Otávio Bueno

Mathematical concepts play at least three roles in the application of mathematics: an inferential role, a representational role, and an expressive role. In this paper, I argue that, despite what has often been alleged, platonists do not fully accommodate these features of the application of mathematics. At best, platonism provides partial ways of handling the issues. I then sketch an alternative, anti-realist account of the application of mathematics, and argue that this account manages to accommodate these features of the application process. In this way, a better account of mathematical applications is, in principle, available.

We argue that standard definitions of ‘vagueness’ prejudice the question of how best to deal with the phenomenon of vagueness. In particular, the usual understanding of ‘vagueness’ in terms of borderline cases, where the latter are thought of as truth-value gaps, begs the question against the subvaluational approach. According to this latter approach, borderline cases are inconsistent (i.e., glutty not gappy). We suggest that a definition of ‘vagueness’ should be general enough to accommodate any genuine contender in the debate over how to best deal with the sorites paradox. Moreover, a definition of ‘vagueness’ must be able to accommodate the variety of forms sorites arguments can take. These include numerical, total-ordered sorites arguments, discrete versions, continuous versions, as well as others without any obvious metric structure at all. After considering the shortcomings of various definitions of ‘vagueness’, we propose a very general non-question-begging definition.

In this paper, I provide an account of scientific representation that makes sense of the notion both at the nanoscale and at the quantum level: the partial mappings account. The account offers an extension of a proposal developed by R. I. G. Hughes in terms of denotation, demonstration, and interpretation (DDI). I first argue that the DDI account needs some amendments to accommodate representation of nano and quantum phenomena. I then introduce a generalized framework with the notions of unsharp denotation, unsharp interpretation, and partial mappings. I conclude by arguing that the resulting view not only accommodates nano and quantum representations but also makes sense of representation by imaging in (scanning tunneling) microscopy.

Newton da Costa and Steven French have argued that the concept of partial truth plays an important role in our understanding of significant aspects of scientific practice: from the status of scientific theories through the understanding of inconsistency in science to the nature of induction. In this paper, I use the concept of partial truth and the associated framework of partial structures to offer a formulation of the concept of visual evidence, and I examine some of the roles that this notion plays in scientific activity.

In this paper we show that any reasoning process in which conclusions can be both fallible and corrigible can be formalized in terms of two approaches: (i) syntactically, with the use of defeasible reasoning, according to which reasoning consists in the construction and assessment of arguments for and against a given claim, and (ii) semantically, with the use of partial structures, which allow for the representation of less than conclusive information. We are particularly interested in the formalization of scientific reasoning, along the lines traced by Lakatos’ methodology of scientific research programs. We show how current debates in cosmology could be put into this framework, shedding light on a very controversial topic

Batterman raises a number of concerns for the inferential conception of the applicability of mathematics advocated by Bueno and Colyvan. Here, we distinguish the various concerns, and indicate how they can be assuaged by paying attention to the nature of the mappings involved and emphasizing the significance of interpretation in this context. We also indicate how this conception can accommodate the examples that Batterman draws upon in his critique. Our conclusion is that ‘asymptotic reasoning’ can be straightforwardly accommodated within the inferential conception. 1 Introduction2 Immersion, Inference and Partial Structures3 Idealization and Surplus Structure4 Renormalization and the Stability of Mathematical Representations5 Explanation and Eliminability6 Requirements for Explanation7 Interpretation and Idealization8 Explanation, Empirical Regularities and the Inferential Conception9 Conclusion.

As part of the development of an epistemology for mathematics, some Platonists have defended the view that we have (i) intuition that certain mathematical principles hold, and (ii) intuition of the properties of some mathematical objects. In this paper, I discuss some difficulties that this view faces to accommodate some salient features of mathematical practice. I then offer an alternative, agnostic nominalist proposal in which, despite the role played by mathematical intuition, these difficulties do not emerge.

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.

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.

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.

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.

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.

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.