On Physics, Physicists and Philosophy by N. Mukunda

Author:N. Mukunda
Language: eng
Format: epub
Publisher: Indian Academy of Sciences
Published: 2018-07-15T00:00:00+00:00

4. Beyond Euclidean Geometry

After this brief excursion into biology, let us come back to mathematics. Euclid’s geometry has been admired and studied for over two thousand years. As we have already said, the approach starts with a very small number of fundamental postulates, and builds upon them by logical argument. But from very early times people were puzzled by the status of the fifth postulate of Euclid, also known as the ‘parallel postulate’.5 Was it independent of the first four postulates, or could it be derived from them (in which case there would be no need for an additional fifth postulate)? Eminent thinkers over the centuries pondered over this problem, from the astronomer and geographer Claudius Ptolemy (circa 85–165 AD) to the famous mathematician Adrian-Marie Legendre (1752–1833). Many equivalent reformulations of this postulate were found, but the unchanging belief was that the geometry of space had to be Euclidean, for the sake of consistency. Along the way, René Descartes (1596–1650) invented the method of coordinates to represent points in space, so that the powerful mathematics of algebra became available for tackling problems in geometry.6 As we have seen, Kant subsequently expressed the view that physical space had necessarily to obey the laws of Euclidean geometry, and that there was no other option.

A few decades after Kant, in the early 1800’s, the final resolution of this long-standing problem occurred. The fifth postulate was indeed independent of the others, since it could be replaced in a self-consistent manner by other assumptions, and one could think of alternatives to Euclid’s geometry. This breakthrough was achieved independently by three mathematicians–the incomparable Carl Friedrich Gauss (1777–1855), around 1824, building on his theory of curved surfaces; Nikolai Ivanovich Lobachevsky (1792–1856), in 1829; and János Bolyai (1802–60) in 1832. Thus was born the subject of non-Euclidean geometry.7

Around the middle of the 19th century, Gauss’ gifted student and one of the greatest of mathematicians, Bernhard Riemann (1826–66), took the next big step: the creation of differential geometry. He presented his ideas in his famous probationary lecture titled On the hypotheses that lie at the foundations of geometry, given at Göttingen on June 10, 1854. The fundamental insight was to determine the geometry of a space starting from the definition of (the square of) the interval or distance between nearby points in the space. From this, the concepts of tensors, parallel transport, intrinsic differentiation, connection, curvature, etc., could all be developed. This was a stupendous achievement, and many great contributions from a galaxy of geometers followed.8 In particular, Riemann’s methods were powerful enough to deal with spaces of any number of dimensions.

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.