Mathematics and Metaphilosophy

Mathematics and Metaphilosophy
Author: Justin Clarke-Doane
Publisher: Cambridge University Press
Total Pages: 105
Release: 2022-06-30
Genre: Science
ISBN: 1009002295
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This Element discusses the problem of mathematical knowledge, and its broader philosophical ramifications. It argues that the challenge to explain the (defeasible) justification of our mathematical beliefs ('the justificatory challenge'), arises insofar as disagreement over axioms bottoms out in disagreement over intuitions. And it argues that the challenge to explain their reliability ('the reliability challenge'), arises to the extent that we could have easily had different beliefs. The Element shows that mathematical facts are not, in general, empirically accessible, contra Quine, and that they cannot be dispensed with, contra Field. However, it argues that they might be so plentiful that our knowledge of them is unmysterious. The Element concludes with a complementary 'pluralism' about modality, logic and normative theory, highlighting its surprising implications. Metaphysically, pluralism engenders a kind of perspectivalism and indeterminacy. Methodologically, it vindicates Carnap's pragmatism, transposed to the key of realism.

Morality and Mathematics

Morality and Mathematics
Author: Justin Clarke-Doane
Publisher: Oxford University Press
Total Pages: 208
Release: 2020-03-12
Genre: Philosophy
ISBN: 0192556800
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To what extent are the subjects of our thoughts and talk real? This is the question of realism. In this book, Justin Clarke-Doane explores arguments for and against moral realism and mathematical realism, how they interact, and what they can tell us about areas of philosophical interest more generally. He argues that, contrary to widespread belief, our mathematical beliefs have no better claim to being self-evident or provable than our moral beliefs. Nor do our mathematical beliefs have better claim to being empirically justified than our moral beliefs. It is also incorrect that reflection on the genealogy of our moral beliefs establishes a lack of parity between the cases. In general, if one is a moral antirealist on the basis of epistemological considerations, then one ought to be a mathematical antirealist as well. And, yet, Clarke-Doane shows that moral realism and mathematical realism do not stand or fall together — and for a surprising reason. Moral questions, insofar as they are practical, are objective in a sense that mathematical questions are not, and the sense in which they are objective can only be explained by assuming practical anti-realism. One upshot of the discussion is that the concepts of realism and objectivity, which are widely identified, are actually in tension. Another is that the objective questions in the neighborhood of factual areas like logic, modality, grounding, and nature are practical questions too. Practical philosophy should, therefore, take center stage.

Metaphysics, Mathematics, and Meaning

Metaphysics, Mathematics, and Meaning
Author: Nathan Salmon
Publisher: Oxford University Press
Total Pages:
Release: 2005-11-24
Genre: Philosophy
ISBN: 019153594X
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Metaphysics, Mathematics, and Meaning brings together Nathan Salmon's influential papers on topics in the metaphysics of existence, non-existence, and fiction; modality and its logic; strict identity, including personal identity; numbers and numerical quantifiers; the philosophical significance of Gödel's Incompleteness theorems; and semantic content and designation. Including a previously unpublished essay and a helpful new introduction to orient the reader, the volume offers rich and varied sustenance for philosophers and logicians.

Lectures on the Philosophy of Mathematics

Lectures on the Philosophy of Mathematics
Author: Joel David Hamkins
Publisher: MIT Press
Total Pages: 350
Release: 2021-03-09
Genre: Mathematics
ISBN: 0262542234
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An introduction to the philosophy of mathematics grounded in mathematics and motivated by mathematical inquiry and practice. In this book, Joel David Hamkins offers an introduction to the philosophy of mathematics that is grounded in mathematics and motivated by mathematical inquiry and practice. He treats philosophical issues as they arise organically in mathematics, discussing such topics as platonism, realism, logicism, structuralism, formalism, infinity, and intuitionism in mathematical contexts. He organizes the book by mathematical themes--numbers, rigor, geometry, proof, computability, incompleteness, and set theory--that give rise again and again to philosophical considerations.

Metaphilosophy

Metaphilosophy
Author: Nicholas Rescher
Publisher: Lexington Books
Total Pages: 256
Release: 2014-10-23
Genre: Philosophy
ISBN: 0739199781
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The definitive mission of metaphilosophy is to facilitate an understanding of how philosophy works—the aim of the enterprise, the instrumental and procedural resources for its work, and the prospect of its success. Nicholas Rescher unites two facets of metaphilosophy to show that historical perspective and forward-thinking normative, or systematic, metaphilosophy cannot be independent of one another. The descriptive, or historical, metaphilosophy provides an account of what has been thought regarding the conduct of philosophical inquiry, and the prescriptive, or normative, metaphilosophy which deliberates about what is to be thought regarding the conduct of philosophizing. Rescher argues that metaphilosophy forms a part of philosophy itself. This is a unique feature of the discipline since the philosophy of biology is not a part of biology and the philosophy of mathematics is not a part of mathematics. Ultimately, the salient features of philosophizing in general—including the inherently controversial and discordant nature of philosophical doctrines—are also bound to afflict metaphilosophy. Thus, only by a careful analysis of the central issues can a plausible view of the enterprise be developed. Metaphilosophy: Philosophy in Philosophical Perspective challenges the static, compartmentalized view of metaphilosophy, providing insight for scholars and students of all areas of philosophy.

Philosophy of Mathematics

Philosophy of Mathematics
Author:
Publisher: Elsevier
Total Pages: 735
Release: 2009-07-08
Genre: Philosophy
ISBN: 0080930581
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One of the most striking features of mathematics is the fact that we are much more certain about the mathematical knowledge we have than about what mathematical knowledge is knowledge of. Are numbers, sets, functions and groups physical entities of some kind? Are they objectively existing objects in some non-physical, mathematical realm? Are they ideas that are present only in the mind? Or do mathematical truths not involve referents of any kind? It is these kinds of questions that have encouraged philosophers and mathematicians alike to focus their attention on issues in the philosophy of mathematics. Over the centuries a number of reasonably well-defined positions about the nature of mathematics have been developed and it is these positions (both historical and current) that are surveyed in the current volume. Traditional theories (Platonism, Aristotelianism, Kantianism), as well as dominant modern theories (logicism, formalism, constructivism, fictionalism, etc.), are all analyzed and evaluated. Leading-edge research in related fields (set theory, computability theory, probability theory, paraconsistency) is also discussed. The result is a handbook that not only provides a comprehensive overview of recent developments but that also serves as an indispensable resource for anyone wanting to learn about current developments in the philosophy of mathematics. -Comprehensive coverage of all main theories in the philosophy of mathematics-Clearly written expositions of fundamental ideas and concepts-Definitive discussions by leading researchers in the field-Summaries of leading-edge research in related fields (set theory, computability theory, probability theory, paraconsistency) are also included

Philosophy of Mathematics

Philosophy of Mathematics
Author: Thomas Bedürftig
Publisher: Walter de Gruyter GmbH & Co KG
Total Pages: 474
Release: 2018-10-26
Genre: Mathematics
ISBN: 3110470772
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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

Ptolemy's Philosophy

Ptolemy's Philosophy
Author: Jacqueline Feke
Publisher: Princeton University Press
Total Pages: 250
Release: 2020-10-13
Genre: Mathematics
ISBN: 069121039X
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A stimulating intellectual history of Ptolemy's philosophy and his conception of a world in which mathematics reigns supreme The Greco-Roman mathematician Claudius Ptolemy is one of the most significant figures in the history of science. He is remembered today for his astronomy, but his philosophy is almost entirely lost to history. This groundbreaking book is the first to reconstruct Ptolemy’s general philosophical system—including his metaphysics, epistemology, and ethics—and to explore its relationship to astronomy, harmonics, element theory, astrology, cosmology, psychology, and theology. In this stimulating intellectual history, Jacqueline Feke uncovers references to a complex and sophisticated philosophical agenda scattered among Ptolemy’s technical studies in the physical and mathematical sciences. She shows how he developed a philosophy that was radical and even subversive, appropriating ideas and turning them against the very philosophers from whom he drew influence. Feke reveals how Ptolemy’s unique system is at once a critique of prevailing philosophical trends and a conception of the world in which mathematics reigns supreme. A compelling work of scholarship, Ptolemy’s Philosophy demonstrates how Ptolemy situated mathematics at the very foundation of all philosophy—theoretical and practical—and advanced the mathematical way of life as the true path to human perfection.

Philosophies of Mathematics

Philosophies of Mathematics
Author: Alexander L. George
Publisher: Wiley-Blackwell
Total Pages: 240
Release: 2001-12-03
Genre: Science
ISBN: 9780631195436
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This book provides an accessible, critical introduction to the three main approaches that dominated work in the philosophy of mathematics during the twentieth century: logicism, intuitionism and formalism.

Wittgenstein's Metaphilosophy

Wittgenstein's Metaphilosophy
Author: Paul Horwich
Publisher: Oxford University Press (UK)
Total Pages: 243
Release: 2012-12-13
Genre: Philosophy
ISBN: 019966112X
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Paul Horwich presents a bold new interpretation of Wittgenstein's later work. He argues that it is Wittgenstein's radically anti-theoretical metaphilosophy - and not his identification of the meaning of a word with its use - that underpins his discussions of specific issues concerning language, the mind, mathematics, knowledge, art, and religion.