Joe Heafner <heafnerj@CTC.NET> raised a number of questions. Some of the
questions are a bit open to interpretation.
1) One way of looking at one of the issues is to rephrase it:
-- classically, mass is conserved.
-- in SR, energy is conserved. Mass is one component of this energy.
That is a reasonably concise way of describing the difference between
SR and what went before.
This process -- whereby something that was previously believed to be
separately conserved is later seen to be only part of a more-complex
conservation law -- has been played out repeatedly in the history of
physics. Feynman described this in simple yet incisive terms in
Volume I Chapter 4 ... the parable of Dennis and the blocks.
2) A less-charitable reading of the question would pick on the
alleged connection between atoms and mass. That's not right.
It's not true that all atoms classically have the same mass.
It's not even true that all carbon atoms classically have the
same mass. The point is that classically mass was associated
with things, and if Thing A and Thing B were together in a region,
the mass in that region would be the total mass of the things.
This leads to the notion of conservation of mass mentioned in
item (1) above.
3) Nowadays few experts will argue in favor of the term "rest mass".
Calling it "mass energy" or "existence energy" is mostly harmless
but quite unnecessary. Nowadays when somebody says "mass", that
means the invariant mass, so calling it the rest-mass would be a
redundant pleonasm :-)
Note that conventions have changed in this department. This is a
bit of an embarrassment, because despite the tremendous stature of
the E=mc^2 equation in the popular imagination, the current
recommendation is *not* to write things that way. The recommended
equation is E^2 + p^2 c^2 = m^2 c^4.
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In any case it is a fact that the mass of a helium atom is markedly
less than the mass of two deuterium atoms: 4.002602(2) versus 2 * 2.0140.
Mass is not defined in terms of counting particles. In both cases
(He and D2) we have four protons, four neutrons, and four electrons.
But the mass is different.
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If that doesn't answer the question, please clarify the question.