Most philosophers of mathematics try to show either that the sort of knowledge mathematicians have is similar to the sort of knowledge specialists in the empirical sciences have or that the kind of knowledge mathematicians have, although apparently about objects such as numbers, sets, and so on, isn't really about those sorts of things as well. Jody Azzouni argues that mathematical knowledge really is a special kind of knowledge with its own special means of gathering evidence. He analyses the linguistic (...) pitfalls and misperceptions philosophers in this field are often prone to, and explores the misapplications of epistemic principles from the empirical sciences to the exact sciences. What emerges is a picture of mathematics both sensitive to mathematical practice, and to the ontological and epistemological issues that concern philosophers. (shrink)

Mathematics as a Science of Patterns is the definitive exposition of a system of ideas about the nature of mathematics which Michael Resnik has been elaborating for a number of years. In calling mathematics a science he implies that it has a factual subject-matter and that mathematical knowledge is on a par with other scientific knowledge; in calling it a science of patterns he expresses his commitment to a structuralist philosophy of mathematics. He links this to a defence of realism (...) about the metaphysics of mathematics--the view that mathematics is about things that really exist. Resnik's distinctive philosophy of mathematics is here presented in an accessible and systematic form: it will be of value not only to specialists in this area, but to philosophers, mathematicians, and logicians interested in the relationship between these three disciplines, or in truth, realism, and epistemology. (shrink)

Science Without Numbers caused a stir in 1980, with its bold nominalist approach to the philosophy of mathematics and science. It has been unavailable for twenty years and is now reissued in a revised edition with a substantial new preface presenting the author's current views and responses to the issues raised in subsequent debate.

Numbers and other mathematical objects are exceptional in having no locations in space or time or relations of cause and effect. This makes it difficult to account for the possibility of the knowledge of such objects, leading many philosophers to embrace nominalism, the doctrine that there are no such objects, and to embark on ambitious projects for interpreting mathematics so as to preserve the subject while eliminating its objects. This book cuts through a host of technicalities that have obscured previous (...) discussions of these projects, and presents clear, concise accounts of a dozen strategies for nominalistic interpretation of mathematics, thus equipping the reader to evaluate each and to compare different ones. The authors also offer critical discussion, rare in the literature, of the aims and claims of nominalistic interpretation, suggesting that it is significant in a very different way from that usually assumed. (shrink)

This book expounds a system of ideas about the nature of mathematics which Michael Resnik has been elaborating for a number of years. In calling mathematics a science he implies that it has a factual subject-matter and that mathematical knowledge is on a par with other scientific knowledge; in calling it a science of patterns he expresses his commitment to a structuralist philosophy of mathematics. He links this to a defense of realism about the metaphysics of mathematics--the view that mathematics (...) is about things that really exist. (shrink)

Mathematicians 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? Offering a scrupulously fair treatment of both mathematical and philosophical concerns, Penelope Maddy here delineates and defends a novel version of mathematical realism. (...) She answers the traditional questions and poses a challenging new one, refocusing philosophical attention on the pressing foundational issues of contemporary mathematics. (shrink)

This book does three main things. First, it defends mathematical platonism against the main objections to that view (most notably, the epistemological objection and the multiple-reductions objection). Second, it defends anti-platonism (in particular, fictionalism) against the main objections to that view (most notably, the Quine-Putnam indispensability objection and the objection from objectivity). Third, it argues that there is no fact of the matter whether abstract mathematical objects exist and, hence, no fact of the matter whether platonism or anti-platonism is true.

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

In this book, Balaguer demonstrates that there are no good arguments for or against mathematical platonism. He does this by establishing that both platonism and anti-platonism are defensible views. Introducing a form of platonism ("full-blooded platonism") that solves all problems traditionally associated with the view, he proceeds to defend anti-platonism (in particular, mathematical fictionalism) against various attacks, most notably the Quine-Putnam indispensability attack. He concludes by arguing that it is not simply that we do not currently have any good argument (...) for or against platonism, but that we could never have such an argument and, indeed, that there is no fact of the matter as to whether platonism is correct. (shrink)

In _Realistic Rationalism_, Jerrold J. Katz develops a new philosophical position integrating realism and rationalism. Realism here means that the objects of study in mathematics and other formal sciences are abstract; rationalism means that our knowledge of them is not empirical. Katz uses this position to meet the principal challenges to realism. In exposing the flaws in criticisms of the antirealists, he shows that realists can explain knowledge of abstract objects without supposing we have causal contact with them, that numbers (...) are determinate objects, and that the standard counterexamples to the abstract/concrete distinction have no force. Generalizing the account of knowledge used to meet the challenges to realism, he develops a rationalist and non-naturalist account of philosophical knowledge and argues that it is preferable to contemporary naturalist and empiricist accounts. The book illuminates a wide range of philosophical issues, including the nature of necessity, the distinction between the formal and natural sciences, empiricist holism, the structure of ontology, and philosophical skepticism. Philosophers will use this fresh treatment of realism and rationalism as a starting point for new directions in their own research. (shrink)

In _Realistic Rationalism_, Jerrold J. Katz develops a new philosophical position integrating realism and rationalism. Realism here means that the objects of study in mathematics and other formal sciences are abstract; rationalism means that our knowledge of them is not empirical. Katz uses this position to meet the principal challenges to realism. In exposing the flaws in criticisms of the antirealists, he shows that realists can explain knowledge of abstract objects without supposing we have causal contact with them, that numbers (...) are determinate objects, and that the standard counterexamples to the abstract/concrete distinction have no force. Generalizing the account of knowledge used to meet the challenges to realism, he develops a rationalist and non-naturalist account of philosophical knowledge and argues that it is preferable to contemporary naturalist and empiricist accounts. The book illuminates a wide range of philosophical issues, including the nature of necessity, the distinction between the formal and natural sciences, empiricist holism, the structure of ontology, and philosophical skepticism. Philosophers will use this fresh treatment of realism and rationalism as a starting point for new directions in their own research. (shrink)

This is the first systematic survey of modern nominalistic reconstructions of mathematics, and for this reason alone it should be read by everyone interested in the philosophy of mathematics and, more generally, in questions concerning abstract entities. In the bulk of the book, the authors sketch a common formal framework for nominalistic reconstructions, outline three major strategies such reconstructions can follow, and locate proposals in the literature with respect to these strategies. The discussion is presented with admirable precision and clarity, (...) and should be accessible even to readers with only minimal background in logic and mathematics. There will be many who will turn directly to these pages and use them as a brief manual on the state of the art of nominalism in mathematics. But the most intriguing parts of this elegant book—at least in my view—are the introduction and the conclusion, where the authors examine the significance of reconstructive nominalism. (shrink)