Riemannian Geometry

Abstract

Mathematics enjoys special esteem above all the other sciences for several reasons. Its laws are absolutely certain and indisputable. This priority would not be momentous if the laws of mathematics referred only to objects of one’s imagination. However, it is mathematics which affords the exact natural sciences a certain measure of security, to which without mathematics they could not attain. A fitting exemplification of this role of mathematics can be seen in the mathematics of relativity, one aspect of which is treated in this paper. The explanations of Riemannian geometry and of the general theory of relativity given herein are by no means complete. However, it is hoped that they will provide the reader with a general insight into the relation between geometry and physics. Werner Heisenberg’s statement regarding the law of mathematics adequately sums up this thesis: The elemental particles of modern physics, like the regular bodies of Plato’s philosophy, are defined by the requirements of mathematical symmetry. They are not eternal and unchanging, and they can hardly, therefore, strictly be termed real. Rather, they are simple expressions of fundamental mathematical constructions which one comes upon in striving to break down matter even further, and which provide the content for the underlying laws of nature. In the beginning, therefore, for modem science, was the form, pattern, not the material thing.Mathematics enjoys special esteem above all the other sciences for several reasons. Its laws are absolutely certain and indisputable. This priority would not be momentous if the laws of mathematics referred only to objects of one’s imagination. However, it is mathematics which affords the exact natural sciences a certain measure of security, to which without mathematics they could not attain. A fitting exemplification of this role of mathematics can be seen in the mathematics of relativity, one aspect of which is treated in this paper. The explanations of Riemannian geometry and of the general theory of relativity given herein are by no means complete. However, it is hoped that they will provide the reader with a general insight into the relation between geometry and physics. Werner Heisenberg’s statement regarding the law of mathematics adequately sums up this thesis: The elemental particles of modern physics, like the regular bodies of Plato’s philosophy, are defined by the requirements of mathematical symmetry. They are not eternal and unchanging, and they can hardly, therefore, strictly be termed real. Rather, they are simple expressions of fundamental mathematical constructions which one comes upon in striving to break down matter even further, and which provide the content for the underlying laws of nature. In the beginning, therefore, for modem science, was the form, pattern, not the material thing.