Independence and the Development of the American Identity and Mathemati

During the 1800s, we find the theme of independence, or freedom from outside constraints, in the development of two different frontiers. We find it in the American West through Manifest Destiny, freedom from caste, and in the chance that homesteaders had to acquire virtually free land. We find independence in math through in the building of stronger theoretical foundations, non-Euclidean geometries, and Cantor's infinities. Independence involves breaking from the commonly accepted, traditional views in order to explore the new. It is not necessarily individual people working alone. We can see independence in a community of thought as well as in the work of a single person. Independence is an important part of the Western culture as a concept. We find this in the concept of Manifest Destiny. This concept began when the Puritan immigrants "interpreted their victories over the Indians as part of God's plan." (Hine, 65) O'Sullivan first defined this term. He "coined one of the most famous phrases in American history when he insisted on 'our manifest destiny to overspread the continent.'"(Hine 199) Although Manifest Destiny meant different things to different people, the general definition was that God ordained the United States should expand to cover some undefined area. (Merk p.24) Some thought that the United States should cover the entire continent, and perhaps even South America, but others were more conservative in their views. Manifest Destiny was exemplified by the politicians. Stephen A. Douglas of Illinois declared that he "would blot out the lines on the map which now marked our national boundaries . . .and make the area of liberty as broad as th e continent itself." (Hine 199) The concept of Manifest Destiny is a... ...eings perceive the world. (Quote Delvin p.163) (Euclidean Geometry) "The distinction between common notions and postulates was made on the basis that common notions are truths common to all knowledge, all the sciences, whereas postulates are truths within the field of geometry.(Witter p.232) (Geometry)Klein generalized geometries even further by exploring patterns of different geometries. (Delvin p.199) (Geometry) Make pictures of the parallel postulate and it's replacements. (Calculus) Use idea of tearing a piece of paper in half for the limit. (Calculus) Newton and Leibniz developed calculus in the 1600s.(Eves p.21) (Calculus) Little was done for a century after this toward logically strengthening the underpinnings of calculus. (Eves p.132) (Calculus) Karl Weinstraous (1815-1897) introduced epsilon delta notation. ( Eves p. 137, 139)