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

understanding proofs 325explain the ability to continue the sequence in terms of knowing the correctformula. But then we are left with the task of explaining how he is able toapply the algebraic formula correctly. It is not enough to say that the formulasimply ‘occurs to him’ as he produces the desired behavior; perhaps he willcontinue to think of the formula and do something entirely unexpected atthe next step. So we have simply replaced the problem of explaining whatit means to understand how to continue the sequence with the problem ofexplaining what it means to understand how to apply a formula correctly. Tomake matters worse, there may be other ways in which we can account forthe person’s ability to continue the sequence according to the pattern we havein mind; or we may find that the person is simply able to do it, without beingable to explain how.We are trying to get hold of the mental process of understanding which seems to behidden behind those coarser and therefore more readily visible accompaniments.But we do not succeed; or, rather, it does not get as far as a real attempt.For even supposing I had found something that happened in all those cases ofunderstanding,—why should it be the understanding? ... (§153)The solution is, in a sense, just to give up. In other words, we simply need toresist the temptation to find a suitable ‘source’ for the behavior.If there has to be anything ‘behind the utterance of the formula’ it is particularcircumstances, which justify me in saying that I can go on—when the formulaoccurs to me.Try not to think of understanding as a ‘mental process’ at all.—For that is theexpression which confuses you. But ask yourself: in what sort of case, in whatkind of circumstances, do we say, ‘Now I know how to go on,’ when, that is,the formula has occurred to me?—In the sense in which there are processes (including mental processes) whichare characteristic of understanding, understanding is not a mental process. (§154)If our goal is to explain what it means to say that someone has understooda particular word, formula, or command, we simply need to describe thecircumstances under which we are willing to make this assertion. In doing so,we may find that there is a good deal that we can say that will clarify ourmeaning. Giving up the attempt to identify understanding as some sort of thingdoesn’t mean that we cannot be successful, by and large, in explaining whatunderstanding amounts to.Thus, from a Wittgensteinian perspective, the philosopher’s task is not toexplain the feeling of having understood, or any underlying mental or physicalprocesses. The challenge, rather, is to clarify the circumstances under we whichwe make our ascriptions.