New Progress in Arithmetic Geometry

1990 ◽  
pp. 339-365
Author(s):  
Kenneth Ireland ◽  
Michael Rosen
Keyword(s):  
Arithmetic Geometry

scholarly journals A Joint Assessment of Reasoning about General Statements in Mathematics and Biology

2021 ◽  
Vol 14 (4) ◽  
pp. 270-287
Author(s):  
Libuše Samková ◽  
Lukáš Rokos ◽  
Lukáš Vízek
Keyword(s):  
Empirical Study ◽  
Formative Assessment ◽  
Deductive Reasoning ◽  
Professional Preparation ◽  
Arithmetic Geometry ◽  
Primary School Teachers ◽  
School Subjects ◽  
Conversion Table ◽  
And Mathematics ◽  
Joint Coding

This contribution belongs to a larger empirical study that focuses on issues related to the implementation of inquiry-based learning and formative assessment in science and mathematics education, while it also refers to the issue of STEM education. Here, we discuss the two topics from the perspective of professional preparation of primary school teachers. We employ an educational tool called Concept Cartoons and perceive it as a common diagnostic tool for investigating modes of reasoning about general statements in arithmetic, geometry and biology. The presented qualitative exploratory empirical study maps and codes various kinds of reasoning that can be identified with the tool and investigates possibilities of a joint coding procedure. As a result, it provides a conversion table between various modes of reasoning in the three subject domains. The arisen code categories cover the field of generic examples, including the initial stages so that they can be used for scaffolding the process of learning the foundations of deductive reasoning. The joint approach to reasoning in mathematics and biology shows how argumentation and formative assessment can be understood equally and developed simultaneously in both school subjects. It helps us to see how the two school subjects can be integrated didactically.

scholarly journals Arithmetic gauge theory: A brief introduction

2018 ◽  
Vol 33 (29) ◽  
pp. 1830012 ◽  
Author(s):  
Minhyong Kim
Keyword(s):  
Moduli Spaces ◽  
Automorphic Forms ◽  
Galois Representations ◽  
Arithmetic Geometry ◽  
Quantum Field ◽  
Principal Bundles ◽  
Langlands Program ◽  
Effective Version ◽  
Diophantine Geometry ◽  
Higher Genus

Much of arithmetic geometry is concerned with the study of principal bundles. They occur prominently in the arithmetic of elliptic curves and, more recently, in the study of the Diophantine geometry of curves of higher genus. In particular, the geometry of moduli spaces of principal bundles holds the key to an effective version of Faltings’ theorem on finiteness of rational points on curves of genus at least 2. The study of arithmetic principal bundles includes the study of Galois representations, the structures linking motives to automorphic forms according to the Langlands program. In this paper, we give a brief introduction to the arithmetic geometry of principal bundles with emphasis on some elementary analogies between arithmetic moduli spaces and the constructions of quantum field theory.

Music and the Origins of Ancient Science

2014 ◽  
Author(s):  
Peter Pesic
Keyword(s):  
Higher Education ◽  
Eighteenth Century ◽  
Seventeenth Century ◽  
Natural Philosophy ◽  
Modern Science ◽  
Arithmetic Geometry ◽  
Web Browsers ◽  
Ancient Greek ◽  
Crucial Element ◽  
Ancient Science

Music entered deeply into the making of modern science because it was a crucial element of ancient natural philosophy, through which it thereafter remained active well into the formation of the “new philosophy” during the seventeenth century. The Pythagorean connection between music, numbers, and the sensual world remained potent in the quadrivium, the four-fold study of arithmetic, geometry, astronomy, and music that was the centerpiece of higher education until about the eighteenth century. This chapter surveys the ongoing connection between music and its sister sciences in the quadrivium from Plato and the Pythagoreans to Nicomachus and Boethius. The mythical story of Pythagoras in the blacksmith shop arguably represents the earliest recorded experiment, in the later sense of that word. Ancient Greek distinctions between number and magnitude were crucial elements in the unfolding interaction between arithmetic, geometry, and music. Throughout the book where various sound examples are referenced, please see http://mitpress.mit.edu/musicandmodernscience (please note that the sound examples should be viewed in Chrome or Safari Web browsers).

Higher Education

2019 ◽  
pp. 159-180
Author(s):  
Nicholas D. Smith
Keyword(s):  
Higher Education ◽  
Empirical Study ◽  
Arithmetic Geometry ◽  
Mathematical Study ◽  
Mathematical Objects ◽  
The Future ◽  
Relationship Of ◽  
The Relationship

Explains the curricula included in the proposed higher education of the future rulers: arithmetic, geometry, stereometry, astronomy, harmony, and dialectic. Once again addresses questions of what Plato thought about mathematical objects and how he talks about these in Book VII of the Republic. Considers debates about just how and why Plato assigned such an important role to mathematical studies in the training of the power of knowledge for the future rulers. Considers the relationship of “formal” as opposed to “empirical” study, particularly in Plato’s requirement of astronomy as the penultimate mathematical study. Discusses what we can discern about Plato’s conception of dialectic and how that fits as the final element in the “highest studies” that prepare the future rulers to begin to engage in political rule. Shows how in spite of these studies culminating in the highest cognitive achievements, they must be followed by fifteen years of political apprenticeship, and why only after this training can Plato’s best students become philosopher rulers.

2. Schoolrooms, science, female intuition

2019 ◽  
pp. 22-35
Author(s):  
David Wallace
Keyword(s):  
Arithmetic Geometry

‘Schoolrooms, science, female intuition’ describes the teaching in the medieval schoolroom and explains that few women at the time of Chaucer learned to read and write. Schoolboys entered the educational scheme at the trivium, which was dedicated to grammar, rhetoric, and logic. Beyond this lay the higher arts of the quadrivium (arithmetic, geometry, astronomy, and music). Chaucer’s engagement with these levels is shown in his works the Boece and Treatise on the Astrolabe. In The Wife of Bath’s Tale and Man of Law’s Tale, Chaucer shows how he is sympathetic to women, depends upon their patronage and protection, imitates their voices, and tries to please them.

Medieval Latin Arts of Poetry and Prose

2019 ◽  
Author(s):  
Douglas Kelly
Keyword(s):  
Middle Ages ◽  
Arithmetic Geometry ◽  
Chronological Order ◽  
13Th Century ◽  
Elementary Composition ◽  
Medieval Latin ◽  
Late Middle Ages ◽  
Medieval Poetics ◽  
The University ◽  
Level Of Instruction

Modern scholarship on the medieval Latin arts of poetry and prose has focused on a number of treatises written in the 12th and 13th centuries: Matthew of Vendôme’s Ars versificatoria; Geoffrey of Vinsauf’s Poetria nova, Documentum, and Summa de coloribus; Gervase of Melkley’s Ars versificaria or poetica; John of Garland’s Parisiana Poetria; and Eberhard the German’s Laborintus. Other documents have received attention as well, notably commentaries and glosses. The art of composition in both verse and prose also evolved as new conceptions of the art emerged. In the 13th century, Latin translations and commentaries on Aristotle’s Poetics led to revisions of the Horatian art in the 14th century; treatises that reflect this development begin with the anonymous Long Documentum, renamed Tria sunt, and Mathias of Linköping’s Poetria, based on instruction Mathias received while studying at the University of Paris. The traditional conception of the art of poetry was derived from rhetorical treatises attributed to Cicero, notably the De inventione and the Rhetorica ad Herennium. The medieval treatises adapted a traditional order of parts in rhetoric: topical invention, disposition based on natural chronological order or artificial rearrangement of the chronological order, amplification and abbreviation using figures and tropes common in ornamentation, and eventually Aristotelian notions of imagination as a poetic faculty. Inclusion of these parts indicates the scope and level of instruction in the treatises. Accordingly, the study and practice of poetic composition in classrooms progressed from elementary composition and study to imitation of exemplary masterpieces. Such instruction fit well into the stages in medieval pedagogy from grammar, rhetoric, and logic on to arithmetic, geometry, astronomy, and music, and beyond to philosophy and theology. The scope of the art on these virtually graded levels of instruction led to study, interpretation, imitation, and, ultimately, emulation of perceived ancient and medieval masterpieces like Virgil’s Aeneid, Horace’s lyrics, Bernardus Silvestris’s Cosmographia, and Alan of Lille’s Anticlaudianus and De planctu Naturae, among others. Introductions to specific works (accessus ad auctores) include model works that exemplify the art’s evolution from Antiquity to the Late Middle Ages. Classbooks and other anthologies collected poems and passages for study and imitation on the student level. Finally, the Latin art found its way into some treatises written for vernacular languages. These diverse documents—commentaries, model works, accessus, classbooks and anthologies, authorial statements inserted into their own writings, vernacular treatises, and other documents—enhance our understanding of medieval poetics.

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