The Principles of Mathematics Revisited
Author : Jaakko Hintikka
Publisher :
Page : 288 pages
File Size : 34,28 MB
Release : 1998
Category : First-order logic
ISBN :
The Principles of Mathematics Revisited
Author : Hintikka Jaakko
Publisher :
Page : 288 pages
File Size : 12,7 MB
Release : 1996
Category : First-order logic
ISBN :
Book Description
The Principles of Mathematics Revisited
Author : Jaakko Hintikka
Publisher : Cambridge University Press
Page : 308 pages
File Size : 33,90 MB
Release : 1998-04-28
Category : Mathematics
ISBN : 9780521624985
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.
The Principles of Mathematics Revisited
Author : Kaarlo Jaakko Juhani Hintikka
Publisher :
Page : 288 pages
File Size : 36,65 MB
Release : 1998
Category :
ISBN :
Book Description
Principles of Mathematics
Author : Vladimir Lepetic
Publisher : John Wiley & Sons
Page : 672 pages
File Size : 42,29 MB
Release : 2015-11-30
Category : Mathematics
ISBN : 1119131650
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.
Numbers and Shapes Revisited
Author : Judita Cofman
Publisher :
Page : 334 pages
File Size : 20,66 MB
Release : 1995
Category : Education
ISBN :
Book Description
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.
Authentic Science Revisited
Author : Wolff-Michael Roth
Publisher : BRILL
Page : 245 pages
File Size : 21,80 MB
Release : 2019-02-11
Category : Education
ISBN : 9087906722
Book Description
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.
Lingua Universalis vs. Calculus Ratiocinator:
Author : Jaakko Hintikka
Publisher : Springer Science & Business Media
Page : 290 pages
File Size : 20,84 MB
Release : 2013-04-09
Category : Philosophy
ISBN : 9401586012
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.
Language, Truth and Logic in Mathematics
Author : Jaakko Hintikka
Publisher : Springer Science & Business Media
Page : 339 pages
File Size : 46,74 MB
Release : 2013-03-09
Category : Mathematics
ISBN : 9401720452
Book Description
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.
From Music to Mathematics
Author : Gareth E. Roberts
Publisher : JHU Press
Page : 320 pages
File Size : 30,99 MB
Release : 2016-02-15
Category : Mathematics
ISBN : 1421419181
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.
