1.1 Outline

This Element is structured as follows. Section 2 explores the conceptual question of what ‘models’ are and what it means to be engaged in ‘modelling’. Section 3 examines the different types of models that can be used to model evolution. Section 4 investigates the problem of how such models can be tested. Finally, Section 5 concludes this Element by offering a summary of the arguments advanced in this work and proposes further directions for future research in this area.

2.1 Models as Representations

Modellers often talk about using their models to represent some phenomenon, but what does that actually amount to? Reflecting the difficulty of working this idea out in detail, there is now a fairly large philosophical literature on the ‘models as representations’ view (see Frigg & Nguyen, Reference Frigg, Nguyen, Magnani and Bertolotti2017 for an overview). Hughes, whom I mentioned before as a defender of a representationalist view, himself admits that ‘the concept of representation is as slippery as that of a model’ (Hughes, Reference Hughes1997, p. S325). But while there is ongoing disagreement over how models achieve their representational purpose, there is broad agreement that representations require both a target phenomenon to represent and a specific purpose for which the representations are put to use. Let us now examine both of these ideas in order to understand the models-as-representations view better.

What is a model target? The target of a model is typically a mechanism or process that scientists aim to understand better, but which is difficult or impossible to investigate directly. This scenario is often the case in evolutionary biology. Since evolutionary processes often unfold over timescales far exceeding the lifespans of scientists, models become indispensable tools for advancing our understanding of evolutionary biology. I say ‘often’ rather than ‘always’, because microbiology provides an important exception to this trend. As we were able to observe during the COVID-19 pandemic, the SARS-CoV-2 kept (and still keeps) evolving new strains that made it more resistant to vaccines, putting the idea of evolvability into public discourse. Mathematical models predicting the likely evolution of new strains were common during the pandemic (Yagan et al., Reference Yagan, Sridhar, Eletreby, Levin, Plotkin and Poor2021) and nicely reflect how evolutionary biologists are not just interested in modelling the past. Similarly, microbial populations in laboratories receive a lot of attention from evolutionary biologists since they allow us to observe evolution in real time (O’Malley et al., Reference O’Malley, Travisano, Velicer and Bolker2015). Indeed, as I shall argue in Section 3, such microbial populations can constitute experimental models or so-called ‘model organisms’ (though I prefer the ‘model populations’ since the target isn’t individuals) to represent more general evolutionary dynamics. But microbiology is not the only area where we evolutionary biologists study non-geological timescales of evolution. In the twenty-first century, evolutionary biology gradually shifted towards a recognition of evolution even within our own timescales, studying rapid evolutionary changes even in single generations, often due to events such as climate change (Aguirre-Liguori et al., Reference Aguirre-Liguori, Ramírez-Barahona and Gaut2021).

What is the representational relationship between such models and their targets? One of the most intuitive views in the literature spells out this connection in terms of similarity or resemblance between models and their targets (Giere, Reference Giere1988; Weisberg, Reference Weisberg2013). A successful model or representation is then one that shares enough similarity (or rather, relevant similarities) with its target to be used as a placeholder or surrogate system that can be studied in its stead. While a simple mathematical model may be strikingly different from the complex biological phenomenon it is trying to represent, what matters is that the relevant properties are captured in the variables of the model (e.g. mutation rate, survival chance, and reproductive output in an evolutionary model), to be useful to the modeller. Indeed, some models in evolutionary biology are incredibly simple, representing only a few key aspects of the target.

Let us look at one of the best examples to illustrate this point: John Maynard Smith and George R. Price’s Hawk–Dove Model, that pioneered the field of evolutionary game theory (Maynard Smith & Price, Reference Maynard Smith and Price1973). Imagine a population in which individuals have an interest in procuring a territory, which yields a fitness payoff of V if they do. When two individuals meet, they have a choice between acting aggressively (the hawk strategy) or passively (the dove strategy) as shown in Table 1.

If both individuals are aggressive they split the territory but have to pay a fitness cost C for the fight (e.g. injury or energy loss from the fight). If both are passive they split the territory without having to pay that cost. If a hawk meets a dove, however, the entire territory goes to the aggressive individual. Where would evolution lead such a population of hawks and doves? Maynard Smith and Price (Reference Maynard Smith and Price1973) proposed the solution through the concept of an evolutionarily stable strategy (ESS). They define an ESS as follows: If an incumbent strategy i is an ESS, then there cannot be another invading strategy/mutant m whose fitness u is higher such that it would increase its proportion in the population. If the payoff of an incumbent is equal to that of a mutant, mutants must do worse when they play against each other than if an incumbent plays against a mutant. This would allow the incumbent to take over a mutant population. This can be represented as follows:

$U(i)>U(m)$

Or

$U(i,i)=U(m,i)\text{and}U(i,m)>U(m,m)$

If V > C, hawk becomes the dominant strategy, but if C > V, there will be a mixed population of hawks and doves (or individuals will themselves mix their strategy) because a population of pure doves or hawks could always be invaded by the other. We will look at this model in more detail in Section 4, but we can see here that this simple idealized model nicely illustrates why resource conflicts in nature persist but are typically resolved with minor complications. Furthermore, the model helps to illustrate the general importance of frequency dependence in the evolution of traits in a population. This model is useful despite leaving out many features of the environment, which is not uncommon in evolutionary models. As Neto et al. observe, environmental features in evolutionary models are often ‘highly idealized or abstracted away. If they are included in mathematical models they are typically subsumed within a single fitness coefficient’ (Neto et al., Reference Neto, Meynell and Jones2023, p. 18). What matters, despite all these idealizations and omissions of features of real target systems, is that our model is similar enough to the targets being represented to yield explanatory insights.

One of the reasons why the similarity account has gained popularity is that it accounts for misrepresentation. While models are typically designed to be faithful to their target systems, they are also full of idealizations and omissions, sparking extensive debate about how falsehoods in models can play useful roles rather than always being negative. Potochnik (Reference Potochnik2017), for instance, has argued that ‘idealizations aid in representation not simply by what they eliminate, such as noise or non-central influences, but in virtue of what they add, that is, their positive representational content’ (p. 50). Indeed, a perfectly faithful model with complete similarity could only be the target itself, which would make the testing of models irrelevant and the model more generally pointless.

Here philosophers of science often use the analogy of a map offering a model of a city, subway network, or hiking paths (Nguyen & Frigg, Reference Nguyen, Frigg, Lawler, Khalifa and Shech2022). Maps necessarily idealize to be useful. A one-to-one mirroring would be too complex to be usable, which is why we need to think about fit-for-purpose. The same applies to the use of maps within evolutionary biology, which I shall discuss in Section 3. What matters for models then is not whether or not they are true, which isn’t even a question that can be answered, but rather whether they are useful. Indeed, this leads us to another crucial issue: the purposes of models.

2.2 The Purposes of Models

What is a model’s purpose? In order to evaluate whether a model is useful, good, or successful, we must determine the purpose for which it is put to use. As Wendy Parker argues, we need an ‘adequacy-for-purpose view’ of models, where they are evaluated not on how accurate their representations are, but whether they succeed in what they have been built for (Parker, Reference Parker2020, p. 457). We will return to this idea in Section 4 where we’ll examine how models in evolutionary biology can be tested. Let us start now by asking: what is the goal of the modeller? Depending on what goal one has in mind for a model, it will achieve its representational function in different ways. What makes something a relevant similarity between model and target will come from the purpose we have for that model. Importantly, the goals of models can vary radically.

Probably the first function many will think of is explanation. Models in evolutionary biology can explain why certain patterns or traits evolved and under which conditions they did so. In doing so, models make assumptions about their target system that allow us to reason about the system, which is why one often hears the phrase model-based reasoning in discussions about the role of models. In principle this could enable evolutionary biologists to reason about evolutionary processes in the abstract, without any direct target. This is sometimes described as targetless modelling or the modelling of a fictional system, such as a population with three or four sexes (Frigg & Nguyen, Reference Frigg and Nguyen2016, Reference Frigg, Nguyen and Zalta2021).

Indeed, some philosophers have argued for the importance of the so-called exploratory models that help scientists to explore potential scenarios without having to intervene in the target system itself (Gelfert, Reference Gelfert, Nepomuceno-Fernández, Magnani, Salguero-Lamillar, Barés-Gómez and Fontaine2019; Massimi, Reference Massimi2019). This is especially true for mathematical models, where assumptions and parameters can easily be changed due to the more abstract representations of the target. Such counterfactual reasoning can help us to understand, for example, why certain traits did not evolve, for example.

While models in evolutionary biology are typically concerned with the past, as I have already mentioned, how evolutionary biologists sometimes also use their models to predict the future, such as the evolvability of new strains of viruses. With climate change constituting a threat to many species, evolutionary biologists are now becoming more interested in offering predictions for the extinction of species or adaptations species may undergo (Chown et al., Reference Chown, Hoffmann, Kristensen, Angilletta, Stenseth and Pertoldi2010; Munday et al., Reference Munday, Warner, Monro, Pandolfi and Marshall2013; Pau et al., Reference 59Pau, Wolkovich, Cook, Davies, Kraft, Bolmgren, Betancourt and Cleland2011; Waldvogel et al., Reference Waldvogel, Feldmeyer, Rolshausen, Exposito-Alonso, Rellstab, Kofler, Mock, Schmid, Schmitt and Bataillon2020; Wortel et al., Reference Wortel, Agashe, Bailey, Bank, Bisschop, Blankers, Cairns, Colizzi, Cusseddu, Desai, Van Dijk, Egas, Ellers, Groot, Heckel, Johnson, Kraaijeveld, Krug, Laan and Pennings2023). This has made models in evolutionary biology much more future oriented than it used to be, but this has not yet entered into the public perception of the field.

Another purpose of many models is to directly represent or convey important (and often complex) information condensed into a simpler form. Since humans are visual creatures, many such models take the form of diagrams, such as tree of life models, which ease the communication and understanding of ideas and information. I will discuss these models in more detail in Section 3. Many of these models also have an educational function and are often found in textbooks, such as phylogenetic trees to represent the relationships of different species.

Finally, modellers often also aim at intervention to study causation. When real-world systems cannot be intervened on (e.g. most evolutionary systems are too slow to study in the real world), models can illuminate the possible effects of intervention. For example, microbiologists frequently design experimental setups that are meant to model evolutionary processes in the real world and which allow for causal intervention, to determine processes of cause and effect. I will also discuss these in more detail in Section 3.

This list is far from exhaustive (see Frigg, Reference Frigg2022), but offers an elegant illustration of the great variety of purposes models may have. Notably, many of these goals need not conflict with each other and a single model can be used for multiple ends. Even when they do conflict, modellers may nevertheless be interested in building a model that at least partially satisfies several goals that may trade-off against each other. Rather than just thinking about a single goal/purpose a model is designed for it is therefore often more useful to think about a set of goals for which a model is designed.

Based on this consideration of the purposes of models, we can thus amend our tentative definition of the terms ‘models’ and ‘modelling’ as follows:

• Models: To be a scientific model is to represent a target for a scientific goal or set of goals.

• Modelling: To engage in scientific modelling is to construct representations of a target for a scientific goal or set of goals.

These definitions offer valuable insights to anyone wondering about how models work and what they are – be they philosophers of science or practicing scientists. When examining what makes for a good model, or how to improve one, we can now explicitly ask: (i) what the purpose is for which our model is meant to be used/designed? as well as (ii) whether it has relevant similarities with its target system? However, philosophical inquiry rarely just settles on a single solution – as wonderful as that may be! Before we move on to discuss the different types of models employed in evolutionary biology, let me first offer a challenge to the representationalist view of models.

2.3 Challenges to the Representationalist Consensus

So far, I have presented and defended a representationalist view of models and modelling activities. Yet this view can be challenged. Although the representationalist view of models is widely shared, in recent years it has come under increasing fire. In this subsection, I will examine these arguments and ultimately defend a weaker version of the representationalism according to which many models in evolutionary biology have the role of presentation, but not all.

Ironically, I find myself here in an odd position, as I have myself previously argued not only that the consensus view of models as representations is mistaken, but also that no alternative account can succeed in its place. I called this position ‘model anarchism’, to capture the spirit of Paul Feyerabend’s critical view of philosophers’ attempts to offer a unified analysis of science while neglecting its diversity. This position can be summarized in a single sentence.

If this view is correct, one may wonder what the point would be of an Element that sought to inspire a thriving literature in the philosophy of models of evolution. I will therefore take this opportunity to critically examine my own arguments before ultimately demonstrating why they need not constitute a threat to the goals of this Element.

Earlier in this section, I emphasized that models are so diverse – and of so many different types (see discussion in Section 3) – that the only commonality to tie them together that has been accepted in the literature is the notion of representation. But this representationalist account of models can be regarded as both too broad and too narrow (Veit, Reference Veit2023, p. 226). We will look at the argument for each of these in turn.

Firstly, the account can be challenged as being too broad because it fails to demarcate a unique scientific category. As Callender and Cohen (Reference 53Callender and Cohen2006) have argued in this debate, there is no unique sense of representation within science. This makes it puzzling as to why there has been relatively little engagement from philosophers of science with the larger philosophical literature on representation. Indeed, I have recently argued elsewhere that philosophers have still paid too little attention to the similarities between representations within science and the arts (Veit & Milan, Reference Veit and Milan2021). Without such a fundamental demarcation between scientific and other kinds of representation, however, it becomes less convincing that we can understand the special epistemic (i.e. knowledge-generating) roles of models within science simply by understanding representations.

The broader we stretch and deflate the concept of model to capture as many instances as possible of scientists’ use of the term, the less clear it is that such a highly abstract account could be informative and useful. As I’ve outlined above, the notion of ‘representation’ is hardly more precise than that of ‘model’. Many accounts have been proposed but there is little agreement on how we should understand representations. Much work in this field ‘begins with the assumption that there is a single relationship that bears between models and the world’ (O’Connor & Weatherall, Reference O’Connor and Weatherall2016, p. 626), such as similarity or isomorphism (i.e. structure-preserving identity between model and target), but it is questionable that such a reductive account can succeed. The diversity of alternative views may simply capture genuine differences or multiplicities of ways in which models can be related to their targets.

Representationalist views in philosophy have also faced more general criticisms in the philosophy of mind (see for instance Myin & Hutto, Reference Myin and Hutto2015). This work has been used to argue that the representationalist account of models is by extension also ‘untenable and unnecessary, a philosophical dead end’ (de Oliveira, Reference de Oliveira2021, p. 209). While we need not go so far, the promises of the representationalist account of models – to resolve the philosophical puzzles about how models work, explain, and succeed in science – have not materialized. The view is too deflationary, without any meaningful commitments to how these representations are meant to be realized. If anything at all can be a model, it seems that the success of models within science becomes more philosophically puzzling rather than less so. As I argued in my original article:

Indeed, the view becomes even more problematic when it fails to capture even those things that are purported to be models, as we shall now examine.

Secondly, the representationalist view can be challenged as being too narrow by showing that there are models that play important roles in science without any representational function. As Downes has perhaps most forcefully argued over the years, ‘the role of models in science is by no means exhausted by representation’ with there being ‘far more expansive epistemic role[s] that models can play in forwarding scientific work’ (Downes, Reference Downes2011, p. 760). As the philosophical literature on models grows, more and more of these epistemic roles are uncovered. A now popular distinction in the literature between ‘models of ’ a target system and ‘models for’ intervention provides a nice example (Downes, Reference Downes2020; Keller, Reference Keller2000; Ratti, Reference Ratti2020). While interventionist models are admittedly less common in evolutionary biology, the experimental models in microbiology I mentioned above can be understood as such models. As I will illustrate in Section 3, an evolutionary biologist could be seen as intervening in the evolutionary process itself, rather than as trying to represent evolution in a different system, thus blurring the boundary between model and experiment. Such models are deliberately designed for the purpose of interventions, rather than to merely represent a target.

The crux here seems to be a dependence on the goals of the biologist. Are they trying to understand a different system through the means of what is sometimes called a surrogate (or replacement) system, acting in place of some other target (Bolker, Reference Bolker2009; Mäki, Reference Mäki2009), or are they trying to more specifically understand evolution in this bacterial population and how it can be influenced? The single use of the term ‘model’ by Darwin in On the Origin of Species comes closer to this latter understanding: ‘Breeders habitually speak of an animal’s organisation as something quite plastic, which they can model almost as they please’ (Darwin, Reference Darwin1859, p. 31). Here the word ‘model’ is used analogously to how one might model clay into desirable shapes. While it may be possible to detect some representational relationships in models built for interventions, the reduction of all models to their representational roles is bound to obstruct crucial differences between the things we call models. Indeed, with such vastly different usages of the term ‘model’ it becomes less clear that scientists are even trying to capture a unique form of scientific activity when they use the word.

Perhaps there is no account that could capture the diversity of different kinds of models. This sentiment has been forcefully expressed by O’Connor and Weatherall, who note that: ‘The term [“]modeling,[”] much like the term [“]science,[”] picks out a set of practices that do not constitute any sort of natural category’ (O’Connor & Weatherall, Reference O’Connor and Weatherall2016, p. 614). Frigg has also recently expressed such scepticism after working within the representationalist framework of models for a long time:

Future work may well find a unifying umbrella under which to put all models and modelling activities, but it has not been found yet. I contend that we can leave this issue unresolved for the time being. We do not need to decide whether model anarchism is correct in order to agree that the representationalist account and definition of models offers a highly useful and often accurate conceptualization (dare I say, model) for thinking about the roles of models (see Veit, Reference Veit2023 for a more extensive discussion of model anarchism). This is even more true within evolutionary biology, where non-representational purposes are typically rarer compared to other sciences. But it is good to keep in mind that these other purposes exist and that we should remain vigilant not to overgeneralize from the use of models in one area of evolutionary biology to another. We will explore many of these additional roles in the coming sections of this Element.

To conclude, philosophers have mounted powerful challenges to the representationalist consensus, though we were only able to scratch the surface of this debate here. The important takeaway from this section as a whole is that models in evolutionary biology serve important functions (most often of a representational kind) and that we need to think about the goals for which these models are intended in order to evaluate them.

2.4 Summary

The goal of this section was to answer the questions of what ‘models’ are and what it means to ‘model’ evolution. While newcomers to these debates may have hoped for a quick and easy resolution to these questions, in this rough overview of the literature on these terms I hope to have successfully shown that this debate is far from settled and deserves more attention than I was able to give it here. Nevertheless, there are several key insights and takeaways.

Firstly, models in evolutionary biology are typically representations of a target. For evolutionary biology, these targets will typically be real or imagined counterfactual populations. While the representationalist view may not always apply, it nonetheless remains a good starting point. As I have tried to emphasize in this section, in order to understand the roles of models within science we will need to pay special attention to the goals for which they are put to use. There is no independent sense of model evaluation to be had and Section 4 will be dedicated to discussing how best to test models of evolution. Unfortunately, this Element will not settle the question of what models are once-and-for-all. But instead of trying to find a general account of models, we can simply turn to the perhaps less ambitious, but scientifically more grounded approach of trying to understand the philosophical particularities of different types of models within evolutionary biology. This is the task to which we shall now turn.

3.1 Mental Models

Although evolutionary biologists often lament the misunderstandings of evolution by members of the public (and even within sciences outside of evolutionary biology), it is clear that almost anyone familiar with evolution now has something like a mental model of the process of evolution (d’Apollonia et al., Reference d’Apollonia, Charles and Boyd2004). This doesn’t mean that those models are sophisticated or accurate. The folk understanding of evolution may involve many mistaken ideas and assumptions (see Abenes & Caballes, Reference Abenes and Caballes2020; De Santis, Reference De Santis2021), such as evolution being goal directed and leading to greater perfection, or the idea that natural selection is an agent selecting the best and discarding the rest. But that doesn’t change the fact that evolution is in some way represented in the minds of specialists and non-specialists alike. It is therefore interesting to understand these mental models, since they are ultimately involved in the building of diagrammatic, concrete, or mathematical models of evolution.

Recalling the difference between foreground and background assumptions, it should be clear that mental models rely the most on background assumptions; that is on the implicit knowledge and assumptions of the ‘modeller’. If you have ever had someone trying to explain a scientific concept to you, you may have noticed gaps in their explanation coming from implicit knowledge and assumptions on the part of the explainer. Indeed, the main hallmark of a good teacher is the ability to make these implicit assumptions explicit, in order to be understood by students. But this may not be necessary for your own understanding and only becomes required when you are trying to teach others and are trying to examine the assumptions of your own thinking in more detail.

When Darwin endeavoured to understand the origin of species, he did not have a detailed and worked out idea of how long it would take for species to evolve. Parallels with artificial selection gave him a comparison class but his theorizing was largely vague on the question of timescales, tying himself to current estimates in geology regarding the age of the Earth as well as the durations of other time periods. This vagueness afforded him the flexibility to model the gradual process of evolution by natural selection in his mind, but suffered from an inability to make precise predictions about how long it would take. Both due to his lack of mathematical ability and fear of making his theory easily dismissible by including premature calculations about time, the Origin of Species has largely remained free of precise predictions. Indeed, some of the few calculations and speculations Darwin did make about time prompted religious and scientific opponents of his theory to offer damning attacks (Gunawardena, Reference Gunawardena2014, p. 3442). Similarly, Darwin had little understanding of the mechanisms behind inheritance, treating the similarity between offspring and parent largely as a background assumption in his reasoning. This almost led to the rejection of the theory, if it hadn’t been for the merger between Mendelian genetics and Darwinism that eventually filled this black box in Darwin’s theory (Otsuka, Reference Otsuka2019). Yet, by leaving these aspects of his theory fairly vague, and including an element of chance, Darwin was able to come up with many ideas and explanations for previously disparate and strange biological phenomena that were later confirmed. Indeed, if Darwin had left inheritance and time out of the informal models he presented in the Origin, he might have faced less criticism. However, if Darwin had been forced to explicitly bring out all of his assumptions in the form of even more precise and testable predictions, his theory could easily have been rejected, without allowing for further refinements. Background and implicit assumptions can thus play crucial roles for the creation of entirely new research programmes. As Gunawardena argues:

The development of novel scientific theories, as exemplified by Darwin’s work, hinges on mental modelling. Depending on what aspects of nature one is trying to understand, some assumptions come into the foreground whereas others move into the background. This is a fundamental feature of how our minds cognitively represent things to model the world (Nehm et al., Reference Nehm, Poole, Lyford, Hoskins, Carruth, Ewers and Colberg2009). It is easy to notice that even in one’s own theorizing some assumptions are delegated so far to the background that one isn’t even consciously aware of them as soon as you are trying to explain your implicit model-based reasoning to others. This is where communication with others is crucial for refining mental models. In order to communicate mental models effectively to others, whether verbally or in writing, many implicit assumptions need to be made explicit and that allows one to recognize and describe them. As we shall see now, this is what Darwin did when he tried to work out the general process of evolution by natural selection in his writing.

3.2 Verbal and Linguistic Models

If there are mental models, then it should be easy to see that models can also be verbal or written. This is not to suggest that words themselves are models in virtue of representing states in the world, but rather that language can be used to create models. Indeed, as Dimech points out: ‘Darwin used models in the form of invented, idealized narratives to display how the process of natural selection works’ (2017, p. 20). The most prominent case in The Origin of Species is perhaps Darwin’s usage of artificial selection as a model for natural selection. His mental model of the similarities between the two types of selection enabled him to offer the following linguistic model:

Here, Darwin makes the case that varieties have always existed within species, but that we typically didn’t notice the minor ones. It is only with the advent of agriculture that even the most minor variations became significant to us, triggering artificial selection and yielding marked phenotypic changes within just a few years. With this mental model constructed in the mind of his readers, Darwin moves on to use this model as a surrogate by which to make us consider whether heritable variations in nature could lead to similar accumulations of gradual change, giving rise to natural as opposed to artificial selection:

This is more than just the provision of arguments in favour of his view. Darwin deliberately builds an informal model with the ingredients of variation and selection within a population, using artificial selection as a surrogate for natural selection to provide a proof of concept. Even today, numerous evolutionary biology papers include stories of this kind to offer informal models representing target phenomena of interest, before moving on to offer experiments and formal models. As Gunawardena puts it: ‘Informal models do not necessarily lead to fast thinking, but they encourage intuitive plausibility over logical deduction’ (Gunawardena, Reference Gunawardena2014, p. 3442).

Even philosophers of biology themselves sometimes admit to the use of such models. Pierrick Bourrat for instance, in his recent Element on the levels of selection, dedicates several pages to the elaboration of a ‘simple verbal model to show that the conditions under which fitness [between individuals and the groups constituted by them] could become indefinitely decoupled cannot be met in most situations’ (Bourrat, Reference Bourrat2021, p. 56). Yet, because of the complexity of evolution, verbal models also allow a lot of room ‘for error and oversight in verbal chains of logic’ (Servedio et al., Reference Servedio, Brandvain, Dhole, Fitzpatrick, Goldberg, Stern, Van Cleve and Yeh2014, p. 2). While the assumptions are made more explicit in verbal models than in mental models, the models nevertheless still rely on many unspoken background assumptions. In this, they are similar to diagrammatic models, which (while admittedly relying on a different medium) often do not make their background assumptions explicit.

But before we turn to diagrammatic and pictorial models, let me emphasize that the apparent weakness of verbal models can also be a strength. Indeed, they ‘often derive their influence by functioning as lightning rods for debate about exactly which biological factors and processes are (or should be) under consideration and how they will interact over time’ (Servedio et al., Reference Servedio, Brandvain, Dhole, Fitzpatrick, Goldberg, Stern, Van Cleve and Yeh2014, p. 2). Thus, verbal and written models often function as a tool for social communication and refinement, that can lead background assumptions of mental models to be made explicit and testable in more precise mathematical models.

Unfortunately, philosophers have paid scant attention to the roles of such models. Some philosophers of models, such as (Weisberg, Reference Weisberg2013), only allow for verbal descriptions of models, not the existence of verbal models themselves, but this artificial limitation on the typology of models has been questioned (Dimech, Reference Dimech2017; O’Connor & Weatherall, Reference O’Connor and Weatherall2016; Veit, Reference Veit2023). As Darwin’s models nicely illustrate, verbal argument is no mere placeholder for the description of some other model, whether mathematical or otherwise. They constitute models in their own right that may or may not also be turned into a different medium, such as a mathematical model. As Dimech puts it, ‘what matters for determining whether types of objects are models or not is whether scientists use such objects as idealized intermediaries to study the world’ (Dimech, Reference Dimech2017, p. 24). For a longer in-depth discussion of verbal models within Darwin’s On the Origin of Species, see Dimech’s (Reference Dimech2017) dedicated article on the topic. Let us now turn to another informal type of models: diagrammatic and pictorial models.

3.3 Diagrammatic and Pictorial Models

Modern evolutionary biology abounds with diagrammatic and pictorial models that fulfil their representational roles through two-dimensional images. This is evident in contemporary doctoral studies in the field, which lead many students to become experts at using modern software to create such models; a stark contrast to earlier evolutionists who could only provide rough and ready drawings by hand unless they had significant artistic talent. It has become increasingly common to offer diagrammatic models in leading journals such as Nature and Science, reflecting the unique capacity of such models to represent a lot of information in a concise and useful manner.

In Section 2, I mentioned that philosophers of science often use maps, as a paradigmatic case for understanding how models can represent. Evolutionary biologists interested in the distribution and evolution of species naturally also have an interest in maps. In the fields of phylogeography and biogeography, evolutionary biologists often use geographic maps to explain the distribution and ancestry of species (Loughney et al., Reference Loughney, Badgley, Bahadori, Holt and Rasbury2021). This is perhaps most strikingly illustrated by the use of maps showing the drift of continents, to explain species distributions such as the striking prevalence of marsupials in Australia and South America (Cotner & Wassenberg, Reference Cotner and Wassenberg2020, pp. 37–38).

However, when it comes to the dense forest of diagrammatic models within evolutionary biology, one tree stands out among rest. While Darwin didn’t include any mathematical models in The Origin of Species, he included a single diagram that visually represented the tree of life (see Figure 1).

Figure 1 Darwin’s Phylogenetic Tree from On the Origin of Species (Darwin, Reference Darwin1859, p. 116a(v)).

Reproduced by the kind permission of the Syndics of Cambridge University Library.

In Figure 1, the labels A to L on the horizontal axis represent different populations, with time being represented on the vertical axis. This illustrates the branching species may undergo over time, aiming to explain the diversity of species we find today. Some of the branches (representing populations) go extinct due to variations in their traits (or chance events such as a forest fire), whereas others are favoured by natural selection and continue. For Darwin, it seems clear that this diagrammatic model was useful to visualize and build a mental model for himself of the origin of species diversity. Indeed, already in 1837 he had sketched his first version of this tree (Darwin, 1837, p. 36).

Today, such figures are ubiquitous in evolutionary biology to help visualize the relationships between species. We call them phylogenetic trees. To use a thought experiment, we might imagine early evolutionists hypothesizing about the evolutionary relationships between hippopotamuses and elephants, creating phylogenetic trees depicting close relatedness based on their physiological similarities. However, subsequent evidence from genetics, embryology, behaviour, and other areas could quickly undermine such a model and lead to an updated phylogenetic tree, as shown in Figure 2. We will use this example later in Section 4 to illustrate different methods of testing such models, but for now, it is important to emphasize that such models are always partial, only offering a snapshot of the tree of life.

Figure 2 A very simplified evolutionary tree model depicting the phylogenetic relationships between large aquatic mammals and their closest relatives. The anthracotheres family of hippopotamus-like ungulates went extinct, which is why their branch does not reach the present. This diagrammatic model was kindly created by Erola Fenollosa for this Element.

Figure 2Long description

The figure depicts a simple evolutionary tree with branches splitting upwards and downwards.

At the top right, one branch ends with the label “Cetaceans” and a silhouette of a whale. Below them, a branch connects to “Hippopotamuses” with a hippo silhouette, and a closely related extinct group labeled “Anthracotheres” with a small hippo-like silhouette.

Further down, another branch leads to “Ruminants” with a silhouette of a deer.

Below that, “Pigs” are shown with a pig silhouette.

At the bottom, two final branches split into “Elephants” (elephant silhouette) and “Manatees” (manatee silhouette).

The evolutionary tree illustrated that hippos have a more recent last common ancestor with deers than with manatees.

While we could imagine a tree that specifies the histories of all species from the present back through to the origin of life, such a tree would not only be impossible to create (due to gaps in the data), but also useless in the sense of being visually impenetrable except as depicting the tree of life as a whole. Consider, for instance, the immense task of mapping out the millions of contemporary species that would have to be presented at the tips of such a tree. In any such model, there is a choice of how many taxa to include, giving us a trade-off between a simple model depicting only a limited number of taxa, or a more expansive one that is perhaps more accurate, but less useful to make a specific argumentative point, which highlights how different goals for models can trade off against each other.

Unfortunately, misconceptions about evolutionary trees are quite common, such as the belief that time represents progress or that some species are more ‘evolved’ than others. This may in part be due to the way they represent information (Gregory, Reference Gregory2008; Meir et al., Reference Meir, Perry, Herron and Kingsolver2007). Thus, we should also be careful to also recognize the ways diagrammatic models may hinder science. As Griesemer early on recognized:

Some researchers have made just this point with the tree of life model, arguing that it fits only poorly with microbiology for example, where speciation does not branch in the same way and gene exchange between lineages is common (O’Malley & Koonin, Reference O’Malley and Koonin2011). This is why it is important to recognize that phylogenetic tree models have a fit for purpose and are designed with specific goals in mind, such as to depict the split between humans and our last common ancestors with the other Great apes. Their usefulness will always be relative to this purpose and should not be applied beyond this context.

Computational advances, however, can partially reduce these limits and the trade-offs between alternative models of the tree of life. The first attempt at this has been Open Tree of Life (Hinchliff et al., Reference Hinchliff, Smith, Allman, Burleigh, Chaudhary, Coghill, Crandall, Deng, Drew, Gazis, Gude, Hibbett, Katz, Laughinghouse, McTavish, Midford, Owen, Ree, Rees and Cranston2015), with the most sophisticated version found in OneZoom (Rosindell & Harmon, Reference Rosindell and Harmon2012; Sack, Reference Sack2018; Wong & Rosindell, Reference Wong and Rosindell2022) allowing it to represent on a single website over 2.2 million species via more than a hundred thousand pictures that one can zoomed in and out from.Footnote ¹ In this case, is this still a single model or should we speak of a combination of thousands of diagrammatic models?

With mathematical models in evolutionary biology, such as Maynard Smith’s Hawk–Dove Game, one could easily be led to believe that it is only a single model. Typically, however, even named models such as The Hawk–Dove Model often refer to a broad cluster of models with different parameters and idealizations, rather than one particular model, to show how a result or mechanism can be robust.

Admittedly, this is not what goes on in the OneZoom model of the tree of life. This collection of individual models is not about testing the robustness of a particular hypothesis, but rather about offering a comprehensive and interactive model that can help illustrate the idea that all life has a common origin.

Some computational models, such as those constructed in NetLogo (a programming language for agent-based models), allow one to simulate different evolutionary games, such as the prisoner’s dilemma (Wilensky, Reference Wilensky2002) while changing parameters in a very easy to understand and accessible manner and accompanying visual models demonstrating changes in a population. This could be similarly seen as offering an interactive model to understand evolutionary dynamics. Playing around with models such as OneZoom and these evolutionary game theoretic models, one cannot help but feel a sense of deeper understanding and appreciation for evolution than with static models, whether in the form of a single diagram or mathematical model. These models effectively refine our mental models of evolution, by offering visual models that represent more abstract relations or processes, which is precisely why they are useful as teaching tools. Thus, while we can immediately observe striking differences between the goals behind pictorial and computational models and the means through which they are achieved, such as their representational means, we can nevertheless discover important similarities between them, such as representing the same target system.

This also highlights some of the questions we will try to address in Section 4 of this Element: What is the relationship between models of different types? And how can we test models of evolution? But before we turn to these questions, there are still a few types of models that have yet to be discussed.

3.4 Concrete Models

While mental, verbal, diagrammatic, and mathematical models are all fairly abstract, this group of models are concrete. Following Weisberg, we can define such concrete models as ‘real, physical objects that are intended to stand in representational relationship to some real or imagined system, set of systems, or generalized phenomenon’ (Weisberg, Reference Weisberg2013, p. 24). Can we build tangible models of evolution? Concrete models typically rely on real-world equivalents of a target-system, such as a miniaturized version of the San Francisco Bay, with real water. Surely, many will say that this is not possible for evolutionary models. Creationist critics of evolution often accuse that no one has seen evolution with their own eyes. Yet… Could we imagine a special creator that plays around with the evolutionary process? In their textbook on evolution, Bergstrom and Dugatkin ask students to imagine an evolutionary biologist with special powers:

This may seem like a dream outside of computer simulations, but this is in fact what is frequently done in the field of experimental evolution. As they point out, no better example of this has been provided than the Long-Term Evolution Experiment (LTEE) by Richard Lenski and his collaborators, who have created such a model system with Escherichia coli bacteria. As of their recent announcement on their dedicated LTEE website, they have now passed the incredible number of 80,000 generations.Footnote ² The LTEE thus ‘encompasses more generations than there have been in the entire history of our species, Homo sapiens’ (Bergstrom & Dugatkin, Reference Bergstrom and Dugatkin2012, p. 79). While it would take many lifetimes to observe evolutionary change in a species like ours, microbial populations in a lab may undergo evolutionary change in mere days, allowing evolution to be observed, simulated, and intervened on right in front of our eyes. E. coli, for instance, can reproduce around once every 20 minutes under ideal conditions. Populations can be frozen at any stage during this process to allow a replaying of the evolutionary tape (Gould, Reference Gould1990), such that we can observe whether evolution will lead to the same or different outcomes from the same starting positions. The LTEE has shown, for instance, that genetic drift plays a major role in shaping populations, where future generations are strongly influenced by chance and not just past beneficial adaptations (Plutynski, Reference Plutynski2001). Such physical simulations of evolution do not require the use of mathematical models, but can operate directly on biological populations.

Nevertheless, we should be careful to uphold the distinction between models, simulations, and experiments at least somewhat. Not every experimental population undergoing evolution is necessarily a model; otherwise, we might as well call all experiments models. This would render the distinction between modeller and experimentalist obsolete. However, sometimes experimental populations do indeed constitute models, when they are intended to represent another target rather than examined for their own sake. Biologists sometimes employ model organisms that are intended as surrogate systems for other targets, such as the use of chimpanzees or mice in human medical research (Ankeny & Leonelli, Reference Ankeny and Leonelli2020). But these are not evolutionary models. Evolution is not about individuals, but about populations. Thus, we can analogously use the term model populations to better describe experimental populations used to understand evolutionary processes in another specific target or more generally evolutionary processes as such wherever they may occur.

Thanks to several efforts by philosophers of biology, the importance of such concrete models was recognized early on within the philosophy of biology (Griesemer, Reference Griesemer1990; Plutynski, Reference Plutynski2001). But concrete models also have a special role within the philosophy of models because of their unique position as models operating in the same ontological realm as their target systems. [For the non-philosophers reading this Element, ontology is the metaphysical study of which category of the world things belong to. In the case of a concrete model and its target system, it is the real world for both.] Indeed, whereas all other models (especially mathematical) are often viewed with a more critical stance by philosophers of science, concrete models seem to get a pass. Frigg even calls all other models ‘non-material models’, which can include both formal and informal models (Frigg, Reference Frigg2022, p. 248). This gives us another way of categorizing different types of models. But should such models be exempt from ontological criticisms?

Frigg goes so far as to state that ‘these models are commonplace material objects and as such similarity is no problem for them’ from the perspective of ontology. Even if the similarity problem does exist for concrete models ‘they do not give rise to ontological questions over and above the questions that one can ask about every other material object’ (Frigg, Reference Frigg2022, p. 248). Griesemer similarly states that material models ‘are robust to some changes of theoretical perspective because they are literally embodiments of phenomena’ (Griesemer, Reference Griesemer1990, p. 80). This is because other models are assumed to necessarily have higher degrees of abstraction and idealization, due to their different mediums. Even biologists using model populations express the idea that ‘only studies on natural populations can address the issue of whether populations in the real world tend to show stable dynamic behavior’ (Mueller & Joshi, Reference Mueller and Joshi2000, p. 5). But model populations in evolutionary biology are still highly idealized models when they are used to model general evolutionary dynamics or dynamics within other species, making this difference one of degree and not of kind.

For instance, Paul Rainey and his collaborators have spent a lot of time studying the potential emergence of multicellular individuality in the lab, using the bacterium Pseudomonas f luorescens (Hammerschmidt et al., Reference Hammerschmidt, Rose, Kerr and Rainey2014). The goal of this work was not to argue that P. f luorescens was at the origin of multicellularity individuality. This transition happened several times in evolutionary history, but rather to investigate plausible evolutionary scenarios and dynamics for how it might happen (Black et al., Reference Black, Bourrat and Rainey2020; Hammerschmidt et al., Reference Hammerschmidt, Rose, Kerr and Rainey2014; Rose et al., Reference Rose, Hammerschmidt, Pichugin and Rainey2020). This approach was explicitly interventionist, with the goal of biasing selection towards mutations that may be conducive towards multicellular group formation, reproduction, and cohesion. This reflects the observation by Keller that there are ‘models for’ intervention rather than just ‘models of’ the target (Keller, Reference Keller2000).

It is hardly surprising, then, that the work of Rainey and his collaborators has received a lot attention by philosophers (Neto et al., Reference Neto, Meynell and Jones2023; Veit, Reference Veit2019b, Reference Veit2022), including collaborative work that offers models for how this transition from single cells to genuine higher-level individuals may have happened (Bourrat et al., Reference Bourrat, Takacs, Doulcier, Nitschke, Black, Hammerschmidt and Rainey2024). The goal of such research is not necessarily to explain how things actually happened, but to offer a plausible explanation for how something may possibly have happened. Philosophers of science have introduced the distinction between how-actually and how-possibly explanations to emphasize that explanations can usefully focus on plausible scenarios, without those necessarily having occurred in the real world (Bokulich, Reference Bokulich2014; Resnik, Reference Resnik1991). Indeed, due to the sparsity of historical data, this is often what evolutionary biologists aim for instead. As Plutynski points out, such model populations are not necessarily meant ‘to prove the relative significance of one or another factor in the evolutionary process, but to provide a plausibility argument’ (Plutynski, Reference Plutynski2001, p. 233). This will be of relevance in Section 4, when we discuss the ways in which models of evolution can be tested.

In conclusion, model populations play incredibly important roles within evolutionary biology. Indeed, they blur the supposed hard distinction between models and experiments that is commonly made. The interaction between these models and mathematical models is of central importance in much of evolutionary biology, but has not yet been given sufficient attention by philosophers of biology, and I expect this literature to grow significantly in the future. For further reading, I recommend the book Stability in Model Populations by Mueller & Joshi (Reference Mueller and Joshi2000), who offer a detailed analysis of laboratory studies on model populations (with a special focus on flies such as Drosophila), as well as Plutynski’s (Reference Plutynski2001) analysis of Lenski’s LTEE and similar model populations of Drosophila.

Let us now turn to the final type of models I discuss in this Element: mathematical models.

3.5 Mathematical Models

We have moved our way from the most informal to the most formal types of models. Philosophers of biology, as well as philosophers of models, have spent most of their time thinking about mathematical models. This applies even to this Elements series, with Otsuka’s (Reference Otsuka2019) Element on the role of mathematics in evolutionary biology and Odenbaugh’s (Reference Odenbaugh2019) Element on ecological models, both of which focus on mathematical models and which I recommend as further readings for this section. But it is precisely because of a comparative neglect of non-mathematical models that I wanted to first give the others some attention. While I haven’t dedicated the whole Element to mathematical models, they nevertheless now play an irreplaceable role within modern evolutionary biology. This is a significant change from Darwin’s time. While Darwin provided verbal models of evolution by natural selection, it is often said that it was only with the advancement of mathematical models that the theory became precise, leading to the development of a mature science of evolutionary biology (Bergstrom & Dugatkin, Reference Bergstrom and Dugatkin2012; Otsuka, Reference Otsuka2019). In the transition to mathematical models, assumptions implicit in verbal and mental models come to be specified, to allow for collective usage. Indeed, as Gunawardena argues, in ‘marked contrast to their informal counterparts, they cannot tolerate any form of ambiguity: all assumptions are in the foreground, and there is no background. If it is not in the equations, it is not in the model’ (Gunawardena, Reference Gunawardena2014, p. 3442).

Philosophers of science readily acknowledge that mathematical models are now the primary tool in evolutionary theory. Lloyd, for instance, argues that the ‘main reason that models should be considered in any description of the structure of evolutionary theory is that models themselves are the primary theoretical tools used by evolutionary biologists’ (Lloyd, Reference Lloyd1994, p. 9). It is simply not possible to create concrete models for the many purposes for which mathematical models are designed and used. Biological reality is many ways less flexible than mathematically constructed systems. As Plutynski notes, ‘mathematical models can answer questions that laboratory models cannot’ (Plutynski, Reference Plutynski2001, p. 233). For instance, microbial or insect populations may simply not be similar enough to bird populations to allow them to stand in a useful model–target relationship for some questions. Here, a mathematical model may feature many more of the relevant similarities, despite the target and model being different kinds of objects altogether. In their textbook on evolution, Bergstrom and Dugatkin praise college education in economics and physics for preparing students to translate real-world questions into mathematics and figuring out how to solve them, and take this as something to be emulated by evolutionary biologists (Bergstrom & Dugatkin, Reference Bergstrom and Dugatkin2012, p. xxii). We can thus firmly assert that mathematical models play irreplaceable roles within evolutionary biology and are therefore worth philosophical analysis.

To do their work, formal models typically rely on equations and mathematical symbols. This mathematical structure is generally meant to stand in a similarity relationship with a real-world target. Recall the Hawk–Dove Model discussed in Section 2. Here we had the mathematical symbols V and C representing the fitness value of a resource in nature and the cost of a resource conflict in a particular population. The payoff matrix shown in Table 1 gave us the payoff values for different interactions in the game, intended to represent real conflict situations as they may occur in nature, but idealized so as to exclude other irrelevant factors. The expected payoff of each strategy depends on the probability of encountering hawks or doves.

We can now break down this model mathematically. Let p be the proportion of hawks in the population and $1-p$ be the proportion of doves.

Expected Fitness for hawk: $(U_H)$

Conditional fitness of a hawk facing another hawk: $\frac{V-C}{2}$ (probability p)

Conditional fitness of a hawk facing a dove: V (probability $1-p$)

$U_H=p\cdot (V-C)/2+(1-p)\cdot V$

$U_H=p\frac{V-C}{2}+V-pV$

$U_H=V-pV+p\frac{V-C}{2}$

Expected Fitness for dove: $(U_D)$

Conditional fitness of a dove facing a hawk: $0$ (probability p)

Conditional fitness of a dove facing another dove: $\frac{V}{2}$ (probability $1-p$)

$U_D=p\cdot 0+(1-p)\cdot V/2$

$U_D=(1-p)V/2$

Both strategies should be equal in fitness in an equilibrium:

$U_H=U_D$

$V-pV+p\frac{V-C}{2}=(1-p)\frac{V}{2}$

Solve for $p$:

$V-pV+\frac{p(V-C)}{2}=\frac{V}{2}-\frac{pV}{2}$

$\frac{V}{2}=pV-\frac{pV}{2}-\frac{pV}{2}+\frac{pC}{2}$

$\frac{V}{2}=\frac{pC}{2}$

$p=\frac{V}{C}$

This shows that if the payoff V is higher than the cost C, hawks will dominate, whereas the reverse situation will result in a mixed strategy. If V = 1 and C = 2, there will be an even split of hawks and doves. While the Hawk Dove Game is highly idealized and only consists of simple equations, it nevertheless provides a sense of explanation: a how-possibly explanation (Resnik, Reference Resnik1991) that can in principle explain potential cases of conflicts over resources (or, for that matter, the relative absence of them), even if the actual explanation for real cases might differ. Indeed, as Gunawardena points out, mathematical models are not typically used to ‘test’ assumptions, but rather to aid us in precise and rigorous reasoning:

This is one of the reasons why Maynard Smith (Reference Maynard Smith1982) dedicates a large proportion of his book to discussing more complex versions of basic games such as the Hawk–Dove Game. The initial model offers a so-called proof-of-concept, a mathematical proof for an initially verbal argument about the dynamics of resource conflicts. This is far from trivial. As Servedio et al. point out, when ‘hidden assumptions are altered or removed, the predicted outcomes and resulting inferences of the formal model may differ from, or even contradict, those of the verbal model’ (Servedio et al., Reference Servedio, Brandvain, Dhole, Fitzpatrick, Goldberg, Stern, Van Cleve and Yeh2014, p. 3). Many attempts to strengthen a verbal argument through a mathematical model will result in undermining the original argument, either by highlighting mistaken assumptions or by revealing a failure in the logic of the argument. But one of their most useful roles is to make explicit the hidden assumptions that may differ between scientists disagreeing on an issue. Such proof-of-concept models are incredibly common in evolutionary biology and often do have little connection to data, when long timescales are concerned (see discussion in Servedio et al., Reference Servedio, Brandvain, Dhole, Fitzpatrick, Goldberg, Stern, Van Cleve and Yeh2014).

Nevertheless, by exploring a variety of more complex versions of the same basic game, we can learn how different conditions may change basic dynamics within evolutionary game theory, as well as which dynamics are robust against different kinds of perturbations, thereby moving us beyond a mere proof-of-concept. One of the most obvious changes that can be made to such a model is the inclusion of genetic details or combination with population genetics models (McGlothlin et al., Reference McGlothlin, Akçay, Brodie, Moore and Van Cleve2022), though such models inevitably become more complex without the simple ‘like begets like’ assumption that parents will give rise to identical offspring. As we shall discuss in the next section, this kind of robustness analysis, which changes aspects of a model to examine whether and how they change the result, is fundamental to ‘tests’ of mathematical models of evolution. They help us to reason through different biological scenarios without the need to design new experiments altogether.

Let me now turn to statistical models and computer simulations. Earlier in this Element, I identified them as instances of formal models, but we may wonder whether they constitute different categories from mathematical models or should be considered as instances of them. I will first consider the former option.

Statistical models are clearly mathematical in nature. Nevertheless, they also function quite differently from the typical mathematical model scientists might build to understand a system. As Sober notes, unlike statistical models, mathematical models are usually understood by scientists to refer to ‘a simplified hypothesis; it purports to explain or predict a set of observations without trying to represent all the factors that are relevant’ (Sober, Reference Sober2008, p. 80). Statistical models are instead largely used to analyse large amounts of data and make inferences from them. This can then allow us to find important relationships between variables, for instance through a regression model. Frigg describes statistical models as ‘a mathematical representation of the observed data’ from scientific observations (Frigg, Reference Frigg2022, p. 473). This needs to be distinguished from the more general modelling of the influence of chance in evolutionary processes, where evolutionary biologists ubiquitously draw on many statistical methods, such as Bayesian methods and maximum likelihood methods (Romeijn, Reference Romeijn, Zalta and Nodelman2022). Indeed, it is worth emphasizing that evolutionary biologists have innovated many statistical methods to deal with their unique scientific problems, especially in population genetics models (Bergstrom & Dugatkin, Reference Bergstrom and Dugatkin2012).

While statistical models do typically have different goals from mathematical models, they often operate in tandem with them, such as when data is used to test predictions of mathematical models of a target system. However, they are still worth distinguishing in a philosophical analysis of models. Statistical data models of the genetic code of different species are, after all, often at the foundation for the construction of phylogenetic trees, as discussed above, to represent such data visually. Indeed, they are both data models, though in one case represented mathematically and in the other visually.

Just as with other models, statistical models still rely on idealizations, such as the assumption that relationships between variables will be linear or that there are no variables other than those. While statistical models are typically closer to reality due to their use of real-world data, they can in some cases be more idealized than mathematical models. Furthermore, the line between these types of models can be further blurred when theoretical models are filled in with data in order to test them, as we shall discuss in Section 4. Moreover, mathematical models are not constructed in a vacuum, but are often implicitly guided by the experience modellers have gained from dealing with various biological data sets such as in statistical models. Nevertheless, compared to some other fields of biology that do not have to deal with the distant past, such as biomedical research, evolutionary biology makes understandably less use of statistical models due to a sparsity of data. But statistical methods themselves could not be removed from evolutionary biology without collapsing the entire field, emphasizing their importance and showing that there is no way of drawing a sharp boundary between mathematical and statistical models within the field. Statistical models are therefore best analysed as a special subtype of mathematical models, rather than a separate category.

Finally, we can look at computational models. Weisberg (Reference Weisberg2013) distinguishes between mathematical models and computational models. Whereas mathematical models can be seen as consisting of equations that can be solved analytically, computational models are meant to simulate change across time through stepwise processes or computer algorithms. However, O’Connor and Weatherall (Reference O’Connor and Weatherall2016) argue that the replicator dynamics, which is a tool in evolutionary game theory for modelling the change of frequency of strategies across time, can be described both as a discrete trajectory or as a smooth trajectory than can easily be transformed from one to the other, with modellers using them interchangeably (pp. 620–621).Footnote ³ Is there a crucial difference here between the mathematical and computational version of the model? Consider the following continuous-time equation from Weibull’s textbook in evolutionary game theory (Weibull, Reference Weibull1995, p. 72):

$\frac{d{x}_i}{dt}=\left[u\left(i,x\right)-u\left(x,x\right)\right]\cdot x_i$

Here, $\frac{d{x}_i}{dt}$ stands for the rate of change of $x_i$ over time, which in turn stands for the proportion of individuals within a population playing the strategy i. The payoff of an individual playing i is represented by $u(i,x)$ whereas $u(x,x)$ stands for the average population fitness. If the difference is positive, i does better than the population average and will thus increase in frequency. This in turn is impacted by how many individuals already play the strategy, with the reverse happening when the difference is negative. This formulation can easily be turned into the following discrete-time equation:

$x_i(t+1)=x_i(t)+\Delta x_i$

Here, $x_i(t)$ stands for the proportion of individuals playing i at time t. $\Delta x_i$ represents the change in frequencies of strategies across one discrete step in time. As O’Connor and Weatherall (Reference O’Connor and Weatherall2016) argue, nothing in a choice like this rests on a difference in actual representational content, instead of ease of analysis, with the distinctions between mathematical and computational models actually obscuring ‘the most important aspects about how the models are used’ (p. 621).

This is a stronger argument than the more common one that computational models simply constitute a special case of mathematical models. Consider, for instance, Giere’s reductive explanation of computer simulations as just a special instance of mathematical modelling:

Agreeing with Giere, Frigg maintains that ‘computational models are also mathematical models and the class of formal models in fact coincides with the class of mathematical models’ (Frigg, Reference Frigg2022, p. 248). This is the view I adopt within the context of this Element. While different applications of formal models may require different approaches, that does not itself justify the claim that there is some deeper categorical difference here. Weisberg himself considers this option, admitting that ‘computational operations are at base mathematical operations’ and ‘are formally described in terms of states and transitions’ that are explained within the context of ‘discrete mathematics’ (Weisberg, Reference Weisberg2013, p. 19). However, he maintains that we should make an epistemological distinction, that is, in terms of how they generate knowledge. Whereas the explanans in one case is an algorithm, in the other it is the mathematical structure (Weisberg, Reference Weisberg2013, p. 20). But as O’Connor and Weatherall (Reference O’Connor and Weatherall2016) show, it is easy to turn one into the other, thus making it questionable how much explanatory weight really hangs on the way the model is specified. In the spirit of the model anarchism we examined earlier, we should be careful not to confuse useful distinctions for some purpose or other in a specific context where they may be useful, with the establishment of genuine categorical differences that apply across different contexts. In practice, mathematical models can often be solved with parameters treated as symbols, whereas computational models need to give these symbols numerical values. It thus seems best to also treat computational models as a subcategory of mathematical models.

3.6 Summary

In this section, I distinguished between five types of models: mental models, verbal and linguistic models, diagrammatic and pictorial models, concrete models, and mathematical models. However, I have no doubt that some will prefer a different typology of models. One could, for instance, follow Weisberg’s distinction between mathematical, computational, and concrete models (Weisberg, Reference Weisberg2013), though I have offered some criticism of it here. My goal was neither to offer an exhaustive typology nor to claim that there are some ultimate distinctions to be made within the many things called ‘models’. For the purposes of this Element, I simply found the categories presented here to be the most useful way of discussing the diversity of models within evolutionary biology. They serve both as a useful introduction to the philosophy of models for biologists interested in methodology and an illustration of the richness of models within evolutionary biology to philosophers of science. While I haven’t discussed models in cultural evolution here, it is worth mentioning that the difficulty of modelling cultural evolution has given rise to fruitful collaborations between philosophers and modellers. The typology offered here may prove useful for this context, though I suspect that cultural evolution comes with its own challenges. For an overview, consider the Element by Lewens (Reference Lewens2024) on cultural selection.

I must also emphasize that this typology is not hierarchical. Different goals and contexts may require different models with level of formality being one continuum along which different goals, benefits, and downsides can be identified. Yet, even restricted to the context of evolutionary biology, it appears that such answers would have to be context-specific to the goals of the model. But some typologies are even more simplistic than this, giving a universal ranking of different types of models. As Downes observes, the categorization in Futuyma’s (Reference Futuyma2006) evolutionary biology textbook into ‘verbal models, physical models, graphical models, mathematical models, and computer models’ is just one instance of the common idea that ‘alternate models that can be improved upon by being presented mathematically’ (Downes, Reference Downes2020, pp. 35–36). While I presented mathematical models last, I hope to have made clear that they are by no means inherently superior to other kinds of models. Their increased usage within evolutionary biology is due to improvements in mathematical skill among evolutionary biologists, not because mathematical models are the final endpoint of scientific modelling activity. Different models serve different goals and are complementary to other kinds of goals. While it is true that a mature science will be full of mathematical models to deal with a wealth of data and the complexity of the world, this by no means implies that informal models have lost their purpose(s). Indeed, as we shall see in the next section, different types of models play crucial roles within the context of testing hypotheses.

4.1 Disciplinary Norms and Modelling Practices

Often, the philosophy of models literature is prone to overgeneralizations: a model is analysed in one field, giving rise to generalizations about models as a whole (including in other scientific disciplines). But, as I argued earlier, such extrapolation should be done with more caution. As Parker makes clear, evaluation of models must always be sensitive to the purpose of the models at hand (Parker, Reference Parker2020). Scientific disciplines (and even sub-fields) can differ greatly in their disciplinary norms and modelling practices, especially regarding the proper goals of models.

This can be illustrated with an easy example. Fields like economics are full of modellers who state that they are only interested in the predictive accuracy of their models, not whether their assumptions are true. This is not true of evolutionary biology, where assumptions are commonly defended as being close enough to reality so as to not undermine the model. As systems biologist Gunawardena emphasizes, formal models ‘are only as good as their assumptions; if you make the wrong assumptions, correct mathematics can still produce wrong science; it is more important to understand the assumptions than to believe the conclusions’ (Gunawardena, Reference Gunawardena2014, p. 3444). For instance, Maynard Smith writes in his 1982 book on evolutionary game theory that the ‘basic assump-tion of evolutionary game theory – that like begets like – corre-sponds to what we actually know about heredity in most cases’ (Maynard Smith, Reference Maynard Smith1982). Detailed information about genetic inheritance can be idealized away to assume that parents will simply transfer all of their traits to their offspring (in the asexual populations typically represented in evolutionary game theory models) because it is realistic enough even when applied to sexual populations at long time horizons. This assumption, however, has not gone without criticism and evolutionary biologists have recently proposed ways to integrate evolutionary game theory models and quantitative genetic models (McGlothlin et al., Reference McGlothlin, Akçay, Brodie, Moore and Van Cleve2022), reflecting that biologist do care significantly about the accuracy of their assumptions.

By contrast, compare economists such as Milton Friedman, who won the Nobel Memorial Prize in Economic Sciences, and was described by the Economist as the ‘most influential economist of the second half of the twentieth century (Keynes died in 1946), possibly of all of it’ (The Economist, 2006). His work turned many economists into militant instrumentalists:

This methodological approach could not be more different from the approach described by Gunawardena: one emphasizes the correctness of assumptions, the other the success of predictions.

Expectedly, Friedman has received a lot of criticism for his radical stance. Rosenberg even argued that his naive instrumentalism ‘single-handedly justified’ the need for a distinct academic field of philosophy of economics (Rosenberg, Reference Rosenberg, Ross and Kincaid2009, p. 57). Gunawardena has similarly harsh words about the modelling approach of economics:

Informal models play a much larger role in biology than they do in economics, but is that necessarily a flaw in economic science? Despite these criticisms, and advances in empirical economic methods (such as behavioural economics and econometrics), the norms in economic modelling have remained influenced by Friedman’s approach that exclusively cares about predictive power.

What to make of this? Must economists learn from evolutionary biologists? Perhaps so, but we should at least allow for the possibility that distinct disciplinary norms may reflect genuine epistemic differences between models and modelling in different sciences. As Brown and Thomson opened with in their article on the evaluation of models within evolutionary biology: ‘Modeling is an exercise in explanation’ (Brown & Thomson, Reference Brown and Thomson2018, p. 96). The predictions made by models within evolutionary biology typically serve primarily to arrive at better explanations and improve our models, rather than to inform real-world action. This is why this Element is restricted to models in evolutionary biology. While some of the conclusions may extend to other subfields of biology and perhaps even different sciences, I do not treat this as a given, nor do we have the space here to engage with this question in more detail. Thus, whereas economists make a strict distinction between the testing of assumptions and making predictions, I do not see this difference reflected in the practice of evolutionary biologists, where models are in the business of prediction as well as explanation. The assumptions and predictions of models equally stand against the tribunal of evidential support.

This is especially relevant when we consider the different types of models already discussed. In the previous section, we saw that models lend themselves to the formation of new hypotheses. Admittedly, some of the models we’ve discussed so far are not designed to make predictions. Mental and linguistic models, for example, are often not precise enough to be tested in a satisfactory manner, but they can inspire more formal models for that purpose. Some evolutionary game theory models are general proof-of-principle models without any particular target in mind. Others use these basic games and apply them to explain the phenotypes of actual species, offering a pathway for tests. As Maynard Smith put it, ‘there is a contrast between simple models, which are not testable but which may be of heuristic value, and applications of those models to the real world, when testability is an essential requirement’ (Maynard Smith, Reference Maynard Smith1982, p. 9). Again, even with the same type of models, we can see that their design will differ drastically depending on the goals we have for them.

Phylogenetic tree models, by contrast, may be better understood as summaries of existing data, which are to be updated if new evidence undermines our current best hypotheses regarding the relationships of descent between different species. For instance, it used to be common to think that hippopotamuses, elephants, and rhinoceroses constituted a joint clade called Pachydermata (meaning thick-skinned). We now know that they are not a monophyletic group, relegating the Pachydermata classification to a relic of the past (Bergstrom & Dugatkin, Reference Bergstrom and Dugatkin2012, p. 111). This came from genetic research that revealed hippopotamuses to be the closest relative of cetaceans, such as whales and dolphins (Gatesy et al., Reference Gatesy, Hayashi, Cronin and Arctander1996; Nikaido et al., Reference Nikaido, Rooney and Okada1999). As Figure 2 illustrates, their last common ancestors with elephants lies much deeper in the animal branch of the tree of life. Similarly, it may be surprising to many that manatees share a closer common ancestor with elephants than with cetaceans.

While some models, such as these phylogenetic trees, are effectively data models, we can nevertheless treat them as explanatory and predictive in nature, because their assumptions, as well as their representations of the relations between taxa, remain continually subject to empirical revision. Constructing a particular branching between different species is equivalent to a hypothesis or at least assumption of a model that can be tested. While the public may often think of phylogenetic trees as a largely settled matter, Gregory points out that such models remain the subject of current research:

Due to gaps in the historical record, it is not uncommon for certain branchings in phylogenetic trees to be conjectural and open to future revisions. The assumptions of biological models are thus just as open to tests as their predictions. While models may not be deliberately designed to make predictions, they can nevertheless generate hypotheses or reflect evidence-based estimates that make them testable.

But the role of models is by no means restricted to the generation of new hypotheses. Indeed, the interplay of models with empirical work is quite a bit more complex within evolutionary biology than the common contrasts made by theoreticians and experimentalists may make it seem. As we shall see shortly, it is a mistake to think that the former researchers are in the business of generating hypotheses, whereas the latter are committed to testing them. I will argue that models themselves can be used to test the predictions of other models, while experimental work can lead to new predictions. For simplicity, however, let us begin by surveying how empirical work can test models of evolution.

4.2 Empirical Tests of Evolution

The simplest way models are typically tested is to look at their predictions, using empirical results to examine whether the predictions are correct. However, for the complex systems studied by evolutionary biologists it is often far from straightforward to derive predictions that can prove a model to be ‘correct’ or ‘incorrect’. As mentioned above, models in evolutionary biology are simultaneously examined for both their assumptions and their predictions. One might thus conclude that the most straightforward way of empirically testing a range of alternative models in evolutionary biology is to assess how accurately they describe their target system. Models will be silent on many aspects of the real world, but the relevant features they include should map onto the features of the real-world target they are employed to understand. This may also be described as model fit, although we have to be careful to distinguish fit to a target system from fit to data (Downes, Reference Downes2020, p. 73). Target systems, such as SARS-CoV-2, may provide us with data that our models can be fitted towards. This provides us with data fit, but not necessarily fit with the target system. After all, data gathering may have been inconclusive, partial, or simply faulty. Thus, comparing data with the predictions and assumptions of our models is far from a neutral enterprise. The data itself has to be assessed in terms of how relevant it is to our models.

Perhaps due to the sparsity of data that Darwin and his contemporaries had available to them, evolutionary biology has become quite adept at uncovering plenty of resources for generating data to validate their models. However, there is one source of evidence that largely trumps all others, viz genetics. It was long hypothesized that chimpanzees (Pan troglodytes) were our closest living relatives, but research in comparative genetics has suggested bonobos (Pan paniscus) as a possible contender for this spot, thus updating their position in the tree of life to sit equally as close to humans (Gibbons, Reference 55Gibbons2012; Mitchell & Gonder, Reference Mitchell and Gonder2013; Prüfer et al., Reference Prüfer, Munch, Hellmann, Akagi, Miller, Walenz, Koren, Sutton, Kodira, Winer, Knight, Mullikin, Meader, Ponting, Lunter, Higashino, Hobolth, Dutheil, Karakoç and Pääbo2012). Molecular genetics effectively invalidated the assumption made in many phylogenetic tree models that bonobos were more distantly related to us than chimpanzees. The primatologist Frans de Waal aptly commented that the ‘[t]he story that the bonobo can be safely ignored or marginalized from debates about human origins is now off the table’ (quoted in Gibbons, Reference 55Gibbons2012). The revolution in molecular genetics has been perhaps the most significant change in evolutionary biology, allowing challenges to many phylogenetic tree models that were built based on data from the fossil record, behavioural and morphological similarities, and ecological data on the regional habitats of species, among other forms of evidence. No matter how strong the combined evidence from these sources may have been, molecular genetics had the power to undermine even the most well-supported evolutionary hypotheses and assumptions due to its reliability of preserving evolutionary history in the genetic code.

For instance, returning to the previous example, it was found that cetaceans and hippos communicate underwater using similar clicking noises (Barklow, Reference Barklow1995), as well as having many other behavioural and physiological similarities (Gatesy et al., Reference Gatesy, Hayashi, Cronin and Arctander1996). But despite this, the most significant line of evidence was the genetic analyses showing the similarity of hippos and cetaceans (such as their milk casein genes) that ultimately settled the case for their close relatedness (Gatesy et al., Reference Gatesy, Hayashi, Cronin and Arctander1996). But what gives genetics such a privileged role?

In classifying different species, biologists had to routinely learn that the visual similarity between different species, such as birds, is no guarantee of their close relatedness. For instance, the Australian magpie (Gymnorhina tibicen) was so named by Europeans due their close similarity in appearance to Eurasian magpies (Pica pica), but has turned out not to be closely related at all, not even belonging to the Corvid family. Because of convergent evolution, different species that are only distantly related to each other can exhibit striking parallels in physiology and behaviour. Genetic sequencing studies do not suffer from this ambiguity found in other sources of evidence, providing more objective evidentiary support. While it is important to gather evidence from a variety of sources to test and evaluate models, evidence from molecular genetics can nevertheless often trump other sources of evidence. We can consider it as a kind of gold standard for testing evolutionary models, though this by no means implies that other sources of evidence have become irrelevant.

This point is perhaps the most easily illustrated within the context of palaeontology, which is the study of the fossil record and relies primarily on non-genetic data. Unfortunately, although modern advances in molecular biology have made it possible to gather of some genetic material from fossilized remains (Abdelhady et al., Reference Abdelhady, Seuss, Jain, Fathy, Sami, Ali, Elsheikh, Ahmed, Elewa and Hussain2024; Briggs et al., Reference Briggs, Evershed and Lockheart2000; Dobson, Reference 54Dobson2012; Oskam et al., Reference Oskam, Haile, McLay, Rigby, Allentoft, Olsen, Bengtsson, Miller, Schwenninger, Jacomb, Walter, Baynes, Dortch, Parker-Pearson, Gilbert, Holdaway, Willerslev and Bunce2010; Rohland et al., Reference Rohland, Glocke, Aximu-Petri and Meyer2018), success is rare and estimated to be restricted to a time horizon of 1.5 million years, due to degradation of DNA strands over time (Kaplan, Reference 56Kaplan2012). Thus, fossilized bones have largely remained as the only remnants of many extinct species.

Consequently, many phylogenetic tree models describing species such as dinosaurs have to rely on the fossil record as the primary line of evidence to support their models. The discovery of feathered dinosaur fossils, for instance, provided increasing support over the years for models that considered birds to have evolved from dinosaurs (Chatterjee, Reference Chatterjee2015; Chiappe, Reference Chiappe2009; G. Mayr, Reference 57Mayr2016; Rashid et al., Reference Rashid, Chapman, Larsson, Organ, Bebin, Merzdorf, Bradley and Horner2014; Zhou, Reference Zhou2004). Indeed, because the fossil record is incomplete, new findings often lead to revisions, or indeed to further support for our best hypotheses about the relationships between species.

The location of fossils is also used to gather information about the distribution and migration of species in the past, so it is unsurprising that the study of the distributions of contemporary living species (biogeography) can also be a source of evidence. Additionally, evidence for homologies (similarities) between species can come from embryological and developmental data, anatomical studies, and even behavioural observations and tests (such as the clicking noises made by cetaceans and hippos), to support or undermine hypotheses in tree of life models.

While these data sources are useful for testing phylogenetic models, what about the concrete models we discussed earlier, where evolution itself is simulated in a population, in order to model some other target or a general evolutionary dynamic? Often, these systems are used precisely because no experiment could be run in the target population and thus no data gathered. So how are these models validated? What modellers do in such circumstances is to study the relevant similarities between their model populations and target populations using various sources of data. When they are similar enough for the purposes at hand, that gives us reason to think that our model populations can help us to explain and predict evolutionary patterns in the target population.

In some concrete models designed to examine more theoretical aspects of evolution, such as microbial models of the evolution of multicellularity, there is often no clear target. It is precisely because we do not know how this transition happened, that we use model populations to examine the possible scenarios for how things might have happened and to examine the plausibility of alternative scenarios. Recall the earlier distinction between models for and models of. The latter are assessed in terms of accurate representations, but the same cannot be said for the former (Ratti, Reference Ratti2020). Concrete models in such cases often function very similarly to highly idealized mathematical models that do not have clear biological targets that would allow for traditional empirical tests. Both of these types of models need to rely instead on other methods of testing. Indeed, as we shall now see, such concrete models are tested in a very similar way to mathematical models: theoretically through possibility proofs and robustness analysis.

4.3 Theoretical Tests of Evolution

When one thinks of testing, it will typically be experiments, or at least observational studies, that come to mind. But not all kinds of testing operate in this manner. As was shown in our discussion of concrete models of the evolution of multicellularity, some models do not have a respective target system from which we could derive data to test the models. Some models seem to not be testable through any of these empirical means. This applies especially to mathematical models that are often so highly abstracted and idealized that it becomes hard to see what would even constitute a test.

Maynard Smith’s book on models of evolutionary game theory provides a striking example. He makes little reference to actual data and yet the book is considered a founding text for an important subfield of evolutionary biology. Mathematical models and simulations of the evolution of cooperation, for instance, involve so many idealizations in their game-theoretic models that it would be hard to see how they can be tested empirically. Though we know that cooperation has evolved in many lineages, this cannot serve as confirmation for these models, since testing has to happen after a model is created and not before. Indeed, doing the opposite is often considered close to scientific fraud. Scientists who change their models after running experiments, in order to make them better fit the data and/or because their hypothesis was not supported by the results, are often criticized for engaging in ‘HARKing: Hypothesizing after the results are known’ (N. L. Kerr, Reference Kerr1998). Admittedly, there can be a blurry boundary here between refining models to improve their accuracy and retroactively fitting them to the data for the purposes of publication. One involves retrofitting models while pretending that this was the original form of the model, just to gain a publication. The other makes this refinement process explicit, to allow for transparency.

In our new age of artificial intelligence and machine learning, it will become more and more common to feed vast amounts of empirical data into machine learning algorithms to come up with new predictive and explanatory models, which raises unique scientific and philosophical challenges (Greener et al., Reference Greener, Kandathil, Moffat and Jones2022). The use of AI and machine learning algorithms to help evolutionary biologists to create better models is only in its infancy (though see Callier, Reference Callier2022; Feltes et al., Reference Feltes, Grisci, de Faria Poloni and Dorn2018; Kuzenkov et al., Reference Kuzenkov, Morozov and Kuzenkova2020; Lürig et al., Reference Lürig, Donoughe, Svensson, Porto and Tsuboi2021; Tarca et al., Reference Tarca, Carey, Chen, Romero and Drӑghici2007), but I am confident that this is an area where the philosophy of models will need to pay a lot of attention in the future. One of the most promising uses of this technology is in training machine learning algorithms to distinguish the effects of drift and selection in shaping changes of the genome of a species (Callier, Reference Callier2022, p. 2). Nevertheless, while it is worth mentioning these tools here due their plausible future importance, it is currently not yet significant enough to give them more space within the confines of this short Element.

What is significant, however, is that despite the lack of direct empirical confirmation for most models of the evolution of cooperation, scientists continue to create new models and find this activity meaningful. This might seem mysterious if not for the fact that models can receive theoretical support through other models, which is an idea I shall now turn towards:

Mathematical models can show us possibilities, as well as allow us to examine how stable/robust the dynamics or effects in our models are. Whether we should really call such theoretical work testing is up for debate. As I mentioned before, in the minds of many, testing implies empirical tests. Ever since Quine’s attack on logical positivists in his article ‘Two Dogmas of Empiricism’ (Quine, Reference Quine1951), however, the seemingly sharp boundary between analytic statements (true by definition) and synthetic statements (true in virtue of correct descriptions of the world) statements, or theoretical and empirical work, has been contested. The results of theoretical models, just like those from empirical work, can become parts of our network of beliefs. If modellers create a broad range of models with altered assumptions and parameters to show that a particular process or outcome, such as the evolution of cooperation, is robust, then this can strengthen our confidence just as experimental work would. Whether we ultimately call this ‘testing’ or not, we have to acknowledge that models can be supported through theoretical work. This doesn’t deny that empirical tests may still play unique and perhaps more important roles in the scientific validation of models, only that we shouldn’t draw too sharp a boundary here between the types of support. As Lloyd notes, a mathematical model is often used ‘as a substitute for a real system on which experiments are done’ (Lloyd, Reference Lloyd1994, p. 9). In this discussion, I have mentioned the idea of robustness analysis a couple of times without going into more detail, but we shall turn to this now.

In an influential 1966 article, titled ‘The Strategy of Model Building in Population Biology’, Richard Levins brought robustness analysis to the attention of biologists and philosophers of models. He argued that if models ‘despite their different assumptions, lead to similar results we have what we can call a robust theorem which is relatively free of the details of the model’ (Levins, Reference Levins1966, p. 423). This was initially puzzling to philosophers of science, who struggled with the idea that models involving idealizations and literal recognized falsehoods could provide us with true insights into reality. Levins described robustness analysis as providing us with truths at ‘the intersection of independent lies’ (Levins, Reference Levins1966, p. 423). This sense of testing models no longer considers only single models but moves us towards sets or collectives of models. While a lot of highly idealized models may not by themselves be considered very useful or epistemically insightful, it is when they come together and collectively point at a stable mechanism or prediction that is immune to variations in the assumptions of a model, that they can be considered to be representing something real. But, as I alluded to before, this is no unique feature of mathematical models. Experimental model systems of the evolution of multicellularity are also engaged in changing the conditions within the model to see if they influence the aggregation of individual cells. While these results are typically not robust and often break down fairly quickly, experimental modellers are nevertheless also engaged in robustness analysis. Does this undermine the distinction between theoretical and empirical tests?

Not necessarily, but once again we should highlight that the similarities between the two cases are more important here than the differences between theory and empirical work in modelling. Both concrete and mathematical models are used to explore how-possibly scenarios as well as to examine how robust a result is within one model once we vary the assumptions. Indeed, sometimes robustness analysis involves creating a concrete version of a mathematical model, or vice versa; such as the mathematical models created by Rainey and his collaborators to further explore hypotheses arising from their experimental work (Black et al., Reference Black, Bourrat and Rainey2020). However, some may criticize the idea that this experimental work could constitute robustness analysis since the models differ in kind. This may well be true if we conceive of robustness analysis in a narrow way. As I have argued in an article titled ‘Model Pluralism’, the epistemic benefits of multiple models go far beyond robustness analysis (Veit, Reference Veit2020).

It’s not an extreme claim that some phenomena may have aspects that are too complex to be understood or explained within a single model. After all, science makes progress through a combination of the many models we discussed in Section 3. Scientists don’t restrict themselves only to the use of mathematical models when working to understand the origins of multicellular organisms. They also employ mental models, verbal models, diagrammatic models, and concrete models. We can distinguish this moderate view on the usefulness of a plurality of models from a more radical thesis that I dubbed strong model pluralism: ‘For almost any aspect x of phenomenon y, scientists require multiple models to achieve scientific goal z’ (Veit, Reference Veit2020, p. 96). It would be a naive and overly reductive picture of the philosophy of science if we tried to restrict science to the creation of individual model, each corresponding to a particular goal. After all, as we have discussed, mental and verbal models precede the construction of other kinds of models. Similar to how evolutionary biologists think of populations rather than individuals as the centre of their attention, philosophers of modelling need to engage in population thinking, that is, by treating sets of related models as changing populations of models, with particular names of models counterintuitively referring to sets of models rather than just one particular model (Veit, Reference Veit2020, p. 107).

The analogy to population thinking is, of course, a loose one since such sets of models used for a particular purpose may have no ancestral relationship and may have been independently created. However, I believe that as this metaphor is familiar to evolutionary biologists, it will be helpful in avoiding too much emphasis on individual models. Variation of populations – just like for models – should be the starting point of our investigations. If we talk of the scientific importance of the Hawk–Dove Game, for instance, we do not mean just a particular form of the game, but hundreds of variations of the basic setup. A verbal model that comes to be explicated in precise mathematical form without losing its core argument may be considered at least as a soft confirmation of a theory. And verbal and mental models may be used to check whether slight alterations will significantly change the result of a (fairly simple) model without necessarily requiring it to be modelled mathematically. Scientific articles in evolutionary biology frequently feature sets of models (e.g. concrete, verbal, diagrammatic, and mathematical, see Kerr et al., Reference Kerr, Riley, Feldman and Bohannan2002) of very different types to argue for a particular explanation and we should therefore be careful not to reduce all progress in evolutionary biology only to specific mathematical and concrete models that have undergone tests.

Philosophers of models have only scratched the surface of the ways in which different kinds of models interact with each other to solve epistemological problems, but we can at least attest that we should not prioritize formal models in a sort of hierarchy over more informal models, even when it comes to their testability. As Gunawardena notes, if the assumptions in a formal model ‘are at odds with your informal model, then life gets interesting, and you have to decide which to change; formal and informal models ought to work together, each influencing the other in a virtuous cycle; but this may be harder than it looks, if evolutionary biology is any guide’ (Gunawardena, Reference Gunawardena2014, p. 3442). With this in mind let me now summarize this section.

4.4 Summary

This section highlighted many of the ways in which models in evolutionary biology can be tested and yet we have only scratched the surface. Creating testable models is uniquely challenging within evolutionary biology, due to the slow and hidden nature of the processes of interest. As Maynard Smith once put it, ‘Evolution is a historical process; it is a unique sequence of events. This raises special difficulties in formulating and testing scientific theories, but I do not think the difficulties are insuperable’ (Maynard Smith, Reference Maynard Smith1982, p. 8). As we have seen in this section, evolutionary biologists have come up with many ingenious ways to provide data to test their models. Models are not just products of science, and they also play fundamental roles in scientific practice. As Plutynski argues, ‘the diverse types and functions of models and modelling in biology are not easily categorized as either product or process’ (Plutynski, Reference Plutynski2001, p. 225). However, is the purpose of models in evolutionary biology simply to generate new hypotheses that are then tested through empirical work?

As I have shown here, this is a far too reductive take, with many models being supported through theoretical means such as robustness analysis, as well as through their relationships to whole sets of models more generally. Indeed, while the importance of testing is often emphasized by modellers themselves, they often have very liberal views on what testing consists in. This deflates the notion almost to the extent that we are merely capturing all the methods for evaluating whether a model is good or bad (or at least better than its alternatives). If we restrict the term ‘testing’ to specific empirical tests of particular hypotheses, we may well want to distinguish it from various forms of data fitting, robustness analysis, and the more general fit of a model with other models. Depending on the particular purpose(s) of a model, we will be interested in different forms of evaluation. Are we interested in explaining the evolution of a particular trait in the real world? Or might we be interested in modelling a fictional trait that has never been observed in the biological world? How general is our model meant to be? All of these questions will change how the model is being evaluated, without there being a general methodological answer for how to proceed. A general model, of cooperation and the evolution of altruism, for instance, can draw on significantly more data than a model restricted to altruism in honeybees. The particular goals of the modeller, as well as context of the problem at hand will determine how models should be tested, and this will almost always lead to the creation of further models.