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Understanding Proofs JEREMY AVIGAD

Summary: 12
Understanding Proofs
`Now, in calm weather, to swim in the open ocean is as easy to the practised
swimmer as to ride in a spring-carriage ashore. But the awful lonesomeness is
intolerable. The intense concentration of self in the middle of such a heartless
immensity, my God! who can tell it? Mark, how when sailors in a dead calm
bathe in the open sea--mark how closely they hug their ship and only coast
along her sides.' (Herman Melville, Moby Dick, Chapter 94)
12.1 Introduction
What does it mean to understand mathematics? How does mathematics help
us understand?
These questions are not idle. We look to mathematics for understanding,
we value theoretical developments for improving our understanding, and we
design our pedagogy to convey understanding to students. Our mathematical
practices are routinely evaluated in such terms. It is therefore reasonable to ask
just what understanding amounts to.
The issue can be addressed at different levels of generality. Most broadly,
we need to come to terms with the sort of thing that understanding is,
and the sort of thing that mathematics is, in order to discuss them in an


Source: Avigad, Jeremy - Departments of Mathematical Sciences & Philosophy, Carnegie Mellon University


Collections: Multidisciplinary Databases and Resources; Mathematics