The Epistemology of Geometry I: the Problem of Exactness

ASCS 2009 Proceedings of the 9th Australasian Society for Cognitive Science Conference (2010)
Abstract
We show how an epistemology informed by cognitive science promises to shed light on an ancient problem in the philosophy of mathematics: the problem of exactness. The problem of exactness arises because geometrical knowledge is thought to concern perfect geometrical forms, whereas the embodiment of such forms in the natural world may be imperfect. There thus arises an apparent mismatch between mathematical concepts and physical reality. We propose that the problem can be solved by emphasizing the ways in which the brain can transform and organize its perceptual intake. It is not necessary for a geometrical form to be perfectly instantiated in order for perception of such a form to be the basis of a geometrical concept.
Keywords problem of exactness  Aristotle  philosophy of mathematics  perception  cognition
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