Structure in Classical Mechanics

How do we learn about the fundamental nature of the world from a mathematically formulated physical theory? To learn about spacetime, we follow this rule: posit the least spacetime structure to the world required by a theory’s dynamical laws. Applied to special relativity, for example, this rule tells us to not posit an absolute simultaneity structure. I suggest that we should use this rule for more than just spacetime structure. We should use the rule for statespace, positing the least statespace structure required by a theory’s dynamical laws. Using this rule, I argue that a classical mechanical world has surprisingly little fundamental structure. Fundamentally, such a world does not have a Euclidean distance structure. This bears on more general questions: what physics tells us about the world; what possibilities are distinguished by a theory; what is in a theory’s fundamental ontology (which I suggest includes the statespace structure); and when two formulations of a theory are mere notational variants
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