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

110 kenneth mandersLet’s spell out the conditions. We write ‘LineSeg(ADB)’ to indicate a linesegment (in the antiquated sense of line found in Euclid, which does not entailthat the ‘line’ is straight, rather than, say, a circle, ellipse, or other simplenon-self-intersecting curve), with endpoints A and B, and an intermediatepoint D. When the dust has settled on the assumption for reductio, we havethe co-exact conditions indicated in the diagram:(d-i) LineSeg(ADB), (d-ii) LineSeg(AC), (d-iii) LineSeg(BC),(d-iv) LineSeg(DC) ,(d-v) NonNull(AD);and the exact conditions indicated in the discursive text:(t-i) Straight(ADB), (t-ii) Straight(AC), (t-iii) Straight(BC),(t-iv) Straight(DC), (t-v) angle(ABC) = angle(ACB), (t-vi) DB = AC.Of these exact conditions, the diagram we find in Euclid is subject to(t-i)–(t-v), but, reflecting the unsatisfiability of the assumptions in forcefor reductio, not (t-vi). Any other diagram, however, indicating precisely(d-i)–(d-v) subject to (t-i)–(t-iv)—that is, with straight segments ADB, BC,DC, AC—would be topologically equivalent; any such diagram would dojust as well to display all geometrical objects and their inclusions required forthe proof. The exact conditions here do contribute to the topology, to someextent: if AC were curved, say, it might intersect CD. Of all the constraints onthe diagram, at most (d-v) and the straight-line combinations (of (d-i)–(d-iv)and (t-i)–(t-iv) respectively) are relevant to the conclusions which the argumentreads off from the diagram as it proceeds:(d-vi) Region(DBC), with precisely three angles: Angle(BCD), Angle(CDB), Angle(DBC)–hence, because the sides are straight, Triangle(DBC);(d-vii) region(DBC) IsPartOf region(ABC)—hence, because they aretriangles, triangle(DBC) IsPartOf triangle(ABC) = triangle(ACB).From (d-vi) and (t-i)–(t-vi), the text applies the side-angle-side congruencecriterion I.4, giving:(t-vii) triangle(DBC) = triangle(ACB),which is incompatible with (d-vii)—parts are proper parts.That the diagram is not subject to (t-vi) is thus not terribly significant. Theequalities (t-v)–(t-vii) are irrelevant to the inferential use of the diagram here:they would in any case directly license moves only because they are indicatedin the discursive text. Indeed, the diagram for the reductio proof of Euclid I.7appears subject to neither segment equality in question there, nor to eitherangle equality obtained in the course of the proof.