A primordial, mathematical, logical and computable, demonstration (proof) of the family of conjectures known as Goldbach´s

licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.In this document, by means of a novel system model and first order topological, algebraic and geometrical free-­‐context formal language (NT-­‐FS&L), first, we describe a new signature for a set of the natural numbers that is rooted in an intensional inductive de-­‐embedding process of both, the tensorial identities of the known as “natural numbers”, and the abstract framework of theirs locus-­‐positional based symbolic representations. Additionally, we describe that NT-­‐FS&L is able to: i.-­‐ Embed the De Morgan´s Laws and the FOL-­‐Peano´s Arithmetic Axiomatic. ii.-­‐ Provide new points of view and perspectives about the succession, precede and addition operations and of their abstract, topological, algebraic, analytic geometrical, computational and cognitive, formal representations. Second, by means of the inductive apparatus of NT-­‐FS&L, we proof that the family of conjectures known as Glodbach’s holds entailment and truth when the reasoning starts from the consistent and finitary axiomatic system herein describedWe wish to thank the Organic Chemistry Institute of the Spanish National Research Council (IQOG/CSIC) for its operative and technical support to the Pedro Noheda Research Group (PNRG). We also thank the Institute for Physical and Information Technologies (ITETI/CSIC) of the Spanish National Research Council for their hospitality. We also thank for their long years of dedicated and kind support Dr. Juan Martínez Armesto (VATC/CSIC), Belén Cabrero Suárez (IQOG/CSIC, Administration), Mar Caso Neira (IQOG/CENQUIOR/CSIC, Library) and David Herrero Ruíz (PNRG/IQOG/CSIC). We wish to thank to Bernabé-­‐Pajares´s brothers (Dr. Manuel Bernabé-­‐Pajares, IQOG/CSIC Structural Chemistry & Biochemistry; Magnetic Nuclear Resonance and Dr. Alberto Bernabé Pajares (Greek Philology and Indo-­‐European Linguistics/UCM), for their kind attention during numerous and kind discussions about space, time, imaging and representation of knowledge, language, transcription mistakes, myths and humans always holding us familiar illusion and passion for knowledge and intellectual progress. We wish to thank Dr. Carlos Cativiela Marín (ISQCH/UNIZAR) for his encouragement and for kind listening and attention. We wish to thank Miguel Lorca Melton for his encouragement and professional point of view as Patent Attorney. Last but not least, our gratitude to Nati, María and Jaime for the time borrowed from a loving husband and father. Finally, we apologize to many who have not been mentioned today, but to whom we are grateful. Finally, let us point out that we specially apologize to many who have been mentioned herein for any possible misunderstanding regarding the sense and intension of their philosophic, scientific and/or technical hard work and milestone ideas; we hope that at least Goldbach, Euler and Feymann do not belong to this last human´s collectivity.Peer reviewe

Enacting Inquiry Learning in Mathematics through History

International audienceWe explain how history of mathematics can function as a means for enacting inquiry learning activities in mathematics as a scientific subject. It will be discussed how students develop informed conception about i) the epistemology of mathematics, ii) of how mathematicians produce mathematical knowledge, and iii) what kind of questions that drive mathematical research. We give examples from the mathematics education at Roskilde University and we show how (teacher) students from this program are themselves capable of using history to establish inquiry learning environments in mathematics in high school. The realization is argued for in the context of an explicit-reflective framework in the sense of Abd-El-Khalick (2013) and his work in science education

History of Mathematics in Mathematics Education: Recent devlopments

International audience<p>This is a concise survey on the recent developments (since 2000) concerning research on the relations between History and Pedagogy of Mathematics (the <i>HPM domain</i>). Section 1 explains the rationale of the study and formulates the key issues. Section 2 gives a brief historical account of the development of the <i>HPM domain</i> with focus on the main activities in its context and their outcomes. Section 3 provides a sufficiently comprehensive bibliographical survey of the work done in this area since 2000. Finally, section 4 summarizes the main points of this study.</p

Original Sources in the Mathematical Classroom

International audienceThis discussion group seeks to bring together individuals who are interested in the use of original sources in the mathematics classroom, from the perspective of a classroom teacher or a mathematics education researcher, for a discussion of issues and concerns related to their educational potential and effects. Each of the two sessions will focus on a different theme related to the use of original sources in the mathematics classroom. The two sessions will structured around a common framework but sufficiently independent of each other to allow interested individuals to participate in the second session, even if they did not participate in the first session. Both novice and more experienced users of original sources are strongly encouraged to participate in both sessions

Studia Lulliana 1977, vol. 21, fasc. 3

Abstract not availabl

Ragione, razionalità e razionalizzazione in età moderna e contemporanea

[Italiano]: “Ragione” e “razionalita” sono oggi nozioni problematiche: considerate a lungo, nella storia del pensiero occidentale, come tratti distintivi dell’uomo e della civiltà euro-americana, da un certo punto in poi sono state messe in discussione e lacerate dalla crisi e dal malessere di quest’ultima. Alla luce delle controversie più recenti e delle riflessioni sul tema sviluppate nella prima metà del Novecento, questo libro si propone di riaprire la querelle sulla ratio con l’intento di mettere in luce – attraverso una serie di “casi di studio” – la grana sottile dei molteplici significati e usi della nozione di ragione, ma anche di interrogarsi sulle differenti “epoche” della ragione provando a ricostruirne la “storia”. I saggi contenuti in questo volume, dunque, contestualizzano e analizzano le accezioni e le ambiguità del termine ratio e dei suoi derivati nel quadro delle molteplici discussioni che ne hanno ritmato la storia a partire dalla prima modernità, quando il concetto assunse nuove configurazioni rispetto agli usi attestati nel pensiero antico e medioevale, sino al dibattito contemporaneo ./[English]: “Reason” and “rationality” are, today, problematic notions: considered for a long time, in the history of Western thought, as distinctive traits of man and of the Euro-American civilization, from a certain point onwards they have been called into question and ripped apart by the crisis and the malaise of the latter. In the light of the most recent controversies and reflections on the subject developed over the first half of the 20th century, this book aims to riegnite the debate on “ratio” with the intention of highlighting – through a series of “case studies” – the fine texture of the multiple meanings and uses of the notion of reason, but also to question the different “eras” of reason by attempting to reconstruct its “history”. The essays included in this volume, therefore, contextualise and analyse the meanings and ambiguities of the term “ratio” and its derivatives within the framework of the many discussions that have marked its history from the earliest modernity, when the concept assumed new configurations with respect to the uses attested in Ancient and Medieval thought, to the contemporary debate

Ragione, razionalità e razionalizzazione in età moderna e contemporanea

“Ragione” e “razionalita” sono oggi nozioni problematiche: considerate a lungo, nella storia del pensiero occidentale, come tratti distintivi dell’uomo e della civiltà euro-americana, da un certo punto in poi sono state messe in discussione e lacerate dalla crisi e dal malessere di quest’ultima. Alla luce delle controversie più recenti e delle riflessioni sul tema sviluppate nella prima metà del Novecento, questo libro si propone di riaprire la querelle sulla ratio con l’intento di mettere in luce – attraverso una serie di “casi di studio” – la grana sottile dei molteplici significati e usi della nozione di ragione, ma anche di interrogarsi sulle differenti “epoche” della ragione provando a ricostruirne la “storia”. I saggi contenuti in questo volume, dunque, contestualizzano e analizzano le accezioni e le ambiguità del termine ratio e dei suoi derivati nel quadro delle molteplici discussioni che ne hanno ritmato la storia a partire dalla prima modernità, quando il concetto assunse nuove configurazioni rispetto agli usi attestati nel pensiero antico e medioevale, sino al dibattito contemporaneo