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

the euclidean diagram (1995) 127branching control is a relational matter. Given the statement of I.7, Proclus’alternative is a different-appearance diagram arising from different metricchoices (but not limitations on control in making the diagram from thosemetric choices) satisfying the verbal statement of hypothesis for reductio.Hence it is a case arising in the course of the argument, by our test of differentdegrees of uniformity in text and diagram. On the other hand, both diagramsin I.7 can be regarded as arising from appearance control probing; not in I.7but in earlier constructions such as that of I.1.4.6.3 Probing and knowledgeTraditional geometrical practice has no access, except through diagrammingand proof, to ulterior standards of geometrical truth, by which it might back upits standards for handling text and diagram in demonstration. Not far under thesurface of Proclus’ frequent praise of the subtlety of Euclid’s formulations lieburied hosts of objections based on variant readings and earlier texts, objectionswhich are only avoided through what must long have seemed open-endedrefinement of expressive means and exploration of diagrammatic possibilities.Using diagrams is a much more effective strategy in gaining an intellectualgrip on space than we have been able to show here; but it does come at theprice of a certain open-endedness in geometrical reasoning: although ‘exact’conclusions are not read off from the diagram, diagram use also remains inneed of probing, due to the limitations of diagram control.Because of this, the long-range stability of traditional geometry—acrosscenturies, participants, and the extended body of accumulated geometricalargument—cannot be understood without taking into account the constructivecritical attitude towards geometrical argument expressed through probing. Inour philosophical account of geometrical knowledge, probing must thereforeshare center stage with reasons.But if probing is conceptually central to traditional geometric practice, wecan no longer treat that practice as requiring unqualified assent to geometricclaims. Instead, geometrical practice must cultivate dissent in order to avoiddisarray! This dissent takes the form of vigorous critical scrutiny of geometricalargument. The normative structure of this dissent, however, differs from thatof dissent in other practices, in which dissenting parties may stay clear ofeach other while they seek power (dissent in the body politic) or in whichthey may sometimes agree to disagree (dissent in the body philosophic). Thegeometer’s obligation is to assert (the protagonist), to probe, and to seek aresolution of differences; suited to stand up in that very process continued byother participants bound only by that same obligation. Taken all together, thisis a structure of unqualified responsibility, with the widest possible scope, for