Abstract:

Many mathematicians, from antiquity to the present, have expressed a preference for proofs of statements that are "pure". Roughly, a proof of a statement is "pure" if it draws only on what is "close" or "intrinsic'' to that statement, rather than what is "distant", "extraneous" or "alien" to it. This preference raises many questions—for instance, what reasons, if any, are there for preferring pure proofs over impure proofs?—but a precondition of answering these is getting clearer on what grounds the judgments of "distance" underpinning purity talk. A promising starting point for carrying out this latter task is to investigate how such judgments arise in practice. This talk considers one such case from geometry, concerning the search for a purely projective proof of Desargues' theorem on homological triangles. In the late nineteenth century geometers showed that while purely projective proofs of Desargues' theorem are available, such proofs necessarily draw on spatial considerations, despite the statement's formulation in purely planar terms. If proofs of planar statements using spatial considerations are impure (as they have been judged by many geometers), then we seem to have a statement for which no pure proof is possible. In a recent article, though, Michael Hallett has argued, drawing on a key result of Hilbert, that Desargues' theorem has tacit spatial content, and so spatial considerations ought to be judged acceptable in pure proofs of it. Hallett's argument has several problems, but the central issue is with Hallett's conception of content. I will make the case that Hallett's conception of content, while grounded in particular important mathematical practices, is different than the conception at play in the practice of seeking pure proofs. Thus this case brings out a variety of types of content at play in mathematical practice, and provides a springboard for the investigation of these different conceptions of content, a matter of mainstream philosophical interest.