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Author: Viktor Blasjo
Publisher: Academic Press
ISBN: 0128132981
Category : Mathematics
Languages : en
Pages : 284
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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
Author: Viktor Blasjo
Publisher: Academic Press
ISBN: 0128132981
Category : Mathematics
Languages : en
Pages : 284
Get Book Here
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
Author: Richard C. Brown
Publisher: World Scientific
ISBN: 9814390798
Category : Mathematics
Languages : en
Pages : 333
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Book Description
1. Evolution or revolution in mathematics -- 2. Issues in seventeenth century mathematics -- 3. Isaac Barrow: a foil to Leibniz -- 4. A young central European polymath -- 5. First steps in mathematics -- 6. The creation of calculus -- 7. Logic -- 8. The universal characteristic -- 9. The baroque cultural context -- 10. Epilogue -- 11. Some concluding remarks on mathematical change -- Appendices.
Author: Edward Harrington Lockwood
Publisher: Cambridge University Press
ISBN: 9781001224114
Category : Curves
Languages : en
Pages : 290
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Book Description
Describes the drawing of plane curves, cycloidal curves, spirals, glissettes and others.
Author: Maria Rosa Antognazza
Publisher: Oxford University Press
ISBN: 0190913630
Category : Philosophy
Languages : en
Pages : 825
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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.
Author: Niccolo Guicciardini
Publisher: MIT Press
ISBN: 0262291657
Category : Mathematics
Languages : en
Pages : 449
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Book Description
An analysis of Newton's mathematical work, from early discoveries to mature reflections, and a discussion of Newton's views on the role and nature of mathematics. Historians of mathematics have devoted considerable attention to Isaac Newton's work on algebra, series, fluxions, quadratures, and geometry. In Isaac Newton on Mathematical Certainty and Method, Niccolò Guicciardini examines a critical aspect of Newton's work that has not been tightly connected to Newton's actual practice: his philosophy of mathematics. Newton aimed to inject certainty into natural philosophy by deploying mathematical reasoning (titling his main work The Mathematical Principles of Natural Philosophy most probably to highlight a stark contrast to Descartes's Principles of Philosophy). To that end he paid concerted attention to method, particularly in relation to the issue of certainty, participating in contemporary debates on the subject and elaborating his own answers. Guicciardini shows how Newton carefully positioned himself against two giants in the “common” and “new” analysis, Descartes and Leibniz. Although his work was in many ways disconnected from the traditions of Greek geometry, Newton portrayed himself as antiquity's legitimate heir, thereby distancing himself from the moderns. Guicciardini reconstructs Newton's own method by extracting it from his concrete practice and not solely by examining his broader statements about such matters. He examines the full range of Newton's works, from his early treatises on series and fluxions to the late writings, which were produced in direct opposition to Leibniz. The complex interactions between Newton's understanding of method and his mathematical work then reveal themselves through Guicciardini's careful analysis of selected examples. Isaac Newton on Mathematical Certainty and Method uncovers what mathematics was for Newton, and what being a mathematician meant to him.
Author: Richard C Brown
Publisher: World Scientific
ISBN: 9814401617
Category : Mathematics
Languages : en
Pages : 333
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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.
Author: Davide Crippa
Publisher: Springer
ISBN: 3030016382
Category : Mathematics
Languages : en
Pages : 189
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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.
Author: Maria Rosa Antognazza
Publisher: Cambridge University Press
ISBN: 1316154742
Category : Philosophy
Languages : en
Pages : 668
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Book Description
Of all the thinkers of the century of genius that inaugurated modern philosophy, none lived an intellectual life more rich and varied than Gottfried Wilhelm Leibniz (1646–1716). Maria Rosa Antognazza's pioneering biography provides a unified portrait of this unique thinker and the world from which he came. At the centre of the huge range of Leibniz's apparently miscellaneous endeavours, Antognazza reveals a single master project lending unity to his extraordinarily multifaceted life's work. Throughout the vicissitudes of his long life, Leibniz tenaciously pursued the dream of a systematic reform and advancement of all the sciences. As well as tracing the threads of continuity that bound these theoretical and practical activities to this all-embracing plan, this illuminating study also traces these threads back into the intellectual traditions of the Holy Roman Empire in which Leibniz lived and throughout the broader intellectual networks that linked him to patrons in countries as distant as Russia and to correspondents as far afield as China.
Author: Vincenzo De Risi
Publisher: Springer Nature
ISBN: 3030255727
Category : Science
Languages : en
Pages : 304
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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.
Author: Colin MacLaurin
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 480
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Book Description
