I'm not sure the distinction between a force and a symmetry is a useful one. Any use of "force" in modeling physics can be equivalently expressed via conservation of linear momentum, which itself is equivalent to the fact that the laws of physics are symmetric in translation (i.e. translation-invariant, i.e. have the same form when the origin of the coordinate system everything is expressed in is moved around).
Literally, for any force, you can say, "that's just the playing out of a necessary symmetry in the laws of physics".
Any use of "force" in modeling physics can be equivalently expressed via conservation of linear momentum
I don't see how. Either you're misunderstanding something, or you have a higher background in quantum mechanics than I do (I've had one in-depth class, and I've read the quantum physics sequence), and it works out like this for reasons I do not currently understand. Which is it?
In any case, force is clearly defined in the simplified version of quantum physics I've learned. It's the gradient of potential energy, which must be specified in the...
I thought this video was a really good question dissolving by Richard Feynman. But it's in 240p! Nobody likes watching 240p videos. So I transcribed it. (Edit: That was in jest. The real reasons are because I thought I could get more exposure this way, and because a lot of people appreciate transcripts. Also, Paul Graham speculates that the written word is universally superior than the spoken word for the purpose of ideas.) I was going to post it as a rationality quote, but the transcript was sufficiently long that I think it warrants a discussion post instead.
Here you go: