"But you can't measure the force that way."

Says who?

I could, were I so minded measure the distance between my hindquarters and the ground as a function of time, calculate the acceleration (it will be close to 8.9 m/s/s) and since I know that M is 70 Kg, I can calculate F which will be close to 700N).

Says me.

You are in "free fall" accelerating towards the Earth. Your measurements always produce the same result, even if you vary your mass. What do you suppose you are measuring?

Nonsense.

I can see the ground coming towards me if I fall towards it.

As I already said "I could, were I so minded measure the distance between my hindquarters and the ground "

so why do you ask "What do you suppose you are measuring?"?

If the chair is there I stay at the same height and I can feel the force on my butt.

If the chair goes away those facts change.

What you are saying is that I cannot tell if the chair is there or not.

How absurd are you prepared to be?

Incidentally, the fact that gravitational mass and inertial mass are the same (as far as we know ) is just an oddity of the universe.

Have a look at electrostatics.

I can get two conductors of different masses, but the same charge. I can put them near the earth (presumed to be much heavier and also conducting).

They are attracted to the earth with a force proportional to the charge squared and the reciprocal of the distance above the (approximately) flat surface of the earth.

Because they have different masses they have different accelerations in spite of the fact that they have the same charge.

If we were to look at electrons instead then, because they all have the same charge and the same mass they would all have the same acceleration (from a given height).

Now imagine that I (magically) come up with some other material that has the same charge^2 to mass ratio as an electron.

It will fall at the same speed as the electrons.

The important factor for rate of fall is the ratio of the charge (squared) to the mass.

It's not that the mass has been written out of the equation; it's still there. It's just that when you look at the ratio of the rates of fall, it cancels out.

The thing is that, unlike electrostatic charges; with gravity, the ratio of "propensity to move" to "difficulty of getting it to move" is the same for any object.

It's called mass.

Just because it cancels out in some pair of equations doesn't mean it isn't real.

Look at de Broglie's equation.

He took E=MC^2 and E=h nu

and said that the E in both cases is the energy so

MC^2 =h nu

and he got the Nobel prize for it (after rearranging it in terms of momentum).

Just because the E term has been eliminated doesn't mean that energy has ceased to exist.

" our chairs are continuously producing a force on our bums that counteracts gravitational acceleration."

Wrong, by dimensional analysis.

A force (like that from the chair) can counteract another force (like weight) but it cannot counter an acceleration because you can't do the vector sum on quantities with different units.

The fact that we don't fully understand one of those forces doesn't matter.