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Physics

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    $\begingroup$ Excellent answer. I'm not sure if this covers the feature that I was going to comment on -- that many of these laws describe spatial relationships, and space comes in countable dimensions. So this explains the relationship between one-dimensional and two-dimensional measures of regular shapes (circles, spheres, squares, cubes). I suspect that it also accounts for how the intensity of forces decreases with distance. $\endgroup$ Commented Feb 3, 2015 at 15:55
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    $\begingroup$ Good point, geometry by itself also comes all in integer powers (rational, with small-powered roots if you play with norms). And indeed the inverse square law and its relatives come directly from that. $\endgroup$ Commented Feb 3, 2015 at 17:36
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    $\begingroup$ This answer hits the nail on the head at the beginning: the reason is that the relationships in question are all linear. Why are they linear relationships? For the geometrical measurements, the answer is proportional scale. For the physical measurements, the answer is (usually) conservation of energy. $\endgroup$ Commented Feb 5, 2015 at 13:05
  • $\begingroup$ "An example is F=ma (just defines what force is - it's convenient to define it like that)" There is no universal agreement on this point of view, there are others. For example, in one view force is not defined by $a$ but by spring deformation or weight measurements. Also, in special relativity this equation is not used any more, not because the definition of force was changed, but because Newtonian mechanics is approximate to special relativistic mechanics. $\endgroup$ Commented Feb 7, 2015 at 10:44
  • $\begingroup$ Sure, the more proper definition of force would be the derivative of a particular conserved value -- the linear momentum. But that doesn't change the fact it's linear and that it's not even strictly needed as a physical quantity. Physics works quite well without ever talking about forces specifically. $\endgroup$ Commented Feb 7, 2015 at 11:23