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Answer To Paul R. Martin About Math Origin Of All Natural Laws.

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Posted by Alexander on June 28, 2001 20:04:01 UTC

It is obvious that all laws come from mathematical symmetries. For example, lets consider shape of a body in symmetric 3-D space. Let's assume that the body consists of many somehow interacting particles, that they are free to move (so shape can change), and that this mutual interaction of particles is completely symmetric in space (does not depend on location in space and on direction in space). This assumption is actually completely equivalent to the assumption that SPACE ITSELF is symmetric: NOTHING depends on position in this space nor on orientation in this space - all places and all directions in THIS space are equal. Of course, interaction between particles may depend on MUTUAL position between particles (it may be inverse square, or any other dependance - it does not matter), but this interaction should NOT depend on location in space or on direction in space. Othervise you can NOT call such space positionally or directionally symmetric.

Now, let's ask ourselves: what do we expect to be the shape of the body due to any interaction between particles and taking into consideration symmetry of space (invariance or independance of interaction from location in space and from direction in space)? We expect the shape to be the same at any place in such space wherever we put those interacting particles. Moreover, due to symmetry of all directions we also expect the shape to be a sphere (the only shape which is the same in all directions). And that is exactly what you have in symmetric space. You may label it as a "law of nature": all celestial bodies which are in symmetric 3-D environment (and which are free to change their shape) should be spherical regardless the nature of interaction between their parts.

I hope this example also answers your previous question about how natural laws can originate fom mathematical symmetries without writing the equation on paper (in this case without writing x^2+y^2+x^2 = const). There is no need in writing the equation of sphere to create a sphere. There is no need in paper as well as in pen or ink. Moreover, I can assure you that there is no need in someone to understand relationship between directional symmetry and spherical shape in order for the body to shape itself according to the equation of a sphere above. There is no need even in someone to see (observer) the body. Ther is no even need in someone to exist at all (for the body to be spherical).

Basicly, all fundamental laws of nature and all fundamental forces are derived similar way from possible (allowed by math) symmetries of space and time (gauge symmetries for forces). For details you should see any good physics textbook, or look up the internet. Good site to start is Emmy Noether theorem (equivalence of any symmetry to some conserved quantity), like:

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