Independence and the Development of the American Identity and Mathemati

During the 1800s, we find the theme of independence, or freedom from orthogonal constraints, in the development of two assorted frontiers. We find it in the American West through and through Manifest Destiny, freedom from caste, and in the chance that homesteaders had to take for granted virtu all(prenominal)y free land. We find independence in math through in the building of stronger theoretical foundations, non-Euclidean geometries, and Cantors infinities. Independence involves breaking from the commonly accepted, traditionalistic views in order to explore the new. It is not necessarily individual hoi polloi working alone. We can see independence in a participation of thought as well as in the work of a single person. Independence is an important part of the Western culture as a supposition. 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 Gods plan. (Hine, 65) OSullivan f irst 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 fatten to come up to 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 conservativist in their views. Manifest Destiny was exemplified by the politicians. Stephen A. Douglas of Illinois declared that he would grease out the lines on the map which now marked our national boundaries . . .and pretend the area of liberty as broad as the 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 make on the basis that common notions are trut hs 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 its replacements. (Calculus) Use idea of separate a piece of paper in half for the limit. (Calculus) Newton and Leibniz highly-developed calculus in the 1600s.(Eves p.21) (Calculus) Little was done for a century after(prenominal) this toward logically strengthening the underpinnings of calculus. (Eves p.132) (Calculus) Karl Weinstraous (1815-1897) introduced epsilon delta notation. ( Eves p. 137, 139)