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

diagram-based geometric practice 73ensure that general claims are justified? The key point is how proceedingsin UG scopes are un-responsive, at the level of inference rules, to distinctivefeatures of instances.3. Modern logic employs what one might call representation-enforced (forshort, representational) unresponsiveness: the UG letter, placeholder for aninstance, lacks features on which differential responses to non-shared features ofinstances might be based. This ensures that formal inferences cannot depend onsuch distinctive features, and hence apply to all instances uniformly. Algebraicmethods in geometry since Descartes also exploit representational unresponsiveness:a letter, originally standing for a Euclidean line segment, lacks featuresto preclude it holding the place of a negative or even complex quantity, allowingone to display uniformities lacking uniform traditional geometric proof.Drawn geometric diagrams, however, are not mere placeholders; theydisplay endless distinctive features to which geometrically pertinent responsesmight be made. It is therefore sometimes suggested, assimilating Euclideanreasoning generality to the representational unresponsiveness model, thatproper Euclidean proof requires generic diagrams, ones only displaying featuresthat the demonstration indeed attributes.Such a conception does not allow uniform treatment of diagrams: thosewith, for example, irrelevant right angles or equalities of sides could not beused in demonstrations. But if there are generic diagrams, this objection wouldnot undercut the ability of demonstrations using them to justify their claimsalso for instances with exceptional (but non-singular) diagrams that do haveall features the demonstration attributes; even if they have further features aswell. (That Locke somehow lacks this flexibility would threaten to render hisidea-of-a-triangle paradoxally equilateral.)Nonetheless, among further objections to generic-diagram conceptions ofdemonstration (medieval manuscripts notoriously use non-generic diagrams,which may or may not reflect earlier practice; how generic must a genericdiagram be?), the emergence of multiple non-exceptional cases seems fatal:because each case demonstration requires attributions that are false in the othercases, such proofs have no generic diagram spanning their UG scope.4. Representation-enforced unresponsiveness to distinctive features of instances,however, is not the only appropriate way to justify general conclusions!Even if the (placeholder for the) instance has distinctive features to whichhumans might respond with claims of the very sort being justified, standardsof reasoning may sucessfully prohibit such differential response.This is fundamental to the generality- and understanding-generating qualityof a great range of reasoning (including the use of examples in ethics). Evensome formal systems of quantifier logic enforce uniformity across instances