Sambodhi 1977 Vol 06

Author(s): Dalsukh Malvania, H C Bhayani, Nagin J Shah
Publisher: L D Indology Ahmedabad

Page 34

S. S. Pokharna
material system in a homogeneous space it becomes inhomogencouis (say because of the gravitational field which is present because of the material system itself). Hence the space in the presence of a mass is not homogeneous and the definition of linear momentum becomes ambiguous. Furthermore, our universe is inhomogeneous, so the law of conservation of momentum is not strictly true. (The ordinary definition of lincar momentum of a body as a product of its mass and velocity is derivable from the above general definition.) . A similar story can be written for other fundamental conserved quantities. As far as velocity and position measurements are concerned we have already discussed that they can be only measured approximately with some uncertainity.
It can be easily realized that all other properties of any system are derivable from tha above conserved quantities e. g, temperature of a systeni is related with the kinetic energy of the particles of the systems, Thus the quantities themselves which are used for description of different systems are not properly defined.
(e) Godel's incompleteness theorems :
The most attractive aspect of scientific knowledge is its mathematical basis. We generally feel that this mathematical representation of various scientific facts make our knowledge more precise and accurate. However, from the following theorems which have been put forward by the great mathematician Kurtz Godel, we find that any mathematical representation of any physical reality limits our knowledge of that reality. Not only this but the theorem also imply that none of the languages or representation can express the reality of nature with perfection. Complete knowledge must necessarily have its foundation in an inexpressed, unmanifest field of intelligence. Let us begin with the theorems.
(i) Golde's first in-completeness theorem This theorem says that the truth of a formalism (which describes any phenomenon) cannot be proved. Thus no finite expression of mathematical knowledge can ever provide a basis for comprehensive knowledge even of the elementary properties of the counting numbers. Thus if one starts with a collection C of symbolic mathematical (or any other) axioms which is specifiable by a finite number of mechanical rules, and if C is consistent, then there will be a true statement about the counting numbers which cannot be proved from the axioms C, using the standard rules of mathematical logic. The proof of this theorem shows that from C one can constructa sentences in the simple mathematical language of elementary number theory whose meaning is : This sentence is not provable from C. Once s