On the classification of convex quadrilaterals

2016 ◽  
Vol 100 (547) ◽  
pp. 68-85 ◽  
Author(s):  
Martin Josefsson
Keyword(s):  
Golden Age ◽  
Euclidean Geometry ◽  
History Of Mathematics ◽  
Dynamic Geometry ◽  
Computer Programs ◽  
Primary Reason ◽  
Triangle Geometry ◽  
History Of

We live in a golden age regarding the opportunities to explore Euclidean geometry. The access to dynamic geometry computer programs for everyone has made it very easy to study complex configurations in a way that has never been possible before. This is especially apparent in the study of triangle geometry, where the flow of new problems, properties, and papers is probably the highest it has ever been in the history of mathematics. Even though it has increased a bit in recent years, the interest in quadrilateral geometry is significantly lower. Why are triangles so much more popular than quadrilaterals? In fact, we think it would be more logical if the situation were reversed, since there are so many classes of quadrilaterals to explore. This is the primary reason we think that quadrilaterals are a lot more interesting to study than triangles.

scholarly journals Um Passeio pelo Labirinto da Lógica Matemática em companhia de Malba Tahan

2018 ◽  
Vol 15 (19) ◽  
pp. 247-264
Author(s):  
Inocêncio Fernandes Balieiro Filho
Keyword(s):  
Mathematical Logic ◽  
Euclidean Geometry ◽  
History Of Mathematics ◽  
Logical Structure ◽  
Deductive System ◽  
Axiomatic Method ◽  
Concept Definition ◽  
History Of ◽  
Contemporary Approach

O presente artigo tem por objetivo discutir numa perspectiva contemporânea os conteúdos de Lógica, Matemática, Filosofia da Matemática e História da Matemática presentes no livro A Lógica na Matemática, escrito por Malba Tahan. Para isso, mediante o uso da historiografia, foram selecionados temas concernentes com os assuntos da pesquisa. Foram tratados os seguintes temas: a base lógica da Matemática, a definição de conceito, os princípios para se definir um objeto, as definições e a natureza dos axiomas em Matemática, o método axiomático e as diversas axiomáticas para a geometria euclidiana, a estrutura lógica de um sistema dedutivo, os métodos de demonstração em Matemática, a indução, analogia e dedução em Matemática.   Palavras-chave: Lógica Matemática; História da Matemática; Filosofia da Matemática.   A TOUR BY THE LABYRINTH OF MATHEMATICAL LOGIC IN THE COMPANY OF MALBA TAHAN   Abstract   In this paper we discuss the Mathematics, the Logic of Mathematics, the Philosophy and History of Mathematics that presents in the book A Lógica na Matemática of the Malba Tahan, in a contemporary approach. For that, we use the historiography to select matters in adherence with the research. Are treated this topics: the basis of the Logic of Mathematics; the concept definition; principles to define an object; definitions and nature of the axioms in Mathematics; the axiomatic method and the diverse axiomatic to the Euclidean Geometry; the logical structure of a deductive system; demonstration methods in mathematics; the induction, analogy and deduction in mathematics.  

scholarly journals That Was Then…This is Now: Utilizing the History of Mathematics and Dynamic Geometry Software

2021 ◽  
Vol 9 (2) ◽  
pp. 198-212
Author(s):  
Michelle Meadows ◽  
Joanne Caniglia
Keyword(s):  
Mathematics Teacher ◽  
History Of Mathematics ◽  
Dynamic Geometry ◽  
Dynamic Geometry Software ◽  
Dynamic Nature ◽  
Pedagogical Approach ◽  
History Of ◽  
Toulmin’S Model

Pre-service mathematics teacher (PST) education often addresses within Geometry Classes how to utilize Dynamic Geometric Software (DGS). Other classes may also incorporate teaching pre-service teachers about the history of mathematics. Although research has documented the use of Dynamic Geometric Software (DGS) in teaching the history of mathematics (HoM) (Zengin, 2018), the focus of this research specifically targets the development of proof for pre-service teachers by utilizing DGS to revisit historical proofs with a modern lens. The findings concur with Fujita et.al. (2010), Zengin (2018), and Conners (2007) work on proof. The novelty of this article was the combination of incorporating the history of mathematics (HoM), dynamic geometry software (DGS), and Toulmin’s model of argumentation. A pedagogical approach appeared to emerge: DGS’s dynamic nature allowed PSTs to see several examples of a method to provide them with an illustration that may be used in proofs.   

Visual Thinking in Mathematics

Philosophy ◽  
2019 ◽  
Author(s):  
Jessica Carter
Keyword(s):  
20Th Century ◽  
Euclidean Geometry ◽  
18Th Century ◽  
History Of Mathematics ◽  
Mathematical Practice ◽  
Visual Representations ◽  
Visual Thinking ◽  
General Acceptance ◽  
History Of ◽  
Euclid’S Elements

In contemporary philosophy, “visual thinking in mathematics” refers to studies of the kinds and roles of visual representations in mathematics. Visual representations include both external representations (i.e., diagrams) and mental visualization. Currently, three main areas and questions are being investigated. The first concerns the roles of diagrams, or the diagram-based reasoning, found in Euclid’s Elements. Second is the epistemic role of diagrams: the question of whether reasoning based on diagrams can be rigorous. This debate includes the question of whether beliefs based on visual input can be justified, and whether visual perception may lead to mathematical knowledge. The third observes that diagrams abound in (contemporary) mathematical practice, and so tries to understand the role they play, going beyond the traditional debates on the legitimacy of using diagrams in mathematical proofs. Looking at the history of mathematics, one will find that it is only recently that diagrammatic proofs have become discredited. For about 2,000 years, Euclid’s Elements was conceived as the paradigm of (mathematical) rigorous reasoning, and so until the 18th century, Euclidean geometry served as the foundation of many areas of mathematics. One includes the early history of analysis, where the study of curves draws on results from (Euclidean) geometry. During the 18th and 19th centuries, however, diagrams gradually disappear from mathematical texts, and around 1900 one finds the famous statements of Pasch and Hilbert claiming that proofs must not rely on figures. The development of formal logic during the 20th century further contributed to a general acceptance of a view that the only value of figures, or diagrams, is heuristic, and that they have no place in mathematical rigorous proofs. A proof, according to this view, consists of a discrete sequence of sentences and is a symbolic object. In the latter half of the 20th century, philosophers, sensitive to the practice of mathematics, started to object to this view, leading to the emergence of the study of visual thinking in mathematics.

scholarly journals Logical and historical determination of the Arrow and Sen impossibility theorems

2007 ◽  
Vol 52 (172) ◽  
pp. 7-20 ◽  
Author(s):  
Branislav Boricic
Keyword(s):  
Social Choice ◽  
History Of Mathematics ◽  
Direct Proof ◽  
Mathematical Economics ◽  
General Classification ◽  
Logical Equivalence ◽  
History Of

General classification of mathematical statements divides them into universal, those of the form xA , and existential ?xA ones. Common formulations of impossibility theorems of K. J. Arrow and A. K. Sen are represented by the statements of the form "there is no x such that A". Bearing in mind logical equivalence of formulae ??xA and x?A, we come to the conclusion that the corpus of impossibility theorems, which appears in the theory of social choice, could make a specific and recognizable subclass of universal statements. In this paper, on the basis of the established logical and methodological criteria, we point to a sequence of extremely significant "impossibility theorems", reaching throughout the history of mathematics to the present days and the famous results of Arrow and Sen in field of mathematical economics. We close with specifying the context which makes it possible to formulate the results of Arrow and Sen accurately, presenting a new direct proof of Sen?s result, with no reliance on the notion of minimal liberalism. .

scholarly journals Identificação e classificação de estudos históricos correlatos à organização curricular de cursos de matemática no Brasil

2019 ◽  
Vol 3 (3) ◽  
pp. 823
Author(s):  
Suélen Rita Andrade Machado ◽  
Lucieli Maria Trivizoli
Keyword(s):  
History Of Mathematics ◽  
Social Practices ◽  
Mathematical Education ◽  
Mathematics Courses ◽  
Mathematics History ◽  
Global View ◽  
History Of ◽  
Object Of Study ◽  
Higher Institutions

Resumo: Neste artigo de revisão bibliográfica, identificamos e classificamos estudos relacionados à organização histórica e curricular de cursos de Matemática de instituições superiores no Brasil. A identificação dos trabalhos ocorreu no Catálogo de Teses e Dissertações do banco de dados da Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES). Na primeira etapa selecionamos trabalhos relacionados à área da Matemática, e submetemos esses a uma pré-leitura ou leitura de reconhecimento com a finalidade de obter uma visão global do tema tratado. Escolhemos então, trabalhos correlatos e procuramos elencar seu objeto de estudo, objetivos, métodos e resultados. Posteriormente, classificamos os trabalhos nos campos investigativos em História da Matemática, História da Educação Matemática e História na Educação Matemática. Por fim, identificamos a relevância da classificação desses trabalhos, enquanto produto do conhecimento de práticas sociais investigativas relativas à História da Matemática, uma vez que reafirmam a autonomia desse campo de pesquisa.Palavras-chave: Cursos de Matemática; Currículos dos Cursos de Matemática; História da Educação Matemática. Describing and classification of historical studies related to the organization of mathematics courses in BrazilAbstract: In this bibliographical review, we identify and classify historical studies related to the historical and curricular organization of Mathematics courses of higher institutions in Brazil. The identification of researches published in the Catalog of Dissertations and Theses of the database of the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES). In the first step we selected researches related to the field of Mathematics and subjected them to a pre-reading or recognitional reading in order to get a global view of the theme. We then choose related researches and seek to list their object of study, objectives, methods and results. Subsequently, we classified the researches in the investigative fields of History of Mathematics, History of Mathematical Education and History in Mathematical Education. Finally, we understand the relevance of the identification and classification of these researches, as a product of knowledge of investigative social practices related to the History of Mathematics, while they reaffirm the autonomy of this field of research.Keywords: Math courses; Mathematics Courses Curriculum; History of Mathematical Education. 

scholarly journals REFLECTIONS ON LEARNING MATHEMATICS IN PHYSICS PHENOMENOLOGY AND HISTORICAL CONCEPTUAL STREAMS

2012 ◽  
Vol 46 (1) ◽  
pp. 93-100
Author(s):  
Maria Mellone ◽  
Raffaele Pisano
Keyword(s):  
Science Education ◽  
History Of Science ◽  
Euclidean Geometry ◽  
History Of Mathematics ◽  
Point Of View ◽  
Mathematical Education ◽  
Epistemology Of Science ◽  
History Of ◽  
Mathematical Sciences ◽  
Mathematics And Physics

Although several efforts produced by new mathematical education approaches for improving education systems and teaching, yet the results are not sufficient to adsorb the totality of innovations proposed, both in the contents and management. In this sense constructive debates and new ideas were welcomed and appreciated upon new aspects of science education, side new learning and Cognitive Modelling, for our interests. A parallel effort was produced by scientist-epistemologist-historians concerning the history of science and its foundations in science education. Historical foundations represent the most important intellectual part of the science, even if sometimes they were avoided or limited to specialist disciplines such as history of mathematics, history of physics, only. Nevertheless some results, such as the operative concept of mass by Mach, rather the coherence and validity of an algebraic–geometric group in a Euclidean geometry and in non-Euclidean geometry was firstly appointed by epistemological point of view by (e.g.,) Poincaré, etc... Thus, what kind of concrete relationship between science education (mathematics and physics) and history of science (idem) one can discuss correlated with foundations of science? and above all, how this relationship can be appointed? The history and epistemology of science help to understand evolution/involution of mathematical and physical sciences in the interpretation-modelling of a phenomenon and its interpretation-didactic-modelling, and how the interpretation can change for a different use of mathematical: e.g., mathematics à la Cauchy, non-standard analysis, constructive mathematics in physics. Based on previous studies, a discussion concerning mathematics education and history of science is presented. In our paper we will focus on learning modelling to discuss its efficacy and power both from educational point of view and the need of mathematics and physics teachers education. Some case–studies on the relationship between physics and mathematics in the history are presented, as well. Particularly we focus on a possible learning modelling activity within physics phenomenology to create a resonance among the above poles and mathematical modelling cycle to argue its efficacy, power and related with historical foundations of physical, mathematical sciences. Key words: modelling, mathematics, physics, history of foundations, epistemology of science.

To Understand the Informal (on the Book BY V. L. Tambovtsev “Economics of Informal Institutions”)

2015 ◽  
pp. 151-158
Author(s):  
A. Zaostrovtsev
Keyword(s):  
Social Psychology ◽  
Detailed Analysis ◽  
Human Behavior ◽  
Interdisciplinary Approach ◽  
Informal Institutions ◽  
Economic Thought ◽  
History Of ◽  
Original Classification

The review considers the first attempt in the history of Russian economic thought to give a detailed analysis of informal institutions (IF). It recognizes that in general it was successful: the reader gets acquainted with the original classification of institutions (including informal ones) and their genesis. According to the reviewer the best achievement of the author is his interdisciplinary approach to the study of problems and, moreover, his bias on the achievements of social psychology because the model of human behavior in the economic mainstream is rather primitive. The book makes evident that namely this model limits the ability of economists to analyze IF. The reviewer also shares the author’s position that in the analysis of the IF genesis the economists should highlight the uncertainty and reject economic determinism. Further discussion of IF is hardly possible without referring to this book.

Riemann–Weyl in Deleuze's Bergsonism and the Constitution of the Contemporary Physico-Mathematical Space

2015 ◽  
Vol 9 (1) ◽  
pp. 59-87 ◽  
Author(s):  
Martin Calamari
Keyword(s):  
Field Theory ◽  
Gauge Field ◽  
Fibre Bundle ◽  
History Of Mathematics ◽  
Gauge Field Theory ◽  
Contemporary Science ◽  
Explicit Mention ◽  
History Of ◽  
Key Features ◽  
Bernhard Riemann

In recent years, the ideas of the mathematician Bernhard Riemann (1826–66) have come to the fore as one of Deleuze's principal sources of inspiration in regard to his engagements with mathematics, and the history of mathematics. Nevertheless, some relevant aspects and implications of Deleuze's philosophical reception and appropriation of Riemann's thought remain unexplored. In the first part of the paper I will begin by reconsidering the first explicit mention of Riemann in Deleuze's work, namely, in the second chapter of Bergsonism (1966). In this context, as I intend to show first, Deleuze's synthesis of some key features of the Riemannian theory of multiplicities (manifolds) is entirely dependent, both textually and conceptually, on his reading of another prominent figure in the history of mathematics: Hermann Weyl (1885–1955). This aspect has been largely underestimated, if not entirely neglected. However, as I attempt to bring out in the second part of the paper, reframing the understanding of Deleuze's philosophical engagement with Riemann's mathematics through the Riemann–Weyl conjunction can allow us to disclose some unexplored aspects of Deleuze's further elaboration of his theory of multiplicities (rhizomatic multiplicities, smooth spaces) and profound confrontation with contemporary science (fibre bundle topology and gauge field theory). This finally permits delineation of a correlation between Deleuze's plane of immanence and the contemporary physico-mathematical space of fundamental interactions.

Sign in / Sign up

Export Citation Format

Share Document