This book considers the manifold possible approaches, past and present, to our understanding of the natural numbers. They are treated as epistemic objects: mathematical objects that have been subject to epistemological inquiry and attention throughout their history and whose conception has evolved accordingly. Although they are the simplest and most common mathematical objects, as this book reveals, they have a very complex nature whose study illuminates subtle features of the functioning of our thought. Using jointly history, mathematics and philosophy to grasp the essence of numbers, the reader is led through their various interpretations, presenting the ways they have been involved in major theoretical projects from Thales onward. Some pertain primarily to philosophy (as in the works of Plato, Aristotle, Kant, Wittgenstein...), others to general mathematics (Euclid's Elements, Cartesian algebraic geometry, Cantorian infinities, set theory...). Also serving as an introduction to the works and thought of major mathematicians and philosophers, from Plato and Aristotle to Cantor, Dedekind, Frege, Husserl and Weyl, this book will be of interest to a wide variety of readers, from scholars with a general interest in the philosophy or mathematics to philosophers and mathematicians themselves.

This work documents the history of techniques that statisticians have used to manipulate economic, meteorological, biological and physical data taken from observations recorded over time. The manipulation tools include per cent change, index numbers, moving averages and 'first differences', i.e., subtracting one observation from the previous value. Professor Klein argues that nineteenth-century business journals, such as The Economist, were as important to the development of time series analysis as Latin treatises on probability theory. While examining the roots of mathematical statistics in commercial practice, she traces changes in analytical forms from table to graph to equation. Klein cautions that we risk measurement without history in unduly mechanistic blending of stationary probability theory with the practical dynamics of commercial traders. This history is accessible to students with a basic knowledge of statistics as well as financial analysts, statisticians and historians of economic thought and science.

As Eugene Wigner stressed, mathematics has proven unreasonably effective in the physical sciences and their technological applications. The role of mathematics in the biological, medical and social sciences has been much more modest but has recently grown thanks to the simulation capacity offered by modern computers.

This book traces the history of population dynamics---a theoretical subject closely connected to genetics, ecology, epidemiology and demography---where mathematics has brought significant insights. It presents an overview of the genesis of several important themes: exponential growth, from Euler and Malthus to the Chinese one-child policy; the development of stochastic models, from Mendel's laws and the question of extinction of family names to percolation theory for the spread of epidemics, and chaotic populations, where determinism and randomness intertwine.

The reader of this book will see, from a different perspective, the problems that scientists face when governments ask for reliable predictions to help control epidemics (AIDS, SARS, swine flu), manage renewable resources (fishing quotas, spread of genetically modified organisms) or anticipate demographic evolutions such as aging."

What is the best way to photograph a speeding bullet? Why does light move through glass in the least amount of time possible? How can lost hikers find their way out of a forest? What will rainbows look like in the future? Why do soap bubbles have a shape that gives them the least area?

By combining the mathematical history of extrema with contemporary examples, Paul J. Nahin answers these intriguing questions and more in this engaging and witty volume. He shows how life often works at the extremes--with values becoming as small (or as large) as possible--and how mathematicians over the centuries have struggled to calculate these problems of minima and maxima. From medieval writings to the development of modern calculus to the current field of optimization, Nahin tells the story of Dido's problem, Fermat and Descartes, Torricelli, Bishop Berkeley, Goldschmidt, and more. Along the way, he explores how to build the shortest bridge possible between two towns, how to shop for garbage bags, how to vary speed during a race, and how to make the perfect basketball shot.

Written in a conversational tone and requiring only an early undergraduate level of mathematical knowledge, "When Least Is Best" is full of fascinating examples and ready-to-try-at-home experiments. This is the first book on optimization written for a wide audience, and math enthusiasts of all backgrounds will delight in its lively topics.

The transformation of mathematics from its ancient Greek practice to its development in the medieval Arab-speaking world is approached by focusing on a single problem proposed by Archimedes and the many solutions offered. From a practice of mathematics based on the localized solution (originating in the polemical practices of early Greek science), we see a transition to a practice of mathematics based on the systematic approach (grounded in the deuteronomic practices of Late Antiquity and the Middle Ages). A radically new interpretation is accordingly offered of the historical trajectory of pre-modern mathematics.

Charles Babbage (1791–1871) is today remembered mainly for his attempt to complete his difference and analytical engines, the principles of which anticipate the major ideas of the modern digital computer. This book describes the evolution of Babbage’s work on the design and implementation of the engines by means of a detailed study of his early mathematical investigations. Babbage is an almost legendary figure of the Victorian era, yet relatively little is known about him and no authoritative account of his life and work has appeared. He was primarily a mathematician and his early working life was devoted mainly to the study of pure mathematics. While containing much biographical information, this book concentrates on this crucial aspect of Babbage’s work.

By any measure, the Pythagorean theorem is the most famous statement in all of mathematics. In this book, Eli Maor reveals the full story of this ubiquitous geometric theorem. Maor shows that the theorem, although attributed to Pythagoras, was known to the Babylonians more than a thousand years earlier. Pythagoras may have been the first to prove it, but his proof--if indeed he had one--is lost to us. The theorem itself, however, is central to almost every branch of science, pure or applied. Maor brings to life many of the characters that played a role in the development of the Pythagorean theorem, providing a fascinating backdrop to perhaps our oldest enduring mathematical legacy.

This book provides a way to understand a momentous development in human intellectual history: the phenomenon of deductive argument in classical Greek mathematics. The argument rests on a close description of the practices of Greek mathematics, principally the use of lettered diagrams and the regulated, formulaic use of language.

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? Experience shows that disentangling scientific knowledge from opinion is harder than one might expect. Full of illuminating examples and quotations, and with a scope ranging from psychology and evolution to quantum theory and mathematics, this book brings alive issues at the heart of all science.

The logician Kurt Goedel (1906-1978) published a paper in 1931 formulating what have come to be known as his 'incompleteness theorems', which prove, among other things, that within any formal system with resources sufficient to code arithmetic, questions exist which are neither provable nor disprovable on the basis of the axioms which define the system. These are among the most celebrated results in logic today. In this volume, leading philosophers and mathematicians assess important aspects of Goedel's work on the foundations and philosophy of mathematics. Their essays explore almost every aspect of Godel's intellectual legacy including his concepts of intuition and analyticity, the Completeness Theorem, the set-theoretic multiverse, and the state of mathematical logic today. This groundbreaking volume will be invaluable to students, historians, logicians and philosophers of mathematics who wish to understand the current thinking on these issues.

One of the greatest revolutions in mathematics occurred when Georg Cantor (1845-1918) promulgated his theory of transfinite sets. This revolution is the subject of Joseph Dauben's important studythe most thorough yet writtenof the philosopher and mathematician who was once called a "corrupter of youth" for an innovation that is now a vital component of elementary school curricula.

Set theory has been widely adopted in mathematics and philosophy, but the controversy surrounding it at the turn of the century remains of great interest. Cantor's own faith in his theory was partly theological. His religious beliefs led him to expect paradoxes in any concept of the infinite, and he always retained his belief in the utter veracity of transfinite set theory. Later in his life, he was troubled by recurring attacks of severe depression. Dauben shows that these played an integral part in his understanding and defense of set theory.

An era of sweeping cultural change in America, the postwar years saw the rise of beatniks and hippies, the birth of feminism, and the release of the first video game. It was also the era of new math. Introduced to US schools in the late 1950s and 1960s, the new math was a curricular answer to Cold War fears of American intellectual inadequacy. In the age of Sputnik and increasingly sophisticated technological systems and machines, math class came to be viewed as a crucial component of the education of intelligent, virtuous citizens who would be able to compete on a global scale. In this history, Christopher J. Phillips examines the rise and fall of the new math as a marker of the period's political and social ferment. Neither the new math curriculum designers nor its diverse legions of supporters concentrated on whether the new math would improve students' calculation ability. Rather, they felt the new math would train children to think in the right way, instilling in students a set of mental habits that might better prepare them to be citizens of modern society a world of complex challenges, rapid technological change, and unforeseeable futures. While Phillips grounds his argument in shifting perceptions of intellectual discipline and the underlying nature of mathematical knowledge, he also touches on long-standing debates over the place and relevance of mathematics in liberal education. And in so doing, he explores the essence of what it means to be an intelligent American by the numbers.

One of the paradoxes of the physical sciences is that as our knowledge has progressed, more and more diverse physical phenomena can be explained in terms of fewer underlying laws, or principles. In Hidden Unity, eminent physicist John Taylor puts many of these findings into historical perspective and documents how progress is made when unexpected, hidden unities are uncovered between apparently unrelated physical phenomena. Taylor cites examples from the ancient Greeks to the present day, such as the unity of celestial and terrestrial dynamics (17th century), the unity of heat within the rest of dynamics (18th century), the unity of electricity, magnetism, and light (19th century), the unity of space and time and the unification of nuclear forces with electromagnetism (20th century). Without relying on mathematical detail, Taylor's emphasis is on fundamental physics, like particle physics and cosmology. Balancing what is understood with the unestablished theories and still unanswered questions, Taylor takes readers on a fascinating ongoing journey. John C. Taylor is Professor Emeritus of Mathematical Physics at the University of Cambridge. A student of Nobel laureate Abdus Salam, Taylor's research career has spanned the era of developments in elementary particle physics since the 1950s. He taught theoretical physics at Imperial College, London, and at the Universities of Oxford and Cambridge, and he has lectured worldwide. He is a Fellow of the Royal Society and a Fellow of the Institute of Physics.

This book is at once an analytical study of one of the most important mathematical texts of antiquity, the Mathematical Collection of the fourth-century AD mathematician Pappus of Alexandria, and also an examination of the work's wider cultural setting. This is one of very few books to deal extensively with the mathematics of Late Antiquity. It sees Pappus' text as part of a wider context and relates it to other contemporary cultural practices and opens new avenues to research into the public understanding of mathematics and mathematical disciplines in antiquity.

2013 Reprint of 1931 edition. Full facsimile of the original edition, not reproduced with Optical Recognition Software. Frank Plumpton Ramsey (1903-1930) was a British mathematician who also made significant and precocious contributions in philosophy and economics before his death at the age of 26. He was a close friend of Ludwig Wittgenstein, and was instrumental in translating Wittgenstein's "Tractatus Logico-Philosophicus" into English, and in persuading Wittgenstein to return to philosophy and to Cambridge. This volume collects Ramsey's most important papers. Contents: The foundations of mathematics.--Mathematical logic.--On a problem of formal logic.--Universals.--Note on the preceding paper.--Facts and propositions.--Truth and probability.--Further considerations.--Last papers.

A FINANCIAL TIMES AND TLS BOOK OF THE YEAR An exhilarating new biography of John von Neumann: the lost genius who invented our world 'A sparkling book, with an intoxicating mix of pen-portraits and grand historical narrative. Above all it fizzes with a dizzying mix of deliciously vital ideas. . . A staggering achievement' Tim Harford The smartphones in our pockets and computers like brains. The vagaries of game theory and evolutionary biology. Self-replicating moon bases and nuclear weapons. All bear the fingerprints of one remarkable man: John von Neumann. Born in Budapest at the turn of the century, von Neumann is one of the most influential scientists to have ever lived. His colleagues believed he had the fastest brain on the planet - bar none. He was instrumental in the Manhattan Project and helped formulate the bedrock of Cold War geopolitics and modern economic theory. He created the first ever programmable digital computer. He prophesied the potential of nanotechnology and, from his deathbed, expounded on the limits of brains and computers - and how they might be overcome. Taking us on an astonishing journey, Ananyo Bhattacharya explores how a combination of genius and unique historical circumstance allowed a single man to sweep through so many different fields of science, sparking revolutions wherever he went. Insightful and illuminating, The Man from the Future is a thrilling intellectual biography of the visionary thinker who shaped our century.

Symbols, Impossible Numbers, and Geometric Entanglements is the first history of the development and reception of algebra in early modern England and Scotland. Not primarily a technical history, this book analyses the struggles of a dozen British thinkers to come to terms with early modern algebra, its symbolic style, and negative and imaginary numbers. Professor Pycior uncovers these thinkers as a 'test-group' for the symbolic reasoning that would radically change not only mathematics but also logic, philosophy and language studies. The book furthermore shows how pedagogical and religious concerns shaped the British debate over the relative merits of algebra and geometry. Positioning algebra firmly in the Scientific Revolution and pursue Newton the algebraist, it highlights Newton's role in completing the evolution of algebra from an esoteric subject into a major focus of British mathematics. Other thinkers covered include Oughtred, Harriot, Wallis, Hobbes, Barrow, Berkeley and MacLaurin.

The history of mathematics is filled with major breakthroughs resulting from solutions to recreational problems. Problems of interest to gamblers led to the modern theory of probability, for example, and surreal numbers were inspired by the game of Go. Yet even with such groundbreaking findings and a wealth of popular-level books, research in recreational mathematics has often been neglected. The Mathematics of Various Entertaining Subjects now returns with a brand-new compilation of fascinating problems and solutions in recreational mathematics. This latest volume gathers together the top experts in recreational math and presents a compelling look at board games, card games, dice, toys, computer games, and much more. The book is divided into five parts: puzzles and brainteasers, geometry and topology, graph theory, games of chance, and computational complexity. Readers will discover what origami, roulette wheels, and even the game of Trouble can teach about math. Essays contain new results, and the contributors include short expositions on their topic's background, providing a framework for understanding the relationship between serious mathematics and recreational games. Mathematical areas explored include combinatorics, logic, graph theory, linear algebra, geometry, topology, computer science, operations research, probability, game theory, and music theory. Investigating an eclectic mix of games and puzzles, The Mathematics of Various Entertaining Subjects is sure to entertain, challenge, and inspire academic mathematicians and avid math enthusiasts alike.

Max Tegmark leads us on an astonishing journey through past, present and future, and through the physics, astronomy and mathematics that are the foundation of his work, most particularly his hypothesis that our physical reality is a mathematical structure and his theory of the ultimate multiverse. In a dazzling combination of both popular and groundbreaking science, he not only helps us grasp his often mind-boggling theories, but he also shares with us some of the often surprising triumphs and disappointments that have shaped his life as a scientist. Fascinating from first to last--this is a book that has already prompted the attention and admiration of some of the most prominent scientists and mathematicians.

Social revolutions--critical periods of decisive, qualitative change--are a commonly acknowledged historical fact. The publication of Kuhn's The Structure of Scientific Revolutions in 1962 led to an exciting discussion of revolutions in the natural sciences; an off-shoot of this was a debate in the United States in the mid-1970's as to whether the concept of revolution could be applied to mathematics as well as science. This book is the first comprehensive examination of the question. It reprints the original papers of leading supporters and opponents, together with additional chapters giving their current views. To this are added new contributions from nine other experts in the history of mathematics, who each discuss an important episode and consider whether it was a revolution. The whole question of mathematical revolutions is thus examined comprehensively and from a variety of perspectives, and will interest mathematicians, philosophers, and historians alike.