History Of Mathematics Research Articles

Generation of Polynomial Automorphisms Appropriate for the Generalization of Fuzzy Connectives

Fuzzy logic is becoming one of the most-influential fields of modern mathematics with applications that impact not only other sciences, but society in general. This newly found interest in fuzzy logic is in part due to the crucial role it plays in the development of artificial intelligence. As a result, new tools and practices for the development of the above-mentioned field are in high demand. This is one of the issues this paper was composed to address. To be more specific, a sizable part of fuzzy logic is the study of fuzzy connectives. However, the current method used to generalize them is restricted to the use of basic automorphisms, which hinders the creation of new fuzzy connectives. For this reason, in this paper, a new method of generalization is conceived of that aims to generalize the fuzzy connectives using polynomial automorphism functions instead. The creation of these automorphisms is achieved through numerical analysis, an endeavor that is supported with programming applications that, using mathematical modeling, validate and visualize the research. Furthermore, the automorphisms satisfy all the necessary criteria that have been established for use in the generalization process and, consequently, are used to successfully generalize fuzzy connectives. The result of the new generalization method is the creation of new usable and flexible fuzzy connectives, which is very promising for the future development of the field.

John Stillwell: The Story of Proof: Logic and the History of Mathematics

John Stillwell: The Story of Proof: Logic and the History of Mathematics

A study of the strategies of the combinatorial game

The combinatorial game theory is not only a new branch of modern mathematics but also an important subject of operations research. One basic combinatorial game can develop many different rules and different strategies to solve the problems, and both the rules and strategies follow the fundamental game theory to give the answer for the new version of the combinatorial game, such as the Nim game, one of the most basic combinatorial games, which has been researched for many years by a large number of researchers. This paper shows and extends the strategies of two combinatorial games, Tic-Tac-Toe and Green Hackenbush, based on the regular version which is connected with the combinatorial game theory.

History of mathematics at the University of Freiburg

AbstractThis paper is about the history of the Mathematical Institute of the University of Freiburg with special reference to the applied mathematician Henry Görtler.

Ptolemy’s Science of the Stars in the Middle Ages edited by David Juste, Benno van Dalen, Dag Nikolaus Hasse, and Charles Burnett

This exciting collection of articles was a result of a 2015 conference by the same name, “Ptolemy’s Science of the Stars in the Middle Ages”, sponsored by the Ptolemaeus Arabus et Latinus (PAL). Like the entire PAL project, this work highlights Arabic and Latin scholars’ criticisms of, inquiries into, and refutations of, the Ptolemaic corpus and astrological conceptions of the universe. Consisting of 15 articles organized into three sections—“The Greek and Near Eastern Traditions”, “The Arabic Tradition”, and “The Latin Tradition”—this collection is comprehensive. It features novel, important research on Ptolemy’s works and the influence of Ptolemaic thought by top scholars in the field. The volume will be of particular value to specialists but is accessible to others interested in the history of astronomy and mathematics.

On the Fourier transform of functions from a Lorentz space L2,r with a mixed metric

The classical inequalities of Bochkarev play a very important role in harmonic analysis. The meaning of these inequalities lies in the connection between the metric characteristics of functions and the summability of their Fourier coefficients. One of the most important directions of harmonic analysis is the theory of Fourier series. His interest in this direction is explained by his applications in various departments of modern mathematics and applied sciences, as well as the availability of many unsolved problems. One of these problems is the study of the interrelationships of the integral properties of functions and the properties of the sum of its coefficients. The solution of these problems was dedicated to the efforts of many mathematicians. And further research in this area are important and interesting problems and can give new, unexpected effects. In the article we receive a two-dimensional analog of the Bochkarev type theorem for the Fourier transform.

Ancient Indian mathematics needs an honorific place in modern mathematics celebration

Ancient Indian mathematics needs an honorific place in modern mathematics celebration

Encouraging Mathematical Explorations Through Reasoning by Analogy in Abstract Algebra

Analogy has played an important role in developing modern mathematics. However, it is unclear to what extent students are granted opportunities to productively reason by analogy. This article proposes a set of lessons for introducing topics in ring theory that allow students to engage with the process of reasoning by analogy while exploring new (to the students) mathematics. In this way, students come to creatively establish new concepts that they may take ownership of. I provide insights from previous implementations and conclude by reflecting on what has (and has not) worked well in my experience with implementing the lesson.

Transdisciplinary role of physical and mathematical disciplines in modern natural science and engineering education

Introduction. In recent decades, a gradual degradation of mathematics and physics training in schools and universities has begun. Obviously, this has caused a significant decrease in the quality of professional training of students in the conditions of commercialisation and continuous reforms of education, and its chaotic digitalisation. At the same time, modern mathematics and physics have become the leader of the transdisciplinary trend in the natural, engineering, and other sciences of the digital era. The transdisciplinary trend generates a universal methodology capable of solving complex multifactorial interdisciplinary problems of nature and society. As a result of the transdisciplinary trend in science, such scientific fields as cybernetics, general systems theory, catastrophe theory, synergetics, artificial intelligence, big data, etc. have emerged. All these concepts were developed on the basis of the achievements of mathematics and physics over the past 70–80 years, which gave rise to the most unique practical achievements in modern, natural, and technical sciences.Aim. The present research aims to explore transdisciplinarity role of physical and mathematical disciplines in modern natural science and engineering education.Methodology and research methods. The system-based approach was used to analyse the role of mathematical and physical science systems in modern education. The synergetic approach became the basis for the study of the transdisciplinary trend of these systems in historical retrospect. In the methodology of these approaches, an important role was played by the methods of students’ holistic scientific worldviews formation, including ideas about the modern picture of the world of mathematics and physics. The authors applied the methods to develop student systems thinking (with its important quality of nonlinearity), which underlies the solution of multifactorial interdisciplinary tasks of their professional activity.Results. The results of the analysis of the transdisciplinary role of physical and mathematical disciplines indicate the fundamental importance of the unique potential of these disciplines in the natural science and engineering education in the era of the computer revolution.Scientific novelty. The authors identified, justified and characterised transdisciplinary ideas and methods (in mathematics and physics), which bring natural and engineering sciences to a higher level of cognition, thus contributing to improving the quality of natural science and engineering training of students at universities using a computer.Practical significance. The research materials emphasise considerable practical importance to implement a transdisciplinary approach in didactics and methods of teaching physical and mathematical disciplines in the system of natural science, and engineering education. The research findings will be of interest to educational theorists and teachers, who conduct professional training of students of natural science and engineering fields and to everyone who is interested in the prosperous future of the educational system.

In the Shadow of the Palms: The Selected Works of David Eugene Smith

In the Shadow of the Palms: The Selected Works of David Eugene Smith

Recurrence and linear sequences: a three-dimensional visualization with support of GeoGebra software

"This article is an excerpt from a master's thesis developed in Brazil, in which we approach recurrent and linear sequences, given some intriguing particularities in their definitions and the scarcity of discussion of this topic in the literature of the History of Mathematics, especially with regard to its geometric representation. Thus, we aim to present the identities of Fibonacci, Lucas, Jacobsthal and Padovan in a three-dimensional visualization with the contribution of GeoGebra software. The research methodology chosen was bibliographical, exploratory in nature, where we have theoretical support in works such as Oliveira and Alves (2019), Silva (2017), Souza and Alves (2018), Vieira and Alves (2020). This research brings as results a set of geometric constructions of the identities of the proposed sequences, in three-dimensional perspective, being a support for future works developed around this theme. GeoGebra was essential in the process of constructing and visualizing the sequences, as it provided strategies for understanding the recurrence relations and the properties of the Fibonacci, Lucas, Jacobsthal and Padovan sequences, through the behavior of the visual representations of these identities."

The Exact Measurement of Pi

Abstract: This research study gives the actual value of PI (π), a mathematical constant that describes the relationship between a circular structure's circumference and diameter. It also highlights the lacuna in the most commonly used approximations of PI (π), such as 3.14159…, which were supplied by Archimedes. It also rejecting all the values of PI calculated by various mathematicians in almost 4000 years of mathematical history, and presents the absolute 100% true value of PI through various mathematical process and conclusions

In Search of Comrade Agronomof: Some Tribonacci History

A note from May 1914 by a certain Agronomof in the Belgian journal Mathesis is frequently cited in the context of the mathematical history of the Tribonacci number sequence. We round out the historical perspective by providing details on Agronomof’s life and work, and highlight two late-nineteenth-century occurrences of this number sequence. One occurrence is connected to Charles Darwin’s On the Origin of Species. Furthermore, we show that the main identity in Agronomof’s note can be leveraged to derive several nontrivial identities with surprising ease and elegance. Some of these identities have only recently been proven by different and more laborious methods; others are new.

목표로서의 수학사를 위한 이차곡선의 교수·학습 자료개발

Objectives The purpose of this study is to develop a teaching and learning material of “Quadratic curve” using history of Mathematics as a Goal.
 Methods The direction of data development was determined after examining the type of mathematics history utilization and the method of mathematics history utilization in Korean textbooks. Materials were developed through the extraction of mathematical history contents that are acceptable for subject and purpose. Twelve math teachers with experience teaching high school students were tested for verifying the appropriateness of the developed material. After analyzing the test result, the material was revised and completed.
 Results The developed teaching and learning materials are organized according to the flow of time from ancient times to the present, in the order of etymological analysis of terms, discovery of ancient quadratic curves, relevance of modern analytic geometry and quadratic curves, development of modern astronomy, and use in modern earth science. While revealing the connection with other studies, the human side and usefulness of mathematics were revealed. A suitability test was conducted on the development materials, and positive responses were obtained from teachers of related subjects that they were developed well for the purpose of research.
 Conclusions In an environment where students' interest in mathematics is declining, the development type of using mathematical history as a goal can be a way to cope with the overall change in the educational paradigm of strengthening the curriculum centered on choice. It will be helpful for students' interest and choices if the human side of mathematics is emphasized through mathematics history.

From Zeno ad infinitum: Iterative Reasonings in Early Greek Philosophy

Abstract This paper considers some aspects of the early conception and use of the infinite in ancient Greece, in the spirit of recent results in the history of ancient mathematics. It follows aspects of the practice of reasoning ad infinitum from the extant corpus of and about Zeno of Elea up to early Hellenistic examples in Aristotle and Euclid. Starting with the idea of ‘reasoning from indefinite iteration’, based on the metalogical recognition of the unachievability of an inference process, it identifies several different classes of more or less sophisticated arguments that make use of this idea, and examines the logical devices and notions required for their acceptance in the philosophical practice. Those include ‘Infinite regress’ properly speaking, where Non-Contradiction is used in the formation of indirect infinitary arguments.

History of mathematics (and mathematics) with gender perspective: towards a social history of mathematics

Our official mathematics subject curricula are full of names next to rules, laws, theorems and results. Usually, most of these names refer to male mathematicians whose works had a high impact in the history of mathematics. However, there is the possibility of giving a more coeducative view to this science, achieving a less unequal curriculum with respect to the gender perspective. The first step on this way is the paradigm shift which must occur in university curricula, and history of mathematics is a very good example of this. This article shows how it is possible to coeducate from the history of mathematics and what acceptance this new look in gender perspective can have among our students.

Triangulations of grassmannians and flag manifolds

MacPherson (in: Topological methods in modern mathematics: a symposium in Honor of John Milnor’s Sixtieth Birthday Stony Brook NY 1991, Perish, Houston, 1993. https://doi.org/10.2307/1970177) conjectured that the Grassmannian textrm{Gr}(2, mathbb {R}^n) has the same homeomorphism type as the combinatorial Grassmannian text{ MacP }(2,n), while Babson (in: A combinatorial flag space, MIT, 1993. https://doi.org/10.2307/1970177) proved that the spaces text{ Gr }(2,mathbb {R}^n) and text{ Gr }(1,2,mathbb {R}^n) are homotopy equivalent to their combinatorial analogs, the simplicial complexes Vert text{ MacP }(2,n)Vert and Vert text{ MacP }(1,2,n)Vert respectively. We will prove that text{ Gr }(2, mathbb {R}^n) and text{ Gr }(1,2, mathbb {R}^n) are homeomorphic to Vert text{ MacP }(2,n)Vert and Vert text{ MacP }(1,2,n)Vert respectively.

History of mathematics through collaboration: Toward a composite portrait of Oswald Veblen

Oswald Veblen played a pivotal role in the history of American mathematics in the twentieth century. His life, however, remains largely unstudied. This conference was designed to redress this issue by exploring Oswald Veblen and his contributions to the history of American and international mathematics in an interactive workshop that used the Veblen Papers from the US Library of Congress as a foundational and shared resource. With this frame, the conference raised queries and discussed issues related to Veblen, his mathematical contributions, and his collaborative initiatives, including his critical work aiding refugee mathematicians in WWII that helped establish long standing programs at American institutions that continue to advance mathematics at the highest level. The workshop echoed Veblen’s collaborative focus and brought together historians of mathematics and mathematicians to work alongside one another during the conference. This content and collaborative approach combined to advance our understanding of Veblen’s collaborations and the history of twentieth-century mathematics more broadly.

History of mathematics (HM) in secondary school mathematics textbook

History of mathematics (HM) in secondary school mathematics textbook

MOTIVATIONAL COMPONENT OF MATHEMATICS TEXTBOOKS FOR PRIMARY SCHOOL (THE BEGINNING OF THE 21ST CENTURY)

The article analyzes the motivational component of mathematics textbooks for students of grades 2–4 at the beginning of the 21st century, the author of which is the well-known Ukrainian researcher and methodologist M. Bohdanovych. It is significant that these textbooks were used as alternative textbooks by most primary school teachers in Ukraine during the specified period. The analysis was carried out according to our elaborated methodology for studying the motivational component of textbooks designed for younger schoolchildren, which includes the following components: emotionality of the content of the educational material; ways of its didactic organization; textbook design. Ways to implement the motivational function of mathematics textbooks at the beginning of the 21st century for primary school have been described. It was done according to the first component which is the most productive and gives a general idea of the motivational component of the textbook. In particular, mathematics textbooks of the beginning of the 21st century were analyzed according to the criterion of “selection of information taking into account students’ cognitive interests”, which is detailed in the corresponding indicators. In the process of analysis, the positive characteristics of these educational books were established (the presence of characters in the plots of mathematical problems, things and activities that are valuable and close to children’s sensory experience, interesting tasks and mathematical problems with fairy-tale plots, etc.). Also, the shortcomings were indicated. These include insufficient number of the previously mentioned tasks, elements of humor, historical information, “curiosity” sections, etc.). On the basis of the obtained results, scientific and methodological recommendations were formulated regarding the strengthening of the motivational component of textbooks designed for younger schoolchildren.
 The specified recommendations offer specific instructions on improving the content of the mathematical problems plots to strengthen their motivational orientation, in particular on increasing the number of characters that are attractive to children, objects that children often use in everyday life and which are valueable for them, and types of personally meaningful activities, performed by the protagonists of the tasks, as well as mathematical problems with original and fairy-tale plots; approximate examples of mathematical problems with a natural plot and information on interesting statistics and agriculture are given. It is also proposed to include material in the “curiosity” rubrics in mathematics textbooks, the author provides the tips on the effective use of other means of motivational influence: elements of humor, historical information, prominent people’s quotes, information on comparative linguistics and etymology, local history material, folk traditions and customs, written and oral folklore. The expediency of using the mentioned recommendations in the process of creating modern mathematics textbooks and books on other educational subjects is substantiated..