History Of Algebra Research Articles

Dialectics, Mathematics and the Cavaillès-Hyppolite Encounter

This essay utilises Suzanne Bachelard's essay ‘Forme et Contenu’ as an entry point to sketch out the possibility (or impossibility) of a dialectic of mathemata following the work of Jean Cavaillès and Jean Hyppolite. Firstly, I outline what Bachelard sees as a ‘dialectic’ of form and content in the history of abstract algebra, highlighting a parallel between Bachelard's reading of algebra and Hyppolite reading of the relation of the human subject to logos. Secondly, I move towards Jean Hyppolite's discussion of Lautman in ‘Mathematical Thought’ which allows for a revealing of convergences between the work of Hyppolite and Cavaillès. I conclude by noting that the similarities between Cavaillès and Hyppolite – despite possible protestations by Hyppolite – go against Hegel's understanding of mathematics, leading to the possibility of opening up a ‘Hegelian mathematics against Hegel’.

Composition Series of the Solvable Multiplicative Abelian Groups over a Regular Even nth Roots of Unity: A Classical Approach

Solubility of algebraic structures is what gleaned the introduction of group theory, which later stems the other realms of abstract algebra viz: rings, fields and semigroup theories. The nth roots of unity is found in the most sensitive texts ever in the history of abstract algebra: Cauchy’s, Galois’ and Cayley’s. These three giant group theorists had the common ground of the roots of unity in even the title of their works. The idea is that if the nth roots of unity are solvable by radicals and so do the composition series approach, then all other products of the nth roots of unity - which the unity itself is part of - will automatically be solvable. Hence, all equations that dissolve to the least of the nth roots of unity are solvable by the composition series. This article penciled down how the congruence modulo of arithmetics due to Gauss and Leibnitz were used to break down the nth roots of unity, so that the recursive process can generate the composition series of normal subgroups between the unity and the group itself. Since they are P-Groups, they have normal P-Sylow Subgroups. The normality comes from the Index Theorem. Because they all have index 2 in their P-Groups, they are the maximal proper normal P-Sylow Subgroups and their factor groups are abelian accounting to the solubility of nth roots of unity by composition series. We combine the classical Euler Formula and the De Moivre Theorem to present the solvability of nth roots of unity. The P-Groups over nth roots of unity are multiplicative. nth roots of unity are subsequences of nth roots of unity and it converges to the limit point of the nth roots of unity.

Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century

Reviewed Title: Katz, Victor J. and Karen Hunger Parshall. Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century. Princeton, NJ: Princeton University Press, 2014. 485 pages. ISBN 9780691149059

A plurality of algebras, 1200–1600: Algebraic Europe from Fibonacci to Clavius

In memory of Jackie Stedall, friend and colleagueAs Jackie Stedall argued in her 2011 book, From Cardano's great art to Lagrange's reflections: filling a gap in the history of algebra, there was a ‘transition from the traditional algebra of equation-solving in the sixteenth and seventeenth centuries to the emergence of “modern” or “abstract” algebra in the mid nineteenth century’ (page vii). This paper traces the evolution from the thirteenth-century work of the Pisan mathematician, Leonardo Fibonacci, to the early seventeenth-century work of the German Jesuit Christoph Clavius of what came to be considered ‘traditional algebra’. It contends that rather than a single ‘traditional algebra’, in fact, a plurality of intimately related yet subtly different algebras emerged over the course of those four centuries in different yet interacting national settings.

Book Review: Taming the Unknown - A History of Algebra from Antiquity to the Early Twentieth Century

Book Review: Taming the Unknown - A History of Algebra from Antiquity to the Early Twentieth Century

Book Review: Taming the unknown. A history of algebra from antiquity to the early twentieth century

Book Review: Taming the unknown. A history of algebra from antiquity to the early twentieth century

Taming the unknown: history of algebra from antiquity to the early twentieth century

Taming the unknown: history of algebra from antiquity to the early twentieth century

From Cardano's great art to Lagrange's reflections: filling a gap in the history of algebra (Heritage of European Mathematics)

<i>From Cardano's great art to Lagrange's reflections: filling a gap in the history of algebra</i> (Heritage of European Mathematics)

Review of an introduction to the history of algebra solving equations from Mesopotamian Times to the Renaissance by Jacques Sesiano

Review of an introduction to the history of algebra solving equations from Mesopotamian Times to the Renaissance by Jacques Sesiano

From Cardano's great art to Lagrange's reflections: filling a gap in the history of algebra

From Cardano's great art to Lagrange's reflections: filling a gap in the history of algebra

William Kingdom Clifford Mathématicien et Philosophe

In this paper, we want to present W.K. Clifford as a mathematician and a philosopher. After a brief overview of his life and work, we say a few words about his main contributions to the history of algebra. Of course, we could not study the whole scientific work of the English mathematician: among the numerous “Mathematical papers” of Clifford, we especially expose his lectures on biquaternions (1873) and a new kind of algebras (1878). These discoveries came after the researches of Hamilton (1843) and Grassmann (1844-62), and are very important to understand Clifford’s views about geometry. Then we study some ideas of the author in epistemology (for instance, about the question of the nature of space or the mind-body problem) and also in ethics and politics, showing that Clifford’s philosophy is closely connected with the main aspects of his mathematics.

By Way of Introduction: School Algebra in the Start of the New Century

Algebra entered America's school curriculum in the early part of the nineteenth century. Since that time, it has often been the centerpiece of educational debate and discussion. Kilpatrick and Izsák provide an excellent overview of the evolution of school algebra in the United States in “A History of Algebra in the School Curriculum,” the first chapter in Algebra and Algebraic Thinking in School Math.

Unknown quantity: a real and imaginary history of algebra

Unknown quantity: a real and imaginary history of algebra

Stages in the History of Algebra with Implications for Teaching

In this article, we take a rapid journey through the history of algebra, noting the important developments and reflecting on the importance of this history in the teaching of algebra in secondary school or university. Frequently, algebra is considered to have three stages in its historical development: the rhetorical stage, the syncopated stage, and the symbolic stage. But besides these three stages of expressing algebraic ideas, there are four more conceptual stages which have happened along side of these changes in expressions. These stages are the geometric stage, where most of the concepts of algebra are geometric ones; the static equation-solving stage, where the goal is to find numbers satisfying certain relationships; the dynamic function stage, where motion seems to be an underlying idea, and finally, the abstract stage, where mathematical structure plays the central role. The stages of algebra are, of course not entirely disjoint from one another; there is always some overlap. We discuss here high points of the development of these stages and reflect on the use of these historical stages in the teaching of algebra.

A Faulty Survey of Algebra's Roots

Unknown Quantity . A Real and Imaginary History of Algebra. By John Derbyshire . Joseph Henry Press, Washington, DC, 2006. 382 pp. $27.95, C$31.95. ISBN 0-309-09657-X. The author describes his account of what algebraic ideas are like, how they developed historically, and who were responsible for them as written for the nonmathematician, but that has not stopped him from including numerous formulae.

The History of Algebra and the Development of the Form of its Language

The History of Algebra and the Development of the Form of its Language

Algebra at the Turn of the Century

At the turn of the century it seems to be appropriate to pause and to try to envision future possibilities. We want to discuss the prospects of algebra. To look into the future requires an understanding of the past, the longstanding aims, but also the difficulties which have been encountered. We are going to review part of the history of algebra in order to outline its present state. It is important to notice the missed opportunities and to analyze the reasons. To recognize the present possibilities requires to be aware of the tools which now are available and which may not yet have been used in an optimal way. We urge the reader to focus attention to the need for algebraic considerations in all parts of mathematics but also outside of mathematics. Of course, a view back should also strengthen the interest in classical open problems which now may be feasible to attack.

Historical Roots of Confusion Among Beginning Algebra Students: A Newly Discovered Manuscript

and the related freedom of the algebraist. The history of algebra can facilitate students' adjustment to the spirit and direction of not only abstract algebra but contemporary mathematics in general. The perplexed student might find comfort, for example, in the realization that arbitrariness and freedom are youthful additions to mathematics, born of the early nineteenth-century research on abstract algebra and non-Euclidean geometries, and widely accepted only later in the century. Around 1800 there was but one algebra. In this algebra, frequently called universal arithmetic, letters stood for numbers or quantities, and the laws of arithmetic, such as commutativity of addition and multiplication, prevailed. By the middle of the nineteenth century, however, mathematicians had created many different algebras, including the quaternions, whose multiplication is noncommutative. From this perspective, the confused modem algebra student may be a victim of the circumstance of having been born in the twentieth century. Additional solace might be derived from the revelation that bewilderment and occasionally even rejection greeted the original formulation of the symbolical approach to algebra. Thus the perplexed student has historical roots! A study of early objections to symbolical algebra can also shed light on and stimulate discussion of the oft-unvoiced concerns of present-day modern algebra students. This paper offers a newly discovered nineteenth-century manuscript which documents confu- sion as an almost immediate response to the emergence of symbolical algebra in its limited form. (Like modern algebra, early nineteenth-century symbolical algebra admitted undefined entities; unlike modern algebra, symbolical algebra, in its original form, was governed by the laws of arithmetic.) The manuscript is a spoof-play on Augustus De Morgan's Elements of Algebra (5), one of the first undergraduate algebra textbooks to incorporate the symbolical (or limited modem) approach to algebra.

On the history of algebra in the USSR during her first twenty-five years

On the history of algebra in the USSR during her first twenty-five years

Mathematics Exhibit – World's Fair, Chicago

The mathematics exhibit at the century of crogress exhibition in chicago during the summer of 1933 consisted of two sections: A) an immense octagonal prism, on the lateral faces of which were notations concerning the various subdivisions of mathematics together with slides, illustrating the development and history of algebra and geometry; B) a collection of nineteen exhibits, illustrating various interesting mathematical principles and practical applications of mathematics.