Mathematics, a Creative Art

Description

Mathematics is a creative art. Words and symbols are used in its compositions and, in all art, it is limited only by the potentialities of its practitioners. Few people possess the gifts required to produce great mathematics, but almost every student can share the creative spark. We will develop some simple mathematics and thereby glimpse the characteristics and the content of certain topics of current interest. We will find that some of our results contrast sharply with others, because the mathematician can construct any world of mathematical imagination he chooses. We will discuss the laws that help ton ensure that each system is consistent within itself, and we will consider some of the criteria bay which the worth of our conclusions may be judged. Most important, we will have an opportunity to experience the excitement and the satisfaction that come from using our minds creatively. 1. Algebraic systems. 1.1 The five-element system: addition. 1.2 The five-element system: multiplication. 1.3 The five-element system: further properties. 1.4 A six-element system. 1.5 A system of pairs of elements. 1.6 A second system of pairs of elements. 1.7 A system of matrices. 2. Logic and sets. 2.1 The algebra of logic. 2.2 The algebra of sets. 3. Geometry. 3.1 Introduction to geometry. 3.2 Hyperbolic geometry. 3.3 Elliptic geometry. 4. Appendix: Geometry. 5. Hints.