3. Limitations of the approach
This mathematical approach for GRT calculates the spatial boundary of any conscious entity in each moment and a simple scalar value for phenomenal capacity of that entity. This framework may provide a useful set of tools to probe the capacity for consciousness at all scales. The main value of this approach will be to provide for comparisons across all levels of consciousness and physical complexity.
There are some limitations in employing this relatively simple approach, however, and I’ll discuss the more obvious ones here.
  1. A scalar value can give no indication of the different types of conscious experience available to comparably complex conscious entities, or of the different types of consciousness present in the same conscious entity in each moment.
Omega is a measure of the capacity for phenomenal consciousness, not a measure or characterization of any actual phenomenal consciousness. It is, rather, a means for measuring comparative cognition and consciousness.
Integrated Information Theory (IIT) does suggest methods for characterizing consciousness, in addition to a quantification framework (Oizumi, et al. 2014). IIT’s tools for characterizing phenomenal experience – depiction of visual “constellations” for each possible quale in a given “complex” – rather than only quantifying the capacity for such experiences, may be compatible with GRT, and may be a useful addition to the tools offered here. That is, IIT’s constellation-qualia characterization tools may be compatible with GRT. This is an area for future work with respect to GRT’s development as an alternative or complement to IIT.
            
            2. Calculating PI and CI in any biologically-complex entity will be difficult
As discussed above, any biologically interesting system examined in the GRT framework will be difficult to quantify accurately because of the complexity of the biological structures involved. Even relatively simple biological systems like drosophila or C. elegans have great depth that will require, at least early in the development of measures of comparative consciousness, many simplifying assumptions. However, this is the case for any theory that attempts to make sense of the biological world, and it will require, like any such theory, dedicated effort by many researchers to develop reliable simplifying tools for making meaning quantifications. Over time, such techniques will improve and more accurate quantifications will become possible. One such example has already been developed with respect to integrated information. Casarotto et al. 2016 (with Tononi as a co-author) employs a simplifying approach, the Perturbational Complexity Index, as a proxy for integrated information.
            3. GRT adopts a process notion of the flow of time
GRT adopts a process time notion of time (Hunt 2014). As described in Hunt 2014, this is a notion of time that matches the human experience of time, which appears to entail a steady passage of time, and resolves various difficulties presented by other notions of time. So while some observers may consider GRT’s adoption of process time to be a difficulty or limitation of the theory, this may be one of its strengths.
4. Conclusion
The mathematical framework offered here may be useful to both researchers and philosophers in probing the nature and extent of consciousness. Various other quantification and characterization approaches have been offered, including in particular the Integrated Information Theory developed by Tononi, Koch and others. Hunt 2011 and 2014 describe how GRT and IIT differ and those differences between the theories generally remain. We acknowledge in the present paper that IIT’s tools for characterizing phenomenal experience – depiction of visual “constellations” for each possible quale in a given complex – rather than only quantifying the capacity for such experiences, as is done in the present paper, may be compatible with GRT. IIT’s constellation-qualia characterization tools may help to fill in key gaps with GRT.
The key difference between GRT and IIT is an explicitly process view of time in GRT, and various types of shared resonance (and thus various phase transitions in the transmission of information) forming, consequently, the key signature of complex consciousness. IIT relies instead on an “exclusion principle” for defining the dominant consciousness (“maximally irreducible conceptual structure” or MICS) present in any collection of items (Oizumi, et al. 2014). GRT results in a nested hierarchy of conscious entities in any complex consciousness, whereas IIT results in the extinction of subsidiary conscious entities as a result of the combination of consciousness into a single larger entity. Hunt 2014 describes these differences in more detail.
This is not the place for a detailed comparison between GRT and other theories of consciousness. However, GRT was developed as a way to mitigate at least some of the difficulties posed by other theories of consciousness. The approach proposed is also a relatively simple mathematical framework, and simplicity presents some advantages.