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Summary Mathematical structuralism is the view on which mathematical theories, rather than being about mathematical objects (like THE number zero, THE number one, etc., if there are such things), are about classes of structures (e.g., all omega sequences) whatever the objects in such structures are and whatever their nature is. The view comes in two important variants. Ante rem structuralism is a type of mathematical platonism, on which structures are abstract mathematical objects existing independently of their instances (called systems). In rebus structuralism is the view on which mathematical theories are about systems, which do not have to be abstract. So the former has to handle the problems that mathematical platonism encounters and the latter has to handle the problems encountered by mathematical nominalism. Apart from that, both approaches face specific challenges related to how the notion of a structure is understood and how it is to be squared with mathematical practice and the applicability of mathematics. 
Key works Benacerraf 1965Hellman 1989Resnik 1997Shapiro 1997Chihara 2004.
Introductions Start with appropriate sections of Horsten 2008 and references therein. Also worth a read are Shapiro 2010 and Shapiro 1996.
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  1. Andrew Aberdein (2012). The Parallel Structure of Mathematical Reasoning. In Alison Pease & Brendan Larvor (eds.), Proceedings of the Symposium on Mathematical Practice and Cognition Ii: A Symposium at the Aisb/Iacap World Congress 2012. Society for the Study of Artificial Intelligence and the Simulation of Behaviour. 7--14.
    This paper proposes an account of mathematical reasoning as parallel in structure: the arguments which mathematicians use to persuade each other of their results comprise the argumentational structure; the inferential structure is composed of derivations which offer a formal counterpart to these arguments. Some conflicts about the foundations of mathematics correspond to disagreements over which steps should be admissible in the inferential structure. Similarly, disagreements over the admissibility of steps in the argumentational structure correspond to different views about mathematical practice. (...)
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  2. Peter Ainsworth (2011). Ontic Structural Realism and the Principle of the Identity of Indiscernibles. Erkenntnis 75 (1):67-84.
    Recently, there has been a debate as to whether or not the principle of the identity of indiscernibles (the PII) is compatible with quantum physics. It is also sometimes argued that the answer to this question has implications for the debate over the tenability of ontic structural realism (OSR). The central aim of this paper is to establish what relationship there is (if any) between the PII and OSR. It is argued that one common interpretation of OSR is undermined if (...)
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  3. Holger Andreas (forthcoming). Carnapian Structuralism. Erkenntnis:1-19.
    This paper aims to set forth Carnapian structuralism, i.e., a syntactic view of the structuralist approach which is deeply inspired by Carnap’s dual level conception of scientific theories. At its core is the axiomatisation of a metatheoretical concept AE(T) which characterises those extensions of an intended application that are admissible in the sense of being models of the theory-element T and that satisfy all links, constraints and specialisations. The union of axiom systems of AE(T) (where T is an element of (...)
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  4. Holger Andreas & Frank Zenker (forthcoming). Perspectives on Structuralism. Erkenntnis:1-1.
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  5. David Aubin (1997). The Withering Immortality of Nicolas Bourbaki: A Cultural Connector at the Confluence of Mathematics, Structuralism, and the Oulipo in France. Science in Context 10 (2).
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  6. S. Awodey (1996). Structure in Mathematics and Logic: A Categorical Perspective. Philosophia Mathematica 4 (3):209-237.
    A precise notion of ‘mathematical structure’ other than that given by model theory may prove fruitful in the philosophy of mathematics. It is shown how the language and methods of category theory provide such a notion, having developed out of a structural approach in modern mathematical practice. As an example, it is then shown how the categorical notion of a topos provides a characterization of ‘logical structure’, and an alternative to the Pregean approach to logic which is continuous with the (...)
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  7. Steve Awodey (2013). Structuralism, Invariance, and Univalence. Philosophia Mathematica 22 (1):nkt030.
    The recent discovery of an interpretation of constructive type theory into abstract homotopy theory suggests a new approach to the foundations of mathematics with intrinsic geometric content and a computational implementation. Voevodsky has proposed such a program, including a new axiom with both geometric and logical significance: the Univalence Axiom. It captures the familiar aspect of informal mathematical practice according to which one can identify isomorphic objects. While it is incompatible with conventional foundations, it is a powerful addition to homotopy (...)
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  8. Steve Awodey (2004). An Answer to Hellman's Question: ‘Does Category Theory Provide a Framework for Mathematical Structuralism?’. Philosophia Mathematica 12 (1):54-64.
    An affirmative answer is given to the question quoted in the title.
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  9. Alan Baker (2006). Book Review: Charles S. Chihara. A Structural Account of Mathematics. [REVIEW] Notre Dame Journal of Formal Logic 47 (3):435-442.
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  10. V. Bakos (2005). Structuralism Redivivus (Situation in the 1960s). Filozofia 60 (10):746-760.
    Structuralism took part in the efforts to revive intellectual development in the former Czecho-Slovakia in the 1960s as an efficient methodological instrument, a scientific meta-theory, as well as an inovation in the field of philosophical theory . Open discussions were focused on the questions such as the relation between methodology and philosophy, system and structure, system and development, problem of the connection of synchronicity and diachronicity, the role of the subject, etc. Czecho-Slovak school applied the dynamic aspect of the structure (...)
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  11. Mark Balaguer (1999). Critical Studies / Book Reviews. Philosophia Mathematica 7 (1):108-126.
  12. Garret Barden (1976). Structuralism. Philosophical Studies 25:324-326.
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  13. R. H. Barnes (2001). Structuralism. In N. J. Smelser & B. Baltes (eds.), International Encyclopedia of the Social and Behavioral Sciences. 15222--15225.
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  14. Paul Benacerraf (1996). Recantation or Any Old W-Sequence Would Do After All. Philosophia Mathematica 4 (2):184-189.
    What Numbers Could Not Be’) that an adequate account of the numbers and our arithmetic practice must satisfy not only the conditions usually recognized to be necessary: (a) identify some w-sequence as the numbers, and (b) correctly characterize the cardinality relation that relates a set to a member of that sequence as its cardinal number—it must also satisfy a third condition: the ‘<’ of the sequence must be recursive. This paper argues that adding this further condition was a mistake—any w-sequence (...)
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  15. J. L. Bermudez (2007). Indistinguishable Elements and Mathematical Structuralism. Analysis 67 (2):112-116.
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  16. José Luis Bermúdez (2007). Indistinguishable Elements and Mathematical Structuralism. Analysis 67 (294):112-116.
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  17. Alisa Bokulich & Peter Bokulich (eds.) (2011). Scientific Structuralism. Springer Science+Business Media.
    This book will be of particular interest to those philosophers, scientists, and mathematicians who are interested in the foundations of science.
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  18. Izabela Bondecka-Krzykowska (2004). Strukturalizm jako alternatywa dla platonizmu w filozofii matematyki. Filozofia Nauki 1.
    The aim of this paper is to analyze structuralism as an alternative view to platonism in the philosophy of mathematics. We also try to find out if ontological and epistemological problems of platonism can be avoided by admitting the principles of structuralism. Structuralism claims that mathematical objects are merely positions in structures and have no identity or in general any important features outside these structures. Such view allows to avoid problems of the nature of numbers and other mathematical objects. But (...)
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  19. Izabela Bondecka-Krzykowska & Roman Murawski (2008). Structuralism and Category Theory in the Contemporary Philosophy of Mathematics. Logique Et Analyse 51 (204):365.
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  20. B. Borstner (2002). Mathematical Structuralism is a Kind of Platonism. Filozofski Vestnik 23 (1):7-24.
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  21. Roy Boyne (2000). Structuralism. In Bryan S. Turner (ed.), The Blackwell Companion to Social Theory. Blackwell Publishers. 2--160.
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  22. John P. Burgess, Putting Structuralism in its Place.
    One textbook may introduce the real numbers in Cantor’s way, and another in Dedekind’s, and the mathematical community as a whole will be completely indifferent to the choice between the two. This sort of phenomenon was famously called to the attention of philosophers by Paul Benacerraf. It will be argued that structuralism in philosophy of mathematics is a mistake, a generalization of Benacerraf’s observation in the wrong direction, resulting from philosophers’ preoccupation with ontology.
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  23. John P. Burgess (2005). Charles S. Chihara. A Structural Account of Mathematics. Oxford: Oxford University Press, 2004. Pp. XIV + 380. ISBN 0-19-926753-. [REVIEW] Philosophia Mathematica 13 (1):78-90.
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  24. Jacob Busch (2003). What Structures Could Not Be. International Studies in the Philosophy of Science 17 (3):211 – 225.
    James Ladyman has recently proposed a view according to which all that exists on the level of microphysics are structures "all the way down". By means of a comparative reading of structuralism in philosophy of mathematics as proposed by Stewart Shapiro, I shall present what I believe structures could not be. I shall argue that, if Ladyman is indeed proposing something as strong as suggested here, then he is committed to solving problems that proponents of structuralism in philosophy of mathematics (...)
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  25. Tim Button (2006). Realistic Structuralism's Identity Crisis: A Hybrid Solution. Analysis 66 (3):216–222.
    Keränen (2001) raises an argument against realistic (ante rem) structuralism: where a mathematical structure has a non-trivial automorphism, distinct indiscernible positions within the structure cannot be shown to be non-identical using only the properties and relations of that structure. Ladyman (2005) responds by allowing our identity criterion to include 'irreflexive two-place relations'. I note that this does not solve the problem for structures with indistinguishable positions, i.e. positions that have all the same properties as each other and exactly the same (...)
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  26. Jessica Carter (2008). Structuralism as a Philosophy of Mathematical Practice. Synthese 163 (2):119 - 131.
    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 (...)
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  27. Jessica Carter (2005). Individuation of Objects – a Problem for Structuralism? Synthese 143 (3):291 - 307.
    . This paper identifies two aspects of the structuralist position of S. Shapiro which are in conflict with the actual practice of mathematics. The first problem follows from Shapiros identification of isomorphic structures. Here I consider the so called K-group, as defined by A. Grothendieck in algebraic geometry, and a group which is isomorphic to the K-group, and I argue that these are not equal. The second problem concerns Shapiros claim that it is not possible to identify objects in a (...)
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  28. Carlos M. Madrid Casado (2009). El realismo estructural a debate: matemáticas, ontología y representación. Revista de Filosofía (Madrid) 33 (2):49-66.
    The aim of this paper is to undermine structural realism by testing the soundness of its three main theses. The first section presents the epistemic and ontic forms of structural realism. The following section defends that if scientific theories represent the structure of the world, structural realism needs a general account of representation. Representation is the crux of structural realism. Section 3 argues that structure/ontology distinction collapses. Mathematical structures are ontologyladen. Lastly, section 4 is devoted to analyse whether there is (...)
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  29. Peter Caws (1997). Sartrean Structuralism? In William Leon McBride (ed.), Sartre's French Contemporaries and Enduring Influences. Garland. 8--297.
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  30. CharlesParsons (2004). Structuralism and Metaphysics. Philosophical Quarterly 54 (214):56–77.
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  31. C. Chihara & Fraser MacBride (2005). REVIEWS-A Structural Account of Mathematics. Bulletin of Symbolic Logic 11 (1):79-82.
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  32. Charles Chihara (2004). A Structural Account of Mathematics. Clarendon Press.
    A Structural Account of Mathematics will be required reading for anyone working in this field.
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  33. Desmond M. Clarke (1976). Structuralism. Philosophical Studies 25:326-328.
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  34. Julian C. Cole (2010). Mathematical Structuralism Today. Philosophy Compass 5 (8):689-699.
    Two topics figure prominently in recent discussions of mathematical structuralism: challenges to the purported metaphysical insight provided by sui generis structuralism and the significance of category theory for understanding and articulating mathematical structuralism. This article presents an overview of central themes related to these topics.
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  35. M. Colyvan (1998). Review. Michael D Resnik. Mathematics as a Science of Patterns. Oxford, Clarendon Press, 1997. Stewart Shapiro. Philosophy of Mathematics: Structure and Ontology, New York, Oxford University Press, 1997. [REVIEW] British Journal for the Philosophy of Science 49 (4):652-656.
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  36. M. Colyvan (1998). Review of M Resnik 'Mathematics as a Science of Patterns' and S Shapiro 'Philosophy of Mathematics'. [REVIEW] British Journal for the Philosophy of Science 49 (4):652-6.
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  37. R. Cook & S. Shapiro (1998). Charles E. Rickart. Structuralism and Structures: A Mathematical Perspective. Singapore: World Scientific Publishing, 1995. Pp. Xiii + 219. ISBN 981-02-1860-5. [REVIEW] Philosophia Mathematica 6 (2):227-231.
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  38. Leo Corry (1992). Nicolas Bourbaki and the Concept of Mathematical Structure. Synthese 92 (3):315 - 348.
    In the present article two possible meanings of the term mathematical structure are discussed: a formal and a nonformal one. It is claimed that contemporary mathematics is structural only in the nonformal sense of the term. Bourbaki's definition of structure is presented as one among several attempts to elucidate the meaning of that nonformal idea by developing a formal theory which allegedly accounts for it. It is shown that Bourbaki's concept of structure was, from a mathematical point of view, a (...)
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  39. Paul Cortois (1996). The Structure of Mathematical Experience According to Jean Cavaillèst. Philosophia Mathematica 4 (1):18-41.
    In this expository article one of the contributions of Jean Cavailles to the philosophy of mathematics is presented: the analysis of ‘mathematical experience’. The place of Cavailles on the logico-philosophical scene of the 30s and 40s is sketched. I propose a partial interpretation of Cavailles's epistemological program of so-called ‘conceptual dialectics’: mathematical holism, duality principles, the notion of formal contents, and the specific temporal structure of conceptual dynamics. The structure of mathematical abstraction is analysed in terms of its complementary dimensions: (...)
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  40. Jairo José da Silva (2010). Structuralism and the Applicability of Mathematics. Axiomathes 20 (2-3):229-253.
    In this paper I argue for the view that structuralism offers the best perspective for an acceptable account of the applicability of mathematics in the empirical sciences. Structuralism, as I understand it, is the view that mathematics is not the science of a particular type of objects, but of structural properties of arbitrary domains of entities, regardless of whether they are actually existing, merely presupposed or only intentionally intended.
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  41. Rafael de Clercq (2012). On Some Putative Graph-Theoretic Counterexamples to the Principle of the Identity of Indiscernibles. Synthese 187 (2):661-672.
    Recently, several authors have claimed to have found graph-theoretic counterexamples to the Principle of the Identity of Indiscernibles. In this paper, I argue that their counterexamples presuppose a certain view of what unlabeled graphs are, and that this view is optional at best.
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  42. W. Dean (2014). Models and Computability. Philosophia Mathematica 22 (2):143-166.
    Computationalism holds that our grasp of notions like ‘computable function’ can be used to account for our putative ability to refer to the standard model of arithmetic. Tennenbaum's Theorem has been repeatedly invoked in service of this claim. I will argue that not only do the relevant class of arguments fail, but that the result itself is most naturally understood as having the opposite of a reference-fixing effect — i.e., rather than securing the determinacy of number-theoretic reference, Tennenbaum's Theorem points (...)
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  43. William Demopoulos (2003). 12 RusselPs Structuralism and the Absolute Description of the World. In Nicholas Griffin (ed.), The Cambridge Companion to Bertrand Russell. Cambridge University Press. 392.
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  44. Werner Diederich (1996). Pragmatic and Diachronic Aspects of Structuralism. In Wolfgang Balzer & Carlos Ulises Moulines (eds.), Structuralist Theory of Science: Focal Issues, New Results. Walter de Gruyter. 75--82.
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  45. A. Díez, Javier Echeverría & Andoni Ibarra (eds.) (1990). Structures in Mathematical Theories: Reports of the San Sebastian International Symposium, September 25-29, 1990. Argitarapen Zerbitzua Euskal, Herriko Unibertsitatea.
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  46. Marcus du Sautoy (2010). 4 Mathematical Identity. In Giselle Walker & E. S. Leedham-Green (eds.), Identity. Cambridge University Press. 88.
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  47. Roman Duda (2000). Integralność matematyki. Filozofia Nauki 1.
    The origins of mathematics, a close connection and interpenetration of its parts, and uniform procedures of dealing with the mathematical matter - all of them speak in favour of the integrality of mathematics. It seems that a strong argument for such a view is a fundamental object of contemporary mathematics; namely a real line, which contains real numbers (so arithmetics as well) and constitutes a basis of geometry, mathematical analysis and all derivative branches. From the basic-structures perspective it is clear (...)
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  48. Javier Eliozondo & Axel Arturo Barceló Aspeitia (2005). Reseña de "Thinking About Mathematics" de Stewart Shapiro. Signos Filosóficos 7 (13):135-137.
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  49. Solomon Feferman, Conceptual Structuralism and the Continuum.
    • This comes from my general view of the nature of mathematics, that it is humanly based and that it deals with more or less clear conceptions of mathematical structures; for want of a better word, I call that view conceptual structuralism.
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  50. John Forge (2002). Reflections on Structuralism and Scientific Explanation. Synthese 130 (1):109 - 121.
    This paper is about structuralism as a form of reconstructing theories, associated with the work Sneed, Balzar and Moulines among others, and not about "structuralism" is any of its other manifold senses. The paper is a reflection in that it looks back on some earlier work of my own on the subject of structuralism and explanation, in which I argued that structuralism and my 'instance view' of explanation go well together, with structuralism providing the means to develop the idea of (...)
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