Mathematical Understanding

The philosophy of mathematics has traditionally been concerned with questions about the proper justification of claims to knowledge. The goals of mathematics, however, are not limited to making true mathematical statements; ultimately, we look to mathematics for understanding. This chapter considers the ways we talk about mathematical understanding in our assessments of historical and contemporary mathematical results, in our discussions of computers in mathematics, and in mathematics education. It argues that a theory of mathematical understanding should be a theory of abilities to think and reason mathematically, and that such a theory should inform, and be informed by, practical applications. It also situates models of reasoning and understanding in the context of symbolic and neural AI, which offer complementary perspectives.