A Puzzle about Laws, Symmetries and Measurability

I describe a problem about the relations among symmetries, laws and measurable quantities. I explain why several ways of trying to solve it will not work, and I sketch a solution that might work. I discuss this problem in the context of Newtonian theories, but it also arises for many other physical theories. The problem is that there are two ways of defining the space-time symmetries of a physical theory: as its dynamical symmetries or as its empirical symmetries. The two definitions are not equivalent, yet they pick out the same extension. This coincidence cries out for explanation, and it is not clear what the explanation could be. 1. The Puzzle: Symmetries, Measurability and Invariance 1.1 The symmetries and the measurable quantities of Newtonian mechanics 1.2 The puzzle2. Two Easy Answers3. Another Unsuccessful Solution: Appeal to Geometrical Symmetries4. Locating the Puzzle5. The Relation between Laws and Measurability6. A Possible Solution The Puzzle: Symmetries, Measurability and Invariance 1.1 The symmetries and the measurable quantities of Newtonian mechanics 1.2 The puzzle Two Easy Answers Another Unsuccessful Solution: Appeal to Geometrical Symmetries Locating the Puzzle The Relation between Laws and Measurability A Possible Solution