This concise book introduces nonphysicists to the core philosophical issues surrounding the nature and structure of space and time, and is also an ideal resource for physicists interested in the conceptual foundations of space-time theory. Tim Maudlin's broad historical overview examines Aristotelian and Newtonian accounts of space and time, and traces how Galileo's conceptions of relativity and space-time led to Einstein's special and general theories of relativity. Maudlin explains special relativity using a geometrical approach, emphasizing intrinsic space-time structure rather than (...) coordinate systems or reference frames. He gives readers enough detail about special relativity to solve concrete physical problems while presenting general relativity in a more qualitative way, with an informative discussion of the geometrization of gravity, the bending of light, and black holes. Additional topics include the Twins Paradox, the physical aspects of the Lorentz-FitzGerald contraction, the constancy of the speed of light, time travel, the direction of time, and more.Introduces nonphysicists to the philosophical foundations of space-time theory Provides a broad historical overview, from Aristotle to Einstein Explains special relativity geometrically, emphasizing the intrinsic structure of space-time Covers the Twins Paradox, Galilean relativity, time travel, and more Requires only basic algebra and no formal knowledge of physics. (shrink)

That there is no substance causation is often treated as the default position. My aim in this paper is primarily one of burden shifting: opponents of substance causation must do more to defend their position. After outlining the thesis I wish to defend, I present a simple argument for substance causation, arguing that opponents of substance causation owe us an explanation of why this argument is unsound. I end by answering objections to the view that substances can be causes.

An analysis of phaedo 96c-606c seeks to demonstrate that when forms are cited as either "safe" or "clever" aitiai they are not meant to function as either final or efficient causes, But as logico-Metaphysical essences which have no causal efficacy whatever, But which do have definite (and far-Reaching) implications for the causal order of the physical universe, For it is assumed that a causal statement, Such as "fire causes heat" will be true if, And only if, The asserted physical bond (...) between fire and heat instantiates the eternal and logically necessary relation of entailment between the instantiated forms, Fire, Heat. (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)

A theory of the causal relate as aspects or property instances is developed. A supposed problem for transitivity is assessed and then resolved with aspects as the causal relata.

This article discusses the argument we cannot have knowledge of abstract entities because they are not part of the causal order. The claim of this article is that the argument fails because of equivocation. Assume that the “causal order” is concerned with contingent facts involving time and space. Even if the existence of abstract entities is not contingent and does not involve time or space it does not follow that no truths about abstract entities are contingent or involve time or (...) space. I argue that it is the latter which is required to obtain the desired conclusion. (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)

In this paper, we develop an alternative strategy, Platonized Naturalism, for reconciling naturalism and Platonism and to account for our knowledge of mathematical objects and properties. A systematic (Principled) Platonism based on a comprehension principle that asserts the existence of a plenitude of abstract objects is not just consistent with, but required (on transcendental grounds) for naturalism. Such a comprehension principle is synthetic, and it is known a priori. Its synthetic a priori character is grounded in the fact that it (...) is an essential part of the logic in which any scientific theory will be formulated and so underlies (our understanding of) the meaningfulness of any such theory (this is why it is required for naturalism). Moreover, the comprehension principle satisfies naturalist standards of reference, knowledge, and ontological parsimony! As part of our argument, we identify mathematical objects as abstract individuals in the domain governed by the comprehension principle, and we show that our knowledge of mathematical truths is linked to our knowledge of that principle. (shrink)

Many philosophers posit abstract entities – where something is abstract if it is acausal and lacks spatio-temporal location. Theories, types, characteristics, meanings, values and responsibilities are all good candidates for abstractness. Such things raise an epistemological puzzle: if they are abstract, then how can we have any epistemic access to how they are? If they are invisible, intangible and never make anything happen, then how can we ever discover anything about them? In this article, I critically examine epistemological objections to (...) belief in abstract objects offered by Paul Benacerraf, Colin Cheyne and Hartry Field. (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)

objects are standardly taken to be causally inert, but this claim is rarely explicitly argued for. In the context of his platonism about musical works, in order for musical works to be audible, Julian Dodd argues that abstracta are causally efficacious in virtue of their concrete tokens participating in events. I attempt to provide a principled argument for the causal inertness of abstracta by first rejecting Dodd’s arguments from events, and then extending and generalizing the causal exclusion argument to the (...) abstract/concrete distinction. For reasons of parsimony, if concrete tokens or instantiations of abstract objects account for all causal work, then there is no reason to attribute causal efficacy to abstracta, and thus reason to maintain their causal inertness. I then consider how one of the main arguments against causal exclusion, namely Stephen Yablo’s notion of “proportionality”, could be modified to support the causal efficacy of abstracta. I argue that from a few simple premises Yablo’s account in fact supports their causal inertness. Having a principled reason for the causal inertness of abstracta appears to entail that the musical platonist must admit that we never literally hear the musical work, but only its performances. I sketch a solution to this problem available to Dodd, so that the musical platonist can maintain that musical works are abstract objects and are causally inert while retaining their audibility. (shrink)

I argue that so-called ‘absence causation’must be treated in terms of counterfactuals about causation such as ‘had a occurred, a would have caused b’. First, I argue that some theories of causation that accept absence causation are unattractive because they undermine the idea of possible causation. And second, I argue that accepting absence causation violates a principle commonly associated with relativity.

According to the discovery model in the ontology of art, the facts concerning the ontological status of artworks are mind-independent and, hence, are facts about which the folk may be substantially ignorant or in error. In recent work Amie Thomasson has claimed that the most promising solution to the ‘ qua problem’—a problem concerning how the reference of a referring-expression is fixed—requires us to give up the discovery model. I argue that this claim is false. Thomasson's solution to the qua (...) problem—a hybrid descriptive/causal theory of reference-fixing—has a superior competitor, in the form of the account of reference-fixing suggested by Gareth Evans; and Evans's theory leaves the discovery model untouched. (shrink)

Attributions of omnipresence, most familiar within the philosophy of religion, typically take the omnipresence of an entity to either consist in that entity's occupation of certain regions or be dependent upon other of that entity's attributes, such as omnipotence or omniscience. This paper defends an alternative conception of omnipresence that is independent of other purported divine attributes and dispenses with occupation. The resulting view repurposes the metaphysics of necessitism and permanentism, taking omnipresent entities to be those entities that exist at (...) all regions. This view is then shown to best accommodate attributions of omnipresence to a diverse range of metaphysical posits, like abstract entities, and a more diverse class of religious posits. (shrink)

Can a musical work be created? Some say ‘no’. But, we argue, there is no handbook of universally accepted metaphysical truths that they can use to justify their answer. Others say ‘yes’. They have to find abstract objects that can plausibly be identified with musical works, show that abstract objects of this sort can be created, and show that such abstract objects can persist. But, we argue, none of the standard views about what a musical work is allows musical works (...) both to be created and to persist. (shrink)

It is widely believed that platonists face a formidable problem: that of providing an intelligible account of mathematical knowledge. The problem is that we seem unable, if the platonist is right, to have the causal relationships with the objects of mathematics without which knowledge of these objects seems unintelligible. The standard platonist response to this challenge is either to deny that knowledge without causation is unintelligible, or to make room for causal interactions by softening the platonism at issue. In this (...) essay I argue that the idea of causal relations with fully platonist objects is unproblematic. (shrink)

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)

In her recent paper, “A Defense of Substance Causation,” Ann Whittle makes a case for substance causation. In this paper, assuming that causation is a generative or productive relation, I argue that Whittle’s argument is not successful. While substances are causally relevant in causal processes owing to outcomes being counterfactually dependent upon their role in such occurrences, the real productive work in causal processes is accomplished by the causal powers of substances.

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)

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.

Benjamin Callard has recently suggested that causation between Platonic objects—standardly understood as atemporal and non-spatial—and spatio-temporal objects is not ‘a priori’ unintelligible. He considers the reasons some have given for its purported unintelligibility: apparent impossibility of energy transference, absence of physical contact, etc. He suggests that these considerations fail to rule out a priori Platonic-object causation. However, he has overlooked one important issue. Platonic objects must causally affect different objects differently, and different Platonic objects must causally affect the same objects (...) differently. How are Platonic objects—ones outside space and time—supposed to do that? CiteULike Connotea Del.icio.us What's this? (shrink)

In this short text, a distinguished philosopher turns his attention to one of the oldest and most fundamental philosophical problems of all: How it is that we are able to sort and classify different things as being of the same natural class? Professor Armstrong carefully sets out six major theories—ancient, modern, and contemporary—and assesses the strengths and weaknesses of each. Recognizing that there are no final victories or defeats in metaphysics, Armstrong nonetheless defends a traditional account of universals as the (...) most satisfactory theory we have.This study is written for advanced students, but as Armstrong goes considerably beyond his earlier work on this topic, it will interest professional scholars as well. Carefully plotted and clearly written, Universals is both a paradigm of exposition and a case study on the value of careful analysis of fundamental issues in philosophy. (shrink)

Think of a number, any number, or properties like fragility and humanity. These and other abstract entities are radically different from concrete entities like electrons and elbows. While concrete entities are located in space and time, have causes and effects, and are known through empirical means, abstract entities like meanings and possibilities are remarkably different. They seem to be immutable and imperceptible and to exist "outside" of space and time. This book provides a comprehensive critical assessment of the problems raised (...) by abstract entities and the debates about existence, truth, and knowledge that surround them. It sets out the key issues that inform the metaphysical disagreement between platonists who accept abstract entities and nominalists who deny abstract entities exist. Beginning with the essentials of the platonist–nominalist debate, it explores the key arguments and issues informing the contemporary debate over abstract reality: arguments for platonism and their connections to semantics, science, and metaphysical explanation the abstract–concrete distinction and views about the nature of abstract reality epistemological puzzles surrounding our knowledge of mathematical entities and other abstract entities. arguments for nominalism premised upon concerns about paradox, parsimony, infinite regresses, underdetermination, and causal isolation nominalist options that seek to dispense with abstract entities. Including chapter summaries, annotated further reading, and a glossary, _Entities_ is essential reading for anyone seeking a clear and authoritative introduction to the problems raised by abstract entities. (shrink)

This challenging study places fiction squarely at the centre of the discussion of metaphysics. Philosophers have traditionally treated fiction as involving a set of narrow problems in logic or the philosophy of language. By contrast Amie Thomasson argues that fiction has far-reaching implications for central problems of metaphysics. The book develops an 'artifactual' theory of fiction, whereby fictional characters are abstract artifacts as ordinary as laws or symphonies or works of literature. By understanding fictional characters we come to understand how (...) other cultural and social objects are established on the basis of the independent physical world and the mental states of human beings. (shrink)

Timothy Williamson gives an original and provocative treatment of deep metaphysical questions about existence, contingency, and change, using the latest resources of quantified modal logic. Contrary to the widespread assumption that logic and metaphysics are disjoint, he argues that modal logic provides a structural core for metaphysics.

This study addresses a central theme in current philosophy: Platonism vs Naturalism and provides accounts of both approaches to mathematics, crucially discussing Quine, Maddy, Kitcher, Lakoff, Colyvan, and many others. Beginning with accounts of both approaches, Brown defends Platonism by arguing that only a Platonistic approach can account for concept acquisition in a number of special cases in the sciences. He also argues for a particular view of applied mathematics, a view that supports Platonism against Naturalist alternatives. Not only does (...) this engaging book present the Platonist-Naturalist debate over mathematics in a comprehensive fashion, but it also sheds considerable light on non-mathematical aspects of a dispute that is central to contemporary philosophy. (shrink)

Introduction -- The type/token theory introduced -- Motivating the type/token theory : repeatability -- Nominalist approaches to the ontology of music -- Musical anti-realism -- The type/token theory elaborated -- Types I : abstract, unstructured, unchanging -- Types introduced and nominalism repelled -- Types as abstracta -- Types as unstructured entities -- Types as fixed and unchanging -- Types II : platonism -- Introduction : eternal existence and timelessness -- Types and properties -- The eternal existence of properties reconsidered -- (...) Types and patterns -- Defending the type/token theory I -- Unstructuredness and analogical predication -- Musical works as fixed and unchanging -- Abstractness and audibility (again) -- Works and interpretations -- Conclusion and resumé -- Defending the type/token theory II : musical platonism -- Platonism it is : replies to Anderson and Levinson -- The existence conditions of works of music -- Composition as creative discovery -- The nature of the compositional process : replies to objections -- Composition and aesthetic appraisal : a reply to Levinson -- Composition and aesthetic appraisal : understanding, interpretation, and correctness -- Musical works as continuants : a theory rejected -- A theory introduced -- Explicating and motivating the continuant view -- The continuant view and repeatability -- Further objections to the continuant view -- Musical works as compositional actions : a critique -- Currie's action-type hypothesis -- Davies's performance theory -- Sonicism I : against instrumentalism -- Sonicism introduced -- Sonicism motivated : moderate empiricism -- Instrumentation : timbral sonicism introduced -- Scores -- Instrumentation, artistic properties, and aesthetic content -- Levinson's rejoinder -- Sonicism II : against contextualism -- Introduction : formulating contextualism -- Contextualist ontological proposals -- Levinson's doppelgänger thought-experiments -- Artistic, representational, and object-directed expressive properties -- Aesthetic and non-object-directed expressive properties -- Conclusion : the place of context. (shrink)

This highly acclaimed book is a major contribution to the philosophy of language as well as a systematic interpretation of Frege, indisputably the father of ...

NOWADAYS, even schoolchildren babble about "null sets" and "singletons" and "one-one correspondences," as if they knew what they were talking about. But if they understand even less than their teachers, which seems likely, they must be using the technical jargon with only an illusion of understanding. Beginners are taught that a set having three members is a single thing, wholly constituted by its members but distinct from them. After this, the theological doctrine of the Trinity as "three in one" should (...) be child's play. (shrink)

In this essay I first outline contemporary Platonism about musical works – the theory that musical works are abstract objects. I then consider reasons to be suspicious of such a view, motivating a consideration of nominalist theories of musical works. I argue for two conclusions: first, that there are no compelling reasons to be a nominalist about musical works in particular, i.e. that nominalism about musical works rests on arguments for thoroughgoing nominalism, and, second, that if Platonism fails, fictionalism about (...) musical works is to be preferred to other nominalist ontologies of musical works. (shrink)