Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

318 jeremy avigadmeans to understand a mathematical definition, a theorem, or a proof; or tounderstand a theory, a method, an algorithm, a problem, or a solution. Wecan, alternatively, focus our task by restricting our attention to particular typesof judgments, such as historical or mathematical evaluations of theoreticaldevelopments, or pedagogical evaluations of teaching practices and lessonplans. We can be even more specific by considering individual objects ofunderstanding and particular evaluatory judgments. For example, we can askwhat it means to understand algebraic number theory, the spectral theoremfor bounded linear operators, the method of least squares, or Gauss’s sixthproof of the law of quadratic reciprocity; or we can try to explain howthe introduction of the group concept in the 19th century advanced ourunderstanding, or why the ‘new math’ initiative of the 1960s did not deliverthe desired understanding to students. The way we deal with the specificexamples will necessarily presuppose at least some conception of the natureof mathematical understanding; but, conversely, the things we find to say inspecific cases will help us establish a more general framework.In this chapter, I will defend the fairly simple claim that ascriptions ofunderstanding are best understood in terms of the possession of certain abilities,and that it is an important philosophical task to try to characterize the relevantabilities in sufficiently restricted contexts in which such ascriptions are made.I will illustrate this by focusing on one particular type of understanding, inrelation to one particular field of scientific search. Specifically, I will explorewhat it means to understand a proof, and discuss specific efforts in formalverification and automated reasoning that model such understanding.This chapter is divided in two parts. In Part I, I will argue that thegeneral characterization of understanding mentioned above provides a coherentepistemological framework, one that accords well with our intuitions and iscapable of supporting rational inquiry in practical domains where notions ofmathematical understanding arise. In Part II, I will present four brief casestudies in formal verification, indicating areas where philosophical reflectioncan inform and be informed by contemporary research in computer science.Part I. The nature of understanding12.1 Initial reflectionsA central goal of the epistemology of mathematics has been to identify theappropriate support for a claim to mathematical knowledge. We show that a