Tuesday, December 05, 2023

The Notion of Generalized Integer

The inspiration for this contribution came from the article "Space Element Reduction Duplication (SERD) model produces photon-like information packets and light-like cosmological horizons" by Thomas L. Wood. published in Metodologia IV B:  Journal of International and Finnish Methodology, expressess the basic assumptions of the SERD approach very coherently and in a systematic way so that it easy easy to criticize them and compare with other views, in my case the TGD view.

My criticism is based on a different interpretation of the discreteness. It would be assignable to cognitive representations based on p-adic numbers fields involving extensions of rationals. Bringing in also the continuous number fields (reals, complex numbers, quaternions, octonions) brings in real space-time as sensory representation and one ends up to a generalization of the standard model proving a number theoretic interpretation for its symmetries.

The approach of Thomas looks to me essentially topological: for instance, the information propagating in the hypergraph is assumed to be topological. In TGD, discrete structures analogs define cognitive representations of the continuous sensory world and are basically number theoretic. The description of the sensory world involves both topology and geometry.

The articulation of this view led to the main result of this article, which is a generalization of the number concept as a fusion of all p-adic number fields and rationals to a single structure that I call generalized integers. Besides being useful in TGD, this framework could be very useful in the modelling of spin glass-like systems.

See the article The Notion of Generalized Integer or the chapter Trying to fuse the basic mathematical ideas of quantum TGD to a single coherent whole

For a summary of earlier postings see Latest progress in TGD. For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

No comments: