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I think Newton believed in absolute time and space and, at the outset,
fully expected that the best that he might do would be to find a physics
which would work only for an observer at absolute rest. The existence of
a whole class of inertial frames was an unexpected (and, I think,
undesired) conclusion, not an assumption.
He certainly believed in INVARIANT force pairs of interaction between
bodies as the basis of his mechanics. Only the inertial observer can use
observed acceleration as the measure of these forces, since accelerations
are frame dependent, but Newtonian forces are frame invariant,