Abstract
The focus of this chapter is twofold. The first is a semiotic description of the nature of diagrams. The second is a description of the type of reasoning that the transformation of diagrams facilitates in the construction of mathematical meanings. I am guided by the Peircean definition of diagrams as icons of possible relations and his conceptualization of diagrammatic reasoning. When a diagram is actively and intentionally observed, perceptually and intellectually, a manifold of structural relations among its parts emerges. Such relations among the parts of the diagram can potentially unveil the deep structural relations among the parts of the Object that the icon plays to represent. An Interpreter, who systematically observes and experiments with diagrams, mathematical or not, also generates evolving chains of interpretants by means of abductive, inductive and deductive thinking. Using Stjernfelt’s model of diagrammatic reasoning, which is rooted in Peircean semiotics, I illustrate an emergent reasoning process to prove two geometric propositions that were posed by means of diagrams.
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Notes
- 1.
The word SIGN, in capital letters, is used here to refer to the Peircean notion of ‘sign’ defined as a system constituted by a set of three elements and the dyadic relations among the three elements. The Peircean triadic notion of ‘sign’ was and continues to be a historically new conceptualization of ‘sign’ for which he is famously known (see Vasco et al. 2009). In other words, we could symbolize his triadic notion of ‘sign’ as a system constituted by a set and the relations governing the elements of the set in the following way:
SIGN = ({sign vehicle, interpretant, Object}, Dyadic relations among the three elements of the set).
The sign vehicle is only one of the elements of the set that stands as a representation of another element in the set, namely the Object. Most of the time, Peirce used the word ‘sign’ for sign vehicle without advising the reader about the use that he meant; the meaning has to be decoded from the context in which the words were used. However, sometimes he clearly uses the words sign vehicle and representamen to refer to the representation of the Object.
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Sáenz-Ludlow, A. (2018). Iconicity and Diagrammatic Reasoning in Meaning-Making. In: Presmeg, N., Radford, L., Roth, WM., Kadunz, G. (eds) Signs of Signification. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-70287-2_11
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