Dealing with measure theory and Lebesque integration, this is an introductory graduate text.

    It is the mathematical method for the analysis of things that change, and since in the natural world we are surrounded by change, the development of calculus was a huge breakthrough in the history of mathematics. But it is also something of a mathematical adventure, largely because of the way infinity enters at virtually every twist and turn...In The Calculus Story David Acheson presents a wide-ranging picture of calculus and its applications, from ancient Greece right up to the present day. Drawing on their original writings, he introduces the people who helped to build our understanding of calculus. With a step by step treatment, he demonstrates how to start doing calculus, from the very beginning.

    Even after more than three centuries and the revolutions of Einsteinian relativity and quantum mechanics, Newtonian physics continues to account for many of the phenomena of the observed world, and Newtonian celestial dynamics is used to determine the orbits of our space vehicles.This completely new translation, the first in 270 years, is based on the third (1726) edition, the final revised version approved by Newton; it includes extracts from the earlier editions, corrects errors found in earlier versions, and replaces archaic English with contemporary prose and up-to-date mathematical forms. Newton's principles describe acceleration, deceleration, and inertial movement; fluid dynamics; and the motions of the earth, moon, planets, and comets. A great work in itself, the Principia also revolutionized the methods of scientific investigation. It set forth the fundamental three laws of motion and the law of universal gravity, the physical principles that account for the Copernican system of the world as emended by Kepler, thus effectively ending controversy concerning the Copernican planetary system.The illuminating Guide to the Principia by I. Bernard Cohen, along with his and Anne Whitman's translation, will make this preeminent work truly accessible for today's scientists, scholars, and students.

    From the very first attempts by the Greeks to grapple with the complexities of our known world to the latest application of infinity in physics, The Road to Reality carefully explores the movement of the smallest atomic particles and reaches into the vastness of intergalactic space. Here, Penrose examines the mathematical foundations of the physical universe, exposing the underlying beauty of physics and giving us one the most important works in modern science writing.

    A lecture class taught by Wittgenstein, however, hardly resembled a lecture. He sat on a chair in the middle of the room, with some of the class sitting in chairs, some on the floor. He never used notes. He paused frequently, sometimes for several minutes, while he puzzled out a problem. He often asked his listeners questions and reacted to their replies. Many meetings were largely conversation. These lectures were attended by, among others, D. A. T. Gasking, J. N. Findlay, Stephen Toulmin, Alan Turing, G. H. von Wright, R. G. Bosanquet, Norman Malcolm, Rush Rhees, and Yorick Smythies. Notes taken by these last four are the basis for the thirty-one lectures in this book. The lectures covered such topics as the nature of mathematics, the distinctions between mathematical and everyday languages, the truth of mathematical propositions, consistency and contradiction in formal systems, the logicism of Frege and Russell, Platonism, identity, negation, and necessary truth. The mathematical examples used are nearly always elementary.

    In the four main sections of the book, Stein tells the stories of the mathematical thinkers who discerned some of the most fundamental aspects of our universe. From their successes and failures, delusions, and even duels, the trajectories of their innovations—and their impact on society—are traced in this fascinating narrative. Quantum mechanics, space-time, chaos theory and the workings of complex systems, and the impossibility of a "perfect" democracy are all here. Stein's book is both mind-bending and practical, as he explains the best way for a salesman to plan a trip, examines why any thought you could have is imbedded in the number π , and—perhaps most importantly—answers one of the modern world's toughest questions: why the garage can never get your car repaired on time.Friendly, entertaining, and fun, How Math Explains the World is the first book by one of California's most popular math teachers, a veteran of both "math for poets" and Princeton's Institute for Advanced Studies. And it's perfect for any reader wanting to know how math makes both science and the world tick.

    This is the amazing story of how the "map problem" was solved.The problem posed in the letter came from a former student: What is the least possible number of colors needed to fill in any map (real or invented) so that neighboring counties are always colored differently? This deceptively simple question was of minimal interest to cartographers, who saw little need to limit how many colors they used. But the problem set off a frenzy among professional mathematicians and amateur problem solvers, among them Lewis Carroll, an astronomer, a botanist, an obsessive golfer, the Bishop of London, a man who set his watch only once a year, a California traffic cop, and a bridegroom who spent his honeymoon coloring maps. In their pursuit of the solution, mathematicians painted maps on doughnuts and horseshoes and played with patterned soccer balls and the great rhombicuboctahedron. It would be more than one hundred years (and countless colored maps) later before the result was finally established. Even then, difficult questions remained, and the intricate solution--which involved no fewer than 1,200 hours of computer time--was greeted with as much dismay as enthusiasm.Providing a clear and elegant explanation of the problem and the proof, Robin Wilson tells how a seemingly innocuous question baffled great minds and stimulated exciting mathematics with far-flung applications. This is the entertaining story of those who failed to prove, and those who ultimately did prove, that four colors do indeed suffice to color any map.

     This graphic novel recounts the spiritual odyssey of philosopher Bertrand Russell. In his agonized search for absolute truth, he crosses paths with thinkers like Gottlob Frege, David Hilbert & Kurt Gödel, & finds a passionate student in Ludwig Wittgenstein. But his most ambitious goal—to establish unshakable logical foundations of mathematics—continues to loom before him. Thru love & hate, peace & war, he persists in the mission threatening to claim both his career & happiness, finally driving him to the brink of insanity. This story is at the same time a historical novel & an accessible explication of some of the biggest ideas of mathematics & modern philosophy. With rich characterizations & atmospheric artwork, it spins the pursuit of such ideas into a satisfying tale. Probing, layered, the book throws light on Russell’s inner struggles while setting them in the context of the timeless questions he tried to answer. At its heart, Logicomix is a story about the conflict between ideal rationality & the flawed fabric of reality.

    Len Fisher turns his attention to the science of cooperation in his lively and thought-provoking book. Fisher shows how the modern science of game theory has helped biologists to understand the evolution of cooperation in nature, and investigates how we might apply those lessons to our own society. In a series of experiments that take him from the polite confines of an English dinner party to crowded supermarkets, congested Indian roads, and the wilds of outback Australia, not to mention baseball strategies and the intricacies of quantum mechanics, Fisher sheds light on the problem of global cooperation. The outcomes are sometimes hilarious, sometimes alarming, but always revealing. A witty romp through a serious science, Rock, Paper, Scissors will both teach and delight anyone interested in what it what it takes to get people to work together.

    This book investigates what cannot be known. Rather than exploring the amazing facts that science, mathematics, and reason have revealed to us, this work studies what science, mathematics, and reason tell us cannot be revealed. In The Outer Limits of Reason, Noson Yanofsky considers what cannot be predicted, described, or known, and what will never be understood. He discusses the limitations of computers, physics, logic, and our own thought processes.Yanofsky describes simple tasks that would take computers trillions of centuries to complete and other problems that computers can never solve; perfectly formed English sentences that make no sense; different levels of infinity; the bizarre world of the quantum; the relevance of relativity theory; the causes of chaos theory; math problems that cannot be solved by normal means; and statements that are true but cannot be proven. He explains the limitations of our intuitions about the world -- our ideas about space, time, and motion, and the complex relationship between the knower and the known.Moving from the concrete to the abstract, from problems of everyday language to straightforward philosophical questions to the formalities of physics and mathematics, Yanofsky demonstrates a myriad of unsolvable problems and paradoxes. Exploring the various limitations of our knowledge, he shows that many of these limitations have a similar pattern and that by investigating these patterns, we can better understand the structure and limitations of reason itself. Yanofsky even attempts to look beyond the borders of reason to see what, if anything, is out there.

    Notions like denumerability, modal scope distinction, Bayesian conditionalization, and logical completeness are usually only elucidated deep within difficultspecialist texts. By offering simple explanations that by-pass much irrelevant and boring detail, Philosophical Devices is able to cover a wealth of material that is normally only available to specialists.The book contains four sections, each of three chapters. The first section is about sets and numbers, starting with the membership relation and ending with the generalized continuum hypothesis. The second is about analyticity, a prioricity, and necessity. The third is about probability, outliningthe difference between objective and subjective probability and exploring aspects of conditionalization and correlation. The fourth deals with metalogic, focusing on the contrast between syntax and semantics, and finishing with a sketch of Godel's theorem.Philosophical Devices will be useful for university students who have got past the foothills of philosophy and are starting to read more widely, but it does not assume any prior expertise. All the issues discussed are intrinsically interesting, and often downright fascinating. It can be read withpleasure and profit by anybody who is curious about the technical infrastructure of contemporary philosophy.

    The Calculus Lifesaver provides students with the essential tools they need not only to learn calculus, but to excel at it.All of the material in this user-friendly study guide has been proven to get results. The book arose from Adrian Banner's popular calculus review course at Princeton University, which he developed especially for students who are motivated to earn A's but get only average grades on exams. The complete course will be available for free on the Web in a series of videotaped lectures. This study guide works as a supplement to any single-variable calculus course or textbook. Coupled with a selection of exercises, the book can also be used as a textbook in its own right. The style is informal, non-intimidating, and even entertaining, without sacrificing comprehensiveness. The author elaborates standard course material with scores of detailed examples that treat the reader to an inner monologue--the train of thought students should be following in order to solve the problem--providing the necessary reasoning as well as the solution. The book's emphasis is on building problem-solving skills. Examples range from easy to difficult and illustrate the in-depth presentation of theory.The Calculus Lifesaver combines ease of use and readability with the depth of content and mathematical rigor of the best calculus textbooks. It is an indispensable volume for any student seeking to master calculus.Serves as a companion to any single-variable calculus textbookInformal, entertaining, and not intimidatingInformative videos that follow the book--a full forty-eight hours of Banner's Princeton calculus-review course--is available at Adrian Banner lecturesMore than 475 examples (ranging from easy to hard) provide step-by-step reasoningTheorems and methods justified and connections made to actual practiceDifficult topics such as improper integrals and infinite series covered in detailTried and tested by students taking freshman calculus

    It provides a simple, thorough survey of elementary topics, starting with set theory and advancing to metric and topological spaces, connectedness, and compactness. 1975 edition.

    Darrell Huff runs the gamut of every popularly used type of statistic, probes such things as the sample study, the tabulation method, the interview technique, or the way the results are derived from the figures, and points up the countless number of dodges which are used to fool rather than to inform.

    Reveals the important contributions of mathematics to the physical and social sciences, philosophy, religion, literature, and art.