History of Mathematics

Thomas Hardy’s Wessex novels move in circles: characters retrace their steps, and history repeats itself. The limited geography registers and reflects both movements. At the same time, the novels repeatedly figure choice as binary—limited to two seemingly equivalent alternatives. This essay argues that these apparently disparate features of Hardy’s plots are, in fact, structurally linked. Circular movement and binary choice echo through contemporary cultural artifacts, ranging from popular board games to mathematical and biological theories. Drawing on developments in the mathematical study of probability—the drunkard’s walk and the Markov chain—this essay claims that Hardy’s binary choices generate his characters’ roundabout geographic trajectories, transforming Victorian models for human motion and choice into a formal principle for the novel.

Many have argued that Plato’s intermediates are not independent entities. Rather, they exemplify the incapacity of discursive thought (διάνοια) for cognizing Forms. But just what does this incapacity
consist in? Any successful answer will require going beyond the intermediates themselves to another aspect of Plato’s mathematical
thought - his attribution of a quasi-numerical structure to Forms (the ‘eidetic numbers’). For our purposes, the most penetrating account
of eidetic numbers is Jacob Klein’s, who saw clearly that eidetic numbers are part of Plato’s inquiry into the ontological basis for all counting:
the existence of a plurality of formal elements, distinct yet combinable into internally articulate unities. However, Klein’s study of the Sophist
reveals such articulate unities as imperfectly countable and therefore opaque to διάνοια. Only this opacity, I argue, successfully explains
the relationship of intermediates to Forms.

A selection of some of Novalis’s most important reflections on mathematics from the years 1798-1800. Introduced and translated into English by David W. Wood.

In: Symphilosophie: International Journal of Philosophical Romanticism 3 (2021): 273-290
https://symphilosophie.com/issue-3-2021/2/

This short article is the continuation of the “A Sumerian Numerical Palimpsest” where it is tried to find if the nomenclature of Sumerian numerals appears to be the result of the use of hand(s) for counting. That Sumerian counting in multiples of sixty has to begin with the “making” the “first” sixty is obvious, but is seldom stated. For the people who count in concrete and in any mode, a natural model - a counting device, like hands – has to be recognized so that a visual one-to-one correspondence between the individual elements of the device and the counted items became obvious. And that should apply to sixty for which I am suggesting a healthy robust spike of the six rowed barley as a natural model of a morphology that extremely closely reflects the Sumerian 60 based numerals. An idealized six rowed barley spike, še ḡeš, an intellectual achievement, which preliterate Sumerians were quite capable of, may had, or rather, could been the reckoning device on which the counting in multiples of sixty was established and prospered.

David W. Wood. Book Review of Rudolf Steiner, The Fourth Dimension. Sacred Geometry, Alchemy, and Mathematics (trans. Catherine E. Creeger, Anthroposophic Press, 2001).
Review originally published in the Newsletter of the Science Group of the Anthroposophical Society in Great Britain, September 2002, pp. 2-3: https://sciencegroup.org.uk/wp-content/uploads/2021/11/sgnl902.pdf

The article discusses the handwritten revisions and drawn additions by Albrecht Dürer in his own copy of the treatise on geometry, Underweysung der Messung (1525). Situating Dürer’s interest in mathematics within the scholarly milieu of Renaissance Nuremberg, the article addresses the shifts in style and content that Dürer proscribes and offers new perspectives on the artist’s relation to his own late work. The article concludes by displaying several drawings bound into the edition by Dürer that illustrate variations on perspectival apparatuses. These lesser known drawings illuminate the evolution of Dürer’s conceptualization of the dynamics between artist and subject, as well as the tools used to facilitate this interaction.

Nel secolo dei Lumi spicca il nome di Maria Gaetana Agnesi nelle discussioni sulla validità delle teorie newtoniane, sul calcolo infinitesimale di Leibniz e Newton, sull'istruzione femminile e sul ruolo della donna nell'indagine scientifica. Nel confronto a distanza con Algarotti, la scienziata milanese promuove un newtonianesimo, delle donne e non "per le donne", dai tratti marcatamente neostoici. La pubblicazione delle Instituzioni analitiche, che le procurò fama e prestigio in Europa, segnò poi una svolta. Ammirata da Goldoni e definita immortale da Pietro Verri, Agnesi volle costruire una propria identità di scienziata e di donna autonoma nella dimensione etica dell'impegno sociale e dell'esperienza di fede, influenzata dalla spiritualità teatina e dalla religiosità di Muratori.

In copertina: Filippo Morghen (1730-1807), Ritratto di Maria Gaetana Agnesi, incisione [realizzata da Francesco Pisante (1804-1889)], in «L'Omnibus pittoresco», Napoli 30 giugno 1842, anno quinto, n. 11 (collezione privata).
Aracne CADST 39 - 18,00 euro. Il "cannocchIale" dello storIco: mItI e IdeologIe / 39 Collana fondata da Achille Olivieri e diretta da Daniele Santarelli.

Giordano Bruno and the Geometry of Language brings to the fore a sixteenth-century philosopher's role in early modern Europe as a bridge between science and literature, or more specifically, between the spatial paradigm of geometry and that of language. Arielle Saiber examines how, to invite what Bruno believed to be an infinite universe–its qualities and vicissitudes–into the world of language, Bruno forged a system of 'figurative' vocabularies: number, form, space, and word. This verbal and symbolic system in which geometric figures are seen to underlie rhetorical figures, is what Saiber calls 'geometric rhetoric.'

Through analysis of Bruno's writings, Saiber shows how Bruno's writing necessitates a crafting of space, and is, in essence, a lexicon of spatial concepts. This study constitutes an original contribution both to scholarship on Bruno and to the fields of early modern scientific and literary studies. It also addresses the broader question of what role geometry has in the formation of any language and literature of any place and time.

Colin Maclaurin (1698-1746) was appointed professor of mathematics at the University of Edinburgh at the recommendation of Isaac Newton. He died at the early age of 48 as a result of his exertions resisting the Jacobite rebellion in Scotland. He left a body of unpublished manuscripts, now stored in the archives of the University of Edinburgh. I spent an invited sabbatical leave in Edinburgh editing some of these manuscripts, which includes this Journal he kept detailing his attempts to defend Edinburgh against the Jacobites.

We investigate the treatment of fractions in Russell’s 1919 classic Introduction to Mathematical Philosophy. In contrast to rational numbers, every fraction has an integral numerator and a non-zero integral denominator, but usage varies on exactly which integers are involved (. Our interest was first drawn to the topic by the following surprising result—paraphrasing Russell’s page 66].

Russell 1919’s Fraction-Separator Theorem: the fraction whose numerator is the sum of the numerators of two unequal given fractions and whose denominator is the sum of the denominators is between the two given fractions—excluding cases when the sum of the denominators equals zero.

Although Russell gives none, proof is obtainable from page 270 of De Morgan 1831.
REFERENCE LINKS

Russell 1919’s fraction-separator theorem. Bulletin of Symbolic Logic. 24 (2018) 381.
https://www.academia.edu/s/4f60c070ca/russell-1919s-fraction-separator-theorem?source=link
https://www.academia.edu/34438558/2_Russell_1919_s_fraction-separator_theorem_090117.pdf
Semiotic confusions in Russell 1919. Bulletin of Symbolic Logic. 24 (2018) 382–3.
(Co-author: Kevin Tracy) https://www.academia.edu/34195085/Semiotic_confusions_in_Russell_1919.AC
https://www.academia.edu/34195085/Semiotic_confusions_in_Russell_1919.AC
HOW CAN THIS BE IMPROVED?

God is a mathematician and He loves doing it by the numbers. Your future is tomorrow and your past we can not find and today you learn; so your future will be secure in the hands of an amazing God, but all this would not be without numbers!

No creation, no day or night, no sun, moon, or stars; not even you would be reading this today because you would not exist!

Numbers are something; a creation of God and not man. Numbers are more than something created to count to see how much of something or how little of something we have.

Numbers are not something that just exist in our minds; they are a reality of life, of spiritual life and even the scriptures we read and without them even God's written word would not exist.

Numbers are everyone's reality. True you don't see them but they are not arbitrary products of one person's imagination. They are as real as you and

Este volume, resultado de um trabalho desenvolvido juntamente com o grupo de estudos em pesquisa HEEMa (História e Epistemologia na Educação Matemática/PUCSP), aborda a a relação entre música e matemática, privilegiando os contextos de elaboração, transformação e transmissão (bem como de apropriação) dos conhecimentos compartilhados pela música e pela matemática. Esperamos que a leitura deste material, juntamente com outras que versam e discorrem sobre o mesmo assunto, propiciem novas discussões e reflexões que contribuam para introduzir novas abordagens de investigação na articulação entre história e ensino de ciência e de matemática.

An investigative attempt at explaining Euclid's first definition in the "Elements," and answering the question "what is a point"? I assert that a point is an indefinitely small quantity of space, and thus can "have no part" because it is incapable of being quantified in order to be divided into parts.

Vers la fin du 19e siècle, les mathématiciens et ingénieurs qui travaillent pour l'artillerie n'ont plus d'espoir de trouver une loi élémentaire de la résistance de l'air, ni de parvenir à une intégration analytique exacte des équations différentielles de la balistique. La vitesse et la portée croissantes des projectiles, l'apparition de nouvelles situations comme le tir à longue portée sur des objectifs invisibles, le tir à angle élevé à travers des couches d'air de densité très variable ou le tir contre les avions, amènent à remettre en question les anciennes théories et à s'orienter de plus en plus vers des approches numériques et graphiques combinées à des expérimentations sur le terrain. Dans ce chapitre, nous nous intéresserons plus spécifiquement aux méthodes graphiques (constructions géométriques, nomogrammes, instruments graphomécaniques) pratiquées à cette époque par les artilleurs, ainsi qu'aux institutions et aux milieux professionnels qui ont été mobilisés pour les concevoir, notamment pendant la Première Guerre mondiale.

The first half of the 20th century in the history of Russian mathematics is striking with a combination of dramaticism, sometimes a tragedy, and outstanding achievements. The paper is devoted to St. Petersburg-Leningrad Mathematical School. It is based on a chapter in the multi-author monograph [1].

This is the ultimate proof that the Prime Numbers are not Random Numbers as famous Mathematicians believe and claim publicly through presentations you can find on U Tube
Any pupil around the world and a wide public will understand the first 2 pages of the paper below.  Feel free to ask questions by emailing me constantine.adraktas@mit-partners.eu

Abstract:  The Hakluyt Society (founded in 1846) is the premier English historical society when it comes to the Age of Discovery.  Its 2 volume Hakluyt Handbook is a comprehensive guide, first proposed by Dr. R. A. Skelton and Professor D. B. Quinn, consisting of materials and the sources of Richard Hakluyt’s nearly 300 volumes of primary records regarding the Age of Discovery voyages, travels and other geographical material.  In the early 1960’s the Council of the Society set up a committee to work on the project and began enlisting contributors for the guide. After years of hard work by its members, The Hakluyt Handbook was completed and finally published in 1974. The Handbook Preface by C. F. Beckingham, Hakluyt Society President, stated that there had never been a comprehensive study of Rickard Hakluyt’s works along with the background of his sources.  Beckingham’s hope was that the correlation between overlapping disciplines of the handbook be used by various scholars would give rise to further study.  This is one such study.

"Just as Grant Wood portrayed Parson Weems pulling the curtain back on the life of George Washington, we shall illuminate Washington's mathematical education. We are blessed with 179 pages of handwritten manuscript in Washington's youthful hand and while this material has frequently been mentioned by scholars, it has never been analyzed; we shall present an abundance of detail. Little is known about Washington's youth, so these papers provide a way of learning about his education.

The individuals and organizations that have controlled these papers have organized and reorganized them into disorder. Is it reasonable for a thirteen-year-old --- living in plantation Virginia where there were few schools --- to begin his mathematical education with the study of formal geometry? Could he have learned surveying before studying arithmetic and trigonometry? Using physical evidence, handwriting analysis, and mathematical context, we shall present our conjectured order of the pages of the manuscript.

This is a case study in mid-eighteenth century mathematical education in the American Colonies. We shall contrast the surprising depth of his theoretical education --- including logarithms and trigonometry --- with his practical use of mathematics as a field surveyor.

After serving two terms as President, Washington took pains to preserve his papers for he believed that someday they might be "of interest.'' You will be the judge of how interesting these papers are."

We aimed to describe how history of mathematics was framed in teacher education programs to be used by pre-service elementary mathematics teachers in their future teaching within the context of a specific undergraduate course named “History of Mathematics”. Data was collected by the observation of the course setting, interviews with the pre-service elementary mathematics teachers enrolled in the course and the course instructor, and the course materials in the fall semester of 2011-2012 academic year. Considering Jankvist’s (2009) classification, findings revealed that the course fostered the pre-service teachers’ history as a goal ideas rather than history as a tool ideas. The illumination approaches were seemed to be the only way of employing history of mathematics in their teaching. We concluded with discussions on the effectiveness of the course and implications for improving such “History of Mathematics” courses.

Mit Laugiers Urhütte wurden die zeitgenössischen Entwurfsmethoden bezüglich ihres Anspruchs zur totalen Dynamisierung des architektonischen Entwurfsprozesses kritisch beurteilt. Wie ist diese Dynamik als rein rationalisierte Form geometrischer Idealgestalten zu denken, ohne dass auch ein hier klassischer " Analogieschluss " vorliegt? Diese Frage richtet sich auf den Ursprung des Analogiebegriffs in der Architektur, der in Vitruvs Anthropomorphismus wurzelt. Der historische Rückblick auf Vitruvs Analogiebegriff soll zunächst seine genaue Verwendung aufzeigen. Auf dieser Grundlage wird geprüft, warum der Anthropomorphismus zum Streitobjekt moderner Theorien wurde. Schlussendlich steht aber zur Frage, ob die zeitgenössischen Entwurfsmethoden ihren Anspruch, das alte anthropomorphe System und damit die " Metaphysik der Architektur " aufzuheben, eingehalten haben oder vielmehr einen Analogieschluss verbergen, um das eigene System aufrecht zu halten.

The doxography for Anaximander’s account of the rings of the sky gives proportions for them that are discrepant. So a widely accepted hypothesis proposes that, since the circles of the celestial bodies are compared to wheels, we should add the thickness of their ‘rims’ to the measurements for the celestial rings. This paper proposes an entirely different hypothesis which avoids this awkward expedient by suggesting that there is a ‘geometrical’ reading of the numerical data. The discrepancies can then be explained by there being two different methods for calculating the circumference of the circles.

Review of "Concinnitas" (2014), a portfolio of ten aquatints by noted mathematicians and physicists including Michael Atiyah, Enrico Bombieri, Stephen Smale, Murray Gell-Mann, David Mumford, Steven Weinberg, Simon Donaldson, Richard Karp, Peter Lax and Freeman Dyson. "Concinnitas" was printed by Harlan & Weaver, New York, and published by Parasol Press, Portland, OR, in collaboration with the Yale University Art Gallery, New Haven, CT, and Bernard Jacobson Gallery, London.

Citation: Julie Warchol, "Truth, Beauty and Mathematics" Art in Print 6:2 (July-August 2016): 31-33.

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 paper reviews contemporary interpretations of classical Greek geometry, and offers a Deleuzian, post structural alternative. We emphasize the embodied aspect of Greek mathematical practice, and indicate how it conflicts with ideal linear-modular accounts. Our main theoretic tools are the notions of 'catastrophe' and 'haptic eye' as reconstructed by Deleuze in his analysis of the artistic practices of painter Francis Bacon.

The mean value theorem IS the most important theorem in calculus and it should be rightly called the fundamental theorem of calculus because the theorem known by the same name is derived in one step from the mean value theorem.

The mvt is about an an arithmetic mean or a level term that is used to find slopes, areas, volumes, etc.  The mainstream statement of this theorem is unremarkable.

https://www.academia.edu/32016652/How_did_my_idea_of_positional_derivative_come_about

My New Calculus is based only on well-formed concepts that exclude infinity, infinitesimals and the circular rot of limit theory which is not required at all in calculus.

https://www.academia.edu/45154026/Teaching_the_fundamental_theorem_of_calculus

Read an introduction to my New Calculus here:

https://www.academia.edu/41616655/An_Introduction_to_the_Single_Variable_New_Calculus

The LinkedIn links are broken but those same articles can be found here on Academia.edu.