 Solution

Newton's first law identifies a class of inertial frames in which isolated bodies move uniformly, while the second relates changes of momentum to resultant force. Both have definition-like roles: the first helps characterize an inertial frame, and the second can be used to calibrate force given independently measured mass and acceleration. That does not make the whole scheme an empty set of definitions.

If force meant nothing except whatever product $m\mathbf{a}$ happens to be observed, the second law alone would have little independent predictive content. Actual mechanics also specifies gravitational, elastic and other interactions with independently constrained parameters, superposition rules and reaction relations. The combined framework predicts the accelerations of new arrangements. One cannot arbitrarily change one body's mass or redefine every unexplained acceleration as a different force without affecting those cross-system comparisons.

Likewise, an inertial frame cannot be separately tailored to make each trajectory straight while retaining a common interaction account. A rotating frame introduces coordinated velocity-dependent accelerations for different freely moving bodies. The empirical claim is that a common family of frames exists with the prescribed relations, at least over the applicable domain. Tests involve judgments about isolation, clocks, geometry and interaction laws, so they are not independent of all auxiliary assumptions.

Thus definitions organize the theory but do not exhaust its content. Whether one calls $\mathbf{F}=d\mathbf{p}/dt$ a definition or a dynamical postulate depends partly on how the remaining force concepts are introduced. What is testable is the consistent package that connects independently characterized interactions, mass ratios and motion. Relativity and gravitation show how that package can be revised when its domain is exceeded.^[ref]