Craig Spencer, M.S.

An interest in developing a differential version of synthetic geometry is motivated by its potential for gravitational theory, especially a version which might gracefully incorporate spin. This thesis only addresses the needed mathematical foundations; physical applications, if there are any, will have to come later. An axiom set for two dimensions is proposed, some of its consequences explored and, to a certain point, considerable progress is made.

• Front Matter

• Chapter 1 Introduction
  What is the essential nature of physical space? This is a physical question but the answer sought is a mathematical one -- a mathematical description faithful to reality. The classical geometry of the ancient Greeks is at least such an excellent approximation that it surely must contain something of the answer to this question. But what about it is central and universal and what is inessential and particular? How can the physically appropriate generalization be identified?

  The discovery, during the Renaissance, that geometry could be modeled in algebraic terms gave rise to analytic geometry. This provided the basis on which Riemann developed a differential generalization of geometry. But analytic geometry introduces artifacts which pertain to the model and are extraneous to the geometry itself. It is, as a result, characteristically difficult, in this form of geometry, to separate out the essential from the extraneous. The question then arises as to the nature of the Riemannian generalization. Is it purely a generalization of the geometry or is it, at least in part, a generalization of inessential aspects of the particular model? Is it inadvertently specialized in an essential way (at least for use as a description of physical space) by the peculiarities of the algebraic model or its Euclidean basis? Such imponderable questions cannot, of course, be answered definitively. However, it might be possible to demonstrate other paths leading to alternative generalizations.

• Chapter 2 Synthetic Geometry
  Synthetic geometry is that kind of geometry which deals purely with geometric objects directly endowed with geometrical properties by abstract axioms. This is in contrast with a procedure which constructs "geometric objects" from other things; as, for example, analytic geometry which, with the artifice of a coordinate system, models points by n-tuples of numbers. Synthetic geometry is the kind of geometry for which Euclid is famous and that we all learned in high school.

  Modern synthetic geometry, however, has a more logically complete and consistent foundation. In this chapter the pattern of this foundation will be adapted, informed by the previous physical considerations, to develop a synthetic system of axioms which do not entail such things as uniformity or isotropy. This geometry is a global one but it is hoped that its elaboration, like that of Euclidean geometry, will be instructive for the development of a similar, but more general, local theory.

• Chapter 3 Fundamentals
  Some fundamental properties and initial concepts consequent tot he Axioms will be developed in this chapter. These will provide the foundation for introducing concepts concerning direction later.

• Chapter 4 Two Dimensions
  The initial results which have been obtained so far are independent of dimension and it is dimension 4 which is of physical interest. However, most the rest of this paper will be specialized to the case of two dimensions by taking Pasch's Postulate as a new axiom. When doing research it is always methodologically prudent to investigate simple cases because what is learned in solving them is often the key to the harder ones; this is the stage of the present work.

• Chapter 5 Direction
  The concept of direction is a particularly important geometric idea. It is remarkable that, unlike familiar Euclidean geometry where the measurment of angles requires the introduction of a new axiom, direction is already a consequence of the more general axiom system. Direction is no longer an independent concept; it is integrated into the geometry in a more fundamental way.

• Appendix
• Summary
• Bibliography