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

understanding proofs 321theorems, proofs, problems, solutions, definitions, concepts, and methods; atthe same time, we take all these things to contribute to our understanding.This duality is reflected in the two questions I posed at the outset.The way the questions are posed seem to presuppose that understanding isa relationship between an agent, who understands, and mathematics, which isthe object of that understanding. On the surface, talk of knowledge shares thisfeature; it is an agent that knows that a theorem is true. But the role of an agentis usually eliminated from the epistemological account; once knowledge of atheorem is analyzed in terms of possession of a proof, proofs become the centralfocus of the investigation. To adopt a similar strategy here would amount tosaying that an agent understands X just in case the agent is in possession of Y.But what sort of thing should Y be?Suppose I tell you that my friend Paolo understands group theory, and youask me to explain what I mean. In response, I may note that Paolo can statethe definition of a group and provide some examples; that he can recognizethe additive group structure of the integers, and characterize all the subgroups;that he knows Lagrange’s theorem, and can use it to show that the order ofany element of a finite group divides the order of the group; that he knowswhat a normal subgroup is, and can form a quotient group and work with itappropriately; that he can list all the finite groups of order less than 12, up toisomorphism; that he can solve all the exercises in an elementary textbook;and so on.What is salient in this example is that I am clarifying my initial ascriptionof understanding by specifying some of the abilities that I take such anunderstanding to encompass. On reflection, we see that this example is typical:when we talk informally about understanding, we are invariably talking aboutthe ability, or a capacity, to do something. It may be the ability to solve aproblem, or to choose an appropriate strategy; the ability to discover a proof;the ability to discern a fruitful definition from alternatives; the ability to applya concept efficaciously; and so on. When we say that someone understands wesimply mean that they possess the relevant abilities.Ordinary language is sloppy, and it would be foolish to seek sharp accounts ofnotions that are used in vague and imprecise ways. But notions of understandingalso play a role in scientific claims and policy decisions that should be subjectto critical evaluation. In more focused contexts like these, philosophicalclarification can help further inquiry.What I am proposing here is that in such situations it is often fruitfulto analyze understanding in terms of the possession of abilities. This is astraightforward extension of the traditional epistemological view: the ability todetermine whether a proof is correct is fundamental to mathematics, and the