The twentieth century science paradoxes

The isolation of hypothetical theories from the realities of living matter has caused mysticism to penetrate scientific theories. With mystical thinking, the idea of using an analytical method to solve cognitive problems does not occur. Dialectical logic, in contrast to mysticism, states the opposite: any problematic tasks of cognizing the vital processes and phenomena of the universe are solvable exclusively in an analytic way, with the only method. The author created a universal and formal theory of solving intellectual (i.e., having no previously known algorithms for solving) problems associated with the knowledge of the vital functions of natural and man-made processes in any phenomena of the universe - the Kondratenko method of axiomatic modeling, the effectiveness of which is achieved by correctly setting the problem and solving it purely formal method. The correctness of the statement of the problem means, first of all, the recognition of the failure of all hypothetical (not confirmed by the results of full-scale experimentation with the subject of knowledge) theories. This requirement, in particular, to the mathematical tools used to solve problems of cognition, it revealed paradoxes in the foundations of mathematics, which are discussed in the article. At present, in the natural and applied sciences in most publications, i.e. more than 90% associated with the construction of formal theories in these sciences, the proof of theorems is carried out: firstly, in a meaningful way, which contradicts the urgent requirement of philosophers of science to use exclusively formal evidence, which is a criterion for assessing the correctness and reliability of evidence; secondly, in substantive evidence in 95% of cases, an exclusively standard list of tautologies is used, which by definition is incorrect for the purpose of proving theorems on phenomena and processes of the universe based on exclusively true axioms obtained as a result of full-scale experimentation with these phenomena and processes. The article deals with the paradox in the classical approach to proving theorems, which consists in the inappropriateness of generally accepted stereotypical tautologies of classical mathematics for proving theorems.

Operationalism: An Interpretation of the Philosophy of Ancient Greek Geometry

I present a systematic interpretation of the foundational purpose of constructions in ancient Greek geometry. I argue that Greek geometers were committed to an operationalist foundational program, according to which all of mathematics—including its entire ontology and epistemology—is based entirely on concrete physical constructions. On this reading, key foundational aspects of Greek geometry are analogous to core tenets of 20th-century operationalist/positivist/constructivist/intuitionist philosophy of science and mathematics. Operationalism provides coherent answers to a range of traditional philosophical problems regarding classical mathematics, such as the epistemic warrant and generality of diagrammatic reasoning, superposition, and the relation between constructivism and proof by contradiction. Alleged logical flaws in Euclid (implicit diagrammatic reasoning, superposition) can be interpreted as sound operationalist reasoning. Operationalism also provides a compelling philosophical motivation for the otherwise inexplicable Greek obsession with cube duplication, angle trisection, and circle quadrature. Operationalism makes coherent sense of numerous specific choices made in this tradition, and suggests new interpretations of several solutions to these problems. In particular, I argue that: Archytas’s cube duplication was originally a single-motion machine; Diocles’s cissoid was originally traced by a linkage device; Greek conic section theory was thoroughly constructive, based on the conic compass; in a few cases, string-based constructions of conic sections were used instead; pointwise constructions of curves were rejected in foundational contexts by Greek mathematicians, with good reason. Operationalism enables us to view the classical geometrical tradition as a more unified and philosophically aware enterprise than has hitherto been recognised.

Hybrid structures applied to ideals in near-rings

Human endeavours span a wide spectrum of activities which includes solving fascinating problems in the realms of engineering, arts, sciences, medical sciences, social sciences, economics and environment. To solve these problems, classical mathematics methods are insufficient. The real-world problems involve many uncertainties making them difficult to solve by classical means. The researchers world over have established new mathematical theories such as fuzzy set theory and rough set theory in order to model the uncertainties that appear in various fields mentioned above. In the recent days, soft set theory has been developed which offers a novel way of solving real world issues as the issue of setting the membership function does not arise. This comes handy in solving numerous problems and many advancements are being made now-a-days. Jun introduced hybrid structure utilizing the ideas of a fuzzy set and a soft set. It is to be noted that hybrid structures are a speculation of soft set and fuzzy set. In the present work, the notion of hybrid ideals of a near-ring is introduced. Significant work has been carried out to investigate a portion of their significant properties. These notions are characterized and their relations are established furthermore. For a hybrid left (resp., right) ideal, different left (resp., right) ideal structures of near-rings are constructed. Efforts have been undertaken to display the relations between the hybrid product and hybrid intersection. Finally, results based on homomorphic hybrid preimage of a hybrid left (resp., right) ideals are proved.

A method to solve two-player zero-sum matrix games in chaotic environment

This research article proposes a method for solving the two-player zero-sum matrix games in chaotic environment. In a fast growing world, the real life situations are characterized by their chaotic behaviors and chaotic processes. The chaos variables are defined to study such type of problems. Classical mathematics deals with the numbers as static and less value-added, while the chaos mathematics deals with it as dynamic evolutionary, and comparatively more value-added. In this research article, the payoff is characterized by chaos numbers. While the chaos payoff matrix converted into the corresponding static payoff matrix. An approach for determining the chaotic optimal strategy is developed. In the last, one solved example is provided to explain the utility, effectiveness and applicability of the approach for the problem.Abbreviations: DM= Decision Maker; MCDM = Multiple Criteria Decision Making; LPP = Linear Programming Problem; GAMS= General Algebraic Modeling System.

Optimization of the composition of the harvesting and transport complex

The purpose of the article is to obtain additional knowledge, identify patterns of optimization of the harvesting and transport complex and use this knowledge and patterns in further practical activities. The study used methods of classical mathematics, observation, comparison, measurement, and experiment. As a result, the mathematical expectation of “time to fill the hopper” was calculated and justified, the grain yield was calculated and determined by the time of filling the hopper, the number of vehicles was justified by yield and distance to the warehouse from the field. According to the time of loading of the vehicle, coefficients were developed and determined that determine the efficiency of transport work during the transportation of harvest. The obtained research results have theoretical and practical significance for application in agricultural organizations of the Russian Federation, which is implemented in the methodological recommendations for optimization of the composition of harvesting and transport complexes.

Digital Didactics under Distance Learning Constraints

The article considers methodological approaches to the process of eliminating the problems of digitalization of education using the example of the academic disciplines of mathematics and computer science. It is shown that the use of specially designed complex mathematical tasks provides a harmonious combination of analytical research inherent in classical mathematics and constantly progressing methods of numerical analysis and computer modeling. The substantive filling of educational tasks with elements of production situations from the future profession of students or from the process of their training on the principles of a contextual approach has been substantiated. The essence of the ongoing process of digitalization of education and its effective use in the context of the limitations of contact work with students is considered from a constructive point of view.

Math Education in Intercultural Contexts: a logical-mathematical interpretation for duality Kaingang Kamé-Kairu

The aim of this paper is to present the results of research carried out in a Mathematics and Science training program for indigenous teachers of the Kaingang ethnic group by the Unochapecó in the municipality of Chapecó, Santa Catarina, Brazil. Based on the paradigm of ethno mathematics, one of the objectives of the program is to look for relations between traditional indigenous thinking and classical mathematics. We propose an analogy between logical-mathematical binary calculation systems and the Kaingang thought system expressed in its social organization and supported by two opposite and complementary dual categories: Kamé-Kairu. Our hypothesis postulates a certain similarity between the binary logic used for example, in logical circuits, and the binary system with which the traditional Kaingang culture encodes its social, natural, and even supernatural world. The binary pair Kaingang, Kamé-Kairu serves to codify relations of kinship, exchange, and alliances of the tribe, as well as the natural world that surrounds them. The analogy formulated provides interesting information for mathematical and scientific education in intercultural contexts.

On the approximate inverse Laplace transform of the transfer function with a single fractional order

The history of fractional calculus dates back to 1600s and it is almost as old as classical mathematics. Although many studies have been published on fractional-order control systems in recent years, there is still a lack of analytical solutions. The focus of this study is to obtain analytical solutions for fractional order transfer functions with a single fractional element and unity coefficient. Approximate inverse Laplace transformation, that is, time response of the basic transfer function, is obtained analytically for the fractional order transfer functions with single-fractional-element by curve fitting method. Obtained analytical equations are tabulated for some fractional orders of [Formula: see text]. Moreover, a single function depending on fractional order alpha has been introduced for the first time using a table for [Formula: see text]. By using this table, approximate inverse Laplace transform function is obtained in terms of any fractional order of [Formula: see text] for [Formula: see text]. Obtained analytic equations offer accurate results in computing inverse Laplace transforms. The accuracy of the method is supported by numerical examples in this study. Also, the study sets the basis for the higher fractional-order systems that can be decomposed into a single (simpler) fractional order systems.