth sections briefly outline what is known about the relation between Malthus’ and Darwin’s models and then provide detailed reconstructions of the conceptual structures of both models. Having laid bare the conceptual structures of both theories, section five subsequently makes it possible to demonstrate that Malthus’ core conceptual structure is indeed embedded in that of Darwin’s model of organism populations. This section also shows that the five epistemic strategies offer reasonable hypotheses or models to capture and further investigate the process through which Malthus’ model is incorporated into Darwin’s model. The sixth section contrasts our structural incorporation approach with the template-based approach to model transfer, showing that the latter is not well-adapted for the Malthus-Darwin case and other similar cases, because it does not pay attention to the conceptual aspect of models. However, we can use the expansion strategy to connect the template-based approach with ours. The final section draws some general conclusions about the methodology of structural incorporation.
2. Structural Incorporation of a Conceptual Structure
Our goal in this section is to characterize the conceptual aspect of theoretic models. This involves understanding theoretic models as being conceptual structures, that is, as sets of kernel concepts linked together by connecting or principal propositions according to specific relations. Using propositions to represent a conceptual structure does not require us to endorse the syntactic view of theories, because we characterize theoretic models through their conceptual and semantic components. In our view, a theoretic model is first and foremost a conceptual structure.
The conceptual structure underlying a theoretic model can be seen as a conceptual network in which concepts work as nodes and principal propositions connect these nodes with one another. Conceptual structures will differ if their respective kernel concepts differ. Conceptual structures may also differ if they involve the same kernel concepts but have them connected differently. For example, “force,” “mass,” “distance,” and “time” are kernel concepts of Newtonian mechanics. They are interconnected by the definition of acceleration, the second law of motion, and the law of gravitation in a specific network. Those ideas are also central to Aristotelian dynamics, but they connect with one another in a very different network. In any case, “force,” “distance,” “mass,” and “time” are in no way linked with something like “gravitation.” Thus, the conceptual structure of Aristotelian dynamics is very different from that of Newtonian mechanics. This understanding of theoretic models having both structure and content is in line with that shared by various schools of thoughts, including not only logical empiricism, historicism, scientific realism, and social constructivism (Hempel 1966; Kuhn 1970, 1989; Putnam 1975, 1981; Barnes 1982; Thagard 1990) but also the syntactic view and the semantic view of scientific theories (Suppe 1977; Giere 1990; Humphreys 2004).
“Structural incorporation” so far provides us with a convenient expression to refer to the fact that some models appear to be harbouring the conceptual structure of a previously existing theoretic model. The structural part of the existing model should be straightforwardly found as embedded in a later version of a theoretic model, as we will see in the next section. Thus, the theoretic models sharing the same conceptual structure may be genealogically related. Indeed, and this is stating the obvious, there would not be much point in looking for the incorporation of a conceptual structure if there is not a shred of filiation between the two theoretical models. Conversely, it is possible to imagine that identifying incorporation between two models might be used as a tool to unearth an unknown, most likely indirect, filiation between them.
Under what conditions can we say that the conceptual structure of a model is embedded in that of another broader model? If one finds conceptual components from a model in another more recent model, does this mean there is a relation of being-embedded between the two theoretic models? Is Greek atomism embedded in Dalton’s chemical atomism in the nineteenth century, as the two theoretic models use a common concept of “atom”? According to our perspective, the answer would be negative because it is quite clear that Greek atomism and Dalton’s chemical atomism do not share a common conceptual structure. A second example will help to further characterize the being-embedded relation. Coulomb’s law of electrostatics (F = k|Qq|/r2) and Newton’s law of gravitation (F = G|Mm|/r2) obviously share a common formal structure, but Newton’s law of gravitation is not embedded in Coulomb’s law, because the kernel concepts in the two structures are fundamentally different. Using these two counterexamples we can circumscribe the being-embedded relation and expound it in such a way: A conceptual structure is embedded in another model if the model’s kernel concepts are present in the other theoretic model without any structural change. Given that a conceptual structure is defined by the connecting network of kernel concepts, the expression “without structural change” means that the network (usually represented by a set of principal propositions) must be kept mostly unchanged. This implies that the being-embedded relation does not rule out the possibility that some conceptual components are modified to fit new data in the new theoretic model.
It should be clear, however, that identifying an embedded structure in a case study does not say much about the target model and any of its other components. In the study case we will be using in this paper, many conceptual components of Darwin’s theory of evolution by natural selection will not be discussed. What we shall investigate is how and to what extent Darwin’s model of population relates to Malthus’ model of population. In addition, the methodological approach outlined here does not address how much more a target model is able to explain compared to the source model, or how much closer the explanations it can provide are to some criterion of truth.
It is therefore quite plain that this perspective on theory construction does not enable us to assess the extent to which two theoretic models are historically related by using their conceptual structure alone. However, the notion of structural incorporation does allow us to formulate hypotheses as to how a given conceptual structure may have been co-opted in a particular model. In order to better understand how a particular model was constructed from the suspicion, or limited evidences, that it is genealogically related to another one, it is helpful to investigate the detailed operation through which the conceptual structure of the source model is embedded in the target model. In what follows, we outline five types of epistemic strategies that could have played a significant role in this case of structural incorporation.5 Strategies offer guidelines that help us to make inferences or epistemic operations such as remembering, learning, conceiving, imagining, and thinking.
We are using the term “strategy” in a way similar to Lindley Darden’s general understanding that “a strategy is a method, a procedure, a practice, a principle. Strategies can be codified and taught” (Darden 1991, p. 20). Strategies are a key to Darden’s methodology of theory construction, which is developed in her Theory Change in Science: Strategies from Mendelian Genetics (1991) and ensuing works (Darden and Cain 1989; Darden 1995, 2006). In order to show how classical geneticists used strategies to jointly construct the final and complete version of the classical genetic theory, which culminated with Morgan’s theory of the gene in 1926, Darden identifies three types of strategies: (1) strategies for producing new ideas; (2) strategies for theory assessment; and (3) strategies for anomaly resolution and expansion of scope (Darden 1991: 4).6
The strategies we are discussing here have a different scope than those outlined by Darden. First, they do not belong to Darden’s first type, because they involve whole constructs rather than independent ideas or concepts. Second, the strategies we propose only partly overlap with Darden’s second and third types, because theory construction is not primarily concerned with theory assessment and anomaly solution. From the perspective of the conceptual structure itself, and whether it is embedded is another broader one, theory assessment and anomaly resolution relate more to the empirical underpinnings of theory construction than to its underlying conceptual and structural constitution, and to the specific role played by these constitutive elements in theory construction. The strategies introduced here have a limited scope and are to be construed for cases in which a scientist elaborates a theory from a previous one. The five epistemic strategies of structural incorporation and transfer are:
Conceptual Inheritance: A source model’s conceptual components and underlying structure are introduced into a target model without any significant changes.
Conceptual Modification: Conceptual components of a source model are given novel features and introduced into a target model. The original structure or the connective network in which the modified concepts are embedded is retained. Thus, it is important to note that this strategy does not involve structural change.
Structural Abstraction: The conceptual structure of a source model is abstracted from its original context. As the model is given a more general scope, it comes to cover a broader field.7
Structural Projection: The abstracted conceptual structure is projected onto the particular field on which focuses a target model. Therefore, the conceptual structure in the source model becomes a subset of conceptual structure in the target model. In some philosophers’ view, structural projection is an essential element of analogy (Hesse 1966; Holyoak and Thagard 1994; Chen 2022).
Expansion from Conceptual Modification and Structural Projection: Due to conceptual modification and structural projection, the source model is built within the broader context of a target theoretic model, including additional components not occurring in the source. In the context of the target model, additional principal propositions expand the source model with novel components. In other words, the target model’s conceptual structure articulates old and new kernel concepts.
In order to analyse how these strategies might explain (in the sense of modelling) the incorporation of Malthus’ model of human population into Darwin’s model of organism populations, we first have to present the conceptual structure underlying the source model. Thus, in the following section we describe Malthus’ model and delineate the conceptual structure that is involved in the analysis.
3. The Conceptual Structure of Malthus’ Model of Human Population
It is widely known that the economist Robert Malthus’ theory of population inspired the theory of evolution by natural selection. Charles Darwin and Alfred R. Wallace—the two founders of the theory of natural selection—explicitly acknowledged that Malthus’ Essay on the Principle of Population exerted a key influence on their thinking (Darwin 1969, pp. 98–9). Given these testimonies, one cannot help but wonder what the nature of that influence was. There is a large body of literature dealing with this topic, and in the end there is no overwhelming consensus on the nature and extent of Malthus’ influence on Darwin (Flew [1957] 1986; Vorzimmer 1969; Herbert 1971; Kohn 1980; Mayr 1982; Ospovat [1981] 1993; Young 1985; Bowler 1990; Desmond and Moore 1991; Ruse 1999). Historical evidence is simply lacking. However, no one would reject that Darwin’s theory incorporates at least some features of Malthus’. Here we understand Malthus’ theory as a theoretic model of human population and, as such, it has a conceptual structure that can be described or represented by a set of propositions.
As outlined in An Essay on the Principle of Population (1826), Malthus’ conceptual structure is easily summarized by such a set of propositions. For the needs of our analysis, we delineate the theory’s conceptual structure through seven propositions, identified M1 to M7. The first three propositions introduced in the first chapter of An Essay play a critical role in Malthus’ model.8
M1 Population, when unchecked, increases in a geometrical ratio.
M2 Subsistence increases only in an arithmetical ratio.
M3 This implies a strong and constantly operating check on population from the difficulty of subsistence.9
In the second chapter of An Essay, Malthus introduces three other propositions summarizing the relations between population size, subsistence, and checks through his lengthy presentation of the checks on populations past and present (Malthus 1826, pp. 23–4).
M4 Population is necessarily limited by the means of subsistence.
M5 Population invariably increases where the means of subsistence increase, unless prevented by some very powerful and obvious checks.
M6 These checks, and the checks that repress the superior power of population and keep its effects on a level with the means of subsistence, are all resolvable into moral restraint, vice, and misery.
The six propositions are needed to properly introduce the fundamental principles of Malthus’ model of population. However, to represent Malthus’ whole conceptual structure, the concept of ‘struggle for existence’ also has to be brought up. In chapter 6 of An Essay, Malthus illustrates this concept with lengthy descriptions of the hardships that some primitive tribes must have encountered (Malthus 1798, pp. 47–8; and in Malthus 1826, p. 95). The various hardships faced by a population—or the checks, in Malthus’ vocabulary—connect Malthus’ model with empirical data. This may be expressed according to the following proposition:
M7 The perpetual struggle for room and food causes a prodigious waste of human life.
Let us now consider the conceptual structure harboured in the principal propositions. Propositions M1 and M2 qualify the rate of increase in populations and in their means of subsistence, respectively. In an ideal situation, the rate of increase of a particular population is higher than that of its means of subsistence. It is a third proposition, M4, that connects M1 with M2 by stating how increases in population and subsistence rates relate to one another. M2’s upper limit sets M1 upper limit. Thus, the three propositions together constitute a function or a simple mathematical model without causation and without much explanatory grip. Introducing propositions such as M3 and M5 allows sketching explanations; particularly with regard to the overall stability of populations’ size, which was a topic Malthus was especially interested in. More precisely, M3 and M5 together play four roles in Malthus’ conceptual construct. First, the two propositions reassert the asymmetry between the size of a population (M1) and the available means of subsistence (M2) laid down by M4. Second, they introduce the concept of check on population, explicitly stating that whenever the population size reaches the upper limit set by available resources, there is a strong limitation or check against any further increase in the population’s size. Third, and more straightforwardly in M5 than in M3, they set the relation between population size and the available means of subsistence in a dynamic perspective: to the extent the means of subsistence increase, the size of the population may also increase. Finally, M5 introduces the notion that powerful phenomena other than the amount of available means of subsistence may be involved in a population size’s stability. Thus, the adjunction of M3 and M5 to M1, M2, and M4 allows Malthus to build a full causal model that represents a fundamental process of a population size’s stability and variations.
How can the extended model be applied to specific populations? Before he could fit his theory to actual societies, Malthus still had a riddle to solve. Indeed, why do past or actual populations appear stable in size even though they do not have the same fecundity or the same rate of increase in resources? The economist thought that each population is checked by some constraining factors, and that different societies may be checked by different factors. Throughout chapters 3 to 14 of volume 1 and the entirety of volume 2 of his An Essay, Malthus presents and describes the different checks that impinge on the population size of human societies. Proposition M6 is thus a synthetic statement of those checks on the population size and on its rate of increase. Moreover, M6 also distinguishes three categories of checks. Although what is specifically covered by each category is not relevant here, it is important to note that these categories allow bridging the theoretic causal model with empirical observations of the demeanours within particular societies. In the remainder of our presentation, we will neglect M6 since it is not directly involved in the case of structural incorporation that we are considering.
Before we turn to Darwin’s theory, it is useful to take a more synthetic look at the conceptual structure that we are interested in. While M1, M2, and M4 establish an abstract mathematical model, M3 and M5 bring a dynamical and explanatory character to the relation between population size and the amount of available resources. Proposition M6 adds some flesh to the model and brings it in closer contact with possible empirical observations of past or contemporary human societies. As to M7, the statement of what the struggle for existence is at the scale of human societies, it asserts the most important consequence of Malthus’ conceptual structure. Whether or not such a structure exists in Darwin’s theoretic model of population size regulation will be the subject of the next section.
4. The Conceptual Structure of Darwin’s Model of Organism Populations
Although Darwin never explicitly wrote that he borrowed some determining elements of Malthus’ theory, we cannot rule out the hypothesis that he used a whole conceptual structure. Of course, Darwin’s theory of evolution by natural selection is not a restatement of Malthus’ theory of population for the animal and vegetal kingdoms. Darwin’s theory is more sophisticated and its scope is much broader. Moreover, it is concerned not only with the relative stability of population size, but also with organic functions and features and with the transmutation of species. Nevertheless, the two theories share some common elements. More precisely, it is our claim that Malthus’ model of population is part of Darwin’s theory of evolution, or, said otherwise, that Darwin’s theory harbours a conceptual sub-structure very close to the one proposed by Malthus. To acknowledge the subordinated role of Malthus’ model, it is necessary to construe Darwin’s theory as one built upon constituent models. In fact, it is an umbrella theory made up of at least three independent but interrelated models: a model of population size regulation, a model of natural selection, and a model of species evolution. In order to adequately investigate the possible occurrence of structural incorporation, it is useful to unfold the propositions used to describe the constituent models in the manner Malthus’ model of population was described. Thus, the first model of Darwin’s theory, the model of organism populations, can be synthesized in the following way.
D1 Populations of animals and plants increase in a geometrical ratio under ideal conditions.
D2 The amount of food (and of other resources) for each animal and plant species has an upper limit.10
D3 “[A]s more individuals are produced than can possibly survive, there must in every case be a struggle for existence, between either one individual with another of the same species, or with the individuals of distinct species, or with the physical conditions of life” (Darwin 1859, p. 63).
The two other constituent models we are alluding to can be construed in the same way. The model of natural selection can be described by the following propositions:
D4 Individuals of the same species—and species of the same genera—display variations in some of their traits.
D5 Under natural circumstances, individuals harbouring favourable variations would tend to be preserved, while others having unfavourable ones would tend to disappear.
D6 Some trait variations can be inherited by offspring.
D7 Natural selection (the struggle for existence) always retains the most favourable variations of a trait.
D8 The selection of favourable variations has cumulative effects over time.
D9 Species transmutation occurs over a very large time scale.
In what follows, we shall concern ourselves only with the first constituent model. This description of Darwin’s model of population shares important conceptual elements with Malthus’ model of population as outlined above with propositions M1–M7. Darwin chiefly relates the rate of population size increase and the amount of available means of subsistence (D1–D2). Although Darwin’s presentation of this relation is truncated compared to Malthus’—Darwin does not express the rate at which means of subsistence replenish or deplete—the conclusion is the same: a population cannot maintain itself or increase in size above some level of available subsistence or resources.
It is thus not surprising that Malthus’ proposition about the rate at which resources increase (M2) is left aside. To the extent the size of a population is ultimately bound to the amount of available subsistence, the rate at which resources fluctuate is really a point of detail. In addition, although Darwin says nothing about it, that rate is very unlikely to be constant throughout the natural world. This being said, one should be careful not to over-interpret Darwin’s claims about the relation between the number of organisms and the amount of food (or other resources) available. The dependency of living beings on some resources certainly makes up one of the most widespread checks in the struggle for existence, but obviously it is not the sole one.
The difference between the two theoretical models thus lies in the additional propositions (M3, M5, M7, and D3) delineating the core relation between population size and means of subsistence in each model. As we have seen in Section 3, propositions M3 and M5 introduce an explanatory character to the relation between population size and the amount of available resources. These two propositions allow Malthus to instantiate his model with information on observed or hypothesized checks, thus tailoring his analyses to any society. The ultimate consequence of the principle of population—the struggle for existence in any and each society (M7)—is a veiled but constant threat that only moral restraint may defuse.
Much like proposition M7, Darwin’s proposition D3 outlines the consequence that follows when the number of actual organisms reaches or even surpasses the upper limit set by the available resources: a struggle for existence ensues. Interestingly, the scarcity of resources is only one factor amongst the many others discussed in Origin. Thus, not specifically relating to resources, D3 also allows drawing a strong parallel with Malthus’ propositions M3 and M5 where “checks” other than the shortage of subsistence may easily be pictured. Once more, the correspondence should not be overstated. Although the two authors acknowledge the “pressure” exerted on societies (Malthus) or on species/organisms (Darwin) by limits on resources’ availability, the other checks on the growth of population size exhibit a very distinct nature for the authors. However, this difference is conceptual rather than structural; the content or perimeter of some conceptual kernels differ, but not the relations between these kernels, or the conceptual structure shared by both models.
In Section 4 we have seen that most of Malthus’ checks on population increase take their origin from within the society they impinge upon. Whether or not these checks affect individuals or the society as a whole, they take their origin in the organization of society and in its prevailing moral and political ideas. Other checks such as unsolicited wars, spells of drought, and epidemics are assumed to be caused by factors external or alien to the society they impact. At least for bad weather and epidemics, Malthus even suggests that some checks “… appear to arise unavoidably from the laws of nature …” (Malthus 1826, p. 16). For Darwin, all checks or factors playing a role in the struggle for existence are naturalistic in kind. The distinction between checks that originate within a society and checks that arise from some natural phenomena outside that society is irrelevant. This is why M3, M5, and M7 can be substituted for the single proposition D3 in Darwin’s theoretic model. Thus, Malthus’ and Darwin’s sets of propositions can be compared in the following way.
M1 → D1
M2 and M4 → D2
M3, M5, and M7 → D3
From this comparison of the two theoretical models, the possible incorporation of Malthus’ model of population into a constituent model of Darwin’s theory of evolution appears much clearer. Indeed, kernel concepts such as “population,” “geometric ratio,” “subsistence,” “check,” and “struggle for existence,” as well as the structure connecting them together in Malthus’ model—in short M1–M5 and M7—also occur in Darwin’s model, but under a different guise (D1–D3). In the end, however, a single conceptual structure can be inferred from the two distinct sets of propositions. Differences between these sets manifest both the distinct background from which these theoretic models arose and the specific domain they were designed to cover.
Now that we have shown that Malthus’ model is embedded within Darwin’s, we need to discuss the strategies that can (meta-) model the transfer from one theory to the other.
5. How Could the Conceptual Structure of Malthus’ Model Be Transferred and Incorporated into Darwin’s?
Section 2 introduced five epistemic strategies and suggested that they can play a heuristic role in understanding specific cases of model construction—namely, those in which an existing conceptual structure (or a part of it) may have been transferred and incorporated. Sections 3 and 4 have shown that the being-embedded relation between Malthus’ and Darwin’s models is in principle possible due to the common underlying conceptual structure they share. It is important to emphasize, once again, that our assessment of Darwin’s construction of theoretic model does not involve any claims about the historical process through which Darwin assembled his theory. Said otherwise, it is not an exercise of speculation on the possible course of history. Rather, it is an attempt—based solely on the isomorphism between Darwin’ and Malthus’ models of population—at unveiling some of the strategies that may possibly help understanding Darwin’s progress toward his theory.
The first strategy of structural incorporation that might have been involved in Darwin’s theorizing is that of conceptual inheritance. In what we have referred to as the model of organism populations, Darwin considers the increase of populations’ size much in the same way that Malthus does. In fact, both authors take the geometric progression of the number of individuals under ideal conditions as a basic principle which can model empirical facts: the fluctuation and/or the stability of population. However, not all concepts in Darwin’s model were inherited from Malthus’ model; some underwent significant alteration.
Much like in Malthus’ model of population, Darwin introduces the notion of check when discussing the upper limits to population size. Like his predecessor, Darwin lengthily discusses the most obvious of the checks on an indefinite expansion of a population: the limitation of subsistence and more broadly of available resources. The fact that Malthus and Darwin do not conceptualize the limitation of populations’ size to the same extent can be explained by the role the shortage of subsistence plays in their respective work. While Malthus goes as far as to consider the interplay of population size and means of subsistence increase rates as a mathematical function, thus rendering much weight to a rather limited number of checks, the specific checks associated with a shortage in means of subsistence play a less decisive role in Darwin’s model. Darwin is fully aware that natural selection’s reach is in no way limited to the ability of organisms to access the resources they need. They may well fall prey to other species, suffer from adverse weather, etc. (Darwin 1859, p. 68). This means that the conceptual content of check is a bit different from that in Malthus’ model. To give an account of this slight difference, a second strategy—conceptual modification—may have been employed for concepts such “check” and “struggle for existence” as Darwin developed his theory. In any case, there is an intentional commonality in the way both scientists use the term ‘check’ in their work. After all, a check on a human population or on a species of organism is something that restrains its increase in size or number.
The second epistemic strategy can also be used to understand the appropriation and alteration of the notion of “struggle for existence” by Darwin. Darwin understood that the transmutation of species had to do with the regulation of the numbers of organisms, as much as Malthus acknowledged that most of the changes observed in a society were consequences of the population’s rate of increase in size. However, in order to come up with his model of organism populations, and further to his theory of evolution, Darwin realized that the change in the number of individuals is not sufficient by itself to ground the evolution of species. Indeed, it is not because a species gets to have more (or fewer) members that it persists in time and even further adapts itself. Rather, it is the fact that this persistence in time or adaptation depends on existing differences among members of a given species. Thus, in order to get to his theory, the naturalist had to infer that what matters is not solely the population as a whole, but also the units that make it up. In that sense, Darwin modified the notion of struggle for existence to include both the struggle between populations and within a population. Said otherwise, while Malthus takes societies as the units struggling for their existence, Darwin also underlines the struggle among individual organisms.
The conceptual structure that is comprised of “population,” “geometric ratio,” “subsistence,” “check,” and “struggle for existence” in Darwin’s model of population is applied to the animal and vegetal kingdoms, in which the humankind is naturally included. This means the whole conceptual structure of Darwin’s model is located at a broader and more abstract level than that of Malthus’ model is. The third strategy—structural abstraction—can thus be used to understand why Darwin’s conceptual structure has a broader scope than Malthus’. However, abstraction is not sufficient to explain the fact that Malthus’ conceptual structure reoccurs in Darwin’s model of population. An epistemic strategy is needed to explain the shift from human societies to the animal and vegetal kingdoms.
Using the fourth epistemic strategy, structural projection, we interpret that the conceptual structure is projected from the domain of human societies to all living beings. This strategy gives a good account of this shift in perspective and thus indicate how the conceptual structure of a source model is transferred to a target model.
However, extending observations and speculations on human societies to the whole of nature certainly required some grounding. This is why Darwin’s model of population required that he uses species of interest to man (those on which an artificial selection can be observed) as an intermediary step in his projection of Malthus’ work onto the whole living world. Of course, this step might have only served some argumentative purposes as some have argued (Herbert 1971; Waters 1986), but it could also have played an important bridging role in the process of projection. Another important instance of projection, already introduced above as a potential case of the conceptual modification strategy, can be identified in the specific checks that the species of interest are subjected to. Indeed, while Darwin’s presentation of artificial selection makes it obvious that some populations other than human ones can have their rate of increase controlled, it also keeps the concept of check closely linked to the actions of some agents. Stated otherwise, Darwin’s depiction of artificial selection kept Malthus’ idea that most checks are caused by agents, but then expanded it to the whole range of domestic plants and animals. From Malthus to Darwin, the role of agents in causing the checks and in experiencing them is modified, whereas intentionality is kept.
The last strategy that may have been involved in the development of Darwin’s theory of evolution by natural selection is identified as expansion from conceptual modification and structural projection. By this, we mean that Darwin’s appropriation of Malthus’ model may have had an impact not only on Darwin’s model of population but also beyond it. Below, we provide a preliminary analysis to show the interpretative power of this strategy; a detailed analysis goes beyond the scope of this paper.
Malthus’ model may have had an influence on Darwin’s development of some other constituent models in his theory of evolution. One such influence is implied by proposition D3 of Darwin’s population model. D3 states that individual organisms necessarily struggle with other members of their species, or with individuals of distinct species, or with the physical conditions of life. This struggle implies that there is a sorting of organisms—some go through, some do not—and also that this sorting is not random; otherwise, the term ‘struggle’ would not be adequate. Some individuals get picked off because they differ from others. This hints at some variability among them.
The necessary struggle among species or individuals, realized by the widespread and highly diverse checks on them in nature, entails in turn a distinctive feature of variability. Indeed, individuals are not simply different from one another. Each variation they display, however slight it may be, is something a check can possibly act upon. To use a language foreign to Darwin and his contemporaries, individuals display variations in the way different checks can impair their survival and reproduction. Hence, switching back to the naturalist’s vocabulary, they are more or less adapted or fit to a particular situation. This lends nicely to the construction of a model of sorting through natural selection.
It should be noted that individuals’ varying degree of fitness or of adaptation to particular circumstances directly relates to the notion of design, as construed by natural theology at that time. Although it is not explicitly referred to in the model of population, the topic of adaptation or design raises the question as to how some individual organisms came to be better suited than other members of their species, or than individuals of other species, in their struggles for existence. Following this line of thought, Darwin could only have found an important support for his budding theory of the transmutation of species. Here, considering why individuals came to differ from each other or from individuals of other species, instead of how they actually differ, leads to a notion of species’ evolution. To sum up, using this fifth strategy provides a perspective on how Darwin’s theory of evolution by natural selection could owe something to Malthus’ model of population.
Again, we are not claiming that Darwin constructed his theory of evolution singly based on Malthus’ model of population. In addition to drawing inspiration from Malthus, Darwin most likely used inferential means such as “inference to the best explanation” from the wealth of geological and biological data he observed through his investigations (Thagard 1978; Hodge 1983; Okasha 2000).
6. The Template-Based Approach: A Comparison
To analyse how scientific knowledge is applied to the world, Paul Humphreys (2002, 2004) proposes a philosophical theory using an abstract model or a template as a unit, as opposed to other units such as theories, research programme, or paradigms. In his work, Humphreys investigates both the application of templates in cross-disciplinary and cross-domain contexts (2002) and the relations between template construction/modification and empirical data (Humphreys 2004, pp. 72–9). Humphreys’ template-based analysis offers a generally useful approach to the construction and cross-disciplinary transfer of knowledge that attracted a number of followers (Houkes and Zwart 2019; Knuuttila and Deister 2019; Price 2019; Kaznatcheev and Lin 2022). However, can the template-based approach be applied to the Malthus-Darwin case?
A template, according to Humphreys, is a highly abstract model, symbolized into a mathematical equation, which can be applied to fields with different domains of objects. Typical examples are Newton’s second law of motion, Laplace’s equation, Schrödinger’s equation, the Lotka-Volterra equations, and the like.11 The cross-disciplinary application of templates is direct in the sense that whether or not the assumptions of a template are satisfied in a given case can be determined directly, while Kuhn’s paradigms or exemplars rely on analogical reasoning from successful applications to new cases. Humphreys (2004) further describes the important components of template construction, transfer and application by proposing five construction assumptions and a correction set containing five components. The five construction assumptions are ontology, idealizations, abstractions, constraints (which include laws), and approximations and the correction set contains relaxation of idealizations, relaxation of abstractions, relaxation of constraints, refinement of approximates, and changes in the ontology of the system (Humphreys 2004, pp. 78–9; see also Knuuttila and Loettgers 2012, pp. 3–4). By developing the concept of template and analysing the construction and application of templates, Humphreys claims that he provides a mode alternative to Kuhn’s mode (Humphreys 2019, p. 3).
In their series of articles, Knuuttila and Loettgers (2012, 2014, 2016) develop Humphreys’ template-based framework, including the concept of template and the construction assumptions, to investigate the construction and transfer of two computational templates, the Lotka-Volterra model and the Ising model. They propose the concept of model template and define it as “a mathematical structure that is coupled with a general conceptual idea that is capable of taking on various kinds of interpretations in view of empirically observed patterns in materially different systems” (Knuuttila and Loettgers 2016, p. 396). Addressing the construction of Lotka-Volterra model as a model template, Knuuttila and Loettgers (2012) investigate in detail how Vito Volterra and Alfred Lotka individually established the renowned Lotka-Volterra mathematical equation through different approaches. Addressing the Ising model, Knuuttila and Loettgers (2014, 2016) analyse in detail how it was applied as a model template and transferred from the physical study of magnetic systems to the neuroscientific (the Hopfield model) and socio-economic fields (urban segregation and opinion formation). In their detailed case studies, Knuuttila and Loettgers highlight two points in addition to the template transfer: (1) Analogy is still frequently used in the interpretations of mathematical symbols in the model templates across different disciplines. (2) Conceptual ideas play a crucial role in interpreting the symbols of mathematical equations to mediate between mathematical formalisms with computational methods and the modelled phenomena.
The template-based approach, including Humphreys’ original version and Knuuttila and Loettgers’ improvement, is seemingly applicable to the Malthus-Darwin if one focuses on the simple mathematical model given by the geometric growth of population and the arithmetic growth of subsistence. This model can indeed be identified as a theoretical or model template that, in the way outlined in previous sections, is transferred from one domain to the other and thus incorporated in the target theory. Darwin’s model of population would thus have been developed using some construction assumptions, as well as analogy and a conceptual framework/ideas, by incorporating Malthus’ template. However, this approach to theory construction is not quite satisfactory for the Malthus-Darwin case in a key point, because philosophers’ templates are still mathematical and computational. This falls short of capturing one of the critical features of the theory of population developed by Darwin from the model introduced by Malthus: the broad nature of the checks and the struggle for existence that can possibly modulate the relation between populations and means of subsistence increases. The transfer of these components cannot be captured by the template approach because it is part of a conceptual structure and is not mathematical or computational in nature. It is important to stress that these conceptual components were transferred and incorporated into Darwin’s new structure through conceptual modification and expansion, which are not probed by the template-based approach. There is no formal template from which mathematical symbols could be given alternative interpretations to fit different phenomena or subject-matters in different disciplines. Investigating the possible transfer of a conceptual structure from one theory to the other amounts to assessing whether a theoretical concept in one theory can be modified or expanded to fit new phenomena in a new field. If the concept of template was to be extended to non-computational models such as Malthus’, then Malthus’ model of population could be construed as a template of a conceptual structure. In that sense, our methodology of structural incorporation can be seen as the application of the expansion by conceptual modification epistemic strategy to the template-based approach. Indeed, the relation between the two approaches can simply be modelled as a concept-driven expansion of the range of applicable templates, from strictly computational models to non-computational ones.
Our approach shares two common features with Knuuttila and Loettgers’ series of studies. First, analogy is still inevitable in the transfer of a conceptual structure. This point is shown in the strategy of structural projection, which is an essential element of analogy. Second, our focus on the conceptual aspect of theoretical models is aligned with Knuuttila and Loettgers’ approach in which the conceptual side of model templates bridges “the mathematical/computational forms and their various empirical interpretations” (Knuuttila and Loettgers 2016, p. 396). However, according to our approach, the conceptual components play a central rather than mediating role in the Malthus-Darwin case, because there is not a symbolic formalism that needs to be mediated.
7. Conclusions
In this paper we have argued that the methodology of structural incorporation can help shed a new light on the relation between Malthus’ and Darwin’s theories, or between any other similar pairs of theoretical models. For instance, models in classical genetics and in molecular biology, or models in the economic game theory and evolutionary game theory, would be interesting pairs of candidates to investigate. Our starting point was the fact that Darwin’s theoretic models share conceptual content and some underlying features with Malthus’ much simpler model. First, we showed that a being-embedded relation between Darwin’s and Malthus’ models is at least possible by considering the genealogical lineage and by comparing the two models’ conceptual structures. Second, we introduced five epistemic strategies that help analyse and understand how an incorporating relation may have happened in the step-by-step process of model transfer and construction. Although we do not claim that Darwin actually used any of these epistemic strategies to construct his theory of evolution from Malthus’ work, we nevertheless developed an approach to explain how this could possibly have happened. We have argued that the five strategies can model this incorporation process. The adequacy and the relevance of our approach can be appraised by asking (1) whether the model we propose can be generalized to other similar cases in which a particular conceptual structure is embedded into another descendant model; and (2) whether there are other methodologies that enable a better assessment of conceptual components and logical structure shared between different theoretical models such as those of Malthus and Darwin.
Addressing (2), we have compared our approach with the template-based approach to theory construction and transfer, showing that the latter is not so well-adapted for the Malthus-Darwin case and other non-computational models. However, this approach would likely be applicable if we expand the range of “template” to non-computational models with the set of epistemic strategies we have revealed. Addressing (1), the methodology of structural incorporation offers a new approach to investigate relations among theories and theoretic models. It also offers an alternative mode to investigate knowledge, theory, or model transfer. Furthermore, it offers the opportunity to re-investigate whether there is an incorporating/being-incorporated relation between pairs of theoretic models such as those in classical genetics and molecular biology, supposedly related through a reductive relation, while avoiding criticisms related to reductionism. Hypothesizing a genealogical and isomorphic relation between theoretic models—in other words, considering the possibility that a single model/theory is a prototype or a conceptual template having been transferred and incorporated into a target model/theory—opens up a new perspective for historical enquiries. Whether or not such a connection exists between two theories/models thus becomes the subject of an empirical investigation.
Notes
“To offer evidence” here is a very simplified term for describing this kind of knowledge transfer. According to Chapman and Wylie (2016), this kind of evidential knowledge transfer involves a complex trade of knowledge that needs a large amount of external resources and makes archaeology a “trading zone” in Peter Galison’s sense. Giving an account of Chapman and Wylie’s work is beyond the scope of this paper.
Now, some philosophers are investigating new modes of transfer focusing on different aspects, for instance, measurement (Kaznatcheev and Lin 2022).
Our approach shares some features with that of Post (1971), in which the author develops a methodology of theory construction using heuristic criteria such as flaw, footprint, conservation, simplicity, the general correspondence principle, adding to interpretation, and enlarging the domain as guidelines for constructing theories. However, there are some essential differences between Post’s methodology and ours. First, we explore the construction of theories and models by drawing on cases in cross-disciplinary contexts, thus involving the topic of model transfer; while Post focuses on cases of scientific revolution in the same discipline, mostly physics. Second, we explore the conceptual aspect of theories, and thus understand theories as combinations of models that are, in turn, construed as conceptual structures, while Post’s understanding of theories and models is still in the style of logical empiricism. Therefore, our approach focuses on analyzing the isomorphism between pairs of candidate theories while Post’s interest is on the extent to which both theories conform to specific heuristic criteria.
It is noteworthy that Weisberg himself distinguishes modelling from abstract direct representation as different strategies of theory construction. Without analyzing Darwin’s theory construction, he simply states that Darwin’s theory arose from abstract direct representation rather than modeling (Weisberg 2007, p. 207). If we adopt the perspective that Darwin could have used Malthus’ theory of population as a “model” or a “conceptual template” to develop his theory, then we may adequately attribute part of Darwin’s theory construction to modelling. See the discussion in the following sections.
The five strategies introduced here do not exhaust all likely strategies for structural incorporation. For example, analogy as a reasoning pattern is frequently used in the construction of theories and models.
Later, Darden calls strategies in discoveries of mechanisms and biological discoveries “reasoning strategies” (Darden 2006; Craver and Darden 2013). Here, we call the five strategies of structural incorporation “epistemic strategies,” because we think that not all of them are guidelines for reasoning.
Jürg Niehans claims that Malthus’ theory can be defined by the first three propositions (Niehans 1994, pp. 79–80).
This proposition is extracted from the following paragraph: “The amount of food for each species of course gives the extreme limit to which each can increase; but very frequently it is not the obtaining food, but the serving as prey to other animals, which determines the average numbers of a species” (Darwin 1859, p. 68).
Humphreys (2002, 2004) first proposed the concepts of theoretical template and computational template. A theoretical template describes a very general constraint on the relationship between the quantities represented by symbols in a mathematical equation. If the values of parameters in a mathematical equation can been determined, then the equation as a theoretical template turns into a computational template and thus can be applied to compute and predict. Humphreys makes a revision and distinguishes between theoretical templates such as Newton’s second law of motion and formal templates such as the Lotka-Volterra equation (Humphreys 2019, pp. 3–5).
References
Author notes
We thank the anonymous referees for their insightful reviews and suggestions, which have greatly improved the quality of this manuscript. We very much appreciate their kind and warm comments. We also express our gratitude to Alexandre Guay (UCLouvain) and Chao, Hsiang-Ke (NTHU) for their support and guidance since the inception of this project.