1. Introduction

Science is valuable in many ways, but one important reason for valuing science is that it provides us with understanding of the world. Indeed, the understanding that comes with scientific explanation is regarded as one of the central epistemic aims of science. The nature of scientific understanding is a focus of attention in current philosophy of science, and a variety of philosophical theories of understanding have been proposed in the recent literature. In de Regt (Reference de Regt2017), I argue that scientists achieve understanding of phenomena by basing their explanations on intelligible theories. The intelligibility of theories is related to scientists’ abilities: theories are intelligible if scientists have the skills to use those theories in fruitful ways. Accordingly, intelligibility is a pragmatic value that promotes science’s epistemic aim of understanding.

In this article, I examine the relation between understanding and other epistemic aims of science, and I investigate how these aims in turn relate to particular values. More specifically, I focus on understanding’s relation to the traditionally acknowledged aims of description, prediction, and explanation, and I analyze and compare the roles of the values of intelligibility, empirical adequacy, internal consistency, and representational accuracy in achieving understanding. Section 2 addresses the question of what the epistemic aims of science are and analyzes the relation between these aims and relevant values. Section 3 presents my contextual theory of scientific understanding, in which intelligibility plays a central role. In section 4, this theory is illustrated with a historical case study of the controversy between physicists James Clerk Maxwell and Ludwig Boltzmann over the latter’s molecular model for explaining the so-called specific heat anomaly. Section 5 analyzes the role of scientists’ value judgments in the process of achieving scientific understanding.

2. Values and the Aims of Science

In “The Aim of Science,” chapter 5 of Objective Knowledge, Popper writes: “To speak of ‘the aim’ of scientific activity may perhaps sound a little naïve; for clearly, different scientists have different aims and science itself (whatever that may mean) has no aims.” Popper adds “I admit all this,” but he subsequently suggests that “it is the aim of science to find satisfactory explanations, of whatever strikes us as being in need of explanation” (Reference Popper1972, 191). Indeed, explanation is traditionally regarded as one of the central aims of science, next to description and prediction of phenomena.Footnote ¹ A recent study of the aims of science characterizes the traditional aims as “prediction, explanation, and accurate representation” (Potochnik Reference Potochnik2017, 105; cf. 92). Note that description and representation may refer to observable phenomena only or to unobservable phenomena as well, depending on one’s stance in the realism debate.

To achieve these aims, scientists develop theories and models, a process in which value judgments play a crucial role. So, which values promote the epistemic aims of science? Douglas (Reference Douglas2013, 798) suggests that internal consistency and empirical adequacy are truly epistemic values that need to be respected by scientists. Are these indeed necessary for achieving the above-mentioned aims? Internal consistency may seem to be required for all, but below I qualify this claim in an important way. Achieving the predictive aim of science, the accommodation and successful prediction of empirical evidence regarding a phenomenon, is obviously guaranteed by empirical adequacy, as defined by van Fraassen (Reference van Fraassen1980, 12): “a theory is empirically adequate exactly if what it says about the observable things and events in the world is true.”Footnote ² This is a strong definition, and Bhakthavatsalam and Cartwright (Reference Bhakthavatsalam and Cartwright2017, 446) offer an attenuated version: “a theory (or model or set of scientific claims) is empirically adequate when the claims it makes about empirical phenomena—or at least the bulk of these claims, or the central ones—are correct, or approximately correct enough.” The descriptive aim is realized if theories or models give accurate representations of the phenomena. Hence, it is promoted by the value of representational accuracy, which concerns the goodness of fit between a theory or model and the target system it purports to represent, typically in a realistic way (measured in terms of degree of similarity).Footnote ³ Achieving the aim of explanation, finally, requires theories and models that have explanatory power. Defining explanatory power is not easy, however, as there are diverging views about the nature of explanation. Therefore, I suggest to approach this issue via an investigation of what is usually regarded as the product of explanation: understanding. According to some authors, understanding is inherently related to the explanatory aim of science, while others see it as an additional epistemic aim, and some even suggest that understanding should replace the traditional aims (Potochnik Reference Potochnik2017, 101).

This raises the question of what exactly scientific understanding consists in, an issue that has attracted much attention among philosophers of science and is discussed in several recent philosophical books (de Regt Reference de Regt2017; Elgin Reference Elgin2017; Khalifa Reference Khalifa2017; Potochnik Reference Potochnik2017). An investigation of this problem should start with distinguishing between various types of understanding, illustrated by the following examples:

1. a) Albert understands that the skin of polar bears is white (understanding-that).

2. b) Barbara understands why the skin of polar bears is white (understanding-why).

3. c) Chris understands how natural selection works (understanding-how).

4. d) Dorothy understands the theory of evolution (objectual understanding).

Understanding-that may be identified with knowledge-that, and as such it relates to the descriptive and predictive aims of science. Understanding-why is related to its explanatory aim: scientific explanations typically produce understanding-why. For example, the theory of evolution provides an explanation of the color of polar bears’ skins, which affords Barbara to understand why polar bears are white. Philosophers disagree, however, whether understanding-why can only be produced through explanations: some (notably Lipton Reference Lipton, de Regt, Leonelli and Eigner2009) argue that understanding-why without explanation is possible; others (e.g., Khalifa Reference Khalifa2017, 125–54) deny this. While I believe that understanding-why is a cognitive achievement in its own right, which is not reducible to explanation, in this article I focus on the understanding that is produced by explanations. The difference between understanding and explanation can be gleaned already from the fact that explanation can be, and typically is, regarded as objective, while the understanding it produces is relative to the subject who understands and to the context in which she finds herself. The same objective explanation of the whiteness of polar bears’ skins may afford understanding to Barbara but not to Bernard. The contextual theory of scientific understanding that will be presented in the next section accounts for this difference and explicates the relation between understanding-why and explanation. It hypothesizes that the construction of an explanation requires that scientists understand the theory on which the explanation is based. Hence, objectual understanding is a prerequisite for understanding-why. As I argue in the next section, understanding a theory involves intelligibility, which is in turn related to understanding-how, since intelligibility relates to scientists’ skills. In sum, the value of intelligibility is crucial for promoting the epistemic aim of explanatory understanding.

3. The Contextual Theory of Scientific Understanding

I will now elaborate and substantiate the theses about the nature of understanding presented above. The context dependence of explanatory understanding, one of the fundamental tenets of my theory, is supported not only by conceptual analysis but also by studies of scientific practice and history (see sec. 4 for an example). The theory is built on a model-based account of explanation (see, e.g., Cartwright Reference Cartwright1983, 143–62; Morrison Reference Morrison, Morgan and Morrison1999, 60–64), which implies that scientists achieve understanding of a phenomenon P if they construct an explanation in the form of an appropriate model of P on the basis of relevant theories T. As Cartwright (Reference Cartwright1983, 133) has shown, constructing such models involves pragmatic judgments and decisions, since models do not follow deductively from theories or the empirical data. These judgments concern, for example, the idealizations and approximations—inherent to modeling—that need to be made (the case presented in sec. 4 provides an illustration).

Deciding which idealizations and approximations are appropriate in a particular situation is a matter of expertise and skill on the part of scientists. These skills vary with the relevant scientific theories: to construct an explanatory, understanding-providing model, scientists should be able to work with the theories on which the models are based. It is here that the value of intelligibility comes into play. I submit that the above considerations entail that if scientists want to understand a phenomenon, they first have to understand the relevant theories. In other words, these theories have to be intelligible to them, where intelligibility is a measure of the ‘match’ between scientists’ skills and the theory’s qualities. Only if theories are intelligible to them can scientists use them in fruitful ways. More precisely, intelligibility can be defined as follows:

This definition entails that intelligibility is not an intrinsic property of theories but a value assigned to theories by scientists. What is more, intelligibility is context dependent, since scientists’ value judgments regarding preferred qualities of theories will vary with their (contextually acquired) skills. Visualizability and simplicity, for example, are widely valued theoretical qualities, because many scientists find theories with these qualities more tractable and easier to work with. But visualizability is not a necessary condition for intelligibility: depending on the context, there may be alternative ways to render theories intelligible (see de Regt Reference de Regt2017, chap. 7). And simplicity comes in many different forms, where one particular form of simplicity may enhance intelligibility in particular contexts but not in others.

Although evaluations of intelligibility may vary contextually (e.g., across scientific communities, disciplines, and historical periods and even between individual scientists), we can still give a general characterization of scientific understanding in the form of a Criterion for Understanding Phenomena:

Note that CUP includes not only intelligibility but also empirical adequacy and consistency as values necessary for scientific understanding. Intelligibility of T does not by itself guarantee success in achieving explanatory understanding of phenomena: scientists have to use T to actually construct an explanation of P that is acceptable, which implies that it should respect the values of empirically adequacy and internal consistency. In section 5 the role and interaction of these values will be examined more closely.

That intelligibility is not a sufficient condition for scientific understanding can be illustrated with the theory of intelligent design (ID), defended by its proponents as a superior alternative to the Darwinian theory of evolution, specifically to the idea of natural selection. Adherents of ID suggest that their theory is more intelligible than the theory of evolution, since it may seem difficult to understand how complex functional features of organisms can emerge as an outcome of a process of ‘blind’ variation and selection. ID is presented as an alternative theory of the emergence of such complex functional features, which allegedly provides understanding of this process by postulating an intelligent agent who has brought these features into existence. However, as Kitcher (Reference Kitcher2007, 105–6) shows, ID fails to supply specific theoretical principles that can be used to construct explanations of particular phenomena. While the general idea of ID may appear intuitively intelligible, this intelligibility does not allow one to construct specific, empirically adequate explanations of concrete phenomena. As yet, ID has not yielded scientific understanding.

4. Maxwell versus Boltzmann on Understanding Gases

In the previous section it was argued that scientific understanding is produced by empirically adequate and internally consistent explanations based on intelligible theories, where intelligibility is a pragmatic, context-dependent value. I will illustrate this thesis with a historical case that shows how epistemic and pragmatic values interact in the creation of understanding. The case concerns an episode in the development of the kinetic theory of gases and features a controversy between Ludwig Boltzmann and James Clerk Maxwell over the value of the so-called dumbbell model of diatomic gases. In this section I present the case, and in section 5 the role of value judgments regarding the intelligibility, empirical adequacy, consistency, and representational accuracy will be analyzed.

The kinetic theory of gases, first developed by Rudolf Clausius and James Clerk Maxwell in the mid-nineteenth century, is based on the idea that heat is mechanical motion and that gases consist of tiny particles (molecules) in motion, behaving in accordance with the laws of Newtonian mechanics. With this theory, physicists attempted to explain known experimental facts about gases, such as the laws of Boyle-Charles and Gay-Lussac, the temperature independence of specific heats, and transport phenomena (viscosity, heat conduction, and diffusion). The explanation of these phenomena required specific models of molecules, in line with the model-based account of explanation sketched in section 3. The first model employed by Clausius and Maxwell pictures molecules as elastic spheres. This is clearly an idealization, as it neglects the volume and internal structure of molecules, as well as intermolecular forces. Maxwell was fully aware of this and also suggested an alternative center-of-force (point-particle) model, a different idealization that gives the same results. Application of the kinetic theory to a system of such idealized gas molecules yields equations for its behavior, on the basis of which Maxwell succeeded in explaining the above-mentioned empirical gas laws and phenomena. In addition, an important theoretical result of applying the theory to such a system was an equipartition theorem, which states that the kinetic energy of the system is divided equally over all ‘modes of motion’: translation, rotation, and vibration.

This equipartition theorem generated a prediction for the specific heat ratio γ, that is, the ratio between the specific heat at constant pressure C_p and the specific heat at constant volume C_v:

(1) $\begin{array}{}& \gamma =\frac{C_p}{C_v}=1+\frac{2}{3}\times \frac{K}{H},\end{array}$

where H is the total kinetic energy of the molecules and K is the kinetic energy of the translational motion.

For spherical (or center-of-force) molecules the translational energy is equal to the total kinetic energy, hence the theoretical value $\gamma =1.66$. However, the experimentally measured value for many common gases (e.g., oxygen and nitrogen) was $\gamma =1.41$. Maxwell concluded that these gases are apparently polyatomic, which means that they can rotate. In that case, however, equipartition implies that $K/H=0.5$, so that the specific heat ratio would be $\gamma =1.33$. Thus, the experimental value could not be explained theoretically. The discrepancy that Maxwell noted between theoretical predictions and experimental results for specific heat ratios became known as the specific heat anomaly.

In 1875 Maxwell employed a new tool for modeling molecules: the notion of ‘degree of freedom’, based on the idea that “the position and configuration of the molecule can be completely expressed by a certain number of variables,” of which “three are required to determine the position of the centre of mass of the molecule, and the remaining $n-3$ to determine its configuration relative to its centre of mass. To each of the variables corresponds a different kind of motion” (Maxwell Reference Maxwell, Garber, Brush and Francis Everitt1986, 230). Using the concept of degrees of freedom, Maxwell derived a new formula for the specific heat ratio:

(2) $\begin{array}{}& \gamma =\frac{2+n+e}{n+e},\end{array}$

where n is the number of degrees of freedom and e is a quantity depending on the binding forces in the molecule (e accounts for the relative amount of heat that is stored as potential energy in the molecule). Comparison with experimental values led Maxwell to conclude that $n+e$ “for air and several other gases cannot be more than 4.9” (231), since the experiments have determined that $\gamma =1.41$. Furthermore, these gases cannot consist of monatomic molecules without internal structure, as these have only three degrees of freedom ( $n=3$), which yields $\gamma =1.66$, a value not yet observed for any gas. Moreover, spectroscopic research had shown that most gases emit spectral lines, the best explanation of which seemed to lie in internal vibration of the molecules. This would imply at least three additional degrees of freedom, so that $n\ge 6$ and thereby $\gamma \le 1.33$. The observed ratio of $\gamma =1.41$ was thus still unexplainable. Maxwell concluded that this anomaly is “the greatest difficulty yet encountered by the molecular theory” (232).

One year later, in 1876, Ludwig Boltzmann proposed a simple and straightforward solution of the specific heat anomaly, using the concept of degrees of freedom. Boltzmann’s proposal was inspired by an 1875 experiment of August Kundt and Emil Warburg, who had measured the specific heat ratio of mercury vapor to be $\gamma =1.66$, the value that had been predicted for monatomic gases but had never before been observed experimentally. This result implied that the mercury molecules behave as material points with only three degrees of freedom, while the observed emission of spectral lines by mercury vapor suggests that the molecules possess an internal structure, implying the presence of additional vibrational degrees of freedom. Boltzmann (Reference Boltzmann1968, 105) concluded from this that molecules of mercury vapor “behave almost like material points or, if one prefers, like elastic spheres,” even though this does not accurately describe their real structure. Next, he suggested a similar idealization for the molecules of the gases with the anomalous specific heat ratio: suppose that these consist of two atoms (material points or elastic spheres) that are rigidly connected. Such ‘rigid dumbbells’ possess five degrees of freedom: three of translation and two of rotation; hence $n=5$. Moreover, since the molecules are absolutely rigid, they cannot store potential energy; hence $e=0$. This leads to a value of $\gamma =1.4$, almost precisely in accordance with the experimentally measured value.

Maxwell’s response to Boltzmann’s proposal was negative. While admitting that the dumbbell model accommodates the empirical facts about specific heat ratios, Maxwell criticized the idealizations it contains: “It will not do to take a body formed of continuous matter endowed with elastic properties, and to increase the coefficients of elasticity without limit till the body becomes practically rigid. For such a body, though apparently rigid, is in reality capable of internal vibrations, and these of an infinite variety of types, so that the body has an infinite number of degrees of freedom” (Reference Maxwell1877, 245). Maxwell’s main point is that the model is inconsistent given the laws of mechanics, since rigidity of extended bodies is incompatible with elasticity. Moreover, the hypothesis of rigidity appeared at odds with the experimental evidence regarding spectral lines, which suggests that molecules vibrate.

The fact that Maxwell and Boltzmann could disagree about whether the dumbbell model provided a valid solution for the anomaly, while they were both strongly committed to the kinetic theory of gases, supports the idea that the model is ‘autonomous’ (Morrison Reference Morrison, Morgan and Morrison1999): it does not follow from the kinetic theory or from the experimental data. Instead, approximation and idealizations are needed to construct the model, which ‘mediates’ between theory and data, in order to attain explanatory understanding of the phenomenon. Maxwell and Boltzmann differed in opinion about whether the dumbbell model provides understanding, and this had to do with their opposing attitudes toward the idealizations. In the next section I analyze these attitudes in more detail, by showing how they are rooted in value judgments regarding empirical adequacy and internal consistency.

5. Values and the Creation of Scientific Understanding

In section 3 I argued that understanding-providing explanations should be based on intelligible theories and should be empirically adequate and internally consistent (criterion CUP). So, did Boltzmann’s dumbbell model advance understanding of the behavior of gases? There is no easy answer to this question because, as we saw above, Boltzmann and Maxwell disagreed about whether the model-based explanation was (sufficiently) consistent and empirically adequate. It appears that application of the additional requirements in CUP does not always yield unambiguous results. The reason is that the values of empirical adequacy and consistency do not function as strict rules but as values, just like intelligibility. In the current section, I investigate the role of the various relevant values in achieving the aims of science, especially understanding, on the basis of an analysis of the Maxwell-Boltzmann controversy.

6. Conclusion

Understanding phenomena is a central epistemic aim of science, and intelligibility of theories is a pragmatic value that is crucial for the construction of models that generate explanatory understanding. Such models should also be empirically adequate and internally consistent, but these requirements do not override intelligibility. Rather, they are context-dependent epistemic values that may be ranked and applied differently in specific situations. The value of representational accuracy, which is regarded as essential for understanding by scientific realists, serves a different epistemic aim of science, namely, accurate description. I have argued that it is irrelevant for the aim of understanding. Indeed, achieving understanding often requires the abandonment of representational accuracy. Inaccurate representations (e.g., idealizations) and theoretical falsehoods are oft-used tools for enhancing scientific understanding.