The Cognitive and Epistemic Value of Mathematics: Making the World Intelligible – The Role of Abduction, Diagrams, and Affordances

When dealing with the relationship between mathematics and cognition, we face two main intellectual traditions. First of all the abundant studies about the role of mathematics in the human (and animal) development of cognitive abilities; second, the philosophical reflections upon the various ways provided by mathematics in generating specific kinds of “knowledge.” Among the various perspectives offered by the philosophical studies about the status of mathematics, I think that Immanuel Kant’s ideas represent a valuable and indispensable fil rouge able to furnish a conceptual instrument which can highlight how mathematics and cognition are strictly intertwined. I say that Kantian perspective constitutes a conceptual fil rouge because it is only through it that it is possible to synthetically understand the epistemological nature of the various approaches at play. Kant provides a philosophical anti-metaphysical framework for mathematics that constitutes a fundamental defense of its role in high-level cognitive activities and its capacity to make rational intelligibility of the world, avoiding old-fashioned ontological views: the empirical world becomes a world of mathematical relations. I contend that it is thanks to Kantian philosophy of mathematics that the door to the subsequent studies regarding the cognitive and epistemic value of mathematics is opened up. I will take advantage of this classical perspective to provide new insight into some of the main problems related to the issue: (1) the historicization/naturalization of mathematics, which shows that their cognitive mechanisms of discovery and application and their historical development are strictly interrelated; (2) the role of manipulative abduction, affordances, model-based and diagrammatic reasoning, and distributed cognition as ways for clarifying the cognitive aspects of mathematics in the context of discovery; (3) the emphasis on the cognitive virtues of mathematical modeling in science as an antidote against the recent exaggerated attention to the management of big data, as a way of reaching scientific results; and (4) the lack of a mathematical genuine cognitive schematic effort of creating scientific intelligibility, which often leads to mere surrogate “modeling,” unreasonably supposed to be scientific. Finally, taking advantage of the Lobachevskian discovery of the first non-Euclidean geometry I will exemplify the issue of the abductive, model-based, diagrammatic, heuristic, and the extra-theoretical dimension of geometrical cognition, by illustrating the role played by the so-called mirror and unveiling diagrams.