Diagrammatic reasoning

Abstract

Many types of everyday and specialized reasoning depend on diagrams: we use maps to find our way, we draw graphs and sketches to communicate concepts and prove geometrical theorems, and we manipulate diagrams to explore new creative solutions to problems. The active involvement and manipulation of representational artifacts for purposes of thinking and communicating is discussed in relation to C.S. Peirce’s notion ofdiagrammatical reasoning. We propose to extend Peirce’s original ideas and sketch a conceptual framework that delineates different kinds of diagram manipulation: Sometimes diagrams are manipulated in order to profile known information in an optimal fashion. At other times diagrams are explored in order to gain new insights, solve problems or discover hidden meaning potentials. The latter cases often entail manipulations that either generate additional information or extract information by means of abstraction. Ideas are substantiated by reference to ethnographic, experimental and historical examples.

Naturalised modal epistemology and quasi-realism

“…Consequently, mathematical cognition (Tylén et al 2014) and cognition in general (Gureckis and Goldstone 2006) are no longer understood in terms of individual achievements. Cognitive study of mathematical cognition outstrips facts about the psychology of individuals engaged in mathematical practice, including artifacts that have structured the cognitive niche in which the practice thrives.…”

The making of occupation-based models and diagrams: History and semiotic analysis

“…The three semiotic elements in Peirce's theory of signs are placed into a triadic relationship. The triad consisted of the (a) object or reference context, (b) model or symbol, and (c) concept or symbol's meaning (Atkin, 2010;Stjernfelt, 2000;Tylen, Fusaroli, Bjorndahl, Raczaszek-Leonardi, & Stjernfelt, 2014). By applying Peirce's triadic relationship to an occupational therapy model, we might see something akin to Figure 3.…”

“…In this case, the newcomer to the model could be a student or a therapist seeing a new diagram for the first time. The translation of the sign's meaning or negotiated interpretation was understood by Peirce to be contextual, cultural, and personal (Stjernfelt, 2000;Tylen et al, 2014), in much the same way as described above. However, the complexity of the functions or relationships presented in the diagram may be too hard to sort out and decode.…”

Cognitive Artifacts for Geometric Reasoning

“…Moreover, a diagramat least in principle-can be passed on and further developed by next generations of mathematicians. In other words, a diagram delivers an external representational support to mathematical thinking both in shorter and longer time scales (Tylén et al 2014). The role of diagrams does not end, however, on organizing an argument and facilitation of understanding of argument expressed orally.…”

“…The typical view in cognitive science is that mathematical cognition should be studied as a purely individual achievement. In contrast, we point out that mathematical justificatory practices may be understood in terms of repeatable public procedures that rely on the capacities of cognitive agents to jointly construct, explore and reconfigure representational tokens (Tylén et al 2014). In other words, mathematical practice is shared by many individuals who can publicly use the same artifacts to achieve the same results, for example prove theorems by manipulating diagrams.…”