What Do F=ma and the Definitions of Force and Mass Really Mean?

[Creator]

1. masses are something intrinsic to objects, and additive
...
... It really gets me confused.

I think it will be much easier if you understand the definition of "inertial mass" to be "resistance to acceleration".

By the way; there has been no definitive proof as to the origin (source) of inertial mass; IOW, proof of the source of this "resisitance to acceleration" has not been shown definitively to be intrinsic to the matter itself.
In fact there is plenty of speculation that it is actually EXtrinsic, and could arise in a Machian sort of way from the totality of the rest of the matter in the universe. It has also been postulated that this "inertial resistance" comes from the Quantum vacuum field in the surrounding vacuum, which is also quite EXtrinsic to the material particle itself.

IOW, the origin of inertial mass is still an open question in physics.

Creator

 

[Martin]

F=ma
How do we define m and F? If their definition both come from this equation, then the equation doesn't really mean anything...

But their definitions do not come from this equation. The equation simply expresses how they are related through a.

Physics, and virtually every other field of science, ultimately rests upon a small set of basic concepts that are taken as virtually self-evident (similar to axioms in mathematics). Time is taken as one of those concepts, distance the second, and mass (or force—take your pick) the third. In other words, every other quantity in classical mechanics is defined via a set of rules that, ultimately, involves some combination of time, distance, and mass (or force). But these three "fundamental indefinables" are defined only "operationally"—that is, via procedures for their measurement.