One might ask if the experiments described above could be modified to determine
the ``true'' coordinate system used by motor cortex to describe hand
movement direction. However, I claim that
for certain classes of distributed representation this is not a
well-defined question. Distributed representations of measured variables
can be coordinate-free
in the sense that they do not imply any particular coordinate system.
To see this, let X be any variable represented in cortex (such as hand
movement direction),
and let 
be a vector-valued function representing the outputs of a
large set of basis functions 
which describe the behavior of
(motor) cortical cells. 
is then a
distributed representation of the variable X. Now, consider a vector
function
which measures X in a particular
coordinate system (
might give the 3 Cartesian components of hand
movement direction,
for example). If there exists a matrix H such that 
,
then one can say that the distributed representation D codes the
coordinate system T. Yet this will hold for any 
which is close to
the linear span of the basis functions 
, so we cannot claim that
D encodes any single coordinate system for X within this span better
than another.