Ontic structural realism argues that structure is all there is. In (French, 2014) I argued for an ‘eliminativist’ version of this view, according to which the world should be conceived, metaphysically, as structure, and objects, at both the fundamental and ‘everyday’ levels, should be eliminated. This paper is a response to a number of profound concerns that have been raised, such as how we might distinguish between the kind of structure invoked by this view and mathematical structure in (...) general, how we should choose between eliminativist ontic structural realism and alternative metaphysical accounts such as dispositionalism, and how we should capture, in metaphysical terms, the relationship between structures and particles. In developing my response I shall touch on a number of broad issues, including the applicability of mathematics, the nature of representation and the relationship between metaphysics and science in general. (shrink)

In this paper, I explore Bunge’s fictionism in philosophy of mathematics. After an overview of Bunge’s views, in particular his mathematical structuralism, I argue that the comparison between mathematical objects and fictions ultimately fails. I then sketch a different ontology for mathematics, based on Thomasson’s metaphysical work. I conclude that mathematics deserves its own ontology, and that, in the end, much work remains to be done to clarify the various forms of dependence that are involved in (...) class='Hi'>mathematical knowledge, in particular its dependence on mental/brain states and material objects. (shrink)

A prominent version of mathematical structuralism holds that mathematical objects are at bottom nothing but "positions in structures," purely relational entities without any sort of nature independent of the structure to which they belong. Such an ontology is often presented as a response to Benacerraf's "multiple reductions" problem, or motivated on hermeneutic grounds, as a faithful representation of the discourse and practice of mathematics. In this paper I argue that there are serious difficulties with this kind (...) of view: its proponents rely on a distinction between "essential" and "nonessential" features of mathematical objects, and there's no good way to articulate this distinction which is compatible with basic structuralist commitments. But all is not lost. For I further argue that the insights motivating structuralism (or at least those worth preserving) can be preserved without formulating the view in ontologically committal terms. (shrink)

One kind of structuralism holds that mathematics is about structures, conceived as a type of abstract entity. Another denies that it is about any distinctively mathematical entities at all—even abstract structures; rather it gives purely general information about what holds of any collection of entities conforming to the axioms of the theory. Of these, pure structuralism is most plausibly taken to enjoy significant advantages over platonism. But in what appears to be its most plausible—modalised—version, even restricted (...) to elementary arithmetic, it requires defence of a very strong possibility claim: that there could be a completed w-sequence of concrete objects. There are very serious epistemological difficulties in the way of providing the requisite defence. (shrink)

The paper proposes to amend structuralism in mathematics by saying what places in a structure and thus mathematical objects are. They are the objects of the canonical system realizing a categorical structure, where that canonical system is a minimal system in a specific essentialistic sense. It would thus be a basic ontological axiom that such a canonical system always exists. This way of conceiving mathematical objects is underscored by a defense of an essentialistic version of Leibniz’ principle (...) according to which each object is uniquely characterized by its proper and possibly relational essence (where “proper” means “not referring to identity"). (shrink)

Despite some surface similarities, Charles Peirce’s philosophy of mathematics, pragmaticism, is incompatible with both mathematical structuralism and fictionalism. Pragmaticism has to do with experimentation and observation concerning the forms of relations in diagrammatic and iconic representations ofmathematical entities. It does not presuppose mathematical foundations although it has these representations as its objects of study. But these objects do have a reality which structuralism and fictionalism deny.

Recent work in the philosophy of mathematics has suggested that mathematical structuralism is not committed to a strong form of the Identity of Indiscernibles (II). José Bermúdez demurs, and argues that a strong form of II can be warranted on structuralist grounds by countenancing identity properties, or haecceities, as legitimately structural. Typically, structuralists dismiss such properties as obviously non-structural. I will argue to the contrary that haecceities can be viewed as structural but that this concession does not warrant (...) Bermúdez’s version of II but, rather, another easily falsified version. I close with some reflections on reference vis-à-vis structurally indiscernible objects. (shrink)

The present work is a systematic study of five frameworks or perspectives articulating mathematical structuralism, whose core idea is that mathematics is concerned primarily with interrelations in abstraction from the nature of objects. The first two, set-theoretic and category-theoretic, arose within mathematics itself. After exposing a number of problems, the book considers three further perspectives formulated by logicians and philosophers of mathematics: sui generis, treating structures as abstract universals, modal, eliminating structures as objects in favor of freely entertained (...) logical possibilities, and finally, modal-set-theoretic, a sort of synthesis of the set-theoretic and modal perspectives. (shrink)

This paper has two goals. The first goal is to show that the structuralists’ claims about dependence are more significant to their view than is generally recognized. I argue that these dependence claims play an essential role in the most interesting and plausible characterization of this brand of structuralism. The second goal is to defend a compromise view concerning the dependence relations that obtain between mathematical objects. Two extreme views have tended to dominate the debate, namely the view (...) that all mathematical objects depend on the structures to which they belong and the view that none do. I present counterexamples to each of these extreme views. I defend instead a compromise view according to which the structuralists are right about many kinds of mathematical objects (roughly, the algebraic ones), whereas the anti-structuralists are right about others (in particular, the sets). I end with some remarks about how to understand the crucial notion of dependence, which despite being at the heart of the debate is rarely examined in any detail. (shrink)

According to the realist rendering of mathematical structuralism, mathematical structures are ontologically prior to individual mathematical objects such as numbers and sets. Mathematical objects are merely positions in structures: their nature entirely consists in having the properties arising from the structure to which they belong. In this paper, I offer a bundle-theoretic account of this structuralist conception of mathematical objects: what we normally describe as an individual mathematical object is the mereological bundle of (...) its structural properties. An emerging picture is a version of mereological essentialism: the structural properties of a mathematical object, as a bundle, are the mereological parts of the bundle, which are possessed by it essentially. (shrink)

Gerald Holton has famously described Einstein’s career as a philosophical “pilgrimage”. Starting on “the historic ground” of Machian positivism and phenomenalism, following the completion of general relativity in late 1915, Einstein’s philosophy endured (a) a speculative turn: physical theorizing appears as ultimately a “pure mathematical construction” guided by faith in the simplicity of nature and (b) a realistic turn: science is “nothing more than a refinement ”of the everyday belief in the existence of mind-independent physical reality. Nevertheless, Einstein’s (...) class='Hi'>mathematical constructivism that supports his unified field theory program appears to be, at first sight, hardly compatible with the common sense realism with which he countered quantum theory. Thus, literature on Einstein’s philosophy of science has often struggled in finding the thread between ostensibly conflicting philosophical pronouncements. This paper supports the claim that Einstein’s dialog with Émile Meyerson from the mid 1920s till the early 1930s might be a neglected source to solve this riddle. According to Einstein, Meyerson shared (a) his belief in the independent existence of an external world and (b) his conviction that the latter can be grasped only by speculative means. Einstein could present his search for a unified field theory as a metaphysical-realistic program opposed to the positivistic-operationalist spirit of quantum mechanics. (shrink)

This paper compares the statement ‘Mathematics is the study of structure’ with the actual practice of mathematics. We present two examples from contemporary mathematical practice where the notion of structure plays different roles. In the first case a structure is defined over a certain set. It is argued firstly that this set may not be regarded as a structure and secondly that what is important to mathematical practice is the relation that exists between the structure and the set. (...) In the second case, from algebraic topology, one point is that an object can be a place in different structures. Which structure one chooses to place the object in depends on what one wishes to do with it. Overall the paper argues that mathematics certainly deals with structures, but that structures may not be all there is to mathematics. (shrink)

Three principal varieties of mathematical structuralism are compared: set-theoretic structuralism (‘STS’) using model theory, Shapiro's ante rem structuralism invoking sui generis universals (‘SGS’), and the author's modal-structuralism (‘MS’) invoking logical possibility. Several problems affecting STS are discussed concerning, e.g., multiplicity of universes. SGS overcomes these; but it faces further problems of its own, concerning, e.g., the very intelligibility of purely structural objects and relations. MS, in contrast, overcomes or avoids both sets of problems. Finally, it (...) is argued that the modality of MS or a close cousin appears at crucial junctures in both STS and SGS, so that the above outcome is not obviously tendentious. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Husserl and Mathematics by Mirja HartimoAndrea StaitiMirja Hartimo. Husserl and Mathematics. Cambridge: Cambridge University Press, 2021. Pp. 214. Hardback, $99.99.Mirja Hartimo has written the first book-length study of Husserl's evolving views on mathematics that takes his intellectual context into full consideration. Most importantly, Hartimo's historically informed approach to the topic benefits from her extensive knowledge of Husserl's library. Throughout the book, she provides references to texts and articles (...) that Husserl read, marked, and annotated, although regrettably, very few of these sources are quoted explicitly in his published works. The overall picture that emerges from Hartimo's book shows that Husserl continued to read and engage with mathematics throughout his career, and, even though extensive discussions of mathematical issues are not frequent in his later works, some of the ideas he develops, for instance, in The Crisis of the European Sciences and Transcendental Phenomenology, can be read against the backdrop of the developments of mathematics in his time.Chapter 1 sets the method for the rest of the book. Hartimo emphasizes the importance of a methodological device that takes center stage in Husserl's work from the 1920s onward: Besinnung. Hartimo claims that Besinnung is a method to reactivate the "goals and purposes" (21) that originally motivated scientists engaging in a particular line of research. According to Husserl, as specialized research progresses and becomes standard practice, such original goals and purposes might be blurred by other concerns. When applied to mathematics, Besinnung thus amounts to a reflection "on what mathematicians should do, on what the genuine goal of their activities should be" (21–22). Hartimo argues that this kind of approach to mathematics comes close to Penelope Maddy's naturalism, which takes its departure from what mathematicians are actually doing and then proceeds to inquire into the ontological status of mathematical objects and theories. Hartimo identifies particularly in David Hilbert, a colleague of Husserl's at Göttingen, a vivid embodiment of what Husserl considered to be exemplary work oriented toward the ideal goal of mathematics.In chapters 2 and 3, Hartimo expands upon Husserl's conception of what the true goal of mathematics should be. She does so, first, by addressing the classical topic of Husserl's psychologism in his first (and only) work entirely devoted to mathematics, The Philosophy of Arithmetic, and Frege's well-known critical review of it. In Hartimo's rendition, Husserl criticizes Frege for his attempt to provide formal definitions of basic notions, such as equality. For Husserl, such basic notions are intuitively available and require no definition—rather, they should be put to work to clarify vague notions. Frege, in return, attacks what he takes to be Husserl's wishy-washy psychological account of numbers, which revolves around the concrete act of collecting items and abstracting from their specific contents. Ultimately, Hartimo argues, Husserl takes Frege's criticism to heart and in his later Prolegomena no longer pursues a psychological foundation of number, but rather emphasizes the distinction between the subjective and the objective sides of mathematics, with philosophers studying the former and mathematicians focusing on the latter. While Hartimo's verdict on the philosophical yield of the debate is in line with the standard interpretation of Frege's review as prompting Husserl's antipsychologistic turn, she makes a remark that might be worthy of further discussion: "While for Husserl the debate was about logicism, Frege shifts the debate to be about psychologism and here commentators have followed Frege's example" (50). It would be interesting to imagine what a Fregean reply to Husserl's view about the uselessness of formal definitions for basic notions like equality might look like. In any case, Hartimo's reconstruction makes it clear that, however motivated, Husserl's rejection of psychologism does not amount to an acceptance of logicism.In chapter 3, Hartimo puts forward two of her central claims: (1) Husserl was a structuralist about mathematics; and (2) for Husserl, the goal of modern mathematics is encapsulated in the concept of definiteness. According to (1), mathematics is about the formal structures of systems, which can be isolated and compared. Definiteness, in turn, is the property of systems... (shrink)

The concept of process is ubiquitous in science, engineering and everyday life. Category theory, and monoidal categories in particular, provide an abstract framework for modelling processes of many kinds. In this paper, we concentrate on sequential and parallel decomposability of processes in the framework of monoidal categories: We will give a precise definition, what it means for processes to be decomposable. Moreover, through examples, we argue that viewing parallel processes as coupled in this framework can be seen as a category (...) mistake or a misinterpretation. We highlight the suitability of category theory for a structuralistic interpretation of mathematical modelling and argue that for appliers of mathematics, such as engineers, there is a pragmatic advantage from this. (shrink)

The structuralist approach represents the relation between a model and physical system as a relation between two mathematical structures. However, since a physical system is _prima facie_ _not_ a mathematical structure, the structuralist approach seemingly fails to represent the fact that science is about concrete, physical reality. In this paper, I take up this _problem of lost reality_ and suggest how it may be solved in a purely structuralist fashion. I start by briefly introducing both the structuralist approach (...) and the problem of lost reality and discussing the various (non-structuralist) solutions that have been proposed in the literature. Following this, I decompose the problem into the _ontological mismatch_ and _specification_ problems. In response to the former, I present a _metascientific dissolution argument_, according to which the difference in kind between mathematical structures and physical systems poses no deep obstacle to the structuralist approach, and consider some upshots of this argument for our views on representation. By way of conclusion, I argue that the metascientific dissolution argument paves the way for a solution to the specification problem as well. (shrink)

We investigate the form of mathematical structuralism that acknowledges the existence of structures and their distinctive structural elements. This form of structuralism has been subject to criticisms recently, and our view is that the problems raised are resolved by proper, mathematics-free theoretical foundations. Starting with an axiomatic theory of abstract objects, we identify a mathematical structure as an abstract object encoding the truths of a mathematical theory. From such foundations, we derive consequences that address the (...) main questions and issues that have arisen. Namely, elements of different structures are different. A structure and its elements ontologically depend on each other. There are no haecceities and each element of a structure must be discernible within the theory. These consequences are not developed piecemeal but rather follow from our definitions of basic structuralist concepts. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Is Literature Self-Referential?Eric MillerIIs literary language necessarily self-referential? And does this put paradox at the heart of literature? For at least two decades now, affirmative answers to both questions have been articles of faith among critics in the structuralist and poststructuralist mainstream. Literature’s ineluctable paradoxicality attracts us so because a paradox suggests that there are limits to human rationality, and thus strikes a blow for literature and against science. (...) Paradox ensures literature its own special realm, safe from the culturally imperialistic inroads of science’s orderly, rational-empirical constructions. Literature thus gets to be seen as “bigger” than reason, because only literature can live with paradox, and thus only literature reveals those “deeper” truths of contradiction-ridden human existence, whereas science can never penetrate beyond rational manipulations of phenomenal surfaces. Indeed, for anyone who holds all human existence to be fundamentally paradoxical, this special capacity makes literature the only genuine, demystified species of human knowledge. In the war between C. P. Snow’s Two Cultures, a war humanists are now losing badly, we like to regard paradox as our ultimate weapon.Belief in the paradoxicality of literature may also be found among the New Critics, 1 and in fact goes back at least to the Romantics, 2 but they did not derive this belief from a necessary self-referentiality of literary language. This newer way of deriving paradox appears to offer several competitive advantages. Self-reference seems to be the ultimate version of “art for art’s sake”: if all literary language is necessarily self-referential, then literature must be something totally self-contained, and is thus legitimized solely in and through itself. In addition, deriving [End Page 475] literature’s paradoxicality from self-reference seems to ground it in the same conceptual realm as mathematical logic, and thus tempts us to claim for literature a “rigor” normally conceded only to logic, mathematics, and theoretical physics. Self-referentiality would thus seem both to protect literature’s autonomy and to raise its cultural status into the company of mathematics and science.But it is a trap. For behind this new strategy lurks a capitulation to the underlying conceptual scheme of natural science, and literature thus comes to be conceived, in effect, as a science. Self-reference is, after all, still a kind of reference; paradox is still a kind of truth-relational construction;—and truth-relations built around reference constitute the correspondence theory of truth. From such a foundational commitment to reference and correspondence-truth, flows the rest of the scientific worldview: the dualism of referring expression and referent, word and object, sentence and fact, theory and data, language and world, culture and nature. The presentation of literature as a quasi-negation of this ontology changes nothing. The dualism of reference still functions as literature’s conceptual starting point, as its arche. We only appear to be defining a peculiar and radically distinct sphere for literature when we insist on its self-referentiality, on the (recursive) identity, for it, of referring expression and referent. For, this is still to treat the conceptual distinction between the two as foundational, and literature thereby tacitly assimilates science’s fundamental categorial division: the referential language-world duality of correspondence truth. Indeed, literature thus comes to live even more completely in the margins of science: it will have no essence of its own; it will differ at most in being a photo-negative image of science’s positiv(istic) project. In Snow’s war, self-reference is a Trojan horse.With self-reference as the essence of literature, we are thus staking our reputations on literature’s being a special field of formalizable “knowledge about...,” rather than, say, a quality or kind or realm of possible experience—which is to adopt science’s understanding of what is important. Matters are only made worse by that other attempt to define literary studies as a negation of the theory-data dualism of natural science: the poststructuralist denial of the distinction between criticism and literature. For the institutional need professional critics have to be seen as a discipline with a method and a subject, as a form of “knowledge about,” hardly disappears with... (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)

This volume covers a wide range of topics in the most recent debates in the philosophy of mathematics, and is dedicated to how semantic, epistemological, ontological and logical issues interact in the attempt to give a satisfactory picture of mathematical knowledge. The essays collected here explore the semantic and epistemic problems raised by different kinds of mathematical objects, by their characterization in terms of axiomatic theories, and by the objectivity of both pure and applied mathematics. They investigate controversial (...) aspects of contemporary theories such as neo-logicist abstractionism, structuralism, or multiversism about sets, by discussing different conceptions of mathematical realism and rival relativistic views on the mathematical universe. They consider fundamental philosophical notions such as set, cardinal number, truth, ground, finiteness and infinity, examining how their informal conceptions can best be captured in formal theories. The philosophy of mathematics is an extremely lively field of inquiry, with extensive reaches in disciplines such as logic and philosophy of logic, semantics, ontology, epistemology, cognitive sciences, as well as history and philosophy of mathematics and science. By bringing together well-known scholars and younger researchers, the essays in this collection – prompted by the meetings of the Italian Network for the Philosophy of Mathematics (FilMat) – show how much valuable research is currently being pursued in this area, and how many roads ahead are still open for promising solutions to long-standing philosophical concerns. Promoted by the Italian Network for the Philosophy of Mathematics – FilMat. (shrink)

Plato himself would be pleased at the recent emergence of a certain highly Platonic variety of platonism concerning mathematics, viz., the structuralism of Michael Resnik and Stewart Shapiro. In fact, this species of platonism is so Platonic that it is susceptible to an objection closely related to one raised against Plato by Parmenides in the dialogue of that name. This is the Third Man Argument against a view about the relation of Forms to particulars. My objection is not a (...) TMA against structuralism; the position avoids that objection, but is vulnerable to a different one precisely at the point where it avoids the TMA. The way structuralism avoids the TMA has in fact been considered, as one of Plato’s options, by at least one commentator on the Parmenides, Colin Strang, who explicitly rejects it on logical grounds. In the course of the discussion, I shall clarify the reason that I believe led Strang to reject this option, and shall modify his own statement of that reason. (shrink)

Structuralism is a view about the subject matter of mathematics according to which what matters are structural relationships in abstraction from the intrinsic nature of the related objects. Mathematics is seen as the free exploration of structural possibilities, primarily through creative concept formation, postulation, and deduction. The items making up any particular system exemplifying the structure in question are of no importance; all that matters is that they satisfy certain general conditions—typically spelled out in axioms defining the structure or (...) structures of interest—characteristic of the branch of mathematics in question. Thus, in the basic case of arithmetic, the famous “axioms” of Richard Dedekind (taken over by Giuseppe Peano, as he acknowledged) were conditions in a definition of a “simply infinite system”, with an initial item, each item having a unique next one, no two with the same next one, and all items finitely many steps from the initial one. (The latter condition is guaranteed by the axiom of mathematical induction.) All such systems are structurally identical, and, in a sense to be made more precise, the shared structure is what mathematics investigates. (In other cases, multiple structures are allowed, as in abstract algebra with its many groups, rings, fields, and so forth.) This structuralist view of arithmetic thus contrasts with the absolutist view, associated with Gottlob Frege and Bertrand Russell, that natural numbers must in fact be certain definite objects, namely classes of equinumerous concepts or classes. (shrink)

Charles Parsons examines the notion of object, with the aim to navigate between nominalism, denying that distinctively mathematical objects exist, and forms of Platonism that postulate a transcendent realm of such objects. He introduces the central mathematical notion of structure and defends a version of the structuralist view of mathematical objects, according to which their existence is relative to a structure and they have no more of a 'nature' than that confers on them. Parsons also analyzes the (...) concept of intuition and presents a conception of it distantly inspired by that of Kant, which describes a basic kind of access to abstract objects and an element of a first conception of the infinite. (shrink)

In this paper we put a thesis that it is possible to perceive mathematics as a science of structures, where the difference between structure as the object of study and theory as something which describes this object is blurred. We discusses the view of set-theoretical structuralism with a special emphasis placed on a certain gradual development of set theory as a formal theory. We proposes a certain view concerning the methodology of formal sciences, which is an attempt at describing (...) precisely and capturing in a scheme the ways of development of deductive sciences in general, set theory including. We show that the standpoint of sui generis structuralism has some features characteristic of purely formal theory. (shrink)

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)

The first of six chapters in which rival views are critically evaluated and compared with the Constructibility view described in earlier chapters. The views considered here are those of Stewart Shapiro and Michael Resnik. A number of difficulties with these two views are detailed and it is explained how the Constructibility Theory is not troubled by the problems that Structuralism was explicitly developed to resolve.

The group of mathematicians known as Bourbaki persuasively proclaimed the isolation of its field of research – pure mathematics – from society and science. It may therefore seem paradoxical that links with larger French cultural movements, especially structuralism and potential literature, are easy to establish. Rather than arguing that the latter were a consequence of the former, which they were not, I show that all of these cultural movements, including the Bourbakist endeavor, emerged together, each strengthening the public appeal (...) of the others through constant, albeit often superficial, interaction. This codependency is partly responsible for their success and moreover accounts for their simultaneous fall from favor, which, however, can clearly be seen as also stemming from different internal problems. To understand this dynamics, I argue that Bourbaki’s role can best be captured by using the notion of cultural connector, which I introduce here. (shrink)

An affirmative answer is given to the question quoted in the title.

The philosophy of mathematical practice sometimes investigates the social constitution of mathematics but does not always make explicit the philosophical-normative framework that guides the discussion. This chapter investigates some recent proposals in the philosophy of mathematical practice that compare social facts and mathematical objects, discussing similarities and differences. An attempt will be made to identify, through a comparison with three different perspectives in social ontology, the kind of objectivity attributed to mathematical knowledge, the type of (...) representational or non-representational semantics adopted, and the justificatory or coordinative role entrusted to axiomatics. After a brief introduction to key issues in social ontology, Sect. 3 of the chapter offers a survey of contributions by Feferman, Ferreirós, and Cole, highlighting a difference between approaches based on rules, practices, and intentional states, respectively. Section 4.1 also discusses results by Carter, Collin, Giardino, and Pantsar, focusing on differences between the objectivity of knowledge and objectivity of objects and showing that many authors combine a realist, structuralist, and pragmatist perspective, as well as the idea that objectivity comes in degrees. Section 4.2 focuses on the kind of semantics that is adopted in socially oriented approaches. A representational semantics is preferred by authors grounding their views on mental states, whereas a non-representational semantics best fits with views based on practices. Section 5 considers how axiomatics can be understood in a social perspective. Axiomatics generally plays a justificatory role also in theories that aim to explain the social constitution of mathematical knowledge. Yet, if one shifts from approaches based on mental states to approaches anchored on rules or habits, axiomatics can be viewed as an institution based on obligations, functions, coordination problems, and agent’s actions and roles, thus playing an organizational and coordination role. (shrink)

The primary justification for mathematical structuralism is its capacity to explain two observations about mathematical objects, typically natural numbers. Non-eliminative structuralism attributes these features to the particular ontology of mathematics. I argue that attributing the features to an ontology of structural objects conflicts with claims often made by structuralists to the effect that their structuralist theses are versions of Quine’s ontological relativity or Putnam’s internal realism. I describe and argue for an alternative explanation for these features (...) which instead explains the attributes them to the mathematical practice of representing numbers using more concrete tokens, such as sets, strokes and so on. (shrink)

Recent semantic approaches to scientific structuralism, aiming to make precise the concept of shared structure between models, formally frame a model as a type of set-structure. This framework is then used to provide a semantic account of (a) the structure of a scientific theory, (b) the applicability of a mathematical theory to a physical theory, and (c) the structural realist’s appeal to the structural continuity between successive physical theories. In this paper, I challenge the idea that, to be (...) so used, the concept of a model and so the concept of shared structure between models must be formally framed within a single unified framework, set-theoretic or other. I first investigate the Bourbaki-inspired assumption that structures are types of set-structured systems and next consider the extent to which this problematic assumption underpins both Suppes’ and recent semantic views of the structure of a scientific theory. I then use this investigation to show that, when it comes to using the concept of shared structure, there is no need to agree with French that “without a formal framework for explicating this concept of ‘structure-similarity’ it remains vague, just as Giere’s concept of similarity between models does ...” (French, 2000, Synthese, 125, pp. 103–120, p. 114). Neither concept is vague; either can be made precise by appealing to the concept of a morphism, but it is the context (and not any set-theoretic type) that determines the appropriate kind of morphism. I make use of French’s (1999, From physics to philosophy (pp. 187–207). Cambridge: Cambridge University Press) own example from the development of quantum theory to show that, for both Weyl and Wigner’s programmes, it was the context of considering the ‘relevant symmetries’ that determined that the appropriate kind of morphism was the one that preserved the shared Lie-group structure of both the theoretical and phenomenological models. (shrink)

The paper outlines a novel version of mathematical structuralism related to invariants. The main objective here is twofold: first, to present a formal theory of structures based on the structuralist methodology underlying work with invariants. Second, to show that the resulting framework allows one to model several typical operations in modern mathematical practice: the comparison of invariants in terms of their distinctive power, the bundling of incomparable invariants to increase their collective strength, as well as a heuristic (...) principle related to the search for new invariants. (shrink)

In her paper 'Does the Mathematical Nature of Physics Undermine Physicalism?' Susan Schneider draws attention to a much neglected challenge to physicalism, arising from its mathematical vocabulary. Whilst I agree with Schneider that the mathematical nature of physics is a concern for the physicalist, I disagree with her concerning the essence of the problem. I argue on the basis of Newman's problem that a purely mathematical description cannot entirely characterize concrete reality. The physicalist can avoid Newman's (...) problem by emphasizing the causal or nomic elements of physics, which ensure that it is not purely mathematical. However, physicalism on such a construal is a form of causal structuralism, which I argue involves either a vicious regress or a vicious circularity. (shrink)

The debate on structuralism in the philosophy of mathematics has brought into focus a question about the status of meta-mathematics. It has been raised by Shapiro (2005), where he compares the ongoing discussion on structuralism in category theory to the Frege-Hilbert controversy on axiomatic systems. Shapiro outlines an answer according to which meta-mathematics is understood in structural terms and one according to which it is not. He finds both options viable and does not seem to prefer one over (...) the other. The present paper reconsiders the nature of the formulae and symbols meta-mathematics is about and finds that, contrary to Charles Parsons' influential view, meta-mathematical objects are not "quasi-concrete". It is argued that, consequently, structuralists should extend their account of mathematics to meta-mathematics. (shrink)

I argue that we need not accept Quine's holistic conception of mathematics and empirical science. Specifically, I argue that we should reject Quine's holism for two reasons. One, his argument for this position fails to appreciate that the revision of the mathematics employed in scientific theories is often related to an expansion of the possibilities of describing the empirical world, and that this reveals that mathematics serves as a kind of rational framework for empirical theorizing. Two, this holistic conception (...) does not clearly demarcate pure mathematics from applied mathematics. In arguing against Quine, I present a formal account of applied mathematics in which the mathematics employed in an empirical theory plays a role that is analogous to the epistemological role Kant assigned synthetic a priori propositions. According to this account, it is possible to insulate pure mathematics from empirical falsification, and there is a sense in which applied mathematics can also be labeled as a priori. (shrink)

When dealing with a certain class of physical systems, the mathematical characterization of a generic system aims to describe the phase portrait of all its possible states. Because they are defined only up to isomorphism, the mathematical objects involved are “schematic structures”. If one imposes the condition that these mathematical definitions completely capture the physical information of a given system, one is led to a strong requirement of individuation for physical states. However, we show there are not (...) enough qualitatively distinct properties in an abstract Hilbert space to fulfill such a requirement. It thus appears there is a fundamental tension between the physicist’s purpose in providing a mathematical definition of a mechanical system and a feature of the basic formalism used in the theory. We will show how group theory provides tools to overcome this tension and to define physical properties. (shrink)

Structuralism, the view that mathematics is the science of structures, can be characterized as a philosophical response to a general structural turn in modern mathematics. Structuralists aim to understand the ontological, epistemological, and semantical implications of this structural approach in mathematics. Theories of structuralism began to develop following the publication of Paul Benacerraf’s paper ‘What numbers could not be’ in 1965. These theories include non-eliminative approaches, formulated in a background ontology of sui generis structures, such as Stewart Shapiro’s (...) ante rem structuralism and Michael Resnik’s pattern structuralism. In contrast, there are also eliminativist accounts of structuralism, such as Geoffrey Hellman’s modal structuralism, which avoids sui generis structures. These research projects have guided a more systematic focus on philosophical topics related to mathematical structuralism, including the identity criteria for objects in structures, dependence relations between objects and structures, and also, more recently, structural abstraction principles. Parallel to these developments are approaches that describe mathematical structure in category-theoretic terms. Category-theoretic approaches have been further developed using tools from homotopy type theory. Here we find a strong relationship between mathematical structuralism and the univalent foundations project, an approach to the foundations of mathematics based on higher category theory. (shrink)

If structuralism is a true view of mathematics on which the statements of mathematicians are taken ‘at face value’, then there are both structures on which classical second-order arithmetic is a correct report, and structures on which intuitionistic second-order arithmetic is correct. An argument due to Dedekind then proves that structures and structures are isomorphic. Consequently, first- and second-order statements true in structures must hold in , and conversely. Since instances of the general law of the excluded third fail (...) in structures but hold in , a contradiction ensues. (shrink)

This paper considers the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, in relation to the "algebraic" approach to mathematical structuralism. My aim is to show that category theory has as much to say about an algebraic consideration of meta-mathematical analyses of logical structure as it does about mathematical analyses of mathematical structure, without either requiring an assertory mathematical or (...) meta-mathematical background theory as a "foundation", or turning meta-mathematical analyses of logical concepts into "philosophical" ones. Thus, we can use category theory to frame an interpretation of mathematics according to which we can be structuralists all the way down. (shrink)

It seems that the theories of language of the present century can be classified into two basic groups. The approaches of the first group perceive language as a mathematical structure and understand any theory of language as a kind of application of mathematics or logic. Their ideological background is furnished by logical positivism and analytical philosophy (esp. by Russell, Carnap, Wittgenstein and their followers); and their practical output is Chomskian formal syntax and subsequent formal semantics. The approaches of (...) the other group do not approve of formalization and consider a theory of language closer to psychology than to mathematics. The specific position within this group is occupied by the so-called structuralists (de Saussure, Hjelmslev, Derrida). (shrink)

Dedekind’s structuralism is a crucial source for the structuralism of mathematical practice—with its focus on abstract concepts like groups and fields. It plays an equally central role for the structuralism of philosophical analysis—with its focus on particular mathematical objects like natural and real numbers. Tensions between these structuralisms are palpable in Dedekind’s work, but are resolved in his essay Was sind und was sollen die Zahlen? In a radical shift, Dedekind extends his mathematical approach (...) to “the” natural numbers. He creates the abstract concept of a simply infinite system, proves the existence of a “model”, insists on the stepwise derivation of theorems, and defines structure-preserving mappings between different systems that fall under the abstract concept. Crucial parts of these considerations were added, however, only to the penultimate manuscript, for example, the very concept of a simply infinite system. The methodological consequences of this radical shift are elucidated by an analysis of Dedekind’s metamathematics. Our analysis provides a deeper understanding of the essay and, in addition, illuminates its impact on the evolution of the axiomatic method and of “semantics” before Tarski. This understanding allows us to make connections to contemporary issues in the philosophy of mathematics and science. (shrink)

Causal structuralism is the view that, for each natural, non-mathematical, non-Cambridge property, there is a causal profile that exhausts its individual essence. On this view, having a property’s causal profile is both necessary and sufficient for being that property. It is generally contrasted with the Humean or quidditistic view of properties, which states that having a property’s causal profile is neither necessary nor sufficient for being that property, and with the double-aspect view, which states that causal profile is (...) necessary but not sufficient. Shoemaker’s (1998) and Hawthorne’s (2001) arguments in favor of causal structuralism primarily focus on problematic consequences of the other two views. I argue, however, that causation does not provide an appropriate framework within which to characterize all physical properties for two main reasons. First, there are physical properties that do not have causal profiles and properties whose causal profiles do not exhaust their essences. Second, there is no unified notion of causation across the sciences. After distinguishing between the causal and the nomological, I suggest that what is needed is a structuralist view of properties that is not merely causal but that incorporates a physical property’s higher-order mathematical and nomological properties into its identity conditions. Such a view retains the naturalistic motivations for causal structuralism while avoiding the problems it faces. (shrink)

There are many ways of understanding the nature of philosophical questions. One may consider their morphology, semantics, relevance, or scope. This article introduces a different approach, based on the kind of informational resources required to answer them. The result is a definition of philosophical questions as questions whose answers are in principle open to informed, rational, and honest disagreement, ultimate but not absolute, closed under further questioning, possibly constrained by empirical and logico-mathematical resources, but requiring noetic resources to (...) be answered. The article concludes with a discussion of some of the consequences of this definition for a conception of philosophy as the study (or “science”) of open questions, which uses conceptual design to analyse and answer them. (shrink)

Some recent discussions of A. N. Whitehead's treatment of the problem of value have stressed the point that his work in this field is open to serious objection. For example, Professor John Goheen claims that Whitehead's attempt to indicate distinguishing characteristics of experience of “the Good”, is too general to be adequate. He also suggests that this generality of approach makes it impossible for Whitehead to differentiate between different species of value. Further, according to Goheen, Whitehead involves himself in confusion (...) when he claims that satisfaction is achieved when experience is characterized by order ; and yet he also suggests that in some cases, the presence of disorder makes possible a higher type of satisfaction. Professor P. A. Schilpp agrees with Goheen in feeling that Whitehead's explanation of the good life in terms of “pattern” is too vague. In Schilpp's opinion, Whitehead does not answer the basic question: “What kind of pattern?” There is a further objection to Whitehead's “theory of the close connection between morality and beauty,” because at times it looks like an actual identification. Schilpp also objects to what appears to be Whitehead's identification of “good” with “interest”. Finally, it is claimed by Schilpp that Whitehead subordinates goodness to beauty. This interpretation is apparently supported by Prof. George Morgan who states: “Truth and moral values are instrumental except in so far as they enhance beauty.” Professor B. Morris objects to Whitehead's attempt to apply the mathematical method to aesthetic experience. He suggests that the method of symbolic logic is too abstract to deal adequately with concrete aesthetic data. Professor J. W. Blyth, as the result of his examination of Whitehead's theory of truth, reaches the conclusion that “the various elements involved in Whitehead's theory of truth cannot be brought together in one coherent and consistent system. (shrink)

The prime purpose of this paper is, first, to restore to discourse-bound occasion sentences their rightful central place in semantics and secondly, taking these as the basic propositional elements in the logical analysis of language, to contribute to the development of an adequate logic of occasion sentences and a mathematical foundation for such a logic, thus preparing the ground for more adequate semantic, logical and mathematical foundations of the study of natural language. Some of the insights elaborated in (...) this paper have appeared in the literature over the past thirty years, and a number of new developments have resulted from them. The present paper aims atproviding an integrated conceptual basis for this new development in semantics. In Section 1 it is argued that the reduction by translation of occasion sentences to eternal sentences, as proposed by Russell and Quine, is semantically and thus logically inadequate. Natural language is a system of occasion sentences, eternal sentences being merely boundary cases. The logic hasfewer tasks than is standardly assumed, as it excludes semantic calculi, which depend crucially on information supplied by cognition and context and thus belong to cognitive psychology rather than to logic. For sentences to express a proposition and thus be interpretable and informative, they must first be properly anchored in context. A proposition has a truth value when it is, moreover, properly keyed in the world, i.e. is about a situation in the world. Section 2 deals with the logical properties of natural language. It argues that presuppositional phenomena require trivalence and presents the trivalent logic PPC3, with two kinds of falsity and two negations. It introduces the notion of -space for a sentence A as the basis of logical model theory, and the notion of P A /, functioning as a `private' subuniverse for A/A. The trivalent Kleene calculus is reinterpreted as a logical account of vagueness, rather than of presupposition. PPC3 and the Kleene calculus are refinements of standard bivalent logic and can be combined into one logical system. In Section 3 the adequacy of PPC3 as a truth-functional model of presupposition is considered more closely and given a Boolean foundation. In a noncompositional extended Boolean algebra, three operators are defined: 1a for the conjoined presuppositions of a, ã for the complement of a within 1a, and â for the complement of 1a within Boolean 1. The logical properties of this extended Boolean algebra are axiomatically defined and proved for all possible models. Proofs are provided of the consistency and the completeness of the system. Section 4 is a provisional exploration of the possibility of using the results obtained for a new discourse-dependent account of the logic of modalities in natural language. The overall result is a modified and refined logical and model-theoretic machinery, which takes into account both the discourse-dependency of natural language sentences and the necessity of selecting a key in the world before a truth value can be assigned. (shrink)

Imagination occupies a central place in philosophy, going back to Aristotle. However, following a period of relative neglect there has been an explosion of interest in imagination in the past two decades as philosophers examine the role of imagination in debates about the mind and cognition, aesthetics and ethics, as well as epistemology, science and mathematics. This outstanding _Handbook_ contains over thirty specially commissioned chapters by leading philosophers organised into six clear sections examining the most important aspects of the philosophy (...) of imagination, including: Imagination in historical context: Aristotle, Descartes, Hume, Kant, Husserl, and Sartre What is imagination? The relation between imagination and mental imagery; imagination contrasted with perception, memory, and dreaming Imagination in aesthetics: imagination and our engagement with music, art, and fiction; the problems of fictional emotions and ‘imaginative resistance’ Imagination in philosophy of mind and cognitive science: imagination and creativity, the self, action, child development, and animal cognition Imagination in ethics and political philosophy, including the concept of 'moral imagination' and empathy Imagination in epistemology and philosophy of science, including learning, thought experiments, scientific modelling, and mathematics. The Routledge Handbook of Philosophy _of Imagination_ is essential reading for students and researchers in philosophy of mind and psychology, aesthetics, and ethics. It will also be a valuable resource for those in related disciplines such as psychology and art. (shrink)

Robert Brandom’s expressivism argues that not all semantic content may be made fully explicit. This view connects in interesting ways with recent movements in philosophy of mathematics and logic (e.g. Brown, Shin, Giaquinto) to take diagrams seriously - as more than a mere “heuristic aid” to proof, but either proofs themselves, or irreducible components of such. However what exactly is a diagram in logic? Does this constitute a semiotic natural kind? The paper will argue that such a natural (...) class='Hi'>kind does exist in Charles Peirce’s conception of iconic signs, but that fully understood, logical diagrams involve a structured array of normative reasoning practices, as well as just a “picture on a page”. (shrink)

A mathematical matrix is usually defined as a two-dimensional array of scalars. And yet, as I explain, matrices are not in fact two-dimensional arrays. So are we to conclude that matrices do not exist? I show how to resolve the puzzle, for both contemporary and older mathematics. The solution generalises to the interpretation of all mathematical discourse. The paper as a whole attempts to reinforce mathematical structuralism by reflecting on how best to interpret mathematics.