History Of Mathematics Research Articles

Why Austrian Economists Can’t Get No Respect: It Is Political

Austrians are the Rodney Dangerfields of economics. We can’t get no respect. Why not? On an intellectual level, our adherence for the necessary a posteriori (often termed synthetic a priori) is strongly off putting to the majority of this profession, who are in thrall to logical positivism, which adamantly rejects this type of claim. There are also divergences as regards the causes and cures for the business cycle, anti-trust, market failures such as public goods, externalities and utility. From an economic self-interest level, were this praxeological school ever to take root within the dismal science, thousands of jobs would be lost and salaries would be greatly lowered. It may be that for both of these reasons, Austrianism is deemed a “cult”. We argue that Austrian economics is far more based on modern mathematics and physics than is mainstream economics.

Из русского «Марбурга» в США: старое вино в новые меха?

The conceptualization of ideological influences in philosophy is usually unidi­rectional: from tradition-source to tradition-recipient. The history of Russian neo-Kantianism demonstrates other possibilities for conceptualizing philosophi­cal reception: documenting the facts of reverse influence forces us to look for new ways of interpreting this philosophical movement. The uniqueness of the his­torical situation, which cannot be ignored when analyzing the fate of Russian Neo-Kantianism, makes it necessary for the historian of ideas to turn to the study of intellectual biographies, professional connections, and in general the “sphere of conversation” in which the transformation of the ideological horizon took place. The emigration of intellectuals from Russia in the late 1910s and early 1920s, termed the “philosophers’ ships”, not only provoked unprecedented exis­tential shifts in individual personal trajectories, but also gave rise to new intel­lectual phenomena, conceptual networks and cultural-institutional clusters. Re­constructing the intellectual biographies of two Russian thinkers, Simon Kaplan and Jacob Klein, whose philosophical development took place in the symbolic Marburg, we uncover difficult problems of self-identity, intellectual relevance, and creative fulfillment in emigration. Kaplan, fascinated by the philosophy of religion of Hermann Cohen, head of the Marburg School of Neo-Kantianism, found himself in demand in the United States in the innovative “Great Books” educational program as an expert on Kant and the Bible and as the translator of Cohen’s major work on the philosophy of religion into English. Klein brought to the New World a reputation as an expert in the history of ancient mathematics and Renaissance science, revealed himself as an expert in Plato’s philosophy and an excellent organizer of the educational process, and contemporary research in­terest turns to him as a phenomenologist.

A Case Study of the Diversity and Optimization of Mathematical Definitions and Symbolization

Taking the idea of “reflective and critical optimization thinking”, this article first examines “the outlook on development of big mathematics” in International Mathematics Day in 2003 from the perspective of the history of mathematics. Then according to the requirements of mathematical core literacy and mathematical beauty, a deep analysis is conducted on the limitations of the teaching methods in the Turing mathematics and statistics series “Terence Tao Analysis”. The case study shows that (1) the sets, relations, functions and operations present a concentric circle relationship of layer-by-layer proper subsets; (2) rejecting mathematical notation damages the beauty of simplicity and learning efficiency; and (3) forcing the first mapping function to be a surjection unnecessarily limits the wider range of applications that composite functions can have, and it lacks the unified beauty of mathematical definitions. Finally, itemized strategies and suggestions for improvement and optimization are proposed.

Value Research of Integrating History of Mathematics into Primary School Mathematics Teaching

Value Research of Integrating History of Mathematics into Primary School Mathematics Teaching

ПЕРИОДИЗАЦИЯ А.Н. КОЛМОГОРОВА КАК ОСНОВА ПРЕДСТАВЛЕНИЙ ОБ ИСТОРИИ РАЗВИТИЯ МАТЕМАТИЧЕСКОЙ НАУКИ

The periodization of the development of mathematical science was proposed by A.N. Kolmogorov. He identifies four periods in the development of mathematics. The basis of periodization includes assessment of the content, level of achievements and features of mathematical research: its most important methods, results, ideas. Particular attention is paid to the development of probability theory structured according to Kolmogorov periods. The prerequisites for the formation of probability theory began to appear during the second millennium BC — the first period. During the period of accumulation of knowledge (before 600 BC), the concept of “chance” — the main concept in the science in question – began to develop. Until the third period, the concept was invariably associated with the philosophical category “fate,” that is, a certain predetermined process, which contradicts modern ideas about chance. Probability Theory received its unofficial name precisely from the period of elementary mathematics (before the 15–16th centuries) — the second period. The next period was called by Kolmogorov “Mathematics of Variables” (XVII–XVIII centuries). Since the beginning of the period of modern mathematics (since the 19th century), random variables and the patterns associated with them have come to the fore in probability theory. This is due to the general scientific growth in the 19th century. The main problems that arose by the end of the third period are being solved: all the basic concepts, rules for their application, and theorems are clearly defined. By the beginning of the 20th century — the fourth period–there was a need to formalize the acquired knowledge. Andrei Nikolaevich built a system based on modern and already developed by that time set theory and measure theory. The periodization of the development of mathematical science, proposed by A.N. Kolmogorov, with the rapid progress of mathematical knowledge and the emergence of information technologies in the future may require adjustments to the last stage of development of the history of mathematics or will lead to the emergence of a new stage of periodization.

数学史融入弧度制的教学设计研究

数学史融入弧度制的教学设计研究

Between Heaven and Earth! A Poem-Collage Pair About Hypatia of Alexandria

The poem-collage pair presented here is a work of collaboration between the mathematician and poet, Sarah Glaz, and the collage and ceramic artist, Mark Sanders. The piece is part of their larger joint poem-collage project involving the history of mathematics. Included as background is a brief discussion on the history and mathematics involved, and a reflection on several landmark locations and some of the relevant imagery appearing in the poem and the collage.

A Study on the Integration of the History of Mathematics into University Mathematics Courses

A Study on the Integration of the History of Mathematics into University Mathematics Courses

Investigation and Analysis of Pre-service Primary School Mathematics Teachers' Knowledge of History of Mathematics

Investigation and Analysis of Pre-service Primary School Mathematics Teachers' Knowledge of History of Mathematics

Legal regulation of digital technologies in the agricultural sector

The article is devoted to the legal regulation of digital technologies in the agricultural sector. In the modern period of development of civil relations, the digitalization of personal and social relations, including in agriculture, is widespread. The regulation of these relations is carried out with the help of civil law. The legislator considers only certain types of regulation. Covering the current problems of digital technology regulation, the author draws attention to the similarities and differences between digital currency and digital financial assets. Digital technology requires proper legislative support and the development of diverse jurisprudence. The introduction of mathematical methods in biology, including biometrics and statistics, as specific sections of modern mathematics reflects the need to intensify cognition in the era of scientific and technological progress. These methods will be used both within the framework of state management and for improving the efficiency of production and marketing processes of enterprises of agro-industrial and fishery complexes. The author unequivocally concludes that digital technology is a good that must be protected by law. Digital technologies are increasingly entering all spheres of human activity, and agriculture, as a key sector of our country's economy, is no exception.

Mutation of DNA and RNA sequences through the application of topological spaces

<abstract><p>Topology is branch of modern mathematics that plays an important role in applications of biology. The aim of this paper is to study DNA sequence mutations using multisets, relations, metric functions, topology and association indices. Moreover, we use association indices to study the similarity between DNA sequences. These different ways of identifying a mutation help biologists to make a decision. A decision of mutation that depends on metrics between two sequences of genes and the topological structure produced by their relationship is presented.</p></abstract>

МОДЕЛЬ СОВРЕМЕННОГО УЧИТЕЛЯ МАТЕМАТИКИ

Research and practice have shown that the teacher is an organizer, methodologist, and researcher, who directs the process of acquiring knowledge by students in a studying subject. The role of a teacher today is the role of a facilitator, trainer, and mentor, who assists in organizing the process of self-education using new technologies, including distance technologies. The article deals with issues related to the main qualitative indicators of a mathematics teacher. Modern mathematics teachers, in addition to the requirements of state educational standards, should be proficient in modern information technologies, be creative organizers of the learning process and develop their methodology and qualifications.

PSYCHODIDACTIC APPROACH TO THE CREATION OF TEXTBOOK TEXTS FOR THE PROMOTION OF INTELLECTUAL DEVELOPMENT OF STUDENTS (ON THE EXAMPLE OF SCHOOL MATHEMATICS)

Introduction. The most important factor for students’ intellectual development is the teaching content. A textbook text is a didactic unit of teaching content. Therefore, the question arises about the requirements for modern textbook texts in the framework of the psycho-didactic approach. Materials and Methods. The study was conducted based on an analysis of the psychological features of the intellectual development of students within the psycho-didactic approach, which is based on the psychological and pedagogical rationale of the multifunctionality of modern mathematics teaching materials. Results and Discussion. The concept of development-focused educational text, which should be constructed as a multidimensional semantic space (hypertext), is introduced. Classification of development-focused educational texts is presented (using school mathematics as an example) based on the ontological theory of intelligence, targeting the enrichment of the main components of students’ mental experience – cognitive, conceptual, metacognitive, and intensional experience. Typology of pedagogical developmental texts of different types in teaching mathematics in middle school is given. Conclusion. The use of developmental textbook texts contributes to understanding mathematical material and developing students’ intellectual resources.

Laudatio Martina Hofmanová and her random walk in the world of modern mathematics

Article Laudatio Martina Hofmanová and her random walk in the world of modern mathematics was published on December 30, 2022 in the journal Mitteilungen der Deutschen Mathematiker-Vereinigung (volume 30, issue 4).

Research on the Development of Knowledge in the Core Literacy of High School Mathematics Teachers Based on History of Mathematics

Research on the Development of Knowledge in the Core Literacy of High School Mathematics Teachers Based on History of Mathematics

On the boundaries of the $m=2$ amplituhedron

Amplituhedra $\mathcal{A}{n,k}^{(m)}$ are geometric objects of great interest in modern mathematics and physics: for mathematicians, they are combinatorially rich generalizations of polygons and polytopes, based on the notion of positivity; for physicists, the amplituhedron $\mathcal{A}^{(4)}{n,k}$ encodes the scattering amplitudes of the planar $\mathcal{N}=4$ super Yang–Mills theory. In this paper, we study the structure of boundaries for the amplituhedron $\mathcal{A}\_{n,k}^{(2)}$. We classify all boundaries of all dimensions and provide their graphical enumeration. We find that the boundary poset for the amplituhedron is Eulerian and show that the Euler characteristic of the amplituhedron equals one. This provides an initial step towards proving that the amplituhedron for $m=2$ is homeomorphic to a closed ball.

Math Teacher and e-Learning Courses. Report on the Project “Modern Mathematics Teacher”

The article is a report on the project conducted by the author, co-financed by the National Center for Research and Development from European funds POWR.03.01.00-IP.08-00-MOC/18 entitled: A modern mathematics teacher. It was dedicated to teachers of mathematics from all over the country, teaching in primary and secondary schools, and future mathematics teachers – gaining qualifications to teach mathematics. The project aimed to improve the quality and efficiency of mathematics teachers’ work through active participation in innovative e-learning courses – offered on the navoica.pl platform, concerning new teaching methods based on information technologies. A short description of the four proposed e-learning courses is presented, as well as an analysis of the choices made by the participants depending on the subject.

Thinking and Exploration of Mathematics Teaching Reform Under Ideological and Political

Ideological and political course is an important link of ideological and political work in universities in the new era, a necessary way to cultivate people by virtue, and also the historical mission of the construction of talent training system in Colleges and universities in the new era. Therefore, this paper discusses how to integrate ideological and political elements into mathematics teaching. Based on the teaching practice of "advanced mathematics", "linear algebra" and "matrix theory", this paper first analyzes the advantages and disadvantages of carrying out ideological and political education in the mathematics course, then excavates the ideological and political elements of the course from the perspective of philosophy, the history of mathematics, the culture of mathematics and the application of mathematics, and puts forward some specific measures in combination with some teaching cases. Finally, it is point out that, in order to implement the ideological and political education of the curriculum, one should improve teachers' cognition, excavate the combination of mathematics and politics, and innovate teaching methods with the help of new media. These can provide reference for ideological and political teaching.

The Teaching of Highlighting Mathematical Culture from the Perspective of HPM ——Take the Teaching for "Letters Represent Numbers" as an Example

"Compulsory Education Mathematics Curriculum Standards (2022 Edition)" clearly states that attention should be paid to mathematics culture, but at present, mathematics textbooks and classrooms still do not attach importance to mathematics culture, and mathematics culture is not really integrated into mathematics classroom teaching. For further exploring how to integrate mathematical culture into mathematical classroom teaching, the traditional teaching design of Letters Represent Numbers is transformed from the perspective of HPM by taking Letters Represent Numbers as a case, and the history of mathematics is used to highlight the mathematical culture embedded in mathematical knowledge, and based on this, ideas and suggestions are put forward for teaching to highlight mathematical culture from the perspective of HPM. Keywords: HPM; Mathematical Culture; Letters Represent Numbers; Teaching Design DOI: 10.7176/JEP/13-35-07 Publication date: December 31 st 2022

On removal of one undetected paradox in foundations of probability theory

The paper points out a logical contradiction that arises in the modern approach to teaching the foundations of the probability theory. This contradiction is of the following form: \U0001d74e = {\U0001d74e}, where \U0001d74e is an arbitrary elementary outcome. In modern mathematics the formation of expressions of the form \U0001d499 = {\U0001d499} is unacceptable. Here {\U0001d499} is a unit set, i.e., a set whose sole element is some element \U0001d499 of an arbitrary nature. This paradox \U0001d74e = {\U0001d74e} is a consequence of the fact that random events and between them also elementary outcomes \U0001d74e are considered as some subsets in the set of all elementary outcomes. It is shown how this contradiction can be easily eliminated by applying the axiomatic method. It is based on the introduction of two simple axioms that impose restrictions on elementary outcomes and on the class of random events under consideration. In addition from these two axioms on the basis of strictly logical reasoning the authors derived the representation of an arbitrary random event in the form of a sum of elementary outcomes favorable to it. Moreover, another non-axiomatic approach to the elimination of this paradox is proposed. In the final part of the paper other variants of the second of the introduced axioms are considered as well.