A distinction between the epistemic practices in mathematics and in the empirical sciences is rehearsed to motivate the epistemic role puzzle. This is distinguished both from Benacerraf's 1973 epistemic puzzle and from sceptical arguments against our knowledge of an external world. The stipulationist position is described, a position which can address this puzzle. Methods of avoiding the stipulationist position by using pure logic to provide knowledge of mathematical abstracta are discussed and criticized.

ABSTRACT The idea that two words can be instances of the same word is a central intuition in our conception of language. This fact underlies many of the claims that we make about how we communicate, and how we understand each other. Given this, irrespective of what we think words are, it is common to think that any putative ontology of words, must be able to explain this feature of language. That is, we need to provide criteria of identity for (...) word-types which allow us to individuate words such that it can be the case that two particular word-instances are instances of the same word-type. One solution, recently further developed by Irmak, holds that words are individuated by their history. In this paper, I argue that this view either fails to account for our intuitions about word identity, or is too vague to be a plausible answer to the problem of word individuation. (shrink)

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 \textbf{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 and hyperintensional cognitivism and modal and hyperintensional expressivism. I also develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic two-dimensional semantics grounded in two-dimensional hyperintensional Turing machines. Chapter \textbf{3} provides an abstraction principle for epistemic (hyper-)intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-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 $\mu$-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal $\mu$-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 \textbf{6} advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{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 \textbf{2} and \textbf{4} are availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. -/- Chapters \textbf{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 \textbf{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 $\Omega$-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 \textbf{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 higher 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 \textbf{10} examines the philosophical significance of hyperintensional $\Omega$-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter \textbf{11} provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter \textbf{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 interpretational and objective modalities and the 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 in the category of Set in category theory. Chapter \textbf{13} examines modal responses to the alethic paradoxes. I provide a counter-example to epistemic closure for logical deduction. Chapter \textbf{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)

Words are indispensable linguistic tools for beings like us. However, there is not much philosophical work done about what words really are. In this paper, I develop a new ontology for words. I argue that words are abstract artifacts that are created to fulfill various kinds of purposes, and words are abstract in the sense that they are not located in space but they have a beginning and may have an end in time given that certain conditions are met. What (...) follows from this two-fold argument is that words, from an ontological point of view, are more like musical works, fictional characters or computer programs, than numbers or sets. (shrink)

The debate over whether we should believe that mathematical objects exist quickly leads to the question of how to determine what we should believe. Indispensabilists claim that we should believe in the existence of mathematical objects because of their ineliminable roles in scientific theory. Eleatics argue that only objects with causal properties exist. Mark Colyvan’s recent defenses of Quine’s indispensability argument against some contemporary eleatics attempt to provide reasons to favor the indispensabilist’s criterion. I show that Colyvan’s argument is not (...) decisive against the eleatic and sketch a way to capture the important intuitions behind both views. (shrink)

Over the decades, the claim that everything is revisable (defended by Quine and others) has played an important role in Epistemology and Philosophy of Science. Some time ago, Katz (1988) argued that this claim is paradoxical. This paper does not discuss this objection but rather argues that the claim of universal revisability allows for two different readings but in each case leads to a contradiction and is false.

This thesis offers a naturalist revision of Alain Badiou’s philosophy. This goal is pursued through an encounter of Badiou’s mathematical ontology and theory of truth with contemporary trends in philosophy of mathematics and philosophy of science. I take issue with Badiou’s inability to elucidate the link between the empirical and the ontological, and his residual reliance on a Heideggerian project of fundamental ontology, which undermines his own immanentist principles. I will argue for both a bottom-up naturalisation of Badiou’s philosophical approach (...) to mathematics and a top-down naturalisation. Articulating my particular understanding of what realism and naturalism should commit us to, I propose a creative fusion of Badiou’s attention to metamathematical results with a structural-informational metaphysics, proposing a ‘matherialism’ uniting the more daring speculative insights of the former with the naturalist and empiricist commitments motivating the latter. (shrink)

Quine's holistic empiricist account of scientific inquiry can be characterized by three constitutive principles: *noncontradiction*, *universal revisability* and *pragmatic ordering*. We show that these constitutive principles cannot be regarded as statements within a holistic empiricist's scientific theory of the world. This claim is a corollary of our refutation of Katz's [1998, 2002] argument that holistic empiricism suffers from what he calls the Revisability Paradox. According to Katz, Quine's empiricism is incoherent because its constitutive principles cannot themselves be rationally revised. Using (...) Gärdenfors and Makinson's logic of belief revision based on epistemic entrenchment, we argue that Katz wrongly assumes that the constitutive principles are *statements* within a holistic empiricist's theory of the world. Instead, we show that constitutive principles are best seen as *properties* of a holistic empiricist's theory of scientific inquiry and we submit that, without Katz's mistaken assumption, the paradox cannot be formulated. We argue that our perspective on the status of constitutive principles is perfectly in line with Quinean orthodoxy. In conclusion, we compare our findings with van Fraassen's [2002] argument that we should think of empiricism as a stance, rather than as a doctrine. (shrink)

It is our contention that an ontological commitment to propositions faces a number of problems; so many, in fact, that an attitude of realism towards propositions—understood the usual “platonistic” way, as a kind of mind- and language-independent abstract entity—is ultimately untenable. The particular worries about propositions that marshal parallel problems that Paul Benacerraf has raised for mathematical platonists. At the same time, the utility of “proposition-talk”—indeed, the apparent linguistic commitment evident in our use of 'that'-clauses (in offering explanations and making (...) predictions)—is also in need of explanation. We account for this with a fictionalist analysis of our use of 'that'-clauses. Our account avoids certain problems that arise for the usual error-theoretic versions of fictionalism because we apply the notion of semantic pretense to develop an alternative, pretense-involving, non-error-theoretic, fictionalist account of proposition-talk. (shrink)

It is commonly held that our intuitive judgements about imaginary problem cases are justified a priori, if and when they are justified at all. In this paper I defend this view — ‘rationalism’ — against a recent objection by Timothy Williamson. I argue that his objection fails on multiple grounds, but the reasons why it fails are instructive. Williamson argues from a claim about the semantics of intuitive judgements, to a claim about their psychological underpinnings, to the denial of rationalism. (...) I argue that the psychological claim — that a capacity for mental simulation explains our intuitive judgements — does not, even if true, provide reasons to reject rationalism. (More generally, a simulation hypothesis, about any category of judgements, is very limited in its epistemological implications: it is pitched at a level of explanation that is insensitive to central epistemic distinctions.) I also argue that Williamson’s semantic claim — that intuitive judgements are judgements of counterfactuals — is mistaken; rather, I propose, they are a certain kind of metaphysical possibility judgement. Several other competing proposals are also examined and criticized. (shrink)

Many philosophers and psychologists hold that intuitions are, or reduce to, beliefs. The argument from intuition without beliefs threatens to undercut any such doxastic account: since there are clear cases of intuition without belief, intuitions cannot be beliefs. Advocates of the intellectual seeming account conclude that intuitions belong to the basic mental kind of intellectual seeming. I argue that rightly understood, apparent cases of intuition without belief are cases of someone having the inclination to believe that p whilst believing that (...) not-p. These can be accommodated by a disjunctive doxastic account holding that to have an intuition is to either have a belief, or to have an inclination to believe. I conclude that intuitions reduce to beliefs, and that there is no need to acknowledge intellectual seemings as basic states in our mental taxonomy. (shrink)

This paper examines the paradox of revisability. This paradox was proposed by Jerrold Katz as a problem for Quinean naturalised epistemology. Katz employs diagonalisation to demonstrate what he takes to be an inconsistency in the constitutive principles of Quine's epistemology. Specifically, the problem seems to rest with the principle of universal revisability which states that no statement is immune to revision. In this paper it is argued that although there is something odd about employing universal revisability to revise itself, there (...) is nothing paradoxical about this. At least, there is no paradox along the lines suggested by Katz. (shrink)

It is widely believed that platonists face a formidable problem: that of providing an intelligible account of mathematical knowledge. The problem is that we seem unable, if the platonist is right, to have the causal relationships with the objects of mathematics without which knowledge of these objects seems unintelligible. The standard platonist response to this challenge is either to deny that knowledge without causation is unintelligible, or to make room for causal interactions by softening the platonism at issue. In this (...) essay I argue that the idea of causal relations with fully platonist objects is unproblematic. (shrink)

The target article by Christiansen & Chater (C&C) offers an integrated framework for the study of language acquisition and, possibly, a novel role for internal speech in language acquisition. However, the “Now-or-Never bottleneck” raises a paradox for language evolution. It seems to imply that language complexity has been either reduced over time or has remained the same. How, then, could languages as complex as ours have evolved in prelinguistic ancestors? Linguistic Platonism could offer a solution to this paradox.

Without a doubt, one of the main reasons Platonsim remains such a strong contender in the Foundations of Mathematics debate is because of the prima facie plausibility of the claim that objectivity needs objects. It seems like nothing else but the existence of external referents for the terms of our mathematical theories and calculations can guarantee the objectivity of our mathematical knowledge. The reason why Frege – and most Platonists ever since – could not adhere to the idea that mathematical (...) objects were mental, conventional or in any other way dependent on our faculties, will or other historical contingencies was that objects whose properties of existence depended on such contingencies could not warrant the objectivity required for scientific knowledge. This idea gained currency in the second half of the 19th Century and remains current for the most part today. However, it was not always like that. Objectivity, after all, has a history, and according to its historians (Daston 2001), the view that scientific knowledge need be objective is a fairly recent one. Up until mid-19th Century, science was not so much concerned with objectivity, as it was concerned with truth. Before the rise of the modern university and the professional scientist, science had the discovery of truths as its ultimate goal. In contrast, modern science now aims at the production and acquisition of objective knowledge. The difference might seem.. (shrink)

In a recent article, Mark Colyvan has criticized Jerrold Katz's attempt to show that Quinean holism is self-refuting. Katz argued that a Quinean epistemology incorporating a principle of the universal revisability of beliefs would have to hold that that and other principles of the system were both revisable and unrevisable. Colyvan rejects Katz's argument for failing to take into account the logic of belief revision. But granting the terms of debate laid down by Colyvan, the universal revisability principle still commits (...) Quineans to holding that one belief is both revisable and unrevisable: the belief that some beliefs are revisable. (shrink)

One version of the Julius Caesar problem arises when we demand assurance that expressions drawn from different theories or stretches of discourse refer to different things. The counter‐Caesar problem arises when assurance is demanded that expressions drawn from different theories. refer to the same thing. The Julio César problem generalises from the counter‐Caesar problem. It arises when we seek reassurance that expressions drawn from different languages refer to the same kind of things. If the Julio César problem is not resolved (...) then the Fregean account of numbers as objects is cast into doubt, the notion of number left relative to a language. Wright introduced this problem by asking whether there can be such a thing as ‘International Platonism’. After rejecting Hale's attempt to resolve it I argue that the threat posed by the Julio César problem diminishes – even though it cannot be made to logically disappear – once it is recognised that the radical interpretation of an unfamiliar language is inevitably holistic, the evidence available invariably defeasible and consequently Cartesian certainty about the significance of the utterances of a foreign tongue neither to be sought after nor attained. (shrink)

The access problem for mathematics arises from the supposition that the referents of mathematical terms inhabit a realm separate from us. Quine’s approach in the philosophy of mathematics dissolves the access problem, though his solution sometimes goes unrecognized, even by those who rely on his framework. This paper highlights both Quine’s position and its neglect. I argue that Michael Resnik’s structuralist, for example, has no access problem for the so-called mathematical objects he posits, despite recent criticism, since he relies on (...) an indispensability argument. Still, Resnik’s structuralist does not provide an account of our access to traditional mathematical objects, and this may be seen as a problem. (shrink)

The concern of this paper is not with the truth of any particular realist or anti-realist view, but rather with determining what it is to be a realist or anti-realist in the first place. While much skepticism has been voiced in recent years about the viability of such a project, my goal is to articulate interesting and informative conditions whereby any view in any domain of experience can count as either a realist or an anti-realist position.

A number of examples of studies from the field ‘The Philosophy of Mathematical Practice’ (PMP) are given. To characterise this new field, three different strands are identified: an agent-based, a historical, and an epistemological PMP. These differ in how they understand ‘practice’ and which assumptions lie at the core of their investigations. In the last part a general framework, capturing some overall structure of the field, is proposed.

There is an apparent conflict in Quine’s work between, on the one hand, his clear commitment to the rational revisability of logic and, on the other, his principle of charitable translation and ‘change of logic, change of subject’ argument. I argue that the apparent conflict is mostly resolved under close exegesis, but that the translation argument normatively rules out collaborative revision and allows only revision by individuals. However, I articulate a Neo-Quinean view that preserves the rational acceptability of collaborative revision. (...) On that view, everything is rationally revisable in some manner or other — it’s just that the logical principles that laypersons find (actually or potentially) obvious and tacitly use to govern their everyday inferences can’t be rationally revised quickly and all at once. Since, in Quine’s view, what most people find obvious resists change, rational changes to laymen’s logic must be gradual. (shrink)

The following essay sets out the background developments in mathematics and set theory that inform Alain Badiou’s Being and Event in order to provide some context both for the original text and for comment on Chris Norris’s excellent exploration of Badiou’s work. I also provide a summary of Badiou’s overall approach.

In this article, I present a novel account of a priori warrant, which I then use to examine the relationship between a priori and a posteriori warrant in mathematics. According to this account of a priori warrant, the reason that a posteriori warrant is subordinate to a priori warrant in mathematics is because processes that produce a priori warrant are reliable independent of the contexts in which they are used, whereas this is not true for processes that produce a posteriori (...) warrant. Following this, I show how this difference explains why a priori warranting processes, such as proof (and mathematical intuition) can, and a posteriori warranting processes cannot, definitively establish mathematical results. I then show why, in the event of a conflict between a priori and a posteriori warranting processes, the former undermine, and cannot be undermined by, the latter. Finally, I explain the connection between apriority and mathematical necessity, but do so in a way that allows for the existence of contingent a priori knowledge. (shrink)

Mark Balaguer ha elaborado una peculiar variante del platonismo matemático –denominada ‘full-blooded platonism’ o ‘FBP’– para solucionar el problema de Benacerraf sobre la inaccesibilidad de las entidades abstractas. Según FBP, todos los objetos matemáticos consistentemente caracterizables existen, aunque de modo contingente. En este trabajo quisiera mostrar que la plenitud ontológica y la contingencia modal no pueden converger en una teoría de objetos matemáticos filosóficamente respetable. Para esto argumento que FBP no cubre algunos factores elementales de confiabilidad epistémica y que envuelve (...) un criterio de plenitud tanto matemática como metafísicamente implausible. (shrink)