Reasoning with visual metaphors

Research into visual reasoning up to now has focused on images that are literal depictions of their objects. I argue in this article that an important further mode of visual reasoning operates on images that depict objects metaphorically. Such images form part of the class of expressive symbols: they are found, for example, in allegorical representations in works of visual art, studied by iconology. They were also a common way of encapsulating insights about the universe in natural philosophy in the Renaissance. Many writers assume that expressive symbols have vanished from modern science, but I argue in the second part of the article that mathematical law statements in present-day physics should be seen, in part, as images that constitute expressive symbols of the world. In support of this view, I offer evidence that law statements relate to their objects metaphorically and that physicists engage with them primarily through visual inspection and visual reasoning. REASONING WITH VISUAL METAPHORS 2 1 Visual reasoning with literal depictions This article has two aims. The first is to broaden the scope of discussions of visual reasoning by arguing that visual reasoning operates on a larger class of images than hitherto acknowledged. Whereas discussions of visual reasoning have so far focused on images taken to be literal depictions of the objects of the reasoning, I will seek to extend the domain to cover images that constitute—for want of a better word at this stage—metaphorical depictions of objects. I will proceed partly by bringing into contact bodies of literature that are usually kept apart. The second aim, to which I will turn in section 4, is to argue that this account applies also to law statements in present-day sciences such as physics. I will argue both that law statements partly constitute metaphorical depictions of their objects, and that physicists achieve understanding of them mainly by visual inspection and visual reasoning. Research into visual reasoning by cognitive scientists, philosophers, and others has proceeded along various lines. The ground has been prepared by work seeking to revalue visual thinking by comparison with propositional thinking (Arnheim 1969), analyses of the roles of illustrations in science (Mazzolini 1993), and discussions of the similarities and intersections of science and visual art (Baigrie 1996; Gooding 2004). Further research has focused on the use of images in concrete media, such as diagrams on paper. Topics of this research have included the epistemology of diagrams in mathematical arguments (Brown 1999; Giaquinto 2007) and graphical modelling techniques in physics, such as Feynman diagrams (Kaiser 2005; Meynell 2008). A parallel line of research has studied the apprehension and manipulation of images conceived in the mind. Research in this vein has studied, among other phenomena, the mental rotation of representations to ascertain whether a given object is congruent to another (Shepard and Metzler REASONING WITH VISUAL METAPHORS 3 1971; Shepard and Cooper 1982), the mental scanning of representations to compare spatially separated properties of objects (Kosslyn 1980), visualizations of the dynamics of physical systems, such as projectile motion (Gentner and Stevens 1983), the manipulation of mental images in thought experiments (Gendler 2004), and visual thinking in the design of technical artefacts (Ferguson 1992). These otherwise diverse lines of research share a notable feature: almost without exception, they discuss images taken to be literal depictions of their objects. This assumption holds on two levels: researchers portray the cognitive agents whose visual reasoning they model as conceiving images to be literal depictions of objects, but they also tacitly endorse the conception of those images as literal depictions, building the property of literalness into their accounts of visual reasoning. “Literalness” is notoriously hard to define. If one accepts a resemblance theory of representation, one is inclined to think that a literal representation shows the highest possible degree of resemblance to its object: this degree is obtained when the representation is isomorphic to its object, i.e. when there is a one-to-one correspondence between elements of the object and elements of the representation. However, this proposal fails to take sufficiently into account the fact that even what we call a literal depiction of an object exhibits a degree of simplification, abstraction, and idealization: indeed, insofar as a depiction differs from its object in any respect, it may be regarded as simplifying, abstracting from, and idealizing it. An improved proposal, therefore, is to say that a depiction is literal if it can be smoothly and cumulatively transformed into another depiction that has any desired degree of isomorphism to its object. A map is a literal depiction of a landscape in this sense: whereas any usable map incorporates substantial simplification, abstraction, and idealization, we can smoothly and cumulatively augment a REASONING WITH VISUAL METAPHORS 4 map by increasing the scale and inserting detail to achieve as high a degree of isomorphism to the landscape as we wish. The images studied up to now in research into visual reasoning share this property of literalness. Take as an example the mental images used in classic thought experiments in physics, such as Galileo’s tower, Newton’s bucket, Maxwell’s demon, Einstein’s lift, Heisenberg’s gamma-ray microscope, and Schrödinger’s cat (Brown 1991): each of these images is presented as a literal depiction of a physical effect. The effect may be encountered in the actual world (free fall) or be deemed impossible (a demon sorting molecules by velocity), and the image may depict familiar articles (a bucket) or imaginary machinery (a lift falling freely in empty space). Nonetheless, the image in each case is intended to model a physical effect literally, thereby supporting the inference that some relation holds or does not hold in the actual world. Thought experiments of this kind owe their evidential value to the literalness of the images that they employ. Some other classes of depiction discussed in research into visual reasoning count as literal on the grounds that, rather than reproducing preexisting objects, they call objects into being. A diagram used in a visual proof of Pythagoras’s theorem, for example, creates a particular geometrical construction—or, for some Platonists, picks out one element from the plenitude of all existing mathematical entities—and thus is necessarily isomorphic to the object that it is intended to depict. Similarly, when an engineer conceives a mental image of an artefact that he or she is designing, that image is isomorphic to the intended object by virtue not of faithfully reproducing it, but of being a mental blueprint of it. Even writers who differ on other fundamental issues pertaining to visual images seem to agree not to widen the scope of the discussion beyond images that constitute literal depictions. For example, both sides in the soREASONING WITH VISUAL METAPHORS 5 called analog–propositional debate about mental imagery accept this restriction. This debate turns on the question whether mental representations that we experience as imagery have intrinsically spatial properties, like pictures, or lack such properties, and thus resemble more closely linguistic descriptions of visual scenes (Kosslyn et al. 1979; Block 1981). Broadening the scope to encompass metaphorical depictions would seem likely to alter materially the terms of this debate. While research into visual reasoning has placed the category of the literal at the centre of discussion, other academic disciplines concerned with representation attribute less importance to it. Researchers in linguistics, literary studies, rhetoric, and allied disciplines call into question the assumption that there is a sharp dividing line between literal and non-literal depictions. Many researchers in these disciplines take the concept of metaphor as the crucial organizing principle: they devote attention to the seeming ubiquity of metaphor, the hypothesis that all meaning is metaphorical, and the roles of metaphor in cognition (Black 1962; Shibles 1971; Lakoff and Johnson 1980; Ortony 1993). Philosophers and historians of science too have increasingly acknowledged the use of metaphor in scientific discourse and practice, especially in the formation of new concepts and the construction of models and explanations (Hesse 1966; Bailer-Jones 2002), and particularly in sciences such as psychology (Leary 1990). Feminist scholars in science studies have shown the role of scientific metaphors in maintaining gender hierarchies (Keller 1995). These lines of research into metaphor are partial in another way, however: the metaphors that they study are conceptual rather than visual. The metaphorical depiction of the brain as a computer, for example, which became popular in the 1970s, was intended to suggest conceptualizations of the brain as an information-processing device, but did not consist primarily of visual depictions or involve visual reasoning. REASONING WITH VISUAL METAPHORS 6 Yet another group of academic disciplines addresses visual metaphors. The branch of art history and criticism known as iconology has flourished since the 1930s (Panofsky 1939; Gombrich 1965, 1972). Iconology studies the meanings of allegorical symbols in works of visual art, such as the personifications of moral virtues and human attributes found in ancient, Renaissance, and Baroque painting and sculpture. It also studies emblems as encapsulations of wisdom (Manning 2004). Symbols of these kinds constitute visual metaphors of their referents. This brief and selective survey reveals the existence of at least three distinct research programmes: an intensive investigation of visual reasoning restricted to images taken to be literal depictions of objects, an expl