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

the euclidean diagram (1995) 85of sentences in rigorous informal argument. Notably, they model restrictedreasoning contexts—hypothetical, reductio, and quantificational contexts—byassigning distinct roles to sentences depending on their position in thosecontexts. These distinct roles have the effect of keeping straight the hypothesesunder which an assertion is made, and conversely, what prior entries are inforce at any given stage of an argument. (This in contrast to systems such asFrege’s and Hilbert’s, which treat all entries in the inferential sequence equallyas assertions to which the reasoner is committed, differentiated as to inferentialrole only by their displayed structure and their order of occurrence. Thisrequires some re-casting, perfectly straightforward, of patterns of occurrencein informal reasoning, such as Frege’s explicit carrying along all hypotheses inforce as antecedents of each sentence.) Although we do not take all inferencesin the geometrical text to be purely logical, modeling patterns of sentenceoccurrence in restricted reasoning contexts seems the most illuminating formalapproach to geometric argument.With reference to this understanding, explicated in natural deduction systems,of being in force in a context, the claim becomes: proofs by contradictionnever admit of semantics in which each claim in force in the reductio context istrue (in any sense which entails their joint compatibility). In particular, neitheran imperfect diagram nor a perfect counterpart can picture or represent, inany sense which entails their joint compatibility, all claims in force within areductio context.²It does not follow that there could not be a picturing-like relationshipbetween the diagram and some claims in force within a reductio context.Rather, these facts put a different pressure on semantic conceptions of therelationship between diagram and text than the usual notice of imperfections(judged by the inferences licensed in the text) in the diagram, or (as in Leibniz)our ability to live with those imperfections. If diagram imperfections only werein play, one might well hold that the function of diagrams could fruitfullybe approached by first elaborating a notion of perfect geometricals of which² Juxtaposition of Euclidean diagram usage with natural deduction formalism suggests anotherremarkable conclusion. Euclid, by and large, lives by ‘one proof, one diagram’. But reductio contextor not, any broadly semantic conception of the relation between diagram and claims in force withina context would suggest instead: ‘one reasoning context, one diagram’. Here, however, the upshot isnot a restriction on the sense in which the relation between text and diagram could be a semanticone. Rather, the appropriate conclusion seems to be the narrowly logical one that Euclidean argumentprefers ‘one proof, one genuinely geometric reasoning context’: except for purely logical unpacking ofpropositions (which in standard natural deduction formalism would require multiple nested contexts)there is preferably only one reasoning context per demonstration in which a diagram plays a specialrole. We have to allow disjunctive case analysis with distinct diagrams (see later), but that too appearsnot preferred.