The Principles of Mathematics Revisited

by Jaakko Hintikka
The Principles of Mathematics Revisited
Author: Jaakko Hintikka
Publisher: Cambridge University Press
Total Pages: 308
Release: 1998-04-28
Genre: Mathematics
ISBN: 9780521624985
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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.

Principles of Mathematics

by Vladimir Lepetic
Principles of Mathematics
Author: Vladimir Lepetic
Publisher: John Wiley & Sons
Total Pages: 672
Release: 2015-11-30
Genre: Mathematics
ISBN: 1119131650
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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.

Lingua Universalis vs. Calculus Ratiocinator:

by Jaakko Hintikka
Lingua Universalis vs. Calculus Ratiocinator:
Author: Jaakko Hintikka
Publisher: Springer Science & Business Media
Total Pages: 290
Release: 2013-04-09
Genre: Philosophy
ISBN: 9401586012
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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

by Jaakko Hintikka
Language, Truth and Logic in Mathematics
Author: Jaakko Hintikka
Publisher: Springer Science & Business Media
Total Pages: 339
Release: 2013-03-09
Genre: Mathematics
ISBN: 9401720452
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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.

The Arché Papers on the Mathematics of Abstraction

by Roy T. Cook
The Arché Papers on the Mathematics of Abstraction
Author: Roy T. Cook
Publisher: Springer Science & Business Media
Total Pages: 480
Release: 2007-11-27
Genre: Mathematics
ISBN: 1402042655
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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.

Rethinking Logic: Logic in Relation to Mathematics, Evolution, and Method

by Carlo Cellucci
Rethinking Logic: Logic in Relation to Mathematics, Evolution, and Method
Author: Carlo Cellucci
Publisher: Springer Science & Business Media
Total Pages: 391
Release: 2013-10-09
Genre: Philosophy
ISBN: 9400760914
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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

All the Math You Missed

by Thomas A. Garrity
All the Math You Missed
Author: Thomas A. Garrity
Publisher: Cambridge University Press
Total Pages: 417
Release: 2021-07-01
Genre: Mathematics
ISBN: 1009006207
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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.

The Structure of Thinking

by Laura E. Wood
The Structure of Thinking
Author: Laura E. Wood
Publisher: Andrews UK Limited
Total Pages: 297
Release: 2013-06-17
Genre: Philosophy
ISBN: 1845405862
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Analytic philosophers and cognitive scientists have long argued that the mind is a computer-like syntactical engine, and that all human mental capacities can be described as digital computational processes. This book presents an alternative, naturalistic view of human thinking, arguing that computers are merely sophisticated machines. Computers are only simulating thought when they crunch symbols, not thinking. Human cognition - semantics, de re reference, indexicals, meaning and causation - are all rooted in human experience and life. Without life and experience, these elements of discourse and knowledge refer to nothing. And without these elements of discourse and knowledge, syntax is vacant structure, not thinking.