Wittgenstein Finitism and the Foundations of Mathematics

Author: Mathieu Marion
Publisher: Oxford University Press
ISBN: 9780198235163
Format: PDF, Mobi
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Mathieu Marion offers a careful, historically informed study of Wittgenstein's philosophy of mathematics. This area of his work has frequently been undervalued by Wittgenstein specialists and philosophers of mathematics alike; but the surprising fact that he wrote more on this subject than any other indicates its centrality in his thought. Marion traces the development of Wittgenstein's thinking from the 1920s through to the 1950s, in the context of themathematical and philosophical work of the times, to make coherent sense of ideas that have too often been misunderstood because they have been presented in a disjointed and incomplete way. He shows that study of Wittgenstein's writings on mathematics is essential to a proper understanding of his philosophy,and also that it can do much to illuminate current debates about the foundations of mathematics.

The Oxford Handbook of Philosophy of Mathematics and Logic

Author: Stewart Shapiro
Publisher: Oxford University Press
ISBN: 0198033990
Format: PDF, ePub, Mobi
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Mathematics and logic have been central topics of concern since the dawn of philosophy. Since logic is the study of correct reasoning, it is a fundamental branch of epistemology and a priority in any philosophical system. Philosophers have focused on mathematics as a case study for general philosophical issues and for its role in overall knowledge- gathering. Today, philosophy of mathematics and logic remain central disciplines in contemporary philosophy, as evidenced by the regular appearance of articles on these topics in the best mainstream philosophical journals; in fact, the last decade has seen an explosion of scholarly work in these areas. This volume covers these disciplines in a comprehensive and accessible manner, giving the reader an overview of the major problems, positions, and battle lines. The 26 contributed chapters are by established experts in the field, and their articles contain both exposition and criticism as well as substantial development of their own positions. The essays, which are substantially self-contained, serve both to introduce the reader to the subject and to engage in it at its frontiers. Certain major positions are represented by two chapters--one supportive and one critical. The Oxford Handbook of Philosophy of Math and Logic is a ground-breaking reference like no other in its field. It is a central resource to those wishing to learn about the philosophy of mathematics and the philosophy of logic, or some aspect thereof, and to those who actively engage in the discipline, from advanced undergraduates to professional philosophers, mathematicians, and historians.

Category Mistakes

Author: Ofra Magidor
Publisher: OUP Oxford
ISBN: 0191662844
Format: PDF, Docs
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Category mistakes are sentences such as 'Green ideas sleep furiously', 'Saturday is in bed', and 'The theory of relativity is eating breakfast'. Such sentences strike most speakers as highly infelicitous but it is a challenge to explain precisely why they are so. Ofra Magidor addresses this challenge, while providing a comprehensive discussion of the various treatments of category mistakes in both philosophy of language and linguistics. The phenomenon of category mistakes is particularly interesting to both these fields because a plausible case can be (and has been) made for explaining it in terms of syntax, semantics, and pragmatics—making it a fruitful case for exploring the relations between and nature of these three fundamental realms of language. Category Mistakes follows this division. After an introduction which explains the aims and motivations for the project and provides a brief historical survey of the (modern) treatment of category mistakes in each of philosophy, linguistics, and computer science, Magidor discusses four approaches in turn: first, the syntactic approach, which maintains that category mistakes are syntactically ill-formed; then two semantic approaches, though ones that appeal to different semantic facets: the meaninglessness view, which maintains that category mistakes are meaningless, and the MBT view, according to which category mistakes are meaningful but truth-valueless; and finally the pragmatic approach, according to which category mistakes are syntactically well-formed, meaningful, truth-valued but nevertheless pragmatically inappropriate. Magidor argues that the first three approaches ought to be rejected, and in the final chapter addresses the main challenge by developing and defending a particular version of the pragmatic approach: a presuppositional account of category mistakes.

The Philosophy of Mathematics Today

Author: Matthias Schirn
Publisher: Oxford University Press
ISBN: 9780199262625
Format: PDF, ePub, Mobi
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The Philosophy of Mathematics Today gives a panorama of the best current work in this lively field, through twenty essays specially written for this collection by leading figures. The topics include indeterminacy, logical consequence, mathematical methodology, abstraction, and both Hilbert's and Frege's foundational programmes. The collection will be an important source for research in the philosophy of mathematics for years to come.

The Autonomy of Mathematical Knowledge

Author: Curtis Franks
Publisher: Cambridge University Press
ISBN: 0521514371
Format: PDF, ePub, Mobi
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This study reconstructs, analyses and re-evaluates the programme of influential mathematical thinker David Hilbert, presenting it in a new light.

Infinity Causation and Paradox

Author: Alexander R. Pruss
Publisher: Oxford University Press
ISBN: 0192538284
Format: PDF, Mobi
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Infinity is paradoxical in many ways. Some paradoxes involve deterministic supertasks, such as Thomson's Lamp, where a switch is toggled an infinite number of times over a finite period of time, or the Grim Reaper, where it seems that infinitely many reapers can produce a result without doing anything. Others involve infinite lotteries. If you get two tickets from an infinite fair lottery where tickets are numbered from 1, no matter what number you saw on the first ticket, it is almost certain that the other ticket has a bigger number on it. And others center on paradoxical results in decision theory, such as the surprising observation that if you perform a sequence of fair coin flips that goes infinitely far back into the past but only finitely into the future, you can leverage information about past coin flips to predict future ones with only finitely many mistakes. Alexander R. Pruss examines this seemingly large family of paradoxes in Infinity, Causation and Paradox. He establishes that these paradoxes and numerous others all have a common structure: their most natural embodiment involves an infinite number of items causally impinging on a single output. These paradoxes, he argues, can all be resolved by embracing 'causal finitism', the view that it is impossible for a single output to have an infinite causal history. Throughout the book, Pruss exposits such paradoxes, defends causal finitism at length, and considers connections with the philosophy of physics (where causal finitism favors but does not require discretist theories of space and time) and the philosophy of religion (with a cosmological argument for a first cause).

Social Constructivism as a Philosophy of Mathematics

Author: Paul Ernest
Publisher: SUNY Press
ISBN: 9780791435878
Format: PDF, Kindle
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Extends the ideas of social constructivism to the philosophy of mathematics, developing a powerful critique of traditional absolutist conceptions of mathematics, and proposing a reconceptualization of the philosophy of mathematics.