Trusting Your Own Eyes: Visual Constructions, Proofs, and Fallacies in Mathematics

This chapter discusses how human biological capability of seeing contributes to the cognitive development of mathematical thinking. Thus, the phenomenon of visualization in mathematics, viewed as practice of handling visual and spatial information with the aid of physical artefacts, diagrams drawn on paper and mental imagery, is examined. It is stressed that visual mental images rely on all human senses; they develop through perception, action, and language, along with the process of conceptualization of mathematical objects. The latter are considered as distinct from the physical objects, from which their properties are abstracted and generalized. One can study mathematical objects through their various semiotic representations, including visual images. Examples of geometric constructions and reasoning with figures in the Euclidean plane are provided. They reveal a possibility of finding and comparing multiple solutions with the aim of noticing new mathematical relations. These examples also illustrate that working with visual material requires and supports the development of mathematical knowledge. This includes knowledge of mathematical properties of figure’s elements and detection of underlying structure. As well, the study of non-Euclidean geometries, while significantly relying on mathematical formalism, calls for supporting visual aid. Geometrical solutions emerge through refocus of viewer’s attention from the external, visual features to internal logical characteristics of the figures. Examples of visual fallacies and paradoxes based on unfaithful images further support the idea of the importance of logical analysis of figures. They necessitate an alternative verification of what could be visually perceived as trustworthy, and call for a rigorous deductive approach in mathematics. While the role of figures in formal proofs may be disputed, visual imagery often empowers work on mathematical discovery.