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

84 kenneth mandersirrelevant, unable to provide its participants a non-trivial grip on life: whateverparticipants have to go on in the game must support distinctions where differentactions are required.If the only response ever required to p 1 and p 2 is q, we could disregardthe difference between p 1 and p 2 ,alwayswritep 1 ,oralwayswritep. Butparticipants typically can’t assess ‘only response ever required’, nor—here ourschematic p 1 , p 2 notation breaks down—can they typically assess what rangeof alternatives might ultimately be relevant. In its diagram use, Euclideangeometry ‘writes p 1 ’ until it becomes clear some ‘p ′ ’ could require a differentresponse. A diagram adequate at one stage of a proof may upon developmentbe ‘un-representative’; this forces a case distinction in which multiple diagramsare considered separately. The peculiar success of various later geometries restsnot only on their surprising reduction of diversity in artifact and response, butalso on the way the correlative collapse of ranges of alternatives, quite as aby-product, often eliminates imponderables that participants in diagram-basedgeometry cannot count on being able to assess.The way its artifacts lie in the game is not innocent to the control its playerscan achieve.4.1.2 Diagrams and semantic roleArtifacts in a practice that gives us a grip on life are sometimes thought of insemantic terms—say, as representing something in life. There is, of course, anage-old debate on how geometrical diagrams are to be treated in this regard.Long-standing philosophical difficulties, on the nature of geometric objectsand our knowledge of them, arise from the assumption that the geometricaltext is in an ordinary sense true of the diagram or a ‘perfect counterpart’. Thesedifficulties aside, a genuinely semantic relationship between the geometricaldiagram and text is incompatible with the successful use of diagrams in proof bycontradiction: reductio contexts serve precisely to assemble a body of assertionswhich patently could not together be true; hence no genuine geometricalsituation could in a serious sense be pictured in which they were.¹ Thissimple-minded objection has nothing to do specifically with geometry: proofsby contradiction never admit of semantics in which each entry in the proofsequence is true (in any sense which entails their joint compatibility).To help be a bit more precise: I take it that among logical formalisms, naturaldeduction systems reconstruct most directly the actual patterns of occurrence¹ David Sherry has directed this argument at Berkeley’s conception of the role of diagrams, anddraws a similar conclusion (Sherry, 1993, p.214). But Sherry makes only incidental proposals on howdiagrams support inferences (p. 217) and expects to invoke Pasch-style axioms (n. 5).