Scientific or mathematical theories or theorems are considered as elegant if their proofs are simple, succinct, and concise. However, elegance could not possibly apply to the underlying postulates or assumptions of every theory. On the other hand, could a theorem with the less number of assumptions or axioms be considered as elegant?
In Euclidean geometry there are 5 assumptions. In non-Euclidean geometries (both Riemannian and Gaussian) there are also 5 assumptions. Their differences were stated in the 5th postulate of parallels: through a point not on a line (1) one and only one parallel, (2) no parallels, and (3) infinite parallels can be drawn. These three geometries are used in cosmology to determine the shape or topology of the whole universe. To date the winner remains undecided as a result of certain dimensional uncertainty. How many dimensions do the whole universe needs? In general relativity there are only 4 dimensions, 3 of space and 1 of time. In quantum mechanics there are only 3 of space with time-dimension reserved as a free parameter. In a quantum field theory, if the same as degrees of freedom then there are infinite number of dimensions. However, it is theoretically logical to formulate a physical theory base only on the 8 directional invariance properties of local infinitesimal uniform accelerated motion.
Time independence: [∂E(g)]²=[∂F(a)×∂r(a)]·[∂F(b)×∂r(b)] and Mass independence: ¶a(t)·¶r(t)=c²
Re: elegant proofs -
07-14-2008, 05:47 PM
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