About this topic
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 Michael 1997Shapiro 1997Charles 2003.
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. Ontic Structural Realism and the Principle of the Identity of Indiscernibles.Peter Ainsworth - 2011 - 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 (...)
  2. Justin Leiber's "Structuralism". [REVIEW]Virgil C. Aldrich - 1979 - Philosophy and Phenomenological Research 39 (4):598.
  3. Carnapian Structuralism.Holger Andreas - 2014 - Erkenntnis 79 (S8):1373-1391.
    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 (...)
  4. Perspectives on Structuralism.Holger Andreas & Frank Zenker - 2014 - Erkenntnis 79 (S8):1365-1365.
  5. PLAGET, J. - "Structuralism". Translated by C. Maschler. [REVIEW]C. Atkinson - 1973 - Mind 82:471.
  6. The Withering Immortality of Nicolas Bourbaki: A Cultural Connector at the Confluence of Mathematics, Structuralism, and the Oulipo in France.David Aubin - 1997 - Science in Context 10 (2).
    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 (...)
  7. Structure in Mathematics and Logic: A Categorical Perspective.S. Awodey - 1996 - 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 (...)
  8. Structuralism, Invariance, and Univalence.Steve Awodey - 2013 - 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 (...)
  9. An Answer to Hellman's Question: ‘Does Category Theory Provide a Framework for Mathematical Structuralism?’.Steve Awodey - 2004 - Philosophia Mathematica 12 (1):54-64.
    An affirmative answer is given to the question quoted in the title.
  10. Book Review: Charles S. Chihara. A Structural Account of Mathematics. [REVIEW]Alan Baker - 2006 - Notre Dame Journal of Formal Logic 47 (3):435-442.
  11. Structuralism Redivivus (Situation in the 1960s).V. Bakos - 2005 - 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 (...)
  12. Critical Studies / Book Reviews.Mark Balaguer - 1999 - Philosophia Mathematica 7 (1):108-126.
  13. Structuralism.Garret Barden - 1976 - Philosophical Studies 25:324-326.
  14. Structuralism.R. H. Barnes - 2001 - In N. J. Smelser & B. Baltes (eds.), Philosophical Books. pp. 15222--15225.
  15. Recantation or Any Old W-Sequence Would Do After All.Paul Benacerraf - 1996 - 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 (...)
  16. Indistinguishable Elements and Mathematical Structuralism.Bermúdez José Luis - 2007 - Analysis 67 (294):112-116.
    The existence of structures with non-trivial authomorphisms (such as the automorphism of the field of complex numbers onto itself that swaps the two roots of – 1) has been held by Burgess and others to pose a serious difficulty for mathematical structuralism. This paper proposes a model-theoretic solution to the problem. It suggests that mathematical structuralists identify the “position” of an n-tuple in a mathematical structure with the type of that n-tuple in the expansion of the structure that has a (...)
  17. Indistinguishable Elements and Mathematical Structuralism.J. L. Bermudez - 2007 - Analysis 67 (2):112-116.
  18. On the Packing of AuCu 3 and Cu 2 Mg Structures.M. Bhatia - 2003 - Philosophical Magazine 83 (19):2293-2300.
    The volumes of formation of AuCu 3 and Cu 2 Mg prototypes in all the binary systems in which the two structures exist have been calculated to show that the contraction experienced per mole is higher for the AuCu 3 structure. The reasons for such behaviour are outlined. The different systems considered have radius ratios varying from 1.078 to 1.372. One normally expects the Cu 2 Mg structure to be better packed at large values of the radius ratio. In the (...)
  19. Is Cusanus the Father of Structuralism.Jasper Blystone - 1972 - Philosophy Today 16 (4):296-305.
  20. Scientific Structuralism.Alisa Bokulich & Peter Bokulich (eds.) - 2011 - 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.
  21. Strukturalizm jako alternatywa dla platonizmu w filozofii matematyki.Izabela Bondecka-Krzykowska - 2004 - 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 (...)
  22. Structuralism and Category Theory in the Contemporary Philosophy of Mathematics.Izabela Bondecka-Krzykowska & Roman Murawski - 2008 - Logique Et Analyse 51 (204):365.
  23. Mathematical Structuralism is a Kind of Platonism.B. Borstner - 2002 - Filozofski Vestnik 23 (1):7-24.
  24. Structuralism and Historical Knowledge.F. Botturi - 1983 - Rivista di Filosofia Neo-Scolastica 75 (4):563-581.
  25. Structuralism.Roy Boyne - 2000 - In Bryan S. Turner (ed.), Philosophical Books. Blackwell. pp. 2--160.
  26. Personality Model.Miro Brada - 2000 - Problem Paradise:42-43.
    In 1995, as a student of psychology inspired by natural science, I defined a logical model of personality explaining psychosis. I created (for my MA thesis, 1998 and grant research, 1999) new kind of tests assessing intelligence, creativity, prejudices, expectations to show more exact methods in psychology. During my Phd study in economics, I developed 'Maximization of Uniqueness (Originality)' model enhancing the classic utility to explain irrational motivations linking economics and psychology. Later I became computer programmer developing functional programming. According (...)
  27. Modelle Mathematikhistorischer Entwicklung.Wolfgang Breidert - 1993 - NTM International Journal of History and Ethics of Natural Sciences, Technology and Medicine 1 (1):193-199.
    In the historiography of mathematics the concept of deviation is relative to a normal way of developing mathematics. Similarly, the concept of abbreviation is necessarily connected to some aim to which this development is directed. We should not speak of deviation or abbreviation without regard for such contects. In the history of mathematics there are some cases of treating impossible objects as possible objects of a new theory. Therefore it may be expected that there are more ways out than in (...)
  28. Structuralism: Moscow, Prague, Paris.Jan M. Broekman - 1974 - D. Reidel Pub. Co..
    THE STRUCTURALISTIC ENDEAVOUR. THE WORLD AS MUSICAL SCORE The recent decades of this century have witnessed unusually rapid and far- reaching changes in the ...
  29. Mathematics, Media, and Cultural Techniques.Jochen Brüning - 2013 - Common Knowledge 19 (2):224-236.
    This contribution, by a mathematician, to the Common Knowledge symposium “Fuzzy Studies” examines some mechanisms that seem essential for the “ratchet effect” that, in Michael Tomasello's use of the term, refers to the ability of human cultures to preserve their achievements even through serious crises and even where preservation entails substantial loss. By taking the word culture to refer to any group of individuals who closely cooperate over an extended period, this article evaluates mathematicians and mathematics as its main example. (...)
  30. Putting Structuralism in its Place.John P. Burgess - unknown
    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.
  31. Charles S. Chihara. A Structural Account of Mathematics. Oxford: Oxford University Press, 2004. Pp. XIV + 380. ISBN 0-19-926753-. [REVIEW]John P. Burgess - 2005 - Philosophia Mathematica 13 (1):78-90.
  32. What Structures Could Not Be.Jacob Busch - 2003 - 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 (...)
  33. Realistic Structuralism's Identity Crisis: A Hybrid Solution.Tim Button - 2006 - 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 (...)
  34. Structure and Categoricity: Determinacy of Reference and Truth Value in the Philosophy of Mathematics.Tim Button & Sean Walsh - 2016 - Philosophia Mathematica 24 (3):283-307.
    This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory.
  35. Structuralism as a Philosophy of Mathematical Practice.Jessica Carter - 2008 - 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 (...)
  36. Individuation of Objects – a Problem for Structuralism?Jessica Carter - 2005 - 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 (...)
  37. El realismo estructural a debate: matemáticas, ontología y representación.Carlos M. Madrid Casado - 2008 - 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 (...)
  38. Structuralism: Origins And Current Developments.Matthieu Casalis - 1973 - Southwest Philosophical Studies.
  39. Sartrean Structuralism?Peter Caws - 1997 - In William Leon McBride (ed.), Sartre's French Contemporaries and Enduring Influences. Garland. pp. 8--297.
  40. Structuralism: The Art of the Intelligible.Peter Caws - 1988 - Humanities Press.
  41. The Recent Literature of Structuralism.Peter Caws - 1972 - Philosophische Rundschau 18:63.
  42. Structuralism and Metaphysics. CharlesParsons - 2004 - Philosophical Quarterly 54 (214):56–77.
  43. REVIEWS-A Structural Account of Mathematics.C. Chihara & Fraser MacBride - 2005 - Bulletin of Symbolic Logic 11 (1):79-82.
  44. L'idée Et la Forme.Jacques Claret - 1979
  45. Structuralism.Desmond M. Clarke - 1976 - Philosophical Studies 25:326-328.
  46. Mathematical Structuralism Today.Julian C. Cole - 2010 - 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.
  47. 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]M. Colyvan - 1998 - British Journal for the Philosophy of Science 49 (4):652-656.
  48. Review of M Resnik 'Mathematics as a Science of Patterns' and S Shapiro 'Philosophy of Mathematics'. [REVIEW]M. Colyvan - 1998 - British Journal for the Philosophy of Science 49 (4):652-6.
  49. Charles E. Rickart. Structuralism and Structures: A Mathematical Perspective. Singapore: World Scientific Publishing, 1995. Pp. Xiii + 219. ISBN 981-02-1860-5. [REVIEW]R. Cook & S. Shapiro - 1998 - Philosophia Mathematica 6 (2):227-231.
  50. Review Of: Hodesdon, K. “Mathematica Representation: Playing a Role”. Philosophical Studies (2014) 168:769–782. Mathematical Reviews. MR 3176431.John Corcoran - 2015 - MATHEMATICAL REVIEWS 2015:3176431.
    This 4-page review-essay—which is entirely reportorial and philosophically neutral as are my other contributions to MATHEMATICAL REVIEWS—starts with a short introduction to the philosophy known as mathematical structuralism. The history of structuralism traces back to George Boole (1815–1864). By reference to a recent article various feature of structuralism are discussed with special attention to ambiguity and other terminological issues. The review-essay includes a description of the recent article. The article’s 4-sentence summary is quoted in full and then analyzed. The point (...)
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