Visual Reasoning in Euclid’s Geometry: An Epistemology of Diagrams

It is widely held that the role of diagrams in mathematical arguments is merely heuristic, or involves a dubious appeal to a postulated faculty of “intuition”. To many
these have seemed to exhaust the available alternatives, and worries about the status
of intuition have in turn motivated the dismissal of diagrams. Thus, on a standard
interpretation, an important goal of 19th Century mathematics was to supersede
appeals to intuition as a ground for knowledge, with Euclid’s geometry—in which
diagrams are ubiquitous—an important target. On this interpretation, Euclid’s presentation is insufficient to justify belief or confer knowledge in Euclidean
geometry. It was only with the work of Hilbert that a fully rigorous presentation of
Euclidean geometry became possible, and such a presentation makes no nonredundant
use of diagrams.
My thesis challenges these claims. Against the “heuristic” view, it argues that
diagrams can be of genuine epistemic value, and it specifically explores the
epistemology of diagrams in Euclid’s geometry. Against the “intuitive” view, it
claims that this epistemology need make no appeal to a faculty of intuition. It
describes in detail how reasoning with diagrams in Euclid’s geometry can be
sufficient to justify belief and confer knowledge. And it shows how the background
dialectic, by assuming that the “heuristic” or “intuitive” views above are exhaustive,
ignores the availability of this further alternative. By using a detailed case study of mathematical reasoning, it argues for the importance of the epistemology of diagrams
itself as a fruitful area of philosophical research.Philosophy of Mathematics, Epistemology & Methodology