Otávio Bueno

Typical accounts of reference demand that referring terms denote existent objects. This assumption is shared by theories across a variety of areas of philosophy, in particular, direct reference views in philosophy of language; neo-Fregean conceptions in the philosophy of mathematics, and easy-ontology approaches in metaphysics. In this paper, this assumption is resisted and the significance and the possibility of referring to the nonexistent is highlighted. After identifying difficulties in all these three theories and resisting a free-logic approach, ontologically neutral quantifiers, which do not require the existence of what is quantified over, are suggested as providing a better conception. It is concluded that the difficulties raised to the previous theories do not affect the ontologically neutral approach, while the approach, properly conceived, allows for nonexistent objects to have properties.

This comprehensive collection of original essays written by an international group of scholars addresses the central themes in Latin American philosophy. Represents the most comprehensive survey of historical and contemporary Latin American philosophy available today Comprises a specially commissioned collection of essays, many of them written by Latin American authors Examines the history of Latin American philosophy and its current issues, traces the development of the discipline, and offers biographical sketches of key Latin American thinkers Showcases the diversity of approaches, issues, and styles that characterize the field.

Proofs are central to mathematical practice in large part due to the heuristic role that some of them play. Not only do they help establish a result, but often provide new avenues of mathematical research. Jody Azzouni has argued that underlying the practice of creating mathematical proofs there is a very specific norm: to each proof there should be a corresponding algorithmic derivation, a derivation in an algorithmic system. Here a framework is provided to classify and assess mathematical proofs. It is argued that there is a plurality of kinds of proofs in mathematics and a plurality of roles these proofs play. In the end, mathematical practice is far less unified in this respect than it may seem to be.

Crucial to Hilary Putnam’s realism in the philosophy of mathematics is to maintain the objectivity of mathematics without the commitment to the existence of mathematical objects. Putnam’s indispensability argument was devised as part of this conception. In this paper, I reconstruct and reassess Putnam’s argument for the indispensability of mathematics, and distinguish it from the more familiar, Quinean version of the argument. Although I argue that Putnam’s approach ultimately fails, I develop an alternative way of implementing his form of realism about mathematics that, by using different resources than those Putnam invokes, avoids the difficulties faced by his view.

Thirteen up-and-coming researchers in the philosophy of mathematics have been invited to write on what they take to be the right philosophical account of mathematics, examining along the way where they think the philosophy of mathematics is and ought to be going. A rich and diverse picture emerges. Some broader tendencies can nevertheless be detected: there is increasing attention to the practice, language and psychology of mathematics, a move to reassess the orthodoxy, as well as inspiration from philosophical logic.

In this paper, I shall discuss the heuristic role of symmetry in the mathematical formulation of quantum mechanics. I shall first set out the scene in terms of Bas van Fraassen’s elegant presentation of how symmetry principles can be used as problem-solving devices (see van Fraassen [1989] and [1991]). I will then examine in what ways Hermann Weyl and John von Neumann have used symmetry principles in their work as a crucial problem-solving tool. Finally, I shall explore one consequence of this situation to recent debates about structural realism (SR) and empiricism in physics (Worrall [1989], Ladyman [1998], and French [1999]).

In this paper, I offer an inferential conception of computer simulations, emphasizing the role that simulations play as inferential devices to represent empirical phenomena. Three steps are involved in a simulation: an immersion step, a derivation step, and an interpretation and correction step. After presenting the view, I mention some cases, such as simulations of the current flow between silicon atoms and buckyballs as well as of genetic regulatory systems. I argue that the inferential conception accommodates the integration of empirical and theoretical data; it makes sense of the role that is played by false traits in a simulation, andhighlights the similarities and differences between simulations and scientific instruments.

A number of people have recently argued for a structural approach to accounting for the applications of mathematics. Such an approach has been called "the mapping account". According to this view, the applicability of mathematics is fully accounted for by appreciating the relevant structural similarities between the empirical system under study and the mathematics used in the investigation ofthat system. This account of applications requires the truth of applied mathematical assertions, but it does not require the existence of mathematical objects. In this paper, we discuss the shortcomings of this account, and show how these shortcomings can be overcome by a broader view of the application of mathematics: the inferential conception.

There has always been interest in inconsistency in science, not least within science itself as scientists strive to devise a consistent picture of the universe. Some important early landmarks in this history are Copernicus’s criticism of the Ptolemaic picture of the heavens, Galileo’s claim that Aristotle’s theory of motion was inconsistent, and Berkeley’s claim that the early calculus was inconsistent. More recent landmarks include the classical theory of the electron, Bohr’s theory of the atom, and the on-going difficulty of reconciling Einstein’s general relativity and quantum theory. But over the past few decades philosophers have taken a particular and increasing interest in inconsistency in science. In 2002 this culminated in the first collection of articles specifically dedicated to the topic: Inconsistency in Science, edited by Joke Meheus, published by Kluwer, and featuring twelve articles on a range of topics in the philosophy of science and mathematics.Since then philosophic.

In this paper, I shall provide a defence of second-order logic in the context of its use in the philosophy of mathematics. This shall be done by considering three problems that have been recently posed against this logic: (1) According to Resnik [1988], by adopting second-order quantifiers, we become ontologically committed to classes. (2) As opposed to what is claimed by defenders of second-order logic (such as Shapiro [1985]), the existence of non-standard models of first-order theories does not establish the inadequacy of first—order axiomatisations (Melia [1995]). (3) In contrast with Shapiro’s suggestion (in his [1985]), second-order logic does not help us to establish referential access to mathematical objects (Azzouni [1994]). As I shall argue, each of these problems can be neatly solved by the second-order theorist. As a result, a case for second-order logic can be made. The first two problems will beconsidered rather briefly in the next section. The rest of the paper is dedicate to a discussion of the third.

One major motivation for nominalism, at least according to Hartry Field, is the desirability of intrinsic explanations: explanations that don’t invoke objects that are causally irrelevant to the phenomena being explained. There is something right about the search for such explanations. But that search must be carefully implemented. Nothing is gained if, to avoid a certain class of objects, one only introduces other objects and relations that are just as nominalistically questionable. We will argue that this is the case for two alleged nominalist views: Field’s fictionalism, and Frank Arntzenius and Cian Dorr’s geometricalism. Central to our competing approach to nominalism is a distinction between terms that refer to objects and ones that instead code empirical phenomena while being referentially empty. We next contrast our approach to nominalism, which uses this term-grained distinction between coding and referring, with approaches that instead attempt to make a sentence-grained distinction between mathematical and non-mathematical content. We show the latter approach fails to be responsive to objections raised by van Fraassen. In the end, only one last approach to nominalism is left standing. 1 Introduction2 Troubles for Mathematical Fictionalism2.1 A distinction2.2 Field’s programme2.3 Parts of nominalistically acceptable entities are nominalistically acceptable2.4 Structural assumptions: How far can these be pushed?2.5 Cases of indispensable theories where the mathematical posits are known not to be real2.6 How does science distinguish between the indispensable structural presuppositions taken to be real and those that aren’t?2.7 Do physicists distinguish between mathematical posits and non-mathematical posits?2.8 Mathematical coding versus genuine metaphysics. Case study: Fields2.9 The upshot3 Troubles for Topologism4 Contrasts: Kitcher, Maddy, Sober, and van Fraassen4.1 Sober’s approach4.2 Maddy’s approach4.3 Kitcher’s approach4.4 Where do we go from here?4.5 Van Fraassen’s objection to Maddy’s approach to ontological commitment5 Conclusion.

An account of scientific representation in terms of partial structures and partial morphisms is further developed. It is argued that the account addresses a variety of difficulties and challenges that have recently been raised against such formal accounts of representation. This allows some useful parallels between representation in science and art to be drawn, particularly with regard to apparently inconsistent representations. These parallels suggest that a unitary account of scientific and artistic representation is possible, and our article can be viewed as laying the groundwork for such an account—although, as we shall acknowledge, significant differences exist between these two forms of representation.

In order to develop an account of scientific rationality, two problems need to be addressed: (i) how to make sense of episodes of theory change in science where the lack of a cumulative development is found, and (ii) how to accommodate cases of scientific change where lack of consistency is involved. In this paper, we sketch a model of scientific rationality that accommodates both problems. We first provide a framework within which it is possible to make sense of scientific revolutions, but which still preserves some (partial) relations between old and new theories. The existence of these relations help to explain why the break between different theories is never too radical as to make it impossible for one to interpret the process in perfectly rational terms. We then defend the view that if scientific theories are taken to be quasi-true, and if the underlying logic is paraconsistent, it's perfectly rational for scientists and mathematicians to entertain inconsistent theories without triviality. As a result, as opposed to what is demanded by traditional approaches to rationality, it's not irrational to entertain inconsistent theories. Finally, we conclude the paper by arguing that the view advanced here provides a new way of thinking about the foundations of science. In particular, it extends in important respects both coherentist and foundationalist approaches to knowledge, without the troubles that plague traditional views of scientific rationality.

It is easy to show that in many natural axiomatic formulations of physical and even mathematical theories, there are many superfluous concepts usually assumed as primitive. This happens mainly when these theories are formulated in the language of standard set theories, such as Zermelo–Fraenkel’s. In 1925, John von Neumann created a set theory where sets are definable by means of functions. We provide a reformulation of von Neumann’s set theory and show that it can be used to formulate physical and mathematical theories with a lower number of primitive concepts very naturally. Our basic proposal is to offer a new kind of set-theoretic language that offers advantages with respect to the standard approaches, since it doesn’t introduce dispensable primitive concepts. We show how the proposal works by considering significant physical theories, such as non-relativistic classical particle mechanics and classical field theories, as well as a well-known mathematical theory, namely, group theory. This is a first step of a research program we intend to pursue