"You believe in a God who plays dice, and I in complete law and order", Albert Einstein. The science of chaos is forcing scientists to rethink Einstein's fundamental assumptions regarding the way the universe behaves. Chaos theory has already shown that simple systems, obeying precise laws, can nevertheless act in a random manner. Perhaps God plays dice within a cosmic game of complete law and order. "Does God Play Dice?" reveals a strange universe in which nothing may be as it seems. Familiar geometrical shapes such as circles and ellipses give way to infinitely complex structures known as fractals, the fluttering of a butterfly's wings can change the weather, and the gravitational attraction of a creature in a distant galaxy can change the fate of the solar system. This revised and updated edition includes three completely new chapters on the prediction and control of chaotic systems. It also incorporates new information regarding the solar system and an account of complexity theory. This text aims to make the complex mathematics of chaos accessible and entertaining.

"I found the book to be fascinating, and the author's presentations and illustrations of the contrasts bewtween mathematical reasoning and scientific reasoning were especially appealing. The book is rich in history, which is carefully integrated into the discussions, and it includes wonderful illustrations and stories." --The Mathematics Teacher "Zebrowski is a wonderful storyteller, and his choices of topics reveal not only the depth of explanation afforded by the available mathematics but the beauty in the explanations; he succeeds in keeping the explanations accessible to the most general audience." --Choice The concept of the circle is ubiquitous. It can be described mathematically, represented physically, and employed technologically. The circle is an elegant, abstract form that has been transformed by humans into tangible, practical forms to make our lives easier. And yet no one has ever discovered a true mathematical circle. Rainbows are fuzzy, car tires are flat on the bottom, and even the most precise roller bearings have measurable irregularities. Ernest Zebrowski, Jr., discusses how investigations into the circle have contributed enormously to our current knowledge of the physical universe. Beginning with the ancient mathematicians and culminating in twentieth-century theories of space and time, the mathematics of the circle has pointed many investigators in fruitful directions in their quests to unravel nature's secrets. Johannes Kepler, for example, triggered a scientific revolution in 1609 when he challenged the conception of the earth's circular motion around the sun. Arab and European builders instigated a golden age of mosque and cathedral building when they questioned the Roman structural arches that were limited to geometrical semicircles. Throughout his book, Zebrowski emphasizes the concepts underlying these mathematicians' calculations, and how these concepts are linked to real-life examples. Substantiated by easy-to-follow mathematical reasoning and clear illustrations, this accessible book presents a novel and interesting discussion of the circle in technology, culture, history, and science. Ernest Zebrowski, Jr., hold professorships in science and mathematics education at Southern University in Baton Rouge, and in physics at Pennsylvania College of Technology of the Pennsylvania State University. He is the author of Perils of a Restless Planet: Scientific Perspectives on Natural Disasters and The Last Days of St. Pierre: The Volcanic Disaster That Claimed 30,000 Lives.

This collection of new essays offers a "state-of-the-art" conspectus of major trends in the philosophy of logic and philosophy of mathematics. A distinguished group of philosophers addresses issues at the center of contemporary debate: semantic and set-theoretic paradoxes, the set/class distinction, foundations of set theory, mathematical intuition and many others. The volume includes Hilary Putnam's 1995 Alfred Tarski lectures published here for the first time. The essays are presented to honor the work of Charles Parsons.

Widespread interest in Frege's general philosophical writings is, relatively speaking, a fairly recent phenomenon. But it is only very recently that his philosophy of mathematics has begun to attract the attention it now enjoys. This interest has been elicited by the discovery of the remarkable mathematical properties of Frege's contextual definition of number and of the unique character of his proposals for a theory of the real numbers.

This collection of essays addresses three main developments in recent work on Frege's philosophy of mathematics: the emerging interest in the intellectual background to his logicism; the rediscovery of Frege's theorem; and the reevaluation of the mathematical content of" The Basic Laws of Arithmetic," Each essay attempts a sympathetic, if not uncritical, reconstruction, evaluation, or extension of a facet of Frege's theory of arithmetic. Together they form an accessible and authoritative introduction to aspects of Frege's thought that have, until now, been largely missed by the philosophical community.

Most contemporary work in the foundations of mathematics takes its start from the groundbreaking contributions of, among others, Hilbert, Brouwer, Bernays, and Weyl. This book offers an introduction to the debate on the foundations of mathematics during the 1920s and presents the English reader with a selection of twenty five articles central to the debate which have not been previously translated. It is an ideal text for undergraduate and graduate courses in the philosophy of mathematics.

This Set contains: Reality Rules, Picturing the World in Mathematics, Volume 1, The Fundamentals by John Casti; Reality Rules, Picturing the World in Mathematics, Volume 2, The Frontier by John Casti

Dramatic changes or revolutions in a field of science are often made by outsiders or 'trespassers, ' who are not limited by the established, 'expert' approaches. Each essay in this diverse collection shows the fruits of intellectual trespassing and poaching among fields such as economics, Kantian ethics, Platonic philosophy, category theory, double-entry accounting, arbitrage, algebraic logic, series-parallel duality, and financial arithmetic.

"The book is indeed a classic. Virtually every philosopher of science now writing about probabilistic inference has been influenced by Edwards' book, and his ideas are now as alive and relevant as they were when the book first appeared. Edwards is an absolutely seminal thinker in the foundations of statistics and scientific inference." -- Elliott Sober, University of Wisconsin-Madison.

"Full of appropriate examples (especially from genetics) and historical commentary, this monograph offers a rare simultaneous treatment of both mathematical and philosophical foundations." -- American Mathematical Monthly.

This new and expanded edition of A. W. F. Edwards' classic volume on scientific inference presents his most important published articles on the subject. Edwards argues that the appropriate axiomatic basis for inductive inference is not that of probability, with its addition axiom, but that of likelihood, the concept introduced by Fisher as a measure of relative support among different hypotheses. Starting from the simplest considerations and assuming no more than a basic acquaintancewith probability theory, the author sets out to reconstruct a consistent theory of statistical inference in science. Using the likelihood approach, he explores estimation, tests of significance, randomization, experimental design, and other statistical topics. Likelihood is important reading for students and professionals in biology, mathematical sciences, and philosophy.

"This book is commended to all philosophers of science who are interested in the problems of scientific inference." -- Search.

"This book, by a well-known geneticist, will do much to publicize the generality of the likelihoodmethod as a foundation for statistical procedure. It is both smoothly written and persuasive." -- Operations Research.

"Likelihood is an important text and, in addition, is a joy to read, being a paragon of lucid and witty exposition." -- Mathematical Gazette

"History and Philosophy of Modern Mathematics " was first published in 1988. Minnesota Archive Editions uses digital technology to make long-unavailable books once again accessible, and are published unaltered from the original University of Minnesota Press editions.

The fourteen essays in this volume build on the pioneering effort of Garrett Birkhoff, professor of mathematics at Harvard University, who in 1974 organized a conference of mathematicians and historians of modern mathematics to examine how the two disciplines approach the history of mathematics. In "History and Philosophy of Modern Mathematics," William Aspray and Philip Kitcher bring together distinguished scholars from mathematics, history, and philosophy to assess the current state of the field. Their essays, which grow out of a 1985 conference at the University of Minnesota, develop the basic premise that mathematical thought needs to be studied from an interdisciplinary perspective.

The opening essays study issues arising within logic and the foundations of mathematics, a traditional area of interest to historians and philosophers. The second section examines issues in the history of mathematics within the framework of established historical periods and questions. Next come case studies that illustrate the power of an interdisciplinary approach to the study of mathematics. The collection closes with a look at mathematics from a sociohistorical perspective, including the way institutions affect what constitutes mathematical knowledge.

First published in 1990, this book consists of a detailed exposition of results of the theory of "interpretation" developed by G. Kreisel - the relative impenetrability of which gives the elucidation contained here great value for anyone seeking to understand his work. It contains more complex versions of the information obtained by Kreisel for number theory and clustering around the no-counter-example interpretation, for number-theorectic forumulae provide in ramified analysis. It also proves the omega-consistency of ramified analysis. The author also presents proofs of Schutte's cut-elimination theorems which are based on his consistency proofs and essentially contain them - these went further than any published work up to that point, helping to squeeze the maximum amount of information from these proofs.

A philosopher, scholar of the natural world, and gifted mathematician, Thomas Reid holds a distinctive place in the Scottish Enlightenment. This volume reconstructs Reid's lifelong engagement with the physical sciences and makes clear why these fields were central to his epistemology and moral and social philosophy. Placing Reid's "Essay on Quantity" alongside his previously unpublished writings on mathematics and the physical sciences, Paul Wood shows that, in contrast to Francis Hutcheson and David Hume, Reid was a philosopher rooted not only in the science of man but also in the sciences of nature. A self-professed Newtonian, Reid honed his observational and experimental skills while investigating a broad range of theoretical problems in astronomy, mechanics, optics, electricity, and chemistry. He championed the practical application of mathematics, immersed himself in Newton's mathematical corpus, and addressed foundational questions such as the conceptual basis of Euclidean geometry. Comprehensive and invaluable, this volume demonstrates that Reid built on his own early precociousness in mathematics to become one of the leading mathematicians and natural philosophers of the Scottish Enlightenment.

Cognitive mathematics provides insights into how mathematics works inside the brain and how it is interconnected with other faculties through so-called blending and other associative processes. This handbook is the first large collection of various aspects of cognitive mathematics to be amassed into a single title, covering decades of connection between mathematics and other figurative processes as they manifest themselves in language, art, and even algorithms. It will be of use to anyone working in math cognition and education, with each section of the handbook edited by an international leader in that field.

The use of diagrams in logic and geometry has encountered resistance in recent years. For a proof to be valid in geometry, it must not rely on the graphical properties of a diagram. In logic, the teaching of proofs depends on sentenial representations, ideas formed as natural language sentences such as "If A is true and B is true...." No serious formal proof system is based on diagrams.
This book explores the reasons why structured graphics have been largely ignored in contemporary formal theories of axiomatic systems. In particular, it elucidates the systematic forces in the intellectual history of mathematics which have driven the adoption of sentential representational styles over diagrammatic ones. In this book, the effects of historical forces on the evolution of diagrammatically-based systems of inference in logic and geometry are traced from antiquity to the early twentieth-century work of David Hilbert. From this exploration emerges an understanding that the present negative attitudes towards the use of diagrams in logic and geometry owe more to implicit appeals to their history and philosophical background than to any technical incompatibility with modern theories of logical systems.

Alain Badiou has claimed that Quentin Meillassoux's book After Finitude (Bloomsbury, 2008) "opened up a new path in the history of philosophy." And so, whether you agree or disagree with the speculative realism movement, it has to be addressed. Lacanian Realism does just that. This book reconstructs Lacanian dogma from the ground up: first, by unearthing a new reading of the Lacanian category of the real; second, by demonstrating the political and cultural ingenuity of Lacan's concept of the real, and by positioning this against the more reductive analyses of the concept by Slavoj Zizek, Alain Badiou, Saul Newman, Todd May, Joan Copjec, Jacques Ranciere, and others, and; third, by arguing that the subject exists intimately within the real. Lacanian Realism is an imaginative and timely exploration of the relationship between Lacanian psychoanalysis and contemporary continental philosophy.

Since antiquity, opposed concepts such a s t he One and the Many, the Finite and the Infinite, and the Absolute and the Relative, have been a driving force in philosophical, scientific, and mathematical thought. Yet they have also given rise to perplexing problems and conceptual paradoxes which continue to haunt scientists and philosophers. In Oppositions and Paradoxes, John L. Bell explains and investigates the paradoxes and puzzles that arise out of conceptual oppositions in physics and mathematics. In the process, Bell not only motivates abstract conceptual thinking about the paradoxes at issue, he also offers a compelling introduction to central ideas in such otherwise-di cult topics as non-Euclidean geometry, relativity, and quantum physics. These paradoxes are often as fun as they are flabbergasting. Consider, for example, the Tristram Shandy paradox: an immortal man composing an autobiography so slowly as to require a year of writing to describe each day of his life-he would, if he had infinite time, never complete the work, although no individual part of it would remain unwritten ... Or imagine an English professor who time-travels back to 1599 to offer a printing of Hamlet to William Shakespeare, so as to help the Bard overcome writer's block and author the play which will centuries later inspire an English professor to travel back in time ... These and many other of the book's paradoxes straddle the boundary between physics and metaphysics, and demonstrate the hidden difficulty of many of our most basic concepts.

What was the basis for the adoption of mathematics as the primary mode of discourse for describing natural events by a large segment of the philosophical community in the seventeenth century?

In answering this question, this book demonstrates that a significant group of philosophers shared the belief that there is no necessary correspondence between external reality and objects of human understanding, which they held to include the objects of mathematical and linguistic discourse. The result is a scholarly reliable, but accessible, account of the role of mathematics in the works of (amongst others) Galileo, Kepler, Descartes, Newton, Leibniz, and Berkeley.

This impressive volume will benefit scholars interested in the history of philosophy, mathematical philosophy and the history of mathematics.

What do Bach's compositions, Rubik's Cube, the way we choose our mates, and the physics of subatomic particles have in common? All are governed by the laws of symmetry, which elegantly unify scientific and artistic principles. Yet the mathematical language of symmetry-known as group theory-did not emerge from the study of symmetry at all, but from an equation that couldn't be solved.
For thousands of years mathematicians solved progressively more difficult algebraic equations, until they encountered the quintic equation, which resisted solution for three centuries. Working independently, two great prodigies ultimately proved that the quintic cannot be solved by a simple formula. These geniuses, a Norwegian named Niels Henrik Abel and a romantic Frenchman named Evariste Galois, both died tragically young. Their incredible labor, however, produced the origins of group theory.
The first extensive, popular account of the mathematics of symmetry and order, "The Equation That Couldn't Be Solved" is told not through abstract formulas but in a beautifully written and dramatic account of the lives and work of some of the greatest and most intriguing mathematicians in history.

This volume brings together a collection of essays on the history and philosophy of probability and statistics by one of the eminent scholars in these subjects. Written over the last fifteen years, they fall into three broad categories. The first deals with the use of symmetry arguments in inductive probability, in particular, their use in deriving rules of succession (Carnap's 'continuum of inductive methods'). The second group deals with four outstanding individuals who made lasting contributions to probability and statistics in very different ways: Frank Ramsey, R. A. Fisher, Alan Turing, and Abraham de Moivre. The last group of essays deals with the problem of 'predicting the unpredictable' - making predictions when the range of possible outcomes is unknown in advance. The essays weave together the history and philosophy of these subjects and document the fascination that they have exercised for more than three centuries.

This book analyzes the different ways mathematics is applicable in the physical sciences, and presents a startling thesis--the success of mathematical physics appears to assign the human mind a special place in the cosmos.

Mark Steiner distinguishes among the semantic problems that arise from the use of mathematics in logical deduction; the metaphysical problems that arise from the alleged gap between mathematical objects and the physical world; the descriptive problems that arise from the use of mathematics to describe nature; and the epistemological problems that arise from the use of mathematics to discover those very descriptions.

The epistemological problems lead to the thesis about the mind. It is frequently claimed that the universe is indifferent to human goals and values, and therefore, Locke and Peirce, for example, doubted science's ability to discover the laws governing the humanly unobservable. Steiner argues that, on the contrary, these laws were discovered, using manmade mathematical analogies, resulting in an anthropocentric picture of the universe as "user friendly" to human cognition--a challenge to the entrenched dogma of naturalism.

The amazing story of one of the greatest math problems of all time and the reclusive genius who solved it
In the tradition of "Fermatas Enigma" and "Prime Obsession," George Szpiro brings to life the giants of mathematics who struggled to prove a theorem for a century and the mysterious man from St. Petersburg, Grigory Perelman, who fi nally accomplished the impossible. In 1904 Henri PoincarA(c) developed the PoincarA(c) Conjecture, an attempt to understand higher-dimensional space and possibly the shape of the universe. The problem was he couldnat prove it. A century later it was named a Millennium Prize problem, one of the seven hardest problems we can imagine. Now this holy grail of mathematics has been found.
Accessibly interweaving history and math, Szpiro captures the passion, frustration, and excitement of the hunt, and provides a fascinating portrait of a contemporary noble-genius.