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.
Naturalism in Mathematics
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.
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.
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.
Pragmatism and Deflationism
Summary: A contemporary defense of a pragmatist account of truth, which contrasts the view with various versions of deflationism. Misak defends the claim that to grasp the concept of truth by exploring its connections with practices we engage in – including assertion, believing, reason-giving, and inquiry. The pragmatist conception of truth, it is argued, helps to elucidate realism/anti-realism: inquiry is truth-apt when it aims at establishing propositions that are indefeasible.
The American Pragmatists (The Oxford History of Philosophy)
Publisher’s Note: Cheryl Misak presents a history of the great American philosophical tradition of pragmatism, from its inception in the Metaphysical Club of the 1870s to the present day. She identifies two dominant lines of thought in the tradition: the first begins with Charles S. Peirce and Chauncey Wright and continues through to Lewis, Quine, and Sellars; the other begins with William James and continues through to Dewey and Rorty. This ambitious new account identifies the connections between traditional American pragmatism and twentieth-century Anglo-American philosophy, and links pragmatism to major positions in the recent history of philosophy, such as logical empiricism. Misak argues that the most defensible version of pragmatism must be seen and recovered as an important part of the analytic tradition.
Where Do Laws of Nature Come From?
Summary: Cartwright explains and defends the view that causal capacities are more fundamental than laws of nature. She does this by considering scientific practice: the kind of knowledge required to make experimental setups and predictions is knowledge of the causal capacities of the entities in those systems, not knowledge of laws of nature.
“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.
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.
Truth deserves to be believed
Abstract: Science seems generally to aim at truth. And governmental support of science is often premised on the instrumental value of truth in service of advancing our practical objectives, both as individuals and as communities, large and small. While there is some political expediency to this view, it is not correct. The value of truth is nowise that it helps us achieve our aims. In fact, just the contrary: truth deserves to be believed only on the condition that its claim upon us is orthogonal to any utility it might have in the service of (any and all) practical ends