This is a conceptual understanding of Newtonian mechanics. What the laws mean, h

Question

This is a conceptual understanding of Newtonian mechanics. What the laws mean, how we know they're true, etc. I'm looking for criticism. I know this is really border line on the "don't ask questions that can't be answered" rule, but here we go anyway.

The Laws
First Law: A body retains its velocity unless acted upon by an outside force.
This first law is actually a definition, not an empirical statement. Body can be defined based on sense data, as can be velocity. But force is as yet undefined. As it is the only undefined term in the statement, the statement must be a definition. A force is defined as "that which is said to 'act upon' a body when that body's velocity changes, the immediate cause for a change of velocity". This contrasts with the Aristotelian definition, which is "that which is said to 'act upon' a body when that body has non-zero velocity, the immediate cause for a change in position".

Second Law: A body's acceleration is a function of its mass and the force acting upon it, according to the relation F = ma.
This second one is still just a definition. It's implied that mass is a function of the specific body. We can apply this to predict accelerations (see Applications, below).

Third Law: When one body exerts a force on another, that other exerts an equal and opposite force on the first. This is the only of the three laws that is empirical. It is not a definition as force was already defined by the first two laws, and it cannot be proven logically from the first two. It would have to be proven by some sort of experiment.

Empirical observations
The most glaring omission from the Laws is what can cause a force. Newton probably just meant it to be implied that forces were caused by collisions between bodies. In any case, you can demonstrate experimentally that collisions cause forces. This is the first apparent strength of Newtonian physics over Aristotelian physics. The Aristotelian definition of a force is valid (no such thing as a false definition), but in Newtonian mechanics it's much easier to express the relationship between collisions and forces.

Applications
We can apply these laws to calculate masses, forces, and accelerations.

Define a unit mass. Mass is an unchanging property of a body and therefore we may simply take an arbitrary body and define its mass as the unit.
Notice that when the same object collides with the same object in the same way, it has the same acceleration. For example, roll the same ball into it from the same height down the same slope, or hit it with a pendulum dropped from the same height. This provides inductive evidence that that sort of collision always produces the same force (F = ma, m is unchanging, a is unchanging, therefore F is unchanging).
Now apply that collision to other objects to deduce their mass. If an object accelerates x times as much as the object of unit mass, it has 1/x mass.
You can now measure mass.

Explanation / Answer

The sentences you set in italic are probably ment to be qualitative explainations. You can do that, but I feel without using mathematics (and thereby taking a shortcut by implying the standard definitions in mathematics, e.g. regarding geometric quantities) the laws are very difficlult or impossible to put into a precise form.

Anyway, from the perspective that what you want is a clear prose, here are some remarks:

First Law

Body can be defined based on sense data, as can be velocity.

Here you eighter want to have already an understanding what space, time and trajectories are and you identify these in some way with the notion of body (are you thinking of point particles or maybe something more complicated?). Otherwise I don't know what you mean by "define" here. If you take data, e.g. your shoesize, then you must already have an association what a shoes is. The shoe is what you want to define here in the sentece right? The sentence "something can defined based on sense data" seems confusing to me.

But force is as yet undefined.

You use the word But here, implying that body and velocity have already been defined.

This first law is actually a definition, not an empirical statement.

I saw this thread here recently discussing related topics. The threads referenced in the question might also be of interest.

A force is defined as "that which is said to 'act upon' a body when that body's velocity changes, the immediate cause for a change of velocity".

Well, as the notion of absolute space is pretty much discarded today, there are some major complications relating to what I mentioned about math being a better suited language here, and some answer might also be found in the question I posted. Problem is that velocity is relative and you can't just take the definition you gave and leave it at that. There is a quote by Einstien going in this direction

"The weakness of the principle of inertia lies in this, that it involves an argument in a circle: a mass moves without acceleration if it is sufficiently far from other bodies; we know that it is sufficiently far from other bodies only by the fact that it moves without acceleration."

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