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

I show that extant attempts to capture and generalize empirical adequacy in terms of partial structures fail. Indeed, the motivations for the generalizations in the partial structures approach are better met by the generalizations via approximation sets developed in “Generalizing Empirical Adequacy I”. Approximation sets also generalize partial structures.

In the last decade, structural realism has been presented as the most promising strategy for developing a defensible realist view of science. Nevertheless, controversy still continues in relation to the exact meaning of the proposed structuralism. The stronger version of structural realism, the so-called ontic structural realism, has been argued for on the basis of some ideas related to quantum mechanics. In this paper, I will first outline these arguments, mainly developed by Steven French and James Ladyman, then challenge them, (...) putting a particular emphasis on a metaphysical principle which, even though it is crucial for the whole argument, hasn't been, in my opinion, clearly stated and examined yet. My overall view will be that a weaker version of the form of realism we are considering is more plausible – namely, epistemic structural realism. (shrink)

It is argued that mathematics is unreasonably effective in fundamental physics, that this is genuinely mysterious, and that it is best explained by a version of Pythagorean metaphysics. It is shown how this can be reconciled with the fact that mathematics is not always effective in real world applications. The thesis is that physical structure approaches isomorphism with a highly symmetric mathematical structure at very high energy levels, such as would have existed in the early universe. As the universe cooled, (...) its underlying symmetry was broken in a sequence of stages. At each stage, more forces and particles were differentiated, leading to the complexity of the observed world. Remnant structure makes mathematics effective in some real world applications, but not all. (shrink)

Combinations of molecules, of biological individuals, or of chemical processes can produce effects that are not simply attributable to the constituents. Such non-redundant causality warrants recognition of those coherences as ontologically significant whenever that efficacy is relevant. With respect to such interaction, the effective coherence is more real than are the components. This ontological view is a variety of structural realism and is also a kind of process philosophy. The designation ‘process structural realism’ (PSR) seems appropriate.

One of the important challenges in the philosophy of mathematics is to account for the semantics of sentences that express mathematical propositions while simultaneously explaining our access to their contents. This is Benacerraf’s Dilemma. In this dissertation, I argue that cognitive science furnishes new tools by means of which we can make progress on this problem. The foundation of the solution, I argue, must be an ontologically realist, albeit non-platonist, conception of mathematical reality. The semantic portion of the problem can (...) be addressed by accepting a Chomskyan conception of natural languages and a matching internalist, mentalist and nativist view of semantics. A helpful perspective on the epistemic aspect of the puzzle can be gained by translating Kurt G ̈odel’s neo-Kantian conception of the nature of mathematics and its objects into modern, cognitive terms. (shrink)

Structural realism first emerged as an epistemological thesis aimed to avoid the socalled pessimistic metainduction on the history of science. Some authors, however, have suggested that the preservation of structure across theory change is best explained by endorsing the metaphysical thesis that structure is all there is. Although the possibility of this latter, ‘ontic’ form of structural realism has been extensively debated, not much has been said concerning its justification. In this article, I distinguish between two arguments in favor of (...) ontic structural realism that can be reconstructed from the literature and find both of them wanting. (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.

Whitehead’s cosmology centers on the self-creation of actual occasions that perish as they come to be, but somehow do combine to constitute societies that are persistent agents and/or patients. “Instance Ontology” developed by D.W. Mertz concerns unification of relata into facts of relatedness by specific intensions. These two conceptual systems are similar in that they both avoid the substance-property distinction: they differ in their understanding of how basic units combine to constitute complex unities. “Process Structural Realism” (PSR) draws from both (...) of these approaches in developing an account of how combinations of processes may produce ontologically significant coherences. When a group of processes achieves such closure that a set of states recurs continually, the effects of that coherence differ from what would occur in the absence of that closure. Such altered effectiveness is an attribute of the system as a whole, and would have consequences. This indicates that the network of processes, as a unit, has ontological significance. The closed network of processes, together with the conditions that prevail, constitute the form of definiteness of the coherence. That form continues to obtain as long as the coherence persists. Constituents contribute to, rather than share, that characteristic. Aspects of some recent research in systems biology, microeconomics, and social psychology illustrate the application of PSR. (shrink)

We outline Ladyman's 'metaphysical' or 'ontic' form of structuralrealism and defend it against various objections. Cao, in particular, has questioned theview of ontology presupposed by this approach and we argue that by reconceptualisingobjects in structural terms it offers the best hope for the realist in thecontext of modern physics.

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)

This paper explores varieties of scientific structuralism. Central to our investigation is the notion of `shared structure'. We begin with a description of mathematical structuralism and use this to point out analogies and disanalogies with scientific structuralism. Our particular focus is the semantic structuralist's attempt to use the notion of shared structure to account for the theory-world connection, this use being crucially important to both the contemporary structural empiricist and realist. We show why minimal scientific structuralism is, at the very (...) least, a powerful methodological standpoint. Our investigation also makes explicit what more must be added to this minimal structuralist position in order to address the theory-world connection, namely, an account of representation. (shrink)

This unpublished paper, written in 2005 in the PhD philosophy program at Notre Dame, argues that algebraic structural realism faces a potentially fatal difficulty raised by the existence of inequivalent representations in quantum field theory.

This thesis offers a naturalist revision of Alain Badiou’s philosophy. This goal is pursued through an encounter of Badiou’s mathematical ontology and theory of truth with contemporary trends in philosophy of mathematics and philosophy of science. I take issue with Badiou’s inability to elucidate the link between the empirical and the ontological, and his residual reliance on a Heideggerian project of fundamental ontology, which undermines his own immanentist principles. I will argue for both a bottom-up naturalisation of Badiou’s philosophical approach (...) to mathematics and a top-down naturalisation. Articulating my particular understanding of what realism and naturalism should commit us to, I propose a creative fusion of Badiou’s attention to metamathematical results with a structural-informational metaphysics, proposing a ‘matherialism’ uniting the more daring speculative insights of the former with the naturalist and empiricist commitments motivating the latter. (shrink)

Benacerraf has presented two problems for the philosophy of mathematics. These are the problem of identification and the problem of representation. This paper aims to reconstruct the latter problem and to unpack its undermining bearing on the version of Ontic Structural Realism that frames scientific representations in terms of abstract structures. I argue that the dichotomy between mathematical structures and physical ones cannot be used to address the Benacerraf problem but strengthens it. I conclude by arguing that versions of OSR (...) that do not rely on mathematical frameworks for representational purposes need not be vulnerable to Benacerraf’s second problem. (shrink)

This paper aims to reassess the philosophical puzzle of the “applicability of mathematics to physical sciences” as a misunderstood disciplinary interplay. If the border isolating mathematics from the empirical world is based on appropriate criteria, how does one explain the fruitfulness of its systematic crossings in recent centuries? An analysis of the evolution of the criteria used to separate mathematics from experimental sciences will shed some light on this question. In this respect, we will highlight the historical influence of three (...) major disciplinary paradigms. According to the Aristotelian classification of the sciences, the separation of mathematics from physics is based on their respective objects of study. The Baconian system distinguishes these sciences by the type of knowledge involved in each field. Finally, the Whewellian disciplinary layout categorises these disciplines by their respective methods. In this paper, we argue that the cascading effect of such disciplinary categorisations—based successively on ontological, epistemological, and heuristic criteria—resulted in a profound redefinition of the border between mathematics and physics, with the consequence of obscuring the foundations of the interplay between these fields. Our approach intends to put forward such a mechanism as a constitutive piece of the puzzle of the applicability of mathematics. (shrink)

The Relational Blockworld (RBW) interpretation of non-relativistic quantum mechanics (NRQM) is introduced. Accordingly, the spacetime of NRQM is a relational, non-separable blockworld whereby spatial distance is only defined between interacting trans-temporal objects. RBW is shown to provide a novel statistical interpretation of the wavefunction that deflates the measurement problem, as well as a geometric account of quantum entanglement and non-separability that satisfies locality per special relativity and is free of interpretative mystery. We present RBW’s acausal and adynamical resolution of the (...) so-called “quantum liar paradox,” an experimental set-up alleged to be problematic for a spacetime conception of reality, and conclude by speculating on RBW’s implications for quantum gravity. (shrink)

Within the philosophy of science, the realism debate has been revitalised by the development of forms of structural realism. These urge a shift in focus from the object oriented ontologies that come and go through the history of science to the structures that remain through theory change. Such views have typically been elaborated in the context of theories of physics and are motivated by, first of all, the presence within such theories of mathematical equations that allow straightforward representation of the (...) relevant structures; and secondly, the implications of such theories for the individuality and identity of putative objects. My aim in this talk is to explore the possibility of extending such views to biological theories. An obvious concern is that within the context of the latter it is typically insisted that we cannot find the kinds of highly mathematised structures that structural realism can point to in physics. I shall indicate how the model-theoretic approach to theories might help allay such concerns. Furthermore, issues of identity and individuality also arise within biology. Thus Dupre has recently noted that there exists a ‘General Problem of Biological Individuality’ which relates to the issue of how one divides ‘massively integrated and interconnected’ systems into discrete components. In response Dupre advocates a form of ‘Promiscuous Realism’ that holds, for example, that there is no unique way of dividing the phylogenetic tree into kinds. Instead I shall urge serious consideration of those aspects of the work of Dupre and others that lean towards a structuralist interpretation. By doing so I hope to suggest possible ways in which a structuralist stance might be extended to biology. (shrink)

In this paper we show that any reasoning process in which conclusions can be both fallible and corrigible can be formalized in terms of two approaches: (i) syntactically, with the use of defeasible reasoning, according to which reasoning consists in the construction and assessment of arguments for and against a given claim, and (ii) semantically, with the use of partial structures, which allow for the representation of less than conclusive information. We are particularly interested in the formalization of scientific reasoning, (...) along the lines traced by Lakatos’ methodology of scientific research programs. We show how current debates in cosmology could be put into this framework, shedding light on a very controversial topic. (shrink)

I argue that a recent version of the doctrine of mathematical naturalism faces difficulties arising in connection with Wigner's old puzzle about the applicability of mathematics to natural science. I discuss the strategies to solve the puzzle and I show that they may not be available to the naturalist.

The volume includes twenty-five research papers presented as gifts to John L. Bell to celebrate his 60th birthday by colleagues, former students, friends and admirers. Like Bell’s own work, the contributions cross boundaries into several inter-related fields. The contributions are new work by highly respected figures, several of whom are among the key figures in their fields. Some examples: in foundations of maths and logic ; analytical philosophy, philosophy of science, philosophy of mathematics and decision theory and foundations of economics. (...) Most articles are contributions to current philosophical debates, but contributions also include some new mathematical results, important historical surveys, and a translation by Wilfrid Hodges of a key work of arabic logic. (shrink)

One of the most unsettling problems in the history of philosophy examines how mathematics can be used to adequately represent the world. An influential thesis, stated by Eugene Wigner in his paper entitled "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," claims that "the miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve." Contrary to this view, this thesis delineates and implements (...) a strategy to show that the applicability of mathematics is very reasonable indeed. I distinguish three forms of the problem of the applicability of mathematics, and focus on one I call the problem of uncanny accuracy: Given that the construction and manipulation of mathematical representations is pervaded by uncertainty, error, approximation, and idealization, how can their apparently uncanny accuracy be explained? I argue that this question has found no satisfactory answer because our rational reconstruction of scientific practice has not involved tools rich enough to capture the logic of mathematical modelling. Thus, I characterize a general schema of mathematical analysis of real systems, focusing on the selection of modelling assumptions, on the construction of model equations, and on the extraction of information, in order to address contextually determinate questions on some behaviour of interest. A concept of selective accuracy is developed to explain the way in which qualitative and quantitative solutions should be utilized to understand systems. The qualitative methods rely on asymptotic methods and on sensitivity analysis, whereas the quantitative methods are best understood using backward error analysis. The basic underpinning of this perspective is readily understandable across scientific fields, and it thereby provides a view of mathematical tractability readily interpretable in the broader context of mathematical modelling. In addition, this perspective is used to discuss the nature of theories, the role of scaling, and the epistemological and semantic aspects of experimentation. In conclusion, we argue for a method of local and global conceptual analysis that goes beyond the reach of the tools standardly used to capture the logic of science; on their basis, the applicability of mathematics finds itself demystified. (shrink)

The aim of the paper is to assess the relative merits of two formal representations of structure, namely, set theory and category theory. The purpose is to articulate ontic structural realism. In turn, this will facilitate a discussion on the strengths and weaknesses of both concepts and will lead to a proposal for a pragmatics-based approach to the question of the choice of an appropriate framework. First, we present a case study from contemporary science—a comparison of the formulation of quantum (...) mechanics in the language of Hilbert spaces and abstract C* algebras. It is then shown how the method of structural representation can be determined based on the pragmatics of goal-oriented research, not a dogmatic choice. We investigate a hypothesis stating that the use of the interplay between the powers of abstraction and detail of different representational methods results in adopting a pluralistic, as opposed to standard, unificatory, perspective on the role of structural representation in OSR. (shrink)

In this paper I reconstruct and critically examine the reasoning leading to the famous prediction of the ‘omega minus’ particle by M. Gell-Mann and Y. Ne’eman (in 1962) on the basis of a symmetry classification scheme. While the peculiarity of this prediction has occasionally been noticed in the literature, a detailed treatment of the methodological problems it poses has not been offered yet. By spelling out the characteristics of this type of prediction, I aim to underscore the challenges raised by (...) this episode to standard scientific methodology, especially to the traditional deductive-nomological account of prediction. (shrink)

In this paper, I argue that symmetry principles in physics (in particular, in quantum mechanics) have a methodological character, rather than an ontological or an epistemological one. First, I provide a framework to address three related issues regarding the notion of symmetry: (i) how the notion can be characterized; (ii) one way of discussing the nature of symmetry principles, and (iii) a tentative account of some types of symmetry in physics. To illustrate how the framework functions, I then consider the (...) case of the early formulation of quantum mechanics, examining the different roles played by symmetry in this context. Finally, I raise difficulties for ontological and purely episte. (shrink)

Philosophical analysis of mathematical knowledge are commonly conducted within the realist/antirealist dichotomy. Nevertheless, philosophers working within this dichotomy pay little attention to the way in which mathematics evolves and structures itself. Focusing on mathematical practice, I propose a weak notion of objectivity of mathematical knowledge that preserves the intersubjective character of mathematical knowledge but does not bear on a view of mathematics as a body of mind-independent necessary truths. Furthermore, I show how that the successful application of mathematics in science (...) is an important trigger for the objectivity of mathematical knowledge. (shrink)

We outline a framework for analyzing episodes from the history of science in which the application of mathematics plays a constitutive role in the conceptual development of empirical sciences. Our starting point is the inferential conception of the application of mathematics, recently advanced by Bueno and Colyvan. We identify and discuss some systematic problems of this approach. We propose refinements of the inferential conception based on theoretical considerations and on the basis of a historical case study. We demonstrate the usefulness (...) of the refined, dynamical inferential conception using the well-researched example of the genesis of general relativity. Specifically, we look at the collaboration of the physicist Einstein and the mathematician Grossmann in the years 1912--1913, which resulted in the jointly published ``Outline of a Generalized Theory of Relativity and a Theory of Gravitation,'' a precursor theory of the final theory of general relativity. In this episode, an independently developed mathematical theory, the theory of differential invariants and the absolute differential calculus, was applied in the process of physical theorizing aiming at finding a relativistic theory of gravitation. We argue that the dynamical inferential conception not only provides a natural framework to describe and analyze this episode, but it also generates new questions and insights. We comment on the mathematical tradition on which Grossmann drew, and on his own contributions to mathematical theorizing. We argue that the dynamical inferential conception allows us to identify both the role of heuristics and of mathematical resources as well as the systematic role of problems and mistakes in the reconstruction of episodes of conceptual innovation and theory change. (shrink)

Today most philosophers of science believe that models play a central role in science and that one of the main functions of scientific models is to represent systems in the world. Despite much talk of models and representation, however, it is not yet clear what representation in this context amounts to nor what conditions a certain model needs to meet in order to be a representation of a certain system. In this thesis, I address these two questions. First, I will (...) distinguish three senses in which something, a vehicle, can be said to be a representation of something else, a target—which I will call respectively denotation, epistemic representation, and faithful epistemic representation—and I will argue that the last two senses are the most important in this context. I will then outline a general account of what makes a vehicle an epistemic representation of a certain target for a certain user—which, according to the account I defend, is the fact that a user adopts what I call an interpretation of the vehicle in terms of the target—and of what makes an epistemic representation of a certain target a faithful epistemic representation of it—which, according to the account I defend, is a specific sort of structural similarity between the vehicle and the target. (shrink)

In this paper I focus on the impact on structuralism of the quantum treatment of objects in terms of symmetry groups and, in particular, on the question as to how we might eliminate, or better, reconceptualise such objects in structural terms. With regard to the former, both Cassirer and Eddington not only explicitly and famously tied their structuralism to the development of group theory but also drew on the quantum treatment in order to further their structuralist aims and here I (...) sketch the relevant history with an eye on what lessons might be drawn. With regard to the latter, Ladyman has explicitly cited Castellani's work on the group-theoretical constitution of quantum objects and I indicate both how such an approach needs to be understood if it is to mesh with Ladyman's 'ontic' form of structural realism and how it might accommodate permutation symmetry through a consideration of Huggett's recent account. (shrink)