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'''Mathematical Consciousness Science''' (MCS) is an interdisciplinary field in the intersection between the scientific study of consciousness and applied mathematics. Many Over the centuries, many mathematicians have taken an interest in consciousness over the centuries , including [https://en.wikipedia.org/wiki/Ren%C3%A9_Descartes René Descartes], [https://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibniz Gottfried Wilhelm Leibniz], [https://en.wikipedia.org/wiki/Bernard_Bolzano Bernard Bolzano], [https://en.wikipedia.org/wiki/Edmund_Husserl Edmund Husserl], [https://en.wikipedia.org/wiki/Alfred_North_Whitehead Alfred North Whitehead], [https://en.wikipedia.org/wiki/Bertrand_Russell Bertrand Russell], [https://en.wikipedia.org/wiki/Alan_Turing Alan Turing] and [https://en.wikipedia.org/wiki/Roger_Penrose Roger Penrose], to name a few, whilst others have developed areas of mathematics that are finding new applications in MCS including [https://en.wikipedia.org/wiki/Thomas_Bayes Thomas Bayes], [https://en.wikipedia.org/wiki/Ludwig_Boltzmann Ludwig Boltzmann], [https://en.wikipedia.org/wiki/Andrey_Markov Andrey Markov] and [https://en.wikipedia.org/wiki/Claude_Shannon 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 ''[https://en.wikipedia.org/wiki/Hard_problem_of_consciousness 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 but any research concerning consciousness. However, some aspect what distinguishes MCS from other forms of consciousness or some issue involving consciousness, where mathematically or computationally formulated models or theories play a research is the central roleof 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.

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 [https://en.wikipedia.org/wiki/Functional_magnetic_resonance_imaging Functional Magnetic Resonance Imaging] (FMRI). In particular, study of the [https://en.wikipedia.org/wiki/Neural_correlates_of_consciousness Neural Correlates of Consciousness] (NCC) was pioneered in the 1990s by Nobel Laureate [https://en.wikipedia.org/wiki/Francis_Crick Francis Crick] and collaborator [https://en.wikipedia.org/wiki/Christof_Koch Christof Koch]. In turn, two series of academic conferences began, [https://en.wikipedia.org/wiki/The_Science_of_Consciousness The Science of Consciousness] (TSC) conference and the conference of the [https://en.wikipedia.org/wiki/Association_for_the_Scientific_Study_of_Consciousness Association for the Scientific Study of Consciousness] (ASSC), which was founded in 1994. During the 1990s and early 2000s two three particularly prominent mathematically , or computationally, formulated theories of consciousness were proposed, [https://en.wikipedia.org/wiki/Orchestrated_objective_reduction Orch OR], [https://en.wikipedia.org/wiki/Global_workspace_theory GNW] and [https://en.wikipedia.org/wiki/Integrated_information_theory IIT]. In general, the scientific study of consciousness now comprises a host of models that collectively include complementary and conflicting assumptions and goalspredictions<ref name=Signorelli2021>Signorelli, C.M; Szczotka, J.; Prentner, R. (2021), Explanatory profiles of models of consciousness: towards a systematic classification, Neuroscience of Consciousness, 7(2), niab021</ref>. 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 [https://seminar.math-consciousness.org/index.html Mathematical Consciousness Science online seminar series] and the [https://omcan.web.ox.ac.uk/home Oxford Mathematics of Consciousness and Applications Network] (OMCAN) began. In September 2019 the first [https://www.models-of-consciousness.org/ Models of Consciousness (MoC1)] academic conference was held at Oxford with an international team of organisers. [https://amcs-community.org/events/moc2-2021/ MoC2] was held online in September 2021, followed by [https://amcs-community.org/events/moc-3-2022/ MoC3] in-person at Stanford University in September 2022. In January 2021 the [https://amcs-community.org/ Association for Mathematical Consciousness Science] (AMCS) was founded in order to pull together the growing number of conferences, workshops and seminars in MCS research.

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 principle principles or hypotheses; as per [https://en.wikipedia.org/wiki/Occam's_razor Occam's razor] and, similarly, Einstein’s quotation “everything should be made as simple as possible, but no simpler”.

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 , Markov processes 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| Mathematizing phenomenology]].

===Higher-level Embodied perception models===There are a number of models in MCS that propose a higher-level wider viewpoint of conscious systems than archetypal models. By higher-level Embodied perception models or viewpoint we are not referring here to Higher-Order Theories of consciousness (HOT) which hypotheses and organisational principles for how systems may develop perception through interacting with their wider environment. As such, embodied perception research has a rather different meaningsignificant overlap with [https://en.wikipedia.org/wiki/Embodied_cognition embodied cognition].

*The '''[https://en.wikipedia.org/wiki/Free_energy_principle Free Energy Principal]''' (FEP)<ref name=Friston2006> Friston, K.; Kilner, J.; Harrison, L. (2006), A free energy principle for the brain. Journal of Physiology-Paris. Elsevier BV. 100 (1–3): 70–87. doi:10.1016/j.jphysparis.2006.10.001</ref> 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 high-level view point wider viewpoint of the potential connections between systems and perception. It has been claimed that FEP has a connection to [https://en.wikipedia.org/wiki/Autopoiesis autopoiesis], but some contest this claim<ref name=Paolo2021> Di Paolo, E. A.; Thompson, E.; Beer, R. D. (2021), Laying down a forking path: Incompatibilities between enaction and the free energy principle. https://psyarxiv.com/d9v8f</ref>.

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<ref name=Yoshimi2007> Yoshimi, J. (2007), Mathematizing phenomenology. Phenomenology and the Cognitive Sciences, 6(3), 271–291. https://doi.org/10.1007/s11097-007-9052-4</ref>, or topological spaces based on particular assumptions about experience<ref name=Stanley1999> Stanley, R. P. (1999), Qualia-Space. Journal of Consciousness Studies, 6(1), 49-60.</ref><ref name=Prentner2019> Prentner, R. (2019), Consciousness and Topologically Structured Phenomenal Spaces. Consciousness and Cognition, 70, 25-38.</ref>. Related proposals along these lines have to do with neurophenomenology<ref name=Varela1996> Varela, F. J. (1996), Neurophenomenology: a methodological remedy for the hard problem. Journal of Consciousness Studies, 3, 330–49.</ref>, mathematical representations of qualia-spaces, or category and process theories of consciousness<ref name=Tsuchiya2021> Tsuchiya, N.; Saigo, H. (2021), A relational approach to consciousness: categories of level and contents of consciousness. Neuroscience of Consciousness, 7(2), niab034.</ref><ref name=SignorelliWangCoecke2021> Signorelli, C. M.; Wang, Q.; Coecke, B. (2021), Reasoning about Conscious Experience with Axiomatic and Graphical ModelsMathematics. Consciousness and Cognition, 95:103168.</ref>.