Greek Mathematics Research Articles

La cisoide de Diocles en el contexto de la matemática antigua

The three classic problems of Greek mathematics are the squaring of the circle, the trisection of the angle and the doubling of the cube. These questions had enormous importance in the progress of Hellenic geometry and acted as true poles of interest, guiding the research of the great mathematicians of Antiquity. This article analyzes in its historical context the genesis of the cissoid, a curve specifically conceived by Diocles to provide a new solution to the problem of doubling the cube, and its influence on the development of Greek mathematics is studied.

Constructing Mathematical Literacy Problems and Assessing Students’ Solving Abilities

The aim of the present study is to investigate students’ abilities in mathematical literacy upon completing their compulsory education in Greece and to elucidate any challenges they may encounter. The research involved 650 students from various regions in Greece, selected based on the urbanization level of their respective school areas. These students had not undergone any training in mathematical literacy problem-solving or in expressing their thoughts in writing while undertaking such tasks. Mathematical literacy performance has been also assessed by the Programme for International Student Assessment (PISA) for the last two decades, in which the mean performance of Greek 15-year-old students has been consistently below the OECD average all these years. This discrepancy has been attributed, in part, to the inadequate alignment of the Greek mathematics curriculum with the PISA mathematics framework. To address this, the real-world problems constructed for this study were aligned with the Greek junior high school's mathematics curriculum. The findings indicate that students' mathematical literacy abilities upon completing compulsory education are limited, with the most significant challenges observed in the areas of mathematical reasoning and the written articulation of their thought processes.

AS PROPRIEDADE DOS NÚMEROS PRIMOS

In the 4th century BC, mathematicians (philosophers of the time) were already studying prime numbers. Aristotle, a Greek mathematician (384–322 BC), created the Aristotle Sieve, a property still used today to identify prime numbers. Euclid, a mathematician who lived in the 4th century BC, wrote that prime numbers are part of an infinite set (the set of prime numbers). Throughout history, several mathematicians have presented works that help in the identification of prime numbers. Those who demonstrated methods or properties that are successful in defining a given number that is prime, at any point on the numerical scale, wrote their expressions in symbols unknown to modern men, which is why such expressions fell into oblivion and were accepted as expressions that did not work in defining prime numbers. The most widely used method to define prime numbers even today is successive division, in which the divisors are prime numbers in sequence. Mathematician Christian Goldebach (1690 – 1764) presented his work entitled Goldebach's Conjecture, which proves that all even numbers  4 can be composed by the sum of two prime numbers. (Our emphasis). French mathematician Marin Mersenne (09/08/1558 – 09/01/1648) wrote that prime numbers would be the predecessors of a power of base two, (2n – 1). We proved that successors of a power of base two (2), for an even exponent n (2n + 1) and n  2, can be prime numbers. And predecessors of a power of base two (2), for an odd exponent n (2n – 1) and n > 2, can also be prime numbers.

Aspecte metodologice privind matematica în filosofia lui Kant

This study analyzes the influence of the mathematical methods of ancient Greek mathematicians (synthetic and analytic) on Kantian (transcendental) philosophy, with a particular focus on the relationship between heuristics and methodology. It examines how Kant employed and engaged with synthetic and analytic methods during his early and critical periods, whether through his appreciation for the rigor of the synthetic method, his “imitation” of mathematical methods, and/or his “borrowing” of “the methods of physicists and chemists” to structure B Critique as a “treatise on method”. In this context, the methodology of B Critique becomes a fundamental thematic framework for transcendental philosophy itself, rooted in the methodological experiment referred to in the preface to the B edition as “the experiment of pure reason”. An achievement of this essay holds that while the mathematical influences on Kantian philosophy have often been considered “heuristic” (as suggested by Loparic), they ultimately lead to a profoundly methodological grounding of the entire program of B Critique. Heuristics, defined as an intuitive way of solving problems, and methodology, a rigorous process of analysis and justification, are reinterpreted here not as opposites but as interdependent. The primary conclusion is that both “produce knowledge,” underscoring the essentially methodological nature of knowledge itself. Through this investigation, the study reaffirms the relevance of mathematical methods to the Kantian critical project, showing how they structure both the heuristic process of discovery and the methodological framework of transcendental philosophy. This provides a deeper understanding of the interaction between heuristics and methodology in the generation of knowledge.

Exact Squaring the Circle with Straightedge and Compass Only

No great theory lasts forever in science, but only specific research and discoveries continuously complement each other. There have been 3 classical problems remaining from ancient Greek mathematics, which are extremely influential in the development of geometry. They are Trisecting an Angle, Squaring the Circle, and Doubling the Cube problems. The problem of Squaring the Circle is stated: Using only a straightedge and a compass, is it possible to construct a square with an area equal to a given circle? From the oldest mathematical documents known up to today's mathematics, the "Squaring the Circle" problem and related problems concerning π have interested professional & non-professional mathematicians for millenniums. I adopted the technique “ANALYSIS” to solve accurately the “Squaring the Circle” problem with only a straightedge & compass by Euclidean Geometry, and did not use the value π/Pi anywhere in this exact solution. Upstream from this method of exact “squaring the circle”, we can deduce, inversely, to get a new Mathematical challenge "Circling the Square" with a straightedge & a compass.

Permutations and Oblong Numbers in the Theravāda-vinaya: A New Intersection of Buddhism and Indian Mathematics

Within the context of Indian religions, Jainism has long been recognized for its extensive use of permutations and combinations. However, the application of these principles within Buddhist scriptures has received relatively little scholarly attention. This paper introduces a new example of the specific application of permutations and combinations in Buddhist scriptures. In this paper, we focus on the first saṅghādisesa rule in the Theravāda-vinaya, which lists a series of element sets and arranges these elements according to a certain pattern known as “ten-roots” (mūla), and we discover that these arrangements form a regular numerical sequence, called “oblong numbers”. Moreover, similar patterns with different quantities are also found in the fourth Pārājika and the fifth saṅghādisesa rules. This indicates that the compilers of the Theravāda-vinaya did not use this mathematical knowledge without basis. Interestingly, we also found the use of this sequence in the Bakhshālī manuscript. Therefore, in this article, after summarizing and verifying the arrangement rules of the Theravāda-vinaya, we discuss whether the oblong numbers were influenced by Greek mathematics.

Circling the Square with Straightedge & Compass in Euclidean Geometry

There are three classical problems remaining from ancient Greek mathematics which are extremely influential in the development of Geometry. They are Trisecting an Angle, Squaring the Circle, and Doubling the Cube problems. I solve the Squaring the Circle problem, of which paper is published in the International Journal of Mathematics Trends and Technology (Volume 69, June 2023). Upstream from this method of exact “Squaring the Circle”, we can deduce, conversely/inversely, to get a new Mathematical challenge "CIRCLING THE SQUARE" with a straightedge & a compass in Euclidean Geometry. This study idea came from my exact solution “Squaring the Circle by Straightedge & compass in Euclidean Geometry”, published by IJMTT in June 2023 at https://ijmttjournal.org/Volume-69/Issue-6/IJMTT-V69I6P506.pdf for this ancient Greek Geometry problem. In this research, I adopt the ANALYSIS method to prove the process of solving this new challenge problem, which has not existed in the Mathematics field till today. The process is an inverse/converse solution solving the ancient Greek Geometry challenge problem of “Squaring the Circle”, using a straightedge & a compass. I hereby commit that this is my own personal research project.

The shaping and evolution of Greek mathematics

The shaping and evolution of Greek mathematics

Too Radical Μέθεξις? Gadamer on Platonic Forms

This paper proposes a new interpretation of Gadamer’s problematic appropriation of Platonic metaphysics. It argues that Gadamer, attempting to respond to the challenge posed by Heidegger’s interpretation of Platonic metaphysics and of its role in the history of Being (Seinsgeschichte), downplayed the transcendence of Platonic Forms. Gadamer achieves a reconfiguration of this transcendence and its transposition into what I call here a plane of immanence through two hermeneutic gestures: 1) interpreting Forms in light of Greek mathematics and especially in light of the structure of the sum-number; 2) introducing temporality and historicity in the Medieval doctrine of transcendentals by giving priority to the Beautiful over the Good. I contend that, Platonically speaking, this amounts to the rejection of νοήσις in favor of διάνοια, and that this raises issues concerning the problem of finitude and the potential limits of linguisticality (Sprachlichkeit).

학교수학에서 무리수 도입으로 인한 담론 변화 분석

This study examined the method of discovering incommensurable quantities, the origin of irrational numbers, and the changes in Greek mathematics resulting from the discovery of these quantities from Sfard's commognitive perspective. In addition, in light of the history of mathematics, this study looked into changes in the discourse of school mathematics due to the introduction of irrational numbers. As a result of the analysis, the discovery of incommensurable quantities in the history of mathematics was an opportunity to change from empirical mathematics to formal and theoretical mathematics, and was an event that brought about changes in various mathematical concepts, proportion theory and roles of mathematical tools. In other words, the discovery of incommensurable magnitude transformed Greek mathematics into an incommensurable discourse. These changes in the history of mathematics appear similarly in the changes in the discourse of school mathematics due to the introduction of irrational numbers. The introduction of irrational numbers into school mathematics resulted in the application of formal definitions of numbers, changed the concepts of number and area, and led to the emergence of narratives that seemed to contradict previous discourses. It is understood that the introduction of irrational numbers has changed the discourse of school mathematics into a discourse that is incommensurable with the previous one. The analysis of this study could be basic data for the design of teaching and learning methods for students facing the task of transitioning to the incommensurable discourse of irrational number learning.

Non-Linear Diophantine Equation

This paper is an important study about these Non-Linear Equation which have integer solution exists. i.e., Non-Linear Diophantine Equation, which have named by the famous Greek Mathematician Diophantus of Alexandria. In this paper we have focused to solve Non-Linear Diophantine Equations.

Автолик Питанский. О восходах и заходах

Autolycus of Pitane was a Greek mathematician and astronomer of the second half of the IV century BC. Two of his treatises are extant: “On the Moving Sphere” and “On Risings and Settings”. They are among the earliest extant mathematical Greek texts. We offer a commented Russian translation of the second treatise. To our knowledge, this is the first Russian publication of this text. Our Russian translation of “On the Moving Sphere” was published earlier.

Comparison of 5th to 8th Grade Mathematics Curricula in Turkey and Greece

This study compares the Turkish secondary school mathematics curriculum and the 5-8th grades mathematics curriculum in Greece, which has an essential place in the history of mathematics in terms of their general structure and content (learning areas, sub-learning areas, achievements, and course hours). In this study, we have used the document analysis method and curricula as a data collection tool. As a result of the research, we have seen that although the Turkish mathematics curriculum is structured as a single document, the Greek mathematics curriculum has a detailed document covering the aims, general objectives, and basic concepts of the interdisciplinary approach. While primary school is 6 years and secondary school is 3 years in Greece, both are 4 years in Turkey. The number of achievements in the Greek mathematics curriculum is higher than in Turkey, but the course hours are less than in Turkey. In addition, sets, linear equations, and similarity learning areas are included in the Turkish curriculum, but not in Greece. Similarly, sub-learning areas such as Functions and Trigonometry are included in the Greek curriculum, but not in Turkey. In line with these results, it can be ensured that the mathematics course hours are different at each grade level, and the course hours increase as the grade level increases in the mathematics curriculum conducted in Turkey, as in the Greek mathematics curriculum. In addition, as in previous curricula, a broader curriculum including activities, educational materials, and mathematical illustrations that will contribute to students' understanding of the subject can be developed in line with re-achievements.

DIOPHANTHO'S ARITHMETIC: A CONTRIBUTION OF GREEK MATHEMATICS AS A STRATEGY FOR TEACHING EQUATIONS IN BASIC EDUCATION

DIOPHANTHO'S ARITHMETIC: A CONTRIBUTION OF GREEK MATHEMATICS AS A STRATEGY FOR TEACHING EQUATIONS IN BASIC EDUCATION

Wagner's Furious Host

Wagner's Furious Host

Doubling the Cube and Constructability in Higher Dimensions

Summary It is known that the three classical geometric construction problems introduced by the ancient Greeks: trisecting an angle, squaring a circle, and doubling a cube, cannot be solved using the Euclidean tools. However, ancient Greek mathematicians solved these three problems using other means. We present solutions to the doubling-the-cube problem using ideas that go beyond the Euclidean tools, and we consider generalizations to higher dimensions.

The place of Archimedes in world history

Did science, as we know it, have to be? The article explores a possible response in the negative, organized around a specific contingency: that of Greek mathematics or, even more specifically, that of the mathematics of the generation of Archimedes. The argument is that (1) Greek mathematics, seen against a cross-cultural comparison, is an anomaly, (2) the scientific revolution, as it in fact unfolded, was directly shaped by the anomaly of Greek mathematics, and (3) it is not clear that, absent Greek mathematics, an equivalent scientific revolution would have taken place. The argument is developed in some detail concerning the history of ancient Greek science, but more is said on the inevitably philosophical questions of counterfactuals in history and on the specific question of the contingency of science.

Archimedes’ legacy for early modern science: Historical-philosophical reflections

The paper presents some reflections concerned with Archimedes’ legacy for the development of physico-mathematics in early modernity. First, I introduce Netz’s main themes concerned with the achievements of two generations of Greek mathematicians in their context, including some comparisons with contemporary non-Greek mathematical cultures. My paper, then, points out the necessary conditions for the historical transmission of Archimedes’ legacy on the one hand, and more generally for the mobility of bodies of knowledge on the other hand. In particular, I elaborate my perception of such transformations while focusing on four parameters for their analysis: (a) Transformations of the objects of knowledge through the gaze, the touch, as well as through linguistic or manual representations; (b) Transformations following a change in the boundaries between domains of knowledge; (c) Transformations related to the authority of particular carriers of knowledge and (d), Tansformations related to the place of knowledge.

The Central Role of Incommensurability in Pre-Euclidean Greek Mathematics and Philosophy

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Mathematics students’ conceptions and reactions to questions concerning the nature of rational and irrational numbers

In this paper, we present parts of an online qualitative research project which lasted a whole semester about the understanding of real numbers by sixty first-year Greek mathematics undergraduate students during the COVID-19 pandemic. We focus on the distinction between rational and irrational numbers and their location on the number line using straightedge and compass. The results, came of questionnaires, focused interviews and tasks, disclosed some gaps regarding real numbers that the students had from school and revealed confusion between irrational numbers and their decimal approximation. The results also led us to group students’ conceptions of rational and irrational numbers into five categories related to the number and pattern of decimal digits. Teaching practices and perspectives as well as the, negative, impact of the pandemic on teaching and learning mathematics are discussed.