Did Gauss Discover That, Too?

1986 ◽  
Vol 79 (4) ◽  
pp. 288-293
Author(s):  
Richard L. Francis
Keyword(s):  
History Of Mathematics ◽  
Remarkable Feature ◽  
History Of ◽  
The Many ◽  
Modern Era ◽  
Long Chain

A remarkable feature of the history of mathematics is the enormous interval that separates the posing of certain problems and their ultimate resolution. Also remarkable is the long chain of attempted solutions that spanned the many centuries. Examples of this phenomenon are the three famous problems of antiquity, which were resolved only in the modern era.

Painless Drilling Not Your Dentist, but the History of Mathematics

1980 ◽  
Vol 27 (8) ◽  
pp. 40-42
Author(s):  
Stephen Krulik
Keyword(s):  
History Of Mathematics ◽  
Theoretical Base ◽  
Body Of Knowledge ◽  
Practical Necessity ◽  
History Of ◽  
Points Of View ◽  
The Many ◽  
Drill And Practice

Many of our students are far from receptive to the many problems of drill materials with which some teachers provide them. And yet, most students need some drill and practice before they can successfully master a new concept or skill. The history of mathematics can play an important role in making these apparently contradictory points of view compatible. Many of the concepts and ideas in the history of mathematics were developed from practical necessity rather than from a theoretical base; it is these same ideas that offer a great deal of practice material for our students. This ancient body of knowledge can provide drill that is interesting, satisfying, challenging. And, at the same time, it offers the necessary drill to achieve competence in fundamental skills and concepts.

The Many Uses of Algebraic Variables

1992 ◽  
Vol 85 (7) ◽  
pp. 557-561 ◽  
Author(s):  
Randolph A. Philipp
Keyword(s):  
Elementary School ◽  
Turning Point ◽  
History Of Mathematics ◽  
History Of ◽  
The Many ◽  
Experience Difficulty

The concept of variable is one of the most fundamental ideas in mathematics from elementary school through college (Davis 1964; Hirsch and Lappan 1989). This concept is so important that its invention constituted a turning point in the history of mathematics (Rajaratnam 1957). However, research indicates that students experience difficulty with the concept of variable, a difficulty that might partially be explained by the fact that within mathematics, variables can be used in many different ways (Rosnick 1981; Schoenfeld and Arcavi 1988; Wagner 1983).

Africa in the World: History and Historiography

2018 ◽  
Author(s):  
Esperanza Brizuela-Garcia
Keyword(s):  
African American ◽  
World War Ii ◽  
Far East ◽  
Western World ◽  
World War ◽  
The World ◽  
History Of ◽  
The Many ◽  
The 19Th Century ◽  
Modern Era

Since antiquity and through the modern era African societies maintained contacts with peoples in Europe, the Near and Far East, and the Americas. Among other things, African peoples developed local forms of Christianity and Islam, contributed large amounts of gold to European medieval economies, and exported millions of slaves through the Sahara, and the Atlantic and Indian Oceans. Despite this, by the 19th century historians and philosophers of history thought Africa was a continent without major civilizations, whose peoples passively rested at the margins of history. These ideas persisted into the 20th century when historians undertook the challenge of writing histories that explained how communities around the world were connected to one another. In their early iterations, however, these “world narratives” were little more than histories of the Western world; Africa continued to be largely absent from these stories. After World War II, increasing interest in the history of African societies and a more generalized concern with the study of communities that were both mis- and under-represented by historical scholarship called for a revision of the goals and methods of world historians. Among the most important critiques were those from Afrocentric, African American, and Africanist scholars. Afrocentric writers argued that Africa had in fact developed an important civilization in the form of Egypt and that Egypt was the foundation of the classical world. African American and Africanist writers highlighted the contributions that peoples of African descent had made to the world economy and many cultures around the globe. Africanists also questioned whether world historical narratives, which meaningfully accounted for the richness and complexity of African experiences, could be achieved in the form of a single universal narrative. Instead, historians have suggested and produced new frameworks that could best explain the many ways in which Africa has been part of the world and its history.

Space

2020 ◽  
Keyword(s):  
Early Modern ◽  
History Of Science ◽  
Natural Science ◽  
Art History ◽  
Cultural Geography ◽  
History Of Mathematics ◽  
Medieval Period ◽  
History Of ◽  
Early Modern Era ◽  
Modern Era

This volume chronicles the development of philosophical conceptions of space from early antiquity through the medieval period to the early modern era, ending with Kant. The chapters describe the interactions between philosophy at different moments in history and various other disciplines, especially geometry, optics, and natural science more generally. Figures from the history of mathematics, science, and philosophy are discussed, including Euclid, Plato, Aristotle, Proclus, Ibn al-Haytham, Nicole Oresme, Kepler, Descartes, Newton, Leibniz, Berkeley, and Kant. A series of shorter essays, or Reflections, characterize perspectives on space found in the disciplines of ecology, mathematics, sculpture, neuroscience, cultural geography, art history, and the history of science.

The Value of René Descartes's Work Devoted to Princess Elisabeth of Bohemia in Psychology

2019 ◽  
Author(s):  
Adib Rifqi Setiawan ◽  
Laila Fariha Zein
Keyword(s):  
Seventeenth Century ◽  
Scientific Inquiry ◽  
French Mathematician ◽  
The Past ◽  
The Bible ◽  
Modern Psychology ◽  
Authority Figures ◽  
History Of ◽  
The Many ◽  
Modern Era

We noted that the seventeenth century saw far-ranging developments in science. Until that time, philosophers had looked to the past for answers, to the works of Aristotle and other ancient scholars, and to the Bible. The ruling forces of inquiry were dogma (the doctrine proclaimed by the established church) and authority figures. In the seventeenth century, a new force became important: empiricism, the pursuit of knowledge through observation and experimentation. Knowledge handed down from the past became suspect. In its place, the golden age of the seventeenth century became illuminated by discoveries and insights that reflected the changing nature of scientific inquiry. Among the many scholars whose creativity marked that period, the French mathematician and philosopher René Descartes contributed directly to the history of modern psychology. His work helped to free scientific inquiry from the control of rigid, centuriesold theological and intellectual beliefs. Descartes symbolized the transition to the modern era of science, and he applied the idea of the clockwork mechanism to the human body. For these reasons, we can say that he inaugurated the era of modern psychology.

Euler, the clothoid and

2016 ◽  
Vol 100 (548) ◽  
pp. 266-273 ◽  
Author(s):  
Nick Lord
Keyword(s):  
History Of Mathematics ◽  
Complex Function ◽  
Real Variable ◽  
Complex Function Theory ◽  
Subsequent Work ◽  
Definite Integrals ◽  
History Of ◽  
The Many ◽  
Image Position

One of the many definite integrals that Euler was the first to evaluate was(1)He did this, almost as an afterthought, at the end of his short, seven-page paper catalogued as E675 in [1] and with the matter-of-fact title,On the values of integrals from x = 0 to x = ∞. It is a beautiful Euler miniature which neatly illustrates the unexpected twists and turns in the history of mathematics. For Euler's derivation of (1) emerges as the by-product of a solution to a problem in differential geometry concerning the clothoid curve which he had first encountered nearly forty years earlier in his paper E65, [1]. As highlighted in the recentGazettearticle [2], E675 is notable for Euler's use of a complex number substitution to evaluate a real-variable integral. He used this technique in about a dozen of the papers written in the last decade of his life. The rationale for this manoeuvre caused much debate among later mathematicians such as Laplace and Poisson and the technique was only put on a secure footing by the work of Cauchy from 1814 onwards on the foundations of complex function theory, [3, Chapter 1]. Euler's justification was essentially pragmatic (in agreement with numerical evidence) and by what Dunham in [4, p. 68] characterises as his informal credo, ‘Follow the formulas, and they will lead to the truth.’ Smithies, [3, p. 187], contextualises Euler's approach by noting that, at that time, ‘a function was usually thought of as being defined by an analytic expression; by the principle of the generality of analysis, which was widely and often tacitly accepted, such an expression was expected to be valid for all values, real or complex, of the independent variable’. In this article, we examine E675 closely. We have tweaked notation and condensed the working in places to reflect modern usage. At the end, we outline what is, with hindsight, needed to make Euler's arguments watertight: it is worth noting that all of his conclusions survive intact and that the intermediate functions of one and two variables that he introduces in E675 remain the key ingredients for much subsequent work on these integrals.

Historically Speaking, - -

1953 ◽  
Vol 46 (8) ◽  
pp. 575-577
Keyword(s):  
Binary System ◽  
High Speed ◽  
History Of Mathematics ◽  
Number System ◽  
Historical Background ◽  
Interesting Phenomenon ◽  
Information And Communication ◽  
History Of ◽  
The Many ◽  
And Algebra

The binary system as a special case of the generalized problem of scales of notation has had a sudden resurgence of popularity. This is largely due to its use in modern high-speed electronic calculators and in new developments in the theory of “information” and “communication.”1 However, this new utility of the binary system arrived at the same time that an even greater emphasis was being placed on “meaning” and “understanding” in the teaching of mathematics. In arithmetic (and algebra) many teachers have felt that understanding of our number system was enhanced, and in some cases first achieved, through a study of numbers written to some base other than ten. These two motives, utilitarian and pedagogical, have led to several articles on the history of the binary system and related topics,2 but it seems that none of them have stressed several additional pedagogical values to be derived from a proper survey of the historical background of scales of notation. This topic is not only intrinsically inter esting, but it also illustrates well the role of generalization and abstraction in mathematics, the roles of necessity and intellectual curiosity in mathematical invention, a few of the many connections between mathematics and philosophy and religion, and the interesting phenomenon of simultaneity in discovery which recurs so often in the history of mathematics.

A 200-Year History of the Study of Childhood Language Disorders of Unknown Origin: Changes in Terminology

2020 ◽  
Vol 5 (1) ◽  
pp. 6-11 ◽  
Author(s):  
Laurence B. Leonard
Keyword(s):  
Language Impairment ◽  
Specific Language Impairment ◽  
Scientific Literature ◽  
Language Disorder ◽  
Language Disorders ◽  
Unknown Origin ◽  
Developmental Language Disorder ◽  
The Past ◽  
History Of ◽  
The Many

Purpose The current “specific language impairment” and “developmental language disorder” discussion might lead to important changes in how we refer to children with language disorders of unknown origin. The field has seen other changes in terminology. This article reviews many of these changes. Method A literature review of previous clinical labels was conducted, and possible reasons for the changes in labels were identified. Results References to children with significant yet unexplained deficits in language ability have been part of the scientific literature since, at least, the early 1800s. Terms have changed from those with a neurological emphasis to those that do not imply a cause for the language disorder. Diagnostic criteria have become more explicit but have become, at certain points, too narrow to represent the wider range of children with language disorders of unknown origin. Conclusions The field was not well served by the many changes in terminology that have transpired in the past. A new label at this point must be accompanied by strong efforts to recruit its adoption by clinical speech-language pathologists and the general public.

Riemann–Weyl in Deleuze's Bergsonism and the Constitution of the Contemporary Physico-Mathematical Space

2015 ◽  
Vol 9 (1) ◽  
pp. 59-87 ◽  
Author(s):  
Martin Calamari
Keyword(s):  
Field Theory ◽  
Gauge Field ◽  
Fibre Bundle ◽  
History Of Mathematics ◽  
Gauge Field Theory ◽  
Contemporary Science ◽  
Explicit Mention ◽  
History Of ◽  
Key Features ◽  
Bernhard Riemann

In recent years, the ideas of the mathematician Bernhard Riemann (1826–66) have come to the fore as one of Deleuze's principal sources of inspiration in regard to his engagements with mathematics, and the history of mathematics. Nevertheless, some relevant aspects and implications of Deleuze's philosophical reception and appropriation of Riemann's thought remain unexplored. In the first part of the paper I will begin by reconsidering the first explicit mention of Riemann in Deleuze's work, namely, in the second chapter of Bergsonism (1966). In this context, as I intend to show first, Deleuze's synthesis of some key features of the Riemannian theory of multiplicities (manifolds) is entirely dependent, both textually and conceptually, on his reading of another prominent figure in the history of mathematics: Hermann Weyl (1885–1955). This aspect has been largely underestimated, if not entirely neglected. However, as I attempt to bring out in the second part of the paper, reframing the understanding of Deleuze's philosophical engagement with Riemann's mathematics through the Riemann–Weyl conjunction can allow us to disclose some unexplored aspects of Deleuze's further elaboration of his theory of multiplicities (rhizomatic multiplicities, smooth spaces) and profound confrontation with contemporary science (fibre bundle topology and gauge field theory). This finally permits delineation of a correlation between Deleuze's plane of immanence and the contemporary physico-mathematical space of fundamental interactions.

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