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    C. Escher and composer Johann Sebastian Bach, discussing common themes in their work and lives. At a deeper level, the book is a detailed and subtle exposition of concepts fundamental to mathematics, symmetry, and intelligence. Through illustration and analysis, the book discusses how self-reference and formal rules allow systems to acquire meaning despite being made of "meaningless" elements. It also discusses what it means to communicate, how knowledge can be represented and stored, the methods and limitations of symbolic representation, and even the fundamental notion of "meaning" itself. In response to confusion over the book's theme, Hofstadter has emphasized that GEB is not about mathematics, art, and music but rather about how cognition and thinking emerge from well-hidden neurological mechanisms.

    The book gives more than it promises, the text as well as the extensive bibliography including contributions of the analysts."-Hibbert Journal

    Hacker leads us into a world of philosophical investigation in which to smell a rat is ever so much easier than to trap it. Wittgenstein defined humans as language-using creatures. The role of philosophy is to ask questions which reveal the limits and nature of language. Taking the expression, description and observation of pain as examples, Hacker explores the ingenuity with which Wittgenstein identified the rules and set the limits of language. (less)

    According to the author, these trends sought to create a unified conceptual apparatus as a common basis for the whole of human knowledge.Because these new developments in logical thought tended to perfect and sharpen the deductive method, an indispensable tool in many fields for deriving conclusions from accepted assumptions, the author decided to widen the scope of the work. In subsequent editions he revised the book to make it also a text on which to base an elementary college course in logic and the methodology of deductive sciences. It is this revised edition that is reprinted here.Part One deals with elements of logic and the deductive method, including the use of variables, sentential calculus, theory of identity, theory of classes, theory of relations and the deductive method. The Second Part covers applications of logic and methodology in constructing mathematical theories, including laws of order for numbers, laws of addition and subtraction, methodological considerations on the constructed theory, foundations of arithmetic of real numbers, and more. The author has provided numerous exercises to help students assimilate the material, which not only provides a stimulating and thought-provoking introduction to the fundamentals of logical thought, but is the perfect adjunct to courses in logic and the foundation of mathematics.

    This seems to have been the case even from the first recognition of the Neanderthals as a species. The first Neanderthal fossil discovery was that of a child’s skull in Belgium in 1829, but it was badly damaged. Another would be discovered in 1856 in a limestone mine of the Neanderthal region of what is present-day Germany, and a skull with differing distinct traits (indicating a different species than the Neanderthals) would be discovered just over a decade later in southwestern France. The latter specimen would come to be recognized as an example of the species Homo Sapiens, and these anatomically modern humans arrived in Europe between 45,000 and 43,000 years ago, around the time the Neanderthals are believed to started going extinct. The Neanderthals are a member of the genus Homo just like Homo sapiens and share roughly 99.7% of their DNA with modern humans (Reynolds and Gallagher 2012). Both species even lived briefly during the same time in Eurasia. However, the Neanderthals evolved separately in Europe, away from modern humans, who evolved in Africa. Physically, the Neanderthal skeleton was much more robust, suggesting that there was more room for muscle attachment. However, while Neanderthals were stronger than modern humans, the average height of the Neanderthal male was shorter, standing at only about 5’5 tall. Other physical characteristics that set the Neanderthals apart from modern humans were certain skull traits. The skull in general was low and elongated, featuring a sloping forehead with an occipital bun (a bone projection at the back of the skull), whereas modern humans have a more vertical forehead with no occipital bun. The cranial capacity of the Neanderthal skull was also greater than the modern human at 1,500–1,740 cc, and it lacked a chin and had more circular eye orbits, in contrast to Homo sapiens, which have a chin and tend to feature more rectangular eye orbits (Wolpoff 1999). Despite these differences, the Neanderthals may have been recognizable enough to interact with Homo sapiens or even blend in with Homo sapiens for the thousands of years they lived together in Europe. The Neanderthals lived in Europe and Asia for nearly 200,000 years and thrived in these regions, but they went extinct between 40,000 and 30,000 years ago, around the same time that modern humans began arriving in Europe. This has prompted much speculation as to the nature of the interactions between Neanderthals and Homo sapiens, especially since some researchers believe they interacted with each other for over 5,000 years before the Neanderthals began going extinct at different times across Europe. One hypothesis is that Homo sapiens displaced the Neanderthals and were better suited for the environment, and it is obviously possible if not likely that these two groups had become competitors for food and other resources, with Homo sapiens being more successful in the end. If such close interactions were taking place, there is also a possibility that the relatively new-to-Europe Homo sapiens brought pathogens from Africa with them that were unknown to the Neanderthal’s immune system. A more recent example of this type of resulting interaction is the European expansion into the Americas, which brought diseases like smallpox that the natives of America had never experienced before, especially diseases resulting from the domestication of animals. It is possible that the domestication of the dog by Homo sapiens may have contributed in spreading foreign diseases among the Neanderthals.

    It aims to reach students who have already done an introductory philosophy course.Topics covered include perception and reflection as grounds of knowledge, and the nature, structure, and varieties of knowledge. The character and scope of knowledge in the crucial realms of ethics, science and religion are also considered.Unique features of Epistemology: * Provides a comprehensive survey of basic concepts and major theories* Gives an up-to-date account of important developments in the field* Contains many lucid examples to support ideas* Cites key literature in an annotated bibliography.

    Here Nozick continues his search for the connections between philosophy and ordinary experience. In the lively and accessible style that his readers have come to expect, he offers a bold theory of rationality, the one characteristic deemed to fix humanity's specialness. What are principles for? asks Nozick. We could act simply on whim, or maximize our self-interest and recommend that others do the same. As Nozick explores rationality of decision and rationality of belief, he shows how principles actually function in our day-to-day thinking and in our efforts to live peacefully and productively with each other.Throughout, the book combines daring speculations with detailed investigations to portray the nature and status of rationality and the essential role that imagination plays in this singular human aptitude.

    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.

    His simple but staggering theory of quantum evolution shows how quantum mechanics gives living organisms the ability to initiate specific actions, including new mutations. As Paul Davies exclaims, "if these ideas are right, they will transform our understanding of the relationship between physics and biology" and may radically revise the notion of random evolution and the debate over consciousness and free will.

    D. Williams wrote this entertaining, witty introduction for the nonscientist, game theory was still a somewhat mysterious subject familiar to very few scientists beyond those researchers, like himself, working for the military. Now, over thirty years after its original publication as a Rand Corporation research study, his light-hearted though thoroughly effective primer is the recognized classic introduction to an increasingly applicable discipline. Used by amateurs, professionals, and students throughout the world in the classroom, on the job, and for personal amusement, the book has been through ten printings, and has been translated into at least five languages (including Russian and Japanese).Revised, updated, and available for the first time in an inexpensive paperback edition, The Compleat Strategyst is a highly entertaining text essential for anyone interested in this provocative and engaging area of modern mathematics. In fully illustrated chapters complete with everyday examples and word problems, Williams offers readers a working understanding of the possible methods for selecting strategies in a variety of situations, simple to complex. With just a basic understanding of arithmetic, anyone can grasp all necessary aspects of two-, three-, four-, and larger strategy games with two or more sets of inimical interests and a limitless array of zero-sum payoffs.As research and study continues not only in this new discipline but in the related areas of statistics, probability and behavioral science, understanding of games, decision making, and the development of strategies will be increasingly important. In the areas of economics, sociology, politics, and the military, game theory is sure to have an even wider impact. For students and amateurs fascinated by game theory's implications there is no better, immediately applicable, or more entertaining introduction to the subject than this engaging text by the late J. D. Williams, Professor of Mathematics at Princeton University and a member of the Research Council of The Rand Corporation.

    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.

    Of particular importance is the examination of Bayesianism and the new experimentalism, as well as new chapters on the nature of scientific laws and recent trends in the realism versus anti-realism debate."Crisp, lucid and studded with telling examples… As a handy guide to recent alarums and excursions (in the philosophy of science) I find this book vigorous, gallant and useful."New Scientist

    Would you believe that these five pieces can be reassembled in such a fashion so as to create two apples equal in shape and size to the original? Would you believe that you could make something as large as the sun by breaking a pea into a finite number of pieces and putting it back together again? Neither did Leonard Wapner, author of The Pea and the Sun, when he was first introduced to the Banach-Tarski paradox, which asserts exactly such a notion. Written in an engaging style, The Pea and the Sun catalogues the people, events, and mathematics that contributed to the discovery of Banach and Tarski's magical paradox. Wapner makes one of the most interesting problems of advanced mathematics accessible to the non-mathematician.

    "Shapin's account is informed, nuanced, and articulated with clarity. . . . This is not to attack or devalue science but to reveal its richness as the human endeavor that it most surely is. . . .Shapin's book is an impressive achievement."—David C. Lindberg, Science"Shapin has used the crucial 17th century as a platform for presenting the power of science-studies approaches. At the same time, he has presented the period in fresh perspective."—Chronicle of Higher Education"Timely and highly readable . . . A book which every scientist curious about our predecessors should read."—Trevor Pinch, New Scientist"It's hard to believe that there could be a more accessible, informed or concise account of how it [the scientific revolution], and we have come to this. The Scientific Revolution should be a set text in all the disciplines. And in all the indisciplines, too."—Adam Phillips, London Review of Books"Shapin's treatise on the currents that engendered modern science is a combination of history and philosophy of science for the interested and educated layperson."—Publishers Weekly"Superlative, accessible, and engaging. . . . Absolute must-reading."—Robert S. Frey, Bridges"This vibrant historical exploration of the origins of modern science argues that in the 1600s science emerged from a variety of beliefs, practices, and influences. . . . This history reminds us that diversity is part of any intellectual endeavor."—Choice"Most readers will conclude that there was indeed something dramatic enough to be called the Scientific Revolution going on, and that this is an excellent book about it."—Anthony Gottlieb, The New York Times Book Review