1. From embodiment to cognitive extension
1.8 Hidden drawings
that although both experts and novices draw diagrams to help them in their thinking, a striking feature of the expert is his or her “ability to represent a problem (often graphically) in such a way that the relation between the initial and final conditions is immediately evident”.
While Latour considers the larger implications of these smaller operations and
manipulations, Simon’s (1982:166) insight regarding the nature of expert skills suggests that these inscriptions are important because they allow “problem solution by recognition … The expert looks at an equation, notices a familiar feature that immediately activates a production, obtains a new equation, notices a third feature, which fires a final production and the equation is solved. Three acts of recognition, are required, each of which is based on stored memories, and can be achieved almost without conscious attention” (Simon, 1982: 166). The expert is thus depicted as using the external inscription in a tightly bound cognitive process. The external inscription shapes and transforms cognitive capacities by providing a different kind of functionality to internal vehicles.
Such operations also support the cognitive integrationist view that the manipulation of external vehicles is a prerequisite for higher cognition and that embodied cognition is a precondition for these manipulative abilities (Menary, 2010b: 232). The cognitive
integrationist view is a branch of the extended mind thesis, which highlights the cognitive role that external vehicles can play. Menary (2010b: 229) suggests that one way to better understand the nature of the integration between elements of a process such as the one described above “is to think of hybrid cognitive processes as enacted skills or capacities for manipulating the environment”. However he warns that “we should not forget that the embodied cognizer is embedded in a physical and social environment, and that
environment contains norms which determine the content of environmental vehicles and how we manipulate them” (2010: 229).
demonstrates that the use of different external formats is conditional on their geographic location, institutional validation and historical path.
Such restrictions are evidence that circumstances are indeed not superfluous to signs or unnecessary in analysis: rather they are “built into the act of use” (Sutton, 2004: 519).
Galison reveals the tension that exists between the de facto use of drawing by individual scientists and the intimidating power of institutional models and mathematical regimes which prescribe the form in which one’s thoughts should be articulated to the world – and drawing is typically proscribed, rather than prescribed.
What Galison (2000: 145) finds fascinating about Dirac is the apparent contradiction between how he presented his work and how he arrived at it. Dirac was viewed as the theorist’s theorist, recognised for the fundamental equation that today bears his name. (It describes the relativistic electron.) Dirac never used diagrams publicly. His books on general relativity and quantum mechanics contain not a single figure or diagram. He was known for his rigorous algebraic solutions and austerity of prose, never showing any schematic diagrams or visualisations in his papers. Galison (2000:146) was therefore astonished at what he found in the Dirac archives. In comments penned in preparation for a 1972 lecture, Dirac confesses: “I prefer the geometric method. Not mentioned in
published work because it is not easy to print diagrams”. This meant that Dirac has
published and worked on both sides of the divide of visualisation and formalism. This split has, for generations, riven both physics and mathematics (Galison, 2000: 146).
In an undated account Dirac explained that projective geometry had a strange beauty and power that fascinated him and had a “profound influence” on him. It gave results
“apparently by magic; theorems in Euclidian geometry which you have been worrying about for a long time drop out by the simplest possible means” under its power (cited in Galison, 2000: 147). Relativistic transformations of the mathematical quantities became surprisingly easy when the geometric reformulation was used. “My research work was based in pictures – I needed to visualise things – and projective geometry was often most useful” and further “when I came to publish the results I suppressed the projective
geometry as the results could be expressed more concisely in analytic from” (Dirac cited in Galison, 2000: 147).
Dirac’s suppressed geometrical work prompts Galison to consider the nature of scientific
grounded in spatial intuition, visualisations, and diagrammatics – collapsed under the language of an autonomous science. His narrative tracks how the status of projective geometry as a state religion at the time of the French revolution entered a precipitous decline so that by the time and in the person of Dirac it had become a repressed and private form of reasoning and knowledge production. Galison compares private scientific visualisation with the schematic and exploratory form of private sketches which precede painting. Private scientific visualisation and sketches would, without requiring rigor, precede the public, published scientific paper. “In such a picture the interior is psychological, aleatory, hermetic, and un-rigorous while the exterior is fixed, formally constrained, communicable, and defensible” (Galison, 2000: 148). Galison (2000: 150) is here reminded of Sigmund Freud, for whom the visual was primary, preceding and
conditioning the development of language. The pictorial, unconscious form of reasoning is of a different species from that of conscious, logical, language-based thought.
As for projective geometry as a state religion: during the late eighteenth century, descriptive geometry (later known as projective geometry) was proclaimed by mathematicians,
engineers and Polytechniciens to be much more than just a useful tool, as they argued that geometry would “hold together reason and the world” (Galison 2000: 152). For the mathematician Gaspard Monge and his Polytechnique School, physical processes including projections, section, duality, and deformation became means of discovery, proof, and generalisation. For many Polytechnique engineers, geometry was more than practical – rather it was valued as towering above all other forms of knowledge as the ideal of well-grounded argumentation. Descriptive geometers insisted that projective geometry could play a central role in improving the French working class, and Dupin (cited in Galison 2000: 153)
proclaimed that geometry: “is to develop, in industrials of all classes, and even in simple workers, the most precious faculties of intelligence, comparison, memory, reflection, judgement, and imagination”.
However, as analysts displaced the geometers, geometry lost its lofty status. A “new, vastly more abstract, rigorous, and algebraic mathematics” was coming into prominence,
emphasising “rigor, axiomatic presentations, and perfect clarity in definitions” (Galison, 2000: 158). This was the prevailing current against which Dirac had to swim. For him the geometric nature of his work was a form of argumentation, an effective structure and a means to explore the unknown (Galison 2000: 160). At a gathering of geometers Dirac expressed his heartfelt sense that pure mathematics had nothing over the applied. He
contended that there was a deep mathematical beauty in the specifics of the “actual world”
that was obscure to the pure mathematician. “To draw diagrams, to picture relationships – these were the starting points for grasping why the universe was as it was” (cited in Galison 2000: 158). In a fragment called “The Physicist and the Engineer,” Dirac maintained that mathematical beauty existed in the approximate reality of the actual world in which we live, not in the realm of pure and exact proof (Galison 2000: 159).
1.9 Image as source of intuition
The mathematician Henri Poincaré (1854 - 1912) had preceded Dirac as an exponent of the counter-current which did see great scientific value in the image. Poincaré saw images as the source of intuition and the means for keeping mathematicians in contact with the real or concrete world (O'Halloran, 2004: 131). Poincaré’s as well as Kekulé’s descriptions of their eureka episodes are filled with images. Arthur Miller’s (1995) research on imagery and the visual imagination led him to claim that many twentieth-century and nineteenth- century physicists were highly visual thinkers. While mathematical symbolism may be seen as more powerful as it divides the world into black and white, large and small, visual images
“can represent shades of grey, ranges of size, and degrees of those external attributes that viewers use in making inferences” (Messaris cited in O’Halloran, 2004: 132). The mathematician Sha Xin Wei (cited in Elkins, 2006: 18) suspects that although the
nondescript little gestures and scribbles that mathematicians draw when they are working out problems together cannot be correlated with the rigorous equations and geometric figures they eventually produce, “they provide just the right level of openness and nuance to help the process of discovery”.