Posts tagged philosophy of mathematics

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 Added by: Jamie Collin, Contributed by:
Publisher’s Note: Charles Chihara’s new book develops and defends a structural view of the nature of mathematics, and uses it to explain a number of striking features of mathematics that have puzzled philosophers for centuries. The view is used to show that, in order to understand how mathematical systems are applied in science and everyday life, it is not necessary to assume that its theorems either presuppose mathematical objects or are even true. Chihara builds upon his previous work, in which he presented a new system of mathematics, the constructibility theory, which did not make reference to, or presuppose, mathematical objects. Now he develops the project further by analysing mathematical systems currently used by scientists to show how such systems are compatible with this nominalistic outlook. He advances several new ways of undermining the heavily discussed indispensability argument for the existence of mathematical objects made famous by Willard Quine and Hilary Putnam. And Chihara presents a rationale for the nominalistic outlook that is quite different from those generally put forward, which he maintains have led to serious misunderstandings. A Structural Account of Mathematics will be required reading for anyone working in this field. generally put forward, which he maintains have led to serious misunderstandings.
Comment: This book, or chapters from it, would provide useful further reading on nominalism in courses on metaphysics or the philosophy of mathematics. The book does a very good job of summarising and critiquing other positions in the debate. As such individual chapters on (e.g.) mathematical structuralism, Platonism and Field and Balaguer's respective developments of fictionalism could be helpful. The chapter on his own contructibility theory is also a good introduction to that position: shorter and less technical than his earlier (1991) book Constructibility and Mathematical Existence, but longer and more developed than his chapter on Nominalism in the Oxford Handbook of the Philosophy of Mathematics and Logic.[This is a stub entry. Please add your comments to help us expand it]

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 Added by: Jamie Collin, Contributed by:
Summary: Introduction to mathematical nominalism, with special attention to Chihara’s own development of the position and the objections of John Burgess and Gideon Rosen. Chihara provides an outline of his constructibility theory, which avoids quantification over abstract objects by making use of contructibility quantifiers which instead of making assertions about what exists, make assertions about what sentences can be constructed.
Comment: This chapter would be a good primary or secondary reading in a course on philosophy of mathematics or metaphysics. Chihara is very good at conveying difficult ideas in clear and concise prose. It is worth noting however that, despite the title, this is not really an introduction to nominalism generally but to Chihara's own (important) development of a nominalist philosophy of mathematics / metaphysics.[This is a stub entry. Please add your comments to help us expand it]

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 Added by: Berta Grimau, Contributed by: Matt Clemens
Publisher’s Note: What is mathematics about? Does the subjectmatter of mathematics exist independently of the mind or are they mental constructions? How do we know mathematics? Is mathematical knowledge logical knowledge? And how is mathematics applied to the material world? In this introduction to the philosophy of mathematics, Michele Friend examines these and other ontological and epistemological problems raised by the content and practice of mathematics. Aimed at a readership with limited proficiency in mathematics but with some experience of formal logic it seeks to strike a balance between conceptual accessibility and correct representation of the issues. Friend examines the standard theories of mathematics – Platonism, realism, logicism, formalism, constructivism and structuralism – as well as some less standard theories such as psychologism, fictionalism and Meinongian philosophy of mathematics. In each case Friend explains what characterises the position and where the divisions between them lie, including some of the arguments in favour and against each. This book also explores particular questions that occupy presentday philosophers and mathematicians such as the problem of infinity, mathematical intuition and the relationship, if any, between the philosophy of mathematics and the practice of mathematics. Taking in the canonical ideas of Aristotle, Kant, Frege and Whitehead and Russell as well as the challenging and innovative work of recent philosophers like Benacerraf, Hellman, Maddy and Shapiro, Friend provides a balanced and accessible introduction suitable for upperlevel undergraduate courses and the nonspecialist.
Comment: This book provides an introduction to the philosophy of mathematics. No previous mathematical skills/knowledge required. Suitable for undergraduate courses on philosophy of mathematics.[This is a stub entry. Please add your comments to help us expand it]

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 Added by: Sara Peppe, Contributed by:
Publisher’s Note: Not limited to merely mathematics, probability has a rich and controversial philosophical aspect. ‘A Philosophical Introduction to Probability’ showcases lesserknown philosophical notions of probability and explores the debate over their interpretations. Galavotti traces the history of probability and its mathematical properties and then discusses various philosophical positions on probability, from the Pierre Simon de Laplace’s ‘classical’ interpretation of probability to the logical interpretation proposed by John Maynard Keynes. This book is a valuable resource for students in philosophy and mathematics and all readers interested in notions of probability
Comment: Very good article for philosophy of science and philosophy of probability courses. It works perfectly to build basic knowledge on the theme of probability.[This is a stub entry. Please add your comments to help us expand it]

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 Added by: Jamie Collin, Contributed by:
Abstract: The goal of the research programme I describe in this article is a realist epistemology for arithmetic which respects arithmetic’s special epistemic status (the status usually described as a prioricity) yet accommodates naturalistic concerns by remaining funda mentally empiricist. I argue that the central claims which would allow us to develop such an epistemology are (i) that arithmetical truths are known through an examination of our arithmetical concepts; (ii) that (at least our basic) arithmetical concepts are accurate mental representations of elements of the arithmetical structure of the inde pendent world; (iii) that (ii) obtains in virtue of the normal functioning of our sensory apparatus. The first of these claims protects arithmetic’s special epistemic status relative, for example, to the laws of physics, the second preserves the independence of arithmetical truth, and the third ensures that we remain empiricists.
Comment: Useful as a primary or secondary reading in an advanced undergraduate course epistemology (in a section on a priori knowledge) or an advanced undergraduate course on philosophy of mathematics. This is not an easy paper, but it is clear. It is also useful within a teaching context, as it provides a summary of the influential neoFregean approach to mathematical knowledge.[This is a stub entry. Please add your comments to help us expand it]

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 Added by: Jamie Collin, Contributed by:
Publisher’s Note: Mary Leng offers a defense of mathematical fictionalism, according to which we have no reason to believe that there are any mathematical objects. Perhaps the most pressing challenge to mathematical fictionalism is the indispensability argument for the truth of our mathematical theories (and therefore for the existence of the mathematical objects posited by those theories). According to this argument, if we have reason to believe anything, we have reason to believe that the claims of our best empirical theories are (at least approximately) true. But since claims whose truth would require the existence of mathematical objects are indispensable in formulating our best empirical theories, it follows that we have good reason to believe in the mathematical objects posited by those mathematical theories used in empirical science, and therefore to believe that the mathematical theories utilized in empirical science are true. Previous responses to the indispensability argument have focussed on arguing that mathematical assumptions can be dispensed with in formulating our empirical theories. Leng, by contrast, offers an account of the role of mathematics in empirical science according to which the successful use of mathematics in formulating our empirical theories need not rely on the truth of the mathematics utilized.
Comment: This book presents the most developed account of mathematical fictionalism. The book, or chapters from it, would provide useful further reading in advanced undergraduate or postgraduate courses on metaphysics or philosophy of mathematics.[This is a stub entry. Please add your comments to help us expand it]

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 Added by: Jamie Collin, Contributed by:
Summary: Surveys the opposition between views of mathematics which take mathematics to represent a independent mathematical reality and views which take mathematical axioms to define or circumscribe their subject matter; and defends the latter view against influential objections.
Comment: A very clear and useful survey text for advanced undergraduate or postgraduate courses on metaphysics or philosophy of mathematics.[This is a stub entry. Please add your comments to help us expand it]

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 Added by: Jamie Collin, Contributed by:
Summary: Defends an account of mathematical knowledge in which mathematical knowledge is a kind of modal knowledge. Leng argues that nominalists should take mathematical knowledge to consist in knowledge of the consistency of mathematical axiomatic systems, and knowledge of what necessarily follows from those axioms. She defends this view against objections that modal knowledge requires knowledge of abstract objects, and argues that we should understand possibility and necessity in a primative way.
Comment: This would be useful in an advanced undergraduate course on metaphysics, epistemology or philosophy of logic and mathematics. This is not an easy paper, but Leng does an excellent job of making clear some difficult ideas. The view defended is an important one in both philosophy of logic and philosophy of mathematics. Any reasonably comprehensive treatment of nominalism should include this paper.[This is a stub entry. Please add your comments to help us expand it]

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 Added by: Sara Peppe, Contributed by:
Abstract: This paper argues that it is scientific realists who should be most concerned about the issue of Platonism and antiPlatonism in mathematics. If one is merely interested in accounting for the practice of pure mathematics, it is unlikely that a story about the ontology of mathematical theories will be essential to such an account. The question of mathematical ontology comes to the fore, however, once one considers our scientific theories. Given that those theories include amongst their laws assertions that imply the existence of mathematical objects, scientific realism, when construed as a claim about the truth or approximate truth of our scientific theories, implies mathematical Platonism. However, a standard argument for scientific realism, the ‘no miracles’ argument, falls short of establishing mathematical Platonism. As a result, this argument cannot establish scientific realism as it is usually defined, but only some weaker position. Scientific ‘realists’ should therefore either redefine their position as a claim about the existence of unobservable physical objects, or alternatively look for an argument for their position that does establish mathematical Platonism.
Comment: Previous knowledge both on Platonism in philosophy of mathematics and scientific realism is needed. Essential paper for advanced courses of philosophy of science.[This is a stub entry. Please add your comments to help us expand it]

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 Added by: Jamie Collin, Contributed by:
Publisher’s Note: Our muchvalued mathematical knowledge rests on two supports: the logic of proof and the axioms from which those proofs begin. Naturalism in Mathematics investigates the status of the latter, the fundamental assumptions of mathematics. These were once held to be selfevident, but progress in work on the foundations of mathematics, especially in set theory, has rendered that comforting notion obsolete. Given that candidates for axiomatic status cannot be proved, what sorts of considerations can be offered for or against them? That is the central question addressed in this book. One answer is that mathematics aims to describe an objective world of mathematical objects, and that axiom candidates should be judged by their truth or falsity in that world. This promising view – realism – is assessed and finally rejected in favour of another – naturalism – which attends less to metaphysical considerations of objective truth and falsity, and more to practical considerations drawn from within mathematics itself. Penelope Maddy defines this naturalism, explains the motivation for it, and shows how it can be helpfully applied in the assessment of candidates for axiomatic status in set theory. Maddy’s clear, original treatment of this fundamental issue is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from both disciplines.
Comment: Good further reading in advanced undergraduate or postgraduate courses on metaphysics, naturalism or philosophy of mathematics. Sections from the book  for instance, the chapters in Part II on indispensability considerations in scientific and mathematical practice  could be profitably read on their own. These sections may also be of interest in philosophy of science courses, as they provide a careful analysis of scientific practice (as it relates to what scientists take themselves to be ontologically committed to).[This is a stub entry. Please add your comments to help us expand it]
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