History of Mathematics (History)

In this paper, we examine Abel’s 1824 proof of the insolubility (in radicals) of the general polynomial of degree five. We give a precise description of what his proof establishes and what is does or does not imply, and we give a sense of the place that this work had in the overall development of algebra. The paper details the overall strategy, and some of the technical machinery, that were used in the proof. It also focuses on one key section of the proof so as to provide a basic sense of the ingenious methods that Abel brought to bear on this problem.

This brief article recalls the life of the English weaver-turned-mathematician Thomas Simpson (Aug. 20, 1710 – May 14, 1761), whose name became attached to Simpson’s Rule for the numerical approximation of integrals. The article takes special note of the fact that during this era, mathematics had special appeal among certain circles of working-class men and upper-middle class women.

The minimum spanning tree is a problem which has become fairly popular among mathematicians after the Second World War. However, already before the War, before the graph-theoretical terminology that is now used in connection with this problem was developed, there existed algorithms for solving this problem. As will become clearer from the description of the individual algorithms, the algorithms of BORŮVKA (1926) and JARNÍK
(1930) were not inferior to those of KRUSKAL (1956), PRIM (1957), LOBERMAN and WEINBERGER (1957), or DIJKSTRA (1959).

While there is general agreement among scholars that the modern numerals with the decimal system in use today hail from India and that these reached medieval Europe via the Arabs, there is no consensus as to whether or not the zero digit was an indigenous Indian invention.  Now a small group is trying to establish the origin of the Zero digit in India.

The Jinkōki's first edition was published in 1627 by Yoshida Mitsuyoshi (1598-1672), when commercial publishing in Kyoto was still in its early stares. Despite its voluminous size and the fact that it was not initially intended for a popular audience, the book met with immediate success, which prompted Yoshida to undertake many revisions. After Yoshida's death, the Jinkōki's success continued to grow. Of the myriad mathematical books that followed, many made use of the expression Jinkōki in their titles and the term came to stand for "manual of computation".

par Marianne Michel.
—
Que nous ont légué les textes des scribes mathématiciens et quelles sont les spécificités de « leurs » mathématiques ?

Les nombreux problèmes et extraits analysés relèvent du corpus mathématique de base datant du Moyen Empire, mais également de documents administratifs et de documents plus récents tels les papyri démotiques. Ce que ces textes nous enseignent dépasse parfois le cadre purement mathématique en donnant des indications sur les valeurs marchandes de produits ou services, les montants de certains salaires ou taxes, les prévisions d’un chantier, la construction d’éléments architecturaux, la gestion des récoltes et du bétail ou la fabrication de la bière.

Rigoureusement scientifique, ce livre se veut aussi pédagogique. La plupart des textes égyptiens sont accompagnés d’une copie hiératique et d’une transcription hiéroglyphique et de nombreuses figures illustrent le propos.

Au fil des chapitres, le lecteur pourra notamment découvrir :
- une nouvelle cartographie du papyrus Rhind,
- un aperçu de l’écriture hiératique,
- une explication des opérations de base (sur les nombres et les fractions)
- et un exposé des systèmes de grandeurs utilisés (métrologie).

Les problèmes d’arithmétique traitent :
- de recherches de quantités inconnues,
- de calculs de racines carrées,
- de progressions arithmétiques ;

les problèmes de géométrie proposent :
- des calculs d’aires,
- de volumes
- et d’inclinaisons.

En outre, les annexes comprennent un lexique des termes mathématiques rencontrés.

Marianne Michel est docteur en égyptologie et titulaire d’un master en langues et littératures anciennes et orientales (spécialisation Égypte et Proche-Orient antiques). Elle est également titulaire d’une licence en sciences mathématiques et agrégée de l’enseignement secondaire supérieur.

Public cible
Ce livre s’adresse à des publics variés. Il intéressera les mathématiciens qui – ils sont nombreux – éprouvent le besoin de remonter aux origines de cette science et à son histoire. Il intéressera les égyptologues qui se spécialisent plus volontiers dans les nombreux autres domaines riches en enseignements de cette discipline tels que l’histoire, la société, la culture, la civilisation, la religion, la littérature, etc. Enfin, il intéressera les étudiants et les amateurs passionnés.

This study aims to find a solution to problem 53 of the "Rhind Mathematical Papyrus" housed in the British Museum since 1864. To date, the problem has not yet been solved and is considered incomplete by Egyptologists who have dealt with it. My hypothesis, completely new, starts from the assumption that the data contained in it are sufficient to identify the procedure and to obtain the result. The triangular shape drawn on the papyrus was divided into three overlapping isosceles triangles having the same vertex, and then the position and size of each of the hieratic numbers indicated on the triangular drawing of the problem 53 was analyzed. The calculations below the drawing seem to have a mathematical correspondence with the measurements of the three triangles having the same vertex, and in this context the numbers indicated by the scribe suggest that the three isosceles triangles superimposed are directly proportional to each other. This solution would confirm what Eric Peet had noted about the Egyptians' ability to calculate "direct proportionality" in Problem 11 of the Moscow Papyrus. The mathematical result is also confirmed by the correspondence between the freehand triangle drawn by the scribe on the papyrus and the geometric drawing made by computer with the measurements resulting from the calculations described in RMP 53. Finally, the geometric drawings resulting from the hypotheses formulated by Egyptologists Chace, Clagett and Miatello were also compared with Ahmose's drawing on Rhind's papyrus.

Phallacies are scientific ideas, hypotheses and theories that may have once been considered true but should no longer be considered completely true. They are more than just fallacies since they control science and its history to such a high degree that they have become 'phallic' symbols that are very difficult to overthrow - Thus the name 'phallacies'. As a matter of historical accuracy and modern scientific truth, physics is now full of such 'phallacies' and some of them are considered our most cherished scientific 'truths'.  For example, common history tells us that the Second Scientific Revolution resulted from two specific experimental crises and the revolution overthrew Newtonian physics. Both of these presently accepted historical truths are 'phallacies'. The twentieth century revolution in science resulted from the successes of Newtonian physics, not its mistakes, and Newtonian science is every bit as alive today as it was one or even two centuries ago. It was never overthrown, but modern quantum theory must say it was overthrown to justify its own existence in the face of severe problems of its own making. The two experiments that scientists cite as the causes of the revolution were nothing more than the 'flotsam and jetsam' of deeper underlying physical principles that raised questions about the Newtonian concepts of space, time, matter and mind. The revolution was as much about mind as it was about matter, but that simple straightforward fact is almost never mentioned in the histories of the revolution and its emergence. The first Scientific Revolution of the seventeenth century was not the first, it was really the second. The period of classical Greek natural philosophy from Thales to Aristotle was actually the first revolution in human scientific thought, so it should now be called the Zeroth Scientific Revolution. These and many more such 'phallacies' shape our modern scientific worldviews and thereby limit the further evolution and development of science. The scientific and academic communities must identify and accept the reality of these 'phallacies' if it is to continue the phenomenal successes of the past.

L'autore dopo una breve introduzione sulle origini dell’astronomia, passa in rassegna le classiche fonti storiche sulla vita e le opere di Talete e sulla cultura dell’antico Egitto, espone un metodo per la misurazione delle altezze delle piramidi che potrebbe essere stato usato dal grande saggio dell'antichità. Immagina anche la realizzazione di un ipotetico distanziometro per misurare dalla costa le distanze delle navi in mare, il quale permette anche di spiegare la scoperta dei teoremi geometrici e la predizione di eventi astronomici che la tradizione attribuisce a Talete.

No mainstream math professor or math educator I've ever known, could tell the difference between C : 2r and C/2r.  In fact, most people mistakenly believe these are the same thing. The difference is both significant and important. One can't understand the concept of number without understanding this difference.

The proof that pi is a transcendental number, first provided by Carl Louis Ferdinand von Lindemann in 1882, was and remains one of the most celebrated results of modern mathematics. The proof was of interest in its own right, and it also resolved a host of questions that had focused the attention of mathematicians since ancient times, including classical problems involving constructions with straightedge and compass. Here, von Lindemann’s result is conveyed in a form that will be accessible to the average undergraduate mathematics major.

This is the third chapter from this book posted on this website.

It concerns the possibility that numbers, writing, and pictures are related to one another in the Western tradition. Writing and picturing are often associated, but counting is rarely spoken about in the same context, so this is an intriguing possibility.

The argument turns on an analysis of Denise Schmandt-Besserat's theories of the origins of counting in the Ancient Middle East. The details of her theories, and their reception, are not of as much interest here as the philosophic possibility that an account like hers lets us posit a common origin for pictures (or images), numbers (or mathematics), and writing.

The idea of the book as a whole is to try to rethink some of the received ideas about pictures and their relation to language. There are a couple of chapters in the book that try to do that by considering prehistoric, or protohistoric, practices. So it's more about contemporary understanding than about what might have happened in the cultures Schmandt-Besserat studied.

This paper will concern itself with the ancient and widespread practice of finger amputation and the symbolic significance of handprints. The historical record provides sufficient evidence to show that fingers or finger joints were originally associated with relatives and that they were amputated in mourning for the deceased. I will discuss and comment on some of the evidence as well as a number of related ideas and practices which have an important bearing on the subject.

Due to the 20th century mathematical and scientific developments of Georg Cantor, Max Karl Planck, Albert Einstein, and Werner Heisenberg, concepts once relegated to obscurity, such as irrationality, infinity, insolvability, and chaos, were brought to mainstream attention, ultimately changing the course of technological and scientific development into the 21st century. Before these seminal thinkers, concepts like numerical irrationality and infinity were considered by many to be worthless if not amoral; such attitudes can be found persisting back to the ancient Greeks under the Pythagoreans. Interestingly, the aesthetic of irrationality follows a similar historical trajectory, mostly finding relegation in peripheral movements and specific artists before the 20th century. However, the 20th century has seen the greatest and longest persisting resurgence in mathematically irrational thought within the arts. This paper compares the visual and musical experiments in irrationality, incommensurability, and infinity in the works of MC Escher and Conlon Nancarrow during the early and mid-twentieth century, showing a correlation between contemporary mathematical and physical innovations and specific aesthetic pursuits in art and music.

A treatment in English can be found in my: Oberwolfach in the French Occupation Zone: 1945 to early 1950s, in: Revue d'histoire des mathématiques 21(2020), 121-172, pp. 127-131
https://www.academia.edu/45585407/Oberwolfach_in_the_French_Occupation_Zone_1945_to_early_1950s_in_Revue_dhistoire_des_math%C3%A9matiques_21_2020_121_172

A partir de alguns textos escritos pelos cosmógrafos do Reino, entre os séculos XVI e XVIII, em Portugal, pretendemos evidenciar as bases técnicas e científicas que possibilitaram a expansão marítima europeia destacando as relações entre elas e a construção e o uso dos instrumentos matemáticos utilizados na navegação oceânica. Nosso objetivo, além de sublinhar os conteúdos científicos relacionados aos artefatos utilizados para as medições, é também verificar como a ciência e o conhecimento adquirem significado prático, estratégico e simbólico no contexto da expansão ultramarina de Portugal.

The fold introduces a double movement: on the one hand, it introduces the “becoming” of an (often marginal) mathematical object inside mathematics, which is, at the same time, marginalized in mathematical discourse; on the other hand, it also represents a certain way in which mathematics is in a process of becoming. What all of this indicates is that the structure of mathematics is constantly in a state of becoming. In the introduction of the book "A History of Folding in Mathematics", I followed Derrida and his logic of supplementarity, but there is another philosopher who also thought of the fold as an endless process of becoming—Gilles Deleuze, in his book "The Fold: Leibniz and the Baroque". This appendix follows the unmathematical figure of the fold of Deleuze and Leibniz.

Le fonds Élie Cartan résulte d’un don fait en 2009 à l’Académie des Sciences.
Nous présentons, ici, une partie de ce fonds, les cahiers de notes de Cartan qui contiennent des recherches de Cartan, des notes de lecture (notamment sur les thèses que Cartan a dirigées ou pour lesquelles il a été rapporteur), des préparations de cours, des notes administratives et personnelles, et des notes rédigées par les étudiants de Cartan lors de ses cours. D’une part, nous donnons une présentation générale de ces quarante-cinq cahiers accompagnant et complétant le catalogue analytique commenté et la base de données Élie Cartan Papers. Nous donnons des éléments quantitatifs permettant de mieux discerner le contenu et l’organisation des cahiers de Cartan et d’en dresser une première cartographie. D’autre part, nous étudions plus précisément certains exemples, afin de mettre en lumière la richesse de ce fonds

--

Élie Cartan's archive was donated in 2009 to the Académie des Sciences. I present, here, a part of the archive, namely Cartan's notebooks. These notebooks contain notes from Cartan's research, reading notes (among which notes on the doctoral dissertations read or advised by Cartan), notes in preparation for Cartan's lectures, and notes written by Cartan's students during his lectures. First, we give a general presentation of these forty-five notebooks, which go with and complete the analytical commented catalogue and database Élie Cartan Papers. We give quantitative elements allowing for a better grasp of the content and organisation of the archive, and to give a first cartography of it. Secondly, I study in more details certain specific points, so as to shed light on the richness of this archive.

Tento popularizačný článok o matematických hádankách z včasného stredoveku vyšiel na stránkach HistoryWeb.sk 19.10. 2016. Článok nájdete na: http://historyweb.dennikn.sk/clanky/detail/tri-matematicke-hadanky-z-vcasneho-stredoveku#.WAkcjSQkw2z

A popularizing article about the early medieval mathematical puzzles preserved in Propositiones ad acuendos iuvenos (in Slovak)

In 1915, Einstein introduced a radical and revolutionary new idea into science and human thought: Gravity was not a force. The observed gravitational attraction between two or more material bodies was the product of the material bodies following the shortest geodesic paths across the curvature of a space-time continuum. The amount or degree of curvature at any location depended on the local presence of matter. As radical as this idea was, the theory became a known and accepted part of physics within just a few years, which was quite rapid relative to the overall scale of changes in science. It was accepted so rapidly because a few scientists had been attempting to find a connection between non-Euclidean geometry (especially Riemannian) and physics for several decades as well as how the geometry of space (or space-time) was related to ultimate nature and structure of matter. However, Einstein’s theory also came as a surprise because no one had detected any specific flaw in Newtonian gravity theory and other researchers working on the problem of matter had expected matter to ultimately be electric (electromagnetic) in nature. Within this historical context, several scientists and mathematicians immediately noticed a problem with the new theory and took advantage of that problem to extend physics further than Einstein originally anticipated. Although a tensor placed at a point in space was used to represent the curvature through that point, the mathematics and physics at the point itself was not completely specified or explained. This flaw allowed others to develop non-Riemannian and pseudo-Riemannian geometries (also called tangent spaces) to explain geometry specifically at or in the individual points of space-time. Since they were independent of the Riemannian metric geometry of extended (metric) space, these new geometries could be used to explain the electromagnetic nature of matter without disturbing the metric or gravitational explanation given by general relativity. This not only began the search for a unified field theory, but it also demonstrated quite clearly that the attempted rigorization of mathematics that had occurred over the past two centuries was inadequate and mathematics was still intimately linked to physics. The story of how these new geometries developed is both interesting and intriguing, yet so much more. The development of physics over the next few decades cannot be completely understood without taking these events into account. This lost history explains fully both Einstein’s search for a unified field theory (with an emphasis on a metric or extension geometry) as well as the later development of quantum mechanics based on the uncertainty principle and how that relates to quantum field theories (using a point non-geometrical space or space-time), thus changing the whole historical context of modern physics.

ABSTRACT: The 3rd BCE Stoic logician “Chrysippus says that the number of conjunctions constructible from ten propositions exceeds one million. Hipparchus refuted this, demonstrating that the affirmative encompasses 103,049 conjunctions and the negative 310,95[4].” After laying dormant for over 2000 years, the numbers in this Plutarch passage were recently identified as the 10th (and a derivative of the 11th) Schröder number, and F.Acerbi showed how the 2nd BCE astronomer Hipparchus could have calculated them. What remained unexplained is why Hipparchus’ logic differed from Stoic logic, and consequently, whether Hipparchus actually refuted Chrysippus. This paper closes these explanatory gaps. (1) I reconstruct Hipparchus’ notions of conjunction and negation, and show how they differ from Stoic logic. (2) Based on evidence from Stoic logic, I reconstruct Chrysippus’ calculations, thereby (a) showing that Chrysippus’ claim of over a million conjunctions was correct; and (b) shedding new light on Stoic logic and – possibly – on 3rd century BCE combinatorics. (3) Using evidence about the developments in logic from the 3rd to the 2nd centuries, including the amalgamation of Peripatetic and Stoic theories, I explain why Hipparchus, in his calculations, used the logical notions he did, and why he may have thought they were Stoic.

This paper reconstructs the relationship between the now, motion, and number in Aristotle to clarify the nature of the now, and, thereby, the relationship between motion and time. Although it is clear that for Aristotle motion, and, more generally, change, are prior to time, the nature of this priority is not clear. But if time is the number of motion, then the priority of motion can be grasped by examining his theory of number. This paper aims to show that, just as numbers are generated by the soul, time is not presupposed by motion, but emerges through the soul’s articulation of motion. Time is co-constituted by the soul and motion. The now is the key to understanding both the contribution of motion and soul to the being of time. The now is part of the soul’s articulation of motion, and sets the stage for an act that distinguishes a unit from its underlying motion. The now, then, sets up an abstraction by which the soul generates the temporal number from motion. Reconstructing this account of abstraction allows us to formulate more strongly Aristotle’s claim to the ontological dependence of time on motion. The paper then gives a systematic overview of the relationships between the now and number in order to address the question of whether the now might be extended. It closes with an examination of the possibility that motion depends on time, and how universal time is possible.

As Ahmed Djebbar has pointed out, 11th-century and earlier al-Andalus produced a “solid
research tradition in arithmetic”. So far, no continuation of this tradition has been known,
but analysis of three sections of two Latin works suggest that they borrow material that
can hardly come from elsewhere:
1. The Liber mahameleth, likely to be a more or less free translation made by Gundisalvi
or somebody close to him of an Arabic original presenting “mu a¯mala¯t mathematics vom
höheren Standpunkt aus” contains systematic variations, for instance of proportions :: p
g
P
G
(claimed to deal with prices and goods), where the givens may be sums, differences,
products, sums or differences of square roots, etc., solved sometimes by means of algebra,
sometimes with appeals to Elements II.5–6, often after reduction by means of proportion
techniques.
2. A passage in Chapter 12 of the Liber abbaci first presents the simple version of the
recreational problem about the “unknown heritage” (likely to be of late Ancient or
Byzantine origin): a father leaves to his first son 1 monetary unit and 1/n of what remains,
to the second 2 units and 1/n of what remains, etc.; in the end, all get the same, and nothing
remains. Next it goes on with complicated cases where the arithmetical series is not
proportional to 1 – 2 – 3 ..., and the fraction is not an aliquot part. Fibonacci gives an
algebraic solution to one variant and also general formulae for all variants – but these
do not come from his algebra, and he thus cannot have derived them himself. A complete
survey of occurrences once again points to al-Andalus.
3. Chapter 15 Section 1 of Fibonacci’s Liber abbaci mainly deals with the ancient theory
of means though not telling so. If M is one such mean between A and B, it is shown
systematically how each of these three numbers can be found if the other two are given –
once more by means of algebra, Elements II.5–6, and proportion techniques. The lettering
shows that Fibonacci uses an Arabic or Greek source, but no known Arabic or Greek work
contains anything similar. However, the structural affinity suggests inspiration from the
same environment as produced the Liber mahameleth.
So, this seems to be a non-narrative, a story that was not revealed by the participants,
and was not discovered by historians so far.

Вторая часть статьи посвящена младшему поколению математиков Петербургской стороны: Якову Викторовичу Успенскому, Герману Мюнцу, Владимиру Ивановичу Смирнову, Григорию Михайловичу Фихтенгольцу, Якову Давидовичу Тамаркину, Сергею Львовичу Соболеву, Георгу Лоренцу, Леониду Витальевичу Канторовичу, Сергею Михайловичу Лозинскому, составивших славу русской математики в России и за рубежом. В статье содержатся адреса и фотографии домов, хранящих память о своих жильцах-математиках.
Ключевые слова: История математики, Петербургская сторона, Я.В. Успенский, Г. Мюнц, А.Я. Шохат, В.И. Смирнов, А.Ф. Гаврилов, Г.М. Фихтенгольц, А.С. Безикович, Я.С. Безикович, Н.С. Кошляков, С.Л. Соболев, Г. Лоренц. The second part of the article is devoted to the next generation of mathematicians from the Petersburg side, who generously enriched mathematics of the 20th century with their results. The article contains addresses and photographs of houses that keep the memory of their mathematicians.
Key words. History of mathematics, Petersburg side, Ya.V. Uspensky, G. Müntz, A. Ya. Shokhat, V.I. Smirnov, A.F. Gavrilov, G.M. Fikhtengolts, A.S. Besikovich, Ya.S. Besikovich, N.S. Koshlyakov, S.L. Sobolev, G. Lorentz.

In 1940 the mathematician Walther Lietzmann published his book ”Frühgeschichte der Geometrie auf germanischem Boden” (Early history of geometry on Germanic ground), in which he investigated the geometric characteristics of prehistoric finds and features. Lietzmann tried to
make use of the characteristics of the ornament for typology, and especially for characterising a „Northern" or „Germanic Kulturkreis” (cultural feld). To the concept of „Northern geometry”, Lietzmann ascribed a „dynamic character”, he missed in Greek geometry. Lietzmann included ”völkische” (racial) and National Socialist ideology in his texts.

1940 veröfentlichte der Mathematikdidaktiker Walther Lietzmann sein Buch „Frühgeschichte der Geometrie auf germanischem Boden“, in welchem er die geometrischen Eigenschaften prähistorischer Funde und Befunde untersuchte. Lietzmann versuchte, die geometrischen Eigenschafen des Ornaments für die Typologie und insbesondere zur Kennzeichnung eines „nordischen“ bzw. „germanischen Kulturkreises“ nutzbar zu machen. Dieser „nordischen Geometrie“ schrieb Lietzmann einen „dynamischen Charakter“ zu, den er in der griechischen Geometrie vermisste. Lietzmann transportierte in seinen Texten völkische und nationalsozialistische Ideologie.

Recorrido crítico por la figura y la obra de Hipócrates de Quíos (fl. mediados del s. V a.C.), matemático (geómetra) y astrónomo griego "para-pitagórico", de cuya obra sólo nos han llegado testimonios indirectos. Esquema del trabajo: Bibliografía general. Biografía y cronología. Filiación filosófica. Obra: El método de análisis. El problema de la duplicación del cubo. La cuadratura de las lúnulas. Los elementos de geometría. La astronomía.

In this paper, I study Richard Dedekind and Heinrich Weber's 1882 Theorie der algebraischen Funktionen einer Veränderlichen, with a focus on the inherently arithmetical aspects of their work. I show that their paper provides an arithmetical rewriting of Riemannian function theory, i.e. a rewriting built on elementary arithmetical notions such as divisibility. I start with contextual elements concerning what is “arithmetical”, to put Dedekind and Weber's works into perspective from that viewpoint. Then, through a detailed analysis of the 1882 paper and using elements of their correspondence, I suggest that Dedekind and Weber deploy a strategy of rewriting parts of mathematics using arithmetic, and that this strategy is essentially related to Dedekind's specific conception of numbers and arithmetic as intrinsically linked to the human mind.