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

118 kenneth mandersdisqualifying the diagram outright (pace Euclid) or by blocking its pentagramproblem.Choosing the first direction, we might insist that broken lines count as madeup of separate segments, and hence here violate the requirement that co-exactconditions assumed for reductio must be be indicated in the diagram. Insupport of this, we might note that the diagram attributions that the pentangleobjection exploits, which are evidently co-exact, are precisely that the regionhas five distinct sides. We might, then, interpret what we see in Euclid as arefusal to read the diagram in regards the separateness of the sides, not via anadditional mechanism for blocking attributions in reductio proof, but simplyin order to disregard a point in which the diagram is known to be defective.In all, this reading is now seen as neutral on the choices we just envisioned.This leaves the question why one would not make a correct diagram,with lines curved, in the first place. There might be several reasons for this.When the diagram cannot be subjected to a straightness condition indicatedin the discursive text for the production of a line, there appears to be apreference to produce the line at an angle (but straight). Perhaps there is ageneral preference for straightness when straight lines are put forward in thetext, perhaps because straightness discipline gives the most powerful control,perhaps because straight-line completion is least likely to produce potentiallyspurious regions and hence give rise to atypicality in the diagram which wouldprovoke a case analysis. We are not in a position to pursue these questions atthis stage.4.5.5 Summary: principal standards for diagrams in Euclidean reductio proof1. The relationship of the diagram to conditions in force upon entering thereductio context is unaffected.2. The diagram indicates all co-exact conditions from the hypothesis forreductio.3. The diagram may be subject to some exact conditions from the hypothesisfor reductio—but not all.4.6 Roles in geometrical practice: cases, objection,probingCurrent philosophical conceptions of mathematics are predominantly agentless:geometry too is cast as a mathematical theory, a body of statements articulated