Throughout the 17th and 18th centuries, talk of infinitesimal line segments and numbers to measure them was commonplace in discussions of the calculus. However, as a result of the conceptual difficulties that arose from the misuse of these conceptions their role became more subdued in the 19th-century calculus discussions and was eventually "banished" therefrom. This is well known to historians and philosophers of mathematics alike. What is not so well known in these communities, however, is that whereas late 19th- and pre-Robinsonian 20th-century mathematicians banished infinitesimals from the calculus, they by no means banished them from mathematics. Indeed, contrary to what is widely believed by historians and philosophers, between the early 1870s and the appearance of Abraham Robinson's work on non-standard analysis in 1961 there emerged a large, diverse, technically deep and philosophically pregnant body of consistent (non-Archimedean) mathematics of the infinitely large and the infinitely small. Unlike non-standard analysis, which is primarily concerned with providing a treatment of the calculus making use of infinitesimals, the bulk of the former work is either concerned with the rate of growth of real-valued functions or with geometry and the concepts of number and of magnitude, or grew out of the natural evolution of such discussions. This important body of work, published under the rubric the theory of non-Archimedean ordered algebraic and geometric systems, has continued to grow since its inception producing a number of mathematically profound and philosophically significant accomplishments including Pejas's and Bachmann's work on the foundations and classification of non-Archimedean geometries, the modern theories of Hahn fields and Hardy Fields, and J.H. Conway's theory of surreal numbers. Unfortunately, like their historical forerunners, these important contributions to non-Archimedean mathematics are relatively unknown among historians and philosophers of mathematics of today. The purpose of this project is produce a monograph that will provide a philosophically sensitive in-depth historical account of the development of the theory of non-Archimedean ordered algebraic and geometric systems during its first golden period, which ranges from the publication of Giuseppe Veronese's Fondamenti di Geometria in 1891 to the appearance of Felix Hausdorff's Grundzuge der Mengenlehre in 1914.
Intellectual Merit. Besides helping to fill an important gap in the historical record, the proposed work will contribute to exposing and correcting the misconceptions regarding non-Archimedean mathematics alluded to above and to familiarizing historians and philosophers of mathematics with the full spectrum of theories of the infinite and the infinitesimal that have emerged from non-Archimedean mathematics since the latter decades of the 19th century.
Broader Impact. It is the author's hope that by drawing attention to the spectrum of theories of the infinite and the infinitesimal that have emerged from non-Archimedean mathematics since the latter decades of the 19th century, it will become clear that the standard 20th-century histories and philosophies of the actual infinite and the infinitesimal that are motivated largely by Cantor's theory of the infinite and by non-standard analysis (as well as by the more recent work in smooth infinitesimal analysis) are not only limited in scope but are inspired by an account of late 19th- and early 20th-century mathematics that is as mathematically myopic as it is historically flawed. Ultimately, it is the author's hope that the proposed work will both help to precipitate and substantially contribute to the development of a more encompassing and fully satisfactory history and philosophy of the actual infinite and the infinitesimal.
