What Should be Computed to Understand and Model Brain Function? From Robotics, Soft Computing, Biology and Neuroscience to Cognitive Philosophy

Book Description

This lively introduction to mathematical logic, easily accessible to nonmathematicians, offers an historical survey, coverage of predicate calculus, model theory, Godel’s theorems, computability and recursive functions, consistency and independence in axiomatic set theory, and much more. Suggestions for Further Reading. Diagrams.

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No-nonsense survey of logic.......2005-05-02

This is an introduction to the main ideas and results of mathematical logic. It is primarily a text for non-logicians but it is still very serious. Practically everything is proved, and the proofs are carefully crafted and not too technical. For a reader with a bit of mathematical background this is far more valuable than the more typical logic-for-casual-readers books, such as for instance "Gödel's Proof" by Nagel & Newman, which are too chatty and trivial and don't really prove anything. By contrast, a high point of this book is a very accessible treatment of the proof of Gödel's incompleteness theorem in a matter of a few pages. On the other hand, this book is perhaps not chatty enough: the clear proofs and discussions of the main results are nicely done, but the discussions of historical background, motivation and context are very sketchy.

Six Rigorous Lectures - Not for the Faint-Hearted.......2003-12-17

Although this book - What is Mathematical Logic? - is written in an informal and entertaining style, it is unlikely to appeal to a reader not familiar with predicate calculus, recursive functions, and set theory. Despite its innocuous title, this little book is surprisingly rigorous.

The six chapters are derived from a series of lectures given by the five authors - J. N. Crossley, C. J. Ash, C. J. Brickhill, J. C. Stillwell, and N. H. Williams - at Monash University and University of Melbourne in 1971. The lectures were substantially revised for publication.

Only the first chapter, a detailed historical survey of mathematical logic, can be readily appreciated by the non-mathematician. The remaining five chapters examine advanced topics in mathematical logic including the Godel-Henkin Completeness Theorem, Model Theory, Turing machines and recursive functions, Godel's Incompleteness Theorem, and advanced set theory.

Chapter 2 introduces the Godel-Henkin Completeness Theorem, a proof that predicate calculus is complete. Chapter 2 is not easy, but it is essential to acquire a reasonable familiarity with predicate calculus before moving forward.

Chapter 3 offers a detailed look at model theory, the study of relations between formal languages and the interpretation of formal languages. Topics include Predicate Calculus with Identity, the Compactness Theorem, and the Lowenheim-Skolem Theorems. I had substantial difficulty with the details, but I did gain a general understanding and appreciation for model theory.

Chapter 4 addressed in considerable detail a more familiar topic, Turing machines and recursive functions. The discussion concludes with a key proof: there is no algorithm which will enable us to decide, given any particular formula of predicate calculus, whether or not this particular formula is deducible from the axioms of predicate calculus.

Chapter 5 was a detailed examination of Godel's Incompleteness Theorem for formal systems that include arithmetic of the natural numbers. I had less difficulty with this topic as I had previously read Godel's Proof by E. Nagel and J. R. Newman. This chapter would very likely be tough going for a reader entirely new to Godel's exceeding complex and abstruse proof.

Chapter 6, titled Set Theory, might be better named Advanced Set Theory. I was entirely new to the Axiom of Choice and the Generalized Continuum Hypothesis.

I highly recommend this intriguing and lively look at mathematical logic to readers with some familiarity with this rather formidable subject. For readers new to mathematical logic, I suggest that the following books might be better starting points.

Foundations and Fundamental Concepts of Mathematics by Howard Eves is outstanding. The chapter titled Logic and Philosophy is an excellent introduction to mathematical logic.

The Advent of the Algorithm by David Berlinski is an eclectic, rather bizarre introduction to a complex mathematical topic. Although many reader reviewers aggressively criticize this book, I enjoyed puzzling my way through Berlinski's discursive discussions.

Godel's Proof by Ernest Nagel and James R. Newman offers a fascinating look at a mind boggling, incredibly complex, inventive mathematical proof.

Dense but readable.......2003-04-15

After a 10-page historical survey of logic from the 1850s through the 1960s, similarly brief chapters on Completeness, Model Theory, Recursion Theory, the Incompleteness Theorems, and Set Theory give an idea of what might be covered in an undergraduate course and the first several graduate courses in mathematical logic. (The last 5 pages of the book are an introduction to forcing arguments and a fairly detailed sketch of the consistency of not-GCH.)

Results are clearly and carefully stated; and while sketches of proofs have a hard time staying nontechnical and still meaningful, most such attempts are admirable.

Book Description

A selection of papers presented at the international conference `Applied Logic: Logic at Work', held in Amsterdam in December 1992. Nowadays, the term `applied logic' has a very wide meaning, as numerous applications of logical methods in computer science, formal linguistics and other fields testify. Such applications are by no means restricted to the use of known logical techniques: at its best, applied logic involves a back-and-forth dialogue between logical theory and the problem domain.
The papers focus on the application of logic to the study of natural language, in syntax, semantics and pragmatics, and the effect of these studies on the development of logic. In the last decade, the dynamic nature of natural language has been the most interesting challenge for logicians. Dynamic semantics is here applied to new topics, the dynamic approach is extended to syntax, and several methodological issues in dynamic semantics are systematically investigated. Other methodological issues in the formal studies of natural language are discussed, such as the need for types, modal operators and other logical operators in the formal framework. Further articles address the scope of these methodological issues from other perspectives ranging from cognition to computation.
The volume presents papers that are interesting for graduate students and researchers in the field of logic, philosophy of language, formal semantics and pragmatics, and computational linguistics.

New Computational Paradigms: Changing Conceptions of What is Computable

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New Computational Paradigms: Changing Conceptions of What is Computable

Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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ASIN: 0387360336

Book Description

In recent years, classical computability has expanded beyond its original scope to address issues related to computability and complexity in algebra, analysis, and physics. The deep interconnection between "computation" and "proof" has originated much of the most significant work in constructive mathematics and mathematical logic of the last 70 years. Moreover, the increasingly compelling necessity to deal with computability in the real world (such as computing on continuous data, biological computing, and physical models) has brought focus to new paradigms of computation that are based on biological and physical models. These models address questions of efficiency in a radically new way and even threaten to move the so-called Turing barrier, i.e. the line between the decidable and the un-decidable.

This book examines new developments in the theory and practice of computation from a mathematical perspective, with topics ranging from classical computability to complexity, from biocomputing to quantum computing. The book opens with an introduction by Alan Hodges, the Turing biographer, who analyzes the pioneering work that anticipated recent developments concerning computationâ€™s allegedly new paradigms. The remaining material covers traditional topics in computability theory such as relative computability, theory of numberings, and domain theory, in addition to topics on the relationships between proof theory, computability, and complexity theory. New paradigms of computation arising from biology and quantum physics are also discussed, as well as the computability of the real numbers and its related issues.

This book is suitable for researchers and graduate students in mathematics, philosophy, and computer science with a special interest in logic and foundational issues. Most useful to graduate students are the survey papers on computable analysis and biological computing. Logicians and theoretical physicists will also benefit from this book.

What Is a Logical System? (Studies in Logic and Computation)

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What Is a Logical System? (Studies in Logic and Computation)

Manufacturer: Oxford University Press, USA
ProductGroup: Book
Binding: Hardcover

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ASIN: 0198538596

Book Description

This extraordinary collection of papers addresses a fundamental question of logic and computation. "What is a logical system?". With contributions from many world famous researchers, it presents a wide spectrum of views on the problem, reflecting mainstream current approaches to logic and how it is applied.

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WHAT IS MATHEMATICAL LOGIC?

Manufacturer: Oxford University Press
ProductGroup: Book
Binding: Hardcover
ASIN: B000GYWU94

Nothing but the truth: An account of what philosophers and others have believed about mathematics

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Nothing but the truth: An account of what philosophers and others have believed about mathematics
Eric Temple Bell
Manufacturer: s.n
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Binding: Unknown Binding

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What is a model?: A consumer's perspective on semantic theory (Technical report. University of Colorado at Boulder. Department of Computer Science)

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What is a model?: A consumer's perspective on semantic theory (Technical report. University of Colorado at Boulder. Department of Computer Science)
Jon Shultis
Manufacturer: University of Colorado, Dept. of Computer Science
ProductGroup: Book
Binding: Unknown Binding

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What is Mathematical Logic?
J. N. Et al Crossley
Manufacturer: Dover
ProductGroup: Book
Binding: Hardcover
ASIN: B000MBCYGU

What Is Mathematical Logic? (Opus Ser., No. 60)

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What Is Mathematical Logic? (Opus Ser., No. 60)
John N. Crossley
Manufacturer: Oxford University Press
ProductGroup: Book
Binding: Paperback
ASIN: B000OLIUMK

Book Description

Ranging from the unfamiliar to the famous--from the inconspicuous, plankton-sucking Elktoe Mussel to the majestic, soaring Bald Eagle--endangered or threatened plants and animals of Kansas continue to play a vital, although diminished, role in the state's ecology.

Providing, for the first time, easily accessible information for the professional naturalist and amateur nature lover alike, this guide highlights the habits and habitats of sixty plants and animals currently listed as endangered or threatened in Kansas and protected by state or federal law. The authors illuminate not only the common bond of these species--their precarious status--but also their widely varying routines, idiosyncrasies, and circumstances. All known Lake Scott Riffle Beetles in the world, they show, spend their lives in one spring area of Western Kansas while the nomadic Whooping Cranes alight only a night or two in the state's central and eastern swamplands during migration.

For each species--three plants, three mammals, nine birds, eight reptiles, ten amphibians, fourteen fishes, and thirteen invertebrates--the guide features a full-color photograph, standard common and current scientific name, range map, threatened or endangered species status, and information on appearance, size, breeding, habits, surroundings, food preference, and natural history. The authors also list possible causes of species reduction, from loss of habitat through land use change, water diversion, and pollution, to hunting and fishing practices and natural selection.

Richly illustrated and informative, this unique guide will be indispensable to anyone wanting to preserve the state's irreplaceable biological diversity.

Book Description

This book represents the most comprehensive compilation of data on threatened vascular plants ever published. It includes the names of some 33,000 plant species determined to be rare or threatened on a global scale. Conservation assessments were provided by the IUCN Species Survival Commission, the National Botanical Institute (South Africa), Environment Australia, and CSIRO, The Nature Conservancy, the Smithsonian Institution, and the Royal Botanic Gardens, Kew, together with hundreds of botanic gardens and botanists throughout the world. The Royal Botanic Gardens Edinburgh and the New York Botanical Garden have made major in-kind contributions.The result of 20 years work by botanists and conservationists around the world, it is intended as a conservation tool, a provider of baseline information to measure conservation progress and as a primary source of data on plant species. Most importantly, however, it provides the building blocks on which to base a worldwide effort to conserve plant species.

Code of Practice for the Conservation of Threatened Animals and Plants and Other Species of International Significance: As Adopted by the Senior Advis

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Code of Practice for the Conservation of Threatened Animals and Plants and Other Species of International Significance: As Adopted by the Senior Advis

Manufacturer: United Nations
ProductGroup: Book
Binding: Paperback

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ASIN: 9211165393

Conservation of Threatened Plants (Nato Conference Ser.I, Ecology: Vol.1)

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Conservation of Threatened Plants (Nato Conference Ser.I, Ecology: Vol.1)

Manufacturer: Springer
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Binding: Hardcover

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ENDANGERED & THREATENED PLANTS
AYENSU E
Manufacturer: Smithsonian
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Endangered and threatened species of Illinois: Status and distribution. Volume 3

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Endangered and threatened species of Illinois: Status and distribution. Volume 3
James R Herkert
Manufacturer: Illinois Endangered Species Protection Board
ProductGroup: Book
Binding: Unknown Binding

Endangered and threatened species recovery program : report to Congress (SuDoc I 49.77/3:)

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Endangered and threatened species recovery program : report to Congress (SuDoc I 49.77/3:)
U.S. Dept of Interior
Manufacturer: The Service
ProductGroup: Book
Binding: Unknown Binding

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ASIN: B000107QB0

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Endangered and threatened species recovery program report to Congress (SuDoc I 49.77/3:)
U.S. Dept of Interior
Manufacturer: The Service
ProductGroup: Book
Binding: Unknown Binding

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ASIN: B00010SXV2

European Red List of Globally Threatened Animals and Plants

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European Red List of Globally Threatened Animals and Plants
Economic Commission for Europe
Manufacturer: United Nations
ProductGroup: Book
Binding: Paperback

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ASIN: 9211165229

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