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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 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 2020-05-18
    Spacetime, Ontology, and Structural Realism.Edward Slowik - 2005 - International Studies in the Philosophy of Science 19 (2):147 – 166.
    This essay explores the possibility of constructing a structural realist interpretation of spacetime theories that can resolve the ontological debate between substantivalists and relationists. Drawing on various structuralist approaches in the philosophy of mathematics, as well as on the theoretical complexities of general relativity, our investigation will reveal that a structuralist approach can be beneficial to the spacetime theorist as a means of deflating some of the ontological disputes regarding similarly structured spacetimes.
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  2. added 2020-03-30
    How Can Mathematical Objects Be Real but Mind-Dependent?Hazhir Roshangar - manuscript
    Taking mathematics as a language based on empirical experience, I argue for an account of mathematics in which its objects are abstracta that describe and communicate the structure of reality based on some of our ancestral interactions with their environment. I argue that mathematics as a language is mostly invented, and it is mind-dependent in a specific sense. However, the bases of mathematics will characterize it as a real, non-fictional science of structures.
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  3. added 2020-01-22
    Gonit Dorshon.Avijit Lahiri - manuscript
    This article briefly reviews a few of the major points of view toward mathematics and the world of mathematical entities, and interprets the philosophy of mathematics as an interaction between these. The existence of these different points of view is indicative that mathematics, in spite of being of universal validity, can nevertheless accommodate alternatives. In particular, I review the alternative viewpoints of Platonism and Intuitionism and present the case that in spite of their great differences, they are not mutually exclusive (...)
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  4. added 2019-09-26
    Review of Michael Resnik, Mathematics as a Science of Patterns. [REVIEW]David Stump - 1998 - History and Philosophy of Logic 19 (3):176-177.
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  5. added 2019-08-31
    The Ontology of Words: A Structural Approach.Ryan M. Nefdt - 2019 - Inquiry: An Interdisciplinary Journal of Philosophy 62 (8):877-911.
    ABSTRACTWords form a fundamental basis for our understanding of linguistic practice. However, the precise ontology of words has eluded many philosophers and linguists. A persistent difficulty for most accounts of words is the type-token distinction [Bromberger, S. 1989. “Types and Tokens in Linguistics.” In Reflections on Chomsky, edited by A. George, 58–90. Basil Blackwell; Kaplan, D. 1990. “Words.” Aristotelian Society Supplementary Volume LXIV: 93–119]. In this paper, I present a novel account of words which differs from the atomistic and platonistic (...)
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  6. added 2019-08-31
    Inferentialism and Structuralism: A Tale of Two Theories.Ryan Mark Nefdt - 2018 - Logique Et Analyse 61 (244):489-512.
    This paper aims to unite two seemingly disparate themes in the philosophy of mathematics and language respectively, namely ante rem structuralism and inferentialism. My analysis begins with describing both frameworks in accordance with their genesis in the work of Hilbert. I then draw comparisons between these philosophical views in terms of their similar motivations and similar objections to the referential orthodoxy. I specifically home in on two points of comparison, namely the role of norms and the relation of ontological dependence (...)
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  7. added 2019-06-29
    Points as Higher-Order Constructs: Whitehead’s Method of Extensive Abstraction.Achille C. Varzi - forthcoming - In Stewart Shapiro & Geoffrey Hellman (eds.), The Continuous. Oxford: Oxford University Press.
    Euclid’s definition of a point as “that which has no part” has been a major source of controversy in relation to the epistemological and ontological presuppositions of classical geometry, from the medieval and modern disputes on indivisibilism to the full development of point-free geometries in the 20th century. Such theories stem from the general idea that all talk of points as putative lower-dimensional entities must and can be recovered in terms of suitable higher-order constructs involving only extended regions (or bodies). (...)
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  8. added 2019-06-07
    Thinking about Mathematics (Stewart Shapiro, Thinking about Mathematics, Nueva York, Oxford University Press, 2002, 328 p. [REVIEW]Javier Eliozondo & Axel Barceló Aspeitia - 2005 - Signos Filosóficos 7:134-137.
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  9. added 2019-06-06
    Aristotelianism in the Philosophy of Mathematics.James Franklin - 2011 - Studia Neoaristotelica 8 (1):3-15.
    Modern philosophy of mathematics has been dominated by Platonism and nominalism, to the neglect of the Aristotelian realist option. Aristotelianism holds that mathematics studies certain real properties of the world – mathematics is neither about a disembodied world of “abstract objects”, as Platonism holds, nor it is merely a language of science, as nominalism holds. Aristotle’s theory that mathematics is the “science of quantity” is a good account of at least elementary mathematics: the ratio of two heights, for example, is (...)
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  10. added 2019-06-06
    II—Bas C. Van Fraassen: Structuralism About Science: Some Common Problems.Bas C. van Fraassen - 2007 - Aristotelian Society Supplementary Volume 81 (1):45-61.
  11. added 2019-06-06
    Categories, Structures, and the Frege-Hilbert Controversy: The Status of Meta-Mathematics.Stewart Shapiro - 2005 - Philosophia Mathematica 13 (1):61-77.
    There is a parallel between the debate between Gottlob Frege and David Hilbert at the turn of the twentieth century and at least some aspects of the current controversy over whether category theory provides the proper framework for structuralism in the philosophy of mathematics. The main issue, I think, concerns the place and interpretation of meta-mathematics in an algebraic or structuralist approach to mathematics. Can meta-mathematics itself be understood in algebraic or structural terms? Or is it an exception to the (...)
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  12. added 2019-06-06
    Computational Structuralism &Dagger.Volker Halbach & Leon Horsten - 2005 - 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 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. On (...)
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  13. added 2019-06-06
    Patterns in the Philosophy of Mathematics.Rieger Adam - 2002 - Philosophical Quarterly 52 (207):247-255.
    Mathematics as a Science of Patterns . By Michael D. Resnik. (Oxford: Clarendon Press, 1997. Pp. xiii + 285. Price £35.00.) Naturalism in Mathematics . By Penelope Maddy. (Oxford: Clarendon Press, 1998. Pp. viii + 254. Price £32.50.) Realistic Rationalism . By Jerrold J. Katz. ( MIT Press, 1998. Pp. xxxiv + 226. Price £22.50.) The Principles of Mathematics Revisited . By Jaakko Hintikka. ( Cambridge UP, 1996. Pp. xii + 288. Price £40.00.).
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  14. added 2019-06-06
    Kitcher’s Circumlocutionary Structuralism.Michael Hand - 1991 - Canadian Journal of Philosophy 21 (1):81-89.
    Philip Kitcher has proposed an account of mathematical truth which he hopes avoids platonistic commitment to abstract mathematical objects. His idea is that the truth-conditions of mathematical statements consist in certain general structural features of physical reality. He codifies these structural features by reference to various operations which are performable on objects: the world is structured in such a way that these operations are possible. Which operations are performable cannot be known a priori; rather, we hypothesize, conjecture, idealize, and eventually (...)
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  15. added 2019-06-06
    Structuralism in Literature. [REVIEW]B. O. G. - 1975 - Review of Metaphysics 29 (1):148-149.
    Structuralism is a contemporary intellectual movement with both methodological and substantive implications. Nowhere has its impact been stronger than in poetics and literary criticism. Scholes book is designed to introduce English speaking audiences to structuralist developments in European literary thought. After detailing the background of structuralism in the work of Saussure and Jakobson and relating formalist and proto-structuralist modes of literary criticism to structuralist methods, the author examines specific micro and macropoetics of fiction. His object is to explain other’s theories, (...)
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  16. added 2019-06-06
    Structuralism and Ethics.William B. Hund - 1973 - Proceedings and Addresses of the American Philosophical Association 47:177.
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  17. added 2019-06-05
    Intensional Mathematics. Stewart Shapiro.Harold T. Hodes - 1989 - Philosophy of Science 56 (1):177-178.
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  18. added 2019-05-06
    Charles Chihara. A Structural Account of Mathematics. Oxford University Press, Oxford, 2004, Xiv + 380 Pp. [REVIEW]Fraser MacBride - 2005 - Bulletin of Symbolic Logic 11 (1):79-83.
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  19. added 2019-04-26
    Stairway to Heaven: The Abstract Method and Levels of Abstraction in Mathematics.Jean Pierre Marquis & Jean-Pierre Marquis - 2016 - The Mathematical Intelligencer 38 (3):41-51.
    In this paper, following the claims made by various mathematicians, I try to construct a theory of levels of abstraction. I first try to clarify the basic components of the abstract method as it developed in the first quarter of the 20th century. I then submit an explication of the notion of levels of abstraction. In the final section, I briefly explore some of main philosophical consequences of the theory.
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  20. added 2019-04-01
    Stewart Shapiro’s Philosophy of Mathematics. [REVIEW]Harold Hodes - 2002 - Philosophy and Phenomenological Research 65 (2):467-475.
    Two slogans define structuralism: contemporary mathematics studies structures; mathematical objects are places in those structures. Shapiro’s version of structuralism posits abstract objects of three sorts. A system is “a collection of objects with certain relations” between these objects. “An extended family is a system of people with blood and marital relationships.” A baseball defense, e.g., the Yankee’s defense in the first game of the 1999 World Series, is a also a system, “a collection of people with on-field spatial and ‘defensive-role’ (...)
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  21. added 2019-01-30
    An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure, by Franklin, James: Hampshire: Routledge, 2014, Pp. X + 308, £63. [REVIEW]Catherine Legg - 2015 - Australasian Journal of Philosophy 93 (4):837-837.
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  22. added 2019-01-23
    Circularities In The Contemporary Philosophical Accounts Of The Applicability Of Mathematics In The Physical Universe.Catalin Barboianu - 2015 - Revista de Filosofie 61 (5):517-542.
    Contemporary philosophical accounts of the applicability of mathematics in physical sciences and the empirical world are based on formalized relations between the mathematical structures and the physical systems they are supposed to represent within the models. Such relations were constructed both to ensure an adequate representation and to allow a justification of the validity of the mathematical models as means of scientific inference. This article puts in evidence the various circularities (logical, epistemic, and of definition) that are present in these (...)
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  23. 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. (...)
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  24. 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 (...)
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  25. 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 (...)
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  26. 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 (...)
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  27. added 2018-06-14
    Modal Structuralism and Reflection.Sam Roberts - 2019 - Review of Symbolic Logic 12 (4):823-860.
    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 (...)
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  28. 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 (...)
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  29. 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 (...)
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  30. added 2018-02-18
    Structuralism, Invariance, and Univalence.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 (...)
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  31. 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 (...)
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  32. 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.
  33. 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 (...)
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  34. added 2017-11-28
    What We Talk About When We Talk About Numbers.Richard Pettigrew - 2018 - Annals of Pure and Applied Logic 169 (12):1437-1456.
    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.
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  35. 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 (...)
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  36. 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 (...)
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  37. added 2017-10-12
    What Constitutes the Numerical Diversity of Mathematical Objects?F. MacBride - 2006 - Analysis 66 (1):63-69.
  38. 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 (...)
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  39. 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 (...)
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  40. added 2017-02-16
    Wittgenstein on Dedekind’s Cut.M. S. Lourenço - 1998 - Disputatio 1 (4):653-66.
  41. added 2017-02-16
    Theory of Mathematical Reality, A.Everett H. Larguier - 1938 - Modern Schoolman 16:88.
  42. added 2017-02-15
    Texts And The Objects Of Mathematics.Paul Ernest - 1997 - Philosophy of Mathematics Education Journal 10.
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  43. added 2017-02-15
    The Peircean Interpretation Of Mathematics.Christopher Ormell - 1995 - Philosophy of Mathematics Education Journal 8.
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  44. 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.
  45. added 2017-02-14
    STRUCTURALISM―Key to Reality and Meaning in Science.Charles W. Lucas Jr - unknown
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  46. 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.
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  47. added 2017-02-14
    Structuralism and Historical Knowledge.F. Botturi - 1983 - Rivista di Filosofia Neo-Scolastica 75 (4):563-581.
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  48. added 2017-02-14
    Michael Resnik, Frege and the Philosophy of Mathematics Reviewed By. [REVIEW]Alan McMichael - 1982 - Philosophy in Review 2 (6):291-294.
  49. added 2017-02-14
    Is Cusanus the Father of Structuralism.Jasper Blystone - 1972 - Philosophy Today 16 (4):296-305.
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  50. 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 (...)
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1 — 50 / 310