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.
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.
“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.
Inductive Risk and Values in Science
Abstract: Although epistemic values have become widely accepted as part of scientific reasoning, non-epistemic values have been largely relegated to the “external” parts of science (the selection of hypotheses, restrictions on methodologies, and the use of scientific technologies). I argue that because of inductive risk, or the risk of error, non-epistemic values are required in science wherever non-epistemic consequences of error should be considered. I use examples from dioxin studies to illustrate how non-epistemic consequences of error can and should be considered in the internal stages of science: choice of methodology, characterization of data, and interpretation of results.
Exploratory Experiments
Abstract: Philosophers of experiment have acknowledged that experiments are often more than mere hypothesis-tests, once thought to be an experiment’s exclusive calling. Drawing on examples from contemporary biology, I make an additional amendment to our understanding of experiment by examining the way that `wide’ instrumentation can, for reasons of efficiency, lead scientists away from traditional hypothesis-directed methods of experimentation and towards exploratory methods.
Making Models Count
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.
Nonreductive physics
Abstract: This paper documents a wide range of nonreductive scientific treatments of phenomena in the domain of physics. These treatments strongly resist characterization as explanations of macrobehavior exclusively in terms of behavior of microconstituents. For they are treatments in which macroquantities are cast in the role of genuine and irreducible degrees of freedom.
Explanation is a genus: An essay on the varieties of scientific explanation
Abstract: I shall endeavor to show that every physical theory since Newton explainswithout drawing attention to causes-that, in other words, physical theories as physical theories aspire to explain under an ideal quite distinctfrom that of causal explanation. If I am right, then even if sometimes theexplanations achieved by a physical theory are not in violation ofthe standard of causal explanation, this is purely an accident. For physicaltheories, as I will show, do not, as such, aim at accommodating the goals oraspirations of causal explanation. This will serve as the founding insightfor a new theory of explanation, which will itself serve as the cornerstoneof a new theory of scientific method.
A modest proposal for interpreting structural explanations
Abstract: Social sciences face a well-known problem, which is an instance of a general problem faced as well by psychological and biological sciences: the problem of establishing their legitimate existence alongside physics. This, as will become clear, is a problem in metaphysics. I will show how a new account of structural explanations, put forward by Frank Jackson and Philip Pettit, which is designed to solve this metaphysical problem with social sciences in mind, fails to treat the problem in any importantly new way. Then I will propose a more modest approach, and show how it does not deserve the criticism directed at a prototype by Jackson and Pettit
Do Newton’s Rules of Reasoning Guarantee Truth … Must They?
Abstract: Newton’s Principia introduces four rules of reasoning for natural philosophy. Although useful, there is a concern about whether Newton’s rules guarantee truth. After redirecting the discussion from truth to validity, I show that these rules are valid insofar as they fulfill Goodman’s criteria for inductive rules and Newton’s own methodological program of experimental philosophy; provided that cross-checks are used prior to applications of rule 4 and immediately after applications of rule 2 the following activities are pursued: (1) research addressing observations that systematically deviate from theoretical idealizations and (2) applications of theory that safeguard ongoing research from proceeding down a garden path.