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I don't have too much trouble with the above; I think most respondents
here have meant to say something like, "We sense internal forces or
pressures that are generated in response to the forces that cause us to
accelerate." This is why I prefer to say simply, "We experience
acceleration."
On the other hand, acceleration is not quite as arbitrary as you imply.
The principle of equivalence teaches us that there are preferred reference
frames--those in which there are no local gravitational fields. We
experience acceleration whenever we are not in one of those reference
frames.
.... For instance, the sensations we experience when
accelerating upward at 2 m/s^2 with respect to the earth's surface are far
from double what we experience when we accelerate at 1 m/s^2. The ratio
is more like (9.8+2)/(9.8+1) = 1.093.
... acceleration with respect to local inertial frames. Now the
correspondence between acceleration with respect to inertial frames and
our sensations is simple; when our acceleration is twice as great we
"experience" twice the sensation.
...
Accelerometers are nonsensical devices indeed within the framework of
Newtonian gravitation. But they are simple to understand and very useful
within the framework of a theory of gravitation based upon the principle
of equivalence like GR.