Towards Ideal Understanding

2.1 Kelp’s Maximal Understanding

Central to Kelp’s theory (and to ours too) is the concept of phenomenon. The phenomenon, for Kelp, counts as the target or object of understanding. It is hard to precisely define what phenomena are, as many different events and processes qualify as phenomena. The most comprehensive analysis of phenomena has been conducted by Bogen and Woodward (1988) and Woodward (2011). They define phenomena as features of the world that in principle could recur under different contexts or conditions. They distinguish phenomena from data, which are public records produced by measurement and experiment that serve as evidence for the existence or features of phenomena. Although Kelp does not refer to these works, his characterization of phenomena, although in terms of examples rather than a general definition, seems to be consistent with them.

Kelp’s goal is then to explain what it means for an agent to understand a phenomenon to the maximal possible degree. In order to do so, he introduces two conditions. First, the agent needs fully comprehensive knowledge about the phenomenon, that is, the agent needs to know everything there is to know about the phenomenon. Consider the phenomenon of a cannonball falling to the ground. There are different ways to explain this phenomenon. One way to explain it is by means of Newton’s law of gravitation. Given the initial conditions of the cannonball (position, velocity, and mass), the law tells us how the cannonball falls to the ground, namely, on a straight trajectory with constant acceleration. But we can explain how the cannonball falls to the ground also by means of conservation of energy. The potential energy of the body at the beginning is transferred into kinetic energy when falling.

But as Kelp rightly points out, one may still lack a certain degree of understanding, even if one has fully comprehensive knowledge. In the case of the falling cannonball, one may know those two explanations without realizing that they are related: it is possible to derive the law of energy conservation from Newton’s law of gravitation (Goldstein et al. 2001: ch. 1.1). Therefore, Kelp also demands maximally well-connected knowledge of the phenomenon; that is, the agent needs to know how the explanations of the phenomena are related to one another (Is one entailed by the other? Are they independent?). Combining his two conditions, Kelp’s account of maximal understanding amounts to (Kelp 2015: 3811):

Maximal Understanding: If one has fully comprehensive and maximally well-connected knowledge of a phenomenon P, then one has maximal understanding of P.

We think that Kelp delivers an important and, in our opinion, correct general framework for maximal understanding. Nonetheless, the framework is too general in this form and a step too far from scientific practice. How does a scientist accomplish fully comprehensive knowledge? What is the role of a scientific theory in providing such knowledge? What criteria does a theory need to fulfill for doing so? Do the relevant connections only need to hold among explanations or propositions regarding the particular phenomenon? We think that the well-connectedness of knowledge needs to go farther than perceived by Kelp: our knowledge of a phenomenon must be also well-connected to our background beliefs. We may well have comprehensive and well-connected knowledge of a phenomenon, but this knowledge may not adequately match our intellectual background, which can include beliefs about metaphysics, about other scientific theories, or about the practice of science. Thus, we would suffer from some cognitive dissonance; such dissonances, however, must be mitigated or amended for ideal understanding to succeed (we discuss this condition in detail in Section 4.3).

2.2 Khalifa’s Ideal Understanding

Inspired by Kelp’s account, Khalifa (2017) develops an account of maximal understanding, which he calls ideal understanding, which shares some features of Kelp’s maximal understanding but also differs in crucial respects. Khalifa (2017: 4) characterizes ideal understanding in the following way:

Ideal Understanding: S ideally understands why P if and only if it is impossible for anyone to understand why P better than S.

In contrast to Kelp, Khalifa bases his ideal understanding on a comparison of degrees of understanding between different agents. Agent S has ideal understanding of a phenomenon P if no other agent (not only in practice but also in principle) can have a better understanding of P than S. How do we quantify that one agent understands better than another one? Khalifa builds his entire theory of understanding on this point. First, he tells us what minimal understanding is, and from there he builds up to ideal understanding by developing a criterion for understanding better.

Minimal Understanding: S has minimal understanding of why P if and only if, for some Q, S believes that Q explains why P, and Q explains why P is approximately true (Khalifa 2017: 14).

An agent S has minimal understanding of a phenomenon, if she has one approximately correct explanation of this phenomenon. Khalifa (2017: 7) demands the following four requirements for an explanation:

Q (correctly) explains why P if and only if:

• (1)  Q is (approximately) true;

• (2)  Q makes a difference to P;

• (3)  Q satisfies your ontological requirements (as long as they are reasonable); and

• (4)  Q satisfies the appropriate local constraints.

Khalifa’s entire account is based on explanations, which some find controversial (Wilkenfeld 2013; Kelp 2015; Dellsén 2020). But given that one may base understanding on explanations, conditions (1) and (2) seem to be uncontroversial. Condition (3), on the other hand, might seem to be too strong a requirement and might also require a stance with respect to realism and antirealism (Chakravartty 2017). In order not to be committed to a particular stance, Khalifa adds the qualifier “as long as they are reasonable”. Whether demanding a certain (unobservable) ontology as part of an explanation depends on whether one regards such ontology reasonable or not, and this depends on where one stands in the realism-antirealism debate. Certain antirealists would abandon any ontological requirements on a scientific explanation, which would make condition (3) vacuous, and Khalifa embraces this option. Pleasing the antirealist with such a broad application of condition (3) makes Khalifa’s theory more attractive to a broader group of scientists and philosophers. Nevertheless, regarding the goal of describing what ideal understanding amounts to, one may re-consider such a loose requirement regarding an unobservable ontology. Even if one may be skeptical with respect to the existence of unobservable entities, one may still grant that those entities deepen one’s understanding of the phenomenon. They may help to foster one’s intuition or at least to memorize or visualize what is going on in the world. For example, mechanistic explanations, which depend in many cases on an unobservable mechanism, do provide a deeper understanding of a phenomenon than explanations that do not rely on such mechanisms (see Hubert 2021, for this argument, and Colombo et al 2015 for an anti-realist interpretation of mechanistic explanations).

It is a bit unclear why Khalifa adds condition (4). He says that the first three conditions are global constraints that are valid for all kinds of scientific explanations, while condition (4) emphasizes that local constraints are needed too. The local constraints Khalifa has in mind are the specific requirements that are imposed on a scientific explanation within a certain scientific discipline or context. Some contexts or disciplines rely on idealizations, others demand causal explanations, etc. Khalifa wants to make sure that specific scientific contexts determine when an explanation is good enough or not (even when it fulfills the first three conditions).

Now having an account of scientific explanations and minimal understanding, we can continue to follow Khalifa in how he develops an account of better understanding. Roughly speaking, the more explanations an agent has at hand the better is her understanding (Khalifa 2017: 14):

Better Understanding I: S₁ understands why P better than S₂ if and only if: Ceteris paribus, S₁ grasps P’s explanatory nexus more completely than S₂.

We agree with Khalifa that the more explanations we know about a phenomenon the better we understand it. Like in Kelp’s case, Khalifa builds into his account a connectedness requirement by describing the set of explanations forming an interrelated nexus. And also, like Kelp, Khalifa does not say if and how the explanatory nexus is anchored in other beliefs we already hold about the world.

Khalifa (2017: 14) presents another criterion, which may be powerful enough to say more about how the explanatory nexus is connected to an agent’s intellectual background:

Better Understanding II: S₁ understands why P better than S₂ if and only if: Ceteris paribus, S1’s grasp of p’s explanatory nexus bears greater resemblance to scientific knowledge than S2’s.

While Khalifa’s first criterion describes better understanding “from below”, that is, by building it up from grasping more explanations, the second criterion describes better understanding “from above” by focusing on how close the grasping of the nexus is to scientific knowledge. This characterization hinges, of course, on what Khalifa means by scientific knowledge. Khalifa does not really define scientific knowledge, and its relation to ideal understanding is not clear either. Although Khalifa writes, “Very roughly, ideal understanding is maximally scientific knowledge of a complete explanatory nexus (2017: 15), it is unclear to us what “maximally” scientific knowledge is. It seems that scientific knowledge comes in degrees. Or Khalifa identifies ideal understanding with scientific knowledge simpliciter, and maximally scientific knowledge refers to the grasping of the explanatory nexus.

Anyway, we think that the notion of scientific knowledge may be able to comprise the connectedness requirement that we miss in Khalifa’s first criterion. He claims that an agent gains scientific knowledge when she evaluates different possible scientific explanations for a particular phenomenon P. In a first step, he considers different candidate explanations and evaluates which are plausible and which are implausible. Although not explicitly mentioned by Khalifa, an agent may use her intellectual background in deciding whether an explanation is plausible or not. Khalifa briefly gives the example that one would immediately rule out an explanation of Newton’s death “by appeal to alien laser guns”. Without conducting an autopsy (which is now impossible anyhow) or historical research about the circumstances of Newton’s death, one can refer to one’s background beliefs about how people die and the likelihood that aliens have visited the Earth to render such an explanation implausible. One crucial ingredient in our account of ideal understanding is exactly this connection with our intellectual background.

Another important criticism of Khalifa’s theory of understanding might be that he ignores too many of the practical skills of a scientist and that he over-emphasizes the importance of explanations (de Regt & Baumberger 2020; de Regt & Höhl 2020). De Regt’s contextualist theory of understanding is supposed to amend this problem, but we will see that he ultimately over-emphasizes the practical aspects required for using a scientific theory.

3.1 De Regt’s Contextualist Theory of Understanding

Central to de Regt’s theory of understanding is a scientific theory by which a scientist gains understanding by grasping an explanation provided by this theory (de Regt 2017). To provide understanding, the scientific theory must instantiate three qualities; it must be:

• i)  intelligible (to the scientist),

• ii)  empirically adequate,

• iii)  internally consistent.

But what does it take for a theory to be “intelligible”, in de Regt’s sense? De Regt defines intelligibility in a pragmatic way: a theory is said to be intelligible if a scientist can easily apply the theory for practical matters (de Regt 2017: 40 and 101). Therefore, the implementation of intelligibility depends on the theory and on what the scientist intends to accomplish with this theory. An instantiation of de Regt’s pragmatic intelligibility criterion that is particularly suited for physics says that a scientist should be able to intuitively apply the theory without making detailed calculations. Since this kind of intelligibility does not only depend on the theory itself but also on the abilities and background knowledge of the scientist (and maybe also on the current state of technology for applying the theory), intelligibility is a non-intrinsic, contextual property of a theory. And because intelligibility in addition depends on the scientist using the theory, a theory that is intelligible to one scientist may be unintelligible (or less intelligible) to another.³

We agree with de Regt’s contextual theory of understanding that successfully applying a theory and having strong intuitions about its consequences, granted the empirical adequacy and consistency requirement, typically results in a certain degree of understanding of the physical world; however, we think that de Regt’s notion of intelligibility incorporates both too much and too little. It incorporates an aspect of understanding that goes beyond the scientist’s skill to use the theory, on the one hand; and on the other, it does not offer the whole story about intelligibility because it disregards crucial aspects of it (particularly, when we aim to work out an ideal).

For de Regt, the pragmatic virtues of a theory do not only depend on the theory but also on the scientist using the theory. Different scientists may have different background knowledge, metaphysical commitments, etc. (de Regt 2009: 592). And because they differ in these commitments, they value different virtues of theories (or even entire theories) differently. De Regt (2017: ch. 5) delves deeper into the metaphysical commitments of physicists (he discusses Newton’s theory of gravity in its historical development, which we will take up in Section 4.3), and he distinguishes two kinds of intelligibility: scientific intelligibility and metaphysical intelligibility. Scientific intelligibility is achieved through using the tools of the theory to find adequate explanations and to make correct predictions. Metaphysical intelligibility is achieved through a metaphysical worldview that makes sense to the scientist and that often provides conceptual tools for scientific intelligibility.⁴ Putting metaphysical intelligibility and scientific intelligibility under the same umbrella, de Regt puts too much weight on the significance of metaphysics for the usability and application of scientific theories. For, a theory that we know to be wrong or that strikes us as implausible, given what we already hold true about reality, can be perfectly scientifically intelligible to us. For example, one may even become an expert in the domain of astrology while thinking that it is fantasy (because of financial incentives, for example, as many great astronomers in history worked also in astrology).⁵ Or an atheist, like Richard Dawkins, might comprehend the reasonings and alleged justifications behind creationism while being committed to evolutionary theory. Even if a theory is metaphysically unintelligible, one may be able to use it, and this usability of a theory, we take it, is an integral part of intelligibility. Therefore, we need to disentangle the scientist’s metaphysical commitments from intelligibility and embed them into a wider network of one’s noetic system, which comprises the scientist’s intellectual background (we discuss all this in more detail in Section 4.3).

Moreover, we think there is an aspect of a theory’s intelligibility (and of understanding) that is not sufficiently appreciated in de Regt’s account. Understanding a theory is not only a matter of being able to use it and to apply it to the phenomena. It is also a matter of being aware of what the world looks like, according to the theory, of having access to the theory’s truth conditions, of knowing which kind of ontology is associated with the theory or postulated by it. A theory with an open interpretation problem, for instance, or with unclear ontological commitments, is a theory that we do not yet understand (fully), even if we are extremely successful in its application. A scientist may successfully apply a theory and have correct intuitions about qualitative results of the theory, but given that she has no clear picture of the ontology postulated by the theory, she would still miss potentially important information about the goings-on in the world. This information may not be particularly important for applying the theory in the particular context and for the particular purposes of the scientists, but it would still expose something about the world and would actually deepen the overall understanding of the scientist. The information we refer to is, for example, about what something is made of: the ontology of a phenomenon or object, or, in other words, the mechanism.

Let us go into some more details on why we think that awareness of mechanisms potentially deepens scientific understanding. What is the starting point and the aim of scientific inquiry in the first place? Grimm (2008) rightly answers that scientific inquiry starts with and aims at answering why something happens the way it happened and not otherwise. This answering-why has been the focus of almost the entire literature on explanation from the deductive-nomological model (Hempel & Oppenheim 1948) to modern causal theories (Woodward 2003). And we see this focus also in both of de Regt’s theories of understanding, as well as in the theory of understanding developed by Khalifa (2017), where the main use of a theory is to explain why some phenomenon happens. The requirements on a theory to be intelligible and successfully usable are mainly to let the scientist efficiently use the theory. De Regt’s theory of understanding, as well as Grimm’s theory of scientific inquiry and much of the literature on scientific explanation, does not appreciate that apart from asking why something happens science also asks what something is made of (Hubert 2021). Contrary to most of his contemporaries, Wesley Salmon pointed to this complementary part of scientific understanding:

We want to know how things work and, it should be added, what they are made of. [. . .] What we want to do is open up the black box and see how it works. (Salmon 1998: 87)

This passage bears two insights: First, we can ask what something is made of independently of why something happens. Second, we can use this information to discover how something works. The how is indeed a combination of what and why: it explains why something happens by means of what it is made of. As Salmon points out, these kinds of explanations are causal-mechanical explanations, which have recently experienced a renaissance (Glennan 2017; Glennan & Illari 2018; Machamer et al. 2000). Glennan (2017: 17) proposes the following minimal characterization of mechanisms: A mechanism for a phenomenon consists of entities (or parts) whose activities and interactions are organized to be responsible for the phenomenon. Glennan specifies very general requirements on the entities that make up the mechanism to apply this definition to a wide range of examples. What is important for our argument is that we can inquire into the entities of a mechanism for basically all kinds of phenomena. And finding out about these entities would give us a deeper understanding of the phenomenon, because we would be able to answer not only “Why?” but also “What?”.

In de Regt’s theory, mechanical explanations are not distinguished from other types of explanations. He does not say that there is no place for mechanical explanations in the contextual theory or that mechanical explanations do not yield understanding; rather, mechanical explanations do lead to understanding (if they are intelligible, etc.), but they provide the same kind or level of understanding as other (intelligible, etc.) explanations. If a scientist uses mechanical explanations in one case and uses another type of explanation in another case, de Regt’s theory says that in both cases the scientist has the same level of understanding, as long as the scientist is successful. As we have argued, mechanical explanations do answer two (instead of one) of the most fundamental questions we can ask about a phenomenon, and this difference, we think, needs to be incorporated into a theory of ideal understanding.

3.2 Wilkenfeld’s MUDy Understanding

We think that de Regt is correct in pointing out that a theory’s intelligibility plays a crucial role in understanding phenomena, but as we discussed in the previous subsection, de Regt leaves out important aspects of understanding, namely, understanding what the world is made of. Wilkenfeld (2017) criticizes de Regt’s theory along similar lines and claims that de Regt ignores the significance of the representational accuracy of the state of understanding (which is mediated in de Regt’s account by a scientific theory).⁶ Wilkenfeld describes representational accuracy in the following way:

We do not have an account of what it means for a representation to be accurate but presumably the general idea is that the actual state of affairs of the world is in some important sense similar to the state of the world as depicted in the representation. Importantly, if we assume a correspondence theory of truth, any true propositions will be representationally accurate. (Wilkenfeld 2017: 1275)

Khalifa’s explanationist theory is a theory of representational accuracy, as the explanatory nexus represents the explanatory relations of the world; mechanistic explanations, as we described them, represent not only causal relations but also represent what the world is made of. On the other hand, theories of understanding based on representational accuracy, such as Khalifa’s, tend to underestimate intelligibility. Therefore, Wilkenfeld (2017: 1276) proposes a multidimensional account of understanding, where intelligibility and representational accuracy form two dimensions:

Multiple Understanding Dimensions (MUD): the quality of a state of understanding is evaluable along multiple orthogonal dimensions, including (but perhaps not limited to) both representational accuracy and intelligibility.

Wilkenfeld leaves it open whether there may be additional dimensions. If we aim at ideal understanding, we think that the two dimensions identified by Wilkenfeld need to be supplemented by (at least) three additional dimensions:

1. Sufficient truth (based on the theory),

2. Reasonable endorsement,

3. Fit into the noetic system (of the scientist).⁷

With these three dimensions, we propose a theory of ideal understanding that comprises five dimensions of understanding evaluation.

Ideal Understanding: An agent S ideally understands a phenomenon P by means of a theory T, only if (i) T represents P accurately enough, (ii) T is intelligible to S, (iii) S is in the position to extract enough truths from T, (iv) S has reasonable grounds to endorse T, and (v) T fits into the noetic system of S.

We take these five dimensions to be necessary conditions for ideal understanding. We are open to the possibility of more dimensions for evaluating ideal understanding; in Section 5, we discuss three candidates for further dimensions (grasping, novel predictions, and transparency), but we conclude that they are inadequate for being included as necessary conditions for ideal understanding.⁸

It should be noted here that we both supplement and digress from Wilkenfeld’s theory. We supplement it, because we argue that besides representational accuracy and intelligibility, we need to postulate three additional dimensions of scientific understanding. We digress from it, because we conceive of representational accuracy differently than Wilkenfeld does. For reasons that will become clear in Section 4.1, we don’t demand from a scientific theory to be perfectly representationally adequate, but only sufficiently so. Moreover, Wilkenfeld is interested in the representational accuracy of the state of understanding. We, on the other hand, take representational accuracy to be a property of a scientific theory, which we regard as the most comprehensive epistemic mediator for scientific understanding. Our notion of sufficient truth, on the other hand, has a pragmatic component and it concerns the truths that a scientist is in the position to extract from a scientific theory.

4.1 Sufficient Truth (Based on the Theory)

Prima facie, it might strike one as natural to assume that ideal understanding of a phenomenon P will be achieved by a theory that contains the whole truth, and nothing but the truth about P. In other words, it seems natural to require that the dependence relations postulated by the theory providing one with ideal understanding will correspond to real dependence relations (Dellsén 2020); that the claims the theory makes about possible worlds will be correct; that the ontology postulated by the theory will correspond to the real ontology, and so on.

And yet here is an argument that might be raised against this idea (see, e.g., Elgin 2017). Our current best science is full of “falsehoods”. More precisely, it is full of representational systems that are known to (partially) misrepresent the way the world actually is. A good example is an idealized model (Cartwright 1983; Giere 2004; Morgan & Morrison 1999; Potochnik 2017). The ideal gas law, for instance, accounts for the behavior of (real) gases by describing the behavior of an ideal gas comprised of molecules that are dimensionless or perfectly spherical and exhibiting no intermolecular forces. Clearly, there is no such thing in the real world. By depicting gases as if they had those features, however, the ideal gas law enables scientists not only to predict, but also to understand how real gases behave. Certain falsehoods, thus, seem to figure centrally in our scientific understanding of the real world (Elgin 2017: 61).

Moreover, Elgin claims, these falsehoods are not simply “necessary evils” in a world that is too complicated for us to deal with; they are not simply tools that scientists use to approach the true description of reality:

Idealization is not taken by scientists to be an unfortunate expedient, but rather to be a powerful tool. Although they expect today’s idealizations to be replaced, they harbor no expectation that in the fullness of time idealizations will be eliminated from scientific theories. [. . .] Elimination of idealization is not a desideratum. Nor is consigning them to the periphery of the theory. [. . .] The ideal gas law lies at the core of thermodynamics, and some such model is likely to lie at the core of any successor to current theories. (2017: 62)

We think we must take the possibility that Elgin is right seriously. Even when the final theory about reality is found, if it exists in the first place, it is plausible to expect that idealizations will remain a crucial part of scientific practice. We thus cannot, or better should not, demand that our scientific theories be perfectly true or perfectly representationally accurate—not even from those providing us with ideal understanding. However, we think we should demand that they be sufficiently representationally accurate, and moreover to work effectively as a source of true beliefs about the phenomena they account for.

As we briefly mentioned at the end of the previous section, in spelling out the relation between understanding and truth there are two levels of adequacy or correctness to distinguish. One is the level of the correctness of the representational systems that scientists deploy. The other is the level of the correctness of the information that scientists believe based on the representational systems that they deploy. The two levels are certainly related, but they are independent. Even a partially false or only partially representationally adequate representational system could in principle work as a source of true information about reality—given that one is equipped with the right background knowledge. Take for example an idealized model. An agent who properly understands the model will be aware that the model is idealized. In the best-case scenario, she will be aware of which information contained in the model is to be interpreted as a realistic representation of its subject matter and which, instead, is merely fictional—and she will believe those parts of the model that are meant to correspond to reality and merely accept or disregard the fictional parts. But given that one has reached this vantage point, everything one will believe about reality on the basis of the model will be true after all, even if the model misrepresents its subject matter to some extent (this point has been appreciated and stressed, among others, by Greco 2014; Lawler 2021; Le Bihan 2021; Mizrahi 2012; Nawar 2019; Rice 2021; Strevens 2008).

With this distinction in the background, we can grant Elgin’s claim that certain falsehoods such as idealizations will be involved in any theory, even in the final theory about P, and hold fast to the intuition that understanding P, as an epistemic state, requires sufficient truth. Our idea is that scientists who will understand P ideally on the basis of an inaccurate, or partially accurate representational system will have the appropriate skills necessary to identify and extract enough true information contained in it. Hence, everything scientists will believe about P and about P’s subject matter on the basis of such a representational system will be true. This weak truth-requirement, we argue, is closer to actual and realistic scientific practice and hence more promising than the initially plausible strong one that we sketched above.

De Regt and Gijsbers controversially claimed that understanding should be conceived as a non-factive cognitive state, because “understanding can be gained from representational devices that are false, and not just slightly false, but wildly so” (de Regt & Gijsbers 2017: 50). We are now in the position to see that this is a non sequitur. Even if the representational systems we use are wildly incorrect, we might have reached the vantage point and developed the skills necessary to identify and to extract the true information contained in them. But if we have, then the understanding we will gain based on these inaccurate representational systems will be factive after all; that is, everything we will believe based on the inaccurate representational systems we use will be true. This may be still hard to accomplish in practice, especially synchronically without the wisdom of hindsight. But at least in model building, scientists are familiar with the problem of deriving truths from idealizations, that is, from intentional falsehoods. The caveats of extracting truths from false theories have been debated between Hasok Chang (2003) and Stathis Psillos (1999) on the plausibility of preservative realism, where Chang emphasizes the problems with such an endeavor.⁹ We want to discuss a more optimistic example in the following.

De Regt and Gijsbers (2017) discuss the phlogiston theory as an example of a theory that, despite being “wildly incorrect”, succeeded in providing scientists with an understanding of phenomena. Pace de Regt and Gijsbers, phlogiston theory is in fact a great example of a theory that can work as to generate (approximately) true beliefs for a scientist who is equipped with the right background knowledge (more precisely: for a scientist who masters modern chemistry). Phlogiston theory explained the process of combustion roughly in this way: when a substance burns, an unobservable matter called “phlogiston” leaves the substance in question—usually in the form of a hot flame or evaporating gas. This has long been considered as “wildly incorrect”, because, according to oxygen theory (the theory that superseded phlogiston theory), when a substance burns, nothing is lost, but something (oxygen) is added to it (this process is called oxidation). Modern chemistry, however, uncovered the mechanism underlying oxidation and made us realize that phlogiston theory was actually not as incorrect as it might seem: in an oxidation process, a chemical bond forms between an electropositive substance (such as metal, coal) and an electronegative substance (such as oxygen). In this chemical bond, the electropositive substance gives up some of its electrons; more precisely, the electropositive substance donates some of its electrons to the electronegative one. Modern chemistry, thus, enables us to see that phlogiston theory was right, or at least on the right track, in claiming that something in the process of combustion “gets lost”.¹⁰

What we want to emphasize through our notion of sufficient truth is the pragmatic aspect of truth seeking in scientific practice, which is part of understanding. It is rarely the case that one can read off the truth of a theory. Even if the theory that we use is representationally accurate, it is often not possible to directly apply the theory. Idealizations and model building are necessary paths in “applying” the theory. And on the other hand, if a theory is (partially) representationally inaccurate, experienced scientists can still extract truths from it. Newtonian mechanics is a good example because we know that it is not the perfectly true theory of the universe, but we still use it successfully in specific domains (for example, when sending a rocket to the moon).

Our notion of sufficient truth differs from Wilkenfeld’s notion of truth. Wilkenfeld works with a (broadly conceived) correspondence theory of truth—one that can be applied also to non-propositional representational systems (Wilkenfeld 2017: 1275). We think that the correspondence theory of truth is roughly correct as a theory of what it means for a representational system to be representationally accurate, but it has to be supplemented by a pragmatic story of how truths are derived from representational systems in scientific practice. Hence, the pragmatic aspect of truth-seeking is not sufficiently appreciated in Wilkenfeld’s MUDy understanding.

Khalifa (2017) also uses a notion of truth that differs from ours. He distinguishes between strict truth and approximate truth. He claims that

Q (correctly) explains why P if and only if:

• (1)  Q is (approximately) true;

• (2)  Q makes a difference to P;

• (3)  Q satisfies your ontological requirements (so long as they are reasonable); and

• (4)  Q satisfies the appropriate local constraints. (Khalifa 2017: 7)

That some proposition Q is (approximately) true can be understood, as Khalifa emphasizes, in two different ways. In the realist sense, that the proposition Q is true means that Q is representationally accurate (as in Wilkenfeld’s case). That is, Q corresponds to facts in the world. If Q is approximately true, Q is close to be representationally accurate. In the anti-realist sense, the proposition Q is true means something weaker, for example, being empirically adequate.

Now, Khalifa defines when explanations are approximately true:

Let’s say that “[Q] explains why [P]” is approximately true if and only if the first of these conditions is satisfied, and, furthermore, some of the terms in the explanans [Q] that purport to make a difference to the explanandum [P] actually do make a difference and also satisfy your preferred ontological requirements. (Khalifa 2017: 55)

An explanation is approximately true, then, if Q (the explanans) contains some, but not all, difference-makers of P (the explanandum). If the explanation contained all difference-makers, it would be strictly true. Because in practice, we cannot and often do not want to identify all difference makers, approximately true explanations are true enough for all practical purposes. How do scientists find out whether an explanation is the right one? Here, Khalifa explicitly mentions a specific truth-seeking process, which he calls SEEing (scientific explanatory evaluation process). Such an evaluation process comprises three steps:

1. consideration,

2. comparison (of the potential explanations),

3. belief-formation.

First, the scientist needs to consider other potential explanations for the phenomenon P. Second, the scientist needs to compare all these explanations according to best scientific practice. After that, the scientist forms a belief about the explanation that turns out to be the best in the set. We think this is a correct and important framework to understand how scientists track explanations that are (most likely) representationally accurate. If our arguments are on the right track, however, this cannot be the whole story about truth-seeking, because what matters for understanding is not only the representational accuracy of the representational systems that we deploy, but also the amount of truth that one is in the position to extract from accurate or even inaccurate representational systems.

4.2 Reasonable Endorsement

We have been working with the implicit assumption that for S to understand P via T, S must entertain a doxastic or noetic attitude of some sort towards T. We briefly mentioned that S must endorse T or commit herself to T.¹¹ We have not yet said anything, however, about the normativity of such endorsement involved in understanding. When is a scientist’s endorsement of a theory justified, or warranted? When is it rational for a scientist to endorse a theory?

Many authors point out that for a subject to be justified in endorsing a theory (or an explanation), such a theory must turn out to be the best of (or “the winner” among) all available alternatives. Dellsén’s optimality model of understanding (Dellsén 2021), Elgin’s reflective equilibrium model (Elgin 2017), and Khalifa’s SEEing model (Khalifa 2017), for example, rest on this idea. It should be noted, however, that this normative principle admits of (at least) two readings, depending on how one chooses to spell out the phrase “the best of all available alternatives”. On what we might call the subjective reading, for a subject to be justified in endorsing a theory T about P, T must turn out to be the best of all the theories about P that the subject took into consideration (and ruled out on evidence-based grounds). On the objective reading, on the other hand, for a subject to be justified in endorsing a theory T about a phenomenon P, T must turn out to be the best of all the alternatives available in the epistemic environment of the subject—regardless of whether the subject actually considered these alternatives or not.

To better grasp the difference between these two readings, imagine the following scenario. It is January 2020. A GP is trying to understand the condition of a patient showing light flu symptoms: cough, sore throat, runny nose. The GP does everything in her ken to make the correct diagnosis (e.g.: she considers the medical history of the patient with great care; she rules out pneumonia after finding out that the patient has no fever; she rules out bronchitis after checking the condition of the patient’s lungs; and so on). The result of her assessment is that the patient has a common seasonal cold she should not really worry about. The diagnosis is correct. However, the very same symptoms could have very easily signalized a much more dangerous health problem: a COVID-19 infection. The WHO sent an emergency alert about this possibility to all registered physicians a couple of hours before the GP made her diagnosis, but the GP did not have access to her email account because of a problem with her Wi-Fi. Now, was the GP justified in assuming that her patient had a seasonal cold (before reading the WHO alert)? The answer here is probably both yes and no. Subjectively, that is, given her best effort and her best judgment, probably yes; objectively, that is, given the objective probability that her diagnosis was the correct one, probably no. The seasonal cold was the best alternative among the alternatives that she considered, but not among all alternatives available in her epistemic environment.

We suggest that in the perfect epistemic situation of ideal understanding that we are describing, the subjective will match the objective; that is: the scientist’s subjective confidence that the theory accounting for P is true will match the theory’s real, objective confirmation. In other words: a scientist understanding a phenomenon P via a theory T (about P) to the highest conceivable extent will be both subjectively and objectively justified in endorsing T.

Within de Regt’s theory of understanding, a scientific theory must be reliably empirically successful to provide a scientist with understanding (de Regt and Gijsbers 2017: 55; see also de Regt 2017: 119). We agree with de Regt on this point. We take this requirement to hold for understanding generally, and to hold a fortiori for ideal understanding. We demand that an “understander” of P be able to successfully predict P’s occurrence and P’s future development. (We probably would not say that one understands the phenomenon of climate change if one predicted that temperature on earth will decrease soon, while instead it ends up increasing.) Moreover, we demand that an understander of P be able to successfully manipulate P’s domain as to influence in some way P’s occurrence or its development. (We probably would not say that a scientist understands the spread of COVID-19 if she devised measures meant to reduce the spread of the virus that end up fostering it instead.) However, we want to point out that the reason why a theory’s empirical success is relevant for understanding is that a theory’s empirical success provides scientists with good (despite prima facie and defeasible) epistemic reasons to endorse the theory. In other words, a theory’s empirical success contributes to the scientist’s justification in endorsing the theory.¹²

4.3 Fit into the Noetic System of the Scientist

“Ontological principles are the basis of intelligibility in any account of reality; the denial of an ontological principle strikes one as more nonsensical than false,” writes Chang (2001), and we want to take the core of this idea to develop our final necessary criterion of ideal understanding: the theory T must also “make sense” relative to or fit to the best possible extent into the scientist’s already established noetic system.

A couple of clarifications are in order here. First, we suggest calling “noetic system” the set of informational units that a particular subject believes, accepts, or endorses at a certain moment in time.¹³ Some of these informational units will be true, some will be false; some will be close enough to the truth to serve a particular epistemic, cognitive, or practical aim. The informational units belonging to a noetic system will not be isolated, that is, they will not form a long conjunction. Rather, they will depend upon one another in many ways (logically, semantically, explanatorily, probabilistically, epistemically, and so on). A noetic system, thus, can be represented as a structured set, and it shows two dimensions: an informational and a relational one (see Malfatti 2021; Schurz & Lambert 1994).

But what does “fitting” exactly require? What does it take for a theory to fit into a scientist’s noetic system? Fitting, as we conceive it, involves a negative and a positive requirement. The negative requirement amounts to the following: how well a theory T fits into a noetic system W is inversely proportional to the number of contradictions and/or cognitive dissonances arising from the conjunction between T and W (or from the incorporation of T in W). So, the more contradictions and/or cognitive dissonances generated by the conjunction of T and W, the poorer the fitting of T into W. Absence of contradictions and/or cognitive dissonances is not enough for fitting, however (as Bartelborth 1999, among others, has argued). Suppose a scientist endorses a theory T that is isolated in her noetic system, and that shows very little positive connection at all with other theories “in the neighborhood” that bear on it. We would probably hardly say that the theory “makes sense” relative to or fits well into the scientist’s noetic system. So, we suggest that how well a theory T fits into a noetic system W will be directly proportional to the number (and strength) of connections bounding T to other theories in its neighborhood and that are relevant to it.

We will not make heavy weather of the claim that, at least typically, resolving contradictions, ironing out cognitive dissonances and enhancing the systematicity of one’s noetic system improve one’s epistemic standing and advance one’s understanding of reality. Of course, specifying exactly which theories and assumptions “in the neighborhood” or “in the background” are relevant for a certain theory (and for understanding phenomena via the theory) will not always be a straightforward matter. It is quite straightforward, however, to assume that whatever scientific theory we formulate for empirical phenomena cannot contradict and must be somehow (explanatorily) connected to our “immanent picture” of the world. In other words, such a theory cannot contradict the deliverances of our perception, and in case it seems to do so, it must have the resources to explain why humans perceive things as they do, given that at a fundamental level, reality is very different from how it appears. It is also plausible that whatever scientific theory we reasonably endorse will be “in equilibrium” (and ideally also properly connected) with our more general metaphysical assumptions about reality. These metaphysical assumptions are partially derived from everyday experiences, but sometimes derive from the historical development of science.

To substantiate our claims, we will now discuss two historical cases in physics that show that a theory needs to fit into a scientist’s noetic system to provide her with understanding. We take this requirement to hold for understanding generally, and to hold a fortiori for ideal understanding. Of course, fitting is not an all-or-nothing matter; it admits of degrees, as understanding does. So, we suggest that for a theory to provide a scientist with the highest conceivable degree of understanding, the theory must fit to the greatest possible extent into the scientist’s noetic system. Lesser degrees of understanding, on the other hand, will be compatible with lesser degrees of fitting.

5.1 Grasping

The talk of “grasping” has become a commonplace in the literature on understanding. Prominent authors argue that whatever understanding is, it crucially involves grasping.¹⁸ Given that this is supposed to hold for degrees of understanding and for understanding generally, it might seem prima facie strange or suspect that grasping has no place whatsoever in our account of ideal understanding. We think, however, that this worry is misguided.

Exactly what is grasping? What does it mean to grasp something—for example, a phenomenon, fact or representational system? There is no agreement in the current literature on how these questions should be answered.

Some authors link grasping to the specific phenomenology of understanding. It is tempting to describe the experience of coming to understand by saying that when understanding succeeds, we “see” or “grasp” something, for example, how things fit together. In other words, there is a way it feels like, when we come to understand—and “grasping”, or so it is argued, is an effective way to describe this process from the first-person perspective. If this explication is correct, grasping might or might not be involved in ideal understanding, but this question is simply uninteresting for us, as it lies outside the scope of epistemology and philosophy of science.

Other authors defend the view that the notion of grasping performs an essential explanatory role in any theory of understanding. Consider the following. We can truly believe, maybe even know, a great deal about something, and yet fail to understand it. Why? How is this possible? The notion of grasping is typically introduced to account for this fact. Understanding, or so it is argued, is not a matter of simply assenting to information or propositions. We can assent to a certain piece of information, we might even have excellent reasons to believe that it is true, while it remains completely “inert” in our noetic system. When, in contrast, we grasp a piece of information, or so it is argued, this becomes like an instrument that we can use—for example, as a basis for inference, explanation, prediction, action (Grimm 2011; 2021; Hills 2016). Grasping, thus, seems to enable one to fill the gap between the theoretical and the practical aspect of understanding.

We think that the ability to use a certain piece of information for a variety of cognitive and practical purposes is indeed an important aspect of understanding. We do think that understanders (and of course, most prominently, ideal understanders) are not successful just theoretically, but also in the practical domain. We just do not think that one needs to postulate a mysterious mental act of grasping to do justice to this aspect of understanding. It is among other things because we felt the pressure of filling the gap between the theoretical and the practical aspect of understanding that we followed de Regt’s path in introducing an intelligibility requirement for understanding. Given that we understand a certain theory or representational system, it is quite natural that we will develop a set of abilities to explain, draw inference, make predictions, maybe even act. But then, the talk of “grasping” turns out to be unnecessary and redundant, as everything epistemologically significant about grasping can be expressed by using more familiar and less controversial epistemological notions. (See also Khalifa 2017, 14 and 79, for a similar reductionist proposal.)

5.2 Novel Predictions

We have claimed that for a theory to provide scientists with ideal understanding, it must work as a reliable basis for action and prediction. In other words, it must be reliably empirically successful. We have also pointed out that the empirical success of a theory contributes to understanding somewhat indirectly, by providing scientists with good reasons to endorse the theory. But as we are concerned with the highest conceivable form of understanding, why not require more than mere reliability in accommodating known phenomena? Why not require the highest conceivable form of empirical confirmation, for example, the one provided by novel predictions? Note that to count as “novel”, a theory’s prediction must be, among other things, informative and a priori improbable (Alai 2014).¹⁹ Suppose that an astronomical theory predicts that “there is an unknown planet somewhere in the universe”. Such an existential claim is so vague and scarcely informative that it is extremely likely to come out true—and this a priori, that is, independently from our particular prior odds. If it does indeed come out true, the degree of confirmation it will provide to the theory from which it follows will thus be very low. The situation would be very different if the theory succeeded in predicting that “there is an unknown planet with mass x with orbit y in the solar system”. Such a prediction would be highly confirming, given how informative it is and how unlikely it is to come out true.

While novel predictions are certainly crucial for scientific progress, we do not think that they play a crucial role in understanding, nor do we think that we should require from a theory providing scientists with ideal understanding to enable novel predictions. To see this, consider the following scenario.²⁰ Suppose that we unify quantum field theory and general relativity into an astonishingly accurate theory T with 23 fundamental constants. T enables us to make multiple novel predictions, so we are very confident that the theory is correct. Moreover, T satisfies all our other criteria for ideal understanding. Then, 1,000 years later, scientists formulate a new theory, T*, which is empirically equivalent to T, but is much simpler, as it has only 3 fundamental constants. T* will arguably make no novel prediction: it will predict exactly what we already expected. And yet, it would be very odd to say that this theory will not provide scientists with better understanding.

5.3 Transparency

Linda Zagzebski famously argues that transparency is the hallmark of understanding. Transparency, in Zagzebski’s view, is also what makes understanding different from knowledge. She writes:

Understanding has internalist conditions for success, whereas knowledge does not. Even when knowledge is defined as justified true belief and justification is construed internalistically, the truth condition for knowledge makes it fundamentally a concept whose application cannot be demonstrated from the inside. Understanding, in contrast, not only has internally accessible criteria, but it is a state that is constituted by a type of conscious transparency. It may be possible to know without knowing that one knows, but it is impossible to understand without understanding that one understands. [. . .] [U]nderstanding is a state in which I am directly aware of the object of my understanding, and conscious transparency is a criterion for understanding. (Zagzebski 2001: 246–47)

Is ideal understanding transparent? There are actually two questions to be tackled here:

• (i)  When a scientist S ideally understands a phenomenon P, is S always also aware that this is the case?

• (ii)  When a scientist S takes him or herself to understand a phenomenon P ideally, does this mean that S also actually understands P in the ideal sense?

We think that both questions should be answered in the negative. It has been widely acknowledged in the literature in epistemology and philosophy of science that agents are not infallible in self-reflection. That is, they are not infallible in their evaluation of their own cognitive states. Think of the so-called “illusions of understanding” (Rozenblit & Keil 2002; Ylikovski 2009). Very often, we might think we understand, while in reality, we do not. Very often, we think to have identified a causal pattern in the real world, while actually, the only pattern we “see” is one within our representations of the world. This holds for understanding generally, and there is no reason to think that ideal understanding will be any different. And on the other hand, it is plausible to assume that agents might think that there is still work to be done to reach ideal understanding, while actually, the top has already been reached. This is because ideal understanding requires a very high degree of justification, but not certainty.

All things considered, then, we have reason to think that ideal understanding is not transparent. Moreover, we would like to point out that transparency might not even be desirable for understanding. If every time we reached understanding (ideal or non-ideal) we were also infallibly aware and certain of this, we would consider things settled and we would tend to stop inquiring. We would lose the flexibility to revise and rearrange our views. We would tend to disregard alternative perspectives and run the risk of becoming closed-minded. In doing so, we might miss opportunities for epistemic improvement and self-correction. Uncertainty, on the other hand, makes us fallibilist and epistemically humble. And as fallible and epistemically humble human beings, we are never certain that our views are correct. Therefore, we keep questioning, analyzing, testing, and challenging our theories. And it is by doing so, by being open to the possibility that, despite our best efforts, we might be fundamentally wrong, that we foster the advancement of science.²¹ As John Stuart Mill puts it, in his famous essay On Liberty:

If even the Newtonian philosophy were not permitted to be questioned, mankind could not feel as complete assurance of its truth as they now do. The beliefs which we have most warrant for, have no safeguard to rest on, but a standing invitation to the whole world to prove them unfounded. (Mill 1859: 23)