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| Author | : Jaakko Hintikka |
| Publisher | : Cambridge University Press |
| Total Pages | : 308 |
| Release | : 1998-04-28 |
| Genre | : Mathematics |
| ISBN | : 9780521624985 |
Download The Principles of Mathematics Revisited Book in PDF, ePub and Kindle
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 | : |
| Total Pages | : 288 |
| Release | : 1996 |
| Genre | : First-order logic |
| ISBN | : |
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| Author | : Kaarlo Jaakko Juhani Hintikka |
| Publisher | : |
| Total Pages | : 288 |
| Release | : 1998 |
| Genre | : |
| ISBN | : |
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| Author | : Vladimir Lepetic |
| Publisher | : John Wiley & Sons |
| Total Pages | : 672 |
| Release | : 2015-11-30 |
| Genre | : Mathematics |
| ISBN | : 1119131650 |
Download Principles of Mathematics Book in PDF, ePub and Kindle
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 | : Judita Cofman |
| Publisher | : |
| Total Pages | : 334 |
| Release | : 1995 |
| Genre | : Education |
| ISBN | : |
Download Numbers and Shapes Revisited Book in PDF, ePub and Kindle
By focusing attention on the links between patterns of numbers and shapes, and on connections between algebraic relations and geometric and combinatorial configurations, the book aims to motivate deeper study of the concepts related to elementary mathematics, emphasize the importance of the interrelations between mathematical phenomena, and foster the interplay of ideas involved in problem solving.
| Author | : Wolff-Michael Roth |
| Publisher | : BRILL |
| Total Pages | : 245 |
| Release | : 2019-02-11 |
| Genre | : Education |
| ISBN | : 9087906722 |
Download Authentic Science Revisited Book in PDF, ePub and Kindle
Since its appearance in 1995, Authentic School Science has been a resource for many teachers and schools to rethink and change what they are doing in and with their science classrooms. As others were trying to implement the kinds of learning environments that we had described, our own thinking and teaching praxis changed in part because of our dissatisfaction with our own understanding.
| Author | : Gareth E. Roberts |
| Publisher | : JHU Press |
| Total Pages | : 320 |
| Release | : 2016-02-15 |
| Genre | : Mathematics |
| ISBN | : 1421419181 |
Download From Music to Mathematics Book in PDF, ePub and Kindle
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 |
| Total Pages | : 290 |
| Release | : 2013-04-09 |
| Genre | : Philosophy |
| ISBN | : 9401586012 |
Download Lingua Universalis vs. Calculus Ratiocinator: Book in PDF, ePub and Kindle
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 | : Jaakko Hintikka |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 339 |
| Release | : 2013-03-09 |
| Genre | : Mathematics |
| ISBN | : 9401720452 |
Download Language, Truth and Logic in Mathematics Book in PDF, ePub and Kindle
One can distinguish, roughly speaking, two different approaches to the philosophy of mathematics. On the one hand, some philosophers (and some mathematicians) take the nature and the results of mathematicians' activities as given, and go on to ask what philosophical morals one might perhaps find in their story. On the other hand, some philosophers, logicians and mathematicians have tried or are trying to subject the very concepts which mathematicians are using in their work to critical scrutiny. In practice this usually means scrutinizing the logical and linguistic tools mathematicians wield. Such scrutiny can scarcely help relying on philosophical ideas and principles. In other words it can scarcely help being literally a study of language, truth and logic in mathematics, albeit not necessarily in the spirit of AJ. Ayer. As its title indicates, the essays included in the present volume represent the latter approach. In most of them one of the fundamental concepts in the foundations of mathematics and logic is subjected to a scrutiny from a largely novel point of view. Typically, it turns out that the concept in question is in need of a revision or reconsideration or at least can be given a new twist. The results of such a re-examination are not primarily critical, however, but typically open up new constructive possibilities. The consequences of such deconstructions and reconstructions are often quite sweeping, and are explored in the same paper or in others.
| Author | : Roy T. Cook |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 480 |
| Release | : 2007-11-27 |
| Genre | : Mathematics |
| ISBN | : 1402042655 |
Download The Arché Papers on the Mathematics of Abstraction Book in PDF, ePub and Kindle
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.
