Otávio Bueno

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.

According to the semantic view, a theory is characterized by a class of models. In this paper, we examine critically some of the assumptions that underlie this approach. First, we recall that models are models of something. Thus we cannot leave completely aside the axiomatization of the theories under consideration, nor can we ignore the metamathematics used to elaborate these models, for changes in the metamathematics often impose restrictions on the resulting models. Second, based on a parallel between van Fraassen’s modal interpretation of quantum mechanics and Skolem’s relativism regarding set-theoretic concepts, we introduce a distinction between relative and absolute concepts in the context of the models of a scientific theory. And we discuss the significance of that distinction. Finally, by focusing on contemporary particle physics, we raise the question: since there is no general accepted unification of the parts of the standard model, we have no theory, in the usual sense of the term. This poses a difficulty: if there is no theory, how can we speak of its models? What are the latter models of? We conclude by noting that it is unclear that the semantic view can be applied to contemporary physical theories.

The mathematical concept of pragmatic truth, first introduced in Mikenberg, da Costa and Chuaqui (1986), has received in the last few years several applications in logic and the philosophy of science. In this paper, we study the logic of pragmatic truth, and show that there are important connections between this logic, modal logic and, in particular, Jaskowski's discussive logic. In order to do so, two systems are put forward so that the notions of pragmatic validity and pragmatic truth can be accommodated. One of the main results of this paper is that the logic of pragmatic truth is paraconsistent. The philosophical import of this result, which justifies the application of pragmatic truth to inconsistent settings, is also discussed

In this paper, I provide an easy road to nominalism which does not rely on a Field-type nominalization strategy for mathematics. According to this proposal, applications of mathematics to science, and alleged mathematical explanations of physical phenomena, only emerge when suitable physical interpretations of the mathematical formalism are advanced. And since these interpretations are rarely distinguished from the mathematical formalism, the impression arises that mathematical explanations derive from the mathematical formalism alone. I correct this misimpression by pointing out, in the cases recently discussed by Mark Colyvan, exactly where the interpretations of the formalism were invoked and the function they played in the resulting explanations. A viable form of easy - road nominalism, which is also sensitive to scientific practice, then arises

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

We formulate Suppes predicates for various kinds of space-time: classical Euclidean, Minkowski's, and that of General Relativity. Starting with topological properties, these continua are mathematically constructed with the help of a basic algebra of events; this algebra constitutes a kind of mereology, in the sense of Lesniewski. There are several alternative, possible constructions, depending, for instance, on the use of the common field of reals or of a non-Archimedian field. Our approach was inspired by the work of Whitehead, though our philosophical stance is completely different from his. The structures obtained are idealized constructs underlying extant, physical space-time

Hartry Field (1980) has developed an interesting nominalization strategy for Newtonian gravitation theory—a strategy that reformulates the theory without quantification over abstract entities. According to David Malament (1982), Field's strategy cannot be extended to quantum mechanics (QM), and so it only has a limited scope. In a recent work, Mark Balaguer has responded to Malament's challenge by indicating how QM can be nominalized, and by “doing much of the work needed to provide the nominalization” (Balaguer 1998, 114). In this paper, I critically assess Balaguer's proposal, and argue that it ultimately fails. Balaguer's strategy is incompatible with a number of interpretations of QM, in particular with Bas van Fraassen's version of the modal interpretation. And given that Balaguer's strategy invokes physically real propensities, it is unclear whether it is even compatible with nominalism. I conclude that the nominalization of QM remains a major problem for the nominalist.

Michael Dummett, Frege and other philosophers. Oxford:Clarendon Press, 1991. xii + 330pp. £35. ISBN W.Balzer and C.U.Moulines (eds,), Structuralist theory of science:focal issues, new results, Berlin; de Gruyter, 1996. xi + 295 pp.DM 210. ISBN 3-11-014075-6 Henry Prakken, Logical tools for modeling legal argument a study of defeasible reasoning in law.Dordrecht, The Netherlands:Kluwer Academic, 1997, xiii + 314pp.£75.00/$125.00 J.Srzednicki and Z.Stachniak (eds.) Lesniewski’s Systems.Protothetic.Nijhoff International Philosophy Series, 54, Dordrecht, Boston and London:Kluwer, 1998. xiv + 310 pp, £99 (cloth). ISBN 0-7923-4504-5

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

The blame for the semantic and set-theoretic paradoxes is often placed on self-reference and circularity. Some years ago, Yablo [1985; 1993] challenged this diagnosis, by producing a paradox that's liar-like but does not seem to involve circularity. But is Yablo's paradox really non-circular? In a recent paper, Beall [2001] has suggested that there are no means available to refer to Yablo's paradox without invoking descriptions, and since Priest [1997] has shown that any such description is circular, Beall concludes that Yablo's paradox itself is circular. In this paper, we argue that Beall's conclusion is unwarranted, given that (1) descriptions are not the only way to refer to Yablo's paradox, and (ii) we have no reason to believe that because the description involves self-reference, the denotation of the description is also circular. As a result, for all that's been said so far, we have no reason to believe that Yablo's paradox is circular

The aim of this paper is to provide a unified account of scientific and mathematical change in a thoroughly empiricist setting. After providing a formal modelling in terms of embedding, and criticising it for being too restrictive, a second modelling is advanced. It generalises the first, providing a more open-ended pattern of theory development, and is articulated in terms of da Costa and French's partial structures approach. The crucial component of scientific and mathematical change is spelled out in terms of partial embeddings. Finally, an application of this second pattern is made, with the examination of the early formulation of set theory.

Perceptual experiences provide an important source of information about the world. It is clear that having the capacity of undergoing such experiences yields an evolutionary advantage. But why should humans have developed not only the ability of simply seeing, but also of seeing that something is thus and so? In this paper, I explore the significance of distinguishing perception from conception for the development of the kind of minds that creatures such as humans typically have. As will become clear, it is crucial to pay careful attention to the different kinds of information that are involved in perceiving and conceiving (including the way such information is gathered and transmitted). By identifying such kinds of information and the role they play, we can then understand an important feature of why creatures like us have the kind of consciousness and mental processes we do

Ask a philosopher what a proof is, and you’re likely to get an answer hii empaszng one or another regimentationl of that notion in terms of a finite sequence of formalized statements, each of which is either an axiom or is derived from an axiom by certain inference rules. (Wecan call this the formal conception of proof) Ask a mathematician what a proof is, and you will rbbl poay get a different-looking answer. Instead of stressing a partic- l uar regimented notion of proof, the answer the mathematician will give ilikl..

The physics and metaphysics of identity and individuality Content Type Journal Article DOI 10.1007/s11016-010-9463-7 Authors Don Howard, Department of Philosophy and Graduate Program in History and Philosophy of Science, University of Notre Dame, Notre Dame, IN 46556, USA Bas C. van Fraassen, Philosophy Department, San Francisco State University, 1600 Holloway Avenue, San Francisco, CA 94132, USA Otávio Bueno, Department of Philosophy, University of Miami, Coral Gables, FL 33124, USA Elena Castellani, Department of Philosophy, University of Florence, Via Bolognese 52, 50139 Florence, Italy Laura Crosilla, Department of Pure Mathematics, School of Mathematics, University of Leeds, Leeds, LS2 9JT UK Steven French, Department of Philosophy, University of Leeds, Leeds, UK Décio Krause, Department of Philosophy, Federal University of Santa Catarina, 88040-900 Campus Trindade, Florianópolis, SC Brazil Journal Metascience Online ISSN 1467-9981 Print ISSN 0815-0796

In this paper we examine Lewis's attempts to provide an epistemology of modality and we argue that he fails to provide an account that properly weds his metaphysics with an epistemology that explains the knowledge of modality that both he and his critics grant. We argue that neither the appeals to acceptable paraphrases of ordinary modal discourse nor parallels with Platonistic theories of mathematics suffice. We conclude that no proper epistemology for modal realism has been provided and that one is needed.