Applications of Non-Standard Analysis in Topoi to Mathematical Neuroscience and Artificial Intelligence: I. Mathematical Neuroscience

Abstract

This work - in two parts - tries to promote a new method in Mathematical Modeling, not ever used before, namely the use of the methods of Non-standard Analysis in Topoi. For the general mathematical theory of topoi (Topoi Theory) we refer to the book [10] . For non-standard analysis in the particular Boolean topos SET (the category of sets) we refer to the book [11]. This first part of the work is related, from an applicative point of view, mainly with Mathematical Neuroscience, more precisely a model of the human thinking, consciousness and subconsciousness. The connections between brain and mind will be also scketched, but a closer study also need some further topics to be studied in Part II: Artificial Intelligence, namely (Quantum) neural networks and (Quantum) Turing machine. However both parts (I. and II.) should be considered together, because they complement each other. The topoi model the intuitionistic logic (multi-valued) and have been used in Quantum Physics (see [9] and its references) while Non-Standard Analysis in SET (introduced by Abraham Robinson[31]) has been applied in Mathematical Economics (see [2] and its references); however, the combination was never used until now in Applied Mathematics. Even more, the Non-standard Analysis in Topoi is not too much studied from the point of view of Pure Mathematics either. One of the main objective of this work is to produce progress in this aria of Mathematics also, mainly for the SET-type topoi and exponential topoi, those topoi used in Quantum Physics and in which we need Non-stantard Analysis for our purposes. Based on the paper [13] we propose the logic of non-standard extensions in topoi as a model of the human thinking (based on infons), these theories representing top and very difficult results in Abstract Mathematics. We will analise the relation between brain and mind via the genetic-epigenetic interplay. More details will be provided in the Introduction, followed by a short presentation of the general theory, later connected with the (Quantum) Yang-Baxter equations in topoi. The connections between these equations and braid groups and knots, with intended applications in Neuroscience and Artificial Intelligence will be further analysed in Part II. Besides promoting a new idea of modeling, the present paper aims to promote the building of a Research Proposal for the European $''$Human Brain Project$''$ (https://www.humanbrainproject.eu/en/), and the NSF-Europe Program $''$Collaborative Research in Computational Neuroscience$''$ (https://www.nsf.gov/funding/pgm\_summ.jsp?pims\_id=5147), jointly with the new founded $''$National Centre for the Research of the Brain$''$ of the Romanian Academy of Science (https://acad.ro/centreAR/CNCC/CNCC.pdf, in Romanian). $''$Blue Brain Project$''$ (https://www.epfl.ch/research/domains/bluebrain/) could also be of big interest. We hope that the journals edited by MDPI (Axioms, in particular) and by the Simion Stoilow Institute of the Romanian Academy - Revue Roumaine de Mathematique Pures et Appliqu (http://imar.ro/journals/Revue\_Mathematique/home\_page.html) - among others - will host special issues related to the research subjects described in both parts of this work, starting with this one. For further details and some of the progress on this research subject, see www.ovidiufpasarescu.com.}

Mathematics and Poetry • Unification, Unity, Union

“…The sections of the current paper contain not only examples of unification structures in mathematics, but also various versions of the Yang-Baxter equation. This paper could be extended to a discussion about Logic [32][33][34], Machine Learning [35,36], transcendence and transcendental numbers [23,37], transdisciplinarity [38][39][40], etc.…”

On Jordan Algebras and Some Unification Results

“…This section was inspired by two papers, [25,42], and it begins with an introduction to the Yang-Baxter equation.…”

Applications of Non-Standard Analysis in Topoi to Mathematical Neuroscience and Artificial Intelligence: Infons, Energons, Receptons (I)

“…For more details concerning the topics from this section, we refer to [7], [33], [36], [13], [34]. See also the extended version [25], which have the same structure as this paper has, but the notions are presented in more detail.…”

“…Preliminary note This is the version 2 of the paper [25] from the References, whose Section 9., entirey written by Dr.…”

“…We hope that we produced an enough self-contained and readable paper by many categories interrested in the topics. In the same time, the preprint [25] represents a much more expanded version of this article, from mathematical point of view, collecting most of what we considered as being the necessary mathematical notions. So, the paper shortly re-explain already known notions and results, and it also contains new and original contributions.…”