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

what structuralism achieves 355Philosophers have pursued their own versions of structural mathematicswithout always specifying which of their ideas aim at purely philosophicalconcerns and which are meant to explicate existing practice.³ Yet even thepurely philosophical projects can benefit from seeing how mathematiciansmake structure work. The philosophical efforts face problems that were solvedlong ago by mathematicians who rely daily on the answers: How can separatestructures relate to one another? Is there a coherent way to define structuresas themselves places in larger structures? Since Emmy Noether, in fact,mathematicians routinely describe structures entirely by the structural relationsamong them. This indeed gives structural definitions such as Benacerraf soughtbut that is incidental to the goals of practice. In practice, structural mathematicsachieves the discovery and proof of great theorems.We might worry that mathematicians are pragmatists, not concerned withour issues, and so mathematical practice might be a poor guide for philosophersin general and structuralists in particular:• Mathematics might lack the conceptual rigor of philosophy.• Practice might violate some favored ontology or epistemology.• Mathematics might be structural in principle yet not in practice.But these concerns underestimate the practical need for rigorous principles.For example Andrew Wiles proved Fermat’s Last Theorem as a step in theLanglands Program, the largest project ever posed in mathematics. He citesscores of theorems so advanced that he cannot assume an audience of expertsknows them or knows offhand where to find them. In mathematics on thisscale it is completely out of the question for one author or even a small numberof authors to give a major proof in full starting from just concepts and theoremsfound in graduate textbooks.⁴So there are feasibility conditions:• Each theorem must be concise and fully explicit, so as to be easily andrigorously detachable from its original context. Proofs are often very long,and for just this reason the statement of each theorem must be brief andrigorously self-contained.• Theorems must be stated structurally. If, per impossibile, authors couldbe made to choose set theoretic definitions of each structure they use,³ Chihara seeks ways mathematics could be ‘nominalist’ although it is not in practice (2004, p.65).Hellman (1989) andShapiro(1997) say little relating their ideas of modality, coherence, and structureto practice. Resnik (1997), Awodey (2004), and Carter (2005) offer explications.⁴ A book aimed at readers already specialized in number theory takes twenty-two authors and 570pages not to prove but merely to introduce the results that Wiles uses (Cornell et al., 1997).