Thinking through drawing - Diagram constructions as epistemic mediators in geometrical discovery

The concept of manipulative abduction is devoted to capture the role of action in many interesting cognitive situations: action provides otherwise unavailable information that enables the agent to solve problems by starting and performing a suitable abductive process of generation or selection of hypotheses. We observe that many external things, usually inert from an epistemological point of view, can be transformed into epistemic mediators. I will present some details derived from the history of the discovery of the non-Euclidean geometries which illustrate the relationships between strategies for anomaly resolution and visual thinking. Geometrical diagrams are external representations which play both a mirror role (to externalize rough mental models), and an unveiling role (as gateways to imaginary entities). I describe them as epistemic mediators able to perform various explanatory, non-explanatory, and instrumental abductive tasks (discovery of new properties or new propositions/hypotheses, provision of suitable sequences of models as able to convincingly verifying theorems, etc.). I am also convinced they can be exploited and studied in everyday non-mathematical applications also to the aim of promoting new trends in artificial intelligence modeling of various aspects of hypothetical reasoning: finding routes, road signs, buildings maps, for example, in connection with various zooming effects of spatial reasoning. I also think the cognitive activities of optical, mirror, and unveiling diagrams can be studied in other areas of manipulative and model-based reasoning, such as the ones involving creative, analogical, and spatial inferences, both in science and everyday situations so that this can extend the epistemological, computational, and the psychological theory. 1 Manipulative Abduction and Epistemic Mediators More than a hundred years ago, the great American philosopher Charles Sanders Peirce coined the term “abduction” to refer to inference that involves the generation and evaluation of explanatory hypotheses. Peirce says that mathematical and geometrical reasoning “consists in constructing a diagram according to a general precept, in observing certain relations between parts of that diagram not explicitly required by the precept, showing that these relations will hold for all such diagrams, and in formulating this conclusion in general terms. All valid necessary reasoning is in fact thus diagrammatic” (Peirce, 1931-1958, 1.54). I contend that geometrical construction is a kind of abductive reasoning, and more precisely, what I call manipulative abduction (cf. Magnani (2001b)). That is a kind of definition that prescribes “what you are to do in order to gain perceptual acquaitance with the object of the world” (Peirce, 1931-1958, 2.330). The analysis of the role of diagrams and visualizations in mathematical reasoning has been recently promoted by philosophers and historians of science (cf. for example the seminal studies by Brown (1997,