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

Author(s):  
Ileana Ruxandra Badea ◽  
Carmen-Elena Mocanu ◽  
Ovidiu Pasarescu

This work promotes new methods in Mathematical Modeling, consisting in the use of the methods of Non-standard Analysis in Topoi, having as its main purpose, the mathematical definitions of the pseudoparticles from the title, with arguments from Biology/Physiology (Mathematical Neuroscience), Physics (String Theory and Emergent Quantum Mechanics), and Cybernetics (Global Brain, including Natural and Artificial Intelligence). The connections between brain and mind will be scketched via the genetic/epigenetic interplay. The topoi model the intuitionistic logic (multi-valued) and have been used in Quantum Physics while Non-Standard Analysis in SET (= the Category of sets) has been applied in Mathematical Economics; topics from Theory of Categories were also used in the study of Consciousness; however, the combination topoi - non-standard analysis was never used until now in Applied Mathematics. Another important objective is to produce progress in this aria of Pure Mathematics also - the build of non-standard analysis in classes of topoi, already used in Physics. We propose the logic of non-standard extensions in topoi as a model of the human thinking (based on infons/receptons), these theories representing new top and very difficult results in Abstract Mathematics either.

Author(s):  
Carmen-Elena Mocanu ◽  
Florin F. Nichita ◽  
Ovidiu Pasarescu

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 ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2048
Author(s):  
Ileana Ruxandra Badea ◽  
Carmen Elena Mocanu ◽  
Florin F. Nichita ◽  
Ovidiu Păsărescu

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.


2012 ◽  
Vol 08 (01) ◽  
pp. 139-152
Author(s):  
K. VELA VELUPILLAI

Non-standard analysis can be harnessed by the recursion theorist. But as a computable economist, the conundrums of the Löwenheim-Skolem theorem and the associated Skolem paradox, seem to pose insurmountable epistemological difficulties against the use of algorithmic non-standard analysis. Discontinuities can be tamed by recursive analysis. This particular kind of taming may be a way out of the formidable obstacles created by the difficulties of Diophantine Decision Problems. Methods of existence proofs, used by the "classical" mathematician — even if not invoking the axiom of choice — cannot be shown to be equivalent to the exhibition of an instance in the sense of a constructive proof. These issues were prompted by the fertile and critical contributions to this special issue.


2019 ◽  
Vol 10 (4) ◽  
Author(s):  
Aleksey Stepanenko ◽  
Diana Stepanenko

In the process of comprehending the prospects for the artificial intelligence development, the authors come to a conclusion that in the scientific learning of the world the problematic issues of the artificial intelligence are connected with problematic issues of recognizing the artificial intelligence systems and ordinary human thinking The article performs an analysis of the concepts of «intelligence» and «artificial intelligence», in the process of which the intelligence is viewed through a systematic approach in its broad sense. The purpose of the article is to present a number of conclusions about the levels of development of scientific studies of the problems under investigation, is there any reason to argue that attempts to implement the epistemological characteristics of thinking in modern artificial intelligence systems have not only been undertaken but also successful, and whether is it possible to talk about full transfer of the intellectual functions to the technical systems, endowing them with epistemological tools (in the context of the discussion about strong and weak versions of the artificial intelligence). The authors study the concept of «phenomenology of intelligence», the perception of intelligence in various historical eras by famous philosophers and scientists of other branches of knowledge; they identify the artificial intelligence as a special branch of science, analyze the existing problems in this field. In writing the article, they use the system approach, the theoretical analysis of and generalization of the scientific information, the historical, predicted, critical and dialectical methods of investigation.


Author(s):  
Volodymyr Vladymyrov

The article discusses the prospects for the development of the mass communication science in the direction of including in its discourse the modern achievements in the field of the relation between quantum physics and consciousness as well as the artificial intelligence. It will give the media new opportunities to influence mass consciousness, and to impact social processes.It allows to forecast the rise of the next, the fourth stage of the information revolution and to recognize the ways for preparation to it. Quantum physics appears as classical physics develops in the study of the world. Scientists explain quantumness as the relations of everything with everything. Quantumism and consciousness are an already mastered topics of modern science. If we talk about mass consciousness, it should be recognized as a scholar problem and the existence of quantumness as an underlying phenomenon in mass communication. Perhaps this will be the content of the new, fourth stage of the information revolution – after its computer, Internet and mobile stages. Therefore, the urgency of the problem we are addressing is the need to “let in” quantum methods into the thinking, creation and dissemination of the mass-information product in the mass audience. Theorists of our scientific discipline are interested in precisely the new knowledge of quantum “confusion” in the depths of mass consciousness, which exists and manifests itself in the processes of mass communication. Then the quantum approach can be extremely fruitful here. Quantumism here appears both as a universal connection of everything with everything, and as the unpredictability of the nature of these connections from the point of view of ordinary and even dialectical logic. Here, the quantum logic of mass communication should be discovered. It should be noted that so far almost no one has noticed that media interference in the natural mental life of large human communities gives rise to a phenomenon similar to quantum “confusion”, when a change in the state of one “mass” thoughts occurs along with a change in the state of another. Quantum “confusion” is what permeates all the “matter” of human communication / existence. All participants in mass communication are “confused” with themselves, like photons in physics. And they’re not just confused, everybody is one and the part of whole in the external, “physical”, existing, and is the part of the ideal world, with its thoughts, hopes, emotions, “confused” in the Internet The introduction of quantumness as the leading principle of influence on mass consciousness is one of the three steps to a new stage in the information revolution. The second is the growing potential of artificial intelligence. The third is the gradual creation of quantum computers. All these together will make it possible to reach each recipient of information, to enter the inner, ideal world of an individual person, and even to see in this communication an intrapersonal discussion of a person with himself – and to control its course. Probably, the result will be the opportunity to learn to predict the unpredictable in behavior of “this” person in his behavior – in personal, group, intergroup, mass ones.


2022 ◽  
Vol 1 ◽  
pp. 01006
Author(s):  
Iurii V. Filatov

Some algorithms, which are often based on the use of elements of higher mathematics, possessing high speed and compact coding in algorithmic languages, are poorly mastered by most students. It can be assumed that this is due to the difficulty of presenting the principles of their work in the form of human actions in ordinary situations. Thus, a certain contradiction arises between the way of solving the problem that a person resorts to without using a computer and the way we force our computer to solve this problem. Comparison of the process of explaining algorithms speaks in favor of algorithms imitating human thinking. The discussion of the advantages of the algorithms themselves is beyond the scope of this article and undoubtedly deserves a separate study. If artificial intelligence is created, then its creator or creators will certainly be ranked among the outstanding geniuses in the history of civilization, no matter what algorithms it uses. However, so far there is no one to solve problems for us and create algorithms, so we will use all available means and try to teach this to children.


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