It is matter of personal opinion, I think, but I do not like stating the first law like
"An object not in motion does not start moving spontaneously" (1)
What about an object that IS in motion?
So, first law defines inertial frame. Perhaps we should even be more broad. Inertial frames are frames in which all laws of physics are equivalent. In this case the law:
dp/dt = F_external is true. As far as we know, all forces are interaction forces.
Imagine someone confined in an accelerating elevator and has no way to communicating with the outside. He would develope physics a lot differently. He found that objects in his world likes to go downward (toward the floor) and it is NOT due to ineraction with anything in his world. So he concludes that his world is anisotropic. Momentum and energy and such are not conserved. And his equation of motion would read: ma=F_interaction
+ F_floor.
(j/k: perhaps the guy can invent a theory of gravity with massive floor?)
So, I think a good definition of inertial frame is as follow:
Inertial Frame are frames in which the world is (or remains) homogeneous and isotropic. And it is precisely the isotropicity that would demand "An object at rest stays at rest" . And also accelerated frame then cannot be inertial frame by definition.
I am not sure if it makes perfect sense though. How does the object "know" where it has been a while ago, so that it can continue to move in the same direction and with same velocity? How does it "remember" that it is moving instead of being at rest? Do the objects have in fact some kind of "memories"?
I don't think the object knows. In its frame, it's merely staying at rest. Since both frames (yours and the objects) are inertial, you can't prefer one frame over another (principle of relativity). Since your frame in inertial, object can't change its motion spontaneously, otherwise there is a preferred direction in your frame (the direction in which the object's motion changed without interaction).
I wonder if one can argue that perhaps the frame can still be inertial (hence isotropic), if an object that spontaneous changes its direction at one point implies an opposite change to another object somewhere else. I suppose homogeneity and isotropic-ness must be locally true as well...
another way of looking at this: you can certianly (classically anyway) plot x(t) in your frame. But it can only be of the form x(t)=x0+v0*t. There can be no acceleration or higher term, otherwise, you can conclude that the object is no in an inertial frame.
A comment on a point of the first post. Observation is the BEST verification that our models and laws of the world are actually correct. A model with well observational support is a theory, those without are hypothesis (string?). Charge quanization is a good example. People has done dilute hydrogen gas exp. for long time hoping to see some charge different between proton and electron. Within experimental error (~2x10^-20e), none has every been observed. On the other hand, I wonder what is (if there is at all) the best experiment data on this topic.