Otávio Bueno

Descartes seems to hold two inconsistent accounts of the ontological status of mathematical essences. Meditation Five apparently develops a platonist view about such essences, while the Principles seems to advocate some form of “conceptualism”. We argue that Descartes was neither a platonist nor a conceptualist. Crucial to our interpretation is Descartes’ dispositional nativism. We contend that his doctrine of innate ideas allows him to endorse a hybrid view which avoids the drawbacks of Gassendi’s conceptualism without facing the difficulties of platonism. We call this hybrid view “quasi-platonism.” Our interpretation explains Descartes’ account of the nature of mathematical essences, dissolves the tension between the two texts, and highlights the benefits of Descartes’ view.

Current versions of nominalism in the philosophy of mathematics face a significant problem to understand mathematical knowledge. They are unable to characterize mathematical knowledge as knowledge of the objects mathematical theories are taken to be about. Oswaldo Chateaubriand’s insightful reformulation of Platonism (Chateaubriand 2005) avoids this problem by advancing a broader conception of knowledge as justified truth beyond a reasonable doubt, and by introducing a suitable characterization of logical form in which the relevant mathematical facts play an important role in the truth of the corresponding mathematical propositions. In this paper, I contrast Chateaubriand’s proposal with an agnostic form of nominalism that is able to accommodate mathematical knowledge without the commitment to mathematical facts.

Philosophers are very fond of making non-factualist claims—claims to the effect that there is no fact of the matter as to whether something is the case. But can these claims be coherently stated in the context of classical logic? Some care is needed here, we argue, otherwise one ends up denying a tautology or embracing a contradiction. In the end, we think there are only two strategies available to someone who wants to be a non-factualist about something, and remain within the province of classical logic. But one of these strategies is rather controversial, and the other requires substantially more work than is often supposed. Being a non-factualist is no easy business, and it may not be the most philosophically perspicuous way to go.

This paper is a historical companion to a previous one, in which it was studied the so-called abstract Galois theory as formulated by the Portuguese mathematician José Sebastião e Silva ). Our purpose is to present some applications of abstract Galois theory to higher-order model theory, to discuss Silva's notion of expressibility and to outline a classical Galois theory that can be obtained inside the two versions of the abstract theory, those of Mark Krasner and of Silva. Some comments are made on the universal theory of structures.

Throughout the last two decades, Newton da Costa and his collaborators have developed some frameworks to help the interpretation of science. Two of them are particularly noteworthy: partial structures and quasi-truth (that provide a way of accommodating the openness and partiality of scientific activity), and quasi-set theory (that allows one to take seriously the idea, put forward by several physicists, that we can't meaningfully apply the notion of identity to quantum particles). In this paper I explore the interconnection between these two frameworks. After reviewing the extant formulations of quasi-truth and quasi-set theory, I suggest a way of combining them, advancing a formulation of quasi-truth in quasi-set theory. In this way, a good sense can be made of the idea that quantum mechanics, if not true, is at least quasi-true. I then explore an application of this combined framework, arguing that it provides a conceptual setting appropriate to overcome two (philosophical) difficulties in van Fraassen's modal interpretation of quantum mechanics.

In this paper, 1 examine the role of the delta function in Dirac’s formulation of quantum mechanics (QM), and I discuss, more generally, the role of mathematics in theory construction. It has been argued that mathematical theories play an indispensable role in physics, particularly in QM [Colyvan, M. (2001). The inrlispensability of mathematics. Oxford University Press: Oxford]. As I argue here, at least in the case of the delta function, Dirac was very clear about its rlispensability. I first discuss the significance of the delta function in Dirac’s work, and explore the strategy that he devised to overcome its use. l then argue that even if mathematical theories turned out to be indispensable, this wouidn’t justify the commitment to the existence of mathematical entities. In fact, even in successful uses of mathematics, such as in Dirac’s discovery of antimatter, there’s no need to believe in the existence of the corresponding mathematical entities. An interesting picture about the application of mathematics emerges from a careful examination of Dirac’s work.

We present an axiomatic framework for semantics that can be applied to natural and formal languages. Our main goal is to suggest a very simple mathematical model that describes fundamental cognitive aspects of the human brain and that can still be applied to artificial intelligence. One of our main results is a theorem that allows us to infer syntactical properties of a language out of its corresponding semantics. The role of pragmatics in semantics in our mathematical framework is also discussed.

As an illustration of the complexities involved in connecting physics and molecular biology at the nanoscale, in this paper I discuss two case studies from nanoscience. The first examines the use of a biological structure to build nanostructures in a controlled way. The second discusses the attempt to build a single molecular wire, and then decide whether such a wire is indeed conducting. After presenting the central features of each case study, I examine the role played in them by microscopic imaging, the different styles of reasoning involved, and the various theoretical, methodological, and axiological differences. I conclude by arguing that, except for the probe microscopes that are used, there is very little in common between the two cases. At the nanoscale, physics and molecular biology seem to meet in a non-unified way.

The issue of what consequences to draw from the existence of non-classical logical systems has been the subject of an interesting debate across a diversity of fields. In this paper the matter of alternative logics is considered with reference to a specific belief system and its propositions :the Azande are said to maintain beliefs about witchcraft which, when expressed propositionally, appear to be inconsistent. When the Azande have been presented with such inconsistencies, they either fail to see them as such or else accept them as non-problematical. Is our knowledge of logical truths a relative and culturally determined phenomenon, or is there some (transcendent) criterion that allows us to adjudicate between alternative logical systems? The authors propose an approach for resolving disputes about the status of Azande reasoning which assumes a paraconsistent framework, thus providing a new perspective on this debate

In this paper we argue that there are no categories of being⎯at least not in the robust metaphysical sense of something fundamental. Central arguments that metaphysicians provide in support of fundamental categories, such as indispensability and theoretical utility arguments, are not adequate to guarantee their existence. We illustrate this point by examining Jonathan Lowe’s [2006] four-category ontology, and indicating its shortcomings. In contrast, we offer an alternative, no-category ontology, which dispenses with any fundamental categories of being, and provides a deflationary understanding of any categorization in terms of concepts. Concepts, we insist, as opposed to fundamental ontological categories, can always be revised, refined, and recast. A distinctive deflationary, no-category ontology then emerges.

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 (with infinitesimals). Our approach was inspired by the work of Whitehead (1919), though our philosophical stance is completely different from his. The structures obtained are idealized constructs underlying extant, physical space-time.

Can a constructive empiricist make sense of scientific representation? Usually, a scientific model is an abstract entity, and scientific representation is conceptualized as an intentional relation between scientific models and certain aspects of the world. On this conception, since both the models and the representation relation are abstract, a constructive empiricist, who is not committed to the existence of abstract entities, would be unable to invoke these notions to make sense of scientific representation. In this paper, instead of understanding representation as a relation between abstract entities, I focus on the activity of representing, and argue that it provides a way of making sense of representation within the boundaries of empiricism. The activity of representing doesn’t deal with abstract entities, but with concrete ones, such as inscriptions, templates, and blueprints. In the end, by examining the practice of representing, rather than an artificially reified product—the representation—the constructive empiricist has the resources to make sense of scientific representation in empiricist terms.

Second-order logic has a number of attractive features, in particular the strong expressive resources it offers, and the possibility of articulating categorical mathematical theories (such as arithmetic and analysis). But it also has its costs. Five major charges have been launched against second-order logic: (1) It is not axiomatizable; as opposed to first-order logic, it is inherently incomplete. (2) It also has several semantics, and there is no criterion to choose between them (Putnam, J Symbol Logic 45:464–482, 1980 ). Therefore, it is not clear how this logic should be interpreted. (3) Second-order logic also has strong ontological commitments: (a) it is ontologically committed to classes (Resnik, J Phil 85:75–87, 1988 ), and (b) according to Quine (Philosophy of logic, Prentice-Hall: Englewood Cliffs, 1970 ), it is nothing more than “set theory in sheep’s clothing”. (4) It is also not better than its first-order counterpart, in the following sense: if first-order logic does not characterize adequately mathematical systems, given the existence of non - isomorphic first-order interpretations, second-order logic does not characterize them either, given the existence of different interpretations of second-order theories (Melia, Analysis 55:127–134, 1995 ). (5) Finally, as opposed to what is claimed by defenders of second-order logic [such as Shapiro (J Symbol Logic 50:714–742, 1985 )], this logic does not solve the problem of referential access to mathematical objects (Azzouni, Metaphysical myths, mathematical practice: the logic and epistemology of the exact sciences, Cambridge University Press, Cambridge, 1994 ). In this paper, I argue that the second-order theorist can solve each of these difficulties. As a result, second-order logic provides the benefits of a rich framework without the associated costs.

In this first paper of a series of works on the foundations of science, we examine the significance of logical and mathematical frameworks used in foundational studies. In particular, we emphasize the distinction between the order of a language and the order of a structure to prevent confusing models of scientific theories with first-order structures, and which are studied in standard model theory. All of us are, of course, bound to make abuses of language even in putatively precise contexts. This is not a problem—in fact, it is part of scientific and philosophical practice. But it is important to be sensitive to the dierent uses that structure, model, and language have. In this paper, we examine these topics in the context of classical logic; only in the last section we touch upon briefly on non-classical ones.

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.

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.