About the concept of 'inertial frame'
Actually, I prefer the expression 'equivalence class of inertial coordinate systems'.
I prefer to emphasize that it always a class of coordinate systems. As we know, the members of this class move with respect to each other with uniform velocity.
I will go through a thought experiment now, and I will conclude by pointing out that the case of that thought experiment generalizes to all of physics.
The thought experiment:
Take our solar system, and make it so that all you can observe from Earth is only the solar system. You can see the planets, but not any of the other stars in the Universe. Under those circumstances would it still have been possible for any astronomer to find Kepler's laws of planetary motion?
As Kepler began the task of figuring out the orbit of Mars, he had, from observations by astronomers, sufficient data to have a rough idea of the shape of Mars' orbit. From observations over many cycles of Mars' orbit the true orbital period of Mars can be inferred. After one Mars year Mars is back at the same spot in the solar system while Earth is at a different location compared to one Mars year before. With that pair of positions Kepler could do a triangulation, so he could calculate the distance of Mars to the Sun at that point in time. Kepler did that for a number of points, and that gave him a rough idea of the shape of Mars' orbit.
In the thought experiment:
Without the fixed stars as background reference the Kepler problem is much, much harder. Without the fixed stars, what can you do?
For one thing, at some point in time it will be established beyond doubt that the 24 hour cycle is indeed the Earth rotating around its own axis. The most direct way of proving that the Earth rotates is by constructing a gyroscope with a high level of precision. Historically this was achieved by Leon Foucault, in 1852.
OK, that establishes an inertial coordinate sytem that is valid for Earth. But in itself that doesn't tell you whether the Earth is rotating around the Sun, or that the Sun is rotating around the Earth.
Since the Earth's axis is tilted with respect to the plane of the Earth's orbit we can infer the length of the year to a high degree of precision.
This gives an interesting hypothesis to explore.
Is it the case that the inertial coordinate system in which the Earth completes a revolution around the Sun in one year also valid for all the planets?
That is: is the Earth inertial coordinate system something that is local to Earth, or does it encompass all of the solar system? You have to treat that as an hypothesis.
At some point in time astronomers will point out the thing that in our history Copernicus pointed out. The motions of all the non-Earth planets can be described as a compound of two periodicities; one that is specific for each planet, and another one that is a period of a year. This suggests that all the planets are orbiting the Sun, not the Earth. If all the other planets are orbiting the Sun, then it is plausible that the Earth is orbiting the Sun too.
Let's pause for a moment here: all of the above can be reached even if the only observations you have is the motions of the planets; al these inferences are without the benefit of fixed stars.
With all of the above in place I think it is possible to find Kepler's laws of planetary motion.
At some point in time some astronomer will point out the following: there is only one solar coordinate system with the property that the orbit of Mars is an ellipse with the Sun at one focus. More generalized: there is one solar coordinate system with the property that the observations of all the planets are consistent with each of those planets orbiting the Sun with the Sun at one focus.
This is strong evidence that indeed the Earth inertial coordinate system is not just local to Earth, but that the Earth inertial coordinate system is part of a solar system inertial coordinate system
Note that invoking Newton's first law is not an option; planetary motion is not a in a straight line. The data that you have is the shapes of the orbits. Whatever you do must be based on the data.
At some point in time the counterpart of Newton's law of universal gravitation will be discovered, and that makes the case beyond reasonable doubt.
There is a solar system inertial coordinate system. When you describe the observed motion with respect to that solar system inertial coordinate system then the Universal law of gravity obtains, otherwise it doesn't.
In this thought experiment it was very hard to find the solar system inertial coordinate system. The concepts of 'inertial coordinate system' and 'law of motion' could only develop hand in hand, each dependent on the other for confirmation.
Not circular reasoning
While the two conceps are co-dependent, it's not circular reasoning. Proof: these concepts are deployed in our technology. When an interplanetary probe needs more speed than our rockets can provide a trajectory is devised that involves one or more gravity assists. Gravity assists work as planned.
Axiomatic approach not feasible
The case I'm presenting is that there is no such thing as first defining the concept of 'inertial coordinate sytem' and from there proceed to find laws of motion.
Each concept needs progress of the other concept in order to make progress itself.
The equivalence class of inertial coordinate systems cannot be observed directly. The laws of motion require the concept, and they are also the definitive validation of the concept.
Generally speaking: when introducing a concept you don't know in advance how far its validity extends. Your starting assumption must be that it is valid only within the realm for which you can validate it.
Our astronomy got quite a jump start: the benefit of the fixed stars. As far as we can tell the inertial coordinate system of our solar system does not rotate with respect to the fixed stars. As far as we can tell: there is an equivalence class of inertial coordinate systems that is valid for the entire Universe. In effect we got the maximum size realm of applicability in one go.
While that jump start was hugely helpful, it also kind of obscures the nature of our laws of motion. It would have been clearer if we would have been forced to begin with a very local realm of applicability, and then meticulously work our way up.
The above thought experiment is illustrative of how all of physics develops over time. Whether the physicists themselves are aware of it or not: there is always a co-development of co-dependent concepts.
A remark about the first law:
I have a preference for the following way of stating the laws of motion (newtonian dynamics):
- Physical space is uniform in the same way that Euclidean geometrical space is uniform, time is uniform.
In the three-laws-version the first law is doing two things at once: asserting the uniformity of space, and asserting that in the absence of a force there will be zero acceleration. Well, the latter part is already covered by the second law. The above two-law version has the same coverage as the three-law-version.
Also, when the time comes to move to, say, General Relativity, you replace the assertion about the nature of space and time with the appropriate one.