Abstract
The purpose of this paper is to promote new methods in mathematical modeling inspired by neuroscience—that is consciousness and subconsciousness—with an eye toward artificial intelligence as parts of the global brain. As a mathematical model, we propose topoi and their non-standard enlargements as models, due to the fact that their logic corresponds well to human thinking. For this reason, we built non-standard analysis in a special class of topoi; before now, this existed only in the topos of sets (A. Robinson). Then, we arrive at the pseudo-particles from the title and to a new axiomatics denoted by Intuitionistic Internal Set Theory (IIST); a class of models for it is provided, namely, non-standard enlargements of the previous topoi. We also consider the genetic–epigenetic interplay with a mathematical introduction consisting of a study of the Yang–Baxter equations with new mathematical results.
Highlights
This paper contains both known and new results, and it addresses an auditorium of great diversity: neuroscientists, computer scientists, physicians, physicists, biologists, mathematicians, etc
We arrive at the pseudo-particles from the title and to a new axiomatics denoted by Intuitionistic Internal Set Theory (IIST); a class of models for it is provided, namely, non-standard enlargements of the previous topoi
One is related to pure mathematics and consist in constructing the non-standard analysis in SET-type topoi (Theorem 1); another one is related to foundations of mathematics; arguments from physics, neuroscience, cybernetics are used in order to justify the importance of introduction of the IIST axiomatics and to provide models for it; and, last but not least, the introduction and justification of the infons, energons, receptons: how they should look like and which is their usefulness (Sections 2–4)
Summary
This paper contains both known and new results, and it addresses an auditorium of great diversity: neuroscientists, computer scientists, physicians, physicists, biologists, mathematicians, etc. (2) Presentation of the axiomatics IIST (Intuitionistic Internal Set Theory) (Section 7.1); (3) Theoretical introduction of non-standard analysis in topoi (until now this was known only for the topos SET), at the level of general definition (having higher order language)) (Section 6.2); (4) Topoi of SET-type and the construction of a model class for IIST on them, based on non-standard analysis in SET (IIST is as consistent as ZF(C) is) (Section 5.2 and Theorem 1); (5) Definitions of the notions of infon, energon, recepton, which would use for a simultaneous approach of the natural and artificial neural (quantum) networks, both part of the global brain (Section 7); this approach is based on a novel study of the global brain (Section 4); and this last-one is based on some novel points of view concerning the human brain, mind, genetics, and epigenetics (Section 2). The topics from (7) are review topics, the topics from (1) and (6) are partly original, partly review, while the topics from (2), (3), (4), (5), are original
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