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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. 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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  2. 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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  3. 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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  4. 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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  5. 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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  6. Mark Balaguer (1999). Critical Studies / Book Reviews. Philosophia Mathematica 7 (1):108-126.
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  7. 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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  8. J. L. Bermudez (2007). Indistinguishable Elements and Mathematical Structuralism. Analysis 67 (2):112-116.
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  9. José Luis Bermúdez (2007). Indistinguishable Elements and Mathematical Structuralism. Analysis 67 (294):112-116.
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  10. 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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  11. B. Borstner (2002). Mathematical Structuralism is a Kind of Platonism. Filozofski Vestnik 23 (1):7-24.
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  12. 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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  13. 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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  14. 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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  15. 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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  16. 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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  17. 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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  18. 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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  19. 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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  20. 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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  21. 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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  22. 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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  23. 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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  24. 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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  25. 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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  26. 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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  27. James Franklin (2014). Aristotelian Realist Philosophy of Mathematics. Palgrave MacMillan.
    An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, are parts of the physical world and (...)
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  28. James Franklin (2009). Aristotelian Realism. In A. Irvine (ed.), The Philosophy of Mathematics (Handbook of the Philosophy of Science series). North-Holland Elsevier.
    Aristotelian, or non-Platonist, realism holds that mathematics is a science of the real world, just as much as biology or sociology are. Where biology studies living things and sociology studies human social relations, mathematics studies the quantitative or structural aspects of things, such as ratios, or patterns, or complexity, or numerosity, or symmetry. Let us start with an example, as Aristotelians always prefer, an example that introduces the essential themes of the Aristotelian view of mathematics. A typical mathematical truth is (...)
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  29. James Franklin (1989). Mathematical Necessity and Reality. Australasian Journal of Philosophy 67 (3):286 – 294.
    Einstein, like most philosophers, thought that there cannot be mathematical truths which are both necessary and about reality. The article argues against this, starting with prima facie examples such as "It is impossible to tile my bathroom floor with (equally-sized) regular pentagonal tiles." Replies are given to objections based on the supposedly purely logical or hypothetical nature of mathematics.
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  30. Simon Friederich (2010). Structuralism and Meta-Mathematics. Erkenntnis 73 (1):67 - 81.
    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. (...)
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  31. Hellman Geoffrey (1996). Structuralism Without Structures. Philosophia Mathematica 4 (2):100-123.
    Recent technical developments in the logic of nominalism make it possible to improve and extend significantly the approach to mathematics developed in Mathematics without Numbers. After reviewing the intuitive ideas behind structuralism in general, the modal-structuralist approach as potentially class-free is contrasted broadly with other leading approaches. The machinery of nominalistic ordered pairing (Burgess-Hazen-Lewis) and plural quantification (Boolos) can then be utilized to extend the core systems of modal-structural arithmetic and analysis respectively to full, classical, polyadic third- and fourthorder number (...)
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  32. M. Giaquinto (1999). Review of M. Resnik, Mathematics as a Science of Patterns. [REVIEW] Mind 108 (432):761-788.
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  33. Volker Halbach & Leon Horsten (2005). Computational Structuralism. Philosophia Mathematica 13 (2):174-186.
    According to structuralism in philosophy of mathematics, arithmetic is about a single structure. First-order theories are satisfied by (nonstandard) models that do not instantiate this structure. Proponents of structuralism have put forward various accounts of how we succeed in fixing one single structure as the intended interpretation of our arithmetical language. We shall look at a proposal that involves Tennenbaum's theorem, which says that any model with addition and multiplication as recursive operations is isomorphic to the standard model of arithmetic. (...)
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  34. Bob Hale (1996). Structuralism's Unpaid Epistemological Debts. Philosophia Mathematica 4 (2):124--47.
    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 (...)
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  35. Michael Hand (1993). Mathematical Structuralism and the Third Man. Canadian Journal of Philosophy 23 (2):179 - 192.
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  36. Adrian Heathcote (2014). On the Exhaustion of Mathematical Entities by Structures. Axiomathes 24 (2):167-180.
    There has been considerable discussion in the literature of one kind of identity problem that mathematical structuralism faces: the automorphism problem, in which the structure is unable to individuate the mathematical entities in its domain. Shapiro (Philos Math 16(3):285–309, 2008) has partly responded to these concerns. But I argue here that the theory faces an even more serious kind of identity problem, which the theory can’t overcome staying within its remit. I give two examples to make the point.
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  37. Geoffrey Hellman, Russell's Absolutism Vs.(?) Structuralism.
    Along with Frege, Russell maintained an absolutist stance regarding the subject matter of mathematics, revealed rather than imposed, or proposed, by logical analysis. The Fregean definition of cardinal number, for example, is viewed as (essentially) correct, not merely adequate for mathematics. And Dedekind’s “structuralist” views come in for criticism in the Principles. But, on reflection, Russell also flirted with views very close to a (different) version of structuralism. Main varieties of modern structuralism and their challenges are reviewed, taking account of (...)
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  38. Geoffrey Hellman, Structuralism.
    With the rise of multiple geometries in the nineteenth century, and in the last century the rise of abstract algebra, of the axiomatic method, the set-theoretic foundations of mathematics, and the influential work of the Bourbaki, certain views called “structuralist” have become commonplace. Mathematics is seen as the investigation, by more or less rigorous deductive means, of “abstract structures”, systems of objects fulfilling certain structural relations among themselves and in relation to other systems, without regard to the particular nature of (...)
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  39. Geoffrey Hellman, Structuralism, Mathematical.
    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 (...)
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  40. Geoffrey Hellman (2006). What is Categorical Structuralism?. In. In Johan van Benthem, Gerhard Heinzman, M. Rebushi & H. Visser (eds.), The Age of Alternative Logics. Springer. 151--161.
  41. Geoffrey Hellman (2003). Does Category Theory Provide a Framework for Mathematical Structuralism? Philosophia Mathematica 11 (2):129-157.
    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 (...)
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  42. Geoffrey Hellman (2001). Three Varieties of Mathematical Structuralism. Philosophia Mathematica 9 (2):184-211.
    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 (...)
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  43. Peter Hertling (1999). A Real Number Structure That is Effectively Categorical. Mathematical Logic Quarterly 45 (2):147-182.
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  44. Harold T. Hodes (2002). Book Review. Philosophy of Mathematics: Structure and Ontology. Stewart Shapiro. [REVIEW] Philosophy and Phenomenological Research 65 (2):467-475.
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  45. Harold T. Hodes (2002). Review of S. Shapiro, Philosophy of Mathematics: Structure and Ontology. [REVIEW] Philosophy and Phenomenological Research 65 (2):467 - 475.
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  46. Harold T. Hodes (2002). Stewart Shapiro's Philosophy of Mathematics. [REVIEW] Philosophy and Phenomenological Research 65 (2):467–475.
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  47. Kate Hodesdon (2014). Mathematical Representation: Playing a Role. Philosophical Studies 168 (3):769-782.
    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 (...)
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  48. A. Irvine (ed.) (2009). The Philosophy of Mathematics (Handbook of the Philosophy of Science Series). North-Holland Elsevier.
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  49. Mathematical Structuralism is A. Kind ofPlatonism (forthcoming). Izvlečki• abstracts. Filozofski Vestnik.
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  50. Mark Eli Kalderon (1996). What Numbers Could Be (and, Hence, Necessarily Are). Philosophia Mathematica 4 (3):238-255.
    This essay explores the commitments of modal structuralism. The precise nature of the modal-structuralist analysis obscures an unclarity of its import. As usually presented, modal structuralism is a form of anti-platonism. I defend an interpretation of modal structuralism that, far from being a form of anti-platonism, is itself a platonist analysis: The metaphysically significant distinction between (i) primitive modality and (ii) the natural numbers (objectually understood) is genuine, but the arithmetic facts just are facts about possible progressions. If correct, modal (...)
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