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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 (...) 

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. 

Jody Azzouni argues that whilst it is indeterminate what the criteria for existence are, there is a criterion that has been collectively adopted to use ‘exist’ that we can employ to argue for positions in ontology. I raise and defend a novel objection to Azzouni: his view has the counterintuitive consequence that the facts regarding what exists can and will change when users of the word ‘exist’ change what criteria they associate with its usage. Considering three responses, I argue Azzouni (...) 

ABSTRACTMetaphysics has a problem with plurality: in many areas of discourse, there are too many good theories, rather than just one. This embarrassment of riches is a particular problem for metaphysical realists who want metaphysics to tell us the way the world is and for whom one theory is the correct one. A recent suggestion is that we can treat the different theories as being functionally or explanatorily equivalent to each other, even though they differ in content. The aim of (...) 

ABSTRACT This paper reveals David Hilbert’s position in the philosophy of mathematics, circa 1900, to be a form of noneliminative structuralism, predating his formalism. I argue that Hilbert withstands the pressing objections put to him by Frege in the course of the FregeHilbert controversy in virtue of this early structuralist approach. To demonstrate that this historical position deserves contemporary attention I show that Hilbertian structuralism avoids a recent wave of objections against noneliminative structuralists to the effect that they cannot distinguish (...) 

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 (...) 

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’ (...) 

In this paper, I describe and motivate a new species of mathematical structuralism, which I call Instrumental Nominalism about SetTheoretic Structuralism. As the name suggests, this approach takes standard SetTheoretic Structuralism of the sort championed by Bourbaki and removes its ontological commitments by taking an instrumental nominalist approach to that ontology of the sort described by Joseph Melia and Gideon Rosen. I argue that this avoids all of the problems that plague other versions of structuralism. 

Philosophers of mathematics agree that the only interpretation of arithmetic that takes that discourse at 'face value' is one on which the expressions 'N', '0', '1', '+', and 'x' are treated as proper names. I argue that the interpretation on which these expressions are treated as akin to free variables has an equal claim to be the default interpretation of arithmetic. I show that no purely syntactic test can distinguish proper names from free variables, and I observe that any semantic (...) 

If numbers were identified with any of their standard settheoretic realizations, then they would have various nonarithmetical properties that mathematicians are reluctant to ascribe to them. Dedekind and later structuralists conclude that we should refrain from ascribing to numbers such ‘foreign’ properties. We first rehearse why it is hard to provide an acceptable formulation of this conclusion. Then we investigate some forms of abstraction meant to purge mathematical objects of all ‘foreign’ properties. One form is inspired by Frege; the other (...) 

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 ClarkeDoane, or does it bolster mathematical realism, as authors such as Joyce and SinnottArmstrong 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 (...) 



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 knowhow. Ascriptions of knowhow 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 knowhow can be used to mount a powerful argument against intellectualism about knowhow—the view (...) 

It is a commonplace of set theory to say that there is no set of all wellorderings 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 (...) 

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 secondorder modal logic. The categorical component asserts the logical possibility of the (...) 

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 (...) 



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 topdown approach to explanation. We argue that these models can be complemented by a bottomup 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 (...) 

Olszewski claims that the ChurchTuring 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 Turingcomputable 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 (...) 

In this paper I defend mathematical nominalism by arguing that any reasonable account of scientific theories and scientific practice must make explicit the empirical nonmathematical grounds on which the application of mathematics is based. Once this is done, references to mathematical entities may be eliminated or explained away in terms of underlying empirical conditions. I provide evidence for this conclusion by presenting a detailed study of the applicability of mathematics to measurement. This study shows that mathematical nominalism may be regarded (...) 

This essay defends the following two claims: (1) liraitationofsize reasoning yields enough sets to meet the needs of most mathematicians; (2) set formation and mereological fusion share enough logical features to justify placing both in the genus composition (even when the components of a set are taken to be its members rather than its subsets). 

This note examines the mereological component of Geoffrey Hellman's most recent version of modal structuralism. There are plausible forms of agnosticism that benefit only a little from Hellman's mereological turn. 

Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous visavis 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 manytopoi view and modalstructuralism. 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 (...) 

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 spacetime continuum. I argue (contrary to Hellman) that these do not. (...) 





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 contradictioninducing 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 (...) 

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 (...) 

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

Crucial to Hilary Putnam’s realism in the philosophy of mathematics is to maintain the objectivity of mathematics without the commitment to the existence of mathematical objects. Putnam’s indispensability argument was devised as part of this conception. In this paper, I reconstruct and reassess Putnam’s argument for the indispensability of mathematics, and distinguish it from the more familiar, Quinean version of the argument. Although I argue that Putnam’s approach ultimately fails, I develop an alternative way of implementing his form of realism (...) 

ABSTRACT Neofregeanism and structuralism are among the most promising recent approaches to the philosophy of mathematics. Yet both have serious costs. We develop a view, structuralist neologicism, which retains the central advantages of each while avoiding their more serious costs. The key to our approach is using arbitrary reference to explicate how mathematical terms, introduced by abstraction principles, refer. Focusing on numerical terms, this allows us to treat abstraction principles as implicit definitions determining all properties of the numbers, achieving a (...) 

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 (...) 

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 takehome message is that understanding of the modal character of explanations (in dynamical systems theory) can undermine platonist arguments from explanatory indispensability. 

Everything you always wanted to know about structural realism but were afraid to ask Content Type Journal Article Pages 227276 DOI 10.1007/s1319401100257 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, HeinrichHeineUniversität Düsseldorf, Universitätsstraße 1, Geb. 23.21/04.86, 40225 Düsseldorf, Germany Journal European Journal for Philosophy of Science Online ISSN 18794920 Print ISSN 18794912 Journal Volume Volume 1 Journal Issue Volume 1, Number 2. 

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 (...) 

This monograph offers a fresh perspective on the applicability of mathematics in science. It explores what mathematics must be so that its applications to the empirical world do not constitute a mystery. In the process, readers are presented with a new version of mathematical structuralism. The author details a philosophy of mathematics in which the problem of its applicability, particularly in physics, in all its forms can be explained and justified. Chapters cover: mathematics as a formal science, mathematical ontology: what (...) 

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 (...) 

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 (...) 

One of the standard views on plural quantification is that its use commits one to the existence of abstract objects–sets. On this view claims like ‘some logicians admire only each other’ involve ineliminable quantification over subsets of a salient domain. The main motivation for this view is that plural quantification has to be given some sort of semantics, and among the two main candidates—substitutional and settheoretic—only the latter can provide the language of plurals with the desired expressive power (given that (...) 

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 (...) 

We develop a pointfree construction of the classical one dimensional continuum, with an interval structure based on mereology and either a weak set theory or logic of plural quantification. In some respects this realizes ideas going back to Aristotle,although, unlike Aristotle, we make free use of classical "actual infinity". Also, in contrast to intuitionistic, Bishop, and smooth infinitesimal analysis, we follow classical analysis in allowing partitioning of our "gunky line" into mutually exclusive and exhaustive disjoint parts, thereby demonstrating the independence (...) 



The concept of linguistic infinity has had a central role to play in foundational debates within theoretical linguistics since its more formal inception in the midtwentieth century. The conceptualist tradition, marshalled in by Chomsky and others, holds that infinity is a core explanandum and a link to the formal sciences. Realism/Platonism takes this further to argue that linguistics is in fact a formal science with an abstract ontology. In this paper, I argue that a central misconstrual of formal apparatus of (...) 

This thesis comprises three main chapters—each comprising one relatively standalone paper. The unifying theme is fragmentalism about truth, which is the view that the predicate “true” either expresses distinct concepts or expresses distinct properties. / In Chapter 1, I provide a formal development of alethic pluralism. Pluralism is the view that there are distinct truth properties associated with distinct domains of subject matter, where a truth property satisfies certain truthcharacterizing principles. On behalf of pluralists, I propose an account of logic (...) 

Quine has famously put forward the indispensability argument to force belief in the existence of mathematical objects (such as classes) due to their indispensability to our best theories of the world (Quine 1960). Quine has also advocated the indeterminacy of reference argument, according to which reference is dramatically indeterminate: given a language, there’s no unique reference relation for that language (see Quine 1969a). In this paper, I argue that these two arguments are in conflict with each other. Whereas the indispensability (...) 

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 nontraditional, holistic tools to achieve this result. In Part Two I delineate an outline of a foundation for logic, employing the new methodology. The (...) 



This dissertation makes two primary contributions. The first three chapters develop an interpretation of Carnap's MetaPhilosophical Program which places stress upon his methodological analysis of the sciences over and above the Principle of Tolerance. Most importantly, I suggest, is that Carnap sees philosophy as contiguous with science—as a part of the scientific enterprise—so utilizing the very same methods and subject to the same limitations. I argue that the methodological reforms he suggests for philosophy amount to philosophy as the explication of (...) 

ABSTRACTWords 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 typetoken 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 (...) 