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

understanding proofs 327possible to do physics in feet and inches as well as in metres and centimetres; thedifference is merely one of convenience. But even this is not true if, for instance,calculations in some system of measurement demand more time and trouble thanis possible for us to give them. (§569)Concepts lead us to make investigations; are the expressions of our interest,and direct our interest. (§570)One finds similar views on the role of specifically mathematical concepts inWittgenstein’s other works. For example, we find the following in the Remarkson the Foundations of Mathematics:The mathematical Must is only another expression of the fact that mathematicsforms concepts.And concepts help us to comprehend things. They correspond to a particularway of dealing with situations.Mathematics forms a network of norms. (Wittgenstein, 1956, VI, §67)This stands in contrast to the traditional view of mathematics as a collection ofdefinitions and theorems. For Wittgenstein, a proposition is not just an objectof knowledge, but, rather, something that shapes our behavior:The mathematical proposition says to me: Proceed like this! (§72)With respect to propositions in general, we find in On Certainty:204. Giving grounds, however, justifying the evidence, comes to an end;—butthe end is not certain propositions’ striking us immediately as true, i.e. it is not akind of seeing on our part, it is our acting, which lies at the bottom of the languagegame. (Wittgenstein, 1969)This way of thinking challenges us to view mathematics in dynamic terms,not as a body of knowledge, but, rather, as a complex system that guides ourthoughts and actions. We will see in Part II of this essay that this provides apowerful and fundamentally useful way of thinking about the subject.12.4 A functionalist epistemologyI have proposed that a theory of mathematical understanding should be atheory of mathematical abilities. In ordinary circumstances, when we say, forexample, that someone understands a particular proof, we may take them topossess any of the following:• the ability to respond to challenges as to the correctness of the proof, andfill in details and justify inferences at a skeptic’s request;