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

102 kenneth mandersconstructions. Examples would be uniqueness results such as I.7, orargumentsusing circumcircle and Simson-line to determine the appearance of the alltriangles-are-isocelesdiagram (Dubnov or Maxwell, pp. 24–25). Arguments ofthis sort shift part of the burden of grasp of construction outcomes, from what‘just happens’ in making diagrams with proper discipline (purely diagram-based)to a mixture of diagram- and discursive text-based moves.Of course, geometrical constructions are in general only quasi-determinate:adding elements to a diagram by a prescribed construction may yield finitelymany diagrams from any given initial diagram, rather than just one; as inthe two circle intersections in I.1. Instead of proving uniqueness, the taskin settling a theoretical question (such as whether all triangles are isoceles)becomes to determine all diagrams that have standing as the result of theconstructions indicated; and verify that they have the properties claimed ofthe construction. Moreover, it is understood that metrically different initialdiagrams with the same appearance may lead, via the same construction, todifferent finite collections of result-diagrams. (We will return to this.)In the example at hand, however, the question involves properties of theinitial diagram only: whether the given triangle is isoceles. It is then intolerable,whether or not the auxiliary constructions are unique in the ordinary sense,as they are here, that after their application to one and the same initial diagram,one result-diagram leads to a proof of the claim and another to its disproof.Because the claim considered involves attributions to the initial diagram only,conflicting conclusions (that the triangle is both isoceles and non-isoceles) fromdistinct outcomes of the constructions indicated, applied to the same initialdiagram, would—except in a reductio context—put singular pressure indeedon any sense that geometric property attribution behaves like a truth predicate(or assertability): holds indifferently of the wider context of the objects ofthe claim.Until somehow relieved, such a situation would constitute an extremeform of disarray; quite distinct from that which one would encounter ifunable to resolve which of two diagrams would be the outcome of oneway of carrying out the construction, and the two potential outcomes led toconflicting conclusions on the main claim: then one would just not have aproof of either conclusion, perhaps a disconcerting situation but a thoroughlyfamiliar one. Indeed, if what looked to us all along as properties of a diagramwere instead implicitly relational qualities with respect to the wider contextof the objects of the claim, this would undercut one of the key contributionsof the discursive text in geometrical practice: it mediates application relationsbetween geometrical arguments, indifferent both to how the configurationto which the lemma applies is embedded in a diagram in the application,