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

the euclidean diagram (1995) 119by starting-points and proofs. Of course, geometry is done by humans, andtheir practice contains a variety of distinguishable roles—author, teacher,student. But these seem beside the philosophical point: they have to dosolely with the interface between mathematical theory and social institutions,or even humans in the flesh; as such they are of no greater philosophicalpertinence to mathematics proper than the role of college president, orof mother.At a minimum, though, bringing out the distinctive character of mathematicswould seem to require that we say something about the kinds of stands thatmathematical practice requires agents to take in order to qualify as properlymathematical; what we, with some oversimplification, above called ‘unqualifiedassent’. We miss something about number theory, for example, if we think thatafter I assert that n is prime and you doubt or even deny it, we can properly justmove on happily; in the way that we properly can and perhaps should, whenIadmiretheWater Lilies in the museum but you respond with indifference.The body of requirements on properly mathematical agents which one mightanalyze out from various such contrast cases are (part of) what in agentlessterms is usually captured by traditional conceptions of mathematical certaintyor mathematical truth. Such conceptions license, it seems, but a single stanceor role within traditional geometry (except perhaps for differences in stancecorresponding to different stages of the science, such as before and after afundamental theorem has been established). The ideally competent geometergrasps the content of geometrical practice up to his time and takes responsibilityfor it personally: being able and willing to affirm geometrical claims and presentgeometrical constructions on his own responsibility, and respond to challengeand question by appropriate performances, including recognizably geometricresearch.Traditional geometry so conceived in effect recognizes only one role,which we might call that of protagonist; and except possibly for the advanceof the science, no differences among protagonists have any standing. Ourusual agentless conception of mathematical theory is, while philosophicallyless informative, nonetheless neatly compatible with such a conception ofgeometrical roles. According to this theoretical conception of mathematics,any discordant response to the protagonist is out of order, or expressesincompetence. That the student is not ready to shoulder responsibility forPythagoras’ theorem after seeing it for the first time just shows he is no geometeryet. One who questions the straightness of a side of a square in the sand hassimply missed the point of the practice. Proclus, however, acknowledgesstanding for roles beyond that of the protagonist, of an arguably theoreticalnature. Admitting these additional roles perhaps modifies the character of the