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Transcendental Curves In The Leibnizian Calculus

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Transcendental Curves in the Leibnizian Calculus

Transcendental Curves in the Leibnizian Calculus Book
Author : Viktor Blasjo
Publisher : Academic Press
Release : 2017-04-22
ISBN : 0128132981
Language : En, Es, Fr & De

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Book Description :

Transcendental Curves in the Leibnizian Calculus analyzes the mathematical and philosophical conflict between Euclidean and Cartesian mathematics. For millennia, mathematical meaning and ontology had been anchored in geometrical constructions, as epitomized by Euclid's ruler and compass. As late as 1637, Descartes had placed himself squarely in this tradition when he justified his new technique of identifying curves with equations by means of certain curve-tracing instruments, thereby bringing together the ancient constructive tradition and modern algebraic methods in a satisfying marriage. But rapid advances in the new fields of infinitesimal calculus and mathematical mechanics soon ruined his grand synthesis. Descartes's scheme left out transcendental curves, i.e. curves with no polynomial equation, but in the course of these subsequent developments such curves emerged as indispensable. It was becoming harder and harder to juggle cutting-edge mathematics and ancient conceptions of its foundations at the same time, yet leading mathematicians, such as Leibniz felt compelled to do precisely this. The new mathematics fit more naturally an analytical conception of curves than a construction-based one, yet no one wanted to betray the latter, as this was seen as virtually tantamount to stop doing mathematics altogether. The credibility and authority of mathematics depended on it. Brings to light this underlying and often implicit complex of concerns that permeate early calculus Evaluates the technical conception and mathematical construction of the geometrical method Reveals a previously unrecognized Liebnizian programmatic cohesion in early calculus Provides a beautifully written work of outstanding original scholarship

Transcendental Curves in the Leibnizian Calculus

Transcendental Curves in the Leibnizian Calculus Book
Author : Viktor Blasjo
Publisher : Academic Press
Release : 2017-04-20
ISBN : 9780128132371
Language : En, Es, Fr & De

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Book Description :

Transcendental Curves in the Leibnizian Calculus analyzes a mathematical and philosophical conflict between classical and early modern mathematics. In the late 17th century, mathematics was at the brink of an identity crisis. For millennia, mathematical meaning and ontology had been anchored in geometrical constructions, as epitomized by Euclid's ruler and compass. As late as 1637, Descartes had placed himself squarely in this tradition when he justified his new technique of identifying curves with equations by means of certain curve-tracing instruments, thereby bringing together the ancient constructive tradition and modern algebraic methods in a satisfying marriage. But rapid advances in the new fields of infinitesimal calculus and mathematical mechanics soon ruined his grand synthesis. Descartes's scheme left out transcendental curves, i.e. curves with no polynomial equation, but in the course of these subsequent developments such curves emerged as indispensable. It was becoming harder and harder to juggle cutting-edge mathematics and ancient conceptions of its foundations at the same time, yet leading mathematicians, such as Leibniz felt compelled to do precisely this. The new mathematics fit more naturally an analytical conception of curves than a construction-based one, yet no one wanted to betray the latter, as this was seen as virtually tantamount to stop doing mathematics altogether. The credibility and authority of mathematics depended on it. Brings to light this underlying and often implicit complex of concerns that permeate early calculus Evaluates the technical conception and mathematical construction of the geometrical method Reveals a previously unrecognized Liebnizian programmatic cohesion in early calculus Provides a beautifully written work of outstanding original scholarship

The Oxford Handbook of Leibniz

The Oxford Handbook of Leibniz Book
Author : Maria Rosa Antognazza
Publisher : Oxford University Press
Release : 2018-10-01
ISBN : 0190913649
Language : En, Es, Fr & De

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Book Description :

The extraordinary breadth and depth of Leibniz's intellectual vision commands ever increasing attention. As more texts gradually emerge from seemingly bottomless archives, new facets of his contribution to an astonishing variety of fields come to light. This volume provides a uniquely comprehensive, systematic, and up-to-date appraisal of Leibniz's thought thematically organized around its diverse but interrelated aspects. Discussion of his philosophical system naturally takes place of pride. A cluster of original essays revisit his logic, metaphysics, epistemology, philosophy of nature, moral and political philosophy, and philosophy of religion. The scope of the volume, however, goes beyond that of a philosophical collection to embrace all the main features of Leibniz's thought and activity. Contributions are offered on Leibniz as a mathematician (including not only his calculus but also determinant theory, symmetric functions, the dyadic, the analysis situs, probability and statistics); on Leibniz as a scientist (physics and also optics, cosmology, geology, physiology, medicine, and chemistry); on his technical innovations (the calculating machine and the technology of mining, as well as other discoveries); on his work as an 'intelligencer' and cultural networker, as jurist, historian, editor of sources and librarian; on his views on Europe's political future, religious toleration, and ecclesiastical reunification; on his proposals for political, administrative, economic, and social reform. In so doing, the volume serves as a unique cross-disciplinary point of contact for the many domains to which Leibniz contributed. By assembling leading specialists on all these topics, it offers the most rounded picture of Leibniz's endeavors currently available.

The Tangled Origins of the Leibnizian Calculus

The Tangled Origins of the Leibnizian Calculus Book
Author : Richard C. Brown
Publisher : World Scientific
Release : 2012
ISBN : 9814390801
Language : En, Es, Fr & De

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Book Description :

This book is a detailed study of Gottfried Wilhelm Leibniz''s creation of calculus from 1673 to the 1680s. We examine and analyze the mathematics in several of his early manuscripts as well as various articles published in the Acta Eruditorum. It studies some of the other lesser known OC calculiOCO Leibniz created such as the Analysis Situs, delves into aspects of his logic, and gives an overview of his efforts to construct a Universal Characteristic, a goal that has its distant origin in the Ars Magna of the 13th century Catalan philosopher Raymond Llull, whose work enjoyed a renewed popularity in the century and a half prior to Leibniz. This book also touches upon a new look at the priority controversy with Newton and a Kuhnian interpretation of the nature of mathematical change. This book may be the only integrated treatment based on recent research and should be a thought-provoking contribution to the history of mathematics for scholars and students, interested in either Leibniz''s mathematical achievement or general issues in the field."

The Tangled Origins of the Leibnizian Calculus

The Tangled Origins of the Leibnizian Calculus Book
Author : Richard C Brown
Publisher : World Scientific
Release : 2012-03-23
ISBN : 9814401617
Language : En, Es, Fr & De

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Book Description :

This book is a detailed study of Gottfried Wilhelm Leibniz's creation of calculus from 1673 to the 1680s. We examine and analyze the mathematics in several of his early manuscripts as well as various articles published in the Acta Eruditorum. It studies some of the other lesser known “calculi” Leibniz created such as the Analysis Situs, delves into aspects of his logic, and gives an overview of his efforts to construct a Universal Characteristic, a goal that has its distant origin in the Ars Magna of the 13th century Catalan philosopher Raymond Llull, whose work enjoyed a renewed popularity in the century and a half prior to Leibniz. This book also touches upon a new look at the priority controversy with Newton and a Kuhnian interpretation of the nature of mathematical change. This book may be the only integrated treatment based on recent research and should be a thought-provoking contribution to the history of mathematics for scholars and students, interested in either Leibniz's mathematical achievement or general issues in the field. Contents:Evolution or Revolution in MathematicsIssues in Seventeenth Century MathematicsIsaac Barrow: A Foil to LeibnizA Young Central European PolymathFirst Steps in MathematicsThe Creation of CalculusLogicThe Universal CharacteristicThe Baroque Cultural ContextEpilogueSome Concluding Remarks on Mathematical ChangeAppendices:A: A Transmutation Theorem of LeibnizB: Leibniz's Series Quadrature of a ConicC: Syllogistic LogicD: The Vis Viva DisputeE: Some Applications of Curves and Neusis in Greek GeometryF: InfinitesimalsA Note on the Author Readership: Advanced undergraduate students, graduate students and researchers in mathematics, history of mathematics or history of science. Keywords:Leibniz;Calculus;Geometry;17th Century MathematicsKey Features:The thoroughness and comprehensiveness of the treatment of this book are based on recent researchTechnical details of the mathematics are carefully dealt with instead of just being summarized for the general readerNo other work on the development of calculus includes a description and analysis of the Baroque/Renaissance atmosphere of fascination with symbols, emblems, Real Characters and philosophical languages which motivated both Leibniz's mathematics and his search for the Universal Characteristic

Leibniz and the Structure of Sciences

Leibniz and the Structure of Sciences Book
Author : Vincenzo De Risi
Publisher : Springer Nature
Release : 2020-01-01
ISBN : 3030255727
Language : En, Es, Fr & De

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Book Description :

The book offers a collection of essays on various aspects of Leibniz’s scientific thought, written by historians of science and world-leading experts on Leibniz. The essays deal with a vast array of topics on the exact sciences: Leibniz’s logic, mereology, the notion of infinity and cardinality, the foundations of geometry, the theory of curves and differential geometry, and finally dynamics and general epistemology. Several chapters attempt a reading of Leibniz’s scientific works through modern mathematical tools, and compare Leibniz’s results in these fields with 19th- and 20th-Century conceptions of them. All of them have special care in framing Leibniz’s work in historical context, and sometimes offer wider historical perspectives that go much beyond Leibniz’s researches. A special emphasis is given to effective mathematical practice rather than purely epistemological thought. The book is addressed to all scholars of the exact sciences who have an interest in historical research and Leibniz in particular, and may be useful to historians of mathematics, physics, and epistemology, mathematicians with historical interests, and philosophers of science at large.

The History of Continua

The History of Continua Book
Author : Stewart Shapiro,Geoffrey Hellman
Publisher : Oxford University Press, USA
Release : 2021
ISBN : 0198809646
Language : En, Es, Fr & De

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Book Description :

Mathematical and philosophical thought about continuity has changed considerably over the ages, from Aristotle's insistence that a continuum is a unified whole, to the dominant account today, that a continuum is composed of infinitely many points. This book explores the key ideas and debates concerning continuity over more than 2500 years.

Historical Scientific Instruments in Contemporary Education

Historical Scientific Instruments in Contemporary Education Book
Author : Anonim
Publisher : BRILL
Release : 2021-11-15
ISBN : 9004499679
Language : En, Es, Fr & De

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Book Description :

When science’s “black boxes” are pried open, its workings become accessible. Like time-travellers into history but grounded in today’s cultures, learners interact directly with authentic instruments and replicas. Chapters describe educational experiences sparked through collaborations interrelating museum, school and university.

The Impossibility of Squaring the Circle in the 17th Century

The Impossibility of Squaring the Circle in the 17th Century Book
Author : Davide Crippa
Publisher : Springer
Release : 2019-03-06
ISBN : 3030016382
Language : En, Es, Fr & De

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Book Description :

This book is about James Gregory’s attempt to prove that the quadrature of the circle, the ellipse and the hyperbola cannot be found algebraically. Additonally, the subsequent debates that ensued between Gregory, Christiaan Huygens and G.W. Leibniz are presented and analyzed. These debates eventually culminated with the impossibility result that Leibniz appended to his unpublished treatise on the arithmetical quadrature of the circle. The author shows how the controversy around the possibility of solving the quadrature of the circle by certain means (algebraic curves) pointed to metamathematical issues, particularly to the completeness of algebra with respect to geometry. In other words, the question underlying the debate on the solvability of the circle-squaring problem may be thus phrased: can finite polynomial equations describe any geometrical quantity? As the study reveals, this question was central in the early days of calculus, when transcendental quantities and operations entered the stage. Undergraduate and graduate students in the history of science, in philosophy and in mathematics will find this book appealing as well as mathematicians and historians with broad interests in the history of mathematics.

Anachronisms in the History of Mathematics

Anachronisms in the History of Mathematics Book
Author : Niccol- Guicciardini
Publisher : Cambridge University Press
Release : 2021-07-22
ISBN : 1108834965
Language : En, Es, Fr & De

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Book Description :

Discover essays by leading scholars on the history of mathematics from ancient to modern times in European and non-European cultures.

The History of Mathematics A Source Based Approach Volume 2

The History of Mathematics  A Source Based Approach  Volume 2 Book
Author : June Barrow-Green,Jeremy Gray,Robin Wilson
Publisher : American Mathematical Society
Release : 2022-05-26
ISBN : 1470443821
Language : En, Es, Fr & De

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Book Description :

The History of Mathematics: A Source-Based Approach is a comprehensive history of the development of mathematics. This, the second volume of a two-volume set, takes the reader from the invention of the calculus to the beginning of the twentieth century. The initial discoverers of calculus are given thorough investigation, and special attention is also paid to Newton's Principia. The eighteenth century is presented as primarily a period of the development of calculus, particularly in differential equations and applications of mathematics. Mathematics blossomed in the nineteenth century and the book explores progress in geometry, analysis, foundations, algebra, and applied mathematics, especially celestial mechanics. The approach throughout is markedly historiographic: How do we know what we know? How do we read the original documents? What are the institutions supporting mathematics? Who are the people of mathematics? The reader learns not only the history of mathematics, but also how to think like a historian. The two-volume set was designed as a textbook for the authors' acclaimed year-long course at the Open University. It is, in addition to being an innovative and insightful textbook, an invaluable resource for students and scholars of the history of mathematics. The authors, each among the most distinguished mathematical historians in the world, have produced over fifty books and earned scholarly and expository prizes from the major mathematical societies of the English-speaking world.

Leibniz and the Invention of Mathematical Transcendence

Leibniz and the Invention of Mathematical Transcendence Book
Author : Michel Serfati
Publisher : Franz Steiner Verlag Wiesbaden GmbH
Release : 2018
ISBN : 9783515120821
Language : En, Es, Fr & De

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Book Description :

The invention of mathematical transcendence in the seventeenth century is linked to Leibniz, who always claimed it to be his own creation. However, Descartes had created a completely new symbolic frame in which one considers plane curves, which was a real upheaval. Leibniz initially appreciated this Cartesian frame. Although, as we see in the book, during his research he was confronted with inexpressible contexts he then called 'transcendent'. The development of a concept of mathematical transcendence is at the core of this book. The description follows a pragmatic path by highlighting how and why aspects of the concept were developed and which obstacles were encountered by mathematicians. Also, there were some dead ends which are described here. Leibniz exceeded Descartes' ideas on a symbolical level (transcendent expressions), a geometrical level (transcendent curves) as well as a numerical level (transcendent numbers) those are also examined in detail.

Research in History and Philosophy of Mathematics

Research in History and Philosophy of Mathematics Book
Author : Maria Zack,Dirk Schlimm
Publisher : Springer Nature
Release : 2020-01-02
ISBN : 3030312984
Language : En, Es, Fr & De

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Book Description :

This volume contains ten papers that have been collected by the Canadian Society for History and Philosophy of Mathematics/Société canadienne d’histoire et de philosophie des mathématiques. It showcases rigorously-reviewed contemporary scholarship on an interesting variety of topics in the history and philosophy of mathematics from the seventeenth century to the modern era. The volume begins with an exposition of the life and work of Professor Bolesław Sobociński. It then moves on to cover a collection of topics about twentieth-century philosophy of mathematics, including Fred Sommers’s creation of Traditional Formal Logic and Alexander Grothendieck’s work as a starting point for discussing analogies between commutative algebra and algebraic geometry. Continuing the focus on the philosophy of mathematics, the next selections discuss the mathematization of biology and address the study of numerical cognition. The volume then moves to discussing various aspects of mathematics education, including Charles Davies’s early book on the teaching of mathematics and the use of Gaussian Lemniscates in the classroom. A collection of papers on the history of mathematics in the nineteenth century closes out the volume, presenting a discussion of Gauss’s “Allgemeine Theorie des Erdmagnetismus” and a comparison of the geometric works of Desargues and La Hire. Written by leading scholars in the field, these papers are accessible not only to mathematicians and students of the history and philosophy of mathematics, but also to anyone with a general interest in mathematics.

The Best Writing on Mathematics 2017

The Best Writing on Mathematics 2017 Book
Author : Mircea Pitici
Publisher : Princeton University Press
Release : 2017-11-14
ISBN : 0691178631
Language : En, Es, Fr & De

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Book Description :

The year's finest mathematics writing from around the world This annual anthology brings together the year’s finest mathematics writing from around the world. Featuring promising new voices alongside some of the foremost names in the field, The Best Writing on Mathematics 2017 makes available to a wide audience many articles not easily found anywhere else—and you don’t need to be a mathematician to enjoy them. These writings offer surprising insights into the nature, meaning, and practice of mathematics today. They delve into the history, philosophy, teaching, and everyday occurrences of math, and take readers behind the scenes of today’s hottest mathematical debates. Here Evelyn Lamb describes the excitement of searching for incomprehensibly large prime numbers, Jeremy Gray speculates about who would have won math’s highest prize—the Fields Medal—in the nineteenth century, and Philip Davis looks at mathematical results and artifacts from a business and marketing viewpoint. In other essays, Noson Yanofsky explores the inherent limits of knowledge in mathematical thinking, Jo Boaler and Lang Chen reveal why finger-counting enhances children’s receptivity to mathematical ideas, and Carlo Séquin and Raymond Shiau attempt to discover how the Renaissance painter Fra Luca Pacioli managed to convincingly depict his famous rhombicuboctahedron, a twenty-six-sided Archimedean solid. And there’s much, much more. In addition to presenting the year’s most memorable writings on mathematics, this must-have anthology includes a bibliography of other notable writings and an introduction by the editor, Mircea Pitici. This book belongs on the shelf of anyone interested in where math has taken us—and where it is headed.

Families of Curves and the Origins of Partial Differentiation

Families of Curves and the Origins of Partial Differentiation Book
Author : S.B. Engelsman
Publisher : Elsevier
Release : 2000-04-01
ISBN : 9780080872049
Language : En, Es, Fr & De

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Book Description :

This book provides a detailed description of the main episodes in the emergence of partial differentiation during the period 1690-1740. It argues that the development of this concept - to a considerable degree of perfection - took place almost exclusively in problems concerning families of curves. Thus, the book shows the origins of the ideas and techniques which paved the way for the sudden introduction of partial differential equations in 1750. The main methodological characteristic of the book is its emphasis on a full understanding of the motives, problems and goals of the mathematicians of that time.

L H pital s Analyse des infiniments petits

L   H  pital s Analyse des infiniments petits Book
Author : Robert E Bradley,Salvatore J. Petrilli,C. Edward Sandifer
Publisher : Birkhäuser
Release : 2015-07-20
ISBN : 3319171151
Language : En, Es, Fr & De

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Book Description :

This monograph is an annotated translation of what is considered to be the world’s first calculus textbook, originally published in French in 1696. That anonymously published textbook on differential calculus was based on lectures given to the Marquis de l’Hôpital in 1691-2 by the great Swiss mathematician, Johann Bernoulli. In the 1920s, a copy of Bernoulli’s lecture notes was discovered in a library in Basel, which presented the opportunity to compare Bernoulli’s notes, in Latin, to l’Hôpital’s text in French. The similarities are remarkable, but there is also much in l’Hôpital’s book that is original and innovative. This book offers the first English translation of Bernoulli's notes, along with the first faithful English translation of l’Hôpital’s text, complete with annotations and commentary. Additionally, a significant portion of the correspondence between l’Hôpital and Bernoulli has been included, also for the fi rst time in English translation. This translation will provide students and researchers with direct access to Bernoulli’s ideas and l’Hôpital’s innovations. Both enthusiasts and scholars of the history of science and the history of mathematics will fi nd food for thought in the texts and notes of the Marquis de l’Hôpital and his teacher, Johann Bernoulli.

Joseph Liouville 1809 1882

Joseph Liouville 1809   1882 Book
Author : Jesper Lützen
Publisher : Springer Science & Business Media
Release : 2012-12-06
ISBN : 1461209897
Language : En, Es, Fr & De

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Book Description :

This scientific biography of the mathematician Joseph Liouville is divided into two parts. The first part is a chronological account of Liouville's career including a description of the institutions he worked in, his relations with his teachers, colleagues and students, and the historical context of his works. It portrays the French scientific community in a period when Germany and England had surpassed France as the leading nations in mathematics and physics. The second part of the book gives a detailed analysis of Liouville's major contributions to mathematics and mechanics. The gradual development of Liouville's ideas, as reflected in his publications and notebooks, are related to the works of his predecessors and his contemporaries as well as to later developments in the field. On the basis of Liouville's unpublished notes the book reconstructs Liouville's hitherto unknown theories of stability of rotating masses of fluid, potential theory, Galois theory and electrodynamics. It also incorporates valuable added information from Liouville's notes regarding his works on differentiation of arbitrary order, integration in finite terms, Sturm-Liouville theory, transcendental numbers, doubly periodic functions, geometry and mechanics.

G W Leibniz Interrelations between Mathematics and Philosophy

G W  Leibniz  Interrelations between Mathematics and Philosophy Book
Author : Norma B. Goethe,Philip Beeley,David Rabouin
Publisher : Springer
Release : 2015-04-20
ISBN : 9401796645
Language : En, Es, Fr & De

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Book Description :

Up to now there have been scarcely any publications on Leibniz dedicated to investigating the interrelations between philosophy and mathematics in his thought. In part this is due to the previously restricted textual basis of editions such as those produced by Gerhardt. Through recent volumes of the scientific letters and mathematical papers series of the Academy Edition scholars have obtained a much richer textual basis on which to conduct their studies - material which allows readers to see interconnections between his philosophical and mathematical ideas which have not previously been manifested. The present book draws extensively from this recently published material. The contributors are among the best in their fields. Their commissioned papers cover thematically salient aspects of the various ways in which philosophy and mathematics informed each other in Leibniz's thought.

Infinitesimal Differences

Infinitesimal Differences Book
Author : Ursula Goldenbaum,Douglas Jesseph
Publisher : Walter de Gruyter
Release : 2008-11-03
ISBN : 3110211866
Language : En, Es, Fr & De

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Book Description :

The development of the calculus during the 17th century was successful in mathematical practice, but raised questions about the nature of infinitesimals: were they real or rather fictitious? This collection of essays, by scholars from Canada, the US, Germany, Japan and Switzerland, gives a comprehensive study of the controversies over the nature and status of the infinitesimal. Aside from Leibniz, the scholars considered are Hobbes, Wallis, Newton, Bernoulli, Hermann, and Nieuwentijt. The collection also contains newly discovered marginalia of Leibniz to the writings of Hobbes.