Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century

Author: Paolo Mancosu
Publisher: Oxford University Press on Demand
ISBN: 0195132440
Category: Drama
Page: 275
View: 8695
The seventeenth century saw dramatic advances in mathematical theory and practice than any era before or since. With the recovery of many of the classical Greek mathematical texts, new techniques were introduced, and within 100 years, analytic geometry, the geometry of indivisibles, the arithmetic of infinites, and the calculus had been developed. Although many technical studies have been devoted to these innovations, Paolo Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. Beginning with the Renaissance debates on the certainty of mathematics, Mancosu leads the reader through the foundational issues raised by the emergence of these new mathematical techniques, including the influence of the Aristotelian conception of science in Cavalieri and Guldin, the foundational relevance of Descartes' Geometrie, the relationship between empiricist epistemology and infinitistic theorems in geometry, and the debates concerning the foundations of the Leibnizian calculus In the process Mancosu draws a sophisticated picture of the subtle dependencies between technical development and philosophical reflection in seventeenth century mathematics.

The Philosophy of Mathematical Practice

Author: Paolo Mancosu
Publisher: Oxford University Press on Demand
ISBN: 0199296456
Category: Philosophy
Page: 447
View: 9489
This book gives a coherent and unified presentation of a new direction of work in philosophy of mathematics. This new approach in philosophy of mathematics requires extensive attention to mathematical practice and provides philosophical analyses of important novel characteristics of contemporary (twentieth century) mathematics and of many aspects of mathematical activity-such as visualization, explanation, understanding etc.-- which escape purely formal logicaltreatment.The book consists of a lengthy introduction by the editor and of eight chapters written by some of the very best scholars in this area. Each chapter consists of a short introduction to the general topic of the chapter and of a longer research article in the very same area. Theeight topics selected represent a broad spectrum of the contemporary philosophical reflection on different aspects of mathematical practice: Diagrammatic reasoning and representational systems; Visualization; Mathematical Explanation; Purity of Methods; Mathematical Concepts; Philosophical relevance of category theory; Philosophical aspects of computer science in mathematics; Philosophical impact of recent developments in mathematical physics.

Visualization, Explanation and Reasoning Styles in Mathematics

Author: Paolo Mancosu,Klaus Frovin Jørgensen,S.A. Pedersen,Stig Andur Pedersen
Publisher: Springer Science & Business Media
ISBN: 9781402033346
Category: Mathematics
Page: 300
View: 7218
This book contains groundbreaking contributions to the philosophical analysis of mathematical practice. Several philosophers of mathematics have recently called for an approach to philosophy of mathematics that pays more attention to mathematical practice. Questions concerning concept-formation, understanding, heuristics, changes in style of reasoning, the role of analogies and diagrams etc. have become the subject of intense interest. The historians and philosophers in this book agree that there is more to understanding mathematics than a study of its logical structure. How are mathematical objects and concepts generated? How does the process tie up with justification? What role do visual images and diagrams play in mathematical activity? What are the different epistemic virtues (explanatoriness, understanding, visualizability, etc.) which are pursued and cherished by mathematicians in their work? The reader will find here systematic philosophical analyses as well as a wealth of philosophically informed case studies ranging from Babylonian, Greek, and Chinese mathematics to nineteenth century real and complex analysis.

The Oxford Handbook of Philosophy in Early Modern Europe

Author: Desmond M. Clarke,Catherine Wilson
Publisher: OUP Oxford
ISBN: 0191654256
Category: Philosophy
Page: 616
View: 6963
In this Handbook twenty-six leading scholars survey the development of philosophy between the middle of the sixteenth century and the early eighteenth century. The five parts of the book cover metaphysics and natural philosophy; the mind, the passions, and aesthetics; epistemology, logic, mathematics, and language; ethics and political philosophy; and religion. The period between the publication of Copernicus's De Revolutionibus and Berkeley's reflections on Newton and Locke saw one of the most fundamental changes in the history of our way of thinking about the universe. This radical transformation of worldview was partly a response to what we now call the Scientific Revolution; it was equally a reflection of political changes that were no less fundamental, which included the establishment of nation-states and some of the first attempts to formulate a theory of international rights and justice. Finally, the Reformation and its aftermath undermined the apparent unity of the Christian church in Europe and challenged both religious beliefs that had been accepted for centuries and the interpretation of the Bible on which they had been based. The Handbook surveys a number of the most important developments in the philosophy of the period, as these are expounded both in texts that have since become very familiar and in other philosophical texts that are undeservedly less well-known. It also reaches beyond the philosophy to make evident the fluidity of the boundary with science, and to consider the impact on philosophy of historical and political events—explorations, revolutions and reforms, inventions and discoveries. Thus it not only offers a guide to the most important areas of recent research, but also offers some new questions for historians of philosophy to pursue and to have indicated areas that are ripe for further exploration.

The Young Leibniz and his Philosophy (1646–76)

Author: Stuart Brown
Publisher: Springer Science & Business Media
ISBN: 9401735077
Category: Philosophy
Page: 314
View: 1274
Despite the importance of Leibniz's mature philosophy, his early work has been relatively neglected. This collection begins with an overview of his formative years and includes 12 original papers by internationally-known scholars. The contributions reflect the wide range of the young Leibniz's philosophical interests and his interests in related subjects, including law, physics and theology. Some chapters explore his relationship to other philosophers, including his teachers in Leipzig and Jena and his Paris friend Tschirnhaus, as well as Hobbes and Spinoza. Others focus on particular periods or texts and deal with themes ranging from ethics and free-will to his philosophically-significant account of transubstantiation and his early monadology. Some of the topics are familiar to Leibniz students - harmony, sufficient reason and possible worlds, for instance - but others are less familiar - for instance, his attitude to historical truth, millenarianism and the relation of mathematics to the natural world. The book provides an introduction to Leibniz's early philosophy and throws light on the development of some of the doctrines with which he is particularly associated.

Cartesian Spacetime

Descartes’ Physics and the Relational Theory of Space and Motion
Author: Edward Slowik
Publisher: Springer Science & Business Media
ISBN: 9401709750
Category: Philosophy
Page: 252
View: 971
Although Descartes' natural philosophy marked an advance in the development of modern science, many critics over the years, such as Newton, have rejected his particular `relational' theory of space and motion. Nevertheless, it is also true that most historians and philosophers have not sufficiently investigated the viability of the Cartesian theory. This book explores, consequently, the success of the arguments against Descartes' theory of space and motion by determining if it is possible to formulate a version that can eliminate its alleged problems. In essence, this book comprises the first sustained attempt to construct a consistent `Cartesian' spacetime theory: that is, a theory of space and time that consistently incorporates Descartes' various physical and metaphysical concepts. Intended for students in the history of philosophy and science, this study reveals the sophisticated insights, and often quite successful elements, in Descartes' unjustly neglected relational theory of space and motion.

The Cambridge Companion to Galileo

Author: Peter Machamer
Publisher: Cambridge University Press
ISBN: 1139825666
Category: Philosophy
Page: N.A
View: 744
Not only a hero of the scientific revolution, but after his conflict with the church, a hero of science, Galileo is today rivalled in the popular imagination only by Newton and Einstein. But what did Galileo actually do, and what are the sources of the popular image we have of him? This 1998 collection of specially-commissioned essays is unparalleled in the depth of its coverage of all facets of Galileo's work. A particular feature of the volume is the treatment of Galileo's relationship with the church. It will be of interest to philosophers, historians of science, cultural historians and those in religious studies.

Mathematical practice and the philosophy of mathematics

Author: Audrey Yap,Stanford University. Dept. of Philosophy
Publisher: N.A
Page: 432
View: 8743

From Discrete to Continuous

The Broadening of Number Concepts in Early Modern England
Author: K. Neal
Publisher: Springer Science & Business Media
ISBN: 940170077X
Category: Mathematics
Page: 175
View: 2553
In the early modern period, a crucial transformation occurred in the classical conception of number and magnitude. Traditionally, numbers were merely collections of discrete units that measured some multiple. Magnitude, on the other hand, was usually described as being continuous, or being divisible into parts that are infinitely divisible. This traditional idea of discrete number versus continuous magnitude was challenged in the early modern period in several ways. This detailed study explores how the development of algebraic symbolism, logarithms, and the growing practical demands for an expanded number concept all contributed to a broadening of the number concept in early modern England. An interest in solving practical problems was not, in itself, enough to cause a generalisation of the number concept. It was the combined impact of novel practical applications together with the concomitant development of such mathematical advances as algebraic notation and logarithms that produced a broadened number concept.

From Brouwer to Hilbert

The Debate on the Foundations of Mathematics in the 1920s
Author: Paolo Mancosu
Publisher: Oxford University Press on Demand
ISBN: 9780195096323
Category: Mathematics
Page: 337
View: 5786
Most contemporary work in the foundations of mathematics takes its start from the groundbreaking contributions of, among others, Hilbert, Brouwer, Bernays, and Weyl. This book offers an introduction to the debate on the foundations of mathematics during the 1920s and presents the English reader with a selection of twenty five articles central to the debate which have not been previously translated. It is an ideal text for undergraduate and graduate courses in the philosophy of mathematics.

Nominalism and Constructivism in Seventeenth-Century Mathematical Philosophy

Author: David Sepkoski
Publisher: Routledge
ISBN: 1136768688
Page: 184
View: 8432
What was the basis for the adoption of mathematics as the primary mode of discourse for describing natural events by a large segment of the philosophical community in the seventeenth century? In answering this question, this book demonstrates that a significant group of philosophers shared the belief that there is no necessary correspondence between external reality and objects of human understanding, which they held to include the objects of mathematical and linguistic discourse. The result is a scholarly reliable, but accessible, account of the role of mathematics in the works of (amongst others) Galileo, Kepler, Descartes, Newton, Leibniz, and Berkeley. This impressive volume will benefit scholars interested in the history of philosophy, mathematical philosophy and the history of mathematics.

Berkeley's Philosophy of Mathematics

Author: Douglas M. Jesseph
Publisher: University of Chicago Press
ISBN: 9780226398983
Category: Mathematics
Page: 322
View: 8243

The Origins of the Infinitesimal Calculus

Author: Margaret E. Baron
Publisher: Courier Corporation
ISBN: 9780486495446
Category: Mathematics
Page: 304
View: 5666
Historical account of infinitesimal calculus, beginning with background mathematical concepts from Greek, Hindu, and Arabic sources, and with particular focus on the geometric techniques and methods developed in the17th century. 1969 edition.

The Oxford Handbook of Philosophy of Mathematics and Logic

Author: Stewart Shapiro
Publisher: OUP USA
ISBN: 0195148770
Category: Mathematics
Page: 833
View: 575
Covers the state of the art in the philosophy of maths and logic, giving the reader an overview of the major problems, positions, and battle lines. The chapters in this book contain both exposition and criticism as well as substantial development of their own positions. It also includes a bibliography.

Perspectives on Mathematical Practices

Bringing Together Philosophy of Mathematics, Sociology of Mathematics, and Mathematics Education
Author: Bart van Kerkhove,Jean Paul van Bendegem
Publisher: Springer Science & Business Media
ISBN: 1402050348
Category: Science
Page: 242
View: 8423
In the eyes of the editors, this book will be considered a success if it can convince its readers of the following: that it is warranted to dream of a realistic and full-fledged theory of mathematical practices, in the plural. If such a theory is possible, it would mean that a number of presently existing fierce oppositions between philosophers, sociologists, educators, and other parties involved, are in fact illusory.

Universities and Science in the Early Modern Period

Author: Mordechai Feingold,Victor Navarro-Brotons
Publisher: Springer Science & Business Media
ISBN: 9781402039744
Category: Education
Page: 309
View: 9934
The ""soul"" of the early modern university was its well-rounded, humanistically informed curriculum. This volume offers synthesis of the fecundity of early modern universities, their receptivity to novel scientific ideas, and their contribution to the critical dialogue that vitalized the emergent European scientific community.

Science after the Practice Turn in the Philosophy, History, and Social Studies of Science

Author: Léna Soler,Sjoerd Zwart,Michael Lynch,Vincent Israel-Jost
Publisher: Routledge
ISBN: 1317935357
Category: Philosophy
Page: 346
View: 5603
In the 1980s, philosophical, historical and social studies of science underwent a change which later evolved into a turn to practice. Analysts of science were asked to pay attention to scientific practices in meticulous detail and along multiple dimensions, including the material, social and psychological. Following this turn, the interest in scientific practices continued to increase and had an indelible influence in the various fields of science studies. No doubt, the practice turn changed our conceptions and approaches of science, but what did it really teach us? What does it mean to study scientific practices? What are the general lessons, implications, and new challenges? This volume explores questions about the practice turn using both case studies and theoretical analysis. The case studies examine empirical and mathematical sciences, including the engineering sciences. The volume promotes interactions between acknowledged experts from different, often thought of as conflicting, orientations. It presents contributions in conjunction with critical commentaries that put the theses and assumptions of the former in perspective. Overall, the book offers a unique and diverse range of perspectives on the meanings, methods, lessons, and challenges associated with the practice turn.

Mathematical Practitioners and the Transformation of Natural Knowledge in Early Modern Europe

Author: Lesley B. Cormack,Steven A. Walton,John A. Schuster
Publisher: Springer
ISBN: 3319494309
Category: Science
Page: 203
View: 6989
This book argues that we can only understand transformations of nature studies in the Scientific Revolution if we take seriously the interaction between practitioners (those who know by doing) and scholars (those who know by thinking). These are not in opposition, however. Theory and practice are end points on a continuum, with some participants interested only in the practical, others only in the theoretical, and most in the murky intellectual and material world in between. It is this borderland where influence, appropriation, and collaboration have the potential to lead to new methods, new subjects of enquiry, and new social structures of natural philosophy and science. The case for connection between theory and practice can be most persuasively drawn in the area of mathematics, which is the focus of this book. Practical mathematics was a growing field in early modern Europe and these essays are organised into three parts which contribute to the debate about the role of mathematical practice in the Scientific Revolution. First, they demonstrate the variability of the identity of practical mathematicians, and of the practices involved in their activities in early modern Europe. Second, readers are invited to consider what practical mathematics looked like and that although practical mathematical knowledge was transmitted and circulated in a wide variety of ways, participants were able to recognize them all as practical mathematics. Third, the authors show how differences and nuances in practical mathematics typically depended on the different contexts in which it was practiced: social, cultural, political, and economic particularities matter. Historians of science, especially those interested in the Scientific Revolution period and the history of mathematics will find this book and its ground-breaking approach of particular interest.

De Motu and the Analyst

A Modern Edition, with Introductions and Commentary
Author: G. Berkeley
Publisher: Springer Science & Business Media
ISBN: 9401125929
Category: Computers
Page: 232
View: 1039
Berkeley's philosophy has been much studied and discussed over the years, and a growing number of scholars have come to the realization that scientific and mathematical writings are an essential part of his philosophical enterprise. The aim of this volume is to present Berkeley's two most important scientific texts in a form which meets contemporary standards of scholarship while rendering them accessible to the modern reader. Although editions of both are contained in the fourth volume of the Works, these lack adequate introductions and do not provide com plete and corrected texts. The present edition contains a complete and critically established text of both De Motu and The Analyst, in addi tion to a new translation of De Motu. The introductions and notes are designed to provide the background necessary for a full understanding of Berkeley's account of science and mathematics. Although these two texts are very different, they are united by a shared a concern with the work of Newton and Leibniz. Berkeley's De Motu deals extensively with Newton's Principia and Leibniz's Specimen Dynamicum, while The Analyst critiques both Leibnizian and Newto nian mathematics. Berkeley is commonly thought of as a successor to Locke or Malebranche, but as these works show he is also a successor to Newton and Leibniz.

The Mathematical Career of Pierre de Fermat, 1601-1665

Author: Michael Sean Mahoney
Publisher: Princeton University Press
ISBN: 9780691036663
Category: Biography & Autobiography
Page: 432
View: 1342
Hailed as one of the greatest mathematical results of the twentieth century, the recent proof of Fermat's Last Theorem by Andrew Wiles brought to public attention the enigmatic problem-solver Pierre de Fermat, who centuries ago stated his famous conjecture in a margin of a book, writing that he did not have enough room to show his "truly marvelous demonstration." Along with formulating this proposition--xn+yn=zn has no rational solution for n > 2--Fermat, an inventor of analytic geometry, also laid the foundations of differential and integral calculus, established, together with Pascal, the conceptual guidelines of the theory of probability, and created modern number theory. In one of the first full-length investigations of Fermat's life and work, Michael Sean Mahoney provides rare insight into the mathematical genius of a hobbyist who never sought to publish his work, yet who ranked with his contemporaries Pascal and Descartes in shaping the course of modern mathematics.