Picture-proofs and platonism

Abstract

This paper concerns the role of intuitions in mathematics, where intuitions are meant in the Kantian sense, i.e. the "seeing" of mathematical ideas by means of pictures, diagrams, thought experiments, etc. The main problem discussed here is whether Platonistic argumentation, according to which some pictures can be considered as proofs (or parts of proofs) of some mathematical facts, is convincing and consistent. As a starting point, I discuss -James Robert Brown's recent book Philosophy of Mathematics, in particular, his primarily examples and analogies. I then consider some alternatives and counterarguments, namely John Norton's opposite view, that intuitions are just pictorially represented logical arguments and are superfluous; and the Kantian transcendental theory of construction in imagination, as it is developed in the works of Marcus Giaquinto and Michael Friedman. Although I support the claim that some intuitions are essential in mathematical justification, I argue that a Platonistic approach to intuitions is partial and one should go further than a Platonist in explaining how some intuitions can deliver new mathematical knowledge.