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.

This is the second volume of philosophical essays by one of the most innovative and influential philosophers now writing in English. Containing thirteen papers in all, the book includes both new essays and previously published papers, some of them with extensive new postscripts reflecting Lewis's current thinking. The papers in Volume II focus on causation and several other closely related topics, including counterfactual and indicative conditionals, the direction of time, subjective and objective probability, causation, explanation, perception, free will, and rational (...) decision. Throughout, Lewis analyzes global features of the world in such a way as to show that they might turn out to supervene on the spatiotemporal arrangement of local qualities. (shrink)

Why do forces compose according to the parallelogram of forces? This question has been controversial; it is one episode in a longstanding, fundamental dispute regarding which facts are not to be explained dynamically. If the parallelogram law is explained statically, then the laws of statics are separate from and “transcend” the laws of dynamics. Alternatively, if the parallelogram law is explained dynamically, then statical laws become mere corollaries to the dynamical laws. I shall attempt to trace the history of this (...) controversy in order to identify what it would be for one or the other of these rival views to be correct. I shall argue that various familiar accounts of natural law not only make it difficult to see what the point of this dispute could have been, but also improperly foreclose some serious scientific options. I will sketch an alternative account of laws that helps us to understand what this dispute was all about. (shrink)

There is an urgent need in philosophy of mathematics for new approaches which pay closer attention to mathematical practice. This book will blaze the trail: it offers philosophical analyses of important characteristics of contemporary mathematics and of many aspects of mathematical activity which escape purely formal logical treatment.

Philosophers who regard some mathematical proofs as explaining why theorems hold, and others as merely proving that they do hold, disagree sharply about the explanatory value of proofs by mathematical induction. I offer an argument that aims to resolve this conflict of intuitions without making any controversial presuppositions about what mathematical explanations would be.

How do we know that God exists? One of Britain's greatest 18th-century philosophers addresses the age-old question in this timeless dialogue. Equally captivating as a philosophical argument and as a work of literature, this classic is particularly relevant in terms of its criticism of the reasoning behind Intelligent Design.

This book not only outlines the indispensability argument in considerable detail but also defends it against various challenges.

It is often presumed that the laws of nature have special significance for scientific reasoning. But the laws' distinctive roles have proven notoriously difficult to identify--leading some philosophers to question if they hold such roles at all. This study offers original accounts of the roles that natural laws play in connection with counterfactual conditionals, inductive projections, and scientific explanations, and of what the laws must be in order for them to be capable of playing these roles. Particular attention is given (...) to laws of special sciences, levels of scientific explanation, natural kinds, ceteris-paribus clauses, and physically necessary non-laws. (shrink)

Mathematical investigation, when done well, can confer understanding. This bare observation shouldn’t be controversial; where obstacles appear is rather in the effort to engage this observation with epistemology. The complexity of the issue of course precludes addressing it tout court in one paper, and I’ll just be laying some early foundations here. To this end I’ll narrow the field in two ways. First, I’ll address a specific account of explanation and understanding that applies naturally to mathematical reasoning: the view proposed (...) by Philip Kitcher and Michael Friedman of explanation or understanding as involving the unification of theories that had antecedently appeared heterogeneous. For the second narrowing, I’ll take up one specific feature (among many) of theories and their basic concepts that is sometimes taken to make the theories and concepts preferred: in some fields, for some problems, what is counted as understanding a problem may involve finding a way to represent the problem so that it (or some aspect of it) can be visualized. The final section develops a case study which exemplifies the way that this consideration – the potential for visualizability – can rationally inform decisions as to what the proper framework and axioms should be. The discussion of unification (in sections 3 and 4) leads to a mathematical analogue of Goodman’s problem of identifying a principled basis for distinguishing grue and green. Just as there is a philosophical issue about how we arrive at the predicates we should use when making empirical predictions, so too there is an issue about what properties best support many kinds of mathematical reasoning that are especially valuable to us. The issue becomes pressing via an examination of some physical and mathematical cases that make it seem unlikely that treatments of unification can be as straightforward as the philosophical literature has hoped. Though unification accounts have a grain of truth (since a phenomenon (or cluster of phenomena) called “unification” is in fact important in many cases) we are far from an analysis of what “unification” is.. (shrink)

Many explanations in science make use of mathematics. But are there cases where the mathematical component of a scientific explanation is explanatory in its own right? This issue of mathematical explanations in science has been for the most part neglected. I argue that there are genuine mathematical explanations in science, and present in some detail an example of such an explanation, taken from evolutionary biology, involving periodical cicadas. I also indicate how the answer to my title question impacts on broader (...) issues in the philosophy of mathematics; in particular it may help platonists respond to a recent challenge by Joseph Melia concerning the force of the Indispensability Argument. (shrink)

The official model of explanation proposed by the logical empiricists, the covering law model, is subject to familiar objections. The goal of the present paper is to explore an unofficial view of explanation which logical empiricists have sometimes suggested, the view of explanation as unification. I try to show that this view can be developed so as to provide insight into major episodes in the history of science, and that it can overcome some of the most serious difficulties besetting the (...) covering law model. (shrink)

What is the nature of mathematical knowledge? Is it anything like scientific knowledge or is it sui generis? How do we acquire it? Should we believe what mathematicians themselves tell us about it? Are mathematical concepts innate or acquired? Eight new essays offer answers to these and many other questions.

This is a review of Marc Lange's _Natural Laws in Scientific Practice<D>.

In an important departure from theories of causation, David Owens proposes that coincidences have no causes, and that a cause is something which ensures that its effects are no coincidence. In Causes and Coincidences, he elucidates the idea of a coincidence as an event which can be analysed into constituent events, the nomological antecedents of which are independent of each other. He also suggests that causal facts can be analysed in terms of non-causal facts, including relations of necessity. Thus, causation (...) is defined in terms of coincidence, and coincidence without reference to causation. David Owens challenges the ideas associated with Hume, Davidson and Lewis, constructing a theory which distinguishes nomological necessity and sufficiency from their logical counterparts. He is able to offer novel solutions to the major problems of causation, including the direction of causation, the logical form of causal statements, the distinction betwen causal connections and logical connections, and the relationship between psychological and physical causation. (shrink)

Argues, contra Dummett, that the platonist need not be any more committed than the intuitionist to the notion that there are arithmetical truths in principle inaccessible to any finite intelligence.

In 1908, Henri Poincar? claimed that: ...the mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a mathematical law, just as experimental facts lead us to the knowledge of a physical law. They are those which reveal to us unsuspected kinship between other facts, long known, but wrongly believed to be strangers to one another. Towards the end of the twentieth century, with many more mathematical (...) facts since discovered, several mathematicians proposed overarching schemes to organise the facts they considered most significant. In this paper, I shall briefly discuss three of these schemes (those of Arnold, Atiyah, and Baez and Dolan), before drawing some philosophical consequences from their attempts. Rather than the kind of claim made by Frege that with the entry of imaginary numbers we reach the ?natural end of the domain of numbers?, we are dealing here with a more open-ended sense of conceptual growth. I shall illustrate this theme by discussing the elaboration of algebraic structures designed to measure symmetry. What emerges is that at any one time mathematicians are operating with a notion very similar to that discussed in the philosophy of science under the heading of ?natural kinds?. We find ?quasi-causal? talk of properties being ?responsible? for a phenomenon, projectability, the transfer of robust mechanisms between domains, and reference to entities not yet fully determined. Debates in philosophy of science prompt further questioning as to the ?naturalness? of these mathematical kinds, whether one should expect them to be dependent on varying human interests, whether there is a distinction between artefactual and real kinds, and whether there is convergence of kinds. I believe that these questions present a wonderful opportunity for a philosophy of mathematics to treat ?real? mathematics, while making powerful points of contact with philosophy of science, and philosophy in general. (shrink)