defending the axioms on the philosophical foundations of set theory

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Defending The Axioms

Author : Penelope Maddy
ISBN : 9780199596188
Genre : Mathematics
File Size : 53. 55 MB
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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.

Realism In Mathematics

Author : Penelope Maddy
ISBN : 9780198240358
Genre : Political Science
File Size : 90. 32 MB
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When engaged in mathematics, most people tend to think of themselves as scientists investigating the features of real mathematical things, and the wildly successful application of mathematics in the physical sciences reinforces this picture of mathematics as an objective study. For philosophers, however, this realism about mathematics raises serious questions: What are mathematical things? Where are they? How do we know about them? Penelope Maddy delineates and defends a novel versionof mathematical realism that answers the traditional questions and refocuses philosophical attention on the pressing foundational issues of contemporary mathematics.

Philosophical Introduction To Set Theory

Author : Stephen Pollard
ISBN : 9780486797144
Genre : Mathematics
File Size : 41. 21 MB
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This unique approach maintains that set theory is the primary mechanism for ideological and theoretical unification in modern mathematics, and its technically informed discussion covers a variety of philosophical issues. 1990 edition.

Second Philosophy

Author : Penelope Maddy
ISBN : 9780199273669
Genre : Mathematics
File Size : 88. 78 MB
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Many philosophers these days consider themselves naturalists, but it's doubtful any two of them intend the same position by the term. In this book, Penelope Maddy describes and practises a particularly austere form of naturalism called 'Second Philosophy'. Without a definitive criterion for what counts as 'science' and what doesn't, Second Philosophy can't be specified directly - 'trust only the methods of science!' or some such thing - so Maddy proceeds instead by illustratingthe behaviours of an idealized inquirer she calls the 'Second Philosopher'. This Second Philosopher begins from perceptual common sense and progresses from there to systematic observation, active experimentation, theory formation and testing, working all the while to assess, correct and improve hermethods as she goes. Second Philosophy is then the result of the Second Philosopher's investigations.Maddy delineates the Second Philosopher's approach by tracing her reactions to various familiar skeptical and transcendental views (Descartes, Kant, Carnap, late Putnam, van Fraassen), comparing her methods to those of other self-described naturalists (especially Quine), and examining a prominent contemporary debate (between disquotationalists and correspondence theorists in the theory of truth) to extract a properly second-philosophical line of thought. She then undertakes to practise SecondPhilosophy in her reflections on the ground of logical truth, the methodology, ontology and epistemology of mathematics, and the general prospects for metaphysics naturalized.

Science Without Numbers

Author : Hartry Field
ISBN : 9780191083778
Genre : Philosophy
File Size : 33. 93 MB
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Science Without Numbers caused a stir in philosophy on its original publication in 1980, with its bold nominalist approach to the ontology of mathematics and science. Hartry Field argues that we can explain the utility of mathematics without assuming it true. Part of the argument is that good mathematics has a special feature ("conservativeness") that allows it to be applied to "nominalistic" claims (roughly, those neutral to the existence of mathematical entities) in a way that generates nominalistic consequences more easily without generating any new ones. Field goes on to argue that we can axiomatize physical theories using nominalistic claims only, and that in fact this has advantages over the usual axiomatizations that are independent of nominalism. There has been much debate about the book since it first appeared. It is now reissued in a revised contains a substantial new preface giving the author's current views on the original book and the issues that were raised in the subsequent discussion of it.

Understanding The Infinite

Author : Shaughan LAVINE
ISBN : 9780674039995
Genre : Philosophy
File Size : 66. 41 MB
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Principia Mathematica

Author : Alfred North Whitehead
ISBN : UOM:39015002922899
Genre : Logic, Symbolic and mathematical
File Size : 71. 74 MB
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Naturalism In Mathematics

Author : Penelope Maddy
ISBN : 0198250754
Genre : Philosophy
File Size : 70. 45 MB
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Naturalism in Mathematics investigates how the most fundamental assumptions of mathematics can be justified. One prevalent philosophical approach to the problem--realism--is examined and rejected in favour of another approach--naturalism--which attends more closely to practical considerations drawn from within mathematics itself. Penelope Maddy defines naturalism, explains the motivation for it, and shows how it can be successfully applied in set theory. Her clear, original discussion is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from both disciplines.

Foundations Without Foundationalism

Author : Stewart Shapiro
ISBN : 0191524018
Genre : Mathematics
File Size : 39. 24 MB
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The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed development of higher-order logic, including a comprehensive discussion of its semantics. Professor Shapiro demonstrates the prevalence of second-order notions in mathematics is practised, and also the extent to which mathematical concepts can be formulated in second-order languages . He shows how first-order languages are insufficient to codify many concepts in contemporary mathematics, and thus that higher-order logic is needed to fully reflect current mathematics. Throughout, the emphasis is on discussing the philosophical and historical issues associated with this subject, and the implications that they have for foundational studies. For the most part, the author assumes little more than a familiarity with logic as might be gained from a beginning graduate course which includes the incompleteness of arithmetic and the Lowenheim-Skolem theorems. All those concerned with the foundations of mathematics will find this a thought-provoking discussion of some of the central issues in this subject.

An Introduction To The Philosophy Of Science

Author : Rudolf Carnap
ISBN : 9780486140865
Genre : Science
File Size : 60. 60 MB
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Stimulating, thought-provoking text by one of the 20th century's most creative philosophers makes accessible such topics as probability, measurement and quantitative language, causality and determinism, theoretical laws and concepts, more.

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