To experience one’s own world, to hold phenomenal content in one’s experience, to know what it is like to be you—these are the murmuring qualities of conscious experience. Though pretty much given from our subjective perspective, it can be maddening to approach scientifically. The problem how conscious experience emerges from the vast amount of electrical activity, which, imperceptibly but surely, is structured within an organic mass of living tissue, is thought-wrenching. Nonetheless, for me at least it is worth spending some time on.
In my own research, I apply practical measures of emergence to complex neural systems and the dynamics that unfold. Emergence can be taken as some coarse-grain regularity that emerges in the system, or, some integrated whole that appears to be greater than the sum of its parts. Ultimately, I am interested in how a practical approach to emergence might inform theories of consciousness, and how we understand brain dynamics associated with phenomenal experience. It is not altogether uncommon that theoretical approaches to consciousness converge on the quality of integrated wholes of brain dynamics as necessary for consciousness. Here, Integrated Information Theory (IIT) is a particularly rigorous example of an approach that contextualises phenomenal experience as the integrated closure of a set of parts that is—in some sense—greater than the sum of its parts.
IIT attempts to develop a formal approach to understanding phenomenal consciousness in physical terms. By construction of a mathematical framework, it outlines a set of self-evident statements (axioms) that are intended to capture the essential qualities of phenomenal experiences. By way of substantiating statements about phenomenal experience in physical terms, these axioms are related to corresponding statements (postulates) on information processing. It is then from postulates that we infer information structures, which, in the IIT formalism, fully determine phenomenal experiences. Grounded in mathematical rigour, it has attracted some criticism of its axiomatic foundations. Though much of the critiques appear to be illuminating and warrant some thought, from my perspective, I find that it would be productive to clarify the axiomatic approach taken by IIT more broadly and offer nuanced but crucial pivots in understanding.
What I will propose here is a generalized definition of IIT’s axioms and the inference procedure from which its postulates are generated. This softens the tethering to strict mathematical formal systems, illustrating that IIT postulates and information structures corresponding to phenomenal experiences are inferred by abduction, or Inference to Best Explanation (IBE), and not deduction. IBE is often defined as the process of selecting a hypothesis or statement that best explains the data that is available. Inferential reasoning by IBE applies to deriving informational structures corresponding to phenomenal experiences from the postulates of IIT.
This offers a nuanced distinction between the axiomatic approach in IIT and other formal mathematical systems. And indeed, I will argue that inferring conscious experiences within IIT stands separate from proving theorems in formal mathematical systems. Next, I will attempt to clarify what utility the axioms and postulates in this current state serve. I also advocate that much of the axiomatisation in mathematics occurs post factum, and can at times be a recursive process between postulates and axioms. My humble hope is that this interpretation will aid in the progress towards understanding how IIT proposes to evaluate phenomenal experience within a mathematical framework.
1. Criticisms of axiomatic structure
A formal system, primarily based on the axiomatisation of mathematical explanations of pattern, structure, and phenomena, is endowed with foundational statements that dictate which statements can be true or false within the system. Simply, axioms are some set of self-evident statements that are true for all other statements that can be generated within the formal system by some set of rules of inference. In 1931, a remarkable mathematician and logician, Kurt Gödel, published a proof of the paradoxical nature of all arithmetical formal systems.
Gödel showed, that once we arithmetise all statements in our formal system, our system cannot be both complete and consistent. He showed that if our system is complete, possessing a set of axioms that are fully determined—i.e., we cannot add extra axioms, then the system will possess truth statements that are not provable by deductive inference from the axioms alone: there might be contradictions in the proof. Simply, provability refers to being able to generate some statement within the system from the axioms alone by the deductive inference rules possible in that system. If Gödel’s proof would apply to IIT’s axioms, this would mean that there are phenomenal experiences that we would not be able to generate from the axioms and postulates alone by deductive inference rules and discern their truth value.
However, and herein lay the paradoxical nature: if every statement can be proved by deductive inference from the axioms of the formal system—i.e., if the system is consistent, then it is fundamentally incomplete! There can always be another axiom that can be added! Again, if this were to apply to IIT’s axioms, this would indicate that if all phenomenal experiences can be proved by deductive inference from the axioms, then the axioms are not complete; there could be an extra axiom of phenomenal experiences that can be added.
Though Gödel’s fundamental proof shook the mathematical world, it did not altogether halt progress. Mathematicians kept working in their respective fields, developing better methods of prediction and postulating useful statements that altogether propelled progress in fields such as electrodynamics, quantum field theory, and evolution. Further, the progress and predictive power cemented the methods used within these fields as robust formal approaches in understanding the reality in which we live and how we fundamentally understand the distinct phenomena associated with electrical and magnetic fields, quantum particles, and living matter.
As the keen reader might already be aware, Gödel’s proof of undecidability is based on formal systems with deductive (and potentially inductive) inference rules. IIT’s axioms, postulates, and generated information structures corresponding to phenomenal experience are not intended in the same mathematical sense. The IIT framework is based on inferring statements (here informational structures) to best explanation. Nevertheless, it seems to me that criticisms of the axiomatic formalism of IIT broadly mirror Gödel’s paradox: either they are arguments about the incompleteness of the axioms, that the axioms are not encompassing of the total repertoire and variations of phenomenal experience; or, they are arguments about the axioms’ inconsistency; that Φ as a measure of consciousness might not be unique and that the Φ-structures are not indicative of phenomenal experiences. Some of these criticisms are to be taken seriously and inherent in the context of characterizing phenomenal experience. Though necessary to discuss, they warrant a standalone work. Here, I address those critiques in the context of formal systems of the kind worked on by Gödel, which might not directly apply to IIT. Indeed, I advocate that criticisms of incompleteness of the axioms or inconsistency within a deductive framework serve as cautionary gestures and not overhaul solutions to scrapping the foundations of IIT. If history is to serve us as a wise teacher, progress can be made even in light of these inherent limitations.
To aid further in this attempt, I use a case example to illustrate another useful point: that the development of axioms in mathematics has, in general, been post factum. I will briefly talk about a branch of mathematics called Group Theory, and how its conception differed from the way we teach and present it. Then, in this light, I hope to elucidate the nature of axioms and postulates in IIT. I propose an inversion of focus from axioms to postulates and with this, a nuanced shift in the terminology used to explain them. With this inversion, it is reasonable to suggest that the axioms can be substantiated post factum. Which reflects how most mathematical and formal approaches to other phenomena have developed axioms after the fact rather than as the starting point. And yet, the axioms may serve as useful guideposts in establishing a mathematical framework for phenomenal experience. Hopefully this shows how IIT parallels the progress made in other mathematical fields.
2. Development of axioms in mathematics
Contrary to how axiomatic approaches are pedagogically presented, the development of formal mathematical systems has not progressed from axioms, but toward them. After the development of a candidate theory has reached a certain threshold of understanding, axioms are refuted or reinforced: similarly, to the way one might write an abstract at the end of finishing the first draft of a paper.
Take for example the much beloved (personal bias here!) mathematical study of groups and the development of Group Theory within Abstract Algebra. A group can be simply defined as a set of constituent elements with an operation used to relate any two elements in that group. This group is then endowed with some basic axioms that we will not divulge into here. What is of illustrative importance is the development of Group theory.
Group theory in itself evolved from a number of parallel sources: Gauss’ modular arithmetic, Euler’s expositions in number theory, Abel’s work on solving polynomial equations, and the list goes on. It all culminated with Évariste Galois who first defined these mathematical expositions as a “group” in the earlier half of the 19th century. A fascinating character himself who died at the age of 20 after he was injured in a duel. Only by 1882 did Walther von Dyck define the axiomatic statements of Group Theory after a century of progress on group structures and their utility in a variety of parallel mathematical systems. It was progress from Galois’ recognition of the similarity of self-evidential statements about groups in different fields, up until von Dyck’s indoctrination of these postulates as generated from corresponding axioms that Group Theory developed as a formal system. Indeed, Group Theory continued to develop after this, pivoting the basic interpretation of the axioms in progressive new discoveries such as Lie and Algebraic groups.
And though we now teach Group Theory by presenting the axioms that define a group first, the evolution of Group Theory has progressed in the opposing direction: toward the axioms. It would seem, that most mathematical systems have evolved in a similar fashion.
3. On the clarity of IIT’s axioms and postulates
Formal mathematical systems, in general, have developed rules of inference based on deductive, and sometimes, inductive reasoning. As mentioned in Section 2, it is these systems that Gödel’s paradoxical proof applies to; particularly deductive systems. However, IIT differs crucially in this regard by being an IBE-based framework. This does not discount IIT’s mathematical framework, which should be distinguished from being a mathematical theory in the deductive and inductive sense. To the best of my knowledge, IIT developers have not referred to IIT as a mathematical theory of consciousness, but rather emphasize that it is a mathematical framework. This is a brief, but important distinction in clarifying the confusion that might conflate IIT with formal systems of the kind confronted by Gödel. Though IIT proposes to be a theory of consciousness, it is not a mathematical theory of consciousness, but a theory of consciousness within a mathematical framework. With this, it allows us to loosen the tethering of how we define our axioms in relation to postulates and vice versa.
IIT takes axioms as self-evident truths regarding intrinsic existence. It is existence from the intrinsic perspective that is of primary interest here. IIT’s assumption then, is that phenomenal experience is defined with recourse to intrinsic existence. These axioms then serve as the basic building blocks that define any structure from its intrinsic perspective. This nuanced but key distinction serves as an important pivoting of perspective. How a structure of-itself is defined intrinsically becomes the starting point. Hence the zeroth axiom–that phenomenal consciousness exists–is the premise on which the other axioms’ self-evidential quality begins to make sense.
For some structure to intrinsically (1st) exist, that said existence (0th) needs to be from the perspective of the structure itself: it exists within, and of, itself. The existence of the structure contains information (2nd); it specifies something. The structure is integrated (3rd); it is a whole and the information it possesses as a whole is not reducible to its constituents. In general, the information cannot be broken down into constituents. This structure, excludes (4th) other structures; it has a defined closure within a specific temporal and spatial scale. Lastly, this structure is compositional (5th); the structure is structured by a collection of constituents. These are the six axioms of IIT.
Simply, IIT defines the existence (0th) of the irreducible (3rd), informational (2nd) closure (4th) of a collection (5th) of interacting parts (1st).
As you might have noticed, I have presented the axioms in terms of intrinsic existence and avoided the use of definitions that evoke phenomenal experience. Consciousness, phenomenology, experience, subjectivity, are all terms that have been avoided in order to illustrate my perspective and bolster intuition on the IIT axioms.
Minor disagreements aside, in science, we tend to agree on the general definition that phenomenal experience is something it is like to be in a given state, or, something it is like to perceive some informational content, such as the azure colour of the sky, the breathlessness after a long run, the touch of a lover’s hand. Though it comes with its own limitations, the ‘something it is like’ fundamentally entails that the ‘something’ is perceived by the thing itself: an intrinsic perspective.
For the experience to exist it must exist from the intrinsic point-of-view.
It is argued that the IIT axioms correspond to fundamental properties of experience, but it is by way of the existence of this experience in physical terms that the axioms define an intrinsic perspective. In order to infer the existence of experience in physical terms we have to infer to the best explanation. As, from the intrinsic perspective, the only element that is given is the experience itself. It is via this route that we develop the postulates in IIT. Assuming we prescribe to existence being defined by the ability to be affected by things in the world, and in turn, effect things in the world, then formalisms of informational structures and information processing lend themselves as natural frameworks in which to situate postulates of intrinsic existence.
To define existence from the intrinsic perspective in physical terms the IBE-inferred self-evidential statements (postulates) assume a cause-effect power; existence in the physical sense. The physical substrate of consciousness must have cause-effect power (0th) that is intrinsic (1st). This cause-effect power must specify some information between the constituent parts (2nd) that is integrated and irreducible to those same constituents (3rd). Importantly, this integrated causal structure must possess some degree of closure from other constituent elements integrated in the substrate (4th). And finally, as is obvious from the other axioms, the information structure is composed of constituents (5th). Therefore, the existence of all properties of an experience should be accounted for in physical terms because they correspond to an informational structure.
From this perspective, the abstraction away from ‘consciousness’ in the definition of the axioms lend themselves directly to their transformation to postulates. Indeed, they could be taken as the postulates themselves if they refer to an information structure; or, to an experience if we rephrase them in terms of phenomenal experience. This isn’t the only time that I attempt to blur the boundaries between the two. As I’ve already mentioned, this cyclical correspondence between axioms and postulates is what mathematical progress has been built on.
4. Reframing the interpretation of IIT’s axioms
From my interpretation, IIT proposes much the same: from postulates that refer to the minimally necessary properties of information structures to exist intrinsically, it attempts to develop a formally grounded approach to the intrinsic existence of experience. A shared set of axioms that might define the existence of phenomenal experience. It is this inversion of the relation between axioms and postulates that clarifies their interpretation, and that I propose here. It is through the process of Inference to the Best Explanation that we infer the postulates and generate the information structures corresponding to any given phenomenal experiences. Consequently, it is how we ascertain the credibility of the axioms. Though the exposition of IIT does begin with axioms, the practical work is done by using the postulates to develop testable predictions and infer (in an IBE sense) the degree of completeness of the axioms.
Ultimately, whether the axioms are complete, or whether IIT is a consistent theory (in the formal systems sense) of consciousness is an empirical question and will be decided by progress towards discerning its empirical validity. If we are to develop a theory of consciousness in physical terms similar to other formal theories of physical phenomena such as quantum fields or electrodynamics, a mathematically grounded approach such as IIT is warranted. Indeed, given that most of science has made progress by defining what things do rather than what things are, it is more likely that this process can always be refined. However, the goal posts of IIT’s six axioms might unfold to be useful guides for progress, similar to Galois’ theory and his definition of a group.
What is proposed here is rather than critiquing the validity of the axioms of IIT, what has proved to be of greater benefit in all mathematically structured theories is to make progress on the correct operationalisation and application of the postulates in order to test and determine whether the postulates correspond to the axioms, the validity of which will unfold from this pursuit. Much of this progress is already proving positive. In my own research I attempt to develop the applicability of measures of emergence to neural systems in order to establish the functional closure of the irreducible information of sets of elements. Which, as already might be clear, evokes much of the postulates of IIT. How measures of emergence relate to IIT is an open question that would yield fruitful outcomes for both: how measures of emergence might inform theories of consciousness, and how IIT can be better framed for testability on neurophysiological data. One of my particular interests lies in whether postulates of IIT and the inferences made from them can be reframed for continuous and discrete dynamics as well as states. Some work has already been done in this direction and I see this as a possibly fruitful endeavour of bringing IIT and dynamical systems approaches to brain dynamics closer together.
In an attempt to reframe how we interpret IIT’s axioms and placing emphasis on the postulates, my hope is that boundaries can be blurred between approaches to aid progress towards scientifically understanding our most maddening and evocative of questions: that of who we are and what we are made of.
I’m currently a Doctoral Researcher in Computational Neuroscience at the University of Melbourne and hold visiting research positions at the Institut du Cerveau (Paris) and the Sussex Centre for Consciousness Science (Brighton, UK). Generally, I explore formal measures of emergence and their application to neurophysiological data and complex neural systems in the context of how they might inform our understanding of brains and minds. Practically, I use large-scale brain modelling and information theory to explore the macroscopic activity unfolding in coupled dynamical systems representing whole-brain activity underlying global states of consciousness. With a background in neuroscience, mathematics, biochemistry, and psychology, I’m a big proponent of transdisciplinary approaches to studying the mind, and have a deep passion for communicating across disciplinary divides. Open science also requires open communication. Amongst other things, in my spare time I write creatively and train martial arts. Sporadically tweeting on @MilinkovBorjan