For more than two thousand years a familiarity with mathematics has been regarded as an indispensable part of the intellectual equipment of every cultured person. Today, unfortunately, the traditional place of mathematics in education is in grave danger. The teaching and learning of mathematics has degenerated into the realm of rote memorization, the outcome of which leads to satisfactory formal ability but does not lead to real understanding or to greater intellectual independence. This new edition of Richard Courant's and Herbert Robbins's classic work seeks to address this problem. Its goal is to put the meaning back into mathematics. Written for beginners and scholars, for students and teachers, for philosophers and engineers, What is Mathematics?, Second Edition is a sparkling collection of mathematical gems that offers an entertaining and accessible portrait of the mathematical world. Covering everything from natural numbers and the number system to geometrical constructions and projective geometry, from topology and calculus to matters of principle and the Continuum Hypothesis, this fascinating survey allows readers to delve into mathematics as an organic whole rather than an empty drill in problem solving. With chapters largely independent of one another and sections that lead upward from basic to more advanced discussions, readers can easily pick and choose areas of particular interest without impairing their understanding of subsequent parts. Brought up to date with a new chapter by Ian Stewart, What is Mathematics?, Second Edition offers new insights into recent mathematical developments and describes proofs of the Four-Color Theorem and Fermat's Last Theorem, problems that were still open when Courant and Robbins wrote this masterpiece, but ones that have since been solved. Formal mathematics is like spelling and grammar--a matter of the correct application of local rules. Meaningful mathematics is like journalism--it t

International reprint edition of ISBN-13: 978-0201558029. Paperback. Published in China. Contents in ENGLISH and totally same as US Edition.

In the late 19th century, a brilliant mathematician languished in an asylum. His greatest accomplishment, the result of a series of leaps of insight, was his pioneering understanding of the nature of infinity. This is the story of Georg Cantor: how he came to his theories and the reverberations of his work, the consequences of which shape our world. Cantor's theory of the infinite is famous for its many seeming contradictions: for example, we can prove there are as many points on a line one inch long as on a line one mile long; we can also prove that in all time there are as many years as there are days. According to Cantor, infinite sets are equal. The mind-twisting, deeply philosophical work of Cantor has its roots in ancient Greek mathematics and Jewish numerology as found in the mystical work known as the Kabbalah. Cantor used the term aleph-the first letter of the Hebrew alphabet, with all its attendant divine associations-to refer to the mysterious number which is the sum of positive integers. It is not the last positive number, because . . . there is no last. It is the ultimate number that is always being approached: just as, for example, there is no last fraction before the number 1.

Please Note That The Following Individual Books As Per Original ISBN and Cover Image In this Listing shall be Dispatched Collectively: Our Mathematical Universe & Life 3.0 Being Human in the Age of Artificial Intelligence By Max Tegmark 2 Books Collection Set: Our Mathematical Universe: In Our Mathematical Universe, Max Tegmark, one of the most original physicists at work today, leads us on an astonishing journey to explore the mysteries uncovered by cosmology and to discover the nature of reality.Part-history of the cosmos, part-intellectual adventure, Our Mathematical Universe travels from the Big Bang to the distant future via parallel worlds, across every possible scale - from the sub-atomic to the intergalactic - showing how mathematics provides the answers to our questions about the world. Where do we come from? What makes the universe the way it is? In essence, why are we here? With dazzling clarity, Max Tegmark ponders these deep mysteries and allows us to grasp the most cutting-edge and mind-boggling theories of physics. Life 3.0 Being Human in the Age of Artificial Intelligence: AI is the future - but what will that future look like? Will superhuman intelligence be our slave, or become our god?Taking us to the heart of the latest thinking about AI, Max Tegmark, the MIT professor whose work has helped mainstream research on how to keep AI beneficial, separates myths from reality, utopias from dystopias, to explore the next phase of our existence.How can we grow our prosperity through automation, without leaving people lacking income or purpose?

Real Analysis is the third volume in the Princeton Lectures in Analysis, a series of four textbooks that aim to present, in an integrated manner, the core areas of analysis. Here the focus is on the development of measure and integration theory, differentiation and integration, Hilbert spaces, and Hausdorff measure and fractals. This book reflects the objective of the series as a whole: to make plain the organic unity that exists between the various parts of the subject, and to illustrate the wide applicability of ideas of analysis to other fields of mathematics and science. After setting forth the basic facts of measure theory, Lebesgue integration, and differentiation on Euclidian spaces, the authors move to the elements of Hilbert space, via the L2 theory. They next present basic illustrations of these concepts from Fourier analysis, partial differential equations, and complex analysis. The final part of the book introduces the reader to the fascinating subject of fractional-dimensional sets, including Hausdorff measure, self-replicating sets, space-filling curves, and Besicovitch sets. Each chapter has a series of exercises, from the relatively easy to the more complex, that are tied directly to the text. A substantial number of hints encourage the reader to take on even the more challenging exercises.