Emotion, Expression, Explanation
Edited By András Benedek and Kristof Nyiri
The Visually Explorative Method of János Bolyai in the Formation of Absolute Geometry
If we consider the approximately 2000-year process of struggling with the problem of parallels from Euclid’s time to the elaboration of Bolyai–Lobachevsky non-Euclidean geometry, then two important observations emerge. First, the geometrical view of the mathematicians working on the question decisively preferred a Euclidean type of space, which raises the question of how an alternative geometry seriously divergent and logically incompatible with our Euclidean-oriented geometrical view could at last be discovered. As a consequence, and secondly, while both János Bolyai and Nikolai Ivanovich Lobachevsky discovered and elaborated hyperbolic geometry, only Bolyai examined absolute geometry.1 This regularly neglected epistemological singularity urges further specification of the first question; is there any methodological explanation for Bolyai’s unique achievement?
In this paper I argue that Bolyai’s heuristic technique of discovering some constituent mathematical propositions, objects, and relationships regarding his absolute geometry was a visually explorative method. In making this argument, I will also investigate how this visually explorative method worked in the exploration of various components of absolute geometry.
Objections to the claim that our geometrical view is largely Euclidean include arguments that our sense of visual space cannot be considered Euclidean. More specifically, Patrick Suppes has argued that the contextual effects of either the presence or absence of some special “extraneous” geometrical objects (e.g., points, lines, etc.) are so robust that their determining influence on perceptual judgment should be seriously considered in order to characterize our sense ← 35 | 36 → of visual...
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