Philip Kitcher
- Published in print:
- 1985
- Published Online:
- November 2003
- ISBN:
- 9780195035414
- eISBN:
- 9780199833368
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0195035410.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
The Nature of Mathematical Knowledge develops and defends an empiricist approach to mathematical knowledge. After offering an account of a priori knowledge, it argues that none of the ...
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The Nature of Mathematical Knowledge develops and defends an empiricist approach to mathematical knowledge. After offering an account of a priori knowledge, it argues that none of the available accounts of a priori mathematical knowledge is viable. It then constructs an approach to the content of mathematical statements, viewing mathematics as grounded in our manipulations of physical reality. From these crude beginnings, mathematics unfolds through the successive modifications of mathematical practice, spurred by the presence of unsolved problems. This process of unfolding is considered in general, and illustrated by considering the historical development of analysis from the seventeenth century to the end of the nineteenth.Less
The Nature of Mathematical Knowledge develops and defends an empiricist approach to mathematical knowledge. After offering an account of a priori knowledge, it argues that none of the available accounts of a priori mathematical knowledge is viable. It then constructs an approach to the content of mathematical statements, viewing mathematics as grounded in our manipulations of physical reality. From these crude beginnings, mathematics unfolds through the successive modifications of mathematical practice, spurred by the presence of unsolved problems. This process of unfolding is considered in general, and illustrated by considering the historical development of analysis from the seventeenth century to the end of the nineteenth.
E. Brian Davies
- Published in print:
- 2007
- Published Online:
- September 2008
- ISBN:
- 9780199219186
- eISBN:
- 9780191711695
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199219186.001.0001
- Subject:
- Physics, History of Physics
How do scientific conjectures become laws? Why does proof mean different things in different sciences? Do numbers exist, or were they invented? Why do some laws turn out to be wrong? This book ...
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How do scientific conjectures become laws? Why does proof mean different things in different sciences? Do numbers exist, or were they invented? Why do some laws turn out to be wrong? This book discusses the basis for scientists' claims to knowledge about the world. It looks at science historically, emphasizing not only the achievements of scientists from Galileo onwards, but also their mistakes. The book rejects the claim that all scientific knowledge is provisional, by citing examples from chemistry, biology, and geology. A major feature of the book is its defence of the view that mathematics was invented rather than discovered. While experience has shown that disentangling knowledge from opinion and aspiration is a hard task, this book provides a clear guide to the difficulties. Including many examples and quotations, and with a scope ranging from psychology and evolution to quantum theory and mathematics, this book aims to bring alive issues at the heart of all science.Less
How do scientific conjectures become laws? Why does proof mean different things in different sciences? Do numbers exist, or were they invented? Why do some laws turn out to be wrong? This book discusses the basis for scientists' claims to knowledge about the world. It looks at science historically, emphasizing not only the achievements of scientists from Galileo onwards, but also their mistakes. The book rejects the claim that all scientific knowledge is provisional, by citing examples from chemistry, biology, and geology. A major feature of the book is its defence of the view that mathematics was invented rather than discovered. While experience has shown that disentangling knowledge from opinion and aspiration is a hard task, this book provides a clear guide to the difficulties. Including many examples and quotations, and with a scope ranging from psychology and evolution to quantum theory and mathematics, this book aims to bring alive issues at the heart of all science.
Philip Kitcher
- Published in print:
- 1985
- Published Online:
- November 2003
- ISBN:
- 9780195035414
- eISBN:
- 9780199833368
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0195035410.003.0006
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
The next stage is to set up the main question for the rest of the book: How is empirical mathematical knowledge possible? The outline of an answer is given: relatively simple experiences provide ...
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The next stage is to set up the main question for the rest of the book: How is empirical mathematical knowledge possible? The outline of an answer is given: relatively simple experiences provide knowledge of elementary mathematics; the historical process extends the basic mathematical knowledge in extraordinary ways.Less
The next stage is to set up the main question for the rest of the book: How is empirical mathematical knowledge possible? The outline of an answer is given: relatively simple experiences provide knowledge of elementary mathematics; the historical process extends the basic mathematical knowledge in extraordinary ways.
Philip Kitcher
- Published in print:
- 1985
- Published Online:
- November 2003
- ISBN:
- 9780195035414
- eISBN:
- 9780199833368
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0195035410.003.0011
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
Concludes with a more sustained look at a single part of the history of mathematics, the development of analysis from the seventeenth century to the work of Dedekind, Cantor and Frege. The discussion ...
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Concludes with a more sustained look at a single part of the history of mathematics, the development of analysis from the seventeenth century to the work of Dedekind, Cantor and Frege. The discussion is linked to the general account of Chapters 7–9.Less
Concludes with a more sustained look at a single part of the history of mathematics, the development of analysis from the seventeenth century to the work of Dedekind, Cantor and Frege. The discussion is linked to the general account of Chapters 7–9.
Paul Erickson
- Published in print:
- 2015
- Published Online:
- May 2016
- ISBN:
- 9780226097039
- eISBN:
- 9780226097206
- Item type:
- chapter
- Publisher:
- University of Chicago Press
- DOI:
- 10.7208/chicago/9780226097206.003.0001
- Subject:
- History, History of Science, Technology, and Medicine
Writing about game theory’s past presents several historiographical challenges. At least prior to its prominent appearances in economics since the 1980s, game theory does not correlate neatly with ...
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Writing about game theory’s past presents several historiographical challenges. At least prior to its prominent appearances in economics since the 1980s, game theory does not correlate neatly with any of the typical objects of analysis in the history of the sciences: an individual life; a defined community of individuals; or a particular academic discipline. Indeed, it frequently appears in the context of “interdisciplines” – operations research, “behavioral science,” “management science” or “general systems,” bolstered by powerful patrons of the Cold War era, especially the United States military. Correspondingly, the search for a stable, unitary identity for and interpretation of game theory during this period is difficult. This chapter therefore follows the work of historians of theory in the physical sciences in interpreting game theory as a heterogeneous, friable, and interpretively flexible collection of “theoretical tools” – roughly speaking, notations and general mathematical frameworks as well as specific results and styles of argument – that can be employed in accomplishing work in a variety of contexts. At the same time, the narrative of the book remains bound together by a set of debates that spilled over from context to context – about the nature of rationality and the uses of “theory” in studying human interaction.Less
Writing about game theory’s past presents several historiographical challenges. At least prior to its prominent appearances in economics since the 1980s, game theory does not correlate neatly with any of the typical objects of analysis in the history of the sciences: an individual life; a defined community of individuals; or a particular academic discipline. Indeed, it frequently appears in the context of “interdisciplines” – operations research, “behavioral science,” “management science” or “general systems,” bolstered by powerful patrons of the Cold War era, especially the United States military. Correspondingly, the search for a stable, unitary identity for and interpretation of game theory during this period is difficult. This chapter therefore follows the work of historians of theory in the physical sciences in interpreting game theory as a heterogeneous, friable, and interpretively flexible collection of “theoretical tools” – roughly speaking, notations and general mathematical frameworks as well as specific results and styles of argument – that can be employed in accomplishing work in a variety of contexts. At the same time, the narrative of the book remains bound together by a set of debates that spilled over from context to context – about the nature of rationality and the uses of “theory” in studying human interaction.
John Fauvel
- Published in print:
- 2013
- Published Online:
- January 2014
- ISBN:
- 9780199681976
- eISBN:
- 9780191761737
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199681976.003.0001
- Subject:
- Mathematics, History of Mathematics
In this introductory chapter we survey various traditions in the 800-year-old story of mathematics at Oxford University. These traditions include the medieval heritage, studies of logic, mathematical ...
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In this introductory chapter we survey various traditions in the 800-year-old story of mathematics at Oxford University. These traditions include the medieval heritage, studies of logic, mathematical instruments, the antiquarian tradition, historical and mathematical research, mathematical printing, popularization, the literary tradition, examinations, and the rivalry between Oxford and Cambridge.Less
In this introductory chapter we survey various traditions in the 800-year-old story of mathematics at Oxford University. These traditions include the medieval heritage, studies of logic, mathematical instruments, the antiquarian tradition, historical and mathematical research, mathematical printing, popularization, the literary tradition, examinations, and the rivalry between Oxford and Cambridge.
David D. Nolte
- Published in print:
- 2018
- Published Online:
- August 2018
- ISBN:
- 9780198805847
- eISBN:
- 9780191843808
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198805847.003.0005
- Subject:
- Physics, History of Physics
This chapter reviews the history of modern geometry with a focus on the topics that provided the foundation for the new visualization of physics. It begins with Carl Gauss and Bernhard Riemann, who ...
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This chapter reviews the history of modern geometry with a focus on the topics that provided the foundation for the new visualization of physics. It begins with Carl Gauss and Bernhard Riemann, who redefined geometry and identified the importance of curvature for physics. Vector spaces, developed by Hermann Grassmann, Giuseppe Peano and David Hilbert, are examples of the kinds of abstract new spaces that are so important for modern physics, such as Hilbert space for quantum mechanics. Fractal geometry developed by Felix Hausdorff later provided the geometric language needed to solve problems in chaos theory. Motion cannot exist without space—trajectories are the tracks of points, mathematical or physical, through it.Less
This chapter reviews the history of modern geometry with a focus on the topics that provided the foundation for the new visualization of physics. It begins with Carl Gauss and Bernhard Riemann, who redefined geometry and identified the importance of curvature for physics. Vector spaces, developed by Hermann Grassmann, Giuseppe Peano and David Hilbert, are examples of the kinds of abstract new spaces that are so important for modern physics, such as Hilbert space for quantum mechanics. Fractal geometry developed by Felix Hausdorff later provided the geometric language needed to solve problems in chaos theory. Motion cannot exist without space—trajectories are the tracks of points, mathematical or physical, through it.
Roger Mathew Grant
- Published in print:
- 2014
- Published Online:
- December 2014
- ISBN:
- 9780199367283
- eISBN:
- 9780199367306
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199367283.003.0005
- Subject:
- Music, Theory, Analysis, Composition
Through the long eighteenth century, the relationship between motion and time was rewritten. Time, in this new view, was no longer a conceptual descendant of motion but was, in its new form, ...
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Through the long eighteenth century, the relationship between motion and time was rewritten. Time, in this new view, was no longer a conceptual descendant of motion but was, in its new form, absolute: a demarcated backdrop against which events were situated. Discourses on meter reflected this shift in time's epistemological grounding. Meter, explained anew, was no longer a motion, the beat and the measure finally parted ways in this transition. Theorists in the eighteenth century shifted the focus of their explanation from the physical act of the beat to the properties of the measure, and the edifice that had once joined meter, character, and tempo began to shatter. Kirnberger's Die Kunst des reinen Satzes in der Musik drew on the particular images and pieces of language associated with absolute time in natural philosophy and mathematics. In this document, Kirnberger reimagined meter as an ongoing, dynamic division of absolute time.Less
Through the long eighteenth century, the relationship between motion and time was rewritten. Time, in this new view, was no longer a conceptual descendant of motion but was, in its new form, absolute: a demarcated backdrop against which events were situated. Discourses on meter reflected this shift in time's epistemological grounding. Meter, explained anew, was no longer a motion, the beat and the measure finally parted ways in this transition. Theorists in the eighteenth century shifted the focus of their explanation from the physical act of the beat to the properties of the measure, and the edifice that had once joined meter, character, and tempo began to shatter. Kirnberger's Die Kunst des reinen Satzes in der Musik drew on the particular images and pieces of language associated with absolute time in natural philosophy and mathematics. In this document, Kirnberger reimagined meter as an ongoing, dynamic division of absolute time.
Dalia Nassar (ed.)
- Published in print:
- 2014
- Published Online:
- April 2014
- ISBN:
- 9780199976201
- eISBN:
- 9780199395507
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199976201.001.0001
- Subject:
- Philosophy, History of Philosophy
In the last two decades, philosophers have become increasingly aware of the fact that, in spite of significant differences between the contemporary and romantic contexts, romanticism continues to ...
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In the last two decades, philosophers have become increasingly aware of the fact that, in spite of significant differences between the contemporary and romantic contexts, romanticism continues to “persist,” and the questions that the romantics raised remain relevant today. The Relevance of Romanticism: Essays on Early German Romantic Philosophy is the first collection of essays that directly considers the reasons why philosophers have recently become deeply interested in romantic thought. Through historical and systematic reconstructions, the volume offers greater understanding of romanticism as a philosophical movement and deeper insight into the role that romantic thought plays—or can play—in contemporary philosophical debates. Sixteen essays by both established and emerging scholars discussing key romantic themes and concerns highlight the diversity within both romantic thought and its contemporary reception. Part 1 consists of the first published encounter between Manfred Frank and Frederick Beiser, in which the two major scholars discuss their differing interpretations of philosophical romanticism. Part 2 draws significant connections between romantic conceptions of history, sociability, hermeneutics, and education and explores the ways in which these views can illuminate questions in contemporary social-political philosophy and theories of interpretation. Part 3 consists in some of the most innovative takes on romantic aesthetics, which seek to bring romantic thought into dialogue, with, for instance, contemporary analytic aesthetics and theories of cognition. Part 4 offers a rare rigorous engagement with romantic conceptions of science, and demonstrates ways in which the romantic view of nature, experimentation, and mathematics need not be relegated to historical curiosities.Less
In the last two decades, philosophers have become increasingly aware of the fact that, in spite of significant differences between the contemporary and romantic contexts, romanticism continues to “persist,” and the questions that the romantics raised remain relevant today. The Relevance of Romanticism: Essays on Early German Romantic Philosophy is the first collection of essays that directly considers the reasons why philosophers have recently become deeply interested in romantic thought. Through historical and systematic reconstructions, the volume offers greater understanding of romanticism as a philosophical movement and deeper insight into the role that romantic thought plays—or can play—in contemporary philosophical debates. Sixteen essays by both established and emerging scholars discussing key romantic themes and concerns highlight the diversity within both romantic thought and its contemporary reception. Part 1 consists of the first published encounter between Manfred Frank and Frederick Beiser, in which the two major scholars discuss their differing interpretations of philosophical romanticism. Part 2 draws significant connections between romantic conceptions of history, sociability, hermeneutics, and education and explores the ways in which these views can illuminate questions in contemporary social-political philosophy and theories of interpretation. Part 3 consists in some of the most innovative takes on romantic aesthetics, which seek to bring romantic thought into dialogue, with, for instance, contemporary analytic aesthetics and theories of cognition. Part 4 offers a rare rigorous engagement with romantic conceptions of science, and demonstrates ways in which the romantic view of nature, experimentation, and mathematics need not be relegated to historical curiosities.
John Fauvel, Raymond Flood, and Robin Wilson (eds)
- Published in print:
- 2013
- Published Online:
- January 2014
- ISBN:
- 9780199681976
- eISBN:
- 9780191761737
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199681976.001.0001
- Subject:
- Mathematics, History of Mathematics
For eight centuries mathematics has been researched and studied at Oxford, and the subject and its teaching have undergone profound changes during that time. This is the story of the intellectual and ...
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For eight centuries mathematics has been researched and studied at Oxford, and the subject and its teaching have undergone profound changes during that time. This is the story of the intellectual and social life of this community, and of its interactions with the wider world. This highly readable and beautifully illustrated book reveals the richness and influence of Oxford’s mathematical tradition and the fascinating characters that helped to shape it. The story begins with the founding of the University of Oxford and the establishing of the medieval curriculum, in which mathematics had an important role. The Black Death, the advent of printing, the Civil War, and the Newtonian revolution all had a great influence on the development of mathematics at Oxford. So too did many well-known figures: Roger Bacon, Henry Savile, Robert Hooke, Christopher Wren, Edmond Halley, Florence Nightingale, Charles Dodgson (Lewis Carroll), and G. H. Hardy, to name but a few. Later chapters bring us to the 20th century, with some entertaining reminiscences by Sir Michael Atiyah of the thirty years he spent as an Oxford mathematician. In this second edition the story is brought right up to the opening of the new Mathematical Institute in 2013 with a foreword from Marcus du Sautoy and recent developments from Peter M. Neumann.Less
For eight centuries mathematics has been researched and studied at Oxford, and the subject and its teaching have undergone profound changes during that time. This is the story of the intellectual and social life of this community, and of its interactions with the wider world. This highly readable and beautifully illustrated book reveals the richness and influence of Oxford’s mathematical tradition and the fascinating characters that helped to shape it. The story begins with the founding of the University of Oxford and the establishing of the medieval curriculum, in which mathematics had an important role. The Black Death, the advent of printing, the Civil War, and the Newtonian revolution all had a great influence on the development of mathematics at Oxford. So too did many well-known figures: Roger Bacon, Henry Savile, Robert Hooke, Christopher Wren, Edmond Halley, Florence Nightingale, Charles Dodgson (Lewis Carroll), and G. H. Hardy, to name but a few. Later chapters bring us to the 20th century, with some entertaining reminiscences by Sir Michael Atiyah of the thirty years he spent as an Oxford mathematician. In this second edition the story is brought right up to the opening of the new Mathematical Institute in 2013 with a foreword from Marcus du Sautoy and recent developments from Peter M. Neumann.
Robin Wilson and John J. Watkins (eds)
- Published in print:
- 2013
- Published Online:
- September 2013
- ISBN:
- 9780199656592
- eISBN:
- 9780191748059
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199656592.001.0001
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics, History of Mathematics
The history of mathematics is a well-studied and vibrant area of research, with books and scholarly articles published on various aspects of the subject. Yet, the history of combinatorics seems to ...
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The history of mathematics is a well-studied and vibrant area of research, with books and scholarly articles published on various aspects of the subject. Yet, the history of combinatorics seems to have been largely overlooked. This book goes some way to redress this and serves two main purposes: it constitutes the first book-length survey of the history of combinatorics, and it assembles, for the first time in a single source, researches on the history of combinatorics that would otherwise be inaccessible to the general reader. Individual chapters have been contributed by sixteen experts. The book opens with an introduction to two thousand years of combinatorics. This is followed by seven chapters on early combinatorics, leading from Indian and Chinese writings on permutations to late-Renaissance publications on the arithmetical triangle. The next seven chapters trace the subsequent story, from Euler’s contributions to such wide-ranging topics as partitions, polyhedra, and latin squares to the 20th-century advances in combinatorial set theory, enumeration, and graph theory. The book concludes with some combinatorial reflections.Less
The history of mathematics is a well-studied and vibrant area of research, with books and scholarly articles published on various aspects of the subject. Yet, the history of combinatorics seems to have been largely overlooked. This book goes some way to redress this and serves two main purposes: it constitutes the first book-length survey of the history of combinatorics, and it assembles, for the first time in a single source, researches on the history of combinatorics that would otherwise be inaccessible to the general reader. Individual chapters have been contributed by sixteen experts. The book opens with an introduction to two thousand years of combinatorics. This is followed by seven chapters on early combinatorics, leading from Indian and Chinese writings on permutations to late-Renaissance publications on the arithmetical triangle. The next seven chapters trace the subsequent story, from Euler’s contributions to such wide-ranging topics as partitions, polyhedra, and latin squares to the 20th-century advances in combinatorial set theory, enumeration, and graph theory. The book concludes with some combinatorial reflections.
