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

the euclidean diagram (1995) 123AFig. 4.10.C B Dand making objection, and may make distinctive arrangements for carryingthem on, such as the division of labor that Proclus records; but we see nooverriding significance to the intellectual function of the practice in suchdivisions. In particular, we just saw how case and objection are easy to confuse,sometimes indistinguishable, sometimes distinguished only by whether theoutcome is comfortable to the protagonist; and in any case tend both to besubmerged in revision of the text.Proclus records ample instances of critical scrutiny of The Elements. Manypoint to the need to state a proposition just right. The ‘Euclidean’ text, whichProclus often praises for its precise formulation, is probably the product ofcenturies of refinement of formulation after Euclid as well as before.For example, consider the statement of I.14: If, with any straight line AB,and at a point B on it, two straight lines BC and BD not lying on the sameside of AB make the adjacent angles ABC and ABD taken together equal totwo right angles, the two straight lines will be in a straight line with another(Fig. 4.10). (Adapted from Heath, Euclid I, p. 276.)Proclus (297–298) gives a counterexample of Porphyry to show that thecondition ‘not lying on the same side’ cannot be omitted: if we do so, aninitial diagram with different appearance will satisfy the remaining conditions,but not the conclusion.To our sensitivity, this makes absolutely clear that the added conditionis needed. It replaces otherwise implicit contributions of the diagram to thestatement of a proposition, by a fuller verbal description of the diagram. Butwhy must one do so? Although this is not the way Euclid’s text works, wecan imagine treating the initial diagram as part of the statement text of aproposition. One could then strip the discursive statement of I.14 down to:Let three line segments as in the diagram have two adjacent angles taken togetherequal to two right angles. Then the two straight lines making the two angles withthe third will be in a straight line with another. Thistooseemstowork.Ifthediagram is treated as part of the statement, we seem to have the same effectas from the verbally more complete Euclidean version. Even in Proclus, a