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 1997Shapiro 1997S. Chihara 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. added 2018-11-21
    What Are Structural Properties?†.Johannes Korbmacher & Georg Schiemer - 2018 - Philosophia Mathematica 26 (3):295-323.
    Informally, structural properties of mathematical objects are usually characterized in one of two ways: either as properties expressible purely in terms of the primitive relations of mathematical theories, or as the properties that hold of all structurally similar mathematical objects. We present two formal explications corresponding to these two informal characterizations of structural properties. Based on this, we discuss the relation between the two explications. As will be shown, the two characterizations do not determine the same class of mathematical properties. (...)
  2. added 2018-11-15
    The Applicability of Mathematics to Physical Modality.Nora Berenstain - 2017 - Synthese 194 (9):3361-3377.
    This paper argues that scientific realism commits us to a metaphysical determination relation between the mathematical entities that are indispensible to scientific explanation and the modal structure of the empirical phenomena those entities explain. The argument presupposes that scientific realism commits us to the indispensability argument. The viewpresented here is that the indispensability of mathematics commits us not only to the existence of mathematical structures and entities but to a metaphysical determination relation between those entities and the modal structure of (...)
  3. added 2018-09-29
    Philosophy of Mathematics.Øystein Linnebo - 2017 - Princeton, NJ: Princeton University Press.
    Mathematics is one of the most successful human endeavors—a paradigm of precision and objectivity. It is also one of our most puzzling endeavors, as it seems to deliver non-experiential knowledge of a non-physical reality consisting of numbers, sets, and functions. How can the success and objectivity of mathematics be reconciled with its puzzling features, which seem to set it apart from all the usual empirical sciences? This book offers a short but systematic introduction to the philosophy of mathematics. Readers are (...)
  4. added 2018-09-04
    Logical Structuralism and Benacerraf’s Problem.Audrey Yap - 2009 - Synthese 171 (1):157-173.
    There are two general questions which many views in the philosophy of mathematics can be seen as addressing: what are mathematical objects, and how do we have knowledge of them? Naturally, the answers given to these questions are linked, since whatever account we give of how we have knowledge of mathematical objects surely has to take into account what sorts of things we claim they are; conversely, whatever account we give of the nature of mathematical objects must be accompanied by (...)
  5. added 2018-06-14
    Modal Structuralism and Reflection.Sam Roberts - forthcoming - Review of Symbolic Logic:1-38.
    Modal structuralism promises an interpretation of set theory that avoids commitment to abstracta. This article investigates its underlying assumptions. In the first part, I start by highlighting some shortcomings of the standard axiomatisation of modal structuralism, and propose a new axiomatisation I call MSST (for Modal Structural Set Theory). The main theorem is that MSST interprets exactly Zermelo set theory plus the claim that every set is in some inaccessible rank of the cumulative hierarchy. In the second part of the (...)
  6. added 2018-06-11
    Mathematical Structural Realism.Christopher Pincock - 2011 - In Alisa Bokulich & Peter Bokulich (eds.), Scientific Structuralism. Springer Science+Business Media. pp. 67--79.
    Epistemic structural realists have argued that we are in a better epistemic position with respect to the structural claims made by our theories than the non-structural claims. Critics have objected that we cannot make the structure/non-structure distinction precise. I respond that a focus on mathematical structure leads to a clearer understanding of this debate. Unfortunately for the structural realist, however, the contribution that mathematics makes to scientific representation undermines any general confidence we might have in the structural claims made by (...)
  7. added 2018-06-06
    Poincaré on the Foundation of Geometry in the Understanding.Jeremy Shipley - 2017 - In Maria Zack & Dirk Schlimm (eds.), Research in History and Philosophy of Mathematics: The CSHPM 2016 Annual Meeting in Calgary, Alberta. Springer. pp. 19-37.
    This paper is about Poincaré’s view of the foundations of geometry. According to the established view, which has been inherited from the logical positivists, Poincaré, like Hilbert, held that axioms in geometry are schemata that provide implicit definitions of geometric terms, a view he expresses by stating that the axioms of geometry are “definitions in disguise.” I argue that this view does not accord well with Poincaré’s core commitment in the philosophy of geometry: the view that geometry is the study (...)
  8. added 2018-02-18
    Structuralism, Invariance, and Univalence†: Articles.Steve Awodey - 2014 - Philosophia Mathematica 22 (1):1-11.
    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. added 2018-02-17
    Univalent Foundations as Structuralist Foundations.Dimitris Tsementzis - 2017 - Synthese 194 (9):3583-3617.
    The Univalent Foundations of Mathematics provide not only an entirely non-Cantorian conception of the basic objects of mathematics but also a novel account of how foundations ought to relate to mathematical practice. In this paper, I intend to answer the question: In what way is UF a new foundation of mathematics? I will begin by connecting UF to a pragmatist reading of the structuralist thesis in the philosophy of mathematics, which I will use to define a criterion that a formal (...)
  10. added 2018-02-17
    The Sensible Foundation for Mathematics: A Defense of Kant's View.Mark Risjord - 1990 - Studies in History and Philosophy of Science Part A 21 (1):123-143.
  11. added 2017-11-28
    What We Talk About When We Talk About Numbers.Richard Pettigrew - manuscript
    In this paper, I describe and motivate a new species of mathematical structuralism, which I call Instrumental Nominalism about Set-Theoretic Structuralism. As the name suggests, this approach takes standard Set-Theoretic Structuralism of the sort championed by Bourbaki and removes its ontological commitments by taking an instrumental nominalist approach to that ontology of the sort described by Joseph Melia and Gideon Rosen. I argue that this avoids all of the problems that plague other versions of structuralism.
  12. added 2017-11-28
    The Semantic Plights of the Ante-Rem Structuralist.Bahram Assadian - 2018 - Philosophical Studies 175 (12):1-20.
    A version of the permutation argument in the philosophy of mathematics leads to the thesis that mathematical terms, contrary to appearances, are not genuine singular terms referring to individual objects; they are purely schematic or variables. By postulating ‘ante-rem structures’, the ante-rem structuralist aims to defuse the permutation argument and retain the referentiality of mathematical terms. This paper presents two semantic problems for the ante- rem view: (1) ante-rem structures are themselves subject to the permutation argument; (2) the ante-rem structuralist (...)
  13. added 2017-10-21
    Mathematics and Its Applications, A Transcendental-Idealist Perspective.Jairo Da Silva - 2017 - Springer.
    This monograph offers a fresh perspective on the applicability of mathematics in science. It explores what mathematics must be so that its applications to the empirical world do not constitute a mystery. In the process, readers are presented with a new version of mathematical structuralism. The author details a philosophy of mathematics in which the problem of its applicability, particularly in physics, in all its forms can be explained and justified. Chapters cover: mathematics as a formal science, mathematical ontology: what (...)
  14. added 2017-10-17
    The Access Problem for Knowledge of Logical Possibility.Sharon Berry - manuscript
    Accepting truth-value realism can seem to raise an explanatory problem: what can explain our accuracy about mathematics, i.e., the match between human psychology and objective mathematical facts? A range of current truth-value realist philosophies of mathematics allow one to reduce this access problem to a problem of explaining our accuracy about which mathematical practices are coherent -- in a sense which can be cashed out in terms of logical possibility. However, our ability to recognize these facts about logical possibility poses (...)
  15. added 2017-10-12
    What Constitutes the Numerical Diversity of Mathematical Objects?F. MacBride - 2006 - Analysis 66 (1):63-69.
  16. added 2017-09-25
    Modal Structuralism Simplified.Sharon Berry - 2018 - Canadian Journal of Philosophy 48 (2):200-222.
    Since Benacerraf’s ‘What Numbers Could Not Be,’ there has been a growing interest in mathematical structuralism. An influential form of mathematical structuralism, modal structuralism, uses logical possibility and second order logic to provide paraphrases of mathematical statements which don’t quantify over mathematical objects. These modal structuralist paraphrases are a useful tool for nominalists and realists alike. But their use of second order logic and quantification into the logical possibility operator raises concerns. In this paper, I show that the work of (...)
  17. added 2017-07-08
    Decision-Making Process and Information.Daegene Song - 2017 - INSPIRE-HEP, High Energy Physics (HEP) Database, CERN Online Publications, EUROPE.
    One of the most important concepts in logic and the foundations of mathematics may be useful in providing an explanation for the cosmological constant problem. A connection between self-reference and consciousness has been previously discussed due to their similar nature of making a reference to itself. Vacuum observation has the property of self-reference and consciousness in the sense that the observer is observing one's own reference frame of energy. In this paper, the cyclical loop model of self-reference is applied to (...)
  18. added 2017-02-16
    Wittgenstein on Dedekind’s Cut.M. S. Lourenço - 1998 - Disputatio 4:653-66.
  19. added 2017-02-16
    Theory of Mathematical Reality, A.Everett Larguier - 1938 - Modern Schoolman 16:88.
  20. added 2017-02-15
    Texts And The Objects Of Mathematics.Paul Ernest - 1997 - Philosophy of Mathematics Education Journal 10.
  21. added 2017-02-15
    The Peircean Interpretation Of Mathematics.Christopher Ormell - 1995 - Philosophy of Mathematics Education Journal 8.
  22. added 2017-02-15
    Frege and the Philosophy of Mathematics by Michael D. Resnik. [REVIEW]J. M. B. Moss - 1982 - Journal of Philosophy 79 (9):497-511.
  23. added 2017-02-14
    STRUCTURALISM―Key to Reality and Meaning in Science.Charles W. Lucas Jr - unknown
  24. added 2017-02-14
    Thematic Files-Mathematics and Knowledge in the Renaissance-Mathematics and Universal Science in Bacon and Descartes.Thierry Gontier - 2006 - Revue d'Histoire des Sciences 59 (2):285.
  25. added 2017-02-14
    Structuralism and Historical Knowledge.F. Botturi - 1983 - Rivista di Filosofia Neo-Scolastica 75 (4):563-581.
  26. added 2017-02-14
    Michael Resnik, Frege and the Philosophy of Mathematics Reviewed By.Alan McMichael - 1982 - Philosophy in Review 2 (6):291-294.
  27. added 2017-02-14
    Is Cusanus the Father of Structuralism.Jasper Blystone - 1972 - Philosophy Today 16 (4):296-305.
  28. added 2017-02-13
    How We Can Apply the Mathematics on the World?A. Ule - 2002 - Filozofski Vestnik 23 (1):25-51.
    In the article are presented the main philosophical explanations of the application of mathematics on the real world (Plato, Aristotle, rationalists, empiricists, Kant, Frege, Husserl, Carnap etc.). They indicate some typical triangular structure of relationships where the mathematical structures somehow correspond to the forms of reality, and thus are possible though something third what bound them. The attempts to solve the question of the application of mathematics by the dispensability of mathematics (e.g. Field) do not success because they do not (...)
  29. added 2017-02-13
    A Critique of TEC Syllabuses in Mathematics and Science.J. R. Nurgombe - 1983 - Educational Studies 9 (1):37-43.
  30. added 2017-02-12
    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. (...)
  31. added 2017-02-12
    Genetic Structuralism and the Analysis of Social Consciousness.William W. Mayrl - 1978 - Theory and Society 5 (1):19-44.
  32. added 2017-02-09
    Elementarity and Anti-Matter in Contemporary Physics: Comments on Michael D. Resnik's "Between Mathematics and Physics".Susan C. Hale - 1990 - PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1990:379 - 383.
    I point out that conceptions of particles as mathematical, or quasi mathematical, entities have a longer history than Resnik notices. I argue that Resnik's attack on the distinction between mathematical and physical entities is not deep enough. The crucial problem for this distinction finds its locus in the numerical indeterminancy of elementary particles. This problem, traced by Heisenberg, emerges from the discovery of antimatter.
  33. added 2017-02-09
    Structuralism and the Challenge of Metaphor.Newton Garver - 1986 - The Monist 69 (1):68-86.
  34. added 2017-02-09
    Mathematics and Science.P. J. M. - 1965 - Review of Metaphysics 18 (4):778-778.
  35. added 2017-02-08
    'Elegance' In Science And Mathematics: A Discussion.Michael Macnamara, Wietske Kistner & Jennifer Wilkinson - 1985 - South African Journal of Philosophy 4 (August):95-99.
  36. added 2017-02-08
    Mathematics and Science: Last Essays.Henri Poincaré - 1963 - Dover Publications.
  37. added 2017-02-08
    The Theory of of Mathematical Form.A. B. Kempe - 1897 - The Monist 7 (3):453-458.
  38. added 2017-02-07
    Modelle Mathematikhistorischer Entwicklung.Wolfgang Breidert - 1993 - NTM Zeitschrift für Geschichte der Wissenschaften, Technik und Medizin 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 (...)
  39. added 2017-02-07
    Structuralism and Beyond: A Critique of Presuppositions.F. Merrell - 1975 - Diogenes 23 (92):67-103.
  40. added 2017-02-02
    Post-Structural Readings of a Logico-Mathematical Text.Roy Wagner - 2008 - Perspectives on Science 16 (2):pp. 196-230.
    This paper will apply post-structural semiotic theories to study the texts of Gödel's first incompleteness theorem. I will study the texts’ own articulations of concepts of ‘meaning’, analyze the mechanisms they use to sustain their senses of validity, and point out how the texts depend (without losing their mathematical rigor) on sustaining some shifts of meaning. I will demonstrate that the texts manifest semiotic effects, which we usually associate with poetry and everyday speech. I will conclude with an analysis of (...)
  41. added 2017-02-01
    Mathematics, Science and Ontology.Thomas Tymoczko - 1991 - Synthese 88 (2):201 - 228.
    According to quasi-empiricism, mathematics is very like a branch of natural science. But if mathematics is like a branch of science, and science studies real objects, then mathematics should study real objects. Thus a quasi-empirical account of mathematics must answer the old epistemological question: How is knowledge of abstract objects possible? This paper attempts to show how it is possible.The second section examines the problem as it was posed by Benacerraf in Mathematical Truth and the next section presents a way (...)
  42. added 2017-01-29
    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 (...)
  43. added 2017-01-28
    Richard Dedekind Et les Fondements des Mathématiques Avec de Nombreux Textes Inédits.Pierre Dugac - 1976 - J. Vrin.
  44. added 2017-01-27
    Say My Name. An Objection to Ante Rem Structuralism.Tim Räz - 2015 - Philosophia Mathematica 23 (1):116-125.
    I raise an objection to Stewart Shapiro's version of ante rem structuralism: I show that it is in conflict with mathematical practice. Shapiro introduced so-called ‘finite cardinal structures’ to illustrate features of ante rem structuralism. I establish that these structures have a well-known counterpart in mathematics, but this counterpart is incompatible with ante rem structuralism. Furthermore, there is a good reason why, according to mathematical practice, these structures do not behave as conceived by Shapiro's ante rem structuralism.
  45. added 2017-01-27
    Michael D. Resnik, Mathematics as a Science of Patterns.B. Larvor - 1998 - International Studies in the Philosophy of Science 12 (3):287-289.
  46. added 2017-01-27
    Structuralism Vs.Operationalism.C. Moulines - 1998 - ProtoSociology 12:78-91.
  47. added 2017-01-27
    The Frankfurt School and Structuralism in Jerzy Kmita's Analysis.Lech Witkowski - 1996 - Poznan Studies in the Philosophy of the Sciences and the Humanities 47:45-64.
  48. added 2017-01-26
    A Structural Account of Mathematics by Charles Chihara. [REVIEW]Fraser MacBride - 2005 - Bulletin of Symbolic Logic 11 (1):79-83.
  49. added 2017-01-24
    The World as a Graph: Defending Graphical Structuralism.Nicholas Shackel - unknown
  50. added 2017-01-23
    Perspectives on Structuralism, Munich, Germany, 16–18 February 2012.Frank Zenker & Holger Andreas - 2013 - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 44 (1):227-234.
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