Between the 1870's and the appearance of A. Robinson's extraordinary work, "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 infinitely small. Unlike non-standard analysis, which is concerned with providing a treatment of the calculus making use of infinitesimals, most of the former work either focused on geometry and the concepts of number and magnitude, or grew out of the natural evolution of such discussions. Moreover, whereas non- standard analysis derives its arithmetic content from an exceptionally narrow, albeit an extremely important, subclass of the class of real-closed ordered number systems, the former work deals with arbitrary ordered number fields as well as with ordered general conception of numbers great and small, as well as with ordered number systems having failures of communitivity, associativity, etc. This so-called non-Archimedean mathematics thereby reveals a more general conception of numbers great and small, as well as a richer theory of the infinitely large and infinitely small. This body of work, published under the rubric The Theory of Non-Archimedean Geometric and (Totally) Ordered Algebraic Systems has continued to grow in recent decades producing a number of major accomplishments including J.H. Conway's work on Surreal Numbers. In this project, Dr. Ehrlich aims to provide a detailed sketch of the historical development of this largely ignored but philosophically important subject, and to develop a number of the philosophical implications of its results for the concepts of number, the infinite, the infinitesimal, and the continuum, as well as for a number of issues in the philosophy of measurement and the philosophy of space and time.
