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

the euclidean diagram (1995) 113What would not work well is a reductio argument starting with aninarticulate denial of a diagram: ‘Things are not like this’; in logical terms, adiagram-based reductio proof of (D 1 &... &D m ), starting with ¬(D 1 &... &D m ).(Of course, there might be such a reductio argument entirely within theresources of the discursive text.) In diagram-based reasoning, reductio gets itsgrip only when it leads determinately to a diagram.4.5.3 Further constraints on the diagram for reductio: I.27Our text of Euclid does not fully conform to the account given so far. I.27considers the straight line EF falling on two straight lines AB, CD such thatthe alternate angles AEF, EFD are equal (Fig. 4.6); and then assumes forreductio, and without loss of generality, that AB and CD produced meet inthe direction of B, D,atG. The diagram in the text would be read as subjectedto the equality of alternate angles with EF, which of course precludes straightlines from meeting a short distance away at G. Thus, the diagram gives up onsubjecting AB and CD produced to being straight, in order to indicate theirintersection at G. (So far, so good.)We moderns might indicate this intersection by producing either or bothAB and CD free-form monotonically curved toward their intersection G.Thiswould allow one to attribute the triangle GEF referred to later in Euclid’sargument (the requisite straightness of GE and GF is indicated in the text eventhough the diagram is not subjected to it). Admittedly, ancient practice doesnot appear to have utilized free-form curves; but instead apparently sought tosatisfy the requirements of argument by a diagram somehow made up fromthe ordinary types of diagram elements. This desideratum can be satisfiedfor any of the proposals just made, however, by substituting a circular arcfor free-form curves. In his diagram for an objection to I.4 (239), ProclusAEBGCFDAEBGFig. 4.6.CFD