The aesthetic theoretical virtues possess an aesthetic shape (fittingness) that is qualitatively different from the logical-conceptual fit of the coherential virtues. Scholars in many fields sometimes appeal to the aesthetic properties of theories in their case for accepting such theories. However, the epistemic status of the aesthetic virtues has been challenged more than that of any other virtues. Besides beauty and simplicity, which are the most frequently purported aesthetic theoretical virtues (Mackonis 2013, p. 979), I will argue that a certain sense of unification also belongs to this aesthetic class of virtues. Unification has more widely recognized intrinsic epistemic value (recall Kitcher’s unificationist account of explanation in Sect. 3.4) than either beauty or simplicity. We will also explore the possibility of finding an epistemic role for beauty by arguing descriptively from the perceptions of some prominent scientists and from the contention that simplicity and unification are epistemically enhanced special cases of beauty. Lipton (2004, p. 68) came close to recognizing this aesthetic class of theoretical virtues when he wrote in regard to his theory of inference:
Moreover, if we do end up selecting Inference to the Best Explanation, it will not simply be because it seems the likeliest explanation, but because it has the features of unification, elegance, and simplicity that make it the loveliest explanation of our inductive behavior.
Beauty as the most general aesthetic theoretical virtue (TV7)
According to my account of aesthetic theoretical virtues, a beautiful theory evokes aesthetic pleasure in properly functioning and sufficiently informed persons (with some degree of cultural and individual variation of aesthetic experience). The properties of theories and mathematical proofs that are among those factors that trigger the experience of beauty include symmetry, aptness (McAllister 1996, pp. 172–173) and surprising inevitability (Montano 2014, pp. 34–36).
In contrast, aesthetic relativism suggests that no judgments about beauty or ugliness (whether in regard to a theory or anything else) are more correct than others. If this view of aesthetics were correct, then it is difficult to see how any aesthetic theoretical virtues could be of rational importance in the theory choice. However, there are good reasons to reject aesthetic relativism. We often make aesthetic judgments and take them to be at least approximately correct, especially as we mature as persons. So aesthetic relativism is out of step with common practice. Moreover, Zangwill (2014) notes that
... one can virtually always catch the professed relativist about judgments of beauty making and acting on non-relative judgments of beauty—for example, in their judgments about music, nature and everyday household objects. Relativists do not practice what they preach.
Zangwill (2014) pinpoints another disturbing feature of aesthetic relativism.
For if “it’s all relative” and no judgment is better than any other, then relativists put their judgments wholly beyond criticism, and they cannot err. Only those who think that there is a right and wrong in judgment can modestly admit that they might be wrong. What looks like an ideology of tolerance is, in fact, the very opposite. Thus relativism is hypocritical and it is intolerant.
Due to a lack of space, we will explore only a few of the many perspectives on aesthetics that are relevant to theoretical virtues. Other views that we will not address include Breitenbach’s (2013) Kantian theory, McAllister’s (1996) largely projectivist account, and Montano’s (2013) major revision of McAllister. Subsequent work could explore how such various understandings of aesthetics might bear upon my taxonomy of theoretical virtues. This endeavor might include better grasping the demarcation of aesthetic properties from non-aesthetic properties. But, as Levinson (2003, p. 12) observes: “It is widely agreed that aesthetic properties are perceptual properties, dependent on lower level perceptual properties, directly experienced rather than inferred, and linked in some way to the aesthetic value of the objects possessing them.” He also identifies substantial agreement on clear examples of aesthetic properties. Note how he begins his “open-ended” list with “beauty” (and its opposite ugliness), and notice also the lurking presence of both simplicity and unification highlighted by my italics.
... beauty, ugliness, sublimity, grace [simplicity or refinement], elegance, delicacy, harmony, balance, unity, power, drive, élan, ebullience, wittiness, vehemence, garishness, gaudiness, acerbity, anguish, sadness, tranquility, cheerfulness, crudity, serenity, wiriness, comicality, flamboyance, languor, melancholy, sentimentality—bearing in mind, of course, that many of the properties on this list are aesthetic properties only when the terms designating them are understood figuratively (p. 6).
Benovsky (2013) argues that all aesthetic evaluative properties of theories (his terminology for aesthetic theoretical virtues) are grounded in the non-aesthetic evaluative properties of theories such as internal consistency, internal coherence (intuitive plausibility), universal coherence, explanatory power, and simplicity. The first three non-aesthetic evaluative properties listed here are the three coherential virtues in the previous section of my essay. Explanatory power is an ambiguous term that often refers to various non-predictive theoretical virtues, but that most often refers to the evidential virtues, especially causal adequacy and explanatory depth—the two evidential virtues that go beyond mere evidential fit. Simplicity is difficult to characterize. Many, including myself for reasons I will articulate, classify it primarily as an aesthetic property of theories, rather than as a non-aesthetic evaluative property as does Benovsky. However, according to my account, persons make partially aesthetic judgments when they differentially weight, or aptly balance, all of the theoretical virtues in particular episodes of theory choice and theory refinement. Indeed, aptness and balance are aesthetic properties. Non-aesthetic considerations also figure into the differential weighting of the virtues in theory choice, but this is beyond the scope of the present study.
Scientists, mathematicians, philosophers, and other scholars sometimes appeal to their aesthetic engagement with theories—experienced most generally as an encounter with beauty—in order to help justify theory choice. Although this descriptive argument for the virtuousness of theoretical beauty is weak on its own, it is worth consideration because later I will argue that simplicity and unification are special cases of beauty that more likely carry at least some epistemic weight. Dirac, one of the founders of quantum mechanics, celebrated general relativity’s elegance.
The foundations of the theory are, I believe, stronger than what one could get simply from the support of experimental evidence. The real foundations come from the great beauty of the theory.... It is the essential beauty of the theory which I feel is the real reason for believing in it (1980, p. 10).
McCartney expresses the affinities of the three major aesthetic theoretical virtues as seen in physics.
Anyone who has dealt with Maxwell’s equations of electromagnetism, or the Schrödinger equation, cannot fail to be impressed with the concise elegance of the formulae. Their compact beauty is made all the more stark when one considers the very broad range of phenomena they ultimately explain and encapsulate (2015, p. 3).
Beauty and (as I shall argue) its two most epistemically important special cases of simplicity and unification are vividly displayed here in their mutual relations. Educated contemplation of the mathematical formulae of physics occasions an encounter with beauty—an experience that is partially describable in terms of both the compactness (simplicity) of the formulae, and the remarkable range of natural phenomena they explain (unification).
Although generic theoretical beauty might have no intrinsic epistemic value, it more likely possesses extrinsic epistemic value to the extent that such a general aesthetic experience inclines researchers toward recognizing and cultivating simplicity and unification as special kinds of beauty that are more epistemically relevant in theory choice. Mackonis (2013, p. 979) remarks regarding “the meaning of beauty or elegance” in theory choice: “simplicity ... is both one of the most often cited explanatory virtues and one of the most often cited features of beauty.” Let us explore this further.
Simplicity (TV8) and unification (TV9) as specific complementary aesthetic theoretical virtues
Now that we have an introductory grasp of beauty in theories as a general aesthetic virtue, we are ready to characterize the specific aesthetic virtues of simplicity and unification as special cases of beauty that are particularly important for theory choice and theory refinement. A theory that exhibits simplicity explains the same facts as rival theories, but with less theoretical content. A unified theory, however, is one that explains more kinds of facts than rival theories with the same amount of theoretical content (Thagard 1978). Simplicity and unification address the same thing, the style of informativeness, from opposite complementary orientations. Simplicity is increased informativeness by means of a comparative reduction (relative to rival theories) of theoretical content. Unification is increased informativeness by means of a comparative increase in the different kinds of data that get explained. A theory can be evaluated (compared to rivals) for its informativeness in proportion to its theoretical content in both stylistic directions—simplicity and unification.Footnote 8 Such comparative evaluations may be difficult to characterize formally due to their subtlety. However, in the case of simplicity, “less theoretical content” means, roughly, fewer entities postulated by the theory (often called parsimony or ontological simplicity) and/or fewer or more concise basic theoretical principles (often called elegance or syntactic simplicity). Note the aesthetic term “elegance,” which reflects simplicity’s aesthetic character.
In what sense are the complementary virtues of simplicity and unification (possibly) intrinsically epistemic in an aesthetic manner? Lipton (2004, p. 124) remarked in connection with our inferential preference for simple and unified explanations: “if the world is a chaotic, disunified place, then I would say it is less comprehensible than if it is simple and unified. Some possible worlds make more sense than others.” The comprehensibility of the world (which supports the epistemic aims of intelligent beings living in such a world) has multiple aesthetic dimensions, including most prominently, simplicity and unification. We prefer to live in a comprehensible world because this is more pleasing to the mind. The history of science attests to this aesthetic preoccupation and its respectable track record in scientific discovery (Glynn 2010). In such a highly comprehensible world, theories ranking higher in simplicity and unification are more likely to be approximately true than rivals that rank lower (the opposite would be true in many more other possible worlds). There is nothing necessary about the existence of a world characterized by simplicity and unification, but our minds are preoccupied by the desire to find such a simple unified world. This aesthetic orientation operates in the broader culture. Simplicity and unification are among the aesthetic qualities celebrated in literature, film, and many other cultural arenas. Humans highly value a story that compactly instantiates a plot (simplicity) while expressing the voice of human experience in a broadly significant manner (unification). Moreover, historians craft stories that likely resemble “what happened” to the degree that they possess many of the theory virtues.
Many versions of simplicity have been identified and they include factors such as a theory that: postulates fewer entities, postulates fewer kinds of entities, raises fewer additional explanatory questions, and posits fewer primitive explanatory ideas (Beebe 2009). Swinburne (1997) adds: postulates fewer laws and postulates laws relating fewer variables. McAllister (1996) observes that this multiplicity of simplicity criteria—many of which might be relevant to a particular theoretical dispute—defies complete reduction to quantified evaluative procedures such as those Sober (2015) and Kelly (2011, 2016) propose.
Sober (2015) recommends simplicity as a consideration in theory choice if it is understood as the number of adjustable model parameters as treated within either the Bayesian or frequentist philosophical traditions of probability theory. He refers to such narrowed conceptions of simplicity as “parsimony” and he illustrates their epistemic value (and instances where they do not apply) in case studies spanning evolutionary biology, psychology, and philosophy. “I am a reductionist about parsimony,” he announces. In short, he considers the aesthetic value of simplicity to be merely subjective and irrelevant to the epistemic value of simplicity.
Sober aims to show that simplicity helps scholars to infer which explanation is best, even if reality is not structured by beautiful simple principles. Several (but not all, says Sober) of the mathematically rigorous treatments of simplicity “agree that parsimony, as measured by the number of adjustable parameters in a model, is relevant to making ... estimates” of either a model’s predictive accuracy or its likelihood (p. 141). In either case simplicity “is not a subjective aesthetic frill. It has an objective epistemic status” (p. 147). While tentatively applauding with Sober these achievements of probability theory, we should remember, as Sober tacitly concedes, that there might also be epistemic-by-means-of-aesthetic value in theoretical simplicity that is not addressed by mathematical analysis. But due to Sober’s reductionist agenda, he dismisses as “subjective aesthetic frill” (sounding like an aesthetic relativist) the purportedly larger aesthetic dimension of rational theory choice (in simplicity and beyond) that has been proclaimed by many influential scientists, including Einstein and Dirac.
Steel (2010, p. 19) notes that, if Sober or Kelly is correct, then simplicity is an extrinsic epistemic value “even if it is not an intrinsic one” (supposing like Sober that its intrinsic epistemic value would require that the world itself is dominated by simplicity; note how DeLancey disputes this assumption below). A theoretical virtue has extrinsic epistemic value, Steel argues, if it promotes the attainment of truth without itself being an indicator or requirement of truth. We have already seen how Sober made such a case for the extrinsic epistemic value of simplicity. Steel summarizes Kelly’s (2011, 2016) alternative to Sober:
A second account of simplicity explains how a preference for simpler hypotheses promotes efficient convergence to the truth, where efficiency is understood in terms of minimizing the maximum number of times the investigator can switch from conjecturing one hypothesis to another.... Both of these accounts, then, defend simplicity as an extrinsic epistemic value. In both cases, a preference for simpler hypotheses is argued to promote the attainment of truth (either of approximately true predictions or efficient convergence to true hypotheses), yet neither approach presumes that the world is simple.
Forster and Sober (1994, p. 14) treat unification’s epistemic role in theory choice in a manner that is similar to their probabilistic analysis of simplicity.
We conclude that estimated accuracy explains why a unified model is (sometimes) preferable to its disunified competitors. At least for cases that can be analyzed in the way just described, it is gratuitous to invoke unification as a sui generis constraint on theorizing.
Forster and Sober concede here that many instances of unification are not captured by mathematical model selection procedures that might have reduced the rationality of favoring more unified theories to merely their higher predictive accuracy estimates. So Forster and Sober concede that not all cases of simplicity and unification are reducible to non-aesthetic predictive accuracy or likelihood. They leave open the possibility that there are irreducibly aesthetic aspects to simplicity and unification that rationally allure us to theories that possess them. Indeed, many scientists have believed that aesthetic properties have a legitimate modest role in rational theory choice, and such belief has guided successful scientific practice in some cases (Glynn 2010). Although the aesthetic branch of my taxonomic argument does not resolve this issue, let us now explore additional pointers that might help guide work in this area.
DeLancey (2011) argues that the simplicity criterion in theory choice could operate reasonably whether or not the world is elegantly simple (and he does so in a manner that differs from Sober and Kelly). His case comes with an ontological stipulation that he thinks is inoffensive to most antirealists: “a minimal realism about the existence of objects and laws, in order to allow that the descriptions of the relevant phenomena contain patterns.” After arguing that we have no reason to expect that the world’s potential complexity has any definite upper limit, DeLancey maintains that “simple” is a relative term, like “small.” So, for any magnitude of complexity “it’s reasonable to call it ‘simple,’ since there are infinitely many alternatives that are more complex” (p. 94). This means that simplicity only operates comparatively with respect to rival theories, which is how I have characterized it. Huemer (2009, p. 219) offers a similar account of simplicity (alongside three other accounts).
The boundary asymmetry account starts from the observation that there is a lower bound but no upper bound to the degree of complexity a theory can have. That is, for any given phenomenon, there is a simplest theory (allowing ties for simplest), but no most complex theory of the phenomenon: however complex a theory is, it is always possible to devise a more complicated one. This is most easily seen if we take a theory’s complexity to be measured by the number of entities that it posits: one cannot posit fewer than zero entities, but for any number n, one could posit more than n entities. Similar points hold for other measures of complexity, such as the number of parameters in an equation.
The boundary asymmetry analysis of simplicity—and its stated rarity of theoretical simplicity compared to theoretical complexity—is supportive of simplicity’s epistemic value (in contrast to the epistemically vicious greater complexity of indefinitely numerous rival theories). This contention stems, in part, from the observation that rarity is a common trait of epistemic and aesthetic value. The analytic project of identifying the necessary and sufficient conditions for a belief to count as knowledge has exposed countless ways for beliefs to fall short of knowledge, and comparatively few ways for beliefs to constitute knowledge. Rarity also is a prominent trait of aesthetic value. Beauty requires a kind of coordination (among musical notes, words on a page, or brush strokes on a canvass) that is rare compared to the vast number of possibilities for disharmony. In sum, DeLancey’s and Huemer’s accounts of theoretical simplicity (both of which are independent of how elegantly simple the world may or may not be) are compatible with the possibility that the value of simplicity in theories consists of interlocked aesthetic-epistemic properties.
Simplicity, although primarily an aesthetic theoretical virtue, has secondary affinities to other virtue classes—we will cover the coherential affinity now and the evidential one in Sect. 7. When we explored internal coherence (TV5), we noted that a theory lacks internal coherence to the extent that it incorporates ad hoc hypotheses that are illegitimate in one or more respects, including insufficient precision and insufficient conceptual fit within the larger theory—often resulting in an awkwardly complex theoretical monstrosity. Note how internal coherence (lack of ad hocness) is similar to simplicity: one way for a theory to score low in simplicity is by the presence of ad hoc hypotheses. So simplicity overlaps with internal coherence. Of course, simplicity also decreases by the addition of non ad hoc hypotheses. So internal coherence and simplicity are quite different.
My account of the aesthetic virtues sits well within Zangwill’s aesthetic realism (2001, 2014). He argues that specific substantive aesthetic properties such as delicacy and dumpiness are ways of being beautiful or ugly—which he calls verdictive aesthetic properties. The latter refer to the degrees of overarching aesthetic merit (beauty) or demerit (ugliness). Adapting Zangwill’s metaphysics to the present study, the theoretical virtue of beauty is the aesthetic property that designates the overall aesthetic value of a theory as grounded in specific substantive aesthetic properties such as symmetry and aptness—which are some of the ways of being theoretically beautiful that may lack the epistemic potency of simplicity and unification. However, simplicity and unification are two substantive aesthetic properties of theories that stand out from the rest due to their interlocking complementarity and (according to many scientists) greater epistemic credentials compared to other substantive aesthetic properties of theories (e.g., symmetry and aptness).
Zangwill does not think that theoretical “beauty” in mathematics and science refers to actual aesthetic judgments, but this is a minor error in an otherwise insightful book (2001, pp. 140–142). He argues that when a scientist describes a theory as beautiful, that such language is metaphorical, and only means that the theory achieves the aim of explaining the data well. But there are counterexamples—beautiful hypotheses that did not explain well the relevant data. For example, Kepler toyed with multiple possible beautiful closed curved figures (geometrical hypotheses) to see which would best fit the astronomical data of planetary positions (he focused on Mars). He finally settled on the ellipse, which exhibits less simplicity than the traditional circular heavenly motions that even Copernicus had retained. Of course Kepler’s new astronomy exhibited heightened simplicity in other respects. Indeed, Kepler often appealed explicitly to the aesthetic properties of theories as partial grounds for their likely truth, a practice that made sense to him because God “introduced nothing into Nature without thoroughly foreseeing not only its necessity but its beauty and power to delight” (Kepler 1981, p. 55). Many other scientists and philosophers, holding a diversity of religious and non-religious views, also have treated beauty in scientific theories as truly aesthetic and epistemic (Breitenbach 2013).
Let us summarize and reinforce the justification for classifying unification as an aesthetic theoretical virtue alongside simplicity. As I have argued above, simplicity and unification address the same thing, style of informativeness, from opposite complementary orientations. Given that simplicity is the most often cited example of an aesthetic theoretical virtue, and given the widely recognized complementary affinity between simplicity and unification, it is plausible to consider unification to be an aesthetic theoretical virtue that assists (in a complementary fashion with simplicity) in rational theory choice. Given that unification is the centerpiece of one of the leading theories of explanation (as outlined earlier), what does this imply regarding unification constituting an aesthetic theoretical virtue?
A unificationist account of explanation would have us give ultimate priority to “using a few patterns of argument in the derivation of many beliefs” because we thereby “minimize the number of types of premises we must take as underived,” which reduces “the number of types of facts we must accept as brute” Kitcher (1981, p. 529) According to this account of explanation, unification is equated with explanation itself, rather than recognizing unification as an aesthetic component to what helps make an explanation more likely true. But if one attends carefully to Kitcher’s description of his own unificationist theory of explanation, some of his descriptive language has an aesthetic ring to it. Furthermore, the different strengths of each of the major theories of explanation (unificationist, causal–mechanical, and causal–counterfactual) might urge us to resist the temptation to reduce all explanation to any one of these explanatory approaches.
Because some scholars are attracted to a unificationist account of explanation (despite its greater weaknesses compared to the other two major accounts), this helps us to recognize unification as a theory virtue that might have some modest epistemic credentials. And given unification’s interlocking complementarity with simplicity (and the arguments for the quantitative correlation between certain facets of simplicity/unification and estimates of predictive accuracy, likelihood, or efficient convergence on truth), this offers some support for the epistemic standing of simplicity. Furthermore, the aesthetic theoretical virtue of beauty might be shown to have more than zero epistemic value if, as I have argued, unification and simplicity are the two main epistemically valuable ways of being beautiful. Even so, in general, the aesthetic virtues are widely considered to have less epistemic value (if any) than either of the first two classes of theoretical virtues: evidential and coherential. If my argument for the possible epistemic role of the aesthetic theoretical virtues lacks sufficient force to move certain readers, I would offer them this aesthetic taxonomic class as merely descriptive of some scientific practice (Glynn 2010)—and I would note how in Sect. 6.2.1 the diachronic sense of unification constitutes a mode of fruitfulness. Finally, even if some readers find my description of these virtues unhelpful, I would still recommend to them the rest of my taxonomy, which stands on its own even in the absence of the aesthetic class.
The fourth class of theoretical virtues possesses a distinctive temporal dimension that is missing in the three previous classes—evidential, coherential, and aesthetic. The diachronic virtues come last in my systematization because they cannot be instantiated in the initial framing of a theory, as is true of the theoretical virtues in the other three classes. Diachronic virtues require additional time after a theory is launched. Their time has come.