Proof and Other Dilemmas

Mathematics and Philosophy

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Author: Roger Simons

Publisher: MAA

ISBN: 9780883855676

Category: Mathematics

Page: 346

View: 2757

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

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Author: Jaakko Hintikka

Publisher: Cambridge University Press

ISBN: 1139465791

Category: Philosophy

Page: N.A

View: 9965

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.

Mathematics and Its Applications

A Transcendental-Idealist Perspective

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Author: Jairo José da Silva

Publisher: Springer

ISBN: 3319630733

Category: Philosophy

Page: 275

View: 7416

This monograph offers a fresh perspective on the applicability of mathematics in science. It explores what mathematics must be so that its applications to the empirical world do not constitute a mystery. In the process, readers are presented with a new version of mathematical structuralism. The author details a philosophy of mathematics in which the problem of its applicability, particularly in physics, in all its forms can be explained and justified. Chapters cover: mathematics as a formal science, mathematical ontology: what does it mean to exist, mathematical structures: what are they and how do we know them, how different layers of mathematical structuring relate to each other and to perceptual structures, and how to use mathematics to find out how the world is. The book simultaneously develops along two lines, both inspired and enlightened by Edmund Husserl’s phenomenological philosophy. One line leads to the establishment of a particular version of mathematical structuralism, free of “naturalist” and empiricist bias. The other leads to a logical-epistemological explanation and justification of the applicability of mathematics carried out within a unique structuralist perspective. This second line points to the “unreasonable” effectiveness of mathematics in physics as a means of representation, a tool, and a source of not always logically justified but useful and effective heuristic strategies.

Naming, Necessity and More

Explorations in the Philosophical Work of Saul Kripke

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Author: Jonathan Berg

Publisher: Springer

ISBN: 1137400935

Category: Philosophy

Page: 245

View: 5505

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.

Meaning in Mathematics

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Author: John Polkinghorne

Publisher: OUP Oxford

ISBN: 0191621293

Category: Mathematics

Page: 176

View: 8797

Is mathematics a highly sophisticated intellectual game in which the adepts display their skill by tackling invented problems, or are mathematicians engaged in acts of discovery as they explore an independent realm of mathematical reality? Why does this seemingly abstract discipline provide the key to unlocking the deep secrets of the physical universe? How one answers these questions will significantly influence metaphysical thinking about reality. This book is intended to fill a gap between popular 'wonders of mathematics' books and the technical writings of the philosophers of mathematics. The chapters are written by some of the world's finest mathematicians, mathematical physicists and philosophers of mathematics, each giving their perspective on this fascinating debate. Every chapter is followed by a short response from another member of the author team, reinforcing the main theme and raising further questions. Accessible to anyone interested in what mathematics really means, and useful for mathematicians and philosophers of science at all levels, Meaning in Mathematics offers deep new insights into a subject many people take for granted.

Hilbert’s Program

An Essay on Mathematical Instrumentalism

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Author: M. Detlefsen

Publisher: Springer Science & Business Media

ISBN: 9401577315

Category: Philosophy

Page: 186

View: 1252

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

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Author: Chad Meister,Paul Copan

Publisher: Routledge

ISBN: 1134180004

Category: Philosophy

Page: 736

View: 9367

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

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Author: Murray Code

Publisher: SUNY Press

ISBN: 9780873959513

Category: Mathematics

Page: 265

View: 1145

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.

Concepts and Approaches in Evolutionary Epistemology

Towards an Evolutionary Theory of Knowledge

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Author: Franz M. Wuketits

Publisher: Springer Science & Business Media

ISBN: 9400971273

Category: Philosophy

Page: 322

View: 7419

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

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Author: Feng Ye

Publisher: Springer Science & Business Media

ISBN: 9789400713475

Category: Science

Page: 272

View: 3045

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.

Philosophy of Mathematics

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Author: Thomas Bedürftig,Roman Murawski

Publisher: Walter de Gruyter GmbH & Co KG

ISBN: 3110468336

Category: Mathematics

Page: 474

View: 6181

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

Philosophy

The Concept and its Manifestations

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Author: Nathan Rotenstreich

Publisher: Springer Science & Business Media

ISBN: 9401029059

Category: Philosophy

Page: 258

View: 9872

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

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Author: György Kampis,L. Kvasz,Michael Stöltzner

Publisher: Springer Science & Business Media

ISBN: 9401707693

Category: Science

Page: 382

View: 5514

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 Oxford Handbook of Philosophy of Mathematics and Logic

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Author: Stewart Shapiro

Publisher: OUP USA

ISBN: 0195148770

Category: Mathematics

Page: 833

View: 8864

Covers the state of the art in the philosophy of maths and logic, giving the reader an overview of the major problems, positions, and battle lines. The chapters in this book contain both exposition and criticism as well as substantial development of their own positions. It also includes a bibliography.

The Revolutionary Kant

A Commentary on the Critique of Pure Reason

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Author: Graham Bird

Publisher: Open Court

ISBN: 0812698789

Category: Philosophy

Page: 736

View: 1615

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.

An Introduction to the Philosophy of Mathematics

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Author: Mark Colyvan

Publisher: Cambridge University Press

ISBN: 0521826020

Category: Mathematics

Page: 188

View: 8672

This introduction to the philosophy of mathematics focuses on contemporary debates in an important and central area of philosophy. The reader is taken on a fascinating and entertaining journey through some intriguing mathematical and philosophical territory, including such topics as the realism/anti-realism debate in mathematics, mathematical explanation, the limits of mathematics, the significance of mathematical notation, inconsistent mathematics and the applications of mathematics. Each chapter has a number of discussion questions and recommended further reading from both the contemporary literature and older sources. Very little mathematical background is assumed and all of the mathematics encountered is clearly introduced and explained using a wide variety of examples. The book is suitable for an undergraduate course in philosophy of mathematics and, more widely, for anyone interested in philosophy and mathematics.