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Added by: Jamie Collin
Summary: Classic presentation of the prosentential theory of truth: an important, though minority, deflationist account of truth. Prosententialists take 'It is true that' to be a prosentence forming operator that anaphorically picks out content from claims made further back in the anaphoric chain (in the same way that pronouns such as 'he', 'she' and 'it' anaphorically pick out referents from nouns further back in the anaphoric chain).Leng, Mary. What’s there to know?2007, In M. Leng, A. Paseau, and M. Potter (eds.), Mathematical Knowledge. OUP-
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Added by: Jamie Collin
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.
Maddy, Penelope. Three Forms of Naturalism2005, in The Oxford Handbook of Philosophy of Mathematics and Logic, (ed.) S. Shapiro. New York: Oxford University Press.-
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Added by: Jamie Collin
Summary: A clear introduction to mathematical naturalism and its Quinean roots; developing and defending Maddy's own naturalist philosophy of mathematics. Maddy claims that the Quinian ignores some nuances of scientific practice that have a bearing on what the naturalist should take to be the real scientific standards of evidence. Historical studies show that scientists sometimes do not take themselves to be committed to entities that are indispensably quantified over in their best scientific theories, hence the Quinian position that naturalism dictates that we are committed to entities that are indispensably quantified over in our best scientific theories is incorrect.Comment: Good primary reading in advanced undergraduate or postgraduate courses on metaphysics, naturalism or philosophy of mathematics. This would serve well both as a clear and fairly concise introduction to Quinean naturalism and to the indispensability argument in the philosophy of mathematics.
Maddy, Penelope. Naturalism in Mathematics1997, Oxford: Oxford University Press.-
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Added by: Jamie Collin
Publisher's Note: Our much-valued 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 self-evident, 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).
Chihara, Charles. Nominalism2005, in The Oxford Hanbook of Philosophy of Mathematics and Logic, ed. S. Shapiro. New York: Oxford University Press.-
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Added by: Jamie Collin
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.
Chihara, Charles. A Structural Account of Mathematics2004, Oxford: Oxford University Press.-
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Added by: Jamie Collin
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.
Leng, Mary. “Algebraic” Approaches to Mathematics2009, In Otávio Bueno & Øystein Linnebo (eds.). New Waves in Philosophy of Mathematics. Palgrave Macmillan.-
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Added by: Jamie Collin
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.
Leng, Mary. Mathematics and Reality2010, Oxford University Press, USA.-
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Added by: Jamie Collin
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.
Alexandrova, Anna. Making Models Count2008, Philosophy of Science 75(3): 383-404.-
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Added by: Nick Novelli
Abstract: What sort of claims do scientific models make and how do these claims then underwrite empirical successes such as explanations and reliable policy interventions? In this paper I propose answers to these questions for the class of models used throughout the social and biological sciences, namely idealized deductive ones with a causal interpretation. I argue that the two main existing accounts misrepresent how these models are actually used, and propose a new account.
Comment: A good exploration of the role of models in scientific practice. Provides a good overview of the main theories about models, and some objections to them, before suggesting an alternative. Good use of concrete examples, presented very clearly. Suitable for undergraduate teaching. Would form a useful part of an examination of modelling in philosophy of science.
Harp, Randall, Kareem Khalifa. Why Pursue Unification? A Social-Epistemological Puzzle2015, Theoria. An International Journal for Theory, History and Foundations of Science 30(3): 431-447.-
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Added by: Nick Novelli
Abstract: Many have argued that unified theories ought to be pursued wherever possible. We deny this on the basis of social-epistemological and game-theoretic considerations. Consequently, those seeking a more ubiquitous role for unification must either attend to the scientific community's social structure in greater detail than has been the case, and/or radically revise their conception of unification.Comment: An interesting argument about how scientific practice influences the rationality of theory choice. Would be suited to any course where these issues are discussed.
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1975, Philosophical Studies 27(1): 73-125.
Comment: Good as a primary reading on a course on truth, philosophy of language, or on deflationism more generally. Any course that treats deflationary accounts of truth in any detail would deal with the prosentential theory of truth, and this is one of the most historically important presentations of that theory. Would be best used in advanced undergraduate or graduate courses.