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

understanding proofs 331to explain, to sum up, what you observe’ (Wittgenstein, 1953, §156). Whatgives our theoretical terms meaning is the role they play in explaining thedesired or observed behaviors; ‘An ‘‘inner process’’ stands in need of outwardcriteria’ (Wittgenstein, 1953,§580). While these outward criteria are necessary,they are also sufficient to give our terms meaning. So, we can once again setour metaphysical qualms aside.What we are left with is essentially a functionalist approach to explainingvarious aspects of mathematical understanding. The fundamental philosophicalchallenge is to develop a language and conceptual framework that is appropriateto our goals. If you want an explanation of how a car works, a description ofthe subsystems and their components, situated against general understanding asto how these interact, may be just what you need to keep your car runningsmoothly, and to diagnose problems when they arise. A more fine-graineddescription is more appropriate if you are studying to be a mechanic or engineer.What we are seeking here are similar explanations of how mathematicalunderstanding works. In this case, however, our intuitions as to how to talkabout the relevant subsystems and components is significantly poorer. I haveargued that a theory of mathematical abilities and their relationships shoulddo the trick, but, at this point, the proposal is vague. The only way to makeprogress is to pay closer attention to the data that we are trying to explain, andto the particular aims that our explanations are to serve.Part II. Formal verification12.5 The nature of proof searchIn Part I, I described a general way of thinking about mathematical understanding.My goal in Part II is to show that this way of thinking is fruitful in at leastone scientific context where informal notions of understanding are used. Indoing so, I will consider only one small aspect of mathematical understanding,with respect to one particular scientific practice. While I expect that the generalperspective will be useful in other domains as well, and that the problems thatarise share enough common structure that they can be supported by a unifiedconceptual framework, I cannot make this broader case here. So I ask you tokeep in mind that, in what follows, we are considering only one restrictedexample.We have seen that understanding an ordinary textbook proof involves, inpart, being able to spell out details that are left implicit in the presentation.I have argued elsewhere (Avigad, 2006) that it is hard to make sense of