Philosophy Colloquium

Logan Fletcher

University of Maryland

Seeing General Truths in Particular Diagrams: the Angle-Sum Theorem

The argument set out in this talk is part of a larger defense of what I call the *diagram-based view*: This is the view that we can come to grasp certain mathematical truths by perceiving spatial relations in suitable visual diagrams. The diagram-based view thus maintains that there is a distinctive *visual* route to genuine mathematical knowledge (for at least some nontrivial body of mathematics). Here I focus on one of the most forceful challenges to the diagram-based view, the *particularity problem*, and on the specific theorem with which this problem is usually associated: the angle-sum theorem, an elementary truth of Euclidean plane geometry. The problem is this: Even granting that the diagram allows us to see that the result holds for the *particular* triangle depicted, how could perception of the diagram ever warrant the judgment that the theorem is true in the general case, that is, for *any* triangle whatsoever? I consider both historical and contemporary solutions to the particularity problem, and conclude that none are satisfactory. I then provide a novel solution, arguing that we can indeed *see* that the angle-sum theorem holds in general, by perceiving the diagram in a way that is animated by the combined application of two different kinds of ‘dynamic’ visual imagery.

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