Dan McAran

As Penrose notes “the Greek insight that it is mathematics that underscores the workings of physical reality has served us extraordinary well”. The conjecture presented here is a reflection on what mathematics is telling us about the nature of reality and concludes that there are aspects of mathematics that provide insight into consciousness: namely Cantor’s slash and Gödel's theorem. Cantor’s slash is a key component of the proof of Gödel's theorem. Cantor’s slash shows that there are more real numbers than natural numbers and is an extension of Euclid’s proof that there are infinitely many prime numbers. The extension in Gödel's theorem is that a set of mathematical axions cannot contain all the procedures used to create mathematical truths. Consciousness is in essence recursive, a looking back upon itself, commonly referred to as subjective experience. Gödel's theorem and other phenomena are evidence of a circular recursive aspect to reality which can be viewed as a prototypical recursive experience and consequently prototypical consciousness. No limit to infinity means no limit to discreteness. Consequently, no limit to complexity as discreteness recombines. This recursive quality can be applied to the hard problem of consciousness. The hard problem of consciousness is the same category of problem as the limits of mathematical axioms as shown by Gödel's theorem. This can also be interpreted as a fundamental aspect of reality: it is unbounded. If reality is unbounded, consciousness must be also be unbounded. Anirban Bandyopadhyay conjectures the prime numbers as information operators that generate a nested structure of reality: a fractal universe. He has connected this framework to the role of microtubules in consciousness as developed by Roger Penrose and Stuart Hameroff. Bandyopadhyay proposes a theory of consciousness, namely the concept of a self-operating mathematical universe (SOMU) “in which In this manner, the integer space is a superposition of many many fractals, each corresponding to a prime number….then the entire universe is a superposition of myriad fractals. So, the resultant universe…will exhibit different patterns — fractal patterns”. This is also consistent with David Bohn’s conjecture of the implicate order and the explicate order when can be seen as unbounded. Prime numbers can be equated to the implicate order and the explicate order be equated to the resulting fractals. Fractals and complex numbers can be seen as reflections of the recursive quality found in Cantor’s slash, Gödel's theorem, and the work of Bandyopadhyay work on primes. Penrose notes the elaborate patterns of the Mandelbrot set are derived from the complex plane. This recursiveness is as fundamental to reality as discreteness. In summary the conjecture is that Cantor’s slash and Gödel's theorem, in that they are recursive, are a prototypical mathematical self-reflection or subjective experience which supports Bandyopadhyay’s work on a self-operating mathematical universe built on fractals originating from prime numbers. This is a step forward in solving the connections between the three worlds Penrose puts forth: The Platonic Mathematical World, the Mental Word, and the Physical World and addressing the hard problem of consciousness.

Dan McAran has been interested in consciousness and its relation to physics for many years.  He as previously presented at one concurrent sessions and two poster sessions of the Science of Consciousness Conferences.  He has a Doctorate in Business Administration degree (University of Reading, UK).