History of Mathematics

The historiography of mathematics of the non-Western world long appeared immune to the influence of cultural studies or of critiques of Orientalism. The turn to “ethno-mathematics” and the interrogation of the historiography of proof as boundary marker (Chemla 2012), the internal challenge to foundationalism, etc., have all played a role in pluralizing conceptions of mathematics—though inklings of the idea can be found in the work of a thinker scholars have, understandably, treated with kid gloves: namely, Oswald Spengler. And yet the practice of mathematics continues to manifest a dynamic and positive relationship with its past and the different ways of doing mathematics: How else is one to understand why mathematicians pursue problems that are 300 years old, or why the theoretical astrophysicist and Nobel laureate S. Chandrasekhar produced a book on Newton’s Principia at the end of the twentieth century (Chandrasekhar 1995)? Furthermore, recent studies on the historiography of non-Western mathematical traditions in Western histories of mathematics produced at the end of the eighteenth and the beginning of the nineteenth century in Europe actually reveal the shaping of the historiographical landscape by the changing representations of the Orient in European historical discourse—splintered, of course, across varying “national interpretative traditions” (Said 1994). Thus, as I have argued elsewhere, the Scottish reading of the Indian mathematical tradition was at variance with the French one (Raina 2001, 2012a). This paper sidesteps the preoccupation of historians with priority disputes and looks more closely at the similarities and differences that might stimulate the development of a cognitively just history of mathematics. This requires closer scrutiny of the historiography of European and Indian mathematics, the modality of construction of the Indian and WestEast mathematical divide, and its relationship with modernity.

Most comprehensive book on the history of computing, consisting of two volumes encompassing some 2,000 pages, with more than 700 illustrations (some 280 of which are new to this edition) and over 150 tables (about 50 of which are new to this edition) New edition (published in the English language for the first time) features a significantly expanded global overview, comprehensive German-English and English-German glossaries of technical terms with over 5,000 entries, and a greatly expanded and updated bibliography with over 6,000 entries Features new information on recently discovered mathematical instruments, including the Multiple Curta (the world's smallest mechanical parallel calculator) an This Third Edition is the first English-language edition of the award-winning Meilensteine der Rechentechnik; illustrated in full color throughout in two volumes. The Third Edition is devoted to both analog and digital computing devices, as well as the world's most magnificient historical automatons and select scientific instruments (employed in astronomy, surveying, clocks, etc.). It also features detailed instructions for analog and digital mechanical calculating machines and instruments, and is the only such historical book with comprehensive technical glossaries of terms not found in print or in online dictionaries. Meticulously researched, the author conducted a worldwide survey of science, technology and art museums with their main holdings of analog and digital calculating and computing machines and devices, historical automatons and selected scientific instrumentsin order to describe a broad range of masterful technical achievements. Also covering the history of mathematics and computer science, this work documents the cultural heritage of technology as well.

By means of interferences of separate prime functions, the article suggests the relation between the Riemann Z function and the distribution of prime numbers in the sense that the -1/2 real part of a nontrivial zero will be placed from the valley (or real zero point) of a cycle of an harmonic prime function different than 3 (only when it's not convergent at that point with the valley of a cycle of the function 3) to the critical strip determined by an extension of that real point in the complex plane to the imaginary point that represents the imaginary part of the non-trivial zero. When two harmonic functions interfere with a a same cycle of the function 3 (that does not have its valley at those points), no prime number will be placed there. A possible hidden asymmetry is examined there. And when there is no interference with the  amplitude of a cycle of the function 3 (there won't be any nontrivial zero) two consecutive prime numbers will be always located there. A visual geometric interpretation of the Riemann critical strip and critical line, and his trivial and non-trivial zeros is attempted to conceptually understand the Riemann Zeta function, its later generalizations,  and the Riemann hypothesis

By means of interferences between prime functions the article shows how an asymmetry between complex conjugates non-trivial zeros inside of the critical strip appears in the Riemann Zeta Function when the prime harmonic functions have a different phase, which could challenge the Riemann Hypothesis while clarifying the relation between prime numbers and the Riemann non-trivial zeros.

Debido a que no es evidente en qué sentido el estudio de la matemática ayuda a los guardianes de la República de Platón a dominar las pasiones y los prepara para ser gobernantes, mostraré la función propedéutica de la matemática; cómo las cinco disciplinas en las que está dividida-aritmética, geometría, estereotomía, astronomía y música-pueden dividirse a su vez en dos: la popular y la filosófica, y es solo esta última la que provoca el giro del alma (περιαγωγή) hacia lo que es siempre. Esta es la razón por la cual se afirma que la matemática es una preparación para la dialéctica, el único método propiamente filosófico y la ciencia máxima, cuyo desarrollo podría haber sido influido por el avance en el conocimiento de los irracionales. Solo tras una década de ejercitación matemática el guardián se convertirá en dialéctico, ascenderá al ámbito inteligible y contemplará las Ideas, divinas y ordenadas. Esto despertará en él el deseo de querer asemejarse a ellas y, por mímesis, volverse él también divino y ordenado (en la medida en la que esto le es posible al hombre), lo que le permitirá ejercer el gobierno de sí (ἐγκράτεια), que en la psicología platónica se traduce en el mandato de lo racional sobre lo irascible y lo apetitivo, el gobierno de lo intelectual sobre lo anímico. A pesar del deseo de permanecer para siempre en el ámbito inteligible, el dialéctico deberá descender nuevamente a la caverna y gobernar de forma justa la pólis e instaurar en las costumbres públicas y privadas de los gobernados el orden contemplado, alejado de la tiranía (el más degenerado de los regímenes políticos) y de las pasiones violentas y los apetitos que tiranizan al propio tirano, según el paralelismo alma-ciudad. Cierro con la respuesta a la pregunta de quién es ese filósofo-gobernante en el siglo XXI.

Outre le calcul mental et le calcul sur les doigts (ou d’autres parties du corps), des instruments de calcul ont été mis au point afin de faciliter les pratiques quotidiennes de calcul. L’idée est donc de se servir de ces instruments que nous offre l’histoire pour faire des mathématiques mais également de les penser dans notre temps, c’est-à-dire en particulier de se les approprier avec les nouvelles technologies éducatives disponibles aujourd’hui. Les instruments décrits sont des abaques à jetons, à boules, « à baguettes ». Un trait commun entre tous ces instruments est le recours à la quinaire qui marque 5, 50, 500, etc. Il est important de rappeler ici cette particularité des instruments qui découle d’une caractéristique de l’oeil humain : notre œil peut dénombrer rapidement jusqu’à quatre, mais pas au-delà, d’où le recours à un marqueur de 5. Ces instruments ne sont pas des anecdotes de l’histoire. En Occident, les abaques à jetons ont été d’usage jusqu’à la fin du 18e siècle. Ensuite s’est imposé, mais non sans mal, le calcul avec la notation en chiffres. La suite de l’histoire introduit calculatrices mécanique puis électronique, puis ordinateurs qui permettent d’automatiser les calculs humains. Reste à savoir si le calcul écrit sera une phase transitoire ou pérenne…

Au 18e siècle, les systèmes de mesure (longueurs, poids, capacités, monnaies) sont complexes. De plus, ils varient d’un pays à l’autre, d’une région à l’autre, d’une ville à l’autre, d’une corporation à l’autre. Le chapitre examine quelques tentatives faites à cette époque pour faciliter les calculs monétaires et métrologiques. On y analyse plusieurs sortes de tables numériques et graphiques conçues pour aider les citoyens, en particulier ceux qui étaient peu instruits en arithmétique, à gérer leurs affaires quotidiennes. Ces tables sont autant d’instruments à exploiter dans l’enseignement d’aujourd’hui, à l’école et au collège, pour aborder dans un contexte historique motivant les questions de numération et de mesure, pour introduire des changements de cadre entre arithmétique et géométrie, et pour favoriser une démarche d’investigation.