His investigations shed light on what we can ultimately know about the universe and the very nature of life. In an infectious and enthusiastic narrative, Chaitin delineates the specific intellectual and intuitive steps he took toward the discovery. He takes us to the very frontiers of scientific thinking, and helps us to appreciate the art—and the sheer beauty—in the science of math.

    In contrast, Karl Popper's seminal book "The Logic of Scientific Discovery" has lapsed into relative obscurity. Although the two men debated the nature of science only once, the legacy of this encounter has dominated intellectual and public discussions on the topic ever since.Almost universally recognized as the modern watershed in the philosophy of science, Kuhn's relativistic vision of shifting paradigms--which asserted that science was just another human activity, like art or philosophy, only more specialized--triumphed over Popper's more positivistic belief in science's revolutionary potential to falsify society's dogmas. But has this victory been beneficial for science? Steve Fuller argues that not only has Kuhn's dominance had an adverse impact on the field but both thinkers have been radically misinterpreted in the process. This debate raises a vital question: Can science remain an independent, progressive force in society, or is it destined to continue as the technical wing of the military-industrial complex? Drawing on original research--including the Kuhn archives at MIT--Fuller offers a clear account of "Kuhn vs. Popper" and what it will mean for the future of scientific inquiry.

    Our eyes and minds are drawn to symmetrical objects, from the pyramid to the pentagon. Of fundamental significance to the way we interpret the world, this unique, pervasive phenomenon indicates a dynamic relationship between objects. In chemistry and physics, the concept of symmetry explains the structure of crystals or the theory of fundamental particles; in evolutionary biology, the natural world exploits symmetry in the fight for survival; and symmetry—and the breaking of it—is central to ideas in art, architecture, and music.Combining a rich historical narrative with his own personal journey as a mathematician, Marcus du Sautoy takes a unique look into the mathematical mind as he explores deep conjectures about symmetry and brings us face-to-face with the oddball mathematicians, both past and present, who have battled to understand symmetry's elusive qualities. He explores what is perhaps the most exciting discovery to date—the summit of mathematicians' mastery in the field—the Monster, a huge snowflake that exists in 196,883-dimensional space with more symmetries than there are atoms in the sun.What is it like to solve an ancient mathematical problem in a flash of inspiration? What is it like to be shown, ten minutes later, that you've made a mistake? What is it like to see the world in mathematical terms, and what can that tell us about life itself? In Symmetry, Marcus du Sautoy investigates these questions and shows mathematical novices what it feels like to grapple with some of the most complex ideas the human mind can comprehend.

    Beginning with the background and very nature of probability theory, the book then proceeds through sample spaces, combinatorial analysis, fluctuations in coin tossing and random walks, the combination of events, types of distributions, Markov chains, stochastic processes, and more. The book's comprehensive approach provides a complete view of theory along with enlightening examples along the way.

    Linear algebra is tightly integrated into the text.

    In the same period, the cross-fertilization of mathematics and philosophy resulted in a new sort of 'mathematical philosophy', associated most notably (but in different ways) with Bertrand Russell, W. V. Quine, and Godel himself, and which remains at the focus of Anglo-Saxon philosophical discussion. The present collection brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers. It is a substantially revised version of the edition first published in 1964 and includes a revised bibliography. The volume will be welcomed as a major work of reference at this level in the field.

    It describes how the laws and discoveries of information theory now support controversial revisions to Darwinian evolution, begin to unravel the mysteries of language, memory and dreams, and stimulate provocative ideas in psychology, philosophy, art, music, computers and even the structure of society. Perhaps its most fascinating and unexpected surprise is the suggestion the order and complexity may be as natural as disorder and disorganization. Contrary to the entropy principle, which implies that order is the exception and confusion the rule, information theory asserts that order and sense can indeed prevail against disorder and nonsense. From the simplest forms of organic life to the words used to express our most complex ideas, from our genes to our dreams, from microcomputers to telecommunications, virtually everything around us follows simple rules of information. Life and the material world, like language, remain "grammatical." Grammatical man inhabits a grammatical universe.

    Never before in history have we been able to acquire, record, communicate, and use information in so many different forms. Never before have we had access to such vast quantities of data of every kind. This revolution goes far beyond the limitless content that fills our lives, because information also underlies our understanding of ourselves, the natural world, and the universe. It is the key that unites fields as different as linguistics, cryptography, neuroscience, genetics, economics, and quantum mechanics. And the fact that information bears no necessary connection to meaning makes it a profound puzzle that people with a passion for philosophy have pondered for centuries.Table of ContentsLECTURE 1The Transformability of Information 4LECTURE 2Computation and Logic Gates 17LECTURE 3Measuring Information 26LECTURE 4Entropy and the Average Surprise 34LECTURE 5Data Compression and Prefix-Free Codes 44LECTURE 6Encoding Images and Sounds 57LECTURE 7Noise and Channel Capacity 69LECTURE 8Error-Correcting Codes 82LECTURE 9Signals and Bandwidth 94LECTURE 10Cryptography and Key Entropy 110LECTURE 11Cryptanalysis and Unraveling the Enigma 119LECTURE 12Unbreakable Codes and Public Keys 130LECTURE 13What Genetic Information Can Do 140LECTURE 14Life’s Origins and DNA Computing 152LECTURE 15Neural Codes in the Brain 169LECTURE 16Entropy and Microstate Information 185LECTURE 17Erasure Cost and Reversible Computing 198LECTURE 18Horse Races and Stock Markets 213LECTURE 19Turing Machines and Algorithmic Information 226LECTURE 20Uncomputable Functions and Incompleteness 239LECTURE 21Qubits and Quantum Information 253LECTURE 22Quantum Cryptography via Entanglement 266LECTURE 23It from Bit: Physics from Information 281LECTURE 24The Meaning of Information 293

    This discussion paves the way for an outline of epistemological methods in which the rationalism of thinkers like Descartes, Leibniz and Kant is compared to the empiricism of Locke and Hume.Over the course of the book, in a manner that is careful and methodic but never dull, Jeans marshals the evidence for his startling conclusion: recent discoveries in astronomy, mathematics, sub-atomic physics and other disciplines have washed away the scientific basis of many older philosophic discussions. Such long-standing problems as causality, free will and determinism, the nature of space and time, materialism and mentalism must be considered anew int he light of new knowledge and information attained by 20th-century physical science. Even then, however, Jeans cautions against drawing any positive conclusions, pointing out that both physics and philosophy are both relatively young and that we are still, in Newton's words, like children playing with pebbles on the sea-shore, while the great ocean of truth rolls, unexplored, beyond our reach.Although first published nearly 40 years ago, nothing in physics has happened to affect Jean's account in this book; it remains remarkably fresh and undated, a classic exposition of the philosophical implications of scientific knowledge.

    Reuben Hersh argues the contrary, that mathematics must be understood as a human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context. Hersh pulls the screen back to reveal mathematics as seen by professionals, debunking many mathematical myths, and demonstrating how the humanist idea of the nature of mathematics more closely resembles how mathematicians actually work. At the heart of his book is a fascinating historical account of the mainstream of philosophy--ranging from Pythagoras, Descartes, and Spinoza, to Bertrand Russell, David Hilbert, and Rudolph Carnap--followed by the mavericks who saw mathematics as a human artifact, including Aristotle, Locke, Hume, Mill, and Lakatos.What is Mathematics, Really? reflects an insider's view of mathematical life, and will be hotly debated by anyone with an interest in mathematics or the philosophy of science.

    The basic techniques and theorems of analysis are presented in such a way that the intimate connections between its various branches are strongly emphasized. The traditionally separate subjects of 'real analysis' and 'complex analysis' are thus united in one volume. Some of the basic ideas from functional analysis are also included. This is the only book to take this unique approach. The third edition includes a new chapter on differentiation. Proofs of theorems presented in the book are concise and complete and many challenging exercises appear at the end of each chapter. The book is arranged so that each chapter builds upon the other, giving students a gradual understanding of the subject.This text is part of the Walter Rudin Student Series in Advanced Mathematics.

    In addition to showing how recent metaphysics has drifted away from connection with all other serious scholarly inquiry as a result of not heeding this restriction, they demonstrate how to build a metaphysics compatible with current fundamental phsyics ("ontic structural realism"), which, when combined with their metaphysics of the special sciences ("rainforest realism"), can be used to unify physics with the other sciences without reducing these sciences to physics intself. Taking science metaphysically seriously, Ladyman and Ross argue, means that metaphysicians must abandon the picture of the world as composed of self-subsistent individual objects, and the paradigm of causation as the collision of such objects. Every Thing Must Go also assesses the role of information theory and complex systems theory in attempts to explain the relationship between the special sciences and physics, treading a middle road between the grand synthesis of thermodynamics and information, and eliminativism about information. The consequences of the author's metaphysical theory for central issues in the philosophy of science are explored, including the implications for the realism vs. empiricism debate, the role of causation in scientific explanations, the nature of causation and laws, the status of abstract and virtual objects, and the objective reality of natural kinds

    Discovered by an 18th century mathematician and preacher, Bayes' rule is a cornerstone of modern probability theory. In this richly illustrated book, intuitive visual representations of real-world examples are used to show how Bayes' rule is actually a form of commonsense reasoning. The tutorial style of writing, combined with a comprehensive glossary, makes this an ideal primer for novices who wish to gain an intuitive understanding of Bayesian analysis. As an aid to understanding, online computer code (in MatLab, Python and R) reproduces key numerical results and diagrams.Stone's book is renowned for its visually engaging style of presentation, which stems from teaching Bayes' rule to psychology students for over 10 years as a university lecturer.

    Worked examples and exercises support the text. An ELBS/LPBB edition is available.

    With Sweet Dreams, Dennett returns to the subject for revision and renewal of his theory of consciousness, taking into account major empirical advances in the field since 1991 as well as recent theoretical challenges.In Consciousness Explained, Dennett proposed to replace the ubiquitous but bankrupt Cartesian Theater model (which posits a privileged place in the brain where it all comes together for the magic show of consciousness) with the Multiple Drafts Model. Drawing on psychology, cognitive neuroscience, and artificial intelligence, he asserted that human consciousness is essentially the mental software that reorganizes the functional architecture of the brain. In Sweet Dreams, he recasts the Multiple Drafts Model as the fame in the brain model, as a background against which to examine the philosophical issues that continue to bedevil the field.With his usual clarity and brio, Dennett enlivens his arguments with a variety of vivid examples. He isolates the Zombic Hunch that distorts much of the theorizing of both philosophers and scientists, and defends heterophenomenology, his third-person approach to the science of consciousness, against persistent misinterpretations and objections. The old challenge of Frank Jackson's thought experiment about Mary the color scientist is given a new rebuttal in the form of RoboMary, while his discussion of a famous card trick, The Tuned Deck, is designed to show that David Chalmers's Hard Problem is probably just a figment of theorists' misexploited imagination. In the final essay, the intrinsic nature of qualia is compared with the naively imagined intrinsic value of a dollar in Consciousness--How Much is That in Real Money?