pinecone/core-2020-05-10-deduplication huggingface dataset:
Soviet Mathematics and Dialectics in the Stalin Era

AbstractStalinist philosophers and other champions of Marxist theory argued against the modern trend in the mathematization of science. They thought that the unrestricted mathematization of the natural sciences and humanities invited “mathematical idealism”, an archenemy of materialistic dialectics. In their opinion, Georg Cantor's set theory was based on speculation unrelated to the real world. Such leading mathematicians as A. N. Kolmogorov and P. S. Aleksandrov had different ideas: they welcomed the modern trend in the accelerated mathematization of science, and they considered set theory the most advanced and promising branch of mathematics. Despite Stalin's unceasing war on free expression in science, Soviet mathematicians made significant contributions to modern mathematical knowledge. Copyright 2000 Academic Press.[formula]Copyright 2000 Academic Press.MSC 1991 subject classifications: 01A60, 01A72, 01A74, 01A8
Justus Grassmann's Contributions to the Foundations of Mathematics: Mathematical and Philosophical Aspects

AbstractThis paper explores Justus Grassmann's ideas concerning the foundations of mathematics as developed between 1806 and 1827. It begins with a discussion of the didactic and philosophical contexts of Grassmann's work. This is followed by an exposition of his ideas on the foundations of geometry and of his 1827 account of the foundations of mathematics. Particular attention is given to Grassmann's concept of construction (synthesis), which represents the basis of his entire thinking. Copyright 2000 Academic Press.Dieser Aufsatz untersucht Justus Grassmanns Ideen zu den Grundlagen der Mathematik, so wie sie zwischen 1806 und 1827 von ihm entwickelt worden sind. Die Arbeit beginnt mit der Darstellung des didaktischen und des philosophischen Kontextes seiner Werke. Anschließend werden Grassmanns Ideen zu den Grundlagen der Geometrie und seine Behandlung der Grundlegung der Mathematik von 1827 dargestellt. Besondere Aufmerksamkeit wird dem Grassmannschen Konstruktionsbegriff (Synthese), welcher die Basis für sein gesamtes Denken bildet, gewidmet. Copyright 2000 Academic Press.AMS 1991 classification numbers: 01A55, 01A72, 03-0
Sources for Danish Mathematics

AbstractThis paper contains a survey of the most important works on Danish mathematics and and a short description of a new archive of mathematical archival material at the Institute for Mathematical Sciences, University of Copenhagen. The archive is especially of interest for the history of Danish mathematics, but some collections contain material which might be of more general interest. Copyright 2000 Academic Press.Denne note består af en oversigt over de vigtigste arbejder om dansk matematik samt en kort beskrivelse af et nyt arkiv med matematisk arkivmateriale ved Institut for Matematiske Fag, Københavns Universitet. Arkivet er isæ r relevant for matematikkens historie i Danmark, men nogle af samlingerne indeholder dog også materiale som er af mere generel interesse. Copyright 2000 Academic Press.Cette note comprend un plan l'ensemble des plus importants travaux dans de domaine des mathématiques danoises ainsi qu'une brève description d'un nouveau système d'archives du matériel d'archives mathématiques à l'Institut des Mathématiques de l'Université de Copenhague. Ces archives son surtout importantes pour l'histoire des Mathématiques au Danemark, mais quelques unes des archives comportent également des informations d'un intérêt plus générel. Copyright 2000 Academic Press.AMS classification numbers: 01A55, 01A60
The Danish Mathematical Society through 125 Years

AbstractThe Danish Mathematical Society celebrated its 125th anniversary in 1998. This work presents an outline of the Society's history and goes into details on some selected episodes. Among other things the society is put in a Scandinavian context. Indirectly, the work also gives an impression of several of the significant Danish mathematicians during the period of time. Copyright 2000 Academic Press.Dansk Matematisk Forening fejrede sin 125-års jubilæum i 1998. Dette arbejde giver et overblik over foreningens historie og går i detalje med udvalgte episoder. Bl.a. sættes foreningen ind i en skandinavisk sammenhæng. Artiklen giver også indirekte et indtryk af flere af de betydeligste danske matematikere i perioden. Copyright 2000 Academic Press.AMS subject classifications: 01A55, 01A60
A Contextualized Historical Analysis of the Kuhn–Tucker Theorem in Nonlinear Programming: The Impact of World War II

AbstractWhen Kuhn and Tucker proved the Kuhn–Tucker theorem in 1950 they launched the theory of nonlinear programming. However, in a sense this theorem had been proven already: In 1939 by W. Karush in a master's thesis, which was unpublished; in 1948 by F. John in a paper that was at first rejected by the Duke Mathematical Journal; and possibly earlier by Ostrogradsky and Farkas. The questions of whether the Kuhn-Tucker theorem can be seen as a multiple discovery and why the different occurences of the theorem were so differently received by the mathematical communities are discussed on the basis of a contextualized historical analysis of these works. The significance of the contexts both mathematically and socially for these questions is discussed, including the role played by the military in the shape of Office of Naval Research (ONR) and operations research (OR). Copyright 2000 Academic Press.En démontrant, en 1950, le théorème qui porte aujourd'hui leur nom, Kuhn et Tucker ont donné naissance à la théorie de la programmation non-linéaire. Cependant, en un sens, ce théorème avait été démontré auparavant, d'abord par W. Karush en 1939 dans un mémoire de maı&#x0302;trise inédit, par la suite par F. John en 1948 dans un article qui avait d'abord été rejeté par le Duke Mathematical Journal, et peut-être même plus tôt par Ostrogradsky et aussi par Farkas. Le présent article cherche à élucider deux questions: Peut-on considérer le théorème Kuhn–Tucker comme un exemple de découverte multiple? Et pourquoi le théorème a-t-il été reçu si différemment dans les diverses communautés mathématiques? Notre discussion se base sur une analyse historique contextuelle des différents ouvrages. Nous examinons ici l'importance du contexte, tant du point de vue des mathématiques que du point de vue social, y compris le rôle joué par le secteur militaire dans le cadre de l'Office of Naval Research et de la recherche opérationnelle. Copyright 2000 Academic Press.MSC 1991 subject classification: 01A60; 49-03; 52-03; 90-03; 90C30
Guido Castelnuovo: Documents for a Biography

AbstractIn recent years it has been possible to begin the study of the archive of Guido Castelnuovo, one of the most important mathematicians of the Italian school of algebraic geometry. The archive contains a large scientific correspondence, only partially examined, and all the notebooks of lectures given by Castelnuovo in the courses of the last two years of the Italian degree (Laurea) in mathematics. In this paper we discuss the notebooks of the lectures. The history of the Italian school of algebraic geometry is greatly illuminated by this recent discovery. Copyright 2001 Academic Press.Da alcuni anni è in corso lo studio dell'archivio di Guido Castelnuovo, uno degli esponenti più importanti della scuola italiana di geometria algebrica. Oltre ad una ricca corrispondenza scientifica, solo in parte esaminata, sono stati ritrovati recentemente tutti i quaderni delle lezioni dei corsi che Castelnuovo tenne per il secondo biennio della Laurea in matematica. In questo articolo diamo una prima descrizione di questi quaderni. Importanti aspetti della storia della scuola italiana di geometria algebrica possono essere chiariti alla luce di questi nuovi documenti. Copyright 2001 Academic Press.MSC 1991 Subject Classifications: 01A55, 01A60, 01A72, 01A73
Neither Sherlock Holmes nor Babylon: A Reassessment of Plimpton 322

AbstractAncient mathematical texts and artefacts, if we are to understand them fully, must be viewed in the light of their mathematico-historical context, and not treated as artificial, self-contained creations in the style of detective stories. I take as a dramatic case study the famous cuneiform tablet Plimpton 322. I show that the popular view of it as some sort of trigonometric table cannot be correct, given what is now known of the concept of angle in the Old Babylonian period. Neither is the equally widespread theory of generating functions likely to be correct. I provide supporting evidence in a strong theoretical framework for an alternative interpretation, first published half a century ago in a different guise. I recast it using regular reciprocal pairs, Høyrup's analysis of contemporaneous “naı̈ve geometry,” and a new reading of the table's headings. In contextualising Plimpton 322 (and perhaps thereby knocking it off its pedestal), I argue that cuneiform culture produced many dozens, if not hundreds, of other mathematical texts which are equally worthy of the modern mathematical community's attention.Wir müssen frühe mathematische Texte und Objekte im Hinblick auf ihre mathematisch-historische Umgebung betrachten und sie nicht als künstliche, vollständige Schöpfungen im Stile von Detektivgeschichten behandeln, wollen wir sie verstehen. Als dramatische Fallstudies dient mir die Keilschrifttafel Plimpton 322. Ich zeige auf, dass die weitverbreitete Ansicht, so etwas wie eine trigonometrische Tabelle vor uns zu haben, nicht richtig sein kann, und zwar aufgrund unseres Wissens über die Vorstellung des Winkels in altbabylonischer Zeit. In gleiche Weise ist die gängige Theorie über erzeugende Funktionen wahrscheinlich falsch. Ich kann meine Neuinterpretation, die in einen stark theoretischen Rahmen eingebettet wird, mit Texten belegen. Hinter meiner Neuinterpretation liegt eine fünfzigjährige Theorie, die auf Bruins zurückgeht. Sie fundiert auf den Gebrauch von regelmassigen, reziproken Paaren, auf Høyrups Analyse der naiven Geometrie und auf eine neue Lesung der Überschriften der Tabelle. Indem ich die Keilschrifttafel Plimpton 322 in ihren historischen Kontext stelle, plädiere ich dafür, dass viele andere mathematische Texte mesopotamischen Ursprungs es ebenso verdienen, von uns beachtet zu werden. Copyright 2001 Academic Press.AMS Subject Classifications: 01A17; 01A85
A Historical Survey of the Fundamental Theorem of Arithmetic

AbstractThe purpose of this article is a comprehensive survey of the history of the Fundamental Theorem of Arithmetic. To this aim we investigate the main steps during the period from Euclid to Gauss. Copyright 2001 Academic Press.Dans cet article nous donnons une vue d'ensemble de l'histoire du Theoréme Fondamental de l'Arithmétique. Pour ce but nous considerons les moments principaux dans la periode de Euclide á Gauss. Copyright 2001 Academic Press.MSC 1991 subject classifications: 01A30, 11-03, 11A5
The Validity of Newton's Lemma 28

AbstractIsaac Newton's Lemma 28 (Principia, Book I) argues that the areas of oval figures are not expressible in finite algebraic equations. Although D. T. Whiteside criticized this lemma, his counterexample is not infinitely smooth and thus would not have been admitted by Newton. Copyright 2001 Academic Press.Le lemme 28 d'Isaac Newton (Principia, livre I) propose que les aires defigures ovales ne sont pas exprimables par des équations algébriques finies. Bien que D. T. Whiteside ait critiqué ce lemme, son contre-exemple n'est pas infiniment dérivable, et Newton ne l'aurait donc pas admis. Copyright 2001 Academic Press.MSC 2000 classification numbers: 01A45, 14M81, 26A27
Soviet Mathematics and Dialectics in the Post-Stalin Era: New Horizons

AbstractThe post-Stalin period in Soviet history (1953–1991) was a time of weakening of official ideological interference with the professional activity of the Soviet scientific community. Many branches of mathematics—particularly cybernetics, mathematical logic, and the foundations of mathematics—played an important role in the struggle against Stalinist ideological and philosophical dogmatism. The Soviet “constructivist” orientation in the foundations of mathematics expressed a conciliatory attitude toward intuitionism, logicism, and formalism, the main currents in Western mathematics. Dialectical materialism became more flexible, but it continued to be the only officially recognized philosophy in the Soviet Union. © 2002 Elsevier Science (USA).[formula]MSC 1991 subject classifications: 01A60, 01A72, 01A74, 01A80
The “Error” in the Indian “Taylor Series Approximation” to the Sine

AbstractIt has been repeatedly noted, but not discussed in detail, that certain so-called “third-order Taylor series approximations” found in the school of the medieval Keralese mathematician Mādhava are inaccurate. That is, these formulas, unlike the other series expansions brilliantly developed by Mādhava and his followers, do not correspond exactly to the terms of the power series subsequently discovered in Europe, by whose name they are generally known. We discuss a Sanskrit commentary on these rules that suggests a possible derivation explaining this discrepancy, and in the process re-emphasize that the Keralese work on such series was rooted in geometric approximation rather than in analysis per se. © 2001 Elsevier Science (USA).Es ist mehrfach festgestellt bisher aber nicht ausführlich diskutiert worden, daß einige sogenannte Taylor-reihennäherungswerte dritter Ordnung, die in der mittelalterlichen Schule keralesischen Mādhava gefunden werden, ungenau sind. Das heißt, diesc Formeln sind den Termen der Potenzreihe, die später in Europa entwickelt wurde und unter dem Namen Taylorreihe bekannt ist, nicht äquivalent, im Gegensatz zu den anderen Entwicklungen von Reihen, die glänzend von Mādhava und seinen Nachfolgern entwickelt werden. Wir behandeln einen Sanskritkommentar zu den Regeln, der eine mögliche Herleitung suggeriert, die diese Diskrepanz erklärt. Dabei betonen wir nochmals, daß die keralesische Arbeit über solche Reihen eher in geometrischen Näherungen als in der Analysis an sich ihre Wurzeln hat. © 2001 Elsevier Science (USA).MSC subject classification: 01A32
The Manuscript of Abel's Parisian Memoir Found in Its Entirety

AbstractThe manuscript of Abel's Parisian memoir on transcendental functions was lost after the printing, and only in 1952 part of this manuscript was found by V. Brun in Florence. Recently, the eight still missing pages of the manuscript have also been found, thus putting an end to more than a century and a half of investigations. © 2002 Elsevier Science (USA).Dopo la stampa, il manoscritto della memoria parigina di Abel sulle funzioni trascendenti andò smarrito, e soltanto nel 1952 parte di esso fu ritrovato da V. Brun a Firenze. Le otto pagine ancora mancanti del manoscritto sono state ritrovate recentemente, ponendo fine a un secolo e mezzo di ricerche. © 2002 Elsevier Science (USA).MSC 1991 subject classifications: 01A50, 01A70, 01A90
Évariste Galois: Principles and Applications

AbstractÉvariste Galois formulated his famous theory in 1831 in the first part of his Mémoire sur les conditions de résolubilité des équations par radicaux. It is titled Principes. Even though the theory is completely understood today, it is hard to follow Galois's original. The style is brief, almost aphoristic and the approach quite different from today's. The aim of this paper is to make Galois's Principes readable for contemporary mathematicians (Sections 1 and 2) and to give a survey of Galois's Applications concerning equations of prime degree, primitive equations, and the modular equation in the theory of elliptic functions (Section 3). Remarks show the relationship to the work of Lagrange and Gauss. © 2002 Elsevier Science (USA).Évariste Galois formulierte seine berühmte Theorie 1831 im ersten Teil seines Mémoire sur les conditions de résolubilité des équations par radicaux. Er trägt den Titel Principes. Auch wenn die Theorie heute vollkommen verstanden ist, ist es bisweilen schwer, Galois' ursprünglichen, äußerst knappen Formulierungen zu folgen. Ziel des Artikels ist es, Galois' Principes für heutige Mathematiker lesbar zu machen (Abschnitte 1 und 2) und einen Überblick über Galois' Applications zu geben, betreffend Gleichungen von Primzahlgrad, primitive Gleichungen und die Modulargleichungen aus der Theorie der elliptischen Funktionen (Abschnitt 3). Bemerkungen verweisen jeweils auf die nahe Beziehung zum Werk von Lagrange und Gauss. © 2002 Elsevier Science (USA).MSC subject classification: 12E10
George David Birkhoff and John von Neumann: A Question of Priority and the Ergodic Theorems, 1931–1932

AbstractBased on a recently discovered letter of John von Neumann to his friend Howard Percy Robertson, the present paper discusses in detail the priority of the discovery and publication of the ergodic theorems by George David Birkhoff and von Neumann in 1931/1932. © 2002 Elsevier Science (USA).À la suite de la découverte récente d'une lettre de John von Neumann à son ami Howard Percy Robertson, cet article examine en détail les priorités de George David Birkhoff et von Neumann dans la découverte et la publication des théorèmes ergodiques, dans les années 1931/1932. © 2002 Elsevier Science (USA).MSC subject classifications: 01A60, 01A70
A Note on Old Babylonian Computational Techniques

AbstractAnalysis of the errors in two Old Babylonian “algebraic” problems shows •that the computations were performed on a device where additive contributions were no longer identifiable once they had entered the computation;•that this device must have been some kind of counting board or abacus where numbers were represented as collections of calculi;•that units and tens were represented in distinct ways, perhaps by means of different calculi.© 2002 Elsevier Science (USA).Eine Analyse der Rechenfehler in zwei altbabylonischen “algebraischen” Aufgaben läßt mehrere Rückschlüsse auf ein Hilfsmittel zu, das zur Durchführung von Rechnungen benutzt worden sein kann: •Additive Beiträge waren nach ihrer Eintragung in die Rechnung nicht länger identifizierbar.•Das Gerät war eine Art Rechenbrett, auf welchem Zahlen als Haufen von Rechensteinen erschienen.•Einer und Zehner wurden in verschiedener Weise, evtl. mittels verschiedener Rechensteine repräsentiert.© 2002 Elsevier Science (USA).MSC subject classification: 01A17
Georg Hamel and Richard von Mises in Brno

AbstractThe article focuses on two outstanding mathematicians who worked at Brno German Technical University, Georg Hamel and Richard von Mises. After their biographies are summarized, their Brno stays are described in detail. A major part of the article is devoted to the process of appointing professors at Brno German Technical University and the difficulties Richard von Mises as a candidate, or those who wanted him to become professor, had to face: he participated in four consecutive competitions for a chair, but without success. © 2002 Elsevier Science (USA).Der Beitrag ist Georg Hamel und Richard von Mises gewidmet—zwei Mathematikern von Weltrang, welche an der deutschen technischen Hochschule in Brünn tätig waren. Auf eine kurze Zusammenfassung ihrer Lebensläufe folgt eine ausführliche Beschreibung ihrer Aufenthalte in Brünn unter besonderer Beachtung der erfolglosen Bemühungen um die Ernennung von Richard von Mises zum Professor für Mathematik und Mechanik. © 2002 Elsevier Science (USA).Tento článek je věnován dvěma mimořádným osobnostem světové matematiky, které působily na německé technice v Brně—Georgu Hamelovi a Richardu von Misesovi. V úvodu jsou krátce zmı́něny jejich životnı́ osudy a poté je podrobně zachycen jejich brněnský pobyt. Je připomenuto úsilı́ o Misesovo jmenovánı́ profesorem matematiky a mechaniky v Brně. © 2002 Elsevier Science (USA).MSC 1991 subject classifications: 01A70, 01A73, 01A60
Two Editions of Ibn al-Haytham's Completion of the Conics

AbstractThe lost Book VIII of the Conics of Apollonius of Perga (ca. 200 B.C.) was reconstructed by the Islamic mathematician Ibn al-Haytham (ca. A.D. 965–1041) in his Completion of the Conics. The Arabic text of this reconstruction with English translation and commentary was published as J. P. Hogendijk, Ibn al-Haytham's Completion of the Conics (New York: Springer-Verlag, 1985). In a new Arabic edition with French translation and commentary (R. Rashed, Les mathématiques infinitésimales du IXe au XIe siécle. Vol. 3., London: Al-Furqan Foundation, 2000), it was claimed that my edition is faulty. In this paper the similarities and differences between the two editions, translations, and commentaries are discussed, with due consideration for readers who do not know Arabic. The facts will speak for themselves. © 2002 Elsevier Science (USA).© 2002 Elsevier Science (USA).© 2002 Elsevier Science (USA).AMS subject classifications: 01A20, 01A30
Algebras, Projective Geometry, Mathematical Logic, and Constructing the World: Intersections in the Philosophy of Mathematics of A. N. Whitehead

AbstractA. N. Whitehead (1861–1947) contributed notably to the foundations of pure and applied mathematics, especially from the late 1890s to the mid 1920s. An algebraist by mathematical tendency, he surveyed several algebras in his book Universal Algebra (1898). Then in the 1900s he joined Bertrand Russell in an attempt to ground many parts of mathematics in the newly developing mathematical logic. In this connection he published in 1906 a long paper on geometry, space and time, and matter. The main outcome of the collaboration was a three-volume work, Principia Mathematica (1910–1913): he was supposed to write a fourth volume on parts of geometries, but he abandoned it after much of it was done. By then his interests had switched to educational issues, and especially to space and time and relativity theory, where his earlier dependence upon logic was extended to an ontology of events and to a general notion of “process,” especially in human experience. These innovations led to somewhat revised conceptions of logic and of the philosophy of mathematics. © 2002 Elsevier Science (USA).A. N. Whitehead (1861–1947) contribuiu de forma marcante para os Fundamentos da Matemática Pura e Aplicada, especialmente entre o fim da década de 1890 e meados da década de 1920. Sendo um algebrista na sua vertente matemática, fez um levantamento de diversas álgebras no seu livro Universal Algebra (1898). Pouco depois de 1900 juntou-se a Bertrand Russell numa tentativa para basear várias partes da matemática sobre a lógica matemática, que se começava então a desenvolver. Nesse âmbito publicou em 1906 um longo artigo sobre geometria, espaço e tempo, e matéria. O principal resultado da colaboração foi um trabalho em três volumes, Principia Mathematica (1910–1913): estava previsto que Whitehead escrevesse um quarto volume sobre aspectos das geometrias, mas abandonou-o depois de uma boa parte já estar escrita. Por essa altura os seus interesses tinham-se voltado para questões educacionais; especialmente para o espaço e o tempo e para a teoria da relatividade, onde a sua anterior dependência da lógica se estendeu a uma ontologia de acontecimentos e a uma noção geral de “processo” especialmente na experiência humana. Estas inovações levaram a concepções um pouco revistas da lógica e da filosofia da matemática. © 2002 Elsevier Science (USA).MSC 1991 subject classifications: 00A30; 01A60; 03-03; 03A05