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

introduction 15classic that finally sees publication—he addresses the problem of the stabilityof diagrammatic reasoning in Euclidean geometry.If mathematicians cared only about the truth of certain results, it would behard to understand why after discovering a certain mathematical truth theyoften go ahead to prove the result in several different ways. This happensbecause different proofs or different presentations of entire mathematical areas(complex analysis etc.) have different epistemic virtues. Explanation is amongthe most important virtues that mathematicians seek. Very often the proof ofa mathematical result convinces us that the result is true but does not tell uswhy it is true. Alternative proofs, or alternative formulations of entire theories,are often given with this explanatory aim in mind. In the introduction, PaoloMancosu (U.C. Berkeley) shows that the topic of mathematical explanationhas far-reaching philosophical implications and then he proceeds, in the jointpaper with Johannes Hafner (North Carolina State), to test Kitcher’s modelof mathematical explanation in terms of unification by means of a case studyfrom real algebraic geometry.Related to the topic of epistemic virtues of different mathematical proofsis the ideal of Purity of methods in mathematics. The notion of purity hasplayed an important role in the history of mathematics—consider, for instance,the elimination of geometrical intuition from the development of analysis inthe 19th century—and in a way it underlies all the investigations concerningissues of conservativity in contemporary proof theory. That purity is oftencherished in mathematical practice is made obvious by the fact that Erdös andSelberg were awarded the Fields Medal for the elementary proof of the primenumber theorem (already demonstrated with analytical tools in the late 19thcentury. But why do mathematicians cherish purity? What is epistemologicallyto be gained by proofs that exclude appeal to ‘ideal’ elements? Proof theoryhas given us a rich analysis of when ideal elements can be eliminated inprinciple (conservativity results), but what proof theory leaves open is thephilosophical question of why and whether we should seek either the useor the elimination of such ideal elements. Michael Detlefsen (University ofNotre Dame) provides a general historical and conceptual introduction tothe topic. This is followed by a study of purity in Hilbert’s work on thefoundations of geometry written by Michael Hallett (McGill University). Inaddition to emphasizing the epistemic role of purity, he also shows that inmathematical practice the dialectic between purity and impurity is often verysubtle indeed.Mathematicians seem to have a very good sense of when a particularmathematical concept or theory is fruitful or ‘natural’. A certain concept mightprovide the ‘natural’ setting for an entire development and reveal this by its