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- TL;DR Summary
- Newton's first law follows from the second, the third (if modified) just defines what an inertial frame is, and the second just defines what the force on an object is.

I have read a bunch of articles online regarding my question, and none have helped.

Newton's Laws:

1. In an inertial reference frame, an object's momentum doesn't change unless acted upon by a force.

2. In an inertial reference frame, the force on an object equals the time derivative of its momentum.

3. In an inertial reference frame, the total momentum of every isolated system is conserved.

I have explicitly mentioned "inertial reference frame" in all three statements since the force on an object is only defined in an inertial reference frame. Also, the law of conservation of momentum is completely equivalent to the usual statement of Newton's third law.

My observations:

[1] follows directly from [2]. It contains no more information than [2] does, so we can scrap it.

[2] is a definition, but it is not complete. We have no way of knowing whether a frame is an inertial reference frame or not.

[3] makes a real statement, but it is incomplete. We still have no way of knowing if a frame is an inertial frame.

If we assume that the total momentum of every isolated system is conserved

But then, [3] gives us no information. It simply defines what an inertial reference frame is. [2] doesn't give us any information, it just defines what force is and it's incomplete without [3].

To my understanding, Newton's laws are just definitions and don't make any real claims about this world.

So, how am I wrong? Also, please note that I have no problems with defining mass through experiment. I have seen many posts where people claim that Newton's laws are "circular" because they don't define mass without using force, but this has nothing to do with that.

Newton's Laws:

1. In an inertial reference frame, an object's momentum doesn't change unless acted upon by a force.

2. In an inertial reference frame, the force on an object equals the time derivative of its momentum.

3. In an inertial reference frame, the total momentum of every isolated system is conserved.

I have explicitly mentioned "inertial reference frame" in all three statements since the force on an object is only defined in an inertial reference frame. Also, the law of conservation of momentum is completely equivalent to the usual statement of Newton's third law.

My observations:

[1] follows directly from [2]. It contains no more information than [2] does, so we can scrap it.

[2] is a definition, but it is not complete. We have no way of knowing whether a frame is an inertial reference frame or not.

[3] makes a real statement, but it is incomplete. We still have no way of knowing if a frame is an inertial frame.

If we assume that the total momentum of every isolated system is conserved

**only**in an inertial frame, then we can use [3] to determine if a frame is an inertial frame. We just check if the total momentum of every isolated system remains constant to determine whether our frame is an inertial reference frame.But then, [3] gives us no information. It simply defines what an inertial reference frame is. [2] doesn't give us any information, it just defines what force is and it's incomplete without [3].

To my understanding, Newton's laws are just definitions and don't make any real claims about this world.

So, how am I wrong? Also, please note that I have no problems with defining mass through experiment. I have seen many posts where people claim that Newton's laws are "circular" because they don't define mass without using force, but this has nothing to do with that.

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