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Moving the Alps: uncovering the mathematics of John Pell

Jacqueline A. Stedall

in A Discourse Concerning Algebra: English Algebra to 1685

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 ... More

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

Why Spin Glasses?

Daniel L. Stein and Charles M. Newman

in Spin Glasses and Complexity

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 ... More

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

From Voyagers to Martyrs: Toward a Storied History of Mathematics

Amir Alexander

in Circles Disturbed: The Interplay of Mathematics and Narrative

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 ... More

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

Narrative and the Rationality of Mathematical Practice

David Corfield

Apostolos Doxiadis and Barry Mazur (eds)

in Circles Disturbed: The Interplay of Mathematics and Narrative

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 ... More

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

Mathematics and Narrative: An Aristotelian Perspective

G. E. R. Lloyd

Apostolos Doxiadis and Barry Mazur (eds)

in Circles Disturbed: The Interplay of Mathematics and Narrative

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 ... More

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

Algebraic Thought in Ancient and Medieval China

Victor J. Katz and Karen Hunger Parshall

in Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century

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 ... More

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

Algebraic Thought in Medieval India

Victor J. Katz and Karen Hunger Parshall

in Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century

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 ... More

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

Mathematical Mysteries in Byzantium: The Transmission of Fermat’S Last Theorem

Judith Herrin

in Margins and Metropolis: Authority across the Byzantine Empire

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 ... More

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

The Early Tradition on Pythagoras and Its Development

Leonid Zhmud

in Pythagoras and the Early Pythagoreans

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 ... More

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

Graph Theory: A Look Back—The Road Ahead

Arthur Benjamin, Gary Chartrand, and Ping Zhang

in The Fascinating World of Graph Theory

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 ... More

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

How to Explain Number Theory at a Dinner Party

Michael Harris

in Mathematics without Apologies: Portrait of a Problematic Vocation

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 ... More

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

Greek Mathematicians: A Group Picture

R. Netz

in Science and Mathematics in Ancient Greek Culture

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 ... More

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

Mathematicians on the geometrization of the Hamilton–Jacobi formalism

JESPER LÜTZEN

in Mechanistic Images in Geometric Form: Heinrich Hertz's 'Principles of Mechanics'

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 ... More

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

No Apologies

Michael Harris

in Mathematics without Apologies: Portrait of a Problematic Vocation

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 ... More

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