AN HISTORICAL INVESTIGATION ABOUT THE DEDEKIND´ S CUTS - SOME IMPLICATIONS FOR THE TEACHING OF MATHEMATICS IN BRAZIL

Oct 15, 2026

Teacher training for mathematics in Brazil requires more than techniques: it depends on the historical and epistemological foundations that enable a real understanding of classic concepts and fundamental definitions. This work discusses the introduction and formulation of Dedekind’s cut, tracing its popularization through Richard Dedekind (1831–1916) while contrasting earlier approaches among contemporaries. It also highlights that Dedekind left key questions unresolved, and argues that teachers need to attend to the intuition- and heuristic-driven epistemological process behind formalization.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages22

Reading Time55 min

AccessFree to download

1. Introduction

2. Dedekind’s cut in historical and epistemological terms

3. Teacher understanding and mathematical formalization

4. Dedekind’s background and teaching influence

5. Proof, intuition, and the number concept

AN HISTORICAL INVESTIGATION ABOUT THE DEDEKIND´ S CUTS: SOME IMPLICATIONS FOR THE TEACHING OF MATHEMATICS IN BRAZIL Francisco Regis Vieira Alves, Marlene Alves Dias Abstract: In the training of mathematics teachers in Brazil we can not disregard the historical and epistemological component aiming the transmission of mathematics through a real understanding of the nature of classic concepts and foundamental definitions to Mathematics, whether in the school context or in the academic context. In this sense, the present work addresses a discussion about the introduction and formulation of Dedekind's cut. Such terminology became popular from the work and pioneering research developed by Richard Dedekind (1831 – 1916), although in theset of his contemporaries, as in the case of A. L. Cauchy (1789 – 1857), the proposition of the construction of the real numbers through other notions and others mathematical methods became known. Thus, a historical and epistemological way for the definition of cut is observed and considered. However, Dedekind did not formally answered mainly some of the questions about this notion. The understanding of this epistemological and mathematical process, on the part of the teacher, in which the mathematical intuition and heuristics has an essencial place and requires more attention. Key words: Historical investigation, Dedekind´s cut, History of Mathematics, Teaching. 1. Introduction The definition of a solid basis for the foundation of the primitive concepts of Mathematical Analysis did not occur in a progressive, unstoppable or linear way (Alves, 2018a) . Indeed, in the context of the History of Mathematics, we recorded a movement of progress and recrudescence of certain assumptions and definitions that have become the object of investigation by several mathematicians over time, because it does not inspire a total recognition and acceptance by researchers and specialists. On the other hand, when we consider the role of the mathematics teacher (Alves, 2018a; 2018b), an understanding of a mathematical, epistemological, and evolutionary process about mathematical ideas, about intuitive reasoning and some primitive arguments used by mathematicians in the past is essential, often en virtue to partially solve a particular problem. However, is recurrent in Mathematics, inasmuch as some time later the constitution of a solid formal and structural foundation in the sense of replacing preliminarily heuristic and intuitive ideas by solid and structured axiomatic foundation. Thus, in the present work we present some elements capable of providing an understanding on the evolutionary, historical and epistemological process of formalization of the notion of real number, especially from the thought of Richard Dedekind (1831 – 1916) . Richard Dedekind was one of the greatest mathematicians ofthe nineteenth-century, as well as one of the most important contributors to algebra and number theory of all time (Ferreirós, 2000) . Certainly, given the existence of other mathematical forms and analytical methods employed for the formal establishment of a primitive notion for the theory of functions in a real variable, the notion of Dedekind's cut provides an indelible chapter in the History of Mathematics which, despite intuitive and heuristic ideas indicated by Richard Dedekind, the notion of cut, produced either by a rational number or by a non-rational number, finds its roots in ancient Greek thought. Corry (2004) describes some of his academic background. Dedekind studied in his native city of Braunschweig and in Göttingen. In 1852 he completed his doctoral dissertation, working under the supervision of Gauss. Later in 1854, he habilitated with Bernhard Riemann (1826-1866) . During his first years as Privatdozent in Göttingen he worked in close collaboration with Peter Lejeune Dirichlet (1805-1859) . Thus, the decisive influence of the leading mathematicians of the early Göttingen tradition marked Dedekind's

FAQs

What is the central focus of the work regarding Dedekind’s cut?

The work addresses how Dedekind’s cut was introduced and formulated, using a historical and epistemological perspective to explain the definition and its implications for teaching mathematics.

Why does the author emphasize historical and epistemological components in teacher training?

The author argues that mathematics teachers in Brazil must not disregard these components, because they help transmit mathematics through genuine understanding of classic concepts and foundational definitions.

What does the paper suggest about teachers’ role in dealing with heuristic and intuition?

It highlights that teachers should give essential attention to the epistemological and mathematical process where intuition and heuristics play a crucial place, especially as ideas move toward solid axiomatic foundations.

Oct 15, 2026

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