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

16 paolo mancosufruitfulness in unifying a wide group of results or by opening unexpectednew vistas. But when mathematicians appeal to such virtues of concepts(fruitfulness, naturalness, etc.) are they simply displaying subjective tastes orare there objective features, which can be subjected to philosophical analysis,which can account for the rationality and objectivity of such judgments? This isthe topic of the introductory chapter on Concepts and Definitions writtenby Jamie Tappenden (University of Michigan) who has already published onthese topics in relation to geometry and complex analysis in the 19th century.His research paper ties the topic to discussions on naturalness in contemporarymetaphysics and then discusses Riemann’s conceptual approach to analysis asan example from mathematical practice in which the natural definitions giverise to fruitful results.The influence of Computer Science on contemporary mathematics hasalready been mentioned in connection with visualization. But uses of thecomputer are now pervasive in contemporary mathematics, and some of theaspects of this influence, such as the computer proof of the four-color theorem,have been sensationally popularized. Computers provide an aid to mathematicaldiscovery; they provide experimental, inductive confirmation of mathematicalhypotheses; they carry out calculations that are needed in proofs; and theyenable one to obtain formal verification of proofs. In the introduction to thissection, Jeremy Avigad (Carnegie Mellon University) addresses the challengesthat philosophy will have to meet when addressing these new developments.In particular, he calls for an extension of ordinary epistemology of mathematicsthat will address issues that the use of computers in mathematics make urgent,such as the problem of characterizing mathematical evidence and mathematicalunderstanding. In his research paper he then goes on to focus on mathematicalunderstanding and here, in an interesting reversal of perspective, he showshow formal verification can assist us in developing an account of mathematicalunderstanding, thereby showing that the epistemology of mathematics caninform and be informed by research in computer science.Some of the most spectacular conceptual achievements in 20th centurymathematics are related to the developments of Category Theory and itsrole in areas such as algebraic geometry, algebraic topology, and homologicalalgebra. Category theory has interest for the philosopher of mathematics bothon account of the claim made on its behalf as an alternative foundationalframework for all of mathematics (alternative to Zermelo–Fraenkel set theory)as well as for its power of unification and its fruitfulness revealed in the abovementionedareas. Colin McLarty (Case Western Reserve University) devoteshis introduction to spelling out how the structuralism involved in muchcontemporary mathematical practice threatens certain reductionist projects