Bernhard Schölkopf (MPI-IS) and I spent time drinking coffee this weekend. Among the many subjects we discussed was language around invariance, equivariance, covariance, coordinate freedom, and symmetry. These words! I have strong opinions, as my loyal reader might know. But during the conversation I had an epiphany in which I understood why Einstein called the symmetry of general relativity “general covariance”: He was probably keying off the mathematicians, who used covariance back then the way we use equivariance now.
I don't like the word “equivariance” at all! Why do we want to write the laws of physics in a rotationally-symmetric (or rotationally equivariant or orientation-free) way? There are two completely different reasons! One is that the laws of physics are observed to be rotationally invariant (which leads to conservation of angular momentum and so on). The other is the theoretical idea that the laws of physics can't depend on investigator choices about coordinates. These are completely different, and the latter is extremely strong. We debated whether there was something to write about all this somewhere.
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