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Philosophy of Mathematics

Edited by Øystein Linnebo (Birkbeck College)
Assistant editor: Sam Roberts (Birkbeck College, University of Oslo)
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  1. added 2015-02-28
    Sean Walsh & Sean Ebels-Duggan (forthcoming). Relative Categoricity and Abstraction Principles. Review of Symbolic Logic.
    Many recent writers in the philosophy of mathematics have put great weight on the relative categoricity of the traditional axiomatizations of our foundational theories of arithmetic and set theory. Another great enterprise in contemporary philosophy of mathematics has been Wright's and Hale's project of founding mathematics on abstraction principles. In earlier work, it was noted that one traditional abstraction principle, namely Hume's Principle, had a certain relative categoricity property, which here we term natural relative categoricity. In this paper, we show (...)
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  2. added 2015-02-27
    Jan Sebestik (forthcoming). Bolzano's Theory of Science Disclosed in EnglishBernard Bolzano. Theory of Science. Volumes I–IV. Paul Rusnock and Rolf George, Trans. Oxford: Oxford University Press, 2014. ISBN: 978-0-19-968438-0. Pp. 2044. [REVIEW] Philosophia Mathematica:nkv001.
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  3. added 2015-02-27
    Alasdair Urquhart (forthcoming). Mathematical Depth. Philosophia Mathematica:nkv004.
    The first part of the paper is devoted to surveying the remarks that philosophers and mathematicians such as Maddy, Hardy, Gowers, and Zeilberger have made about mathematical depth. The second part is devoted to the question of whether we can make the notion precise by a more formal proof-theoretical approach. The idea of measuring depth by the depth and bushiness of the proof is considered, and compared to the related notion of the depth of a chess combination.
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  4. added 2015-02-27
    Alasdair Urquhart (forthcoming). Pavel Pudlák. Logical Foundations of Mathematics and Computational Complexity: A Gentle Introduction. Springer Monographs in Mathematics. Springer, 2013. ISBN: 978-3-319-00118-0 ; 978-3-319-00119-7 . Pp. Xiv + 695. [REVIEW] Philosophia Mathematica:nkv006.
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  5. added 2015-02-25
    Robert Knowles (forthcoming). Heavy Duty Platonism. Erkenntnis:1-16.
    Heavy duty platonism is of great dialectical importance in the philosophy of mathematics. It is the view that physical magnitudes, such as mass and temperature, are cases of physical objects being related to numbers. Many theorists have assumed HDP’s falsity in order to reach their own conclusions, but they are only justified in doing so if there are good arguments against HDP. In this paper, I present all five arguments against HDP alluded to in the literature and show that they (...)
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  6. added 2015-02-16
    Michael Detlefsen (2014). Duality, Epistemic Efficiency and Consistency. In G. Link (ed.), Formalism & Beyond. 1-24.
    Duality has often been described as a means of extending our knowledge with a minimal additional outlay of investigative resources. I consider possible arguments for this view. Major elements of this argument are out of keeping with certain widely held views concerning the nature of axiomatic theories (both in projective geometry and elsewhere). They also require a special form of consistency requirement.
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  7. added 2015-02-16
    Vadim Batitsky (1996). Theories, Theorizers and the World: A Category-Theoretic Approach. Dissertation, University of Pennsylvania
    In today's philosophy of science, scientific theories are construed as abstract mathematical objects: formal axiomatic systems or classes of set-theoretic models. By focusing exclusively on the logico-mathematical structure of theories, however, this approach ignores their essentially cognitive nature: that theories are conceptualizations of the world produced by some cognitive agents. As a result, traditional philosophical analyses of scientific theories are incapable of coherently accounting for the relevant relations between highly abstract and idealized models in science and concrete empirical phenomena in (...)
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  8. added 2015-02-15
    Rowan Grigg, It's Just About Time.
    Presented is a hypothetical model of reality that is consistent with the observational data incompletely addressed by existing models such as general relativity and quantum theory, including non-locality and the accelerating expansion of the universe. The model further suggests a theory of consciousness in which a physical mechanism accounts for interactions with remote agents that were previously categorized as 'spiritual'. I explore the wider implications of this model.
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  9. added 2015-02-14
    Barry Smith (1995). More Things in Heaven and Earth. Grazer Philosophische Studien 50:87-201.
    Philosophers in the field of analytic metaphysics have begun gradually to come to terms with the fact that there are entities in a range of categories not dreamt of in the set-theory and predicate-logic-based ontologies of their forefathers. Examples of such “entia minora” would include: boundaries, places, events, states holes, shadows, individual colour- and tone-instances (tropes), together with combinations of these and associated simple and complex universal species or essences, states of affairs, judgment-contents, and myriad abstract structures of the sorts (...)
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  10. added 2015-02-12
    Juliet Floyd (forthcoming). Depth and ClarityFelix Mühlhölzer. Braucht Die Mathematik Eine Grundlegung? Eine Kommentar des Teils III von Wittgensteins Bemerkungen Über Die Grundlagen der Mathematik [Does Mathematics Need a Foundation? A Commentary on Part III of Wittgenstein's Remarks on the Foundations of Mathematics]. Frankfurt: Vittorio Klostermann, 2010. ISBN: 978-3-465-03667-8. Pp. Xiv + 602. [REVIEW] Philosophia Mathematica:nku037.
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  11. added 2015-02-10
    Catherine Legg & James Franklin (forthcoming). Perceiving Necessity. Pacific Philosophical Quarterly.
    In many diagrams one seems to perceive necessity – one sees not only that something is so, but that it must be so. That conflicts with a certain empiricism largely taken for granted in contemporary philosophy, which believes perception is not capable of such feats. The reason for this belief is often thought well-summarized in Hume’s maxim: ‘there are no necessary connections between distinct existences’. It is also thought that even if there were such necessities, perception is too passive or (...)
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  12. added 2015-02-09
    Linda Wetzel (1992). A.D. Irvine, Ed., Physicalism In Mathematics. [REVIEW] Philosophy in Review 12:260-265.
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  13. added 2015-02-07
    David Lewis (1989). John Bigelow: "The Reality of Numbers: A Physicalist's Philosophy of Mathematics". [REVIEW] Australasian Journal of Philosophy 67:487.
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  14. added 2015-02-05
    Paolo Mancosu & Klaus JØrgensen (2006). Paolo Mancosu, Klaus Frovin JØrgensen, and Stig Andur Pedersen, Eds. Visualization, Explanation and Reasoning Stryles in Mathematics. Synthese Library, Vol. 327. Dordrecht: Springer, 2005. ISBN 1-4020-3334-6 ; 1-4020-3335-4 . Pp. X + 300. [REVIEW] Philosophia Mathematica 14 (2):265.
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  15. added 2015-02-05
    Nathan Salmon (2006). Nathan Salmon, Metaphysics, Mathematics, and Meaning: Philosophical Papers I. Oxford: Clarendon Press, 2005. ISBN-10 0-19-928176-9, ISBN-13 978-0-19-928176-3 ; ISBN-10 0-19-928471-7, ISBN-13 978-0-19-928471-9 . Pp. Xiv + 419. [REVIEW] Philosophia Mathematica 14 (2):267-268.
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  16. added 2015-02-05
    Vann Mcgee (2004). Critical Studies / Book Reviews. [REVIEW] Philosophia Mathematica 12 (3):278-284.
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  17. added 2015-02-05
    Patricia Blanchette (2003). Critical Studies / Book Reviews. [REVIEW] Philosophia Mathematica 11 (3):358-362.
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  18. added 2015-02-05
    Emily Carson (1998). Maoist Mathematics? [REVIEW] Philosophia Mathematica 6 (3):345-350.
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  19. added 2015-02-02
    Thomas Mormann (forthcoming). From Mathematics to Quantum Mechanics - On the Conceptual Unity of Cassirer's Philosophy of Science (1907 - 1937). In J. Tyler Friedman & Sebastian Luft (eds.), The Philosophy of Ernst Cassirer. A Novel Assessment. de Gruyter.
  20. added 2015-01-26
    Justin Clarke-Doane (forthcoming). Objectivity in Ethics and Mathematics. Proceedings of the Aristotelian Society.
    How do axioms, or first principles, in ethics compare to those in mathematics? I argue that while there are similarities between the cases, these are premised on an assumption which can be questioned, and which highlights the peculiarity of normative inquiry.
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  21. added 2015-01-23
    Bart Van Kerkhove (2015). Danielle Macbeth. Realizing Reason: A Narrative of Truth and Knowing. Oxford: Oxford University Press, 2014. ISBN: 978-0-19-870479-1. Pp. Xii + 494. [REVIEW] Philosophia Mathematica 23 (1):140-144.
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  22. added 2015-01-17
    Michael E. Cuffaro (2012). Kant's Views on Non-Euclidean Geometry. Proceedings of the Canadian Society for History and Philosophy of Mathematics 25:42-54.
    Kant's arguments for the synthetic a priori status of geometry are generally taken to have been refuted by the development of non-Euclidean geometries. Recently, however, some philosophers have argued that, on the contrary, the development of non-Euclidean geometry has confirmed Kant's views, for since a demonstration of the consistency of non-Euclidean geometry depends on a demonstration of its equi-consistency with Euclidean geometry, one need only show that the axioms of Euclidean geometry have 'intuitive content' in order to show that both (...)
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  23. added 2015-01-15
    Holger A. Leuz (2009). On the Foundations of Greek Arithmetic. Logical Analysis and History of Philosophy 12:13-47.
    The aim of this essay is to develop a formal reconstruction of Greek arithmetic. The reconstruction is based on textual evidence which comes mainly from Euclid, but also from passages in the texts of Plato and Aristotle. Following Paul Pritchard’s investigation into the meaning of the Greek term arithmos, the reconstruction will be mereological rather than set-theoretical. It is shown that the reconstructed system gives rise to an arithmetic comparable in logical strength to Robinson arithmetic. Our reconstructed Greek arithmetic is (...)
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  24. added 2015-01-13
    Holger A. Leuz, Note on Absolute Provability and Cantorian Comprehension.
    We will explicate Cantor’s principle of set existence using the Gödelian intensional notion of absolute provability and John Burgess’ plural logical concept of set formation. From this Cantorian Comprehension principle we will derive a conditional result about the question whether there are any absolutely unprovable mathematical truths. Finally, we will discuss the philosophical significance of the conditional result.
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  25. added 2015-01-09
    Andrew Arana (forthcoming). On the Depth of Szemerédi's Theorem. Philosophia Mathematica:nku036.
    Many mathematicians have cited depth as an important value in their research. However, there is no single widely accepted account of mathematical depth. This article is an attempt to bridge this gap. The strategy is to begin with a discussion of Szemerédi's theorem, which says that each subset of the natural numbers that is sufficiently dense contains an arithmetical progression of arbitrary length. This theorem has been judged deep by many mathematicians, and so makes for a good case on which (...)
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  26. added 2015-01-09
    Jönne Kriener (2014). Groundedness - Its Logic and Metaphysics. Dissertation, Birkbeck College, University of London
    In philosophical logic, a certain family of model constructions has received particular attention. Prominent examples are the cumulative hierarchy of well-founded sets, and Kripke's least fixed point models of grounded truth. I develop a general formal theory of groundedness and explain how the well-founded sets, Cantor's extended number-sequence and Kripke's concepts of semantic groundedness are all instances of the general concept, and how the general framework illuminates these cases. Then, I develop a new approach to a grounded theory of proper (...)
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  27. added 2015-01-07
    Gabriella Crocco & Eva-Maria Engelen (eds.) (forthcoming). Kurt Gödel: Philosopher-Scientist. Presses Universitaires de Provence.
    This volume represents the beginning of a new stage of research in interpreting Kurt Gödel’s philosophy in relation to his scientific work. It is more than a collection of essays on Gödel. It is in fact the product of a long enduring international collaboration on Kurt Gödel’s Philosophical Notebooks (Max Phil). New and significant material has been made accessible to a group of experts, on which they rely for their articles. In addition to this, Gödel’s Nachlass is presented in a (...)
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  28. added 2015-01-07
    Eva-Maria Engelen (forthcoming). What is the Link Between Aristotle’s Philosophy of Mind, the Iterative Conception of Set, Gödel’s Incompleteness Theorems and God? About the Pleasure and the Difficulties of Interpreting Kurt Gödel’s Philosophical Remarks. In Gabriella Crocco & Eva-Maria Engelen (eds.), Kurt Gödel: Philosopher-Scientist. Presses Universitaires de Provence.
    It is shown in this article in how far one has to have a clear picture of Gödel’s philosophy and scientific thinking at hand (and also the philosophical positions of other philosophers in the history of Western Philosophy) in order to interpret one single Philosophical Remark by Gödel. As a single remark by Gödel (very often) mirrors his whole philosophical thinking, Gödel’s Philosophical Remarks can be seen as a philosophical monadology. This is so for two reasons mainly: Firstly, because it (...)
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  29. added 2015-01-07
    Gabriella Crocco & Eva-Maria Engelen (forthcoming). Kurt Gödel's Philosophical Remarks (Max Phil). In Gabriella Crocco & Eva-Maria Engelen (eds.), Kurt Gödel: Philosopher-Scientist. Presses Universitaires de Provence.
    Kurt Gödel left Philosophical Remarks in his Nachlass that he himself entitled Max Phil (Maximen Philosophie). The opus originally comprised 16 notebooks but one has been lost. The content is on the whole the outline of a rational metaphysics able to relate the different domains of knowledge and of moral investigations to each other. The notebooks were at first started as an intellectual diary in which Gödel writes an account of what he does and especially about what he should do (...)
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  30. added 2014-12-27
    John Stillwell (forthcoming). What Does ‘Depth’ Mean in Mathematics? Philosophia Mathematica:nku033.
    This paper explores different interpretations of the word ‘deep’ as it is used by mathematicians, with a large number of examples illustrating various criteria for depth. Most of the examples are theorems with ‘historical depth’, in the sense that many generations of mathematicians contributed to their proof. Some also have ‘foundational depth’, in the sense that they support large mathematical theories. Finally, concepts from mathematical logic suggest that it may be possible to order certain theorems or problems according to ‘logical (...)
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  31. added 2014-12-27
    Jeremy Gray (forthcoming). Depth — A Gaussian Tradition in Mathematics. Philosophia Mathematica:nku035.
    Mathematicians use the word ‘deep’ to convey a high appreciation of a concept, theorem, or proof. This paper investigates the extent to which the term can be said to have an objective character by examining its first use in mathematics. It was a consequence of Gauss's work on number theory and the agreement among his successors that specific parts of Gauss's work were deep, on grounds that indicate that depth was a structural feature of mathematics for them. In contrast, French (...)
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  32. added 2014-12-27
    Walter Dean (2015). Arithmetical Reflection and the Provability of Soundness. Philosophia Mathematica 23 (1):31-64.
    Proof-theoretic reflection principles are schemas which attempt to express the soundness of arithmetical theories within their own language, e.g., ${\mathtt{{Prov}_{\mathsf {PA}} \rightarrow \varphi }}$ can be understood to assert that any statement provable in Peano arithmetic is true. It has been repeatedly suggested that justification for such principles follows directly from acceptance of an arithmetical theory $\mathsf {T}$ or indirectly in virtue of their derivability in certain truth-theoretic extensions thereof. This paper challenges this consensus by exploring relationships between reflection principles (...)
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  33. added 2014-12-25
    Tim Räz (2015). Say My Name: An Objection to Ante Rem Structuralism. 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.
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  34. added 2014-12-24
    Max Harris Siegel (forthcoming). Ian Hacking, Why Is There Philosophy of Mathematics at All? [REVIEW] Mind 124.
  35. added 2014-12-19
    Luca Incurvati (forthcoming). On the Concept of Finitism. Synthese:1-24.
    At the most general level, the concept of finitism is typically characterized by saying that finitistic mathematics is that part of mathematics which does not appeal to completed infinite totalities and is endowed with some epistemological property that makes it secure or privileged. This paper argues that this characterization can in fact be sharpened in various ways, giving rise to different conceptions of finitism. The paper investigates these conceptions and shows that they sanction different portions of mathematics as finitistic.
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  36. added 2014-12-18
    Steven M. Rosen (2014). How Can We Signify Being? Semiotics and Topological Self-Signification. Cosmos and History: The Journal of Natural and Social Philosophy 10 (2):250-277.
    The premise of this paper is that the goal of signifying Being central to ontological phenomenology has been tacitly subverted by the semiotic structure of conventional phenomenological writing. First it is demonstrated that the three components of the conventional sign as defined by C. S. Peirce—the sign-vehicle, object, and interpretant—bear an external relationship to each other. This is linked to the abstractness of alphabetic language, which objectifies nature and splits subject and object. It is the subject-object divide that phenomenology must (...)
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  37. added 2014-12-17
    Luca Incurvati & Julien Murzi (forthcoming). Maximally Consistent Sets of Instances of Naive Comprehension. Mind.
    Paul Horwich (1990) once suggested restricting the T-Schema to the maximal consistent set of its instances. But Vann McGee (1992) proved that there are multiple incompatible such sets, none of which, given minimal assumptions, is recursively axiomatizable. The analogous view for set theory---that Naïve Comprehension should be restricted according to consistency maxims---has recently been defended by Laurence Goldstein (2006; 2013). It can be traced back to W.V.O. Quine(1951), who held that Naïve Comprehension embodies the only really intuitive conception of set (...)
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  38. added 2014-12-17
    Shay Logan (2015). Category Theory is a Contentful Theory. Philosophia Mathematica 23 (1):110-115.
    Linnebo and Pettigrew present some objections to category theory as an autonomous foundation. They do a commendable job making clear several distinct senses of ‘autonomous’ as it occurs in the phrase ‘autonomous foundation’. Unfortunately, their paper seems to treat the ‘categorist’ perspective rather unfairly. Several infelicities of this sort were addressed by McLarty. In this note I address yet another apparent infelicity.
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  39. added 2014-12-12
    Axel Gelfert (2014). Applicability, Indispensability, and Underdetermination: Puzzling Over Wigner's 'Unreasonable Effectiveness of Mathematics'. Science and Education 23 (5):997-1009.
    In his influential 1960 paper ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’, Eugene P. Wigner raises the question of why something that was developed without concern for empirical facts—mathematics—should turn out to be so powerful in explaining facts about the natural world. Recent philosophy of science has developed ‘Wigner’s puzzle’ in two different directions: First, in relation to the supposed indispensability of mathematical facts to particular scientific explanations and, secondly, in connection with the idea that aesthetic criteria track (...)
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  40. added 2014-12-09
    S. Centrone (2014). Mirja Hartimo Ed. Phenomenology and Mathematics. Phaenomenologia; 195. Dordrecht: Springer, 2010. ISBN 978-90-481-3728-2 ; 978-90-481-3728-2 ; 978-94-007-3196-7 . Pp. Xxv + 222. [REVIEW] Philosophia Mathematica 22 (1):126-129.
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  41. added 2014-12-08
    André Bazzoni (forthcoming). Hintikka on the Foundations of Mathematics: IF Logic and Uniformity Concepts. Journal of Philosophical Logic:1-10.
    The initial goal of the present paper is to reveal a mistake committed by Hintikka in a recent paper on the foundations of mathematics. His claim that independence-friendly logic (IFL) is the real logic of mathematics is supported in that article by an argument relying on uniformity concepts taken from real analysis. I show that the central point of his argument is a simple logical mistake. Second and more generally, I conclude, based on the previous remarks and on another standard (...)
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  42. added 2014-12-08
    Eva-Maria Engelen, Kurt Gödels mathematische Anschauung und John P. Burgess’ mathematische Intuition. XXIII Deutscher Kongress für Philosophie Münster 2014, Konferenzveröffentlichung.
  43. added 2014-12-06
    Peter Simons (2015). Guillermo E. Rosado Haddock. Against the Current: Selected Philosophical Papers. Frankfurt: Ontos, 2012. ISBN: 9783868381481 . Pp. Xii + 456. [REVIEW] Philosophia Mathematica 23 (1):145-148.
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  44. added 2014-12-05
    Paolo Mancosu (2015). William Ewald and Wilfried Sieg, Eds, David Hilbert's Lectures on the Foundations of Arithmetic and Logic, 1917–1933. Heidelberg: Springer, 2013. ISBN: 978-3-540-69444-1 ; 978-3-540-20578-4 . Pp. Xxv + 1062. [REVIEW] Philosophia Mathematica 23 (1):126-135.
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