Philosophy Colloquium

Logan Fletcher

University of Maryland

Can Visual Diagrams Justify Mathematical Beliefs?

The claim that visual diagrams can justify mathematical beliefs faces two main challenges. The first challenge argues that diagrams are inherently unreliable, given a number of well-known cases in which diagrams or visual intuitions have proved to be misleading. I argue that these ‘problem cases’ have been misdiagnosed. In all such cases, the erroneous judgments result, not from any defect inherent to visual thinking as such, but rather from a specific set of cognitive heuristics that operate at an unconscious level. In general, the heuristic-based errors prove, in fact, to be correctable by means of the appropriate use of diagram-based visual understanding. I conclude that there is no reason at all to think that diagrams are inherently unreliable as guides to mathematical truth.

I then turn to the second challenge, which argues that the use of diagrams is not rigorous, and hence cannot justify. Recent work in formal diagrammatic systems shows that it is simply false that reasoning with diagrams cannot be rigorous. Nonetheless, it is true that in many cases of interest, diagrammatic demonstrations do not qualify as mathematically rigorous. I argue, however, that this does not imply that they cannot justify. Here I distinguish between rigorous justification and intuitive justification. I argue that both are properly regarded as kinds of justification, that both constitute central and permanent aims of mathematical practice, and that there is an inherent tension between the two. From this perspective, it is mistaken to say that because ‘visual proofs’ are not rigorous, they fail to justify. It is rather that they pursue the intuitive kind of justification at the expense of the rigorous kind. I conclude that the two main challenges to the justificatory status of mathematical diagrams are both unsuccessful.

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