In Physics, Aristotle starts his positive account of the infinite by raising a problem: “[I]f one supposes it not to exist, many impossible things result, and equally if one supposes it to exist.” His views on time, extended magnitudes, and number imply that there must be some sense in which the infinite exists, for he holds that time has no beginning or end, magnitudes are infinitely divisible, and there is no highest number. In Aristotle's view, a plurality cannot escape having bounds if all of its members exist at once. Two interesting, and contrasting, interpretations of Aristotle's account can be found in the work of Jaako Hintikka and of Jonathan Lear. Hintikka tries to explain the sense in which the infinite is actually, and the sense in which its being is like the being of a day or a contest. Lear focuses on the sense in which the infinite is only potential, and emphasizes that an infinite, unlike a day or a contest, is always incomplete.
L. A. Paul
Counterfactual analyses have received a good deal of attention in recent years, resulting in a host of counterexamples and objections to the simple analysis and its descendants. The counterexamples are often complex and can seem baroque to the outsider (indeed, even to the insider), and it may be tempting to dismiss them as irrelevant or uninteresting. But while we may be able to ignore some counterexamples because the intuitions they evoke are unclear or misguided, the importance of investigating the causal relation via investigating counterexamples should not be underestimated.
If the future is real, and the outcomes of chancy processes are “already” occurrent, then in what sense is the chancy process genuinely chancy? This topic is the fatalism issue. Discussed by Aristotle more than two millennia ago, the question is whether various logical principles, when applied to propositions about future events, imply that the future is in some sense fixed. If the future is like the past, and the past is fixed, then are future events fixed in the same sense? This chapter gives the latest on this traditional topic, carefully surveying various works and trying a new tack that steers away from fatalism. It concludes that it is difficult to find ways to rule out causal loops, and gives an account of the direction of causation.
The topic of identity seems to many of us to be philosophically unproblematic. Identity, it is said, is the relation that each thing has to itself and to nothing else. Of course, there are many disputable claims that one can make using a predicate that expresses the identity relation. For example: there is something that was a man and is identical to God; there is something that might have been a poached egg that is identical to some philosopher. But puzzling as these claims may be, it is not the identity relation that is causing the trouble. The lesson appears to be a general one. Puzzles that are articulated using the word ‘identity’ are not puzzles about the identity relation itself. One may have noticed that our gloss on identity as ‘the relation that each thing has to itself and to nothing else’ was not really an analysis of the concept of identity in any reasonable sense of ‘analysis’, since an understanding of ‘itself’ and ‘to nothing else’ already requires a mastery of what identity amounts to.
Metaontologyis the study of ontology, asking what exactly it is that so-called ontologists are doing when they do ontology. Contemporary debates harken back to one between Quine and Carnap; this article begins by surveying their debate and outlining how various contemporary authors line up with the positions they staked out. The remainder of the article focuses on a contemporary, neo-Carnapian metaontological position inspired by considerations frommetasemantics, or the study of how words get their meanings. The contemporary position insists that any decent theory of how words get their meanings will have the result that many contemporary ontological debates are, in some important sense, without substance. After outlining the position, the article considers several ways a contemporary ontologist might resist this neo-Carnapian position.
Recent work by analytic philosophers on the Trinity takes a mysterious 5th-century document as its starting point, accepting widespread but inaccurate narratives about the history of Trinity theories. This article summarizes the Platonic influence on ancient theologies and describes the rise of transcendent triads, and eventually the idea of a tripersonal God. Recent Trinity theories (positive mysterianism, Trinity monotheism, relative-identity approaches, and “social” theories) are explained as built to respond in various ways to a type of anti-trinitarian argument. But since each recent application of logic and metaphysics to the theology of the Trinity is problematic, it is argued that another look at the minority unitarian report is warranted.
Jeffrey C. King
Propositions have been long thought by many philosophers to play a number of important roles. These include being the information conveyed by an utterance of a sentence, being the primary bearers of truth and falsity, being the possessors of modal properties like being possible and necessary, and being the things we assume, believe, and doubt. This article canvases significant attempts by philosophers to say what sorts of things propositions are. First, the classical views of propositions advanced by Gottlob Frege and Bertrand Russell are considered. Second, the view of propositions as sets of possible worlds is discussed. Next, views of propositions arising out of work on direct reference are discussed. The article closes with a discussion of more recent views of propositions.
Undoubtedly, the most enlightening published work dedicated to giving knowledgeable readers an overview of the topic of nominalism in contemporary philosophy of mathematics is A Subject with No Object by John Burgess and Gideon Rosen. This article begins with a brief description of that work, in order to provide readers with a solidly researched account of nominalism with which the article's own account of nominalism can be usefully compared. The first part, then, briefly presents the Burgess–Rosen account. A contrasting account is given in the longer second part.
Gideon Rosen and John P. Burgess
Nominalism is usually formulated as the thesis that only concrete entities exist or that no abstract entities exist. But where, as here, the interest is primarily in philosophy of mathematics, one can bypass the tangled question of how, exactly, the general abstract/concrete distinction is to be understood by taking nominalism simply as the thesis that there are no distinctively mathematical objects: no numbers, sets, functions, groups, and so on. As to the nature of such objects (if there are any), it can be said that it has come to be fairly widely agreed, under the influence of Frege and others, that they are very different both from paradigmatically physical objects (bricks, stones) and from paradigmatically mental ones (minds, ideas). Modern nominalism emerged in the 1930s as a response to the view of Frege and others that numbers, sets, functions, groups, and so on belong to a “third realm.”
The main types of mathematical structuralism that have been proposed and developed to the point of permitting systematic and instructive comparison are four: structuralism based on model theory, carried out formally in set theory (e.g., first- or second-order Zermelo–Fraenkel set theory), referred to as STS (for set-theoretic structuralism); the approach of philosophers such as Shapiro and Resnik of taking structures to be sui generis universals, patterns, or structures in an ante rem sense (explained in this article), referred to as SGS (for sui generis structuralism); an approach based on category and topos theory, proposed as an alternative to set theory as an overarching mathematical framework, referred to as CTS (for category-theoretic structuralism); and a kind of eliminative, quasi-nominalist structuralism employing modal logic, referred to as MS (for modal-structuralism). This article takes these up in turn, guided by few questions, with the aim of understanding their relative merits and the choices they present.
The properties and relations that perform a role in mathematical reasoning arise from the basic relations that obtain among mathematical objects. It is in terms of these basic relations that mathematicians identify the objects they intend to study. The way in which mathematicians identify these objects has led some philosophers to draw metaphysical conclusions about their nature. These philosophers have been led to claim that mathematical objects are positions in structures or akin to positions in patterns. This article retraces their route from (relatively uncontroversial) facts about the identification of mathematical objects to high metaphysical conclusions. Beginning with the natural numbers, how are they identified? The mathematically significant properties and relations of natural numbers arise from the successor function that orders them; the natural numbers are identified simply as the objects that answer to this basic function. But the relations (or functions) that are used to identify a class of mathematical objects may often be defined over what appear to be different kinds of objects.