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

understanding proofs 345procedures that are immediately available to us, in ordinary mathematics, whenwe recognize one algebraic structure as present in another.The work that has been done is a start, but proof assistants are still a longway from being able to support algebraic reasoning as it is carried out inintroductory textbooks. Thus, a good deal more effort is needed to determinewhat lies behind even the most straightforward algebraic inferences, and howwe understand simple algebraic proofs.12.8 Understanding Euclidean geometryOur third example will take us from modern mathematics to the mathematicsof the ancients. Over the centuries, the style of diagram-based argumentationof Euclid’s Elements was held to be the paradigm of rigor, and presentationsmuch like Euclid’s are still used today to introduce students to the notionof proof. In the 19th century, however, Euclidean reasoning fell from grace.Diagrammatic reasoning came to be viewed as imperfect, lacking sufficientmathematical rigor, and relying on a faculty of intuition that has no place inmathematics. The role of diagrams in licensing inference was gradually reduced,and then, finally, eliminated from most mathematical texts. Axiomatizationsby Pasch (1882) andHilbert(1899), for example, were viewed as ‘corrections’or improvements to the flawed methods of Euclid, filling in gaps that Euclidoverlooked.As Manders points out in his contributions to this volume, this view ofEuclidean geometry belies the fact that Euclidean reasoning was a remarkablystable and robust practice for more than two thousand years. Sustained reflectionon the Elements shows that there are implicit rules in play, and norms governingthe use of diagrams that are just as determinate as the norms governing modernproof. Given the importance of diagrammatic reasoning not just to the historyof mathematics but to geometric thought today, it is important to understandhow such reasoning works.In order to clarify the types of inferences that are licensed by a diagram,Manders distinguishes between ‘exact’ conditions, such as the claim that twosegments are congruent, and ‘coexact’ conditions, such as the claim thattwo lines intersect, or that a point lies in a given region. Coexact conditionsdepend only on general topological features of the diagram, and arestable under perturbations of a diagram, whereas exact conditions are not.Manders observes that only coexact claims are ever inferred from a diagramin a Euclidean proof; text assertions are required to support an inference