III - The Nature of Mathematical Objects and Mathematical Knowledge

Summary

In the last forty years, philosophers of mathematics who are not working strictly in foundations have concentrated on questions about the nature of mathematical objects and how we come to have mathematical knowledge. Because this work has resulted in hundreds of papers and dozens of books, we have four chapters summarizing it. They were written by philosophers with very different perspectives. While there is a common set of questions running through these chapters, each has chosen different aspects in his summary, because of the difference of perspective. Of the philosophers whose chapters are in this section, Chihara has spent his career working on various versions of nominalism, the view that there are no mathematical objects. Shapiro has leaned toward the realist side, currently in a version called structuralism, which has origins in Bourbaki's mother-structures and the view of mathematics as the science of patterns. Balaguer has most recently suggested that there may be no testable distinction between the most appealing versions of platonism (or realism) and nominalism. Linnebo, the youngest of the authors in this section, appears to be working on developing a very minimal version of platonism (that is, a commitment to mathematical objects that involves a minimal “ontological” commitment). Each of these chapters sets forth the general argument overall and then gives the individual author's perspective on where the delicate points are. We end the section with a chapter by a mathematician, offering a very different approach to the question of mathematical objects via category theory.