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Who "invented" the concept of symmetries? This article is quite extensive, but it blurs the history with the modern understanding. https://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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    $\begingroup$ 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. $\endgroup$
    – emarti
    Dec 19, 2012 at 2:03
  • $\begingroup$ As European civilizations concerns and historical documents, symmetry was first formulated and incorporated in geometry by the Ancient Greeks and applied mainly at that time in architecture. The very name of "symmetry" is Greek identifying the historically documented origin of the concept. $\endgroup$
    – Markoul11
    Nov 10, 2021 at 9:48
  • $\begingroup$ This probably goes back to the ancient Greek, who knew about the symmetries of the Platonic solids. The name tells the story! - In terms of groups, the concept of symmetry is due to Galois. $\endgroup$
    – Arnold Neumaier
    Nov 10, 2021 at 11:24
  • $\begingroup$ Galileo's argument about observations in a ship travelling in smooth conditions is a fine example of reasoning from symmetry, whether or not it was so called at the time. $\endgroup$
    – Andrew Steane
    Nov 11, 2021 at 14:04
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    $\begingroup$ @Malavika, you might search for related questions at History of Science and Mathematics, which perhaps didn’t yet exist when this old question was asked. $\endgroup$
    – rob ♦
    Nov 12, 2021 at 3:37

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For any application of symmetries throughout history, one can always find an earlier, more rudimentary application, so there is no first person, at least not as far as records go. The very first mathematics/physics/natural sciences results were geometric in nature, so the symmetries of the objects played a crucial role, if often implicitly.

The modern framework of symmetries as concerns physics is that of group theory, of course. A particularly important class of groups is that of Lie, so a natural candidate is Sophus Lie. He was definitely not the actual first, but he did formalize and generalize many notions of the time. In physics, a symmetry is often synonymous with a Lie group, so this seems a reasonable person to pick, if we must.

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