Godel's Disjunction

The Scope and Limits of Mathematical Knowledge
Author: Leon Horsten,Philip Welch
Publisher: Oxford University Press
ISBN: 0198759592
Category:
Page: 288
View: 3868
The logician Kurt Godel in 1951 established a disjunctive thesis about the scope and limits of mathematical knowledge: either the mathematical mind is not equivalent to a Turing machine (i.e., a computer), or there are absolutely undecidable mathematical problems. In the second half of the twentieth century, attempts have been made to arrive at a stronger conclusion. In particular, arguments have been produced by the philosopher J.R. Lucas and by the physicist and mathematician Roger Penrose that intend to show that the mathematical mind is more powerful than any computer. These arguments, and counterarguments to them, have not convinced the logical and philosophical community. The reason for this is an insufficiency if rigour in the debate. The contributions in this volume move the debate forward by formulating rigorous frameworks and formally spelling out and evaluating arguments that bear on Godel's disjunction in these frameworks. The contributions in this volume have been written by world leading experts in the field.

    • Mathematics

Gödel's Disjunction

The scope and limits of mathematical knowledge
Author: Leon Horsten,Philip Welch
Publisher: Oxford University Press
ISBN: 0191077690
Category: Mathematics
Page: 288
View: 9138
The logician Kurt Gödel in 1951 established a disjunctive thesis about the scope and limits of mathematical knowledge: either the mathematical mind is not equivalent to a Turing machine (i.e., a computer), or there are absolutely undecidable mathematical problems. In the second half of the twentieth century, attempts have been made to arrive at a stronger conclusion. In particular, arguments have been produced by the philosopher J.R. Lucas and by the physicist and mathematician Roger Penrose that intend to show that the mathematical mind is more powerful than any computer. These arguments, and counterarguments to them, have not convinced the logical and philosophical community. The reason for this is an insufficiency if rigour in the debate. The contributions in this volume move the debate forward by formulating rigorous frameworks and formally spelling out and evaluating arguments that bear on Gödel's disjunction in these frameworks. The contributions in this volume have been written by world leading experts in the field.

    • Science

Physical Perspectives on Computation, Computational Perspectives on Physics


Author: Michael E. Cuffaro,Samuel C. Fletcher
Publisher: Cambridge University Press
ISBN: 1316767396
Category: Science
Page: N.A
View: 8813
Although computation and the science of physical systems would appear to be unrelated, there are a number of ways in which computational and physical concepts can be brought together in ways that illuminate both. This volume examines fundamental questions which connect scholars from both disciplines: is the universe a computer? Can a universal computing machine simulate every physical process? What is the source of the computational power of quantum computers? Are computational approaches to solving physical problems and paradoxes always fruitful? Contributors from multiple perspectives reflecting the diversity of thought regarding these interconnections address many of the most important developments and debates within this exciting area of research. Both a reference to the state of the art and a valuable and accessible entry to interdisciplinary work, the volume will interest researchers and students working in physics, computer science, and philosophy of science and mathematics.

    • Science

Interpreting Godel

Critical Essays
Author: Juliette Kennedy
Publisher: Cambridge University Press
ISBN: 1107002664
Category: Science
Page: 288
View: 2101
In this groundbreaking volume, leading philosophers and mathematicians explore Kurt Gödel's work on the foundations and philosophy of mathematics.

    • Analysis (Philosophy).

The Mathematical Analysis of Logic

Being an Essay Towards a Calculus of Deductive Reasoning
Author: George Boole
Publisher: N.A
ISBN: N.A
Category: Analysis (Philosophy).
Page: 82
View: 9757


    • Business & Economics

Model Building in Mathematical Programming


Author: H. Paul Williams
Publisher: John Wiley & Sons
ISBN: 1118506189
Category: Business & Economics
Page: 432
View: 3539
The 5th edition of Model Building in Mathematical Programming discusses the general principles of model building in mathematical programming and demonstrates how they can be applied by using several simplified but practical problems from widely different contexts. Suggested formulations and solutions are given together with some computational experience to give the reader a feel for the computational difficulty of solving that particular type of model. Furthermore, this book illustrates the scope and limitations of mathematical programming, and shows how it can be applied to real situations. By emphasizing the importance of the building and interpreting of models rather than the solution process, the author attempts to fill a gap left by the many works which concentrate on the algorithmic side of the subject. In this article, H.P. Williams explains his original motivation and objectives in writing the book, how it has been modified and updated over the years, what is new in this edition and why it has maintained its relevance and popularity over the years: http://www.statisticsviews.com/details/feature/4566481/Model-Building-in-Mathematical-Programming-published-in-fifth-edition.html

Mathematics for Computer Science


Author: Eric Lehman,F. Thomson Leighton,Albert R. Meyer
Publisher: N.A
ISBN: 9789888407064
Category:
Page: 979
View: 5979
This book covers elementary discrete mathematics for computer science and engineering. It emphasizes mathematical definitions and proofs as well as applicable methods. Topics include formal logic notation, proof methods; induction, well-ordering; sets, relations; elementary graph theory; integer congruences; asymptotic notation and growth of functions; permutations and combinations, counting principles; discrete probability. Further selected topics may also be covered, such as recursive definition and structural induction; state machines and invariants; recurrences; generating functions.

    • Computers

The Influence of Computers and Informatics on Mathematics and Its Teaching

Proceedings From a Symposium Held in Strasbourg, France in March 1985 and Sponsored by the International Commission on Mathematical Instruction
Author: R. F. Churchhouse
Publisher: CUP Archive
ISBN: 9780521311892
Category: Computers
Page: 155
View: 4475
First published in 1986, the first ICMI study is concerned with the influence of computers and computer science on mathematics and its teaching in the last years of school and at tertiary level. In particular, it explores the way the computer has influenced mathematics itself and the way in which mathematicians work, likely influences on the curriculum of high-school and undergraduate students, and the way in which the computer can be used to improve mathematics teaching and learning. The book comprises a report of the meeting held in Strasbourg in March 1985, plus several papers contributed to that meeting.

    • Logic

An Investigation of the Laws of Thought

On which are Founded the Mathematical Theories of Logic and Probabilities
Author: George Boole
Publisher: N.A
ISBN: N.A
Category: Logic
Page: 424
View: 6701

    • Encyclopedias and dictionaries

The Century Dictionary

An Encyclopedic Lexicon of the English Language
Author: William Dwight Whitney
Publisher: N.A
ISBN: N.A
Category: Encyclopedias and dictionaries
Page: 7076
View: 7322

    • Mathematics

Logic and Structure


Author: Dirk van Dalen
Publisher: Springer Science & Business Media
ISBN: 3662029626
Category: Mathematics
Page: 220
View: 6729
New corrected printing of a well-established text on logic at the introductory level.

    • Science

Probability Theory

The Logic of Science
Author: E. T. Jaynes
Publisher: Cambridge University Press
ISBN: 1139435167
Category: Science
Page: N.A
View: 4559
The standard rules of probability can be interpreted as uniquely valid principles in logic. In this book, E. T. Jaynes dispels the imaginary distinction between 'probability theory' and 'statistical inference', leaving a logical unity and simplicity, which provides greater technical power and flexibility in applications. This book goes beyond the conventional mathematics of probability theory, viewing the subject in a wider context. New results are discussed, along with applications of probability theory to a wide variety of problems in physics, mathematics, economics, chemistry and biology. It contains many exercises and problems, and is suitable for use as a textbook on graduate level courses involving data analysis. The material is aimed at readers who are already familiar with applied mathematics at an advanced undergraduate level or higher. The book will be of interest to scientists working in any area where inference from incomplete information is necessary.

    • Mathematics

Six Septembers: Mathematics for the Humanist


Author: Patrick Juola,Stephen Ramsay
Publisher: Lulu.com
ISBN: 1609621115
Category: Mathematics
Page: 422
View: 4303
Scholars of all stripes are turning their attention to materials that represent enormous opportunities for the future of humanistic inquiry. The purpose of this book is to impart the concepts that underlie the mathematics they are likely to encounter and to unfold the notation in a way that removes that particular barrier completely. This book is a primer for developing the skills to enable humanist scholars to address complicated technical material with confidence. This book, to put it plainly, is concerned with the things that the author of a technical article knows, but isn't saying. Like any field, mathematics operates under a regime of shared assumptions, and it is our purpose to elucidate some of those assumptions for the newcomer. The individual subjects we tackle are (in order): logic and proof, discrete mathematics, abstract algebra, probability and statistics, calculus, and differential equations.

    • English language

Webster's New International Dictionary of the English Language

Based on the International Dictionary of 1890 and 1900
Author: William Torrey Harris,Frederic Sturges Allen
Publisher: N.A
ISBN: N.A
Category: English language
Page: 2620
View: 6442

    • Mathematics

Introduction to Mathematical Logic, Sixth Edition


Author: Elliott Mendelson
Publisher: CRC Press
ISBN: 1482237784
Category: Mathematics
Page: 513
View: 2283
The new edition of this classic textbook, Introduction to Mathematical Logic, Sixth Edition explores the principal topics of mathematical logic. It covers propositional logic, first-order logic, first-order number theory, axiomatic set theory, and the theory of computability. The text also discusses the major results of Gödel, Church, Kleene, Rosser, and Turing. The sixth edition incorporates recent work on Gödel’s second incompleteness theorem as well as restoring an appendix on consistency proofs for first-order arithmetic. This appendix last appeared in the first edition. It is offered in the new edition for historical considerations. The text also offers historical perspectives and many new exercises of varying difficulty, which motivate and lead students to an in-depth, practical understanding of the material.

    • Philosophy

The Limitations of the Open Mind


Author: Jeremy Fantl
Publisher: Oxford University Press
ISBN: 0198807953
Category: Philosophy
Page: 256
View: 7000
When we should engage with those we disagree with? Jeremy Fantl argues that sometimes we can know that arguments for controversial ideas go wrong even without engaging critically with them or figuring out where they err. Sometimes we shouldn't engage critically with an argument and, if we do engage, we shouldn't engage open-mindedly.

    • Mathematics

How to Prove It

A Structured Approach
Author: Daniel J. Velleman
Publisher: Cambridge University Press
ISBN: 1139450972
Category: Mathematics
Page: N.A
View: 6588
Many students have trouble the first time they take a mathematics course in which proofs play a significant role. This new edition of Velleman's successful text will prepare students to make the transition from solving problems to proving theorems by teaching 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. The author shows how complex proofs are built up from these smaller steps, using detailed 'scratch work' sections to expose the machinery of proofs about the natural numbers, relations, functions, and infinite sets. To give students the opportunity to 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. This book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and of course mathematicians.

    • Logic, Symbolic and mathematical

Principia Mathematica


Author: Alfred North Whitehead,Bertrand Russell
Publisher: N.A
ISBN: N.A
Category: Logic, Symbolic and mathematical
Page: N.A
View: 1209

    • Computers

Understanding Machine Learning

From Theory to Algorithms
Author: Shai Shalev-Shwartz,Shai Ben-David
Publisher: Cambridge University Press
ISBN: 1107057132
Category: Computers
Page: 409
View: 4222
Introduces machine learning and its algorithmic paradigms, explaining the principles behind automated learning approaches and the considerations underlying their usage.