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

330 jeremy avigadto screen off extraneous beliefs and desires, and assume nothing about anagent’s intent beyond the intent to perform mathematically. I am certainly notclaiming that it is obvious that one can provide adequate characterizations ofthe circumstances under which we tend to ascribe mathematical understanding;only that it is not obvious that attempts to do so are doomed to failure.Perhaps the most compelling criticism of a dispositional account is that evenif we could characterize the behaviors that are correlated with the mentalstates under consideration, identifying the mental states with the associatedbehaviors simply tells the wrong kind of story. We expect a philosophicaltheory to provide some sort of causal explanation that tells us how intelligentand intentional behavior is brought about; it seems unsatisfying to identify‘knowing how to play the piano’ with successful performance, when whatone really wants of a theory is an account of the mental activity that makessuch performance possible. In the case at hand, we would like a theory thatexplains how a proper understanding enables one to function mathematically.This insistence has not only an intuitive appeal, but also a pragmatic one. Forexample, in so far as our theory is to be relevant to mathematical expositionand pedagogy, we would expect it not only to characterize the outward signsof mathematical understanding, but also provide some hints as to how they canbe encouraged and taught. Similarly, I take it that a theory of mathematicalunderstanding should be of service to computer scientists trying to writesoftware that exhibits various types of competent mathematical behavior.Even if we set aside the question as to whether it is appropriate to attribute‘understanding’ to a computer, we might expect a good philosophical theorynot just to clarify and characterize the desired behaviors, but also to providesome guidance in bringing them about.We are therefore tempted to renounce our therapy and try, again, to figureout just what understanding really is. What saves us, however, is the observationthat our theory of mathematical abilities need not degenerate to a laundry listof behavioral cues. The abilities we describe will interact in complex ways,and will not always be cast in terms of behavioral manifestations. Consider ourexplanation of what it means for Paolo to understand group theory. Some ofthe relevant abilities may be cast in terms of behaviors, for example, the abilityto state a theorem or answer a question appropriately. But others may be castin more abstract terms, such as the ability to ‘recognize’ a group structure,‘determine’ subgroups and cosets, ‘apply’ a lemma, or ‘recall’ a fundamentalfact. In fact, we often take these abstract abilities to provide the ‘mechanisms’that explain the observable behaviors. We may be relieved to learn that ourWittgensteinian training does not preclude talk of mechanisms, provided thatwe keep in mind that ‘these mechanisms are only hypotheses, models designed