This thread reminds me of the parable of the blind men and the
elephant. Ten different people have said about twenty different
things ... almost all of which are true!
1) Yes, inertial mass is the invariant norm of the [energy, momentum]
four-vector.
2) Yes, mass has the property that the mass of a compound object
is the sum of the mass of the constituents, corrected for binding
energy (which is usually negligible, unless you are doing nuclear
reactions). This may optionally be considered a consequence of
item (1) and the great [E,p] conservation law.
3) Yes, inertial mass is the thing that shows up in the expression
for kinetic energy:
KE = .5 p^2 / m
valid to second order in v/c.
4) Yes, inertial mass is the thing that shows up in the expression
that relates velocity to momentum:
p = m v
which is either exact or valid to lowest order, depending on
details of how you choose to define "velocity".
5) Yes, inertial mass is the thing that shows up in the expression
F = m a
which is a corollary of item (5) if you define "force" properly.
Note: Items (1), (3), (4), and (5) are _equivalent_ to lowest order
(or better), if you believe in special relativity.
6) As for gravitational mass, it is the thing that shows up (twice)
in the expression
Fg = G M m / r^2
valid provided the gravitation is not too strong and not too
rapidly changing.
7) According to Einstein's equivalence principle, inertial mass
is equal to gravitational mass.
8) Combining (7) with (6) and (5), this is tantamount to
saying that locally, the Newtonian gravitational field is
indistinguishable from an accelerated reference frame.
I consider (7) and (8) to be true but not axiomatic. (If
they were axiomatic, Eötvös experiments would be impossible,
or at best a waste of time, which is not what I believe.)
Indeed you have to be careful how you state the equivalence
principle, lest you say something that's not true. Recently an
experiment was launched to measure gravitomagnetic components
of the gravitational field, terms which cannot be described as
an acceleration. They do not expect a null result.
I suppose it is somewhat a matter of taste, but I find it
pointless and unhelpful to try to axiomatize physics. I
care about knowing useful ideas, and knowing their limits
of validity. You can choose any of the items (1), (3), (4),
(5), or (6) as "the" definition of mass. I don't care.
====
I am not satisfied defining mass in terms of "amount of stuff"
because that just begs the question of what is meant by "amount"
and by "stuff". Many products are sold by volume, which is
definitely not the same as mass, so it is quite likely that
students will arrive with non-mass-related notions of "amount
of stuff".
(Try adding one cup of milk to one cup of cereal. The "amount
of stuff" -- measured by volume -- is not additive.)
I agree with Michael E. that mass is somewhat of an abstraction.
In particular, it would be going too far to define mass in
terms of what any particular instrument indicates. The
instrument might be non-ideal, or it might be broken. In
particular, if the instrument does not uphold the additivity
principle (item 2) then we know something is wrong.
(Example: an ordinary laboratory balance is non-ideal, in
that you must add a correction for buoyancy.)
Meanwhile, I would not be comfortable going to the opposite
extreme and saying that our notion of mass is independent of
how we measure mass. It is pointless to ask what would happen
if we couldn't measure mass, because the fact of the matter is
that we can measure mass. Using items (1) through (8) plus
some engineering skill, we can devise quite a number of
methods for measuring mass (relative to some standard) more or
less accurately, more or less directly. If done right, all
these methods are equivalent.
=====================
The conceptual structure is clear: we have at least eight
major ideas, all of which are true, and all of which are
intimately related.
The question arises, in what order should these ideas be
presented? There are eight-factorial possible ways of
re-ordering these ideas. I see no _physical_ reasons for
declaring any of these ideas to be "the" natural starting
point. There may be _pedagogical_ reasons for preferring
one starting point to another, but the decision will require
tradeoffs and will be sensitive to personal preferences and
other ill-defined details of the situation.
a) For an advanced audience (or even an end-of-term review)
there is some logic in starting with item (1), because it
has the broadest domain of applicability, and serves as
a unifying principle for the next few items. Of course
for an introductory-level audience, it would be madness
to start with item (1).
b) At the introductory level, one might argue for emphasizing
additivity (item 2) very heavily, very early. The principle
of additivity has great generality.
As Hugh Logan pointed out, PSSC takes this approach. They
use the "C-clamp" as a unit of mass, which well-nigh requires
students to think somewhat abstractly about mass: if you
have three C-clamps, the fact that each is physically,
concretely a clamp is obviously not important; what is
important is the fact that there are three units of mass,
and the mass is additive.
Bottom line: We know several things about mass. They are
deeply related. They are all important. They all need to
be covered, but the order in which they are covered is
largely a matter of taste. Asking which is "the" fundamental
notion of mass is pointless and unphysical.