Otávio Bueno

From an empiricist perspective, we argue that physical modality best accounts for the modal status of laws (understood in empiricist terms as suitable empirical regularities). Nomic modality need not be construed in terms of metaphysical modal features. Rather, we submit, we live in a world full of physical possibilities and necessities that are investigated by such processes as gathering evidence, improving inferential practices, and undertaking theory and model construction. Empiricists can safely take distance from Humean accounts of nomic modality, hence maintaining that laws are not restricted to yielding summaries of actual phenomena but inform us about the range of physical possibilities and necessities in their domains. None of this requires the introduction of a metaphysical basis for nomic modality.

In this paper, I develop a new defense of logicism: one that combines logicism and nominalism. First, I defend the logicist approach from recent criticisms; in particular from the charge that a crucial principle in the logicist reconstruction of arithmetic, Hume's Principle, is not analytic. In order to do that, I argue, it is crucial to understand the overall logicist approach as a nominalist view. I then indicate a way of extending the nominalist logicist approach beyond arithmetic. Finally, I argue that nominalist can use the resulting approach to provide a nominalization strategy for mathematics. In this way, mathematical structures can be introduced without ontological costs. And so, if this proposal is correct, we can say that ultimately all the nominalist needs is logic (and, rather loosely, all the logicist needs is nominalism).

Quine has famously put forward the indispensability argument to force belief in the existence of mathematical objects (such as classes) due to their indispensability to our best theories of the world (Quine 1960). Quine has also advocated the indeterminacy of reference argument, according to which reference is dramatically indeterminate: given a language, there’s no unique reference relation for that language (see Quine 1969a). In this paper, I argue that these two arguments are in conflict with each other. Whereas the indispensability argument supports realism about mathematics, the indeterminacy of reference argument, when applied to mathematics, provides a powerful strategy in support of mathematical anti-realism. I conclude the paper by indicating why the indeterminacy of reference phenomenon should be preferred over the considerations regarding indispensability. In the end, even the Quinean shouldn’t be a realist (platonist) about mathematics.

Impressively successful at the empirical level and open to multiple interpretations at the theoretical domain, quantum theory provides a rich source of examples of underdetermination. A more promising line of support for realism about quantum mechanics emerges from experimental physics. Some significant experiments have been conducted that prima facie seem to lead to very natural realist readings. The author first considers the challenge these experiments seem to raise to current forms of empiricism, particularly constructive empiricism. Three arguments are examined: the experiments challenge an important form of underdetermination employed by empiricists; the nature of quantum particles, in particular their identity and individuality, seems unproblematic in the context of these experiments, and the experiments provide an unprecedented form of access to the quantum particles involved. The discussion of an empiricist response to these three arguments is shaped by questions regarding the identity, individuality, and individuation of quantum particles in experimental contexts.

This chapter considers the impact of the Fitch paradox on particular epistemological views about mathematics. It assumes therefore, for the sake of argument, that the reasoning leading to Fitch's paradox is valid (which is indeed the case given the logical assumptions made). Having this focus provides a specific context to assess the nature and limitation of the paradox, while also showing the paradox's significance for current debates in the philosophy of mathematics. The chapter examines two versions of Platonism (standard and full-blooded Platonism) and two versions of nominalism (mathematical fictionalism and agnostic fictionalism). It argues that, given the specific assumptions about mathematical knowledge that these views make, (FP) brings trouble for some, but not for all, of them. In particular, full-blooded Platonism and — to a certain extent — mathematical fictionalism are in trouble with the paradox, but traditional Platonism and agnostic fictionalism don't seem to be. The chapter concludes with a dilemma that this situation poses for Platonism, and the prospects it offers for nominalism.

A common response to those who question the Law of Non-Contradiction (LNC) is that it is impossible to debate such a fundamental law of logic. The reasons for this response vary, but what seems to underlie them is the thought that there is a minimal set of logical resources without which rational debate is impossible. This chapter argues that this response is misguided. First, it defends non-apriorism in logic: the view that logic is in the same epistemic boat as other scientific theories. It then offers an account of logical theory change in terms of this epistemology. The LNC is discussed in terms of this account of logical theory change, and it is shown that rational debate over this law can, and does, proceed. Finally, arguments for and against the LNC are discussed, and how and where non-a priori considerations arise in these arguments are illustrated.

Desde una perspectiva empirista, sostenemos que la modalidad física da cuenta de mejor manera del estatus modal de las leyes (entendidas en términos empiristas, como regularidades empíricas adecuadas). La modalidad nómica no tiene que ser construida en términos de propiedades modales metafísicas. En cambio, argumentaremos que vivimos en un mundo pleno de posibilidades y necesidades físicas que son investigadas por procesos tales como acumulación de evidencia, refinamiento de prácticas inferenciales y procesos de construcción de teorías y modelos. El empirismo puede, sin problemas, tomar distancia de las teorías humeanas de modalidad nómica, defendiendo que las leyes no se restringen a ofrecer resúmenes de fenómenos actuales, sino que nos informan acerca de rangos de posibilidades y necesidades físicas en sus dominios. Nada de esto requiere que introduzcamos un fundamento metafísico para la modalidad nómica.

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

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.

Deflationist approaches to abstract objects typically have a distinctive advantage over non-deflationist counterparts on the epistemological front, given the avoidance of commitment to the existence of these objects (Azzouni, Deflating existential consequence: a case for nominalism. Oxford University Press, 2004; Bueno, Stud Hist Phil Mod Phys, 36:465–490, 2005) or given the light-weight character that is assigned to them, as thin objects (Linnebo, Thin objects: an abstractionist account. Oxford University Press, 2018). But these approaches need to accommodate whatever use abstract objects are put to in mathematics, metaphysics, and in the sciences. Two deflationist approaches should be distinguished: (a) quantificational approaches, which explore the role of quantification in deflating ontological commitment (Azzouni, Deflating existential consequence: a case for nominalism. Oxford University Press, 2004; Bueno, Stud Hist Phil Mod Phys, 36:465–490, 2005), and (b) definitional approaches, which emphasize the role of suitable abstraction principles in the characterization of the relevant objects (Linnebo, Thin objects: an abstractionist account. Oxford University Press, 2018). In this paper, these views are contrasted and assessed. In the end, there is something to be said in favor of quantificational approaches and the thin access to objects they recommend.

Rudolf Carnap developed three central strategies to avoid metaphysics: an eliminativist/verificationist approach; an accommodating/confirmationist strategy, and reconceptualizing/framework-dependent viewpoint. After critically assessing each of them, focusing especially on the framework-dependent one, I outline an alternative that preserves the central elements of Carnap’s overall conception while still maintaining the proper dose of skepticism regarding metaphysics that is crucial to empiricism. A more defensible empiricist stance then results.

What has been called ‘the unreasonable effectiveness of mathematics’ sets a challenge for philosophers. Some have responded to that challenge by arguing that mathematics is essentially anthropocentric in character whereas others have pointed to the range of structures that mathematics offers. Here a middle way is offered that focuses on the moves that have to be made in both the mathematics and the relevant physics in order to bring the two into appropriate relation. This relation can be captured via the inferential conception of the applicability of mathematics which is formulated in terms of immersion inference and interpretation. In particular the roles of idealizations and of surplus structure in science and mathematics respectively are brought to the fore and captured via an approach to models and theories that emphasizes the partiality of the available information: the partial structures approach. The discussion as a whole is grounded in a number of case studies drawn from the history of quantum physics and extended to contest recent claims that the explanatory role of certain mathematical structures in scientific practice supports a realist attitude towards them. The overall conclusion is that the effectiveness of mathematics does not seem unreasonable at all once close attention is paid to how it is actually applied in practice.

Amie Thomasson’s conception of the ontology of fiction in terms of abstract artifacts provides an illuminating account of the existence and persistence conditions for fictional objects and a sophisticated defense of realism about fictional characters. The conception was primarily articulated with a focus on literature, where perceptual experiences play a relatively small role. How well does the conception fare when we consider artforms that rely crucially on perceptual experiences and perceptual imagination, such as film? In this paper, I examine this issue and raise some concerns about the abstract artifact conception when it is extended to this realm. I then articulate an empiricist alternative that does not depend on realism about fictional characters, and which is able to make sense of both films and literary practices in a principled way.

Diagrams are hybrid entities, which incorporate both linguistic and pictorial elements, and are crucial to any account of scientific and mathematical reasoning. Hence, they offer a rich source of examples to examine the relation between model-theoretic considerations (central to a model-based approach) and linguistic features (crucial to a language-based view of scientific and mathematical reasoning). Diagrams also play different roles in different fields. In scientific practice, their role tends not to be evidential in nature, and includes: (i) highlighting relevant relations in a micrograph (by making salient certain bits of information); (ii) sketching the plan for an experiment; and (iii) expressing expected visually salient information about the outcome of an experiment. None of these traits are evidential; rather they are all pragmatic. In contrast, in mathematical practice, diagrams are used as (i) heuristic tools in proof construction (including dynamic diagrams involved in computer visualization); (ii) notational devices; and (iii) full-blown proof procedures (Giaquinto 2005; and Brown in Philosophy of mathematics. Routledge, New York, 2008). Some of these traits are evidential. After assessing these different roles, I explain why diagrams are used in the way they are in these two fields. The result leads to an account of different styles of scientific reasoning within a broadly model-based conception.

A key component of Lorenzo Magnani’s approach to the philosophy of geometry is the use of abduction, especially manipulative abduction, which not only involves thinking through doing but also the integration of theoretical practices. In this paper, I compare manipulative abduction with a different way of implementing mathematical heuristics: a deductivist strategy that highlights the significance of concept refinement and structure preservation. The latter approach interestingly complements the former.

On the heuristic view, mathematics is a fallible, anti-reductionist problem-solving activity that employs the analytic method (i.e., the generation of hypotheses for problem-solving via non-deductive inferences and the assessment of hypotheses’ plausibility by comparing them with experience). Central to Carlo Cellucci’s articulation of the view is the use of non-deductive inferences in the formulation of new hypotheses. Cellucci criticizes Lakatos’s version of the view for being unable to provide a proper heuristics for mathematics. In this paper, I respond to Cellucci’s challenge, advancing a Lakatosian, deductivist version of the proposal that can accommodate the nature and role of heuristics in mathematical practice. A form of pluralism about the heuristic view then emerges.

Central to Imre Lakatos’s philosophy is the methodological continuity between mathematics and the empirical sciences, especially the central role played by counterexamples (refutations) in both fields. It is unclear, however, to what extent the original framework conceived in Proofs and Refutations can be extended beyond the particular case study that motivated it in the first place. In this paper, I argue that, with suitable adjustments, the framework not only illuminates important aspects of the discovery of new logics (such as intuitionistic, paraconsistent and quantum logics), but it also can have a far more extensive role than Lakatos himself seems to have acknowledged in the empirical sciences as well.