Chihara, Charles . A Structural Account of Mathematics

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.
Chihara, Charles . Nominalism

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.
Leng, Mary . Mathematics and Reality

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.
Leng, Mary . “Algebraic” Approaches to Mathematics

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.
Seibt, Johanna . Properties as Processes

Summary: Sellars’ critics have, predominantly, studied single aspects of his work. This essay, on the other hand, is motivated by Sellars’ dictum that “analysis without synopsis is blind” (TWO 527). My intent is to give a synopsis of Sellars’ thought by focusing on the nominalist strands of his scheme. I shall try to draw the reader’s attention to the systematicity and overall coherence of Sellars’ work, since I think that any successful analysis of his writings must heed their systematic context. By presenting Sellars’ logical, semantic, epistemological and metaphysical arguments for the expendability of abstract entities in their systematic connection, I hope to promote both ‘full scope nominalism’ and ‘full scope Sellarsianism.’
Comment: This would be useful in a course on metaphysics or on philosophy of language. The book is not easy, but is unique in being a booklength exploration of metalinguistic nominalism. Recommended for graduate and perhaps advanced undergraduate courses.
Wetzel, Linda . Types and Tokens: On Abstract Objects

Publisher’s Note: There is a widely recognized but infrequently discussed distinction between the spatiotemporal furniture of the world (tokens) and the types of which they are instances. Words come in both types and tokens – for example, there is only one word type ‘the’ but there are numerous tokens of it on this page – as do symphonies, bears, chess games, and many other types of things. In this book, Linda Wetzel examines the distinction between types and tokens and argues that types exist (as abstract objects, since they lack a unique spatiotemporal location). Wetzel demonstrates the ubiquity of references to (and quantifications over) types in science and ordinary language; types have to be reckoned with, and cannot simply be swept under the rug. Wetzel argues that there are such things as types by undermining the epistemological arguments against abstract objects and offering extended original arguments demonstrating the failure of nominalistic attempts to paraphrase away such references to (and quantifications over) types. She then focuses on the relation between types and their tokens, especially for words, showing for the first time that there is nothing that all tokens of a type need have in common other than being tokens of that type. Finally, she considers an oftenoverlooked problem for realism having to do with types occurring in other types (such as words in a sentence) and proposes an important and original solution, extending her discussion from words and expressions to other types that structurally involve other types (flags and stars and stripes; molecules and atoms; sonatas and notes).
Comment: The book, or extracts from the book, could be used in advanced undergraduate or postgraduate courses on metaphysics, nominalism or philosophy of language. Chapter 2 of the book provides a clear account of the ways Quine and Frege thought about ontological commitment and language. Chapters 35 are also useful for students who want to understand nominalism better, though more recent nominalist strategies, such as the kinds of fictionalism developed by Mark Balaguer and Mary Leng, are not addressed.
Wetzel, Linda . The Trouble With Nominalism

Summary: Wetzel raises an important but underdiscussed argument for Platonism. We quantify over types (contrast with tokens) in sentences that we take to be true. This means we are, prima facie, committed to the existence of types. Wetzel considers various ‘nominalization’ strategies to get rid of type discourse and finds them all wanting. As a result, argues Wetzel, nominalism is untenable.
Comment: This would be useful in a course on metaphysics, ontology, or any course in which the debate between nominalists and platonists is an issue. The paper is short, clear, and relatively untechnical. It raises an important dispute in metaphysics which has not received as much attention as it deserves.