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Books

Logic


Gödel, Escher, Bach: An Eternal Golden Braid

Gödel, Escher, Bach: An Eternal Golden Braid Lowest new price: $16.37
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Author: Douglas R. Hofstadter
Brand: Basic Books AZ

Winner of the Pulitzer Prize

A metaphorical fugue on minds and machines in the spirit of Lewis Carroll

Douglas Hofstadter's book is concerned directly with the nature of "maps" or links between formal systems. However, according to Hofstadter, the formal system that underlies all mental activity transcends the system that supports it. If life can grow out of the formal chemical substrate of the cell, if consciousness can emerge out of a formal system of firing neurons, then so too will computers attain human intelligence. Gödel, Escher, Bach is a wonderful exploration of fascinating ideas at the heart of cognitive science: meaning, reduction, recursion, and much more.

Twenty years after it topped the bestseller charts, Douglas R. Hofstadter's Gödel, Escher, Bach: An Eternal Golden Braid is still something of a marvel. Besides being a profound and entertaining meditation on human thought and creativity, this book looks at the surprising points of contact between the music of Bach, the artwork of Escher, and the mathematics of Gödel. It also looks at the prospects for computers and artificial intelligence (AI) for mimicking human thought. For the general reader and the computer techie alike, this book still sets a standard for thinking about the future of computers and their relation to the way we think.

Hofstadter's great achievement in Gödel, Escher, Bach was making abstruse mathematical topics (like undecidability, recursion, and 'strange loops') accessible and remarkably entertaining. Borrowing a page from Lewis Carroll (who might well have been a fan of this book), each chapter presents dialogue between the Tortoise and Achilles, as well as other characters who dramatize concepts discussed later in more detail. Allusions to Bach's music (centering on his Musical Offering) and Escher's continually paradoxical artwork are plentiful here. This more approachable material lets the author delve into serious number theory (concentrating on the ramifications of Gödel's Theorem of Incompleteness) while stopping along the way to ponder the work of a host of other mathematicians, artists, and thinkers.

The world has moved on since 1979, of course. The book predicted that computers probably won't ever beat humans in chess, though Deep Blue beat Garry Kasparov in 1997. And the vinyl record, which serves for some of Hofstadter's best analogies, is now left to collectors. Sections on recursion and the graphs of certain functions from physics look tantalizing, like the fractals of recent chaos theory. And AI has moved on, of course, with mixed results. Yet Gödel, Escher, Bach remains a remarkable achievement. Its intellectual range and ability to let us visualize difficult mathematical concepts help make it one of this century's best for anyone who's interested in computers and their potential for real intelligence. --Richard Dragan

Topics Covered: J.S. Bach, M.C. Escher, Kurt Gödel: biographical information and work, artificial intelligence (AI) history and theories, strange loops and tangled hierarchies, formal and informal systems, number theory, form in mathematics, figure and ground, consistency, completeness, Euclidean and non-Euclidean geometry, recursive structures, theories of meaning, propositional calculus, typographical number theory, Zen and mathematics, levels of description and computers; theory of mind: neurons, minds and thoughts; undecidability; self-reference and self-representation; Turing test for machine intelligence.

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  • Basic Books AZ

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How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library)

How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library) Lowest new price: $13.59
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Author: G. Polya
Brand: Polya G

A perennial bestseller by eminent mathematician G. Polya, How to Solve It will show anyone in any field how to think straight. In lucid and appealing prose, Polya reveals how the mathematical method of demonstrating a proof or finding an unknown can be of help in attacking any problem that can be "reasoned" out―from building a bridge to winning a game of anagrams. Generations of readers have relished Polya's deft―indeed, brilliant―instructions on stripping away irrelevancies and going straight to the heart of the problem.

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  • How to Solve It A New Aspect of Mathematical Method Princeton Science Library

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An Illustrated Book of Bad Arguments

An Illustrated Book of Bad Arguments Lowest new price: $7.32
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Author: Ali Almossawi
Brand: Almossawi Ali

“A flawless compendium of flaws.” —Alice Roberts, PhD, anatomist, writer, and presenter of The Incredible Human Journey

The antidote to fuzzy thinking, with furry animals!

Have you read (or stumbled into) one too many irrational online debates? Ali Almossawi certainly had, so he wrote An Illustrated Book of Bad Arguments! This handy guide is here to bring the internet age a much-needed dose of old-school logic (really old-school, a la Aristotle).

Here are cogent explanations of the straw man fallacy, the slippery slope argument, the ad hominem attack, and other common attempts at reasoning that actually fall short—plus a beautifully drawn menagerie of animals who (adorably) commit every logical faux pas. Rabbit thinks a strange light in the sky must be a UFO because no one can prove otherwise (the appeal to ignorance). And Lion doesn’t believe that gas emissions harm the planet because, if that were true, he wouldn’t like the result (the argument from consequences).

Once you learn to recognize these abuses of reason, they start to crop up everywhere from congressional debate to YouTube comments—which makes this geek-chic book a must for anyone in the habit of holding opinions.


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  • An Illustrated Book of Bad Arguments

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Reverse Mathematics: Proofs from the Inside Out

Reverse Mathematics: Proofs from the Inside Out Lowest new price: $19.00
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Author: John Stillwell

This book presents reverse mathematics to a general mathematical audience for the first time. Reverse mathematics is a new field that answers some old questions. In the two thousand years that mathematicians have been deriving theorems from axioms, it has often been asked: which axioms are needed to prove a given theorem? Only in the last two hundred years have some of these questions been answered, and only in the last forty years has a systematic approach been developed. In Reverse Mathematics, John Stillwell gives a representative view of this field, emphasizing basic analysis―finding the “right axioms” to prove fundamental theorems―and giving a novel approach to logic.

Stillwell introduces reverse mathematics historically, describing the two developments that made reverse mathematics possible, both involving the idea of arithmetization. The first was the nineteenth-century project of arithmetizing analysis, which aimed to define all concepts of analysis in terms of natural numbers and sets of natural numbers. The second was the twentieth-century arithmetization of logic and computation. Thus arithmetic in some sense underlies analysis, logic, and computation. Reverse mathematics exploits this insight by viewing analysis as arithmetic extended by axioms about the existence of infinite sets. Remarkably, only a small number of axioms are needed for reverse mathematics, and, for each basic theorem of analysis, Stillwell finds the “right axiom” to prove it.

By using a minimum of mathematical logic in a well-motivated way, Reverse Mathematics will engage advanced undergraduates and all mathematicians interested in the foundations of mathematics.

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Cut Elimination in Categories (Trends in Logic)

Cut Elimination in Categories (Trends in Logic) Lowest new price: $165.95
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Author: K. Dosen
Brand: Brand: Springer

Proof theory and category theory were first drawn together by Lambek some 30 years ago but, until now, the most fundamental notions of category theory (as opposed to their embodiments in logic) have not been explained systematically in terms of proof theory. Here it is shown that these notions, in particular the notion of adjunction, can be formulated in such as way as to be characterised by composition elimination. Among the benefits of these composition-free formulations are syntactical and simple model-theoretical, geometrical decision procedures for the commuting of diagrams of arrows. Composition elimination, in the form of Gentzen's cut elimination, takes in categories, and techniques inspired by Gentzen are shown to work even better in a purely categorical context than in logic. An acquaintance with the basic ideas of general proof theory is relied on only for the sake of motivation, however, and the treatment of matters related to categories is also in general self contained. Besides familiar topics, presented in a novel, simple way, the monograph also contains new results. It can be used as an introductory text in categorical proof theory.

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  • Used Book in Good Condition


My Best Mathematical and Logic Puzzles (Dover Recreational Math)

My Best Mathematical and Logic Puzzles (Dover Recreational Math) Lowest new price: $4.50
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Author: Martin Gardner
Brand: Dover Publications

Over a period of 25 years as author of the Mathematical Games column for Scientific American, Martin Gardner devoted a column every six months or so to short math problems or puzzles. He was especially careful to present new and unfamiliar puzzles that had not been included in such classic collections as those by Sam Loyd and Henry Dudeney. Later, these puzzles were published in book collections, incorporating reader feedback on alternate solutions or interesting generalizations.
The present volume contains a rich selection of 70 of the best of these brain teasers, in some cases including references to new developments related to the puzzle. Now enthusiasts can challenge their solving skills and rattle their egos with such stimulating mind-benders as The Returning Explorer, The Mutilated Chessboard, Scrambled Box Tops, The Fork in the Road, Bronx vs. Brooklyn, Touching Cigarettes, and 64 other problems involving logic and basic math. Solutions are included.

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  • Dover Publications

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How to Prove It: A Structured Approach, 2nd Edition

How to Prove It: A Structured Approach, 2nd Edition Lowest new price: $28.23
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Author: Daniel J. Velleman
Brand: imusti

Geared to preparing students to make the transition from solving problems to proving theorems, this text teaches them the techniques needed to read and write proofs. The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs. To help students construct their own proofs, this new edition contains over 200 new exercises, selected solutions, and an introduction to Proof Designer software. No background beyond standard high school mathematics is assumed. Previous Edition Hb (1994) 0-521-44116-1 Previous Edition Pb (1994) 0-521-44663-5

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  • Cambridge University Press

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Book of Proof

Book of Proof Lowest new price: $15.64
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Author: Richard Hammack

This book is an introduction to the language and standard proof methods of mathematics. It is a bridge from the computational courses (such as calculus or differential equations) that students typically encounter in their first year of college to a more abstract outlook. It lays a foundation for more theoretical courses such as topology, analysis and abstract algebra. Although it may be more meaningful to the student who has had some calculus, there is really no prerequisite other than a measure of mathematical maturity. Topics include sets, logic, counting, methods of conditional and non-conditional proof, disproof, induction, relations, functions and infinite cardinality.

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Descartes' Error: Emotion, Reason, and the Human Brain

Descartes' Error: Emotion, Reason, and the Human Brain Lowest new price: $9.75
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Author: Antonio Damasio
Brand: Damasio, Antonio R.

Since Descartes famously proclaimed, "I think, therefore I am," science has often overlooked emotions as the source of a person’s true being. Even modern neuroscience has tended, until recently, to concentrate on the cognitive aspects of brain function, disregarding emotions. This attitude began to change with the publication of Descartes’ Error in 1995. Antonio Damasio—"one of the world’s leading neurologists" (The New York Times)—challenged traditional ideas about the connection between emotions and rationality. In this wondrously engaging book, Damasio takes the reader on a journey of scientific discovery through a series of case studies, demonstrating what many of us have long suspected: emotions are not a luxury, they are essential to rational thinking and to normal social behavior.

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A Half-Century of Automata Theory: Celebration and Inspiration

A Half-Century of Automata Theory: Celebration and Inspiration Lowest new price: $78.00
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This volume gathers lectures by 8 distinguished pioneers of automata theory, including two Turing Award winners. In each contribution, the early developments of automata theory are reminisced about and future directions are suggested. Although some of the contributions go into rather intriguing technical details, most of the book is accessible to a wide audience interested in the progress of the age of computers.The book is a must for professionals in theoretical computer science and related areas of mathematics. For students in these areas it provides an exceptionally deep view at the beginning of the new millennium.


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