§ i. After deserting for a time the old Euclidean standards of rigour, mathematics is now returning to them, and even making efforts to go beyond them. ...

In arithmetic, if only because many of its methods and concepts originated in India, it has been the tradition to reason less strictly than in geometry, ...

History and Philosophy of Modern Mathematics was first published in 1988. Minnesota Archive Editions uses digital technology to make long-unavailable books once again accessible, and are published unaltered from the original University of Minnesota Press editions. The fourteen essays in this volume build on the pioneering effort of Garrett Birkhoff, professor of mathematics at Harvard University, who in 1974 organized a conference of mathematicians and historians of modern mathematics to examine how the two disciplines approach the history of mathematics. In (...) History and Philosophy of Modern Mathematics, William Aspray and Philip Kitcher bring together distinguished scholars from mathematics, history, and philosophy to assess the current state of the field. Their essays, which grow out of a 1985 conference at the University of Minnesota, develop the basic premise that mathematical thought needs to be studied from an interdisciplinary perspective. The opening essays study issues arising within logic and the foundations of mathematics, a traditional area of interest to historians and philosophers. The second section examines issues in the history of mathematics within the framework of established historical periods and questions. Next come case studies that illustrate the power of an interdisciplinary approach to the study of mathematics. The collection closes with a look at mathematics from a sociohistorical perspective, including the way institutions affect what constitutes mathematical knowledge. (shrink)

Research on human infants has begun to shed light on early-developing processes for segmenting perceptual arrays into objects. Infants appear to perceive objects by analyzing three-dimensional surface arrangements and motions. Their perception does not accord with a general tendency to maximize figural goodness or to attend to nonaccidental geometric relations in visual arrays. Object perception does accord with principles governing the motions of material bodies: Infants divide perceptual arrays into units that move as connected wholes, that move separately from one (...) another, that tend to maintain their size and shape over motion, and that tend to act upon each other only on contact. These findings suggest that o general representation of object unity and boundaries is interposed between representations of surfaces and representations of objects of familiar kinds. The processes that construct this representation may be related to processes of physical reasoning. (shrink)

Ethics and jurisprudence are liable to the remark in common with logic. Almost every writer having taken a different view of some of the particulars which ...