Mathematical Consciousness Science

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Mathematical Consciousness Science (MCS) is an interdisciplinary field in the intersection between the scientific study of consciousness and applied mathematics. Over the centuries, many mathematicians have taken an interest in consciousness, including René Descartes, Gottfried Wilhelm Leibniz, Bernard Bolzano, Edmund Husserl, Alfred North Whitehead, Bertrand Russell, Alan Turing and Roger Penrose, to name a few, whilst others have developed areas of mathematics that are finding new applications in MCS including Thomas Bayes, Ludwig Boltzmann, Andrey Markov and Claude Shannon.

The term Mathematical Consciousness Science began to be used and recognized from around 2018 onward following a rapid increase in the development of new mathematical and/or computational models and formal theories of consciousness that began from around 2005. Many researchers in the MCS research community anticipate that mathematical approaches are needed to tackle challenges such as: the Hard problem of consciousness, which is the problem of explaining why and how we have phenomenal experience; explaining how consciousness relates to the physical domain, particularly regarding the brain and artificial systems; and, fundamentally, the many questions involving consciousness and the foundations of physics, particularly Quantum Mechanics. Such challenges are at the heart of MCS. However, what distinguishes MCS from other forms of consciousness research is the central role of precise mathematical and computational models in the research program, rather than mathematics just being used as a secondary tool. Different research efforts fall within the field’s scope, since MCS also exists to complement the work of researchers working in the wider field of the scientific study of consciousness which intersects with Neuroscience, Philosophy, Artificial Intelligence and Experimental Psychology, for example.

History of the MCS research community’s development

There have been a number of insightful theories about consciousness over the centuries. The modern generic term for consciousness research is the scientific study of consciousness which began to be used in the 1990s following technological advances such as the development of Functional Magnetic Resonance Imaging (FMRI). In particular, study of the Neural Correlates of Consciousness (NCC) was pioneered in the 1990s by Nobel Laureate Francis Crick and collaborator Christof Koch. In turn, two series of academic conferences began, The Science of Consciousness (TSC) conference and the conference of the Association for the Scientific Study of Consciousness (ASSC), which was founded in 1994. During the 1990s and early 2000s three particularly prominent mathematically, or computationally, formulated theories of consciousness were proposed, Orch OR, GNW and IIT. In general, the scientific study of consciousness now comprises a host of models that collectively include complementary and conflicting assumptions and goals[1]. The term Mathematical Consciousness Science began to be used and recognized from around 2018 onward following a rapid increase in the development of new mathematical and/or computational models and formal theories of consciousness. In 2018 the Mathematical Consciousness Science online seminar series and the Oxford Mathematics of Consciousness and Applications Network (OMCAN) began. In September 2019 the first Models of Consciousness (MoC1) academic conference was held at Oxford with an international team of organisers. MoC2 was held online in September 2021. In January 2021 the Association for Mathematical Consciousness Science (AMCS) was founded in order to pull together the growing number of conferences, workshops and seminars in MCS research.

Mathematical and computational models and theories of consciousness

MCS seeks mathematically formulated models of consciousness that correctly predict the properties of consciousness (and their relation to/interaction with the physical domain) whilst, ideally, following in a natural way from some basic assumptions and relatively simple principles or hypotheses; as per Occam's razor and, similarly, Einstein’s quotation “everything should be made as simple as possible, but no simpler”.

Archetypal models

Being hypotheses about consciousness and its relation to the physical domain, the archetypal model of consciousness arguably has three parts, namely, a mathematical model of salient aspects of the physical system (e.g. circuit models, network models, joint probability distributions etc), a mathematical model for aspects of conscious experience (e.g. topological spaces, metric spaces, matrices of relationships, categories, intensity scales etc) and some mapping between the physical domain model and the consciousness domain model (e.g. a homomorphism, limit or optimal boundary point, functor, scalar function etc). The models make various predictions about, for example, phenomenal perception, the relational content of consciousness, the level and intensity of consciousness, attention, and the unity (and disunity) of consciousness within and between systems. The physical domain models and consciousness domain models are also of interest in their own right and some researchers in MCS focus on the development of these models. In the consciousness domain, this is often referred to as Mathematizing phenomenology.

Examples include:

  • Integrated Information Theory (IIT)[2] models the physical system with Markov processes obtained from a circuit model. The consciousness domain model involves several outputs of the IIT algorithm but most notably includes a non-negative scalar function , related to the intrinsic irreducibility of a network, and an associated “cause-effect structure” (earlier referred to as the “maximally irreducible conceptual structure”) that, according to the theory, respectively correspond to the quantity and quality of consciousness. The algorithm, mapping between the physical domain model and the consciousness domain model, has causation at its heart. Therefore, a key modelling assumption in IIT is that a system’s consciousness is directly related to its causal properties.
  • Expected Float Entropy Minimisation (EFE)[3] and its extension to model unity, models the physical system with a joint probability distribution that represents the system's intrinsic bias to being in certain states over other states. The consciousness domain model involves a hierarchy of relational models in the form of matrices of real valued parameters in the range . According to the theory, the relational models provide an interpretation of system states that gives the relational content of the associated experience. According to the theory, the theory’s extension to model unity then deals with the issue of integration. The mapping between the physical domain model and the consciousness domain model involves the minimisation of expected float entropy so that the resulting relational model gives the minimum expected entropy interpretation of system states. Therefore, two key modelling assumptions in EFE minimisation are that consciousness is a minimum expected entropy interpretation of system states and that the structural content of consciousness comes from the correlations and relationships intrinsically encoded in the bias of a system.

Computational models

At some level of abstraction, computational models in MCS are typically models of the brain together with hypotheses that relate them in some way to consciousness. The hypothetical relation to consciousness may involve properties of the model that are a-priori present by design or are discovered, or confirmed to exist, experimentally. Physical systems used to instantiate computational models vary widely but usually involve Neuromorphic Computing at some level of abstraction. Computational models can also generate synthetic data for use in other models of consciousness such as archetypal models. The development of computational models in MCS has a significant overlap with Artificial Consciousness (AC) research.

Examples include:

  • Cortex-like complex systems of networks-of-networks[4] have been used to model the cortex and involve spiking nodes, that model neurons, and real time lags that model signal transmission delays. The system is a network of sparsely connected modules where each module is a network of densely connected nodes. Instantiation of the model on the SpiNNaker supercomputer revealed a wide set of latent, internal, common dynamical modes of operational behaviour. The hypothesis relating the model to consciousness is that the observed modes are candidates for sensations, feelings and moods in conscious systems. SpiNNaker is being used as one component of the neuromorphic computing platform for the Human Brain Project.
  • The Conscious Turing Machine (CTM)[5] is a computational model that formalizes the Global Workspace Theory (GWT). GWT postulates the existence of a type of working memory in the brain to which various subsystems may gain access and influence the contents of. The hypothesis relating the model to consciousness is that it is the content of this working memory that we are conscious of. The CTM formalization helps to remove ambiguity and allows GWT to be straightforwardly modelled on a digital computer. CTM can be seen as an idealization that aims to synthetically model the essentials of GWT.

Embodied perception models

There are a number of models in MCS that propose a wider viewpoint of conscious systems than archetypal models. Embodied perception models are hypotheses and organisational principles for how systems may develop perception through interacting with their wider environment. As such, embodied perception research has a significant overlap with embodied cognition.

Examples include:

  • The Free Energy Principal (FEP)[6] is a model for how living and non-living systems remain in non-equilibrium steady-states by restricting themselves to a limited number of states. The model provides a principle by which systems may create an internal model of the outside environment in order to maintain their own integrity. The minimisation of free energy is formally related to variational Bayesian methods and was originally introduced as an explanation for embodied perception in neuroscience. Since it is the models that systems internally create under the FEP that may have relevance to perception, and not FEP directly itself, FEP can be thought of as giving a wider viewpoint of the potential connections between systems and perception. It has been claimed that FEP has a connection to autopoiesis, but some contest this claim[7].

Theories involving Quantum Mechanics

Ever since the 1930s, when Schrödinger and Einstein discussed the Schrödinger’s cat thought experiment, there has been the idea that somehow consciousness may have something to do with Quantum Mechanics (QM). Accordingly MCS includes a number of research directions involving QM. Two significant possibilities appear to be that either some kind of consciousness-like property of certain configurations of matter collapses wave functions (a refinement of the role of the observer in the Schrödinger’s cat thought experiment) or that wave function collapse plays a part in how consciousness happens (collapse causes consciousness). Alternatives to these ideas also exist[8] and have been researched by numerous scholars. Researchers in MCS thus continue to narrow down the theoretical possibilities.

Examples include:

  • Orchestrated Objective Reduction (Orch OR)[9] is a theory of consciousness that attributes non-computable quantum processing as the basis of consciousness. It involves an objective process of wave function collapse based on conservation of energy by way of the proposition that if a particle with mass is in a superposition then it ought to give rise to an effective self-energy due to the resulting superposition of distorted spacetime. According to the theory, the objective wave function collapse selects states neither randomly nor algorithmically and plays a part in how consciousness arises from the system. The theory further proposes that the quantum processing may occur within structures called microtubules in the dendrites of neurons.
  • Proto-consciousness induced quantum collapse is a refinement of the idea that the act of conscious observation collapses wave functions. Archetypal models of consciousness tend to predict the weak presence of some aspects of consciousness even for some small simple systems. In principle, these quantities can be incorporated into stochastic differential equations that reduce to the Schrödinger equation in the quantum regime and give classical behaviour for macroscopic objects. At intermediate scales, consciousness-like properties (proto-consciousness) can then have noticeable effects on the predictions of such collapse models which can then be tested experimentally[10]. Examples include Quantum Integrated Information (QII)[11] induced quantum collapse which extends Integrated Information Theory. Model Unity, in the Expected Float Entropy minimisation model of consciousness, has also been suggested for use in collapse models[3]. More fundamentally, it has been shown that if consciousness does have any forward influence on the physical domain then this will at least manifest itself as an influence on quantum collapse[12].
  • The Free will theorem[13] shows that, subject to some minimal assumptions, if we have free will, in the sense that our choices are free rather than being a function of past events, then the behaviour of some elementary particles is also not a function of past events. The experience of actual or apparent free will is an aspect of consciousness and the Free Will Theorem is another example of theory in Quantum Mechanics making a potential connection between consciousness and the physical domain. More generally, there are other, often related, notions of free will.

Mathematizing phenomenology

The phenomenology of experience is perhaps the biggest explanandum for a science of consciousness. One approach to its study concerns the structure of conscious experience. The way this is made precise is very much related to the idea of representing the consciousness domain in terms of a mathematical space such as, for example, a state spaces of a dynamical neural system[14], or topological spaces based on particular assumptions about experience[15][16]. Related proposals along these lines have to do with neurophenomenology[17], mathematical representations of qualia-spaces, or category and process theories of consciousness[18][19].

Such representations could serve as precise statements (or models) of target phenomena for use in archetypal models of consciousness to help establish the relationship between states of neurocomputational systems and states of consciousness. More speculatively, phenomenal experiences could be seen not only as descriptive but as co-emergent with[20], or constitutive of[21], the neuroscientific phenomena that supposedly explain them.

Model validation

By requiring researchers to turn their intuitions into precise mathematical definitions, MCS provides a common language for comparing models of consciousness making them more accessible to scientific scrutiny. Mathematically formulated models of consciousness can be explored using mathematical methods, applied and tested using synthetic or clinical data and, for computationally formulated models, instantiated using neuromorphic computing. However, model validation is a challenge and an active area of research in MCS. More generally, Nagel and Chalmers have produced well known texts[22][23] on the conceptual limitations of the scientific study of consciousness.

Model validation under the closure of the physical

In many cases our own introspection (knowledge of our experiences) can be used for model validation. For example, if a model predicts that you and your next-door neighbours have unified visual perception then, without significant modifications, you know the model is wrong. Such clearly erroneous models of consciousness are sometimes referred to as raspberry jam models. However, model validation may become problematic if we doubt the validity of introspective knowledge or philosophical assumptions. For example, consider the set of all models of consciousness that rule out changes to the theory and laws of physics. Such models of consciousness are said to obey the closure of the physical. But, since measurements are physical events, including human reports, all such models will give identical predictions for the result of measurements[24]. Ultimately then, once all of the raspberry jam models of consciousness have been removed, we may end up with a number of equivalent models that are highly consistent with introspection. Hence, different ways of selecting between empirically indistinguishable models may have to be appealed such as, for example, internal consistency, completeness, beauty, or simplicity. Occam's razor might then be a useful guide in the development of mathematically formulated models of consciousness.

The majority of models of consciousness appeal to the closure of physics, although many are often easily extended to theories where the physical is open, just by involving some quantity given by the model in a collapse term added to the Schrödinger equation.

MCS research has also identified theoretical difficulties with using human report as a method of validating theories of consciousness[25].

Model validation under the physical being open

If nature includes occurrences of consciousness influencing the physical domain (rather than only the physical domain influencing the physical domain and consciousness merely being an epiphenomena) then model validation in MCS also falls within the scope of physics. In this case the current theory and laws of physics, as we know them, would be slightly wrong and in need of adjustment. It has been shown that if consciousness does have an influence on the physical domain then this will at least manifest itself as an influence on quantum collapse[12] and, therefore, proto-consciousness induced quantum collapse is an important line of research in model validation. However, it should be noted that if the physical is open then this does not imply that consciousness is substantially more than an epiphenomena in the sense that the influence of consciousness on the physical domain may merely provide another feedback loop for the physical domain to act on itself. Models that are rooted in non-physicalist philosophical frameworks, such as idealism or neutral monism, are natural candidates when looking for models that assume the physical as open.

Interplay between MCS and philosophy

Philosophical assumptions in models

Formulating mathematical and computational models and theories of consciousness inevitably involves making a number of philosophical assumptions. Typically, some assumptions are stated explicitly as model assumptions whilst others are implicit. It is also the case that models are usually independent of at least some philosophical positions allowing them to be extended in different directions if needed. For example, neither IIT or EFE minimisation exclude the possibility of the physical being open. Examples of philosophical assumptions include, consciousness being an epiphenomena or not, the content of consciousness being defined by the system itself or involving something extrinsic to the system, privacy of experiences, the role of free-will, and other metaphysical assumptions. Philosophical frameworks include, amongst others, physicalism, dualism, idealism, dual-aspect monism, and natural monism. An important argument, sometimes attributed to Bertrand Russell and Arthur Eddington, says that physical properties in nature are extrinsic manifestations of intrinsic properties; that is to say, seen from the outside, the relationships in nature give us physics, chemistry and biology, but seen from the inside they yield conscious experience[26]. Similar intuition has played a part in the formulation of several mathematical models of consciousness including IIT and EFE minimisation.

Philosophical implications of models

Panpsychism of some form is a common implication of archetypal models of consciousness because the physical domain sub-models they employ tend to be given in sufficiently abstract terms as to be applicable to many systems in nature beyond merely the brain. In particular, archetypal models may be applicable everywhere, all be it such that the extent, and aspects, of consciousness they predict will depend on the situation. If proto-consciousness induced quantum collapse were to be confirmed for some archetypal model of consciousness then this would provide evidence of such universality. A successful archetypal model of consciousness may also provide evidence in favour of the Mathematical universe hypothesis if the model is able to be applied to substructures of mathematical structures. This is because such a model implies there are conscience mathematical substructures and, to the extent to which the substructures can interact with the larger mathematical structure in which they are embedded, they will experience that larger structure as their environment, their world, their universe, and this is analogous to the situation we find ourselves in.

Applications of MCS

MCS has potential for disruptive advances in computer science where formal approaches to how some aspects of consciousness emerge at the system level are leading to new ideas about system architectures, neuromorphic computing, AC and AGI[4][27]. The brain learns a huge number of different patterns over the course of our lives and constantly acts like a heuristic machine filling in missing information and anticipating the near future. But whilst current computers are wonderful at crunching numbers they lack the general heuristic abilities of the brain and do not posses human imagination. MCS provides an approach to developing new computing architectures, potentially utilising non-binary and nondeterministic systems, that may narrow the abilities gap[4][27]. Human healthcare may also benefit from MCS. Mathematical models based on theories such as Predictive Processing (PP) may bring new insights into psychological and emotional disorders such as OCD and depression. Moreover, there is the need for a rigorous formal understanding of when consciousness is present in humans. This is important in medical practice regarding anaesthesiology[28] and for those making ethical decisions about patients who are locked-in or comatose. There is also the possibility of future technologies for brain enhancement, or intervention, where the implications for consciousness will need to be properly understood. In physics, experimental confirmation of proto-consciousness induced quantum collapse would create a paradigm shift in science.


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