Proof and Other Dilemmas

Mathematics and Philosophy


Author: Roger Simons

Publisher: MAA

ISBN: 9780883855676

Category: Mathematics

Page: 346

View: 727

For the majority of the twentieth century, philosophers of mathematics focused their attention on foundational questions. However, in the last quarter of the century they began to return to basics, and two new schools of thought were created: social constructivism and structuralism. The advent of the computer also led to proofs and development of mathematics assisted by computer, and to questions concerning the role of the computer in mathematics. This book of sixteen original essays is the first to explore this range of new developments in the philosophy of mathematics, in a language accessible to mathematicians. Approximately half the essays were written by mathematicians, and consider questions that philosophers have not yet discussed. The other half, written by philosophers of mathematics, summarise the discussion in that community during the last 35 years. A connection is made in each case to issues relevant to the teaching of mathematics.

Socratic Epistemology

Explorations of Knowledge-Seeking by Questioning


Author: Jaakko Hintikka

Publisher: Cambridge University Press

ISBN: 1139465791

Category: Philosophy

Page: N.A

View: 8561

Most current work in epistemology deals with the evaluation and justification of information already acquired. In this book, Jaakko Hintikka instead discusses the more important problem of how knowledge is acquired in the first place. His model of information-seeking is the old Socratic method of questioning, which has been generalized and brought up-to-date through the logical theory of questions and answers that he has developed. Hintikka also argues that philosophers' quest for a definition of knowledge is ill-conceived and that the entire notion of knowledge should be replaced by the concept of information. He offers an analysis of the different meanings of the concept of information and of their interrelations. The result is a new and illuminating approach to the field of epistemology.

Naming, Necessity and More

Explorations in the Philosophical Work of Saul Kripke


Author: Jonathan Berg

Publisher: Springer

ISBN: 1137400935

Category: Philosophy

Page: 245

View: 9550

Saul Kripke's Naming and Necessity was one of the most influential philosophical works of the twentieth century. In this collection of essays leading specialists explore issues arising from this and other works of Kripke's.

Hilbert’s Program

An Essay on Mathematical Instrumentalism


Author: M. Detlefsen

Publisher: Springer Science & Business Media

ISBN: 9401577315

Category: Philosophy

Page: 186

View: 563

Hilbert's Program was founded on a concern for the phenomenon of paradox in mathematics. To Hilbert, the paradoxes, which are at once both absurd and irresistible, revealed a deep philosophical truth: namely, that there is a discrepancy between the laws accord ing to which the mind of homo mathematicus works, and the laws governing objective mathematical fact. Mathematical epistemology is, therefore, to be seen as a struggle between a mind that naturally works in one way and a reality that works in another. Knowledge occurs when the two cooperate. Conceived in this way, there are two basic alternatives for mathematical epistemology: a skeptical position which maintains either that mind and reality seldom or never come to agreement, or that we have no very reliable way of telling when they do; and a non-skeptical position which holds that there is significant agree ment between mind and reality, and that their potential discrepan cies can be detected, avoided, and thus kept in check. Of these two, Hilbert clearly embraced the latter, and proposed a program designed to vindicate the epistemological riches represented by our natural, if non-literal, ways of thinking. Brouwer, on the other hand, opted for a position closer (in Hilbert's opinion) to that of the skeptic. Having decided that epistemological purity could come only through sacrifice, he turned his back on his classical heritage to accept a higher calling.

Routledge Companion to Philosophy of Religion


Author: Chad Meister,Paul Copan

Publisher: Routledge

ISBN: 1134180004

Category: Philosophy

Page: 736

View: 5455

The Routledge Companion to Philosophy of Religion is an indispensable guide and reference source to the major themes, movements, debates and topics in philosophy of religion. A team of renowned international contributors provide sixty-five accessible entries organized into nine clear parts: philosophical issues in world religions key figures in philosophy of religion religious diversity the theistic conception of God arguments for the existence of God arguments against the existence of God philosophical theology christian theism recent topics in philosophy of religion. Covering key world religions including Hinduism, Buddhism, and Islam, and key figures such as Augustine, Aquinas and Kierkegaard, the book explores the central topics in theism such as the ontological, cosmological and teleological arguments for God's existence. Three final parts consider Catholicism, Protestantism, Eastern orthodoxy and current debates including phenomenology, reformed epistemology, religious experience, and religion and science. This is essential reading for anyone interested in philosophy, religion and related disciplines.

Order and Organism

Steps Toward a Whiteheadian Philosophy of Mathematics and the Natural Sciences


Author: Murray Code

Publisher: SUNY Press

ISBN: 9780873959513

Category: Mathematics

Page: 265

View: 1820

What is now needed is a way of thinking about the physical that is realistic in outlook but which departs radically from the mechanistic post-Galilean tradition. Since it seems clear that we can no longer take for granted the certainty and absolute objectivity of scientific knowledge, any alternative view must be able to do full justice to subjective modes of knowing. Order and Organism shows how Alfred North Whitehead's thought can reconcile some of the most insistent demands of common sense with the esoteric results of modern physics and mathematics. Whitehead shows a way to resolve the perennial puzzle of why mathematics works. Under his view, it is possible to account for the necessity and uniqueness of mathematical theories without denying the fact that such theories often arise from the mathematician's essentially aesthetic interest in various kinds of pattern.

Philosophy of Mathematics


Author: Thomas Bedürftig,Roman Murawski

Publisher: Walter de Gruyter GmbH & Co KG

ISBN: 3110468336

Category: Mathematics

Page: 474

View: 8736

The present book is an introduction to the philosophy of mathematics. It asks philosophical questions concerning fundamental concepts, constructions and methods - this is done from the standpoint of mathematical research and teaching. It looks for answers both in mathematics and in the philosophy of mathematics from their beginnings till today. The reference point of the considerations is the introducing of the reals in the 19th century that marked an epochal turn in the foundations of mathematics. In the book problems connected with the concept of a number, with the infinity, the continuum and the infinitely small, with the applicability of mathematics as well as with sets, logic, provability and truth and with the axiomatic approach to mathematics are considered. In Chapter 6 the meaning of infinitesimals to mathematics and to the elements of analysis is presented. The authors of the present book are mathematicians. Their aim is to introduce mathematicians and teachers of mathematics as well as students into the philosophy of mathematics. The book is suitable also for professional philosophers as well as for students of philosophy, just because it approaches philosophy from the side of mathematics. The knowledge of mathematics needed to understand the text is elementary. Reports on historical conceptions. Thinking about today‘s mathematical doing and thinking. Recent developments. Based on the third, revised German edition. For mathematicians - students, teachers, researchers and lecturers - and readersinterested in mathematics and philosophy. Contents On the way to the reals On the history of the philosophy of mathematics On fundamental questions of the philosophy of mathematics Sets and set theories Axiomatic approach and logic Thinking and calculating infinitesimally – First nonstandard steps Retrospection

Concepts and Approaches in Evolutionary Epistemology

Towards an Evolutionary Theory of Knowledge


Author: Franz M. Wuketits

Publisher: Springer Science & Business Media

ISBN: 9400971273

Category: Philosophy

Page: 322

View: 805

The present volume brings together current interdisciplinary research which adds up to an evolutionary theory of human knowledge, Le. evolutionary epistemology. It comprises ten papers, dealing with the basic concepts, approaches and data in evolutionary epistemology and discussing some of their most important consequences. Because I am convinced that criticism, if not confused with mere polemics, is apt to stimulate the maturation of a scientific or philosophical theory, I invited Reinhard Low to present his critical view of evolutionary epistemology and to indicate some limits of our evolutionary conceptions. The main purpose of this book is to meet the urgent need of both science and philosophy for a comprehensive up-to-date approach to the problem of knowledge, going beyond the traditional disciplinary boundaries of scientific and philosophical thought. Evolutionary epistemology has emerged as a naturalistic and science-oriented view of knowledge taking cognizance of, and compatible with, results of biological, psychological, anthropological and linguistic inquiries concerning the structure and development of man's cognitive apparatus. Thus, evolutionary epistemology serves as a frame work for many contemporary discussions of the age-old problem of human knowledge.

Strict Finitism and the Logic of Mathematical Applications


Author: Feng Ye

Publisher: Springer Science & Business Media

ISBN: 9789400713475

Category: Science

Page: 272

View: 5832

This book intends to show that radical naturalism (or physicalism), nominalism and strict finitism account for the applications of classical mathematics in current scientific theories. The applied mathematical theories developed in the book include the basics of calculus, metric space theory, complex analysis, Lebesgue integration, Hilbert spaces, and semi-Riemann geometry (sufficient for the applications in classical quantum mechanics and general relativity). The fact that so much applied mathematics can be developed within such a weak, strictly finitistic system, is surprising in itself. It also shows that the applications of those classical theories to the finite physical world can be translated into the applications of strict finitism, which demonstrates the applicability of those classical theories without assuming the literal truth of those theories or the reality of infinity. Both professional researchers and students of philosophy of mathematics will benefit greatly from reading this book.


The Concept and its Manifestations


Author: Nathan Rotenstreich

Publisher: Springer Science & Business Media

ISBN: 9401029059

Category: Philosophy

Page: 258

View: 1005

The present book is concerned with the nature of philosophy and with the scope of philosophical interest. It combines an analysis of the major types of philosophical thinking as they emerged in the history of philosophical ideas with an attempt to examine problems which recurrent ly emerge in philosophical discourse. It is from this point of view that the historical and the systematic approaches are meant to be mutually reinforcing. I am grateful to my friends who helped me to formulate the line of thinking expressed in this book: Z. Bar-On, A. Margalit, E. I. I. Poznanski, Z. Werblovsky and E. Zemach. Some years ago when I visited the Center for the Study of Democratic Institutions in Santa Barbara, Dr. Robert M. Hutchins encouraged me to write the present book. I am dedicating the book to him not only because of that encouragement but more importantly because as an educational thinker Dr. Hutchins represents the position which assigns to the great ideas of the past validity and value in the analysis of topical problems of the present.

Appraising Lakatos

Mathematics, Methodology, and the Man


Author: György Kampis,L. Kvasz,Michael Stöltzner

Publisher: Springer Science & Business Media

ISBN: 9401707693

Category: Science

Page: 382

View: 7436

Imre Lakatos (1922-1974) was one of the protagonists in shaping the "new philosophy of science". More than 25 years after his untimely death, it is time for a critical re-evaluation of his ideas. His main theme of locating rationality within the scientific process appears even more compelling today, after many historical case studies have revealed the cultural and societal elements within scientific practices. Recently there has been, above all, an increasing interest in Lakatos' philosophy of mathematics, which emphasises heuristics and mathematical practice over logical justification. But suitable modifications of his approach are called for in order to make it applicable to modern axiomatised theories. Pioneering historical research in England and Hungary has unearthed hitherto unknown facts about Lakatos' personal life, his wartime activities and his involvement in the political developments of post-war Europe. From a communist activist committed to Györgyi Lukács' thinking, Lakatos developed into a staunch anti-Marxist who found his intellectual background in Popper's critical rationalism. The volume also publishes for the first time a part of his Debrecen Ph.D. thesis and it is concluded by a bibliography of his Hungarian writings.

The Revolutionary Kant

A Commentary on the Critique of Pure Reason


Author: Graham Bird

Publisher: Open Court

ISBN: 0812698789

Category: Philosophy

Page: 736

View: 7038

The Revolutionary Kant offers a new appreciation of Kant’s classic, arguing that Kant's reform of philosophy was far more radical than has been previously understood. The book examines his proposed revolutionary reform — to abandon traditional metaphysics and point philosophy in a new direction — and contends that critics have misrepresented conflicts between Kant and his predecessors. Kant, Bird argues, was not a flawed innovator but an advocate of a new philosophical project, one that began to be appreciated only in the twentieth century.

The Power of Knowledge

How Information and Technology Made the Modern World


Author: Jeremy Black

Publisher: Yale University Press

ISBN: 030019854X

Category: History

Page: 504

View: 2682

Information is power. For more than five hundred years the success or failure of nations has been determined by a country’s ability to acquire knowledge and technical skill and transform them into strength and prosperity. Leading historian Jeremy Black approaches global history from a distinctive perspective, focusing on the relationship between information and society and demonstrating how the understanding and use of information have been the primary factors in the development and character of the modern age. Black suggests that the West’s ascension was a direct result of its institutions and social practices for acquiring, employing, and retaining information and the technology that was ultimately produced. His cogent and well-reasoned analysis looks at cartography and the hardware of communication, armaments and sea power, mercantilism and imperialism, science and astronomy, as well as bureaucracy and the management of information, linking the history of technology with the history of global power while providing important indicators for the future of our world.

Badiou and Philosophy


Author: Sean Bowden

Publisher: Edinburgh University Press

ISBN: 0748668330

Category: Philosophy

Page: 288

View: 8745

This collection of thirteen essays engages directly with the work of Alain Badiou, focusing specifically on the philosophical content of his work and the various connections he established with both his contemporaries and his philosophical heritage.

Modern Logic 1850-1950, East and West


Author: Francine F. Abeles,Mark E. Fuller

Publisher: Birkhäuser

ISBN: 3319247565

Category: Mathematics

Page: 258

View: 8010

This book presents diverse topics in mathematical logic such as proof theory, meta-mathematics, and applications of logic to mathematical structures. The collection spans the first 100 years of modern logic and is dedicated to the memory of Irving Anellis, founder of the journal 'Modern Logic', whose academic work was essential in promoting the algebraic tradition of logic, as represented by Charles Sanders Peirce. Anellis’s association with the Russian logic community introduced their school of logic to a wider audience in the USA, Canada and Western Europe. In addition, the collection takes a historical perspective on proof theory and the development of logic and mathematics in Eastern Logic, the Soviet Union and Russia. The book will be of interest to historians and philosophers in logic and mathematics, and the more specialized papers will also appeal to mathematicians and logicians.


Mathematics, Physics and Philosophy, 1860-1930


Author: Vincent F. Hendricks,Klaus F. Jørgensen,Jesper Lützen,Stig A. Pedersen

Publisher: Springer Science & Business Media

ISBN: 1402051956

Category: Philosophy

Page: 338

View: 6079

The main theme of this anthology is the unique interaction between mathematics, physics and philosophy during the beginning of the 20th century. In this book, ten renowned philosopher-historians probe insightfully into key conceptual questions of pre-quantum mathematical physics. The result is a diverse yet thematically focused compilation of first class papers on mathematics, physics and philosophy, and a source-book on the interaction between them.

The Indispensability of Mathematics


Author: Mark Colyvan

Publisher: Oxford University Press

ISBN: 9780198031444

Category: Philosophy

Page: 192

View: 6146

The Quine-Putnam indispensability argument in the philosophy of mathematics urges us to place mathematical entities on the same ontological footing as other theoretical entities essential to our best scientific theories. Recently, the argument has come under serious scrutiny, with many influential philosophers unconvinced of its cogency. This book not only outlines the indispensability argument in considerable detail but also defends it against various challenges.