Apostolos Doxiadis and Barry Mazur (eds)
- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691149042
- eISBN:
- 9781400842681
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691149042.001.0001
- Subject:
- Mathematics, History of Mathematics
This book brings together important thinkers in mathematics, history, and philosophy to explore the relationship between mathematics and narrative. “Circles disturbed” reflect the last words of ...
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This book brings together important thinkers in mathematics, history, and philosophy to explore the relationship between mathematics and narrative. “Circles disturbed” reflect the last words of Archimedes before he was slain by a Roman soldier—“Don't disturb my circles”—words that seem to refer to two radically different concerns: that of the practical person living in the concrete world of reality, and that of the theoretician lost in a world of abstraction. Stories and theorems are, in a sense, the natural languages of these two worlds—stories representing the way we act and interact, and theorems giving us pure thought, distilled from the hustle and bustle of reality. Yet, though the voices of stories and theorems seem totally different, they share profound connections and similarities. This book delves into topics such as the way in which historical and biographical narratives shape our understanding of mathematics and mathematicians, the development of “myths of origins” in mathematics, the structure and importance of mathematical dreams, the role of storytelling in the formation of mathematical intuitions, the ways mathematics helps us organize the way we think about narrative structure, and much more.Less
This book brings together important thinkers in mathematics, history, and philosophy to explore the relationship between mathematics and narrative. “Circles disturbed” reflect the last words of Archimedes before he was slain by a Roman soldier—“Don't disturb my circles”—words that seem to refer to two radically different concerns: that of the practical person living in the concrete world of reality, and that of the theoretician lost in a world of abstraction. Stories and theorems are, in a sense, the natural languages of these two worlds—stories representing the way we act and interact, and theorems giving us pure thought, distilled from the hustle and bustle of reality. Yet, though the voices of stories and theorems seem totally different, they share profound connections and similarities. This book delves into topics such as the way in which historical and biographical narratives shape our understanding of mathematics and mathematicians, the development of “myths of origins” in mathematics, the structure and importance of mathematical dreams, the role of storytelling in the formation of mathematical intuitions, the ways mathematics helps us organize the way we think about narrative structure, and much more.
Jacqueline A. Stedall
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198524953
- eISBN:
- 9780191711886
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198524953.003.0005
- Subject:
- Mathematics, History of Mathematics
John Pell (1611-1685) has long been considered the most enigmatic of the 17th-century mathematicians. He was well read in both Classical and contemporary mathematics, and there is no doubt that he ...
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John Pell (1611-1685) has long been considered the most enigmatic of the 17th-century mathematicians. He was well read in both Classical and contemporary mathematics, and there is no doubt that he was held in esteem. However, attempts to discover just what Pell's mathematical reputation was based on, create a picture that is strangely unclear. His name is linked with the equation Np2 ± I = q2 (for N, p, q integers), universally known as ‘Pell's equation’ but it was neither proposed nor solved by Pell. His mathematical publications were few and far between: the book for which he is best remembered is An introduction to algebra published in 1668, but other books expected of him failed to appear. There were always hints that he was developing further ideas, but he could never be persuaded to share them.Less
John Pell (1611-1685) has long been considered the most enigmatic of the 17th-century mathematicians. He was well read in both Classical and contemporary mathematics, and there is no doubt that he was held in esteem. However, attempts to discover just what Pell's mathematical reputation was based on, create a picture that is strangely unclear. His name is linked with the equation Np2 ± I = q2 (for N, p, q integers), universally known as ‘Pell's equation’ but it was neither proposed nor solved by Pell. His mathematical publications were few and far between: the book for which he is best remembered is An introduction to algebra published in 1668, but other books expected of him failed to appear. There were always hints that he was developing further ideas, but he could never be persuaded to share them.
Daniel L. Stein and Charles M. Newman
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691147338
- eISBN:
- 9781400845637
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691147338.003.0001
- Subject:
- Sociology, Science, Technology and Environment
Spin glasses are disordered magnetic materials, and it is hard to find a less promising candidate to serve as a focal point of complexity studies, much less as the object of thousands of ...
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Spin glasses are disordered magnetic materials, and it is hard to find a less promising candidate to serve as a focal point of complexity studies, much less as the object of thousands of investigations. On first inspection, they don't seem particularly exciting. Although they're a type of magnet, they're not very good at being magnetic. Metallic spin glasses are unremarkable conductors, and insulating spin glasses are fairly useless as practical insulators. This introductory chapter provides an overview of why spin glasses might be of interest to the reader if they are not a physicist but are interested in any of a variety of other problems outside physics, or more generally in the field of complexity itself. It explores those features of spin glasses that have attracted, in turn, condensed matter and statistical physicists, complexity scientists, and mathematicians and applied mathematicians of various sorts.Less
Spin glasses are disordered magnetic materials, and it is hard to find a less promising candidate to serve as a focal point of complexity studies, much less as the object of thousands of investigations. On first inspection, they don't seem particularly exciting. Although they're a type of magnet, they're not very good at being magnetic. Metallic spin glasses are unremarkable conductors, and insulating spin glasses are fairly useless as practical insulators. This introductory chapter provides an overview of why spin glasses might be of interest to the reader if they are not a physicist but are interested in any of a variety of other problems outside physics, or more generally in the field of complexity itself. It explores those features of spin glasses that have attracted, in turn, condensed matter and statistical physicists, complexity scientists, and mathematicians and applied mathematicians of various sorts.
Amir Alexander
- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691149042
- eISBN:
- 9781400842681
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691149042.003.0001
- Subject:
- Mathematics, History of Mathematics
This chapter traces the history of mathematics, from the late sixteenth century to the present, through the lens of mathematical stories. The period is divided into three main epochs and a possible ...
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This chapter traces the history of mathematics, from the late sixteenth century to the present, through the lens of mathematical stories. The period is divided into three main epochs and a possible fourth, each characterized by a different dominant mathematical story, which in turn is related to a different dominant mathematical style. Each period reveals one or several dominant narratives that enjoyed wide currency not only among the broader population but also among practicing mathematicians. The chapter examines the transition of mathematics from the older narratives, spanning the Enlightenment and the exploration mathematics periods, to the time of tragic mathematical heroes such as Évariste Galois and Georg Cantor.Less
This chapter traces the history of mathematics, from the late sixteenth century to the present, through the lens of mathematical stories. The period is divided into three main epochs and a possible fourth, each characterized by a different dominant mathematical story, which in turn is related to a different dominant mathematical style. Each period reveals one or several dominant narratives that enjoyed wide currency not only among the broader population but also among practicing mathematicians. The chapter examines the transition of mathematics from the older narratives, spanning the Enlightenment and the exploration mathematics periods, to the time of tragic mathematical heroes such as Évariste Galois and Georg Cantor.
David Corfield
Apostolos Doxiadis and Barry Mazur (eds)
- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691149042
- eISBN:
- 9781400842681
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691149042.003.0009
- Subject:
- Mathematics, History of Mathematics
This chapter examines the rationality of mathematical practice in relation to narrative. It begins with a discussion of Alasdair MacIntyre's account of rational enquiry, Three Rival Versions of Moral ...
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This chapter examines the rationality of mathematical practice in relation to narrative. It begins with a discussion of Alasdair MacIntyre's account of rational enquiry, Three Rival Versions of Moral Enquiry, and how this might translate to scientific and mathematical enquiry. It then considers the telos of mathematical enquiry, along with rival claims to truth as the aim of mathematics. The chapter argues that to be fully rational, mathematicians must embrace narrative as a basic tool for understanding the nature of their discipline and research. It also calls for the partial validity of a pre-Enlightenment epistemology of mathematics as a craft whose advance is made possible only through a certain discipleship.Less
This chapter examines the rationality of mathematical practice in relation to narrative. It begins with a discussion of Alasdair MacIntyre's account of rational enquiry, Three Rival Versions of Moral Enquiry, and how this might translate to scientific and mathematical enquiry. It then considers the telos of mathematical enquiry, along with rival claims to truth as the aim of mathematics. The chapter argues that to be fully rational, mathematicians must embrace narrative as a basic tool for understanding the nature of their discipline and research. It also calls for the partial validity of a pre-Enlightenment epistemology of mathematics as a craft whose advance is made possible only through a certain discipleship.
G. E. R. Lloyd
Apostolos Doxiadis and Barry Mazur (eds)
- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691149042
- eISBN:
- 9781400842681
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691149042.003.0011
- Subject:
- Mathematics, History of Mathematics
This chapter explores the relationship between mathematics and narrative from an Aristotelian perspective. Aristotle's philosophy of mathematics is radically different from that of Plato. For ...
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This chapter explores the relationship between mathematics and narrative from an Aristotelian perspective. Aristotle's philosophy of mathematics is radically different from that of Plato. For example, Aristotle did not postulate separate intelligible mathematical objects. For Aristotle, mathematics studied the mathematical properties of physical objects, in abstraction from the physical properties those objects possessed. Another divergence between Aristotle and Plato relates to comments that the former made in the Metaphysics about the activity of mathematicians and the actualization of certain potentialities in their work. In particular, three terms mentioned by Aristotle are problematic: orthe, diagramma, and energeia. The chapter argues that Aristotle challenged the conception of mathematics as atemporal, soon after Plato defended it, by insisting that mathematical proofs are produced by energeia.Less
This chapter explores the relationship between mathematics and narrative from an Aristotelian perspective. Aristotle's philosophy of mathematics is radically different from that of Plato. For example, Aristotle did not postulate separate intelligible mathematical objects. For Aristotle, mathematics studied the mathematical properties of physical objects, in abstraction from the physical properties those objects possessed. Another divergence between Aristotle and Plato relates to comments that the former made in the Metaphysics about the activity of mathematicians and the actualization of certain potentialities in their work. In particular, three terms mentioned by Aristotle are problematic: orthe, diagramma, and energeia. The chapter argues that Aristotle challenged the conception of mathematics as atemporal, soon after Plato defended it, by insisting that mathematical proofs are produced by energeia.
Victor J. Katz and Karen Hunger Parshall
- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691149059
- eISBN:
- 9781400850525
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691149059.003.0005
- Subject:
- Mathematics, History of Mathematics
This chapter takes a look at the scope of mathematics in Ancient and Medieval China. Although the Chinese engaged in numerical calculation as early as the middle of the second millennium BCE, the ...
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This chapter takes a look at the scope of mathematics in Ancient and Medieval China. Although the Chinese engaged in numerical calculation as early as the middle of the second millennium BCE, the earliest detailed written evidence of the solution of mathematical problems in China is the Suan shu shu (or Book of Numbers and Computation), a book discovered in a tomb dated to approximately 200 BCE. The Suan shu shu was part of the Chinese intellectual culture shaped in part by China's tempestuous political history. Within this history, Chinese mathematicians—who seemingly worked in isolation and in widely disparate parts of the country—gradually developed new methods for treating various problems that their works needed to contain. Here, the chapter discusses various mathematical explorations set out by Chinese scholars, such as the Chinese remainder problem.Less
This chapter takes a look at the scope of mathematics in Ancient and Medieval China. Although the Chinese engaged in numerical calculation as early as the middle of the second millennium BCE, the earliest detailed written evidence of the solution of mathematical problems in China is the Suan shu shu (or Book of Numbers and Computation), a book discovered in a tomb dated to approximately 200 BCE. The Suan shu shu was part of the Chinese intellectual culture shaped in part by China's tempestuous political history. Within this history, Chinese mathematicians—who seemingly worked in isolation and in widely disparate parts of the country—gradually developed new methods for treating various problems that their works needed to contain. Here, the chapter discusses various mathematical explorations set out by Chinese scholars, such as the Chinese remainder problem.
Victor J. Katz and Karen Hunger Parshall
- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691149059
- eISBN:
- 9781400850525
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691149059.003.0006
- Subject:
- Mathematics, History of Mathematics
This chapter explores medieval Indian algebraic thought in the works of Āryabhạta, author of the astronomical treatise Āryabhạtīya (ca. 500 CE), and Brahmagupta, who wrote the ...
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This chapter explores medieval Indian algebraic thought in the works of Āryabhạta, author of the astronomical treatise Āryabhạtīya (ca. 500 CE), and Brahmagupta, who wrote the Brāhma-sphụta-siddhānta (ca. 628 CE), before moving on to consider the contributions of later Indian mathematicians to such topics as solving determinate and indeterminate equations, including the so-called Pell equation Dx² + 1 = y², where D, x, and y are positive integers. It closes with a look at some of the more advanced algebraic topics in the work of the Kerala school, which began in the fourteenth century in the south of India and lasted for over two hundred years.Less
This chapter explores medieval Indian algebraic thought in the works of Āryabhạta, author of the astronomical treatise Āryabhạtīya (ca. 500 CE), and Brahmagupta, who wrote the Brāhma-sphụta-siddhānta (ca. 628 CE), before moving on to consider the contributions of later Indian mathematicians to such topics as solving determinate and indeterminate equations, including the so-called Pell equation Dx² + 1 = y², where D, x, and y are positive integers. It closes with a look at some of the more advanced algebraic topics in the work of the Kerala school, which began in the fourteenth century in the south of India and lasted for over two hundred years.
Judith Herrin
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691153018
- eISBN:
- 9781400845224
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153018.003.0015
- Subject:
- History, World Medieval History
This chapter examines how the mathematical mysteries of Diophantus were preserved, embellished, developed, and enjoyed in Byzantium by many generations of amateur mathematicians like Pierre de ...
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This chapter examines how the mathematical mysteries of Diophantus were preserved, embellished, developed, and enjoyed in Byzantium by many generations of amateur mathematicians like Pierre de Fermat, who formulated what became known as Fermat's last theorem. Fermat was a seventeenth-century scholar and an amateur mathematician who developed several original concepts in addition to the famous “last theorem.” One of his sources was the Arithmetika, a collection of number problems written by Diophantus, a mathematician who appears to have flurished in Alexandria in the third century AD. It was through the Greek text translated into Latin that Fermat became familiar with Diophantus's mathematical problems, and in particular the one at book II, 8, which encouraged the formulation of his own last theorem. Fermat's last theorem claims that “the equation xn + yn = zn has no nontrivial solutions when n is greater than 2”.Less
This chapter examines how the mathematical mysteries of Diophantus were preserved, embellished, developed, and enjoyed in Byzantium by many generations of amateur mathematicians like Pierre de Fermat, who formulated what became known as Fermat's last theorem. Fermat was a seventeenth-century scholar and an amateur mathematician who developed several original concepts in addition to the famous “last theorem.” One of his sources was the Arithmetika, a collection of number problems written by Diophantus, a mathematician who appears to have flurished in Alexandria in the third century AD. It was through the Greek text translated into Latin that Fermat became familiar with Diophantus's mathematical problems, and in particular the one at book II, 8, which encouraged the formulation of his own last theorem. Fermat's last theorem claims that “the equation xn + yn = zn has no nontrivial solutions when n is greater than 2”.
John von Neumann (ed.)
- Published in print:
- 2018
- Published Online:
- September 2018
- ISBN:
- 9780691178561
- eISBN:
- 9781400889921
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691178561.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Quantum mechanics was still in its infancy in 1932 when the young John von Neumann, who would go on to become one of the greatest mathematicians of the twentieth century, published Mathematical ...
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Quantum mechanics was still in its infancy in 1932 when the young John von Neumann, who would go on to become one of the greatest mathematicians of the twentieth century, published Mathematical Foundations of Quantum Mechanics—a revolutionary book that for the first time provided a rigorous mathematical framework for the new science. Robert Beyer's 1955 English translation, which von Neumann reviewed and approved, is cited more frequently today than ever before. But its many treasures and insights were too often obscured by the limitations of the way the text and equations were set on the page. This new edition of this classic work has been completely reset in TeX, making the text and equations far easier to read. The book has also seen the correction of a handful of typographic errors, revision of some sentences for clarity and readability, provision of an index for the first time, and prefatory remarks drawn from the writings of Léon Van Hove and Freeman Dyson have been added. The result brings new life to an essential work in theoretical physics and mathematics.Less
Quantum mechanics was still in its infancy in 1932 when the young John von Neumann, who would go on to become one of the greatest mathematicians of the twentieth century, published Mathematical Foundations of Quantum Mechanics—a revolutionary book that for the first time provided a rigorous mathematical framework for the new science. Robert Beyer's 1955 English translation, which von Neumann reviewed and approved, is cited more frequently today than ever before. But its many treasures and insights were too often obscured by the limitations of the way the text and equations were set on the page. This new edition of this classic work has been completely reset in TeX, making the text and equations far easier to read. The book has also seen the correction of a handful of typographic errors, revision of some sentences for clarity and readability, provision of an index for the first time, and prefatory remarks drawn from the writings of Léon Van Hove and Freeman Dyson have been added. The result brings new life to an essential work in theoretical physics and mathematics.
Jürgen Renn and Hanoch Gutfreund
- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691174631
- eISBN:
- 9781400888689
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691174631.001.0001
- Subject:
- Physics, History of Physics
First published in 1922 and based on lectures delivered in May 1921, Albert Einstein's The Meaning of Relativity offered an overview and explanation of the then new and controversial theory of ...
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First published in 1922 and based on lectures delivered in May 1921, Albert Einstein's The Meaning of Relativity offered an overview and explanation of the then new and controversial theory of relativity. The work would go on to become a monumental classic, printed in numerous editions and translations worldwide. Now, this book introduces Einstein's masterpiece to new audiences. The volume contains Einstein's insightful text, accompanied by important historical materials and commentary looking at the origins and development of general relativity. The book provides fresh, original perspectives, placing Einstein's achievements into a broader context for all readers. It tells the rich story behind the early reception, spread, and consequences of Einstein's ideas during the formative years of general relativity in the late 1910s and 1920s. Relativity's meaning changed radically throughout the nascent years of its development, and the book describes in detail the transformation of Einstein's work from the esoteric pursuit of one individual communicating with a handful of colleagues into the preoccupation of a growing community of physicists, astronomers, mathematicians, and philosophers. The book quotes extensively from Einstein's correspondence and reproduces historical documents such as newspaper articles and letters. Inserts are featured in the main text giving concise explanations of basic concepts, and short biographical notes and photographs of some of Einstein's contemporaries are included. The first-ever English translations of two of Einstein's popular Princeton lectures are featured at the book's end.Less
First published in 1922 and based on lectures delivered in May 1921, Albert Einstein's The Meaning of Relativity offered an overview and explanation of the then new and controversial theory of relativity. The work would go on to become a monumental classic, printed in numerous editions and translations worldwide. Now, this book introduces Einstein's masterpiece to new audiences. The volume contains Einstein's insightful text, accompanied by important historical materials and commentary looking at the origins and development of general relativity. The book provides fresh, original perspectives, placing Einstein's achievements into a broader context for all readers. It tells the rich story behind the early reception, spread, and consequences of Einstein's ideas during the formative years of general relativity in the late 1910s and 1920s. Relativity's meaning changed radically throughout the nascent years of its development, and the book describes in detail the transformation of Einstein's work from the esoteric pursuit of one individual communicating with a handful of colleagues into the preoccupation of a growing community of physicists, astronomers, mathematicians, and philosophers. The book quotes extensively from Einstein's correspondence and reproduces historical documents such as newspaper articles and letters. Inserts are featured in the main text giving concise explanations of basic concepts, and short biographical notes and photographs of some of Einstein's contemporaries are included. The first-ever English translations of two of Einstein's popular Princeton lectures are featured at the book's end.
Leonid Zhmud
- Published in print:
- 2012
- Published Online:
- September 2012
- ISBN:
- 9780199289318
- eISBN:
- 9780191741371
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199289318.003.0002
- Subject:
- Classical Studies, Ancient Greek, Roman, and Early Christian Philosophy
This chapter reviews references to Pythagoras by the authors of the pre-Platonic period. It shows that in the course of the fourth century, studies in mathematics, particularly geometry and ...
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This chapter reviews references to Pythagoras by the authors of the pre-Platonic period. It shows that in the course of the fourth century, studies in mathematics, particularly geometry and arithmetic, became a constant element of the tradition of Pythagoras; astronomy and harmonics are less frequently mentioned. Mathematics did not displace metempsychosis and wonders, nor did the tradition of Pythagoras the politician which emerged concurrently with it, yet they did edge them aside, completing the ambivalent, contradictory image of Pythagoras, which was retained by his Neoplatonic biographers and passed from them into modern scholarship. It concludes that Pythagoras the mathematician is as little a product of the Academy as Pythagoras the philosopher.Less
This chapter reviews references to Pythagoras by the authors of the pre-Platonic period. It shows that in the course of the fourth century, studies in mathematics, particularly geometry and arithmetic, became a constant element of the tradition of Pythagoras; astronomy and harmonics are less frequently mentioned. Mathematics did not displace metempsychosis and wonders, nor did the tradition of Pythagoras the politician which emerged concurrently with it, yet they did edge them aside, completing the ambivalent, contradictory image of Pythagoras, which was retained by his Neoplatonic biographers and passed from them into modern scholarship. It concludes that Pythagoras the mathematician is as little a product of the Academy as Pythagoras the philosopher.
Michael Harris
- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691175836
- eISBN:
- 9781400885527
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175836.001.0001
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
What do pure mathematicians do, and why do they do it? Looking beyond the conventional answers, this book offers an eclectic panorama of the lives and values and hopes and fears of mathematicians in ...
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What do pure mathematicians do, and why do they do it? Looking beyond the conventional answers, this book offers an eclectic panorama of the lives and values and hopes and fears of mathematicians in the twenty-first century, assembling material from a startlingly diverse assortment of scholarly, journalistic, and pop culture sources. Drawing on the author's personal experiences as well as the thoughts and opinions of mathematicians from Archimedes and Omar Khayyám to such contemporary giants as Alexander Grothendieck and Robert Langlands, the book reveals the charisma and romance of mathematics as well as its darker side. In this portrait of mathematics as a community united around a set of common intellectual, ethical, and existential challenges, the book touches on a wide variety of questions, such as: Are mathematicians to blame for the 2008 financial crisis? How can we talk about the ideas we were born too soon to understand? And how should you react if you are asked to explain number theory at a dinner party? The book takes readers on an unapologetic guided tour of the mathematical life, from the philosophy and sociology of mathematics to its reflections in film and popular music, with detours through the mathematical and mystical traditions of Russia, India, medieval Islam, the Bronx, and beyond.Less
What do pure mathematicians do, and why do they do it? Looking beyond the conventional answers, this book offers an eclectic panorama of the lives and values and hopes and fears of mathematicians in the twenty-first century, assembling material from a startlingly diverse assortment of scholarly, journalistic, and pop culture sources. Drawing on the author's personal experiences as well as the thoughts and opinions of mathematicians from Archimedes and Omar Khayyám to such contemporary giants as Alexander Grothendieck and Robert Langlands, the book reveals the charisma and romance of mathematics as well as its darker side. In this portrait of mathematics as a community united around a set of common intellectual, ethical, and existential challenges, the book touches on a wide variety of questions, such as: Are mathematicians to blame for the 2008 financial crisis? How can we talk about the ideas we were born too soon to understand? And how should you react if you are asked to explain number theory at a dinner party? The book takes readers on an unapologetic guided tour of the mathematical life, from the philosophy and sociology of mathematics to its reflections in film and popular music, with detours through the mathematical and mystical traditions of Russia, India, medieval Islam, the Bronx, and beyond.
Charles Fefferman, Alexandru D. Ionescu, D.H. Phong, and Stephen Wainger
- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159416
- eISBN:
- 9781400848935
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159416.001.0001
- Subject:
- Mathematics, Numerical Analysis
Princeton University's Elias Stein was the first mathematician to see the profound interconnections that tie classical Fourier analysis to several complex variables and representation theory. His ...
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Princeton University's Elias Stein was the first mathematician to see the profound interconnections that tie classical Fourier analysis to several complex variables and representation theory. His fundamental contributions include the Kunze–Stein phenomenon, the construction of new representations, the Stein interpolation theorem, the idea of a restriction theorem for the Fourier transform, and the theory of Hp Spaces in several variables. Through his great discoveries, through books that have set the highest standard for mathematical exposition, and through his influence on his many collaborators and students, Stein has changed mathematics. Drawing inspiration from Stein's contributions to harmonic analysis and related topics, this book gathers papers from internationally renowned mathematicians, many of whom have been Stein's students. The book also includes expository papers on Stein's work and its influence.Less
Princeton University's Elias Stein was the first mathematician to see the profound interconnections that tie classical Fourier analysis to several complex variables and representation theory. His fundamental contributions include the Kunze–Stein phenomenon, the construction of new representations, the Stein interpolation theorem, the idea of a restriction theorem for the Fourier transform, and the theory of Hp Spaces in several variables. Through his great discoveries, through books that have set the highest standard for mathematical exposition, and through his influence on his many collaborators and students, Stein has changed mathematics. Drawing inspiration from Stein's contributions to harmonic analysis and related topics, this book gathers papers from internationally renowned mathematicians, many of whom have been Stein's students. The book also includes expository papers on Stein's work and its influence.
Arthur Benjamin, Gary Chartrand, and Ping Zhang
- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691175638
- eISBN:
- 9781400852000
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175638.003.0013
- Subject:
- Mathematics, Applied Mathematics
This book concludes with an epilogue, which traces the evolution of graph theory, from the conceptualization of the Königsberg Bridge Problem and its generalization by Leonhard Euler, whose solution ...
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This book concludes with an epilogue, which traces the evolution of graph theory, from the conceptualization of the Königsberg Bridge Problem and its generalization by Leonhard Euler, whose solution led to the subject of Eulerian graphs, to the various efforts to solve the Four Color Problem. It considers elements of graph theory found in games and puzzles of the past, and the famous mathematicians involved including Sir William Rowan Hamilton and William Tutte. It also discusses the remarkable increase since the 1960s in the number of mathematicians worldwide devoted to graph theory, along with research journals, books, and monographs that have graph theory as a subject. Finally, it looks at the growth in applications of graph theory dealing with communication and social networks and the Internet in the digital age and the age of technology.Less
This book concludes with an epilogue, which traces the evolution of graph theory, from the conceptualization of the Königsberg Bridge Problem and its generalization by Leonhard Euler, whose solution led to the subject of Eulerian graphs, to the various efforts to solve the Four Color Problem. It considers elements of graph theory found in games and puzzles of the past, and the famous mathematicians involved including Sir William Rowan Hamilton and William Tutte. It also discusses the remarkable increase since the 1960s in the number of mathematicians worldwide devoted to graph theory, along with research journals, books, and monographs that have graph theory as a subject. Finally, it looks at the growth in applications of graph theory dealing with communication and social networks and the Internet in the digital age and the age of technology.
Michael Harris
- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691175836
- eISBN:
- 9781400885527
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175836.003.0012
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
This chapter continues the discussion began in Chapter α. It presents the third part of author's response to the question, “What is it you do in number theory, anyway?” Working his way up to the ...
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This chapter continues the discussion began in Chapter α. It presents the third part of author's response to the question, “What is it you do in number theory, anyway?” Working his way up to the guiding problem of the Birch–Swinnerton–Dyer conjecture, here he deals with congruences, specifically how to count solutions to congruences. The story of congruences is that a problem where the variables can take infinitely many values can be replaced by one in which the variables can take only finitely many values, and it is sometimes enough to solve the latter problem in order to solve the former.Less
This chapter continues the discussion began in Chapter α. It presents the third part of author's response to the question, “What is it you do in number theory, anyway?” Working his way up to the guiding problem of the Birch–Swinnerton–Dyer conjecture, here he deals with congruences, specifically how to count solutions to congruences. The story of congruences is that a problem where the variables can take infinitely many values can be replaced by one in which the variables can take only finitely many values, and it is sometimes enough to solve the latter problem in order to solve the former.
Leon Horsten
- Published in print:
- 2011
- Published Online:
- August 2013
- ISBN:
- 9780262015868
- eISBN:
- 9780262298643
- Item type:
- book
- Publisher:
- The MIT Press
- DOI:
- 10.7551/mitpress/9780262015868.001.0001
- Subject:
- Philosophy, General
This book investigates the relationship between formal theories of truth and contemporary philosophical approaches to truth. The work of mathematician and logician Alfred Tarski (1901–1983) marks the ...
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This book investigates the relationship between formal theories of truth and contemporary philosophical approaches to truth. The work of mathematician and logician Alfred Tarski (1901–1983) marks the transition from substantial to deflationary views about truth. Deflationism — which holds that the notion of truth is light and insubstantial — can be and has been made more precise in multiple ways. Crucial in making the deflationary intuition precise is its relation to formal or logical aspects of the notion of truth. Allowing that semantical theories of truth may have heuristic value, this book focuses on axiomatic theories of truth developed since Tarski and their connection to deflationism. Arguing that the insubstantiality of truth has been misunderstood in the literature, it proposes and defends a new kind of deflationism, inferential deflationism, according to which truth is a concept without a nature or essence. The book argues that this way of viewing the concept of truth, inspired by a formalization of Kripke’s theory of truth, flows naturally from the best formal theories of truth that are currently available. Alternating between logical and philosophical chapters, the book steadily progresses toward stronger theories of truth. Technicality cannot be altogether avoided in the subject under discussion, but the book attempts to strike a balance between the need for logical precision on the one hand and the need to make his argument accessible to philosophers.Less
This book investigates the relationship between formal theories of truth and contemporary philosophical approaches to truth. The work of mathematician and logician Alfred Tarski (1901–1983) marks the transition from substantial to deflationary views about truth. Deflationism — which holds that the notion of truth is light and insubstantial — can be and has been made more precise in multiple ways. Crucial in making the deflationary intuition precise is its relation to formal or logical aspects of the notion of truth. Allowing that semantical theories of truth may have heuristic value, this book focuses on axiomatic theories of truth developed since Tarski and their connection to deflationism. Arguing that the insubstantiality of truth has been misunderstood in the literature, it proposes and defends a new kind of deflationism, inferential deflationism, according to which truth is a concept without a nature or essence. The book argues that this way of viewing the concept of truth, inspired by a formalization of Kripke’s theory of truth, flows naturally from the best formal theories of truth that are currently available. Alternating between logical and philosophical chapters, the book steadily progresses toward stronger theories of truth. Technicality cannot be altogether avoided in the subject under discussion, but the book attempts to strike a balance between the need for logical precision on the one hand and the need to make his argument accessible to philosophers.
R. Netz
- Published in print:
- 2002
- Published Online:
- February 2010
- ISBN:
- 9780198152484
- eISBN:
- 9780191710049
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198152484.003.0011
- Subject:
- Classical Studies, European History: BCE to 500CE
This chapter examines who the Greek mathematicians were by taking an imaginary group picture taken at a conference of ancient Greek mathematicians — a conference held in heaven, so that everyone is ...
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This chapter examines who the Greek mathematicians were by taking an imaginary group picture taken at a conference of ancient Greek mathematicians — a conference held in heaven, so that everyone is present, from classical times down to late antiquity. For this purpose, a mathematician is defined as one who has written down an original mathematical demonstration, no matter in what context. Two well-documented women in this group are Hypatia and Pandrosion. It is probably relevant that both of these examples are from late Alexandria, a place and a time where many old barriers were brought down. In general ancient women did not live strictly according to the expectation of either classical society or modern scholarship, and they were not always ‘silent’. The Greek mathematicians did not have faculties and conferences, a fact reflected by their form of presentation and ultimately by the contents of their mathematics.Less
This chapter examines who the Greek mathematicians were by taking an imaginary group picture taken at a conference of ancient Greek mathematicians — a conference held in heaven, so that everyone is present, from classical times down to late antiquity. For this purpose, a mathematician is defined as one who has written down an original mathematical demonstration, no matter in what context. Two well-documented women in this group are Hypatia and Pandrosion. It is probably relevant that both of these examples are from late Alexandria, a place and a time where many old barriers were brought down. In general ancient women did not live strictly according to the expectation of either classical society or modern scholarship, and they were not always ‘silent’. The Greek mathematicians did not have faculties and conferences, a fact reflected by their form of presentation and ultimately by the contents of their mathematics.
JESPER LÜTZEN
- Published in print:
- 2005
- Published Online:
- January 2010
- ISBN:
- 9780198567370
- eISBN:
- 9780191717925
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567370.003.0024
- Subject:
- Physics, History of Physics
It has long since been remarked by mathematicians that William Rowan Hamilton's method contains purely geometrical truths, and that a peculiar mode of expression, suitable to it, is required in order ...
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It has long since been remarked by mathematicians that William Rowan Hamilton's method contains purely geometrical truths, and that a peculiar mode of expression, suitable to it, is required in order to express these clearly. But this fact has only come to light in a somewhat perplexing form, namely, in the analogies between ordinary mechanics and the geometry of space of many dimensions. Together with an explicit reference to the work of Eugenio Beltrami, Rudolf Lipschitz, and Jean-Gaston Darboux, Heinrich Hertz made only one other reference to the work on mechanics done by contemporary mathematicians. Hertz correctly connected their work with his own treatment of the Hamiltonian formalism. This chapter gives a short summary of the mathematical developments in Hamilton-Jacobi formalism during the period 1828-1888 and compares them with those of Hertz. The views of Johann Carl Friedrich Gauss and Hamilton on geodesics, optics, and dynamics are discussed, along with those of Joseph Liouville and Lipschitz on the principle of least action, and trajectories as geodesics.Less
It has long since been remarked by mathematicians that William Rowan Hamilton's method contains purely geometrical truths, and that a peculiar mode of expression, suitable to it, is required in order to express these clearly. But this fact has only come to light in a somewhat perplexing form, namely, in the analogies between ordinary mechanics and the geometry of space of many dimensions. Together with an explicit reference to the work of Eugenio Beltrami, Rudolf Lipschitz, and Jean-Gaston Darboux, Heinrich Hertz made only one other reference to the work on mechanics done by contemporary mathematicians. Hertz correctly connected their work with his own treatment of the Hamiltonian formalism. This chapter gives a short summary of the mathematical developments in Hamilton-Jacobi formalism during the period 1828-1888 and compares them with those of Hertz. The views of Johann Carl Friedrich Gauss and Hamilton on geodesics, optics, and dynamics are discussed, along with those of Joseph Liouville and Lipschitz on the principle of least action, and trajectories as geodesics.
Michael Harris
- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691175836
- eISBN:
- 9781400885527
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175836.003.0014
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
This chapter presents the author's reflections about G. H. Hardy's A Mathematician's Apology. Upon rereading Hardy's Apology to prepare for writing this chapter, he was appalled to see what a hearty ...
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This chapter presents the author's reflections about G. H. Hardy's A Mathematician's Apology. Upon rereading Hardy's Apology to prepare for writing this chapter, he was appalled to see what a hearty dose of unapologetic elitism he had imbibed along with Hardy's mathematical idealism when he was fourteen years old. When it began to dawn on him that some of the assumptions he had been taking for granted deserved to be questioned, the reading was buried so deeply in his past that he could not trace these beliefs back to their source. But years later, upon returning to A Mathematician's Apology, there they were.Less
This chapter presents the author's reflections about G. H. Hardy's A Mathematician's Apology. Upon rereading Hardy's Apology to prepare for writing this chapter, he was appalled to see what a hearty dose of unapologetic elitism he had imbibed along with Hardy's mathematical idealism when he was fourteen years old. When it began to dawn on him that some of the assumptions he had been taking for granted deserved to be questioned, the reading was buried so deeply in his past that he could not trace these beliefs back to their source. But years later, upon returning to A Mathematician's Apology, there they were.
