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

the euclidean diagram (1995) 131grasp of its construction) provides some control over what is an appropriatechallenge, this control is singularly inarticulate.The claims and arguments which might be appropriate to the situationprobed may be felt to address a variety of questions, which might or might nothave appeared inarticulately bound up in the original sensation of impotence:when the diagram showed what it did, (i) could that have been a breakdown ofappearance control? (ii) What might have been relevant about the free metricchoices: can we identify features or ranges of these choices that demonstrablylead to the outcome appearance? (iii) What other appearances might arise fromdifferent metric choices? (iv) What is the range of control possibilities andoutcomes?When we have achieved a favorable resolution, acquired a measure ofenhanced intellectual control over the original diagram occurrence, we maychoose to characterize the enhancement as the acquisition of one or more ofseveral broad virtues: we may say we have justified a claim about the outcome,or that we have explained why the construction gave the outcome, or that wehave articulated claims, conditions, reasons, arguments...The responsibility for geometrical claims that comes with being a geometeris thus much broader than a concern for reliability. Probing, too, should now beseen as far more than an element of criticism in an overall justificational strategy.Probing is the form of action within geometrical practice through whichparticipants undertake their responsibility to improve its overall intellectualstanding. Presumably, such a conception is not special to geometry, but insteadcommon to a range of intellectual practices.We tend to fail to recognize probing at work, fail to apply the concept—asit were, miss its unity—in two distinct ways. We can be distracted bythe diverseness or special character of probing actions in particular situations;as when we are tempted to distinguish case and objection by who canconveniently respond. Or we can be distracted by the diverseness or specialcharacter of evaluations that drive probing or praise its accomplishments inparticular situations; as when we take formal proofs in Tarski’s system tosettle justificational issues while asserting that they, taken by themselves, aregeometrically unintelligible.We can learn much, perhaps, by making such discriminations among actiontypes and among evaluation types, in contexts where those distinctions areviable. It is important, however, to insist on the unity of probing overthese two multiplicities because they are incompatible: varieties of probingactions do not match up with varieties of intellectual virtues that arise fromthem. Case proposal does not go with justification while objection goes withexplanation, or vice versa. Probing the two-circle intersection construction by