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.

Wednesday, September 11, 2013

SYM 0215