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

14 paolo mancosuto the two former collections, this book does not trace its inspiration backto Lakatos. It contains a wide range of contributions on visualization, explanationand reasoning styles in mathematics carried out both by philosophersand historians of mathematics. The above-mentioned volumes contain contributionsthat overlap in topic and/or inspiration with those of the presentcollection. However, this collection is more systematic and more focused inits aims.The eight topics studied here are:1) Visualization2) Diagrammatic reasoning3) Explanation4) Purity of methods5) Concepts and definitions6) Philosophical aspects of uses of computer science in mathematics7) Category theory8) Mathematical physicsTaken all together, they represent a broad spectrum of contemporaryphilosophical reflection on different aspects of mathematical practice. Eachauthor (with one exception to be mentioned below) has written a generalintroduction to the subject area and a research paper in that area. I will nothere summarize the single contributions but rather point out why each subjectarea is topical.The first section is on Visualization. Processes of visualization (e.g. bymeans of mental imagery) are central to our mathematical activity and recentlythis has become once again a central topic of concern due to the influence ofcomputer imagery in differential geometry and chaos theory and to the call forvisual approaches to geometry, topology, and complex analysis. But in whatsense can mental imagery provide us with mathematical knowledge? Shouldn’tvisualization be relegated to heuristics? Marcus Giaquinto (University CollegeLondon) argues in his introduction that mathematical visualization can play anepistemic role. Then, in his research paper, he proceeds to examine the role ofvisual resources in cognitive grasp of structures.The second section is entitled Diagrammatic reasoning. In the last twentyyears there has been an explosion of interest in this topic due also to theimportance of such diagrammatic systems for artificial intelligence and theirextended use in certain branches of contemporary mathematics (knot theory,algebraic topology, etc.). Kenneth Manders (University of Pittsburgh) focusesin his introduction on some central philosophical issues emerging from diagrammaticreasoning in geometry and in his research paper—an underground