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  1. added 2019-02-01
    Do Mathematical Explanations Have Instrumental Value?Rebecca Morris - 2019 - Synthese:1-20.
    Scientific explanations are widely recognized to have instrumental value by helping scientists make predictions and control their environment. In this paper I raise, and provide a first analysis of, the question whether explanatory proofs in mathematics have analogous instrumental value. I first identify an important goal in mathematical practice: reusing resources from existing proofs to solve new problems. I then consider the more specific question: do explanatory proofs have instrumental value by promoting reuse of the resources they contain? In general, (...)
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  2. added 2018-12-17
    Independence of the Grossone-Based Infinity Methodology From Non-Standard Analysis and Comments Upon Logical Fallacies in Some Texts Asserting the Opposite.Yaroslav D. Sergeyev - 2019 - Foundations of Science 24 (1):153-170.
    This paper considers non-standard analysis and a recently introduced computational methodology based on the notion of ①. The latter approach was developed with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework and in all the situations requiring these notions. Non-standard analysis is a classical purely symbolic technique that works with ultrafilters, external and internal sets, standard and non-standard numbers, etc. In its turn, the ①-based methodology does not use any of these (...)
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  3. added 2018-12-17
    The Difficulty of Prime Factorization is a Consequence of the Positional Numeral System.Yaroslav Sergeyev - 2016 - International Journal of Unconventional Computing 12 (5-6):453–463.
    The importance of the prime factorization problem is very well known (e.g., many security protocols are based on the impossibility of a fast factorization of integers on traditional computers). It is necessary from a number k to establish two primes a and b giving k = a · b. Usually, k is written in a positional numeral system. However, there exists a variety of numeral systems that can be used to represent numbers. Is it true that the prime factorization is (...)
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  4. added 2018-12-17
    The Olympic Medals Ranks, Lexicographic Ordering and Numerical Infinities.Yaroslav Sergeyev - 2015 - The Mathematical Intelligencer 37 (2):4-8.
    Several ways used to rank countries with respect to medals won during Olympic Games are discussed. In particular, it is shown that the unofficial rank used by the Olympic Committee is the only rank that does not allow one to use a numerical counter for ranking – this rank uses the lexicographic ordering to rank countries: one gold medal is more precious than any number of silver medals and one silver medal is more precious than any number of bronze medals. (...)
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  5. added 2018-12-17
    Higher Order Numerical Differentiation on the Infinity Computer.Yaroslav Sergeyev - 2011 - Optimization Letters 5 (4):575-585.
    There exist many applications where it is necessary to approximate numerically derivatives of a function which is given by a computer procedure. In particular, all the fields of optimization have a special interest in such a kind of information. In this paper, a new way to do this is presented for a new kind of a computer - the Infinity Computer - able to work numerically with finite, infinite, and infinitesimal number. It is proved that the Infinity Computer is able (...)
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  6. added 2018-11-24
    Viewing-as Explanations and Ontic Dependence.William D’Alessandro - forthcoming - Philosophical Studies:1-24.
    According to a widespread view in metaphysics and philosophy of science (the “Dependence Thesis”), all explanations involve relations of ontic dependence between the items appearing in the explanandum and the items appearing in the explanans. I argue that a family of mathematical cases, which I call “viewing-as explanations”, are incompatible with the Dependence Thesis. These cases, I claim, feature genuine explanations that aren’t supported by ontic dependence relations. Hence the thesis isn’t true in general. The first part of the paper (...)
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  7. added 2018-10-19
    The "Artificial Mathematician" Objection: Exploring the (Im)Possibility of Automating Mathematical Understanding.Sven Delarivière & Bart Van Kerkhove - 2017 - In B. Sriraman (ed.), Humanizing Mathematics and its Philosophy. Cham: Birkhäuser. pp. 173-198.
    Reuben Hersh confided to us that, about forty years ago, the late Paul Cohen predicted to him that at some unspecified point in the future, mathematicians would be replaced by computers. Rather than focus on computers replacing mathematicians, however, our aim is to consider the (im)possibility of human mathematicians being joined by “artificial mathematicians” in the proving practice—not just as a method of inquiry but as a fellow inquirer.
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  8. added 2018-09-07
    O nouă filosofie a matematicii?Gabriel Târziu - 2012 - Symposion – A Journal of Humanities 10 (2):361-377.
    O tendinţă relativ nouă în filosofia contemporană a matematicii este reprezentată de nemulţumirea manifestată de un număr din ce în ce mai mare de filosofi faţă de viziunea tradiţională asupra matematicii ca având un statut special ce poate fi surprins doar cu ajutorul unei epistemologii speciale. Această nemulţumire i-a determinat pe mulţi să propună o nouă perspectivă asupra matematicii – una care ia în serios aspecte până acum neglijate de filosofia matematicii, precum latura sociologică, istorică şi empirică a cercetării matematice (...)
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  9. added 2018-02-16
    On Formally Measuring and Eliminating Extraneous Notions in Proofs.Andrew Arana - 2009 - Philosophia Mathematica 17 (2):189-207.
    Many mathematicians and philosophers of mathematics believe some proofs contain elements extraneous to what is being proved. In this paper I discuss extraneousness generally, and then consider a specific proposal for measuring extraneousness syntactically. This specific proposal uses Gentzen's cut-elimination theorem. I argue that the proposal fails, and that we should be skeptical about the usefulness of syntactic extraneousness measures.
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  10. added 2017-12-25
    The Necessity of Mathematics.Juhani Yli‐Vakkuri & John Hawthorne - 2018 - Noûs 52.
    Some have argued for a division of epistemic labor in which mathematicians supply truths and philosophers supply their necessity. We argue that this is wrong: mathematics is committed to its own necessity. Counterfactuals play a starring role.
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  11. added 2017-08-23
    Envisioning Transformations – The Practice of Topology.Silvia De Toffoli & Valeria Giardino - 2016 - In Brendan Larvor (ed.), Mathematical Cultures. Zurich, Switzerland: Birkhäuser. pp. 25-50.
    The objective of this article is twofold. First, a methodological issue is addressed. It is pointed out that even if philosophers of mathematics have been recently more and more concerned with the practice of mathematics, there is still a need for a sharp definition of what the targets of a philosophy of mathematical practice should be. Three possible objects of inquiry are put forward: (1) the collective dimension of the practice of mathematics; (2) the cognitives capacities requested to the practitioners; (...)
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  12. added 2017-08-23
    An Inquiry Into the Practice of Proving in Low-Dimensional Topology.Silvia De Toffoli & Valeria Giardino - 2015 - In Gabriele Lolli, Giorgio Venturi & Marco Panza (eds.), From Logic to Practice. Zurich, Switzerland: Springer International Publishing. pp. 315-116.
    The aim of this article is to investigate specific aspects connected with visualization in the practice of a mathematical subfield: low-dimensional topology. Through a case study, it will be established that visualization can play an epistemic role. The background assumption is that the consideration of the actual practice of mathematics is relevant to address epistemological issues. It will be shown that in low-dimensional topology, justifications can be based on sequences of pictures. Three theses will be defended. First, the representations used (...)
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  13. added 2017-06-14
    The Great Gibberish - Mathematics in Western Popular Culture.Markus Pantsar - 2016 - In Brendan Larvor (ed.), Mathematical Cultures. Springer International Publishing. pp. 409-437.
    In this paper, I study how mathematicians are presented in western popular culture. I identify five stereotypes that I test on the best-known modern movies and television shows containing a significant amount of mathematics or important mathematician characters: (1) Mathematics is highly valued as an intellectual pursuit. (2) Little attention is given to the mathematical content. (3) Mathematical practice is portrayed in an unrealistic way. (4) Mathematicians are asocial and unable to enjoy normal life. (5) Higher mathematics is ...
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  14. added 2017-05-22
    Proof Phenomenon as a Function of the Phenomenology of Proving.Inês Hipólito - 2015 - Progress in Biophysics and Molecular Biology 119:360-367.
    Kurt Gödel wrote (1964, p. 272), after he had read Husserl, that the notion of objectivity raises a question: “the question of the objective existence of the objects of mathematical intuition (which, incidentally, is an exact replica of the question of the objective existence of the outer world)”. This “exact replica” brings to mind the close analogy Husserl saw between our intuition of essences in Wesensschau and of physical objects in perception. What is it like to experience a mathematical proving (...)
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  15. added 2017-05-21
    Arithmetic, Set Theory, Reduction and Explanation.William D’Alessandro - 2018 - Synthese 195 (11):5059-5089.
    Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important particular case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In defense (...)
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  16. added 2017-04-06
    Finite Methods in Mathematical Practice.Peter Schuster & Laura Crosilla - 2014 - In Godehard Link (ed.), Formalism and Beyond: On the Nature of Mathematical Discourse. De Gruyter. pp. 351-410.
    In the present contribution we look at the legacy of Hilbert's programme in some recent developments in mathematics. Hilbert's ideas have seen new life in generalised and relativised forms by the hands of proof theorists and have been a source of motivation for the so--called reverse mathematics programme initiated by H. Friedman and S. Simpson. More recently Hilbert's programme has inspired T. Coquand and H. Lombardi to undertake a new approach to constructive algebra in which strong emphasis is laid on (...)
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  17. added 2017-02-17
    The Interplay Between Mathematical Practices and Results.Mélissa Arneton, Amirouche Moktefi & Catherine Allamel-Raffin - 2014 - In Léna Soler, Sjoerd Zwart, Michael Lynch & Vincent Israel-Jost (eds.), Science after the Practice Turn in the Philosophy, History, and Social Studies of Science. New York - London: Routledge. pp. 269-276.
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  18. added 2017-02-15
    Review of Gabriele Lolli, Numeri. La creazione continua della matematica. [REVIEW]Longa Gianluca - 2016 - Lo Sguardo. Rivista di Filosofia 21 (II):377-380.
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  19. added 2017-02-15
    Mathematical Proofs in Practice: Revisiting the Reliability of Published Mathematical Proofs.Joachim Frans & Laszlo Kosolosky - 2014 - Theoria: Revista de Teoría, Historia y Fundamentos de la Ciencia 29 (3):345-360.
    Mathematics seems to have a special status when compared to other areas of human knowledge. This special status is linked with the role of proof. Mathematicians often believe that this type of argumentation leaves no room for errors and unclarity. Philosophers of mathematics have differentiated between absolutist and fallibilist views on mathematical knowledge, and argued that these views are related to whether one looks at mathematics-in-the-making or finished mathematics. In this paper we take a closer look at mathematical practice, more (...)
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  20. added 2017-02-15
    ‘Juglers Or Schollers?’: Negotiating the Role of a Mathematical Practitioner.Katherine Hill - 1998 - British Journal for the History of Science 31 (3):253-274.
    Until the first quarter of the seventeenth century there was a great deal of agreement about the nature of mathematical practice. Mathematicians, as well as their patrons and clients, viewed all possible aspects of their work, both theoretical and practical, as being included within their discipline. Although the mathematical sciences were a fairly recent foreign import to England, which can barely be traced back beyond the mid-sixteenth century, by the beginning of the seventeenth century there was a large and growing (...)
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  21. added 2017-02-15
    Mathematical Activity and Rhetoric: A Semiotic Analysis of an Episode of Mathematical Activity.Paul Ernest - 1997 - Philosophy of Mathematics Education Journal 10.
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  22. added 2017-02-14
    PhiMSAMP. Philosophy of Mathematics: Sociological Aspects and Mathematical Practice.Benedikt L.öwe & Thomas Müller (eds.) - 2010 - College Publications.
  23. added 2017-02-13
    Mathematical Jujitsu: Some Informal Thoughts About G�Del and Physics.John D. Barrow - 2000 - Complexity 5 (5):28-34.
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  24. added 2017-02-13
    Reasoning with Images in Mathematical Activity.Grayson H. Wheatley - 1997 - In Lyn D. English (ed.), Mathematical Reasoning: Analogies, Metaphors, and Images. L. Erlbaum Associates. pp. 281--297.
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  25. added 2017-02-13
    Mathematical Reasoning.C. Susan Robinson & John R. Hayes - 1978 - In Russell Revlin & Richard E. Mayer (eds.), Human Reasoning. Distributed Solely by Halsted Press. pp. 195.
  26. added 2017-02-09
    Ambiguities of Fundamental Concepts in Mathematical Analysis During the Mid-Nineteenth Century.Kajsa Bråting - 2012 - Foundations of Science 17 (4):301-320.
    In this paper we consider the major development of mathematical analysis during the mid-nineteenth century. On the basis of Jahnke’s (Hist Math 20(3):265–284, 1993 ) distinction between considering mathematics as an empirical science based on time and space and considering mathematics as a purely conceptual science we discuss the Swedish nineteenth century mathematician E.G. Björling’s general view of real- and complexvalued functions. We argue that Björling had a tendency to sometimes consider mathematical objects in a naturalistic way. One example is (...)
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  27. added 2017-02-07
    Mathematical Activity.M. Giaquinto - unknown
    Book description: This book contains groundbreaking contributions to the philosophical analysis of mathematical practice. Several philosophers of mathematics have recently called for an approach to philosophy of mathematics that pays more attention to mathematical practice. Questions concerning concept-formation, understanding, heuristics, changes in style of reasoning, the role of analogies and diagrams, etc. have become the subject of intense interest. The historians and philosophers in this book agree that there is more to understanding mathematics than a study of its logical structure. (...)
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  28. added 2017-01-29
    Commentary On: Begoña Carrascal's "The Practice of Arguing and the Arguments: Examples From Mathematics".Andrew Aberdein - 2013 - In Dima Mohammed & Marcin Lewinski (eds.), Virtues of argumentation: Proceedings of the 10th International Conference of the Ontario Society for the Study of Argumentation (OSSA), May 22–25, 2013. OSSA.
  29. added 2017-01-27
    Structure and Closure of School Mathematical Practice - The Experiences of Kristina.Ann-Sofi Röj-Lindberg - 2011 - Philosophy of Mathematics Education Journal 26.
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  30. added 2017-01-27
    Mancosu, P.-Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century.S. Gaukroger - 1997 - Philosophical Books 38:93-94.
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  31. added 2017-01-26
    For a Philosophy of Mathematical Practice.Gianluigi Oliveri - 2010 - In Bart van Kerkhove, Jean Paul van Bendegem & Jonas de Vuyst (eds.), Philosophical Perspectives on Mathematical Practice. College Publications.
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  32. added 2017-01-26
    What Did the Abbacus Teachers Aim at When They (Sometimes) Ended Up Doing Mathematics? An Investigation of the Incentives and Norms of a Distinct Mathematical Practice.Jens Høyrup - 2009 - In Bart van Kerkhove (ed.), New Perspectives on Mathematical Practices: Essays in Philosophy and History of Mathematics: Brussels, Belgium, 26-28 March 2007. World Scientific. pp. 47--75.
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  33. added 2017-01-25
    Proceedings of the Symposium on Mathematical Practice and Cognition Ii: A Symposium at the Aisb/Iacap World Congress 2012.Alison Pease & Brendan Larvor (eds.) - 2012 - Society for the Study of Artificial Intelligence and the Simulation of Behaviour.
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  34. added 2017-01-24
    What Mathematical Cognition Could Tell Us About the Actual World.Sorin Costreie - 2012 - In Elsevier (ed.), Procedia Social and Behavioral Science 33. pp. 138-142.
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  35. added 2017-01-23
    Early Modern Mathematical Practice in the Round.Richard J. Oosterhoff - 2012 - Studies in History and Philosophy of Science Part A 43 (1):224-227.
  36. added 2017-01-22
    Proceedings of AISB 2010 Symposium on Mathematical Practice and Cognition.Alison Pease, Markus Guhe & Alan Smaill (eds.) - 2010 - AISB.
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  37. added 2017-01-21
    Philosophical Theory and Mathematical Practice in the Seventeenth Century.Douglas M. Jesseph - 1989 - Studies in History and Philosophy of Science Part A 20 (2):215-244.
    It is argued that, contrary to the standard accounts of the development of infinitesimal mathematics, the leading mathematicians of the seventeenth century were deeply concerned with the rigor of their methods. examples are taken from the work of cavalieri and leibniz, with further material drawn from guldin, barrow, and wallis.
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  38. added 2017-01-20
    The Philosophy of Mathematical Practice.Bart Van Kerkhove - 2010 - International Studies in the Philosophy of Science 24 (1):118 – 122.
    This title offers philosophical analyses of important characteristics of contemporary mathematics and of many aspects of mathematical activity which escape purely formal logical treatment.
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  39. added 2017-01-20
    The Philosophy of Mathematical Practice.James Robert Brown - 2010 - International Journal of Philosophical Studies 18 (1):111 – 115.
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  40. added 2017-01-20
    Towards a Theory of Mathematical Argument.Ian J. Dove - 2009 - Foundations of Science 14 (1-2):136-152.
    In this paper, I assume, perhaps controversially, that translation into a language of formal logic is not the method by which mathematicians assess mathematical reasoning. Instead, I argue that the actual practice of analyzing, evaluating and critiquing mathematical reasoning resembles, and perhaps equates with, the practice of informal logic or argumentation theory. It doesn’t matter whether the reasoning is a full-fledged mathematical proof or merely some non-deductive mathematical justification: in either case, the methodology of assessment overlaps to a large extent (...)
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  41. added 2017-01-20
    Mathematical Arguments in Context.Jean Paul Van Bendegem & Bart Van Kerkhove - 2009 - Foundations of Science 14 (1-2):45-57.
    Except in very poor mathematical contexts, mathematical arguments do not stand in isolation of other mathematical arguments. Rather, they form trains of formal and informal arguments, adding up to interconnected theorems, theories and eventually entire fields. This paper critically comments on some common views on the relation between formal and informal mathematical arguments, most particularly applications of Toulmin’s argumentation model, and launches a number of alternative ideas of presentation inviting the contextualization of pieces of mathematical reasoning within encompassing bodies of (...)
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  42. added 2017-01-16
    Mathematics in Kant's Critical Philosophy: Reflections on Mathematical Practice.Lisa Shabel - 2002 - Routledge.
    This book provides a reading of Kant's theory of the construction of mathematical concepts through a fully contextualised analysis. In this work the author argues that it is only through an understanding of the relevant eighteenth century mathematics textbooks, and the related mathematical practice, that the material and context necessary for a successful interpretation of Kant's philosophy can be provided.
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  43. added 2017-01-16
    Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century. Paolo Mancosu.Antoni Malet - 1997 - Isis 88 (1):140-141.
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  44. added 2017-01-15
    Proof and Understanding in Mathematical Practice.Danielle Macbeth - 2012 - Philosophia Scientae 16:29-54.
  45. added 2017-01-14
    Mathematical Practice and Human Cognition.Bernd Buldt - unknown
    Frank Quinn of Jaffe-Quinn fame worked out the basics of his own account of how mathematical practice should be described and analyzed, partly by historical comparisons with 19th century mathematics, partly by an analysis of contemporary mathematics and its pedagogy. Despite his claim that for this task, "professional philosophers seem as irrelevant as Aristotle is to modern physics," this philosophy talk will provide a critical summary of his main observations and arguments. The goal is to inject some of Quinns remarks (...)
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  46. added 2017-01-12
    The Changing Practices of Proof in Mathematics.Andrew Arana - 2017 - Metascience 26 (1):131-135.
    Review of Dowek, Gilles, Computation, Proof, Machine, Cambridge University Press, Cambridge, 2015. Translation of Les Métamorphoses du calcul, Le Pommier, Paris, 2007. Translation from the French by Pierre Guillot and Marion Roman.
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  47. added 2017-01-12
    On the Depth of Szemerédi's Theorem.Andrew Arana - 2015 - Philosophia Mathematica 23 (2):163-176.
    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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  48. added 2017-01-12
    Purity of Methods.Michael Detlefsen & Andrew Arana - 2011 - Philosophers' Imprint 11.
    Throughout history, mathematicians have expressed preference for solutions to problems that avoid introducing concepts that are in one sense or another “foreign” or “alien” to the problem under investigation. This preference for “purity” (which German writers commonly referred to as “methoden Reinheit”) has taken various forms. It has also been persistent. This notwithstanding, it has not been analyzed at even a basic philosophical level. In this paper we give a basic analysis of one conception of purity—what we call topical purity—and (...)
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  49. added 2017-01-12
    L'infinité des nombres premiers : une étude de cas de la pureté des méthodes.Andrew Arana - 2011 - Les Etudes Philosophiques 97 (2):193.
    Une preuve est pure si, en gros, elle ne réfère dans son développement qu’à ce qui est « proche » de, ou « intrinsèque » à l’énoncé à prouver. L’infinité des nombres premiers, un théorème classique de l’arithmétique, est un cas d’étude particulièrement riche pour les recherches philosophiques sur la pureté. Deux preuves différentes de ce résultat sont ici considérées, à savoir la preuve euclidienne classique et une preuve « topologique » plus récente proposée par Furstenberg. D’un point de vue (...)
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  50. added 2017-01-12
    Proof Theory in Philosophy of Mathematics.Andrew Arana - 2010 - Philosophy Compass 5 (4):336-347.
    A variety of projects in proof theory of relevance to the philosophy of mathematics are surveyed, including Gödel's incompleteness theorems, conservation results, independence results, ordinal analysis, predicativity, reverse mathematics, speed-up results, and provability logics.
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