In this paper, I develop a "safety result" for applied mathematics. I show that whenever a theory in natural science entails some non-mathematical conclusion via an application of mathematics, there is a counterpart theory that carries no commitment to mathematical objects, entails the same conclusion, and the claims of which are true if the claims of the original theory are "correct": roughly, true given the assumption that mathematical objects exist. The framework used for proving the safety result has some advantages (...) over existing nominalistic accounts of applied mathematics. It also provides a nominalistic account of pure mathematics. (shrink)

I describe three extant attempts to identify the global external symmetries within a Humean framework with theorems of some or other deductive systematization of the world, respectively, the best system, a meta-best system and a maximally simple system. Each has merits, but also serious flaws. Instead, I propose a view of global external symmetries as consequences of the structure of Humean-consistent world-making relations.

It is now standard to interpret symmetry-related models of physical theories as representing the same state of affairs. Recently, a debate has sprung up around the question when this interpretational move is warranted. In particular, Møller-Nielsen :1253–1264, 2017) has argued that one is only allowed to interpret symmetry-related models as physically equivalent when one has a characterisation of their common content. I disambiguate two versions of this claim. On the first, a perspicuous interpretation is required: an account of the models’ (...) common ontology. On the second, stricter, version of this claim, a perspicuous formalism is required in addition: one whose mathematical structures ‘intrinsically’ represent the physical world, in the sense of Field. Using Dewar’s :485–521, 2019) distinction between internal and external sophistication as a case study, I argue that the second requirement is decisive. This clarifies the conditions under which it is warranted to interpret symmetry-related models as physically equivalent. (shrink)

This book develops a naturalistic aesthetic theory that accounts for aesthetic phenomena in mathematics in the same terms as it accounts for more traditional aesthetic phenomena. Building upon a view advanced by James McAllister, the assertion is that beauty in science does not confine itself to anecdotes or personal idiosyncrasies, but rather that it had played a role in shaping the development of science. Mathematicians often evaluate certain pieces of mathematics using words like beautiful, elegant, or even ugly. Such evaluations (...) are prevalent, however, rigorous investigation of them, of mathematical beauty, is much less common. The volume integrates the basic elements of aesthetics, as it has been developed over the last 200 years, with recent findings in neuropsychology as well as a good knowledge of mathematics. The volume begins with a discussion of the reasons to interpret mathematical beauty in a literal or non-literal fashion, which also serves to survey historical and contemporary approaches to mathematical beauty. The author concludes that literal approaches are much more coherent and fruitful, however, much is yet to be done. In this respect two chapters are devoted to the revision and improvement of McAllister’s theory of the role of beauty in science. These antecedents are used as a foundation to formulate a naturalistic aesthetic theory. The central idea of the theory is that aesthetic phenomena should be seen as constituting a complex dynamical system which the author calls the aesthetic as process theory. The theory comprises explications of three central topics: aesthetic experience, aesthetic value and aesthetic judgment. The theory is applied in the final part of the volume and is used to account for the three most salient and often used aesthetic terms often used in mathematics: beautiful, elegant and ugly. This application of the theory serves to illustrate the theory in action, but also to further discuss and develop some details and to showcase the theory’s explanatory capabilities. (shrink)

I explore the relationship between a prominent analysis of intrinsic properties, due to Langton and Lewis, and the phenomenon of quantum entanglement. As I argue, the analysis faces a puzzle. The full analysis classifies certain properties of entangled particles as intrinsic. But when combined with an extremely plausible assumption about duplication, the main part of the analysis classifies those properties as non-intrinsic instead. I conclude that much of Lewis’s metaphysics is in trouble: Lewis based many of his metaphysical views—his thesis (...) of Humean supervenience, for instance, and his account of recombination—on an analysis of intrinsicality which does not sit well with quantum phenomena. (shrink)

Can we do science without numbers? How much contingency is there? These seemingly unrelated questions--one in the philosophy of math and science and the other in metaphysics--share an unexpectedly close connection. For as it turns out, a radical answer to the second leads to a breakthrough on the first. The radical answer is new view about modality called compossible immutabilism. The breakthrough is a new strategy for doing science without numbers. One of the chief benefits of the new strategy is (...) that, unlike the existing substantialism approach from Field (1980), the new strategy naturally generalizes to theories formulated in terms of state space. (shrink)

The thesis that nothing is true has long been thought to be a self-refuting position not worthy of serious philosophical consideration. Recently, however, the thesis of alethic nihilism—that nothing is true—has been explicitly defended (notably by David Liggins). Nihilism is also, I argue, a consequence of other views about truth that have recently been advocated, such as fictionalism about truth and the inconsistency account. After offering an account of alethic nihilism, and how it purports to avoid the self-refutation problem, I (...) argue that it avoids the problem at the expense of changing the subject. I then present other arguments against nihilism and responses to the considerations offered in defense of it. The only tenable position is that something is indeed true. (shrink)

The main purpose of this dissertation is to examine critically and discuss the role of imagination in science and religion, with particular emphasis on its possible epistemic, creative, and meaning-making functions. In order to answer my research questions, I apply theories and concepts from contemporary philosophy of mind on scientific and religious practices. This framework allows me to explore the mental state of imagination, not as an isolated phenomenon but, rather, as one of many mental states that co-exist and interplay (...) in our cogntive architecture. Based on the philosophical discourse of philosophy of mind, four types of imagination are indentified and conceptualized: sensory, propositional, experiential, and creative imagination. These categories are then employed on five phenomena that can be found in scientific and religious environments: metaphors, models, thought experiments, aspect perception, and - in the religious case - rituals. In relation to the concept of religious "seeing" I consider how imaginings may influence visionary experiences and visualization, and compare these phenomena with cases of scientific visualization and eureka experiences. In regard to scientific and religious models, a distinction is made between, on the one hand, two notions of truth and, on the other hand, truth-independent meaning-making. In light of these categories, I differentiate between, and critically discuss, the use of imagination in doxastic, non-doxastic and fictionalist accounts. In light of this investigation, I formulate and defend the position of interactivism, which acknowledges a constant interplay between different attitudes and mental states. In my examination of rituals and scientific and religious thought experiments, special attention is given to the mental capacity to recreate the experiences that are entailed in an imagined scenario. At the end of the investigation, I consider the possible impact that my study might have on how we view science and religion as well as the dialogue bewteen these two fields. (shrink)

How may we characterize the intrinsic structure of physical quantities such as mass, length, or electric charge? This article shows that group-theoretic methods—specifically, the notion of a free and transitive group action—provide an elegant way of characterizing the structure of scalar quantities, and uses this to give an intrinsic treatment of vector quantities. It also gives a general account of how different scalar or vector quantities may be algebraically combined with one another. Finally, it uses this apparatus to give a (...) simple intrinsic treatment of Newtonian gravitation. (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)

I argue that modal epistemology should pay more attention to questions about the structure and function of modal thought. We can treat these questions from synchronic and diachronic angles. From a synchronic perspective, I consider whether a general argument for the epistemic support of modal though can be made on the basis of modal thoughs’s indispensability for what Enoch and Schechter (2008) call rationally required epistemic projects. After formulating the argument, I defend it from various objections. I also examine the (...) possibility of considering the indispensability of modal thought in terms of its components. Finally, I argue that we also need to approach these issues from a diachronic perspective, and I sketch how to approach this task. (shrink)

Invoking a form of quantifier variance promises to let us explain mathematicians’ freedom to introduce new kinds of mathematical objects in a way that avoids some problems for standard platonist and nominalist views. In this paper I’ll note that, despite traditional associations between quantifier variance and Carnapian rejection of metaphysics, Siderian realists about metaphysics can naturally be quantifier variantists. Unfortunately a variant on the Quinean indispensability argument concerning grounding seems to pose a problem for philosophers who accept this hybrid. However (...) I will charitably reconstruct this problem and then argue for optimism about solving it. (shrink)

Las teorías que usamos para representar el mundo pueden ser extremadamente complejas. Abordan temas tales como electrones, campos cuánticos, estrellas de neutrones, materia oscura, redes neuronales, mercados económicos, la atmósfera y muchas otras entidades que suponemos existen en el universo. Al formular nuestras teorías, recurrimos a lenguajes exactos que nos permiten minimizar la vaguedad y expresarnos lo más precisa y cuantitativamente posible. Recurrimos a la matemática. Cuando formulamos nuestras teorías fácticas en lenguaje matemático, estas se refieren no solamente a objetos (...) que nosotros interpretamos como materiales, tales como partículas o personas, sino también a entidades más raras de un mundo abstracto: conjuntos, números, funciones, espacios algebraicos, variedades, topologías, y otras entidades similares. Estos objetos no son materiales en el sentido en que nosotros decimos que una manzana es material. Ellos no existen en el espacio-tiempo, no interactúan, no cambian o evolucionan. Sin embargo, allí están, profundamente arraigados en nuestras teorías más apreciadas acerca del mundo. (shrink)

Easy-road mathematical fictionalists grant for the sake of argument that quantification over mathematical entities is indispensable to some of our best scientific theories and explanations. Even so they maintain we can accept those theories and explanations, without believing their mathematical components, provided we believe the concrete world is intrinsically as it needs to be for those components to be true. Those I refer to as “mathematical surrealists” by contrast appeal to facts about the intrinsic character of the concrete world, not (...) to explain why our best mathematically imbued scientific theories and explanations are acceptable in spite of having false components, but in order to replace those theories and explanations with parasitic, nominalistically acceptable alternatives. I argue that easy-road fictionalism is viable only if mathematical surrealism is and that the latter constitutes a superior nominalist strategy. Two advantages of mathematical surrealism are that it neither begs the question concerning the explanatory role of mathematics in science nor requires rejecting the cogency of inference to the best explanation. (shrink)

Creationists have long argued that evolutionary science is committed to a dogmatic metaphysics of naturalism and materialism, which is based on faith or ideology rather than evidence. The standard response to this has been to insist that science is not committed to any such metaphysical doctrine, but only to a methodological version of naturalism, according to which science may only appeal to natural entities and processes. But this whole debate presupposes that there is a clear distinction between the natural and (...) the supernatural, and thus that naturalism is a meaningful doctrine. I argue that this assumption is false. The concepts of the natural and the supernatural are in fact hopelessly obscure, such that the claim that science is committed to methodological naturalism cannot be made good. This is no victory for anti-naturalists however; explicitly supernaturalist theories, such as Creationism, can be ruled out of scientific consideration as a priori incoherent, given that they presuppose for their intelligibility that there is a meaningful natural-supernatural distinction. This is not the case for standard scientific theories however, as they are not explicitly naturalistic theories; they do not postulate natural or physical entities or processes as such. (shrink)

Fictionalists maintain that possible worlds, numbers or composite objects exist only according to theories which are useful but false. Hale, Divers and Woodward have provided arguments which threaten to show that fictionalists must be prepared to regard the theories in question as contingently, rather than necessarily, false. If warranted, this conclusion would significantly limit the appeal of the fictionalist strategy rendering it unavailable to anyone antecedently convinced that mathematics and metaphysics concern non-contingent matters. I try to show that their arguments (...) can be resisted by developing and defending a strategy suggested by Rosen, Nolan and Dorr, according to which the fiction-operator is to be analysed in terms of a counterfactual that admits of non-trival truth-values even when the antecedent is impossible. (shrink)

Some proponents of the indispensability argument for mathematical realism maintain that the empirical evidence that confirms our best scientific theories and explanations also confirms their pure mathematical components. I show that the falsity of this view follows from three highly plausible theses, two of which concern the nature of mathematical application and the other the nature of empirical confirmation. The first is that the background mathematical theories suitable for use in science are conservative in the sense outlined by Hartry Field. (...) The second is that the empirical relevance of mathematical statements suitable for use in science is mediated by their non-mathematical consequences. The third is that statements receive additional empirical confirmation only by way of generating additional empirical expectations. Since each of these is a thesis we have good reason to endorse, my argument poses a challenge to anyone who argues that science affords empirical grounds for mathematical realism. (shrink)

I begin by giving reasons to accept the block-universe view, the strongly supported by physics view that we live in a four-dimensional world. According to it, the past and the future are as real as the present. As a result, it seems that the future is determined in the sense that what will be the case will necessarily be the case. In the dissertation, I examine whether we have to accept this consequence. I show that we do not have to (...) bite the bullet: the future might be both real and contingent. I first start to consider modal realism (possible futures are real) as a possible solution. However, I propose then another account, actualist (positing the reality of only one possible world: the space-time we live in). This solution relies on a conventionalist theory about metaphysical modality (Sidelle, 1989). It states that modal modality is purely conventional. In combination with a realist interpretation of natural modality (causal and/or nomological), this framework allows me to propose that the future is conventionally closed and naturally contingent. In this view, the necessity of the future holds in virtue of linguistic conventions and then is conventionally necessary. But this is a distinct phenomenon from the natural contingency of the future: the future is naturally contingent because there are mind-independent probabilistic relations holding between the present and the future. This solution has strong consequences: most importantly, an anti-realist view about ordinary objects and physical particles. I end up by presenting and arguing in favour of the general image of the world that comes out from these views. (shrink)

W.V. Quine is a well-known proponent of naturalism, the view on which reality is described only in science. He is also well-known for arguing that our current scientific theories commit us to the existence of abstract objects. It is tempting to believe that the naturalistic philosopher should think scientists outside of philosophy are in the best position to assess the merits of revising our current commitment to abstract objects. But Quine rejects this deferential view. On the reading of Quine’s philosophical (...) methodology that I defend in this paper, the naturalistic philosopher not only may assess the merits of revising the commitments of our scientific theories, but also will recommend we make such revisions if doing so simplifies and clarifies our science. To develop my reading, I will examine John Burgess and Gideon Rosen’s naturalism in ontology that includes the deferential view Quine rejects. By explaining how Quine’s naturalism differs from the anti-revisionist, deferential naturalism in philosophy of mathematics that Burgess and Rosen advance, I seek to clarify and advance contemporary debates on naturalism. (shrink)

The predisposition of the Indispensability Argument to objections, rephrasing and versions associated with the various views in philosophy of mathematics grants it a special status of a “blueprint” type rather than a debatable theme in the philosophy of science. From this point of view, it follows that the Argument has more an epistemic character than ontological.

Proponents of the explanatory indispensability argument for mathematical platonism maintain that claims about mathematical entities play an essential explanatory role in some of our best scientific explanations. They infer that the existence of mathematical entities is supported by way of inference to the best explanation from empirical phenomena and therefore that there are the same sort of empirical grounds for believing in mathematical entities as there are for believing in concrete unobservables such as quarks. I object that this inference depends (...) on a false view of how abductive considerations mediate the transfer of empirical support. More specifically, I argue that even if inference to the best explanation is cogent, and claims about mathematical entities play an essential explanatory role in some of our best scientific explanations, it doesn’t follow that the empirical phenomena that license those explanations also provide empirical support for the claim that mathematical entities exist. (shrink)

The metaphysics of representation poses questions such as: in virtue of what does a sentence, picture, or mental state represent that the world is a certain way? In the first instance, I have focused on the semantic properties of language: for example, what is it for a name such as ‘London’ to refer to something? Interpretationism concerning what it is for linguistic expressions to have meaning, says that constitutively, semantic facts are fixed by best semantic theory. As here developed, it (...) promises to give a reductive, universal and non-revisionary account of the nature of linguistic representation. -/- Interpretationism in general, however, is threatened by severe internal tension, due to arguments for radical inscrutability. These contend that, given the interpretationist setting, there can be no fact of the matter what object an individual word refers to: for example, that there is no fact of the matter as to whether “London” refers to London or to Sydney. -/- A series of challenges emerge, forming the basis for this thesis. 1. What sort of properties is the interpretationist trying to reduce, and what kind of reductive story is she offering? 2. How are inscrutability theses best formulated? Are arguments for inscrutability effective in their own terms? What kinds of inscrutability arise? 3. Is endorsing radical inscrutability a stable position? 4. Are there theoretical virtues—such as simplicity—that can be appealed to in discrediting the rival (empirically equivalent) theories that underpin inscrutability arguments? -/- In addressing these questions, I concentrate on diagnosing the source of inscrutability, mapping the space of ways of resisting the arguments for radical inscrutability, and examining the challenges faced in developing a principled account of linguistic content that avoids radical inscrutability. -/- The effect is not to close down the original puzzles, but rather to sharpen them into a set of new and deeper challenges. (shrink)

According to ontological nihilism there are, fundamentally, no individuals. Both natural languages and standard predicate logic, however, appear to be committed to a picture of the world as containing individual objects. This leads to what I call the \emph{expressibility challenge} for ontological nihilism: what language can the ontological nihilist use to express her account of how matters fundamentally stand? One promising suggestion is for the nihilist to use a form of \emph{predicate functorese}, a language developed by Quine. This proposal faces (...) a difficult objection, according to which any theory in predicate functorese will be a notational variant of the corresponding theory stated in standard predicate logic. Jason Turner (2011) has provided the most detailed and convincing version of this objection. In the present paper, I argue that Turner's case for the notational variance thesis relies on a faulty metasemantic principle and, consequently, that an objection long thought devastating is in fact misguided. (shrink)

It has been recently debated whether there exists a so-called “easy road” to nominalism. In this essay, I attempt to fill a lacuna in the debate by making a connection with the literature on infinite and infinitesimal idealization in science through an example from mathematical physics that has been largely ignored by philosophers. Specifically, by appealing to John Norton’s distinction between idealization and approximation, I argue that the phenomena of fractional quantum statistics bears negatively on Mary Leng’s proposed path to (...) easy road nominalism, thereby partially defending Mark Colyvan’s claim that there is no easy road to nominalism. (shrink)

The existence of mereological sums can be derived from an abstraction principle in a way analogous to numbers. I draw lessons for the thesis that “composition is innocent” from neo-Fregeanism in the philosophy of mathematics.

How does mathematics apply to something non-mathematical? We distinguish between a general application problem and a special application problem. A critical examination of the answer that structural mapping accounts offer to the former problem leads us to identify a lacuna in these accounts: they have to presuppose that target systems are structured and yet leave this presupposition unexplained. We propose to fill this gap with an account that attributes structures to targets through structure generating descriptions. These descriptions are physical descriptions (...) and so there is no such thing as a solely mathematical account of a target system. (shrink)

Orthodoxy has it that knowledge is absolute—that is, it cannot come in degrees. On the other hand, there seems to be strong evidence for the gradability of know-how. Ascriptions of know-how are gradable, as when we say that one knows in part how to do something, or that one knows how to do something better than somebody else. When coupled with absolutism, the gradability of ascriptions of know-how can be used to mount a powerful argument against intellectualism about know-how—the view (...) that know-how is a species of propositional knowledge. This essay defends intellectualism from the argument of gradability. It is argued that the gradability of ascriptions of know-how should be discounted as a rather superficial linguistic phenomenon, one that can be explained in a way compatible with the absoluteness of the state reported. (shrink)

In this paper, I will attempt to develop and defend a common form of intuitive resistance to the companions in guilt argument. I will argue that one can reasonably believe there are promising solutions to the access problem for mathematical realism that don’t translate to moral realism. In particular, I will suggest that the structuralist project of accounting for mathematical knowledge in terms of some form of logical knowledge offers significant hope of success while no analogous approach offers such hope (...) for moral realism. (shrink)

Homotopy Type Theory is a putative new foundation for mathematics grounded in constructive intensional type theory that offers an alternative to the foundations provided by ZFC set theory and category theory. This article explains and motivates an account of how to define, justify, and think about HoTT in a way that is self-contained, and argues that, so construed, it is a candidate for being an autonomous foundation for mathematics. We first consider various questions that a foundation for mathematics might be (...) expected to answer, and find that many of them are not answered by the standard formulation of HoTT as presented in the ‘HoTT Book’. More importantly, the presentation of HoTT given in the HoTT Book is not autonomous since it explicitly depends upon other fields of mathematics, in particular homotopy theory. We give an alternative presentation of HoTT that does not depend upon ideas from other parts of mathematics, and in particular makes no reference to homotopy theory, and argue that it is a candidate autonomous foundation for mathematics. Our elaboration of HoTT is based on a new interpretation of types as mathematical concepts, which accords with the intensional nature of the type theory. 1 Introduction2 What Is a Foundation for Mathematics?2.1 A characterization of a foundation for mathematics2.2 Autonomy3 The Basic Features of Homotopy Type Theory3.1 The rules3.2 The basic ways to construct types3.3 Types as propositions and propositions as types3.4 Identity3.5 The homotopy interpretation4 Autonomy of the Standard Presentation?5 The Interpretation of Tokens and Types5.1 Tokens as mathematical objects?5.2 Tokens and types as concepts6 Justifying the Elimination Rule for Identity7 The Foundations of Homotopy Type Theory without Homotopy7.1 Framework7.2 Semantics7.3 Metaphysics7.4 Epistemology7.5 Methodology8 Possible Objections to this Account8.1 A constructive foundation for mathematics?8.2 What are concepts?8.3 Isn’t this just Brouwerian intuitionism?8.4 Duplicated objects8.5 Intensionality and substitution salva veritate9 Conclusion9.1 Advantages of this foundation. (shrink)

This paper explains and defends the idea that metaphysical necessity is the strongest kind of objective necessity. Plausible closure conditions on the family of objective modalities are shown to entail that the logic of metaphysical necessity is S5. Evidence is provided that some objective modalities are studied in the natural sciences. In particular, the modal assumptions implicit in physical applications of dynamical systems theory are made explicit by using such systems to define models of a modal temporal logic. Those assumptions (...) arguably include some necessitist principles. -/- Too often, philosophers have discussed ‘metaphysical’ modality — possibility, contingency, necessity — in isolation. Yet metaphysical modality is just a special case of a broad range of modalities, which we may call ‘objective’ by contrast with epistemic and doxastic modalities, and indeed deontic and teleological ones (compare the distinction between objective probabilities and epistemic or subjective probabilities). Thus metaphysical possibility, physical possibility and immediate practical possibility are all types of objective possibility. We should study the metaphysics and epistemology of metaphysical modality as part of a broader study of the metaphysics and epistemology of the objective modalities, on pain of radical misunderstanding. Since objective modalities are in general open to, and receive, natural scientific investigation, we should not treat the metaphysics and epistemology of metaphysical modality in isolation from the metaphysics and epistemology of the natural sciences. -/- In what follows, Section 1 gives a preliminary sketch of metaphysical modality and its place in the general category of objective modality. Section 2 reviews some familiar forms of scepticism about metaphysical modality in that light. Later sections explore a few of the many ways in which natural science deals with questions of objective modality, including questions of quantified modal logic. (shrink)

The paper clarifies and defends the orthodox view that counterfactual conditionals with impossible antecedents are vacuously true against recent criticisms. It argues that apparent counterexamples to orthodoxy result from uncritical reliance on a fallible heuristic used in the processing of conditionals. A comparison is developed between such counterpossibles and vacuously true universal generalizations.

There are various ways of achieving an enlarged understanding of a concept of interest. One way is by giving its proper definition. Another is by giving something else a proper definition and then using it to model or formally represent the original concept. Between the two we find varying shades of grey. We might open up a concept by a direct lexical definition of the predicate that expresses it, or by a theory whose theorems define it implicitly. At the other (...) end of the spectrum, the modelling-this-as-that option also admits of like variation, ranging from models rooted in formal representability theorems to models conceived of as having only heuristic value. There exist on both sides of this divide further differences still. In one of them, both the definiendum and definiens of a definition are words or phrases of some common natural language. In others, the item of interest is a natural language expression and its representation is furnished by the artificial linguistic system that models it. The modern history of these approaches is both very large and growing. Much of this evolution has given too short a shrift to the history of the demotion of ‘intuitive’ concepts in favour of the artificially contrived ones intended to model them. A working assumption of this article is that in the absence of a good understanding of what motivated the modelling-turn in the foundations of mathematics and the intuitive theory of truth, the whole notion of formal representability will have been inadequately understood. In the interests of space, I will concentrate on seminal issues in set theory as dealt with by Russell and Frege, and in the theory of truth in natural languages as dealt with by Tarski. The nub of the present focus is the representational role of model theory in the logics of formalized languages. (shrink)

This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory.

Modal fictionalists face a problem that arises due to their possible-world story being incomplete in the sense that certain relevant claims are neither true nor false according to it. It has recently been suggested that this incompleteness problem generalises to other brands of fictionalism, such as fictionalism about composite or mathematical objects. In this paper, I argue that these fictionalist positions are particularly threatened by a generalised incompleteness problem since they cannot emulate the modal fictionalists’ most attractive response. I then (...) defend mathematical and compositional fictionalism by showing that the reasons for which the incompleteness problem has been thought to affect them are mistaken. This leads to the question of whether there are other fictionalist positions to which the problem does in fact generalise. I give a general account of the features of a fictionalist position that generate the incompleteness problem and argue that whenever a fictionalist position does exemplify these features then the problem can be addressed in analogy to the modal fictionalists’ preferred response. (shrink)

Hilbert’s choice operators τ and ε, when added to intuitionistic logic, strengthen it. In the presence of certain extensionality axioms they produce classical logic, while in the presence of weaker decidability conditions for terms they produce various superintuitionistic intermediate logics. In this thesis, I argue that there are important philosophical lessons to be learned from these results. To make the case, I begin with a historical discussion situating the development of Hilbert’s operators in relation to his evolving program in the (...) foundations of mathematics and in relation to philosophical motivations leading to the development of intuitionistic logic. This sets the stage for a brief description of the relevant part of Dummett’s program to recast debates in metaphysics, and in particular disputes about realism and anti-realism, as closely intertwined with issues in philosophical logic, with the acceptance of classical logic for a domain reflecting a commitment to realism for that domain. Then I review extant results about what is provable and what is not when one adds epsilon to intuitionistic logic, largely due to Bell and DeVidi, and I give several new proofs of intermediate logics from intuitionistic logic+ε without identity. With all this in hand, I turn to a discussion of the philosophical significance of choice operators. Among the conclusions I defend are that these results provide a finer-grained basis for Dummett’s contention that commitment to classically valid but intuitionistically invalid principles reflect metaphysical commitments by showing those principles to be derivable from certain existence assumptions; that Dummett’s framework is improved by these results as they show that questions of realism and anti-realism are not an “all or nothing” matter, but that there are plausibly metaphysical stances between the poles of anti-realism and realism, because different sorts of ontological assumptions yield intermediate rather than classical logic; and that these intermediate positions between classical and intuitionistic logic link up in interesting ways with our intuitions about issues of objectivity and reality, and do so usefully by linking to questions around intriguing everyday concepts such as “is smart,” which I suggest involve a number of distinct dimensions which might themselves be objective, but because of their multivalent structure are themselves intermediate between being objective and not. Finally, I discuss the implications of these results for ongoing debates about the status of arbitrary and ideal objects in the foundations of logic, showing among other things that much of the discussion is flawed because it does not recognize the degree to which the claims being made depend on the presumption that one is working with a very strong logic. (shrink)

This paper maps the landscape for a range of views concerning the metaphysics of natural kinds. I consider a range of increasingly ontologically committed views concerning natural kinds and the possible arguments for them. I then ask how these relate to natural kind essentialism, arguing that essentialism requires commitment to kinds as entities. I conclude by examining the homeostatic property cluster view of kinds in the light of the general understanding of kinds developed.

It has commonly been argued that certain types of mental descriptions, specifically those characterized in terms of propositional attitudes, are part of a folk psychological understanding of the mind. Recently, however, it has also been argued that this is the case even when such descriptions are employed as part of scientific theories in domains like social psychology and comparative psychology. In this paper, I argue that there is no plausible way to understand the distinction between folk and scientific psychology that (...) can support such claims. Moreover, these sorts of claims can have adverse consequences for the neuroscientific study of the brain by downplaying the value of many psychological theories that provide information neuroscientists need in order to build and test neurological models. (shrink)

The paper traces some of the assumptions that have informed conservative naturalism in linguistic theory, critically examines their justification, and proposes a more liberal alternative.

Timothy Williamson’s Knowledge and its Limits has been highly influential since the beginning of this century. It can be read as a systematic response to scepticism. One of the most important notions in this response is the notion of «evidence,» which will be the focus of the present paper. I attempt to show primarily two things. First, the notion of evidence invoked by Williamson does not address the sceptical worry: he stipulates an objective notion of evidence, but this begs the (...) question against his opponent. Second, his related thesis «Evidence equals Knowledge» does not sit well with his own content externalism: he promises to relate epistemology to philosophy of mind, but he fails to live up to this commitment in his crucial chapter on scepticism. Other minor problems concerning evidence will also be discussed in due course. (shrink)

General discussion of grounding, including its formal features, relations to other notions, and applications. (Originally published 2014; revised 2021).

Recent years have seen an explosion of empirical data concerning arithmetical cognition. In this paper that data is taken to be philosophically important and an outline for an empirically feasible epistemological theory of arithmetic is presented. The epistemological theory is based on the empirically well-supported hypothesis that our arithmetical ability is built on a protoarithmetical ability to categorize observations in terms of quantities that we have already as infants and share with many nonhuman animals. It is argued here that arithmetical (...) knowledge developed in such a way cannot be totally conceptual in the sense relevant to the philosophy of arithmetic, but neither can arithmetic understood to be empirical. Rather, we need to develop a contextual a priori notion of arithmetical knowledge that preserves the special mathematical characteristics without ignoring the roots of arithmetical cognition. Such a contextual a priori theory is shown not to require any ontologically problematic assumptions, in addition to fitting well within a standard framework of general epistemology. (shrink)

I develop a non-representationalist account of mathematical thought, on which the point of mathematical theorizing is to provide us with the conceptual capacity to structure and articulate information about the physical world in an epistemically useful way. On my view, accepting a mathematical theory is not a matter of having a belief about some subject matter; it is rather a matter of structuring logical space, in a sense to be made precise. This provides an elegant account of the cognitive utility (...) of mathematics. Further, it makes explicit how the brand of non-representationalism I develop is compatible with there being substantive rationality constraints on our mathematical theorizing. (shrink)

The Bounds of Logic presents a new philosophical theory of the scope and nature of logic based on critical analysis of the principles underlying modern Tarskian logic and inspired by mathematical and linguistic development. Extracting central philosophical ideas from Tarski’s early work in semantics, Sher questions whether these are fully realized by the standard first-order system. The answer lays the foundation for a new, broader conception of logic. By generally characterizing logical terms, Sher establishes a fundamental result in semantics. Her (...) development of the notion of logicality for quantifiers and her work on branching are of great importance for linguistics. Sher outlines the boundaries of the new logic and points out some of the philosophical ramifications of the new view of logic for such issues as the logicist thesis, ontological commitment, the role of mathematics in logic, and the metaphysical underpinning of logic. She proposes a constructive definition of logical terms, reexamines and extends the notion of branching quantification, and discusses various linguistic issues and applications. (shrink)

This article investigates the claim that some truths are fundamentally or really true — and that other truths are not. Such a distinction can help us reconcile radically minimal metaphysical views with the verities of common sense. I develop an understanding of the distinction whereby Fundamentality is not itself a metaphysical distinction, but rather a device that must be presupposed to express metaphysical distinctions. Drawing on recent work by Rayo on anti-Quinean theories of ontological commitments, I formulate a rigourous theory (...) of the notion. In the final sections, I show how this package dovetails with ' interpretationist' theories of meaning to give sober content to thought that some things — perhaps sets, or gerrymandered mereological sums — can be 'postulated into existence'. (shrink)

This thesis articulates the resonances between J. M. Coetzee's lifelong engagement with mathematics and his practice as a novelist, critic, and poet. Though the critical discourse surrounding Coetzee's literary work continues to flourish, and though the basic details of his background in mathematics are now widely acknowledged, his inheritance from that background has not yet been the subject of a comprehensive and mathematically- literate account. In providing such an account, I propose that these two strands of his intellectual trajectory not (...) only developed in parallel, but together engendered several of the characteristic qualities of his finest work. The structure of the thesis is essentially thematic, but is also broadly chronological. Chapter 1 focuses on Coetzee's poetry, charting the increasing involvement of mathematical concepts and methods in his practice and poetics between 1958 and 1979. Chapter 2 situates his master's thesis alongside archival materials from the early stages of his academic career, and thus traces the development of his philosophical interest in the migration of quantificatory metaphors into other conceptual domains. Concentrating on his doctoral thesis and a series of contemporaneous reviews, essays, and lecture notes, Chapter 3 details the calculated ambivalence with which he therein articulates, adopts, and challenges various statistical methods designed to disclose objective truth. Chapter 4 explores the thematisation of several mathematical concepts in Dusklands and In the Heart of the Country. Chapter Five considers Waiting for the Barbarians and Foe in the context provided by Coetzee's interest in the attempts of Isaac Newton to bridge the gap between natural language and the supposedly transparent language of mathematics. Finally, Chapter 6 locates in Elizabeth Costello and Diary of a Bad Year a cognitive approach to the use of mathematical concepts in ethics, politics, and aesthetics, and, by analogy, a central aspect of the challenge Coetzee's late fiction poses to the contemporary literary landscape. (shrink)

David Lewis holds that a single possible world can provide more than one way things could be. But what are possible worlds good for if they come apart from ways things could be? We can make sense of this if we go in for a metaphysical understanding of what the world is. The world does not include everything that is the case—only the genuine facts. Understood this way, Lewis's “cheap haecceitism” amounts to a kind of metaphysical anti-haecceitism: it says there (...) aren't any genuine facts about individuals over and above their qualitative roles. (shrink)

Thomas Hobbes was, rather famously, a nominalist. The core of that nominalism is the belief that the only universal things are universal names: there are no universal objects, or universal ideas. This paper looks at what Hobbes's views about universal names were, how they evolved over time, and how Hobbes argued for them. The remainder of the paper considers two objections to Hobbes's view: a criticism made by several of Hobbes's contemporaries, that Hobbes's view could not account for people saying (...) the same thing in different languages; and a more recently popular criticism of Hobbes, that his nominalism's reliance on similarity implicitly (and inconsistently) involves reliance on a universal. (shrink)