I use my reading of Plato to develop what I call as-ifism, the view that, in mathematics, we treat our hypotheses as if they were first principles and we do this with the purpose of solving mathematical problems. I then extend this view to modern mathematics showing that when we shift our focus from the method of philosophy to the method of mathematics, we see that an as-if methodological interpretation of mathematical structuralism can be used to provide an account of (...) the practice and the applicability of mathematics while avoiding the conflation of metaphysical considerations with mathematical ones. (shrink)

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.

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)

An epistemic formalization of arithmetic is constructed in which certain non-trivial metatheoretical inferences about the system itself can be made. These inferences involve the notion of provability in principle, and cannot be made in any consistent extensions of Stewart Shapiro's system of epistemic arithmetic. The system constructed in the paper can be given a modal-structural interpretation.

We develop a point-free 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 (...) of "indecomposability" from a non-punctiform conception. It is surprising that such simple axioms as ours already imply the Archimedean property and that they determine an isomorphism with the Dedekind-Cantor structure of R as a complete, separable, ordered field. We also present some simple topological models of our system, establishing consistency relative to classical analysis. Finally, after describing how to nominalize our theory, we close with comparisons with earlier efforts related to our own. (shrink)

This compact volume, belonging to the Cambridge Elements series, is a useful introduction to some of the most fundamental questions of philosophy and foundations of mathematics. What really distinguishes realist and platonist views of mathematics from anti-platonist views, including fictionalist and nominalist and modal-structuralist views?1 They seem to confront similar problems of justification, presenting tradeoffs between which it is difficult to adjudicate. For example, how do we gain access to the abstract posits of platonist accounts of arithmetic, analysis, geometry, etc., (...) including numbers, functions, sets, points, lines, spaces, etc., whether it be such objects themselves or the possibilities of such, as postulated by modal and modal structural accounts?2 What real difference does it make, whether it be the existence of such things or the mathematical possibility of such things? Even fictionalist views seem to confront analogous problems. After all, we rely on mathematics for myriad scientific applications; so such mathematics had better at least be coherent, even if not true. But coherence requires at least formal consistency; so we seem implicitly to be committed to things like formal derivations, viz. the absence of any such having contradictory consequences, framed within the appropriate system of axioms and rules. But derivations are themselves akin to mathematical objects, as derivations are strings of strings of symbols. Moreover, derivations provided by axioms and rules form an infinite class, such that, at any future time, infinitely many will not have been written down. Moreover, such strings, as Gödel famously showed, can be coded as natural numbers. Thus, even fictionalist accounts (such as that of Field in his Science without Numbers [2016]) seem to confront problems very much like those plaguing platonist accounts. (shrink)

Probably there is no position in Goodman’s corpus that has generated greater perplexity and criticism than Goodman’s “nominalism”. As is abundantly clear from Goodman’s writings, it is not “abstract entities” generally that he questions—indeed, he takes sensory qualia as “basic” in his Carnap-inspired constructional system in Structure—but rather just those abstracta that are so crystal clear in their identity conditions, so fundamental to our thought, so prevalent and seemingly unavoidable in our discourse and theorizing that they have come to form (...) the generally accepted framework for the most time-honored, exact, sophisticated, refined, central, and secure branch of human knowledge yet devised, mathematics itself! Of all the abstracta to question, why sets? Of course, Goodman gave his “reasons”, the unintelligibility of “generating” an infinitude of “constructed objects” automatically from any given object or objects. But critics have been quick to point out that set theory is intended not as a theory of what can be “generated” or “constructed” from given objects in any literal sense but rather as a theory of a certain realm of objects independently existing in their own right. “Construction” is a metaphor. (shrink)

First we review highlights of the ongoing debate about foundations of category theory, beginning with Fefermantop-down” approach, where particular categories and functors need not be explicitly defined. Possible reasons for resisting the proposal are offered and countered. The upshot is to sustain a pluralism of foundations along lines actually foreseen by Feferman (1977), something that should be welcomed as a way of resolving this long-standing debate.

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)

Ante rem structures were posited as the subject matter of mathematics in order to resolve a problem of referential indeterminacy within mathematical discourse. Nevertheless, ante rem structuralists are inevitably committed to the existence of indiscernible entities, and this commitment produces an exactly analogous problem. If it cannot be sorted out, then the postulation of ante rem structures is futile. In a recent paper, Stewart Shapiro argued that the problem may be solved by analysing some of the singular terms of mathematics (...) not as genuinely referring expressions, but as instantial terms. In this paper, I discuss several competing accounts of the semantics of terms of this kind, and argue that they are all untenable for the ante rem structuralist. Shapiro, then, still owes us an account of the semantics of instantial terms that suits the ante rem structuralist project. Without it, the ante rem structuralist is still unable to determine the reference of the singular terms of mathematics. (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)

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)

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)

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.

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)

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)

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)

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)

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)

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)

ABSTRACT This paper reveals David Hilbert’s position in the philosophy of mathematics, circa 1900, to be a form of non-eliminative structuralism, predating his formalism. I argue that Hilbert withstands the pressing objections put to him by Frege in the course of the Frege-Hilbert 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 non-eliminative structuralists to the effect that they cannot distinguish (...) between structurally identical but importantly distinct mathematical objects, such as the complex roots of $-1$. (shrink)

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 (...) argument supports realism about mathematics, the indeterminacy of reference argument, when applied to mathematics, provides a powerful strategy in support of mathematical anti-realism. I conclude the paper by indicating why the indeterminacy of reference phenomenon should be preferred over the considerations regarding indispensability. In the end, even the Quinean shouldn’t be a realist (platonist) about mathematics. (shrink)

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 dissertation makes two primary contributions. The first three chapters develop an interpretation of Carnap's Meta-Philosophical 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 (...) the concepts of science through the construction and use of suitably robust meta-logical languages. My primary interpretive claim is that Carnap's understanding of logic and mathematics as a set of formal auxiliaries is premised upon this prior analysis of the character of logico-mathematical knowledge, his understanding of its role in the language of science, and the methods used by practicing mathematicians. Thus the Principle of Tolerance, and so Carnap's logical pluralism, is licensed and justified by these methodological insights. This interpretation of Carnap's program contrasts with the popular Deflationary reading as proposed in Goldfarb & Ricketts. The leading idea they attribute to Carnap is a Logocentrism: That philosophical assertions are always made relative to some particular language, and that our choice of syntactical rules for a language are constitutive of its inferential structure and methods of possible justification. Consequently Tolerance is considered the foundation of Carnap's entire program. My third chapter argues that this reading makes Carnap's program philosophically inert, and I present significant evidence that such a reading is misguided. The final chapter attempts to extend the methodological ideals of Carnap's program to the analysis of the ongoing debate between category- and set-theoretic foundations for mathematics. Recent criticism of category theory as a foundation charges that it is neither autonomous from set theory, nor offers a suitable ontological grounding for mathematics. I argue that an analysis of concepts can be foundationally informative without requiring the construction of those concepts from first principles, and that ontological worries can be seen as methodologically unfruitful. (shrink)

Approaches to truth and the Liar paradox seem invariably to face a dilemma: either appeal to some sort of hierarchy, or declare apparently perfectly coherent concepts incoherent. But since both options lead to severe expressive restrictions, neither seems satisfactory. The aim of this paper is a new approach, which avoids the dilemma and the resulting expressive restrictions. Previous approaches tend to appeal to some new sort of semantic value for the truth predicate to take. I argue that such approaches inevitably (...) lead to the dilemma in question. In contrast, the present proposal sticks with classical semantic values, but allows the compositional rules associated with these to admit of exceptions. I show how such an approach can be developed systematically. (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)

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)

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)

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 truth-characterizing principles. On behalf of pluralists, I propose an account of logic (...) and semantics that shows how they can answer central conceptual and logical challenges for their view. -/- In Chapter 2, I motivate and develop a modal account of propositions on the basis of an iterative conception of propositions, where the modality is logico-mathematical. The modal account of propositions takes the conception to motivate an inherently potential hierarchy of propositions. I show that the account helps provide satisfying solutions to the intensional paradoxes of Russell-Myhill, Kaplan, and Prior. -/- In Chapter 3, I propose that “true” is polysemous. I suggest that “true” is initially polysemous between correspondence truth and disquotational truth, and further polysemous between the meanings corresponding to the subconcepts of the concept truth generated by the indefinite extensibility of that concept. I show that the proposal provides satisfying solutions to the semantic paradoxes. (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)