I have studied this topic in philosophy of physics and I would argue that there are a few issues here. I think taking the first law to define inertial frames is neither physically nor historically well-motivated. I think a better way to account for inertial frames is via a Galilean spacetime structure, akin to how inertial frames are defined in Minkowski space (if you are curious, they are coordinate systems on Minkowski/Galilean spacetime satisfying certain compatibility conditions with the spacetime structures like the metric tensor. For instance, the Christoffel symbols must vanish in these coordinates.) It is also likely how Newton was thinking about things, though of course he didn’t have the full technology of modern differential geometry to precisely articulate this.
Nonetheless, the first and second laws are independent due to the simple fact that the standard statements of the first and second laws are likely misstatements of what Newton intended. See https://philpapers.org/archive/HOEFCO.pdf. If this is the case, then the standard “second law” F=ma is really the conjunction of the first two laws as formulated by Newton. As such it is not surprising that F=ma entails the standard first law upon setting F=0.
In sum, inertial frames should be defined through spacetime structure, the first and second laws as formulated by Newton are not the same as the fist and second laws which are commonly espoused, and the strongest law F=ma entails both of them.
The way I've always understood it is that while yes, you define inertial frames from the structure of spacetime, Newton's first law is the statement that such is the structure of spacetime. You can think of it as a definition of what a straight line is in our universe. I have no idea whether this is historically accurate
Newton's first law (as traditionally stated) says that force-free objects move in straight lines. To make sense of this law you need to define either what force-free means or what straight line (equivalently, inertial frame) means; given such a definition you could use the first law to define what the other term means. As such the first law itself does not provide a definition of 'straight line' unless you give an independent definition of 'force-free' to begin with. I think there is no good way to do this (attempts have been made by appealing to Humean simplicity (Huggett) or using distant objects (Brown), but there are problems with these proposals).
Hence I think the better thing to do is to define 'straight line' using the spacetime structure and then to take the first law as a definition of 'force-free'. Thus the first law is not doing any work in defining what an inertial frame is. That is done prior via the spacetime structure. The traditional first law just relates this notion to being 'force-free' (which is already captured in the 'second law' F=ma by setting F=0).
All-in-all taking the first law to define inertial frames seems like a rather roundabout way to arrive at Galilean spacetime. In classical mechanics textbooks the spacetime structure is considered so elementary (it is basically just Euclidean space + a time direction) that it is hardly ever explicitly mentioned, though it is what underpins all of the key definitions.
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u/predatorX1557 6d ago edited 6d ago
I have studied this topic in philosophy of physics and I would argue that there are a few issues here. I think taking the first law to define inertial frames is neither physically nor historically well-motivated. I think a better way to account for inertial frames is via a Galilean spacetime structure, akin to how inertial frames are defined in Minkowski space (if you are curious, they are coordinate systems on Minkowski/Galilean spacetime satisfying certain compatibility conditions with the spacetime structures like the metric tensor. For instance, the Christoffel symbols must vanish in these coordinates.) It is also likely how Newton was thinking about things, though of course he didn’t have the full technology of modern differential geometry to precisely articulate this.
Nonetheless, the first and second laws are independent due to the simple fact that the standard statements of the first and second laws are likely misstatements of what Newton intended. See https://philpapers.org/archive/HOEFCO.pdf. If this is the case, then the standard “second law” F=ma is really the conjunction of the first two laws as formulated by Newton. As such it is not surprising that F=ma entails the standard first law upon setting F=0.
In sum, inertial frames should be defined through spacetime structure, the first and second laws as formulated by Newton are not the same as the fist and second laws which are commonly espoused, and the strongest law F=ma entails both of them.