Defending the Axioms

On the Philosophical Foundations of Set Theory

Author: Penelope Maddy

Publisher: Oxford University Press

ISBN: 0199596182

Category: Mathematics

Page: 150

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Mathematics depends on proofs, and proofs must begin somewhere, from some fundamental assumptions. The axioms of set theory have long played this role, so the question of how they are properly judged is of central importance. Maddy discusses the appropriate methods for such evaluations and the philosophical backdrop that makes them appropriate.
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God and Abstract Objects

The Coherence of Theism: Aseity

Author: William Lane Craig

Publisher: Springer

ISBN: 3319553844

Category: Philosophy

Page: 540

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This book is an exploration and defense of the coherence of classical theism’s doctrine of divine aseity in the face of the challenge posed by Platonism with respect to abstract objects. A synoptic work in analytic philosophy of religion, the book engages discussions in philosophy of mathematics, philosophy of language, metaphysics, and metaontology. It addresses absolute creationism, non-Platonic realism, fictionalism, neutralism, and alternative logics and semantics, among other topics. The book offers a helpful taxonomy of the wide range of options available to the classical theist for dealing with the challenge of Platonism. It probes in detail the diverse views on the reality of abstract objects and their compatibility with classical theism. It contains a most thorough discussion, rooted in careful exegesis, of the biblical and patristic basis of the doctrine of divine aseity. Finally, it challenges the influential Quinean metaontological theses concerning the way in which we make ontological commitments.
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God Over All

Divine Aseity and the Challenge of Platonism

Author: William Lane Craig

Publisher: Oxford University Press

ISBN: 0191090557

Category: Religion

Page: 280

View: 1131

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God Over All: Divine Aseity and the Challenge of Platonism is a defense of God's aseity and unique status as the Creator of all things apart from Himself in the face of the challenge posed by mathematical Platonism. After providing the biblical, theological, and philosophical basis for the traditional doctrine of divine aseity, William Lane Craig explains the challenge presented to that doctrine by the Indispensability Argument for Platonism, which postulates the existence of uncreated abstract objects. Craig provides detailed examination of a wide range of responses to that argument, both realist and anti-realist, with a view toward assessing the most promising options for the theist. A synoptic work in analytic philosophy of religion, this groundbreaking volume engages discussions in philosophy of mathematics, philosophy of language, metaphysics, and metaontology.
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Set Theory

With an Introduction to Real Point Sets

Author: Abhijit Dasgupta

Publisher: Springer Science & Business Media

ISBN: 1461488540

Category: Mathematics

Page: 444

View: 3295

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What is a number? What is infinity? What is continuity? What is order? Answers to these fundamental questions obtained by late nineteenth-century mathematicians such as Dedekind and Cantor gave birth to set theory. This textbook presents classical set theory in an intuitive but concrete manner. To allow flexibility of topic selection in courses, the book is organized into four relatively independent parts with distinct mathematical flavors. Part I begins with the Dedekind–Peano axioms and ends with the construction of the real numbers. The core Cantor–Dedekind theory of cardinals, orders, and ordinals appears in Part II. Part III focuses on the real continuum. Finally, foundational issues and formal axioms are introduced in Part IV. Each part ends with a postscript chapter discussing topics beyond the scope of the main text, ranging from philosophical remarks to glimpses into landmark results of modern set theory such as the resolution of Lusin's problems on projective sets using determinacy of infinite games and large cardinals. Separating the metamathematical issues into an optional fourth part at the end makes this textbook suitable for students interested in any field of mathematics, not just for those planning to specialize in logic or foundations. There is enough material in the text for a year-long course at the upper-undergraduate level. For shorter one-semester or one-quarter courses, a variety of arrangements of topics are possible. The book will be a useful resource for both experts working in a relevant or adjacent area and beginners wanting to learn set theory via self-study.
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A Brief History of Analytic Philosophy

From Russell to Rawls

Author: Stephen P. Schwartz

Publisher: John Wiley & Sons

ISBN: 1118271726

Category: Philosophy

Page: 368

View: 2860

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A Brief History of Analytic Philosophy: From Russell toRawls presents a comprehensive overview of the historicaldevelopment of all major aspects of analytic philosophy, thedominant Anglo-American philosophical tradition in the twentiethcentury. Features coverage of all the major subject areasand figures in analytic philosophy - includingWittgenstein, Bertrand Russell, G.E. Moore, Gottlob Frege, Carnap,Quine, Davidson, Kripke, Putnam, and many others Contains explanatory background material to help make cleartechnical philosophical concepts Includes listings of suggested further readings Written in a clear, direct style that presupposes littleprevious knowledge of philosophy
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Mathematics and the Image of Reason

Author: Mary Tiles

Publisher: Routledge

ISBN: 1134967713

Category: Philosophy

Page: 200

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A thorough account of the philosophy of mathematics. In a cogent account the author argues against the view that mathematics is solely logic.
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Internal Logic

Foundations of Mathematics from Kronecker to Hilbert

Author: Y. Gauthier

Publisher: Springer Science & Business Media

ISBN: 9401700834

Category: Mathematics

Page: 251

View: 2905

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Internal logic is the logic of content. The content is here arithmetic and the emphasis is on a constructive logic of arithmetic (arithmetical logic). Kronecker's general arithmetic of forms (polynomials) together with Fermat's infinite descent is put to use in an internal consistency proof. The view is developed in the context of a radical arithmetization of mathematics and logic and covers the many-faceted heritage of Kronecker's work, which includes not only Hilbert, but also Frege, Cantor, Dedekind, Husserl and Brouwer. The book will be of primary interest to logicians, philosophers and mathematicians interested in the foundations of mathematics and the philosophical implications of constructivist mathematics. It may also be of interest to historians, since it covers a fifty-year period, from 1880 to 1930, which has been crucial in the foundational debates and their repercussions on the contemporary scene.
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