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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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There is at least one philosopher before Plato and he is Anaximander. There are many passages in his works that relate to the concept of symmetry:

The basic elements of nature (water, air, fire, earth) which the first Greek philosophers believed that constituted the universe represent in fact the primordial forces of previous thought. Their collision produced what the mythical tradition had called cosmic harmony.

Anaximander claimed that the cosmic order is not monarchic but geometric and this causes the equilibrium of the earth which is lying in the centre of the universe. This is the projection on nature of a new political order and a new space organized around a centre which is the static point of the system in the society as in nature.

In this space there is isonomy (equal rights) and all the forces are symmetrical and transferrable.

But his most famous reference to symmetry is his statement that:"..the round earth is situated in the midst of empty space but it does not fall...because there is no preferred direction for it to fall in!

Anaximander was the first to conceive a mechanical model of the world. In his model, the Earth floats very still in the centre of the infinite, not supported by anything. It remains "in the same place because of its indifference",... Anaximander's realization that the Earth floats free without falling and does not need to be resting on something has been indicated by many as the first cosmological revolution and the starting point of scientific thinking. Karl Popper calls this idea "one of the boldest, most revolutionary, and most portentous ideas in the whole history of human thinking." Such a model allowed the concept that celestial bodies could pass under the Earth, opening the way to Greek astronomy.

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