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

understanding proofs 323of abilities to supply missing inferences, draw appropriate analogies, discoverother theorems, and so on. And, in turn, these abilities are just the sort ofthing that should explain our pleasure at having understood: it is simply thepleasure of having acquired a new skill, or of finding ourselves capable of doingsomething we could not do before.According to this analysis, when we say that someone understands something—atheorem, a problem, a method, or whatever—what we mean isthat they possess some general ability. Such uses of a particular to stand forsomething more general are familiar. Suppose I tell you that Rebecca, a brightgirl in Ms Schwartz’s second grade class, can multiply 34 by 51. What do Imean by that? Surely I am not just attributing to her the ability to utter ‘1734’in response to the corresponding query; even the dullest student in the classcan be trained to do that. We are often tempted to say that what we mean isthat Rebecca is able to arrive at the answer ‘1734’ by the right process, buta metaphysical commitment to ‘processes’ is easily avoidable. What we reallymean is that Rebecca is capable of carrying out a certain type of arithmeticaloperation, say, multiplying numbers of moderate size. Phrasing that in termsof the ability to multiply 34 by 51 is just a convenient manner of speaking.This way of thinking about understanding is not novel. In the next section, Iwill show that the strategy I have outlined here accords well with Wittgenstein’sviews on language, and I will situate the framework I am describing here withrespect to the general viewpoint of the Philosophical Investigations (Wittgenstein,1953). In the section after that, I will address some of the concerns that typicallyattend this sort of approach.12.3 MathematicsasapracticeAn important segment of the Philosophical Investigations explores what it meansto follow a rule, as well as related notions, like obeying a command or usinga formula correctly.¹ The analysis shows that, from a certain philosophicalperspective, it is fruitless to hope for a certain type of explanation of the‘meaning’ of such judgments. Nor is it necessary: there are philosophical gains¹ I have in mind, roughly, sections 143 to 242, though the topic is foreshadowed in sections 81to 88. The literature on these portions of the Investigations is vast, much of it devoted to critiquingKripke’s interpretation (1982); see, for example, Goldfarb (1985) andTait(1986). The present chapteris essentially devoted to showing that a ‘straightforward’ reading of the Investigations has concreteconsequences for the development of a theory of mathematical understanding. I owe much of myinterpretation of Wittgenstein on rule following, and its use in making sense of a mathematical‘practice’, to discussions with Ken Manders.