Welcome to my home page

I have long been interested in questions concerning the extent of the fundamental significance of mathematical structures. This page gives details of my research in the area of Mathematical Consciousness Science (MCS), and also on spaces and algebras of functions.

Models of Consciousness and the future of AI

Together with Prof. Kobi Kremnitzer, I ran the OMCAN network at the University of Oxford. In September 2019 OMCAN, working together with partners, co-organised and co-supported the inaugural Models of Consciousness (MoC) conference at the Mathematical Institute, Oxford. MoC is a landmark conference bringing together researchers whose activity relates to the theoretical and mathematical foundations of the scientific study of consciousness. I am delighted that the MoC conferences have since become the official conference of the Association for Mathematical Consciousness Science (AMCS).

Today, mathematical and computational modelling in consciousness science is becoming increasingly important as highly sophisticated AI and, ultimately, AGI approach. With the AMCS, I am helping to raise public awareness of the AI consciousness issue. At this time, we do not take a position on whether AI or AGI is, or will become, conscious or not, but do advocate substantially accelerating MCS research and consciousness science in general given the current lack of scientific consensus over which physical systems are capable of consciousness. See the media coverage listed here. Below are my talk videos on Expected Float Entropy Minimisation given at MoC1 (Oxford 2019), MoC3 (Stanford 2022), MoC4 (Oxford 2023).

Expected float entropy minimisation, relationships and consciousness

The question of whether there is a class of mathematical structures that have the property of consciousness may seem bizarre, daunting and yet perhaps on reflection important.

It is relevant to biology, neuroscience, computer science, mathematics and theoretical physics, an example of the latter being Max Tegmark’s "Mathematical Universe Hypothesis" and its variations; see arXiv.org. Whilst it is sensible to approach such questions with caution and humility it is also reasonable for scientists to try to make progress in this area. In this respect I have writer articles proposing that relationships (isolated under expected float entropy minimisation) may play an important role in consciousness.

The theory stems from information theory and introduces a type of conditional entropy that includes relationships as parameters. It turns out that, for non-random systems, certain choices for the relationship parameters are isolated from the rest in the sense that they give much lower expected float entropy values and, hence, the system defines relationships. Since consciousness is awash with relationships and associations, the theory may be relevant to consciousness. The 2021 article Model Unity and the Unity of Consciousness: Developments in Expected Float Entropy Minimisation also introduced the following postulate, where in the mathematical domain the word interpretation means relational model.

(The fundamental postulate of EFE minimisation).If we suppose that consciousness is given by an interpretation or representation of system states then, notwithstanding the possibility that a system may need to satisfy a number of requirements to be conscious, among the infinitely many possible interpretations, consciousness is given by some form of minimum expected entropy model interpretation of system states that yields an experience free of unnecessary discontinuities whilst exhibiting the intrinsic structural regularities of probable system states.

For those interested, below are various resources including a release of the GUI software (runnable .jar files) developed for researching the theory.

• Model Unity and the Unity of Consciousness: Developments in Expected Float Entropy Minimisation. Entropy article 2021. pdf
• From Learning to Consciousness: An Example Using Expected Float Entropy Minimisation. Entropy article 2019. pdf
• Quasi-Conscious Multivariate Systems. Complexity article 2015. pdf
• Quasi-Conscious Multivariate Systems. Preprint 2015. pdf
• Quasi-Conscious Multivariate Systems. ASSC conference poster 2015. pdf
• (Download URFinder3.73 - June 2020: >10x faster than version 3.72)
• (Download URFinder3.73 development source files)
• (Download RasterSampler1.0)
• (Download RasterSampler1.0 development source files)
• Mathematical theories of consciousness. News item and case study on www.maths.ox.ac.uk 2016. Web Page

URFinder is for finding relationship parameters that minimise expected float entropy in the case of small systems. It can also be used to implement Monte Carlo methods and produce expected float entropy histograms. RasterSampler is used for sampling pixels, in a chosen configuration, from digital photographs. When reading the published Complexity article “Quasi-Conscious Multivariate Systems” please note that the copy editing introduced an error on the 19th page (page number 143). The equation in question should say .

Publications and Beamer presentations Also see arXiv.org

• Uniform Algebras Over Complete Valued Fields. PhD. Thesis, University of Nottingham 2012. pdf
• Model Unity and the Unity of Consciousness: Developments in Expected Float Entropy Minimisation. Entropy article 2021. pdf
• From Learning to Consciousness: An Example Using Expected Float Entropy Minimisation. Entropy article 2019. pdf
• Quasi-Conscious Multivariate Systems. Preprint 2015. pdf
• Consciousness and the structuring property of typical data. Complexity article 2012. pdf
• A survey of non-complex analogs of uniform algebras. In Function spaces, Contemp. Math. AMS 2011. pdf
• An inductive proof of the Feinstein-Heath Swiss cheese classicalisation Theorem. Proc. AMS 2010. pdf
• Generalising uniform algebras over complete valued fields. Function Spaces 2010 at SIUE, USA. Beamer
• An intuitive proof of Heath's Theorem on Swiss cheese sets. BMC 2008 at the University of York, UK. Beamer

Useful links

Conference participation

• Models of Consciousness Conference, Oxford UK 2023. Web site.
• Models of Consciousness Conference, Stanford USA 2022. Web site.
• Inaugural Models of Consciousness Conference, Oxford UK 2019. Web site.
• The Nineteenth Annual Conference of the ASSC, Paris France 2015. Web site.
• The Sixth Conference on Function Spaces at SIUE USA 2010. Web site.
• Banach Algebras 2009, Stefan Banach International Mathematical Center, Bedlewo, Poland. Web site.
• The Sixtieth British Mathematical Colloquium 2008, University of York, UK. Web site.

Group conference photo from MoC4 at Oxford 2023.

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