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Who "invented" the concept of symmetries? This article is quite extensive, but it blurs the history with the modern understanding. http://plato.stanford.edu/entries/symmetry-breaking/

Some of the concepts can be traced to Galileo and Newton, but I'm quite certain the modern notion is incompatible with their view of the world. Does the notion come from group theory specifically? Can the first mention be traced accurately?

Although the spatial and temporal invariance of mechanical laws was known and used for a long time in physics, and the group of the global spacetime symmetries for electrodynamics was completely derived by H. Poincaré [7] before Einstein's famous 1905 paper setting out his special theory of relativity, it was not until this work by Einstein that the status of symmetries with respect to the laws was reversed.

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Noether's theorem probably marks our modern understanding of symmetry in physics, though features (such as linear and angular momentum) must have been understood earlier. It's hard to imagine a full appreciation of symmetry in physics without Lagrangian or Hamiltonian mechanics. My guess is that someone must have noticed that physics is (almost) parity symmetric much earlier. –  emarti Dec 19 '12 at 2:03
    
Fascinating. Thanks a lot. –  RParadox Dec 19 '12 at 3:03
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