History and Structure of Mathematics |
Contributions to a Science of Contemporary Mathematics (Draft October 4, 2011). Parts The goal of this essay is a description of modern mathematical practice, with emphasis on differences between the present day and the nineteenth century. I explain how and why these differences greatly increased the effectiveness of mathematics and enabled sweeping developments in the twentieth century. A particular concern is the significance for education: elementary mathematics education remains modeled on the mathematics of the nineteenth century and before, and this may explain why they have had trouble improving on nineteenth-century outcomes. Use of contemporary methodologies might give advantages similar to those seen in the modern profession. Recent changes: October 2011, minor revisions, fragment of chapter on set theory added. <\p> March 2011, Introduction extensively revised, many other revisions. November 4, More consistent emphasis on human-cognition aspects of mathematical practice. July 24, Level 4 of chapter 2 (evolution of methodology) substantially revised. May 5, 2010, section on mathematical objects finished. History of manifolds (incomplete, 2005). Diagram showing the development, with milestones; Outline of conceptual stages (lecture overheads). "Theoretical mathematics": Toward a cultural synthesis of mathematics and theoretical physics (with Arthur Jaffe), Bulletin of the American Math. Soc. Vol. 29 (1993) pp. 1-13 link to journal. Explores the tension between math and physics and considers ways to alleviate it. Initiated an interesting debate in the community. Response to comments on "Theoretical mathematics" , (with Arthur Jaffe), in Bulletin of the American Math. Soc. Vol. 30 (1994) pp. 208-211 link to journal. Responds to a wide variety of reactions solicited by the editors, in the same volume. Cultural adaptation in mathematics and physics, preprint 1996 HTML version, PDF version. Expands on the idea that the subject matter influences the culture and social structure of scientific disciplines. Illustrates this by examining cultural conflicts between pure math and theoretical physics. Atlas of Mathematics A map with regions for major subject divisions in the 1980 MR classification. Areas are proportional to the number of entries in the 1988 Annual Index, and related subjects are more-or-less adjacent. Goat Economics mp3 audio file. |