The notion of potential infinity dominated in mathematical thinking about infinity from Aristotle until Cantor. The coherence and philosophical importance of the notion are defended. Particular attention is paid to the question of whether potential infinity is compatible with classical logic or requires a weaker logic, perhaps intuitionistic.

This article presents an overview of the basic philosophical motivations for, and some recent work in, modal set theory.

I examine explanations’ realist commitments in relation to dynamical systems theory. First I rebut an ‘explanatory indispensability argument’ for mathematical realism from the explanatory power of phase spaces (Lyon and Colyvan 2007). Then I critically consider a possible way of strengthening the indispensability argument by reference to attractors in dynamical systems theory. The take-home message is that understanding of the modal character of explanations (in dynamical systems theory) can undermine platonist arguments from explanatory indispensability.

Drawing an analogy between modal structuralism about mathematics and theism, I o er a structuralist account that implicitly de nes theism in terms of three basic relations: logical and metaphysical priority, and epis- temic superiority. On this view, statements like `God is omniscient' have a hypothetical and a categorical component. The hypothetical component provides a translation pattern according to which statements in theistic language are converted into statements of second-order modal logic. The categorical component asserts the logical possibility of the (...) theism struc- ture on the basis of uncontroversial facts about the physical world. This structuralist reading of theism preserves objective truth-values for theistic statements while remaining neutral on the question of ontology. Thus, it o ers a way of understanding theism to which a naturalist cannot object, and it accommodates the fact that religious belief, for many theists, is an essentially relational matter. (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)

Humans and other animals have an evolved ability to detect discrete magnitudes in their environment. Does this observation support evolutionary debunking arguments against mathematical realism, as has been recently argued by Clarke-Doane, or does it bolster mathematical realism, as authors such as Joyce and Sinnott-Armstrong have assumed? To find out, we need to pay closer attention to the features of evolved numerical cognition. I provide a detailed examination of the functional properties of evolved numerical cognition, and propose that they prima (...) facie favor a realist account of numbers. (shrink)

Much recent philosophy of physics has investigated the process of symmetry breaking. Here, I critically assess the alleged symmetry restoration at the fundamental scale. I draw attention to the contingency that gauge symmetries exhibit, that is, the fact that they have been chosen from an infinite space of possibilities. I appeal to this feature of group theory to argue that any metaphysical account of fundamental laws that expects symmetry restoration up to the fundamental level is not fully satisfactory. This is (...) a symmetry argument in line with Curie’s first principle. Further, I argue that this same feature of group theory helps to explain the ‘unreasonable’ effectiveness of mathematics in physics, and that it reduces the philosophical significance that has been attributed to the objectivity of gauge symmetries. (shrink)

Mathematical concepts play at least three roles in the application of mathematics: an inferential role, a representational role, and an expressive role. In this paper, I argue that, despite what has often been alleged, platonists do not fully accommodate these features of the application of mathematics. At best, platonism provides partial ways of handling the issues. I then sketch an alternative, anti-realist account of the application of mathematics, and argue that this account manages to accommodate these features of the application (...) process. In this way, a better account of mathematical applications is, in principle, available. (shrink)

It is a commonplace of set theory to say that there is no set of all well-orderings nor a set of all sets. We are implored to accept this due to the threat of paradox and the ensuing descent into unintelligibility. In the absence of promising alternatives, we tend to take up a conservative stance and tow the line: there is no universe. In this paper, I am going to challenge this claim by taking seriously the idea that we can (...) talk about the collection of all the sets and many more collections beyond that. A method of articulating this idea is offered through an indefinitely extending hierarchy of set theories. It is argued that this approach provides a natural extension to ordinary set theory and leaves ordinary mathematical practice untouched. (shrink)

Although there is a consensus among philosophers of mathematics and mathematicians that mathematical explanations exist, only a few authors have proposed accounts of explanation in mathematics. These accounts fit into the unificationist or top-down approach to explanation. We argue that these models can be complemented by a bottom-up approach to explanation in mathematics. We introduce the mechanistic model of explanation in science and discuss the possibility of using this model in mathematics, arguing that using it does not presuppose a Platonist (...) view of mathematics and allows one to gain insight into why a theorem is true by answering what-if-things-had-been-different questions. (shrink)

The construction of a systematic philosophical foundation for logic is a notoriously difficult problem. In Part One I suggest that the problem is in large part methodological, having to do with the common philosophical conception of “providing a foundation”. I offer an alternative to the common methodology which combines a strong foundational requirement with the use of non-traditional, holistic tools to achieve this result. In Part Two I delineate an outline of a foundation for logic, employing the new methodology. The (...) outline is based on an investigation of why logic requires a veridical justification, i.e., a justification which involves the world and not just the mind, and what features or aspect of the world logic is grounded in. Logic, the investigation suggests, is grounded in the formal aspect of reality, and the outline proposes an account of this aspect, the way it both constrains and enables logic, logic's role in our overall system of knowledge, the relation between logic and mathematics, the normativity of logic, the characteristic traits of logic, and error and revision in logic. (shrink)

I begin by drawing a parallel between the intuitionistic understanding of quantification over all natural numbers and the generality relativist understanding of quantification over absolutely everything. I then argue that adoption of an intuitionistic reading of relativism not only provides an immediate reply to the absolutist's charge of incoherence but it also throws a new light on the debates surrounding absolute generality.

Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis-a-vis set theory. It is argued here that these theories require a background logic of relations and substantive assumptions addressing mathematical existence of categories themselves. We propose a synthesis of Bell's many-topoi view and modal-structuralism. Surprisingly, a combination of mereology and plural quantification suffices to describe hypothetical large domains, recovering the Grothendieck method of universes. Both topos theory and set theory can be carried out (...) relative to such domains; puzzles about ‘large categories’ and ‘proper classes’ are handled in a uniform way, by relativization, sustaining insights of Zermelo. (shrink)

Abstractionist programs in the philosophy of mathematics have focused on abstraction principles, taken as implicit definitions of the objects in the range of their operators. In second-order logic (SOL) with predicative comprehension, such principles are consistent but also (individually) mathematically weak. This paper, inspired by the work of Boolos (Proceedings of the Aristotelian Society 87, 137–151, 1986) and Zalta (Abstract Objects, vol. 160 of Synthese Library, 1983), examines explicit definitions of abstract objects. These axioms state that there is a unique (...) abstract encoding all concepts satisfying a given formula $$\phi (F)$$, with F a concept variable. Such a system is inconsistent in full SOL. It can be made consistent with several intricate tweaks, as Zalta has shown. Our approach in this article is simpler: we use a novel method to establish consistency in a restrictive version of predicative SOL. The resulting system, RPEAO, interprets first-order PA in extensional contexts, and has a natural extension delivering a peculiar interpretation of PA $$^2$$. (shrink)

The realist versions of mathematical structuralism are often characterized by what I call ‘the insubstantiality thesis’, according to which mathematical objects, being positions in structures, have no non-structural properties: they are purely structural objects. The thesis has been criticized for being inconsistent or descriptively inadequate. In this paper, by implementing the resources of a real-definitional account of essence in the context of Fregean abstraction principles, I offer a version of structuralism – essentialist structuralism – which validates a weaker version of (...) the insubstantiality thesis: mathematical objects have no non-structural essential properties. Next, I show how this rendition of structuralism alleviates a Fregean worry against insubstantiality, which is directed at the explanation of the applicability of mathematics from the structuralist perspective. (shrink)

Pluralist mathematical realism, the view that there exists more than one mathematical universe, has become an influential position in the philosophy of mathematics. I argue that, if mathematical pluralism is true (and we have good reason to believe that it is), then mathematical realism cannot (easily) be justified by arguments from the indispensability of mathematics to science. This is because any justificatory chain of inferences from mathematical applications in science to the total body of mathematical theorems can cover at most (...) one mathematical universe. Indispensability arguments may thus lose their central role in the debate about mathematical ontology. (shrink)

We observe that Putnam’s model-theoretic argument against determinacy of the concept of second-order quantification or that of the set is harmless to the nominalist. It serves as a good motivation for the nominalist philosophy of mathematics. But in the end it can lead to a serious challenge to the nominalist account of mathematical objectivity if some minimal assumptions about the relation between mathematical objectivity and logical objectivity are made. We consider three strategies the nominalist might take to meet this challenge, (...) and we argue that all these strategies are untenable. (shrink)

In ontological debates, realists typically argue for their view via one of two approaches. The _Quinean approach_ employs naturalistic arguments that say our scientific practices give us reason to affirm the existence of a kind of entity. The _Fregean approach_ employs linguistic arguments that say we should affirm the existence of a kind of entity because our discourse contains reference to those entities. These two approaches are often seen as distinct, with _indispensability arguments_ typically associated with the former, but not (...) the latter, approach. This paper argues for a connection between the two approaches on the grounds that the typical arguments of the Fregean approach can be reformulated as indispensability arguments. This connection is significant in at least two ways. First, it implies that indispensability arguments provide a common framework within which to compare the Quinean and Fregean approaches, which allows for a more precise delineation of the two approaches. Second, it implies the possibility of analogical relations that allow proponents and opponents of each approach to draw upon the ideas that have been developed regarding the other. (shrink)

Several natural languages such as English contain prima facie different kinds of referential and quantificational expressions. In particular, natural languages.

This paper provides an exposition of the structuralist approach to underdetermination, which aims to resolve the underdetermination of theories by identifying their common theoretical structure. Applications of the structuralist approach can be found in many areas of philosophy. I present a schema of the structuralist approach, which conceptually unifies such applications in different subject matters. It is argued that two classic arguments in the literature, Paul Benacerraf’s argument on natural numbers and W. V. O. Quine’s argument for the indeterminacy of (...) translation, can be analyzed as instances of the structuralist schema. These two applications illustrate different kinds of conclusions that can be drawn through the structuralist approach; Benacerraf’s argument shows that we can derive an ontological conclusion about the given subject matter, while Quine’s structuralist approach leads to a semantic conclusion about how to determine linguistic meanings given radical translation. Then, as a case study, I review a recent debate in metaphysics between Shamik Dasgupta, Jason Turner, and Catharine Diehl to consider the extent to which different instances of the structuralist schema are conceptually unified. Both sides of the debate can be interpreted as utilizing the structuralist approach; one side uses the structuralist approach for an ontological conclusion, while the other side relies on a semantic conclusion. I argue that this has a strong dialectical consequence, which sheds light on the conceptual unity of the structuralist approach. (shrink)

We contribute to the ongoing discussion on mathematical structuralism by focusing on a question that has so far been neglected: when is a structure part of another structure? This paper is a first step towards answering the question. We will show that a certain conception of structures, abstractionism about structures, yields a natural definition of the parthood relation between structures. This answer has many interesting consequences; however, it conflicts with some standard mereological principles. We argue that the tension between abstractionism (...) about structure and classical mereology is an interesting result and conclude that the mereology of abstract structures is a subject that deserves further exploration. We also point out some connections between our discussion of the mereology of structures and recent work on non-well-founded mereologies. (shrink)

Inspired by Cantor's Theorem (CT), orthodoxy takes infinities to come in different sizes. The orthodox view has had enormous influence in mathematics, philosophy, and science. We will defend the contrary view---Countablism---according to which, necessarily, every infinite collection (set or plurality) is countable. We first argue that the potentialist or modal strategy for treating Russell's Paradox, first proposed by Parsons (2000) and developed by Linnebo (2010, 2013) and Linnebo and Shapiro (2019), should also be applied to CT, in a way that (...) vindicates Countabilism. Our discussion dovetails with recent independently developed treatments of CT in Meadows (2015), Pruss (2020), and Scambler (2021), aimed at establishing the mathematical viability, and therefore epistemic possibility, of Countabilism. Unlike these authors, our goal isn't to vindicate the mathematical underpinnings of Countabilism. Rather, we aim to argue that, given that Countabilism is mathematically viable, Countabilism should moreover be regarded as true. After clarifying the modal content of Countabilism, we canvas some of Countabilism's many positive implications, including that Countabilism provides the best account of the pervasive independence phenomena in set theory, and that Countabilism has the power to defuse several persistent puzzles and paradoxes found in physics and metaphysics. We conclude that in light of its theoretical and explanatory advantages, Countabilism is more likely true than not. (shrink)

Are there genuine mathematical explanations of physical phenomena, and if so, how can mathematical theories, which are typically thought to concern abstract mathematical objects, explain contingent empirical matters? The answer, I argue, is in seeing an important range of mathematical explanations as structural explanations, where structural explanations explain a phenomenon by showing it to have been an inevitable consequence of the structural features instantiated in the physical system under consideration. Such explanations are best cast as deductive arguments which, by virtue (...) of their form, establish that, given the mathematical structure instantiated in the physical system under consideration, the explanandum had to occur. Against the claims of platonists such as Alan Baker and Mark Colyvan, I argue that formulating mathematical explanations as structural explanations in this way shows that we can accept that mathematics can play an indispensable explanatory role in empirical science without committing to the existence of any abstract mathematical objects. (shrink)

Platonists affirm the existence of abstract mathematical objects, and Nominalists deny the existence of abstract mathematical objects. While there are standard arguments in favor of Nominalism, these arguments fail to account for the necessity of Nominalism. Furthermore, these arguments do nothing to explain why Nominalism is true. They only point to certain theoretical vices that might befall the Platonist. The goal of this paper is to formulate and defend a simple, valid argument for the necessity of Nominalism that seeks to (...) precisify the widespread intuition that mathematical objects are somehow ‘spooky’ or ‘mysterious’. (shrink)

In recent years, one of the main thrusts of set-theoretic research has been the investigation of maximality principles for V, the universe of sets. The Hyperuniverse Programme (HP) has formulated several maximality principles, which express the maximality of V both in height and width. The paper provides an overview of the principles which have been investigated so far in the programme, as well as of the logical and model-theoretic tools which are needed to formulate them mathematically, and also briefly shows (...) how optimal principles, among those available, may be selected in a justifiable way. (shrink)

This is Part A of an article that defends non-eliminative structuralism about mathematics by means of a concrete case study: a theory of unlabeled graphs. Part A summarizes the general attractions of non-eliminative structuralism. Afterwards, it motivates an understanding of unlabeled graphs as structures sui generis and develops a corresponding axiomatic theory of unlabeled graphs. As the theory demonstrates, graph theory can be developed consistently without eliminating unlabeled graphs in favour of sets; and the usual structuralist criterion of identity can (...) be applied successfully in graph-theoretic proofs. Part B will turn to the philosophical interpretation and assessment of the theory. (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)

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)

This paper aims to unite two seemingly disparate themes in the philosophy of mathematics and language respectively, namely ante rem structuralism and inferentialism. My analysis begins with describing both frameworks in accordance with their genesis in the work of Hilbert. I then draw comparisons between these philosophical views in terms of their similar motivations and similar objections to the referential orthodoxy. I specifically home in on two points of comparison, namely the role of norms and the relation of ontological dependence (...) in both accounts. Lastly, I show that insights from this purported connection can address certain objections to both theories respectively. (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)

According to the iterative conception of set, each set is a collection of sets formed prior to it. The notion of priority here plays an essential role in explanations of why contradiction-inducing sets, such as the Russell set, do not exist. Consequently, these explanations are successful only to the extent that a satisfactory priority relation is made out. I argue that attempts to do this have fallen short: understanding priority in a straightforwardly constructivist sense threatens the coherence of the empty (...) set and raises serious epistemological concerns; but the leading realist interpretations---ontological and modal interpretations of priority---are deeply problematic as well. I conclude that the purported explanatory virtues of the iterative conception are, at present, unfounded. (shrink)

Quantifier variance faces a number of difficulties. In this paper we first formulate the view as holding that the meanings of the quantifiers may vary, and that languages using different quantifiers may be charitably translated into each other. We then object to the view on the basis of four claims: (i) quantifiers cannot vary their meaning extensionally by changing the domain of quantification; (ii) quantifiers cannot vary their meaning intensionally without collapsing into logical pluralism; (iii) quantifier variance is not an (...) ontological doctrine; (iv) quantifier variance is not compatible with charitable translation and as such is internally inconsistent. In light of these troubles, we recommend the dissolution of quantifier variance and suggest that the view be laid to rest. (shrink)

Stewart Shapiro’s ante rem structuralism recognizes the structural or ‘algebraic’ aspects of mathematical practice while still offering a face-value semantics. Fictionalism, as a purely ‘algebraic’ approach, is held to be at a disadvantage, as compared with Shapiro’s structuralism, in not interpreting mathematics at face value. However, the face-value reading of mathematical singular terms has difficulty explaining how we can use such terms to pick out a unique referent in cases where the relevant mathematical structures admit non-trivial automorphisms. Shapiro offers a (...) solution to this difficulty, but his solution, I argue, evens the score between Shapiro’s structuralism and fictionalism. (shrink)

To the memory of Prof. Grigori Mints, Stanford UniversityBorn: June 7, 1939, St. Petersburg, RussiaDied: May 29, 2014, Palo Alto, California.

Scientific models are important, if not the sole, units of science. This thesis addresses the following question: in virtue of what do scientific models represent their target systems? In Part i I motivate the question, and lay out some important desiderata that any successful answer must meet. This provides a novel conceptual framework in which to think about the question of scientific representation. I then argue against Callender and Cohen’s attempt to diffuse the question. In Part ii I investigate the (...) ideas that scientific models are ‘similar’, or structurally morphic, to their target systems. I argue that these approaches are misguided, and that at best these relationships concern the accuracy of a pre-existing representational relationship. I also pay particular attention to the sense in which target systems can be appropriately taken to exhibit a ‘structure’, and van Fraassen’s recent argument concerning the pragmatic equivalence between representing phenomena and data. My next target is the idea that models should not be seen as objects in their own right, but rather what look like descriptions of them are actually direct descriptions of target systems, albeit not ones that should be understood literally. I argue that these approaches fail to do justice to the practice of scientific modelling. Finally I turn to the idea that how models represent is grounded, in some sense, in their inferential capacity. I compare this approach to anti-representationalism in the philosophy of language and argue that analogous issues arise in the context of scientific representation. Part iii contains my positive proposal. I provide an account of scientific representation based on Goodman and Elgin’s notion of representation-as. The result is a highly conventional account which is the appropriate level of generality to capture all of its instances, whilst remaining informative about the notion. I illustrate it with reference to the Phillips-Newlyn machine, models of proteins, and the Lotka-Volterra model of predator-prey systems. These examples demonstrate how the account must be understood, and how it sheds light on our understanding of how models are used. I finally demonstrate how the account meets the desiderata laid out at the beginning of the thesis, and outline its implications for further questions from the philosophy of science; not limited to issues surrounding the applicability of mathematics, idealisation, and what it takes for a model to be ‘true’. (shrink)

Everything you always wanted to know about structural realism but were afraid to ask Content Type Journal Article Pages 227-276 DOI 10.1007/s13194-011-0025-7 Authors Roman Frigg, Department of Philosophy, Logic and Scientific Method, London School of Economics and Political Science, Houghton Street, London, WC2A 2AE UK Ioannis Votsis, Philosophisches Institut, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, Geb. 23.21/04.86, 40225 Düsseldorf, Germany Journal European Journal for Philosophy of Science Online ISSN 1879-4920 Print ISSN 1879-4912 Journal Volume Volume 1 Journal Issue Volume 1, Number 2.

This paper examines the problem of extending the programme of mathematical constructivism to applied mathematics. I am not concerned with the question of whether conventional mathematical physics makes essential use of the principle of excluded middle, but rather with the more fundamental question of whether the concept of physical infinity is constructively intelligible. I consider two kinds of physical infinity: a countably infinite constellation of stars and the infinitely divisible space-time continuum. I argue (contrary to Hellman) that these do not. (...) pose any insuperable problem for constructivism, and that constructivism may have a useful new perspective to offer on physics. (shrink)

Perhaps the most pressing challenge for singularism—the predominant view that definite plurals like ‘the students’ singularly refer to a collective entity, such as a mereological sum or set—is that it threatens paradox. Indeed, this serves as a primary motivation for pluralism—the opposing view that definite plurals refer to multiple individuals simultaneously through the primitive relation of plural reference. Groups represent one domain in which this threat is immediate. After all, groups resemble sets in having a kind of membership-relation and iterating: (...) we can have groups of groups, groups of groups of groups, etc. Yet there cannot be a group of all non-self-membered groups. In response, we develop a potentialist theory of groups according to which we always can, but do not have to, form a group from any sum. Modalizing group-formation makes it a species of potential, as opposed to actual or completed, infinity. This allows for a consistent, plausible, and empirically adequate treatment of natural language plurals, one which is motivated by the iterative nature of syntactic and semantic processes more generally. (shrink)

Recent work in the philosophy of language attempts to elucidate the elusive notion of aboutness. A natural question concerning such a project has to do with its motivation: why is the notion of aboutness important? Stephen Yablo offers an interesting answer: taking into consideration not only the conditions under which a sentence is true, but also what a sentence is about opens the door to a new style of criticism of certain philosophical analyses. We might criticize the analysis of a (...) given notion not because it fails to assign the right truth conditions to a class of sentences, but because it characterizes those sentences as being about something they are not about. In this paper, I apply Yablo’s suggestion to a case study. I consider meta-fictionalism, the view that the content of a mathematical claim S is ‘according to standard mathematics, S’. I argue, following Woodward, that, on certain assumptions, meta-fictionalism assigns the right truth-conditions to typical assertoric utterances of mathematical statements. However, I also argue that meta-fictionalism assigns the wrong aboutness conditions to typical assertoric utterances of mathematical statements. (shrink)

Attempts at solving what has been labeled as Eugene Wigner’s puzzle of applicability of mathematics are still far from arriving at an acceptable solution. The accounts developed to explain the “miracle” of applied mathematics vary in nature, foundation, and solution, from denying the existence of a genuine problem to designing structural theories based on mathematical formalism. Despite this variation, all investigations treated the problem in a unitary way with respect to the target, pointing to one or two ‘why’ or ‘how’ (...) questions to be answered. In this paper, I argue that two analyses, a semantic analysis ab initio and a metatheoretical analysis starting from the types of unreasonableness involved in this problem, will establish the interdisciplinary character of the problem and reveal many more targets, which may be addressed with different methodologies. In order to address objectively the philosophical problem of applicability of mathematics, a foundational revision of the problem is needed. (shrink)

A core commitment of Bob Hale and Crispin Wright’s neologicism is their invocation of Frege’s Constraint—roughly, the requirement that the core empirical applications for a class of numbers be “built directly into” their formal characterization. According to these neologicists, if legitimate, Frege’s Constraint adjudicates in favor of their preferred foundation—Hume’s Principle—and against alternatives, such as the Dedekind–Peano axioms. In this paper, we consider a recent argument for legitimating Frege’s Constraint due to Hale, according to which the primary empirical application of (...) the naturals is transitive counting, or answering ‘how many’-questions using numerals. We make two claims regarding Hale’s argument. First, it fails to legitimate Frege’s Constraint in virtue of resting on unsupported and highly contentious assumptions. Secondly, even if sound, Hale’s argument would vindicate a version of Frege’s Constraint which fails to adjudicate in favor of Hume’s Principle over alternative characterizations of the naturals. (shrink)

Words form a fundamental basis for our understanding of linguistic practice. However, the precise ontology of words has eluded many philosophers and linguists. A persistent difficulty for most accounts of words is the type-token distinction [Bromberger, S. 1989. “Types and Tokens in Linguistics.” In Reflections on Chomsky, edited by A. George, 58–90. Basil Blackwell; Kaplan, D. 1990. “Words.” Aristotelian Society Supplementary Volume LXIV: 93–119]. In this paper, I present a novel account of words which differs from the atomistic and platonistic (...) conceptions of previous accounts which I argue fall prey to this problem. Specifically, I proffer a structuralist account of linguistic items, along the lines of structuralism in the philosophy of mathematics [Shapiro, S. 1997. Philosophy of Mathematics: Structure and Ontology. Oxford University Press], in which words are defined in part as positions in larger linguistic structures. I then follow Szabò [1999. “Expressions and Their Representations.” The Philosophical Quarterly 49 (195): 145–163] and Parsons [1990. “The Structuralist View of Mathematical Objects.” Synthese 84: 303–346] in further defining words as quasi-concrete objects according to a representation relation. This view aims for general correspondence with contemporary generative linguistic approaches to the study of language. (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 version of the permutation argument in the philosophy of mathematics leads to the thesis that mathematical terms, contrary to appearances, are not genuine singular terms referring to individual objects; they are purely schematic or variables. By postulating ‘ante-rem structures’, the ante-rem structuralist aims to defuse the permutation argument and retain the referentiality of mathematical terms. This paper presents two semantic problems for the ante- rem view: (1) ante-rem structures are themselves subject to the permutation argument; (2) the ante-rem structuralist (...) fails to explain reference in a way that makes her account different to, and privileged over, that of her eliminativist rivals. Both problems undercut the motivation behind ante-rem structuralism. (shrink)

Regardless or independent of any actuality or actualization and exempt from spatiotemporal and causal conditions, each individual possibility is pure. Actualism excludes the existence of individual pure possibilities, altogether or at least as existing independently of actual reality. In this paper, I demonstrate, on the grounds of my possibilist metaphysics—panenmentalism—how some of the most fascinating scientific discoveries in chemistry could not have been accomplished without relying on pure possibilities and the ways in which they relate to each other . The (...) discoveries are the following: Dan Shechtman’s discovery of quasicrystals; Linus Pauling’s alpha helix; the discovery of F. Sherwood Rowland and Mario J. Molina concerning the destruction of the atmospheric ozone layer; and Neil Bartlett’s noble gas compounds. On the grounds of the analysis of these cases, actualism must fail, whereas panenmentalism gains support. (shrink)

Olszewski claims that the Church-Turing thesis can be used in an argument against platonism in philosophy of mathematics. The key step of his argument employs an example of a supposedly effectively computable but not Turing-computable function. I argue that the process he describes is not an effective computation, and that the argument relies on the illegitimate conflation of effective computability with there being a way to find out . ‘Ah, but,’ you say, ‘what’s the use of its being right twice (...) a day, if I can’t tell when the time comes?’ Why, suppose the clock points to eight o’clock, don’t you see that the clock is right at eight o’clock? Consequently, when eight o’clock comes round your clock is right. Lewis Carroll. (shrink)

Structuralism about mathematical objects and structuralist accounts of physical computation both face indeterminacy objections. For the former, the problem arises for cases such as the complex roots i and \, for which a automorphism can be defined, thus establishing the structural identity of these importantly distinct mathematical objects. In the case of the latter, the problem arises for logical duals such as AND and OR, which have invertible structural profiles :369–400, 2001). This makes their physical implementations indeterminate, in the sense (...) that their structural profiles alone cannot establish whether a given physical component is an AND-gate or an OR-gate. Doherty has recently shown both problems to be analogous, and has argued that computational structuralism is threatened with the absurd conclusion that computational digits might be indiscernible, such that, if structural properties are all that we have to go on, the binary digit 0 must be treated as identical to the binary digit 1. However, we think that a solution to the indiscernibility problem for mathematical structuralists, drawing on the work of David Hilbert, can be adapted for the analogous problem in the computational case, thereby rescuing the structuralist approach to physical computation. (shrink)

In this paper, I utilise the growing literature on scientific modelling to investigate the nature of formal semantics from the perspective of the philosophy of science. Specifically, I incorporate the inferential framework proposed by Bueno and Colyvan : 345–374, 2011) in the philosophy of applied mathematics to offer an account of how formal semantics explains and models its data. This view produces a picture of formal semantic models as involving an embedded process of inference and representation applying indirectly to linguistic (...) phenomena. The final aim of the paper is directed at proposing a novel account of the syntax–semantics interface while shedding light on empty categories, semantically null forms, underspecified content and compositionality as a whole. (shrink)