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

328 jeremy avigad• the ability to give a high-level outline, or overview of the proof;• the ability to cast the proof in different terms, say, eliminating or addingabstract terminology;• the ability to indicate ‘key’ or novel points in the argument, and separatethem from the steps that are ‘straightforward’;• the ability to ‘motivate’ the proof, that is, to explain why certain steps arenatural, or to be expected;• the ability to give natural examples of the various phenomena describedin the proof;• the ability to indicate where in the proof certain of the theorem’shypotheses are needed, and, perhaps, to provide counterexamples thatshow what goes wrong when various hypotheses are omitted;• the ability to view the proof in terms of a parallel development, forexample, as a generalization or adaptation of a well-known proof of asimpler theorem;• the ability to offer generalizations, or to suggest an interesting weakeningof the conclusion that can be obtained with a corresponding weakeningof the hypotheses;• the ability to calculate a particular quantity, or to provide an explicitdescription of an object, whose existence is guaranteed by the theorem;• the ability to provide a diagram representing some of the data in theproof, or to relate the proof to a particular diagram;and so on. The philosophical challenge is to characterize these abilities withclarity and precision, and fit them into a structured and informative theory.Thanks to our Wittgensteinian therapy, we will not let the phrase ‘possess anability’ fool us into thinking that there is anything mysterious or metaphysicallydubious about this task. We have serious work to do, and worrying aboutwhat sort of thing is being ‘possessed’ is an unnecessary distraction.And yet we may still be plagued by qualms. Our analysis entails thatunderstanding only becomes manifest in an agent’s behavior across a rangeof contexts, and we seem to have come dangerously close to identifyingunderstanding with the class of relevant behaviors. Such a ‘dispositional’ or‘behavioral’ account of understanding has famously been put forth by GilbertRyle (1949) as part of a more general philosophy of mind. Since Ryle’sapproach is commonly viewed as having failed, it is worth reviewing some ofthe usual criticisms, to see what bearing they have on the more specific issuesaddressed here.²² These criticisms are enumerated, for example, in Carr (1979).