This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. 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 book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable propositions, (...) and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter 2 argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal cognitivism and modal expressivism. I also develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic epistemic two-dimensional hyperintensional semantics. Chapter 3 provides an abstraction principle for epistemic (hyper-)intensions. Chapter 4 advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter 5 applies the fixed points of the modal μ-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal μ-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal μ-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter 6 advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter 7 provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters 2 and 4 are availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. -/- Chapters 8-12 provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter 8 examines the interaction between my hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of Ω-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter 9 examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest, too, that observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter 10 examines the philosophical significance of hyperintensional Ω-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter 11 provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter 12 avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between the interpretational and objective modalities and truthmakers thereof. This yields the first hyperintensional category theory in the literature. I invent a new mathematical trick in which first order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals while bypassing the category of Set in category theory. Chapter 13 examines modal responses to the alethic paradoxes. Chapter 14 examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)

This paper is an exposition and evaluation of the Agustín Rayo's views about the epistemology and metaphysics of mathematics, as they are presented in his book The Construction of Logical Space.

This essay aims to redress the contention that epistemic possibility cannot be a guide to the principles of modal metaphysics. I introduce a novel epistemic two-dimensional truthmaker semantics. I argue that the interaction between the two-dimensional framework and the mereological parthood relation, which is super-rigid, enables epistemic possibilities and truthmakers with regard to parthood to be a guide to its metaphysical profile. I specify, further, a two-dimensional formula encoding the relation between the epistemic possibility and verification of essential properties obtaining (...) and their metaphysical possibility or verification. I then generalize the approach to haecceitistic properties, and examine the Julius Caesar problem as a test case. (shrink)

This volume will be of particular interest to researchers working in the history, and in the philosophy, of logic and mathematics, and more generally, to ...

This book offers an historically-informed critical assessment of Dummett's account of abstract objects, examining in detail some of the Fregean presuppositions whilst also engaging with recent work on the problem of abstract entities.

This volume covers a wide range of topics in the most recent debates in the philosophy of mathematics, and is dedicated to how semantic, epistemological, ontological and logical issues interact in the attempt to give a satisfactory picture of mathematical knowledge. The essays collected here explore the semantic and epistemic problems raised by different kinds of mathematical objects, by their characterization in terms of axiomatic theories, and by the objectivity of both pure and applied mathematics. They investigate controversial aspects of (...) contemporary theories such as neo-logicist abstractionism, structuralism, or multiversism about sets, by discussing different conceptions of mathematical realism and rival relativistic views on the mathematical universe. They consider fundamental philosophical notions such as set, cardinal number, truth, ground, finiteness and infinity, examining how their informal conceptions can best be captured in formal theories. The philosophy of mathematics is an extremely lively field of inquiry, with extensive reaches in disciplines such as logic and philosophy of logic, semantics, ontology, epistemology, cognitive sciences, as well as history and philosophy of mathematics and science. By bringing together well-known scholars and younger researchers, the essays in this collection – prompted by the meetings of the Italian Network for the Philosophy of Mathematics (FilMat) – show how much valuable research is currently being pursued in this area, and how many roads ahead are still open for promising solutions to long-standing philosophical concerns. Promoted by the Italian Network for the Philosophy of Mathematics – FilMat. (shrink)

ABSTRACT Is it possible to effect singular reference to mathematical objects in the abstractionist framework? I will argue that even if mathematical expressions pass the relevant syntactic and inferential tests to qualify as singular terms, that does not mean that their semantic function is to refer to a particular object. I will defend two arguments leading to this claim: the permutation argument for the referential indeterminacy of mathematical terms, and the argument from the semantic idleness of the terms introduced by (...) abstraction principles. (shrink)

This paper aims to provide hyperintensional foundations for mathematical platonism. I examine Hale and Wright's (2009) objections to the merits and need, in the defense of mathematical platonism and its epistemology, of the thesis of Necessitism. In response to Hale and Wright's objections to the role of epistemic and metaphysical modalities in providing justification for both the truth of abstraction principles and the success of mathematical predicate reference, I examine the Necessitist commitments of the abundant conception of properties endorsed by (...) Hale and Wright and examined in Hale (2013), and demonstrate how a two-dimensional approach to the epistemology of mathematics is consistent with Hale and Wright's notion of there being non-evidential epistemic entitlement rationally to trust that abstraction principles are true. A choice point that I flag is that between availing of intensional or hyperintensional semantics. The hyperintensional semantic approach that I advance is a topic-sensitive epistemic two-dimensional truthmaker semantics. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest that observational type theory can be applied to first-order abstraction principles in order to make abstraction principles recursively enumerable. Epistemic and metaphysical states and possibilities may thus be shown to play a constitutive role in vindicating the reality of mathematical objects and truth, and in providing a conceivability-based route to the truth of abstraction principles as well as other axioms and propositions in mathematics. (shrink)

Distinctively mathematical explanations (DMEs) explain natural phenomena primarily by appeal to mathematical facts. One important question is whether there can be an ontic account of DME. An ontic account of DME would treat the explananda and explanantia of DMEs as ontic structures and the explanatory relation between them as an ontic relation (e.g., Pincock 2015, Povich 2021). Here I present a conventionalist account of DME, defend it against objections, and argue that it should be considered ontic. Notably, if indeed it (...) is ontic, the conventionalist account seems to avoid a convincing objection to other ontic accounts (Kuorikoski 2021). (shrink)

I argue that absolutism, the view that absolutely unrestricted quantification is possible, is to blame for both the paradoxes that arise in naive set theory and variants of these paradoxes that arise in plural logic and in semantics. The solution is restrictivism, the view that absolutely unrestricted quantification is not possible. -/- It is generally thought that absolutism is true and that restrictivism is not only false, but inexpressible. As a result, the paradoxes are blamed, not on illicit quantification, but (...) on the logical conception of set which motivates naive set theory. The accepted solution is to replace this with the iterative conception of set. -/- I show that this picture is doubly mistaken. After a close examination of the paradoxes in chapters 2--3, I argue in chapters 4 and 5 that it is possible to rescue naive set theory by restricting quantification over sets and that the resulting restrictivist set theory is expressible. In chapters 6 and 7, I argue that it is the iterative conception of set and the thesis of absolutism that should be rejected. (shrink)

This essay is an attempt to explain Kantian transcendental idealism to contemporary metaphysicians and make clear its relevance to contemporary debates in what is now called ‘meta-metaphysics.’ It is not primarily an exegetical essay, but an attempt to translate some Kantian ideas into a contemporary idiom.

We face reality presented with the data of conscious experience and nothing else. The project of early modern philosophy was to build a complete theory of the world from this starting point, with no cheating. Crucial to this starting point is the data of conscious sensory experience – sense data. Attempts to avoid this project often argue that the very idea of sense data is confused. But the sense-data way of talking, the sense-data language, can be freed from every blemish (...) using ideas from contemporary metaontology. We can adopt a sense-data framework that vindicates the traditional claims of sense-data theories and leads to plausible theories of perception and color. We can, we should, and in a sense, we must. Yet when we do, we face the traditional problem of external world skepticism, head-on. The real challenge of skepticism is forced upon us; it cannot honestly be avoided using externalist tricks, burden of proof shifting, or other razzle dazzle. But the challenge can be met: we are rational to posit the external world as the best explanation of the many synchronic and diachronic patterns over our sense-data. This paper argues for all of these points and ends with a plea for analytic empiricism – a traditional sense-data version of empiricism that uses all of the tools of analytic philosophy while avoiding the more questionable doctrines of the Vienna Circle (phenomenalism, verificationism). Analytic empiricism is our last best hope for completing the great philosophical project of early modern philosophy. (shrink)

The role of possible worlds in philosophy is hard to overestimate. Nevertheless, their nature and existence is very controversial. This is particularly serious, since their standard applications depend on there being sufficiently many of them. The paper develops an account of possible worlds on which it is particularly easy to believe in their existence: an account of possible worlds as pleonastic entities. Pleonastic entities are entities whose existence can be validly inferred from statements that neither refer to nor quantify over (...) them as a matter of conceptual necessity. Definitions are proposed that ensure that this is the case for possible worlds. (shrink)

Abstraction principles and grounding can be combined in a natural way Modality: metaphysics, logic, and epistemology, Oxford University Press, Oxford, pp 109–136, 2010; Schwartzkopff in Grazer philosophische studien 82:353–373, 2011). However, some ground-theoretic abstraction principles entail that there are circles of partial ground :775–801, 2017). I call this problem auto-abstraction. In this paper I sketch a solution. Sections 1 and 2 are introductory. In Sect. 3 I start comparing different solutions to the problem. In Sect. 4 I contend that the (...) thesis that the right-hand side of an abstraction principle is prior to its left-hand side motivates an independence constraint, and that this constraint leads to predicative restrictions on the acceptable instances of ground-theoretic abstraction principles. In Sect. 5 I argue that auto-abstraction is acceptable unless the left-hand side is essentially grounded by the right-hand side. In Sect. 6 I highlight several parallelisms between auto-abstraction and the puzzles of ground. I finally compare my solution with the strategies listed in Sect. 3. (shrink)

Some have argued for a division of epistemic labor in which mathematicians supply truths and philosophers supply their necessity. We argue that this is wrong: mathematics is committed to its own necessity. Counterfactuals play a starring role.

ABSTRACT Four philosophical concerns about higher-order logic in general and the specific demands placed on it by the neo-logicist project are distinguished. The paper critically reviews recent responses to these concerns by, respectively, the late Bob Hale, Richard Kimberly Heck, and myself. It is argued that these score some successes. The main aim of the paper, however, is to argue that the most serious objection to the applications of higher-order logic required by the neo-logicist project has not been properly understood. (...) The paper concludes by outlining a strategy, prefigured in recent work of Øystein Linnebo, for meeting this objection. (shrink)

Are there different sizes of infinity? That is, are there infinite sets of different sizes? This is one of the most natural questions that one can ask about the infinite. But it is of course generally taken to be settled by mathematical results, such as Cantor’s theorem, to the effect that there are infinite sets without bijections between them. These results settle the question, given an almost universally accepted principle relating size to the existence of functions. The principle is: for (...) any sets A and B, if A is the same size as B, then there is a bijection from A to B. The aim of the paper, however, is to argue that this question is in fact wide open: to argue that we are not in a position to know the answer, because we are not in one to know the principle. The aim, that is, is to argue that for all we know there is only one size of infinity. (shrink)

The paper is concerned with the bad company problem as an instance of a more general difficulty in the philosophy of mathematics. The paper focuses on the prospects of stability as a necessary condition on acceptability. However, the conclusion of the paper is largely negative. As a solution to the bad company problem, stability would undermine the prospects of a neo-Fregean foundation for set theory, and, as a solution to the more general difficulty, it would impose an unreasonable constraint on (...) mathematical practice. (shrink)

Fitch's basic logic is an untyped illative combinatory logic with unrestricted principles of abstraction effecting a type collapse between properties (or concepts) and individual elements of an abstract syntax. Fitch does not work axiomatically and the abstraction operation is not a primitive feature of the inductive clauses defining the logic. Fitch's proof that basic logic has unlimited abstraction is not clear and his proof contains a number of errors that have so far gone undetected. This paper corrects these errors and (...) presents a reasonably intuitive proof that Fitch's system K supports an implicit abstraction operation. Some general remarks on the philosophical significance of basic logic, especially with respect to neo-logicism, are offered, and the paper concludes that basic logic models a highly intensional form of logicism. (shrink)

Richard Kimberly Heck and Paolo Mancosu have claimed that the possibility of non-Cantorian assignments of cardinalities to infinite concepts shows that Hume's Principle (HP) is not implicit in the concept of cardinal number. Neologicism would therefore be threatened by the ‘good company' HP is kept by such alternative assignments. In his review of Mancosu's book, Bob Hale argues, however, that ‘getting different numerosities for different countable infinite collections depends on taking the groups in a certain order – but it is (...) of the essence of cardinal numbers that the cardinal size of a collection does not depend upon how its members are ordered'. This paper's goal is to implement Hale's response to the Good Company problem by producing a Cantorian argument for HP. In Section 2, we present the Heck-Mancosu argument against neologicism. In Section 3, we discuss Hale's defence of Hume's Principle. In Section 4, we discuss Cantor's abstractionist definitions of number. In Section 5, we argue that good abstraction must comply with what we call ‘Gödel’s Minimal Account of Abstraction’ (GMAA). We finally show (Sections 5 and 6) that non-Cantorian theories of cardinality fail to satisfy GMAA. (shrink)

In Thin Objects: An Abstractionist Account, Linnebo offers what he describes as a “simple and definitive” solution to the bad company problem facing abstractionist accounts of mathematics. “Bad” abstraction principles can be rendered “good” by taking abstraction to have a predicative character. But the resulting predicative axioms are too weak to recover substantial portions of mathematics. Linnebo pursues two quite different strategies to overcome this weakness in the case of set theory and arithmetic. I argue that neither infinitely iterated abstraction (...) nor abstraction on possible specifications fully resolves the bad company problem. (shrink)

The Caesar problem arises for abstractionist views, which seek to secure reference for terms such as ‘the number of Xs’ or #X by stipulating the content of ‘unmixed’ identity contexts like ‘#X = #Y’. Frege objects that this stipulation says nothing about ‘mixed’ contexts such as ‘# X = Julius Caesar’. This article defends a neglected response to the Caesar problem: the content of mixed contexts is just as open to stipulation as that of unmixed contexts.

'Philosophy arises through misconceptions of grammar', said Wittgenstein. Few people have believed him, and probably none, therefore, working in the area of the philosophy of mathematics. Yet his assertion is most evidently the case in the philosophy of Set Theory, as this paper demonstrates (see also Rodych 2000). The motivation for twentieth century Set Theory has rested on the belief that everything in Mathematics can be defined in terms of sets [Maddy 1994: 4]. But not only are there notable items (...) which cannot be so defined, including numbers and mereological sums, the very notion of a set, as formalized within this tradition, is based on a series of grammatical confusions. (shrink)

Some authors have claimed that ante rem structuralism has problems with structures that have indiscernible places. In response, I argue that there is no requirement that mathematical objects be individuated in a non-trivial way. Metaphysical principles and intuitions to the contrary do not stand up to ordinary mathematical practice, which presupposes an identity relation that, in a sense, cannot be defined. In complex analysis, the two square roots of –1 are indiscernible: anything true of one of them is true of (...) the other. I suggest that ‘i’ functions like a parameter in natural deduction systems. I gave an early version of this paper at a workshop on structuralism in mathematics and science, held in the Autumn of 2006, at Bristol University. Thanks to the organizers, particularly Hannes Leitgeb, James Ladyman, and Øystein Linnebo, to my commentator Richard Pettigrew, and to the audience there. The paper also benefited considerably from a preliminary session at the Arché Research Centre at the University of St Andrews. I am indebted to my colleagues Craige Roberts, for help with the linguistics literature, and Ben Caplan and Gabriel Uzquiano, for help with the metaphysics. Thanks also to Hannes Leitgeb and Jeffrey Ketland for reading an earlier version of the manuscript and making helpful suggestions. I also benefited from conversations with Richard Heck, John Mayberry, Kevin Scharp, and Jason Stanley. CiteULike Connotea Del.icio.us What's this? (shrink)

Foundational projects disagree on whether pure and applied mathematics should be explained together. Proponents of unified accounts like neologicists defend Frege’s Constraint (FC), a principle demanding that an explanation of applicability be provided by mathematical definitions. I reconsider the philosophical import of FC, arguing that usual conceptions are biased by ontological assumptions. I explore more reasonable weaker variants — Moderate and Modest FC — arguing against common opinion that ante rem structuralism (and other) views can meet them. I dispel doubts (...) that such constraints are ‘toothless’, showing they both assuage Frege’s original concerns and accommodate neo-logicist intents by dismissing ‘arrogant’ definitions. (shrink)

A thriving literature has developed over logical and mathematical pluralism – i.e. the views that several rival logical and mathematical theories can be equally correct. These have unfortunately grown separate; instead, they both could gain a great deal by a closer interaction. Our aim is thus to present some novel forms of abstractionist mathematical pluralism which can be modeled on parallel ways of substantiating logical pluralism (also in connection with logical anti-exceptionalism). To do this, we start by discussing the Good (...) Company Problem for neo-logicists recently raised by Paolo Mancosu (2016), concerning the existence of rival abstractive definitions of cardinal number which are nonetheless equally able to reconstruct Peano Arithmetic. We survey Mancosu’s envisaged possible replies to this predicament, and suggest as a further path the adoption of some form of mathematical pluralism concerning abstraction principles. We then explore three possible ways of substantiating such pluralism—Conceptual Pluralism, Domain Pluralism, Pluralism about Criteria—showing how each of them can be related to analogous proposals in the philosophy of logic. We conclude by considering advantages, concerns, and theoretical ramifications for these varieties of mathematical pluralism. (shrink)

The aim of this paper is to explore the possibility that, at least, some metaphysical debates are ‘metalinguistic negotiations’. I will take the dispute between the dominant approaches of realism and the anti-realism ones about the ontological status of scientific models as a case-study. I will argue that such a debate may be better understood as a disagreement, at bottom normatively, motivated, insofar as a normative and non-factual question may be involved in it: how the relevant piece of language ought (...) to be used. Even though I will generally assess the prospects for a broadly deflationist approach, I shall outline a sense in which the dispute can be recast as ‘minimally substantive’. (shrink)

Penelope Maddy advances a purportedly naturalistic account of mathematical methodology which might be taken to answer the question 'What justifies axioms of set theory?' I argue that her account fails both to adequately answer this question and to be naturalistic. Further, the way in which it fails to answer the question deprives it of an analog to one of the chief attractions of naturalism. Naturalism is attractive to naturalists and nonnaturalists alike because it explains the reliability of scientific practice. Maddy's (...) account, on the other hand, appears to be unable to similarly explain the reliability of mathematical practice without violating one of its central tenets. (shrink)

Frege regarded Hume's Principle as insufficient for a logicist account of arithmetic, as it does not identify the numbers; it does not tell us which objects the numbers are. His solution, generally regarded as a failure, was to propose certain sets as the referents of numerical terms. I suggest instead that numbers are properties of pluralities, where these properties are treated as objects. Given this identification, the truth-conditions of the statements of arithmetic can be obtained from logical principles with the (...) help of definitions, just as the logicist thesis maintains. (shrink)

This paper has a two-fold objective: to provide a balanced, multi-faceted account of the origins of logicism; to rehabilitate Richard Dedekind as a main logicist. Logicism should be seen as more deeply rooted in the development of modern mathematics than typically assumed, and this becomes evident by reconsidering Dedekind's writings in relation to Frege's. Especially in its Dedekindian and Fregean versions, logicism constitutes the culmination of the rise of ?pure mathematics? in the nineteenth century; and this rise brought with it (...) an inter-weaving of methodological and epistemological considerations. The latter aspect illustrates how philosophical concerns can grow out of mathematical practice, as opposed to being imposed on it from outside. It also sheds new light on the legacy and the lasting significance of logicism today. (shrink)

Gottlob Frege is often called a "platonist". In connection with his philosophy we can talk about platonism concerning three kinds of entities: numbers, or logical objects more generally; concepts, or functions more generally; thoughts, or senses more generally. I will only be concerned about the first of these three kinds here, in particular about the natural numbers. I will also focus mostly on Frege's corresponding remarks in The Foundations of Arithmetic (1884), supplemented by a few asides on Basic Laws of (...) Arithmetic (1893/1903) and "Thoughts" (1918). My goal is to clarify in which sense the Frege of Foundations and Basic Laws is a platonist concerning the natural numbers.1.. (shrink)

This essay is a study of ontological commitment, focused on the special case of arithmetical discourse. It tries to get clear about what would be involved in a defense of the claim that arithmetical assertions are ontologically innocent and about why ontological innocence matters. The essay proceeds by questioning traditional assumptions about the connection between the objects that are used to specify the truth-conditions of a sentence, on the one hand, and the objects whose existence is required in order for (...) the truth-conditions thereby specified to be satisfied, on the other. This allows one to set forth an assignment of truth-conditions to arithmetical sentences whereby nothing is required of the world in order for the truth-conditions of a truth of pure arithmetic to be satisfied. The essay then argues that such an assignment can be used to account for the a priori knowability of certain arithmetical truths. (shrink)

In this paper, I argue that Amie Thomasson’s Easy Ontology rests on a vicious circularity that is highly damaging. Easy Ontology invokes the idea of application conditions that give rise to analytic entailments. Such entailments can be used to answer ontological questions easily. I argue that the application conditions for basic terms are only circularly specifiable showing that Thomasson misses her self-set goal of preventing such a circularity. Using this circularity, I go on to show that Easy Ontology as a (...) whole collapses. (shrink)

The rather unrestrained use of second-order logic in the neo-logicist program is critically examined. It is argued in some detail that it brings with it genuine set-theoretical existence assumptions and that the mathematical power that Hume’s Principle seems to provide, in the derivation of Frege’s Theorem, comes largely from the ‘logic’ assumed rather than from Hume’s Principle. It is shown that Hume’s Principle is in reality not stronger than the very weak Robinson Arithmetic Q. Consequently, only a few rudimentary facts (...) of arithmetic are logically derivable from Hume’s Principle. And that hardly counts as a vindication of logicism. (shrink)

I will contrast two conceptions of the nature of mathematical objects: the conception of mathematical objects as preconceived objects, and heavy duty platonism. I will argue that friends of the indispensability argument are committed to some metaphysical theses and that one promising way to motivate such theses is to adopt heavy duty platonism. On the other hand, combining the indispensability argument with the conception of mathematical objects as preconceived objects yields an unstable position. The conclusion is that the metaphysical commitments (...) of the indispensability argument should be carefully scrutinized. (shrink)

A central dispute in understanding Frege’s philosophy concerns how the sense of a complex expression relates to the senses of its component expressions. According to one reading, the sense of a complex expression is a whole built from the senses of the component expressions. On this interpretation, Frege is an early proponent of structured propositions. A rival reading says that senses compose by functional application: the sense of a complex expression is the value of the function denoted by its functional (...) component for the arguments denoted by its remaining components. I argue that two non-negotiable Fregean theses entail that senses compose by functional application. One thesis is that referents compose by functional application. The other thesis is that an expression in an indirect context refers to its customary sense. (shrink)

It is commonplace for philosophers to distinguish mere truths from truths that perspicuously represent the world's structure. According to a popular view, the perspicuous truths are supposed to be metaphysically revelatory and to play an important role in the accounts of law-hood, confirmation, and linguistic interpretation. Yet, there is no consensus about how to characterize this distinction. I examine strategies developed by Lewis and by Sider in his Writing the Book of the World which purport to explain this distinction in (...) terms of vocabulary: the truths that represent the world perspicuously have better, joint-carving vocabulary. I argue that the distinction between a perspicuous and mere truth concerns both the vocabulary of the sentence and its grammar. I then show that the collective motivations for distinguishing perspicuous from mere truths do not allow Lewis and Sider to properly impose constraints on grammar. (shrink)

Neo-Fregeans such as Bob Hale and Crispin Wright seek a foundation of mathematics based on abstraction principles. These are sentences involving a relation called the abstraction relation. It is usually assumed that abstraction relations must be equivalence relations, so reflexive, symmetric and transitive. In this article I argue that abstraction relations need not be reflexive. I furthermore give an application of non-reflexive abstraction relations to restricted abstraction principles.

Wright claims that his and Hale’s abstractionist neologicist project is primarily epistemological in aim. Its epistemological aims include establishing the possibility of a priori mathematical knowledge, and establishing the possibility of reference to abstract mathematical objects. But, as Wright acknowledges, there is a question of how neologicist epistemology applies to actual, ordinary mathematical beliefs. I take up this question, focusing on arithmetic. Following a suggestion of Hale and Wright, I consider the possibility that the neologicist account provides an idealised reconstruction (...) of how people acquire cognition of numbers or arithmetical knowledge. I conclude that it is unlikely that this account can be used to reconstruct even idealised answers to pressing questions about our actual epistemology. (shrink)

The standard argument for the existence of distinctively mathematical objects like numbers has two main premises: some mathematical claims are true, and the truth of those claims requires the existence of distinctively mathematical objects. Most nominalists deny. Those who deny typically reject Quine’s criterion of ontological commitment. I target a different assumption in a standard type of semantic argument for. Benacerraf’s semantic argument, for example, relies on the claim that two sentences, one about numbers and the other about cities, have (...) the same grammatical form. He makes this claim on the grounds that the two sentences are superficially similar. I argue that these grounds are not sufficient. Other sentences with the same superficial form appear to have different grammatical forms. I offer two plausible interpretations of Benacerraf’s number sentence that make use of plural quantification. These interpretations appear not to incur ontological commitments to distinctively mathematical objects, even assuming Quine’s criterion. Such interpretations open a new, plural strategy for the mathematical nominalist. (shrink)

We correct a misunderstanding by Hale and Wright of an objection we raised earlier to their abstractionist programme for rehabilitating logicism in the foundations of mathematics.

This article examines whether it is possible to uphold one form of deflationism towards metaphysics, ontological pluralism, whilst maintaining metaphysical realism. The focus therefore is on one prominent deflationist who fits the definition of an ontological pluralist, Eli Hirsch, and his self-ascription as a realist. The article argues that ontological pluralism is not amenable to the ascription of realism under some basic intuitions as to what a “realist” position is committed to. These basic intuitions include a commitment to more than (...) a stuff-ontology, and a view that realism carries with it more than a rejection of idealism. This issue is more than merely terminological. The ascription of realism is an important classification in order to understand what sorts of entities can be the truthmakers within a given theory. “Realism” is thus an important term to understand the nature of the entities that a given theory accepts into its ontology. (shrink)

In this short paper we consider Linnebo's thin/thick dichotomy: first, we show that it does not overlap with the very common one between abstract/concrete objects; second, on the basis of some difficulties with the distinction, we propose, as a possible way out, to move from thin/thick objects to thin/thick concepts.

Neo-Fregean logicists claim that Hume's Principle (HP) may be taken as an implicit definition of cardinal number, true simply by fiat. A longstanding problem for neo-Fregean logicism is that HP is not deductively conservative over pure axiomatic second-order logic. This seems to preclude HP from being true by fiat. In this paper, we study Richard Kimberly Heck's Two-sorted Frege Arithmetic (2FA), a variation on HP which has been thought to be deductively conservative over second-order logic. We show that it isn't. (...) In fact, 2FA is not conservative over $n$-th order logic, for all $n \geq 2$. It follows that in the usual one-sorted setting, HP is not deductively Field-conservative over second- or higher-order logic. (shrink)

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. (shrink)

Neo-Fregean logicism attempts to base mathematics on abstraction principles. Since not all abstraction principles are acceptable, the neo-Fregeans need an account of which ones are. One of the most promising accounts is in terms of the notion of stability; roughly, that an abstraction principle is acceptable just in case it is satisfiable in all domains of sufficiently large cardinality. We present two counterexamples to stability as a sufficient condition for acceptability and argue that these counterexamples can be avoided only by (...) major departures from the existing neo-Fregean programme. (shrink)

ABSTRACTThis paper presents a view of quantities as ‘adverbial’ entities of a certain kind—more specifically, determinate ways, or modes, of having length, mass, speed, and the like. In doing so, it will be argued that quantities as such should be distinguished from quantitative properties or relations, and are not universals but are particulars, although they are not objects, either. A main advantage of the adverbial view over its rivals will be found in its superior explanatory power with respect to both (...) certain fundamental principles of quantity and ordinary quantitative reasoning involving quantitative relations like three times as long as and 2 metres longer than. (shrink)