Dividing by m to make conclusions for m=0

[A.T.]

It is the vector nature that is important to the proof, not the independence.

Okay, it depends on the type of relationship. So your derivation doesn't show that "a is independent of m", but it rules out certain types of relationships.

That said, if you agree with the proof in 42, then I think that the math discussion is complete, and the rest is necessarily a physics discussion.

I have no problem with the physics part. A physicist can always state that the potential relationships not covered by the proof are "too weird to occur in nature", and discard them on that basis.

 

[Dale]

Okay, it depends on the type of relationship. So your derivation doesn't show that "a is independent of m", but it rules out certain types of relationships.

Yes. Or rather, it "rules in" certain types of relationships.

It says that if x and y are vectors*, then then you can factor out the m without any problem including for m=0. It does not say that you definitely cannot factor out m if they are not vectors*, it just says that you definitely can factor it out if they are.

*with all of the detailed requirements on x, y, and m explicitly spelled out in 42

I have no problem with the physics part. A physicist can always state that the potential relationships not covered by the proof are "too weird to occur in nature", and discard them on that basis.

OK, then I think it is done.

 

[A.T.]

OK, then I think it is done.

Yes, thanks for the patient explanations to everyone.