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Author: Jaakko Hintikka
Publisher: Cambridge University Press
ISBN: 9780521624985
Category : Mathematics
Languages : en
Pages : 308
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Book Description
This book, written by one of philosophy's pre-eminent logicians, argues that many of the basic assumptions common to logic, philosophy of mathematics and metaphysics are in need of change. It is therefore a book of critical importance to logical theory. Jaakko Hintikka proposes a new basic first-order logic and uses it to explore the foundations of mathematics. This new logic enables logicians to express on the first-order level such concepts as equicardinality, infinity, and truth in the same language. The famous impossibility results by Gödel and Tarski that have dominated the field for the last sixty years turn out to be much less significant than has been thought. All of ordinary mathematics can in principle be done on this first-order level, thus dispensing with the existence of sets and other higher-order entities.
Author: Jaakko Hintikka
Publisher: Cambridge University Press
ISBN: 9780521624985
Category : Mathematics
Languages : en
Pages : 308
Get Book
Book Description
This book, written by one of philosophy's pre-eminent logicians, argues that many of the basic assumptions common to logic, philosophy of mathematics and metaphysics are in need of change. It is therefore a book of critical importance to logical theory. Jaakko Hintikka proposes a new basic first-order logic and uses it to explore the foundations of mathematics. This new logic enables logicians to express on the first-order level such concepts as equicardinality, infinity, and truth in the same language. The famous impossibility results by Gödel and Tarski that have dominated the field for the last sixty years turn out to be much less significant than has been thought. All of ordinary mathematics can in principle be done on this first-order level, thus dispensing with the existence of sets and other higher-order entities.
Author: Hintikka Jaakko
Publisher:
ISBN:
Category : First-order logic
Languages : en
Pages : 288
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Book Description
Author: Vladimir Lepetic
Publisher: John Wiley & Sons
ISBN: 1119131650
Category : Mathematics
Languages : en
Pages : 672
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Book Description
Presents a uniquely balanced approach that bridges introductory and advanced topics in modern mathematics An accessible treatment of the fundamentals of modern mathematics, Principles of Mathematics: A Primer provides a unique approach to introductory andadvanced mathematical topics. The book features six main subjects, whichcan be studied independently or in conjunction with each other including: settheory; mathematical logic; proof theory; group theory; theory of functions; andlinear algebra. The author begins with comprehensive coverage of the necessary building blocks in mathematics and emphasizes the need to think abstractly and develop an appreciation for mathematical thinking. Maintaining a useful balance of introductory coverage and mathematical rigor, Principles of Mathematics: A Primer features: Detailed explanations of important theorems and their applications Hundreds of completely solved problems throughout each chapter Numerous exercises at the end of each chapter to encourage further exploration Discussions of interesting and provocative issues that spark readers’ curiosity and facilitate a better understanding and appreciation of the field of mathematics Principles of Mathematics: A Primer is an ideal textbook for upper-undergraduate courses in the foundations of mathematics and mathematical logic as well as for graduate-level courses related to physics, engineering, and computer science. The book is also a useful reference for readers interested in pursuing careers in mathematics and the sciences.
Author: Hans Freudenthal
Publisher: Springer Science & Business Media
ISBN: 0306472023
Category : Education
Languages : en
Pages : 202
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Book Description
This book is a product of love and respect. If that sounds rather odd I initially apologise, but let me explain why I use those words. The original manuscript was of course Freudenthal’s, but his colleagues have carried the project through to its conclusion with love for the man, and his ideas, and with a respect developed over years of communal effort. Their invitation to me to write this Preface e- bles me to pay my respects to the great man, although I am probably incurring his wrath for writing a Preface for his book without his permission! I just hope he understands the feelings of all colleagues engaged in this particular project. Hans Freudenthal died on October 13th, 1990 when this book project was well in hand. In fact he wrote to me in April 1988, saying “I am thinking about a new book. I have got the sub-title (China Lectures) though I still lack a title”. I was astonished. He had retired in 1975, but of course he kept working. Then in 1985 we had been helping him celebrate his 80th birthday, and although I said in an Editorial Statement in Educational Studies in Mathematics (ESM) at the time “we look forward to him enjoying many more years of non-retirement” I did not expect to see another lengthy manuscript.
Author: Gareth E. Roberts
Publisher: JHU Press
ISBN: 1421419181
Category : Mathematics
Languages : en
Pages : 320
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Book Description
A guided tour of the mathematical principles inherent in music. Taking a "music first" approach, Gareth E. Roberts's From Music to Mathematics will inspire students to learn important, interesting, and at times advanced mathematics. Ranging from a discussion of the geometric sequences and series found in the rhythmic structure of music to the phase-shifting techniques of composer Steve Reich, the musical concepts and examples in the book motivate a deeper study of mathematics. Comprehensive and clearly written, From Music to Mathematics is designed to appeal to readers without specialized knowledge of mathematics or music. Students are taught the relevant concepts from music theory (notation, scales, intervals, the circle of fifths, tonality, etc.), with the pertinent mathematics developed alongside the related musical topic. The mathematics advances in level of difficulty from calculating with fractions, to manipulating trigonometric formulas, to constructing group multiplication tables and proving a number is irrational. Topics discussed in the book include • Rhythm • Introductory music theory • The science of sound • Tuning and temperament • Symmetry in music • The Bartók controversy • Change ringing • Twelve-tone music • Mathematical modern music • The Hemachandra–Fibonacci numbers and the golden ratio • Magic squares • Phase shifting Featuring numerous musical excerpts, including several from jazz and popular music, each topic is presented in a clear and in-depth fashion. Sample problems are included as part of the exposition, with carefully written solutions provided to assist the reader. The book also contains more than 200 exercises designed to help develop students' analytical skills and reinforce the material in the text. From the first chapter through the last, readers eager to learn more about the connections between mathematics and music will find a comprehensive textbook designed to satisfy their natural curiosity.
Author: Jaakko Hintikka
Publisher: Springer Science & Business Media
ISBN: 9401586012
Category : Philosophy
Languages : en
Pages : 290
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Book Description
R. G. Collingwood saw one of the main tasks of philosophers and of historians of human thought in uncovering what he called the ultimate presuppositions of different thinkers, of different philosophical movements and of entire eras of intellectual history. He also noted that such ultimate presuppositions usually remain tacit at first, and are discovered only by subsequent reflection. Collingwood would have been delighted by the contrast that constitutes the overall theme of the essays collected in this volume. Not only has this dichotomy ofviews been one ofthe mostcrucial watersheds in the entire twentieth-century philosophical thought. Not only has it remained largely implicit in the writings of the philosophers for whom it mattered most. It is a truly Collingwoodian presupposition also in that it is not apremise assumed by different thinkers in their argumentation. It is the presupposition of a question, an assumption to the effect that a certain general question can be raised and answered. Its role is not belied by the fact that several philosophers who answered it one way or the other seem to be largely unaware that the other answer also makes sense - if it does. This Collingwoodian question can be formulated in a first rough approximation by asking whether language - our actual working language, Tarski's "colloquiallanguage" - is universal in the sense of being inescapable. This formulation needs all sorts of explanations, however.
Author: Randall E. Auxier
Publisher:
ISBN: 9780812694635
Category : Biography & Autobiography
Languages : en
Pages : 1004
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Book Description
Dr. Jaakko Hintikka is an influencial philosopher most famous for originating game-theoretical semantics. This book includes an intellectual biography, 29 previously unpublished critical and descriptive essays by famous scholars and a reply to each essay by Hintikka himself.
Author: Carlo Cellucci
Publisher: Springer Science & Business Media
ISBN: 9400760914
Category : Philosophy
Languages : en
Pages : 391
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Book Description
This volume examines the limitations of mathematical logic and proposes a new approach to logic intended to overcome them. To this end, the book compares mathematical logic with earlier views of logic, both in the ancient and in the modern age, including those of Plato, Aristotle, Bacon, Descartes, Leibniz, and Kant. From the comparison it is apparent that a basic limitation of mathematical logic is that it narrows down the scope of logic confining it to the study of deduction, without providing tools for discovering anything new. As a result, mathematical logic has had little impact on scientific practice. Therefore, this volume proposes a view of logic according to which logic is intended, first of all, to provide rules of discovery, that is, non-deductive rules for finding hypotheses to solve problems. This is essential if logic is to play any relevant role in mathematics, science and even philosophy. To comply with this view of logic, this volume formulates several rules of discovery, such as induction, analogy, generalization, specialization, metaphor, metonymy, definition, and diagrams. A logic based on such rules is basically a logic of discovery, and involves a new view of the relation of logic to evolution, language, reason, method and knowledge, particularly mathematical knowledge. It also involves a new view of the relation of philosophy to knowledge. This book puts forward such new views, trying to open again many doors that the founding fathers of mathematical logic had closed historically. trigger
Author: Roy T. Cook
Publisher: Springer Science & Business Media
ISBN: 1402042655
Category : Mathematics
Languages : en
Pages : 480
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Book Description
This volume collects together a number of important papers concerning both the method of abstraction generally and the use of particular abstraction principles to reconstruct central areas of mathematics along logicist lines. Attention is focused on extending the Neo-Fregean treatment to all of mathematics, with the reconstruction of real analysis from various cut- or cauchy-sequence-related abstraction principles and the reconstruction of set theory from various restricted versions of Basic Law V as case studies.
Author: Thomas A. Garrity
Publisher: Cambridge University Press
ISBN: 1009006207
Category : Mathematics
Languages : en
Pages : 417
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Book Description
Beginning graduate students in mathematical sciences and related areas in physical and computer sciences and engineering are expected to be familiar with a daunting breadth of mathematics, but few have such a background. This bestselling book helps students fill in the gaps in their knowledge. Thomas A. Garrity explains the basic points and a few key results of all the most important undergraduate topics in mathematics, emphasizing the intuitions behind the subject. The explanations are accompanied by numerous examples, exercises and suggestions for further reading that allow the reader to test and develop their understanding of these core topics. Featuring four new chapters and many other improvements, this second edition of All the Math You Missed is an essential resource for advanced undergraduates and beginning graduate students who need to learn some serious mathematics quickly.
