  Any theory amenable to scientific inquiry must have testable consequences.
This minimal criterion is uniquely challenging for the study of consciousness,
as we do not know if it is possible to confirm via observation from the outside
whether or not a physical system knows what it feels like to have an inside - a
challenge referred to as the "hard problem" of consciousness. To arrive at a
theory of consciousness, the hard problem has motivated the development of
phenomenological approaches that adopt assumptions of what properties
consciousness has based on first-hand experience and, from these, derive the
physical processes that give rise to these properties. A leading theory
adopting this approach is Integrated Information Theory (IIT), which assumes
our subjective experience is a "unified whole", subsequently yielding a
requirement for physical feedback as a necessary condition for consciousness.
Here, we develop a mathematical framework to assess the validity of this
assumption by testing it in the context of isomorphic physical systems with and
without feedback. The isomorphism allows us to isolate changes in $\Phi$
without affecting the size or functionality of the original system. Indeed, we
show that the only mathematical difference between a "conscious" system with
$\Phi>0$ and an isomorphic "philosophical zombies" with $\Phi=0$ is a
permutation of the binary labels used to internally represent functional
states. This implies $\Phi$ is sensitive to functionally arbitrary aspects of a
particular labeling scheme, with no clear justification in terms of
phenomenological differences. In light of this, we argue any quantitative
theory of consciousness, including IIT, should be invariant under isomorphisms
if it is to avoid the existence of isomorphic philosophical zombies and the
epistemological problems they pose.
