I argue that a recent version of the doctrine of mathematical naturalism faces difficulties arising in connection with Wigner's old puzzle about the applicability of mathematics to natural science. I discuss the strategies to solve the puzzle and I show that they may not be available to the naturalist.

This dissertation concerns the foundations of epistemic modality. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The dissertation demonstrates how phenomenal consciousness and gradational possible-worlds models in Bayesian perceptual psychology relate to epistemic modal space. The dissertation demonstrates, then, how epistemic modality relates to the computational theory of mind; metaphysical modality; deontic modality; logical modality; the types of mathematical modality; to the (...) epistemic status of undecidable propositions and abstraction principles in the philosophy of mathematics; to the apriori-aposteriori distinction; to the modal profile of rational propositional intuition; and to the types of intention, when the latter is interpreted as a modal mental state. Examining the nature of epistemic logic itself, I develop a novel approach to conditions of self-knowledge in the setting of the modal μ-calculus, as well as novel epistemicist solutions to Curry's, the liar, and the knowability paradoxes. Solutions to previously intransigent issues concerning the first-person concept; the distinction between fundamental and derivative truths; and the unity of intention and its role in decision theory, are developed along the way. (shrink)

Crispin Wright and Bob Hale have defended the strategy of defining the natural numbers contextually against the objection which led Frege himself to reject it, namely the so-called ‘Julius Caesar problem’. To do this they have formulated principles (called sortal inclusion principles) designed to ensure that numbers are distinct from any objects, such as persons, a proper grasp of which could not be afforded by the contextual definition. We discuss whether either Hale or Wright has provided independent motivation for a (...) defensible version of the sortal inclusion principle and whether they have succeeded in showing that numbers are just what the contextual definition says they are. (shrink)

This paper is structured around the three elements of the title. Section 2 claims that (a) structures need objects and (b) scientific structuralism should focus on in re structures. Therefore, pure structuralism is undermined. Section 3 discusses whether the world has `excess structure' over the structure of appearances. The main point is that the claim that only structure can be known is false. Finally, Section 4 argues directly against ontic structural realism that it lacks the resources to accommodate causation within (...) its structuralist slogan. (shrink)

Hume famously warned us that the ‘[The] ultimate springs and principles are totally shut up from human curiosity and enquiry’. Or, again, Newton: ‘Hitherto I have not been able to discover the cause of these properties of gravity … and I frame no hypotheses.’ Aristotelian science was concerned with just such questions, the specification of occult qualities, the real essences that answer the question What is matter, etc?, the preoccupation with circular definitions such as dormative virtues, and so on. The (...) rise of modern science is usually seen as a break with the sterility of Aristotelianism, so what exactly is it that modern science does discover, if it is not the essential nature of matter, of force, of energy, of space and time? A famous answer was provided by Poincaré: ‘The true relations between these real objects are the only reality we can attain.’ This is often regarded as the manifesto of so-called structural realism, as espoused in recent years by John Worrall, for example ). In response to the arguments of Larry Laudan against convergent realism, Worrall points to the continuity in the formal relations between elements of reality expressed by mathematical equations, while the intrinsic nature of these elements of reality gets constantly revised. (shrink)

This paper sets out a new framework for discussing a long-standing problem in the philosophy of mathematics, namely the connection between the physical world and a mathematical domain when the mathematics is applied in science. I argue that considering counterfactual situations raises some interesting challenges for some approaches to applications, and consider an approach that avoids these challenges.

We see no grounds for insisting that, because the concept natural number is abstract, its foundations must be innate. It is possible to specify domain general learning processes that feed into more abstract concepts of numerical infinity. By neglecting the messiness of children's slow acquisition of arithmetical concepts, Rips et al. present an idealized, unnecessarily insular, view of number development.

According to the species of neo‐logicism advanced by Hale and Wright, mathematical knowledge is essentially logical knowledge. Their view is found to be best understood as a set of related though independent theses: (1) neo‐fregeanism—a general conception of the relation between language and reality; (2) the method of abstraction—a particular method for introducing concepts into language; (3) the scope of logic—second‐order logic is logic. The criticisms of Boolos, Dummett, Field and Quine (amongst others) of these theses are explicated and assessed. (...) The issues discussed include reductionism, rejectionism, the Julius Caesar problem, the Bad Company objections, and the charge that second‐order logic is set theory in disguise.The irresistible metaphor is that pure abstract objects […] are no more than shadows cast by the syntax of our discourse. And the aptness of the metaphor is enhanced by the reflection that shadows are, after their own fashion, real. (Crispin Wright [1992], p. 181–2)But I feel conscious that many a reader will scarcely recognise in the shadowy forms which I bring before him his numbers which all his life long have accompanied him as faithful and familiar friends; (Richard Dedekind [1963], p. 33)1 Introduction2 Logical‐historical preliminaries3 The linguistic turn4 Reductionism5 Rejectionism6 The ‘Julius Caesar’ problem7 Second‐order logic8 Bad company objections9 Conclusion. (shrink)

In §57 of the Foundations of Arithmetic, Frege famously turns to natural language to support his claim that numbers are ‘self-subsistent objects’:I have already drawn attention above to the fact th...

Is it possible to characterize the sortal essence of Fs for a sortal concept F solely in terms of a criterion of identity C for F? That is, can the question ‘What sort of thing are Fs?’ be answered by saying that Fs are essentially those things whose identity can be assessed in terms of C? This paper presents a case study supporting a negative answer to these questions by critically examining the neo-Fregean suggestion that cardinal numbers can be fully (...) characterized as those things whose identity can be assessed in terms of one-one correspondence between concepts. (shrink)

This article investigates the semantics of sentences that express numerical averages, focusing initially on cases such as 'The average American has 2.3 children'. Such sentences have been used both by linguists and philosophers to argue for a disjuncture between semantics and ontology. For example, Noam Chomsky and Norbert Hornstein have used them to provide evidence against the hypothesis that natural language semantics includes a reference relation holding between words and objects in the world, whereas metaphysicians such as Joseph Melia and (...) Stephen Yablo have used them to provide evidence that apparent singular reference need not be taken as ontologically committing. We develop a fully general and independently justified compositional semantics in which such constructions are assigned truth conditions that are not ontologically problematic, and show that our analysis is superior to all extant rivals. Our analysis provides evidence that a good semantics yields a sensible ontology. It also reveals that natural language contains genuine singular terms that refer to numbers. (shrink)

The syntactic priority thesis (henceforth SP) asserts that the truth of appropriate sentential contexts containing what are, by syntactic criteria, singular terms, is sufficient to justify the attribution of objectual reference to such terms (Wright, 1983, 24). One consequence that the neo-Fregean draws from SP is that it is through an analysis of the syntactic structure of true statements that 'ontological questions are to be understood and settled' (Wright, 1983, 25). Despite the significant literature on SP, little consideration has been (...) given to this bold meta-ontological claim.1 My concern here is accordingly not with specific applications of SP to debates in the philosophy of mathematics, but rather with the .. (shrink)

The paper scrutinizes Frege's Euclideanism - his view of arithmetic and geometry as resting on a small number of self-evident axioms from which non-self-evident theorems can be proved. Frege's notions of self-evidence and axiom are discussed in some detail. Elements in Frege's position that are in apparent tension with his Euclideanism are considered - his introduction of axioms in The Basic Laws of Arithmetic through argument, his fallibilism about mathematical understanding, and his view that understanding is closely associated with inferential (...) abilities. The resolution of the tensions indicates that Frege maintained a sophisticated and challenging form of rationalism, one relevant to current epistemology and parts of the philosophy of mathematics. (shrink)

Can concepts be acquired by testing hypotheses about these concepts? Fodor famously argued that this is not possible. Testing the correct hypothesis would require already possessing the concept. I argue that this does not generally hold for mathematical concepts. I discuss specific, empirically motivated, hypotheses for number concepts that can be tested without needing to possess the relevant number concepts. I also argue that one can test hypotheses about the identity conditions of other mathematical concepts, and then fix the application (...) conditions based on those hypotheses—under the assumption that the neo‐logicist view on abstraction principles is correct. (shrink)

La supervivencia académica de la filosofía está sometida actualmente en Argentina a una presión impuesta por criterios de productividad e impacto alentados a través del rediseño en curso del sistema científico. Si bien este proceso en desarrollo creciente atañe a las ciencias básicas en general, puede advertirse que aspectos específicos de la elaboración del conocimiento filosófico, incluyendo sus períodos de tiempo y criterios de validación tradicionales, son difícilmente compatibles con una idea de conocimiento socialmente dominante cada vez más orientada a (...) la aplicación y al retorno económico de la inversión en ciencia. En este contexto, es dudoso que la filosofía pueda apelar a justificaciones como la de la “potencial utilidad”, que tienen a su alcance otras ciencias básicas. Hay dos cuestiones candentes y conjuntas que considerar aquí: ¿qué hace de la filosofía todavía hoy una forma del saber socialmente aceptable?, y ¿por qué el estado tiene que seguir manteniendo con fondos públicos esa disciplina dentro de las instituciones académicas? En este artículo me propongo esbozar una justificación posible del papel que puede caberle a la filosofía en el contexto actual ; sugiero que esa justificación proviene de la necesidad de revisar críticamente la idea dominante del saber y apunta a reformular el proyecto del ideal humano de racionalidad. (shrink)

In this paper, we will discuss what is called the “Manifestation Challenge” to semantic realism, which was originally developed by Michael Dummett and has been further refined by Crispin Wright. According to this challenge, semantic realism has to meet the requirement that knowledge of meaning must be publically manifested in linguistic behaviour. In this regard, we will introduce and evaluate John McDowell’s response to this anti-realistic challenge, which was put forward to show that the challenge cannot undermine realism. According to (...) McDowell, knowledge of undecidable sentences’ truth-conditions can be properly manifested in our ordinary practice of asserting such sentences under certain circumstances, and any further requirement will be redundant. Wright’s further objection to McDowell’s response will be also discussed and it will be argued that this objection fails to raise any serious problem for McDowell’s response and that it is an implausible objection in general. (shrink)