Algebraic Structures of Mathematical Foundations

In this paper, we continue to examine the role of algebra underlying geome- try. The mathematics that we examine are abstract and are quite far removed from applications. Those who may wish to examine the possible uses of ab- stract geometry in physics might well begin by reading (Schlichenmaier, 2007; Bub, 1997). The method of research that we apply is to use the logic of H. S. Leonard to extract the underlying algebra of mathematical proofs and then use the results for the purpose of finding the algebra that underlies mathe- matical results.