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  1. Reduction – Abstraction – Analysis.H. Leitgeb A. Hieke (ed.) - 2009 - Ludwig Wittgenstein Society.
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  2. A Piagetian Perspective on Mathematical Construction.Michael A. Arbib - 1990 - Synthese 84 (1):43 - 58.
    In this paper, we offer a Piagetian perspective on the construction of the logico-mathematical schemas which embody our knowledge of logic and mathematics. Logico-mathematical entities are tied to the subject's activities, yet are so constructed by reflective abstraction that they result from sensorimotor experience only via the construction of intermediate schemas of increasing abstraction. The axiom set does not exhaust the cognitive structure (schema network) which the mathematician thus acquires. We thus view truth not as something to be defined within (...)
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  3. Jody Azzouni. Tracking Reason: Proof, Consequence and Truth.Conrad Asmus - 2009 - Philosophia Mathematica 17 (3):369-377.
    (No abstract is available for this citation).
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  4. The Turning Point and the Revolution: Philosophy of Mathematics in Logical Empiricism From Tractatus on Logical Syllogism.Steve Awodey & A. W. Carus - unknown
    Steve Awodey and A. W. Carus. The Turning Point and the Revolution: Philosophy of Mathematics in Logical Empiricism from Tractatus on Logical Syllogism.
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  5. Le Concept de Mod Ele, Introduction a Une Épistémologie Matérialiste des Mathématiques.Alain Badiou - 1969 - F. Maspero.
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  6. Le Concept de Modèle Introduction À Une Épistémologie Matérialiste des Mathématiques. --.Alain Badiou - 1969 - F. Maspero.
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  7. Introduction.Carla Bagnoli - 2004 - Croatian Journal of Philosophy 4 (3):311-316.
    This volume collects articles in realism, anti-realism, and constructivism.
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  8. Is There a Problem of Induction for Mathematics?Alan Baker - 2007 - In M. Potter (ed.), Mathematical Knowledge. Oxford University Press. pp. 57-71.
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  9. Structures, Fictions, and the Explanatory Epistemology of Mathematics in Science.Mark Balaguer, Elaine Landry, Sorin Bangu & Christopher Pincock - 2013 - Metascience 22 (2):247-273.
  10. On the Coevolution of Basic Arithmetic Language and Knowledge.Jeffrey A. Barrett - 2013 - Erkenntnis 78 (5):1025-1036.
    Skyrms-Lewis sender-receiver games with invention allow one to model how a simple mathematical language might be invented and become meaningful as its use coevolves with the basic arithmetic competence of primitive mathematical inquirers. Such models provide sufficient conditions for the invention and evolution of a very basic sort of arithmetic language and practice, and, in doing so, provide insight into the nature of a correspondingly basic sort of mathematical knowledge in an evolutionary context. Given traditional philosophical reflections concerning the nature (...)
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  11. Hintikka on the Foundations of Mathematics: IF Logic and Uniformity Concepts.André Bazzoni - 2015 - Journal of Philosophical Logic 44 (5):507-516.
    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 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 fact (...)
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  12. Review of M. Wille, Mathematics and the Synthetic A Priori: Epistemological Investigations Into the Status of Mathematical Axioms[REVIEW]Claus Beisbart - 2008 - Philosophia Mathematica 16 (1):130-132.
    Kant famously thought that mathematics contains synthetic a priori truths. In his book, Wille defends a version of the Kantian thesis on not-so-Kantian grounds. Wille calls his account neo-Kantian, because it makes sense of Kantian tenets by using a methodology that takes the linguistic and pragmatic turns seriously.Wille's work forms part of a larger project in which the statuses of mathematics and proof theory are investigated. The official purpose of the present book is to answer the question: what is mathematics. (...)
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  13. The Epistemology of Abstract Objects: Access and Inference.David Bell & W. D. Hart - 1979 - Proceedings of the Aristotelian Society 53:153-165.
  14. The Mathematical Psychology of Gratry and Boole, Translated From the Language of the Higher Calculus Into That of Elementary Geometry.Mary Everest Boole - 1897
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  15. Visualizations in Mathematics.Kajsa Bråting & Johanna Pejlare - 2008 - Erkenntnis 68 (3):345 - 358.
    In this paper we discuss visualizations in mathematics from a historical and didactical perspective. We consider historical debates from the 17th and 19th centuries regarding the role of intuition and visualizations in mathematics. We also consider the problem of what a visualization in mathematical learning can achieve. In an empirical study we investigate what mathematical conclusions university students made on the basis of a visualization. We emphasize that a visualization in mathematics should always be considered in its proper context.
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  16. Truth and Proof.Otávio Bueno - 2008 - Manuscrito 31 (1):419-440.
    Current versions of nominalism in the philosophy of mathematics face a significant problem to understand mathematical knowledge. They are unable to characterize mathematical knowledge as knowledge of the objects mathematical theories are taken to be about. Oswaldo Chateaubriand’s insightful reformulation of Platonism (Chateaubriand 2005) avoids this problem by advancing a broader conception of knowledge as justified truth beyond a reasonable doubt, and by introducing a suitable characterization of logical form in which the relevant mathematical facts play an important role in (...)
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  17. Accessibility of Reformulated Mathematical Content.Stefan Buijsman - 2017 - Synthese 194 (6).
    I challenge a claim that seems to be made when nominalists offer reformulations of the content of mathematical beliefs, namely that these reformulations are accessible to everyone. By doing so, I argue that these theories cannot account for the mathematical knowledge that ordinary people have. In the first part of the paper I look at reformulations that employ the concept of proof, such as those of Mary Leng and Ottavio Bueno. I argue that ordinary people don’t have many beliefs about (...)
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  18. Referring to Mathematical Objects Via Definite Descriptions.Stefan Buijsman - 2017 - Philosophia Mathematica 25 (1):128-138.
    Linsky and Zalta try to explain how we can refer to mathematical objects by saying that this happens through definite descriptions which may appeal to mathematical theories. I present two issues for their account. First, there is a problem of finding appropriate pre-conditions to reference, which are currently difficult to satisfy. Second, there is a problem of ensuring the stability of the resulting reference. Slight changes in the properties ascribed to a mathematical object can result in a shift of reference (...)
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  19. Philosophy of Mathematics for the Masses : Extending the Scope of the Philosophy of Mathematics.Stefan Buijsman - 2016 - Dissertation, Stockholm University
    One of the important discussions in the philosophy of mathematics, is that centered on Benacerraf’s Dilemma. Benacerraf’s dilemma challenges theorists to provide an epistemology and semantics for mathematics, based on their favourite ontology. This challenge is the point on which all philosophies of mathematics are judged, and clarifying how we might acquire mathematical knowledge is one of the main occupations of philosophers of mathematics. In this thesis I argue that this discussion has overlooked an important part of mathematics, namely mathematics (...)
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  20. Life and Works of Giovanni Vailati.Paola Cantù & De Zan Mauro - 2009 - In Arrighi Claudia, Cantù Paola, De Zan Mauro & Suppes Patrick (eds.), Life and Works of Giovanni Vailati. CSLI Publications.
    The paper introduces Vailati’s life and works, investigating Vailati’s education, the relation to Peano and his school, and the interest for pragmatism and modernism. A detailed analysis of Vailati’s scientific and didactic activities, shows that he held, like Peano, a a strong interest for the history of science and a pluralist, anti-dogmatic and anti-foundationalist conception of definitions in mathematics, logic and philosophy of language. Vailati’s understanding of mathematical logic as a form of pragmatism is not a faithful interpretation of Peano’s (...)
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  21. The Growth of Mathematical Knowledge: An Open World View.Carlo Cellucci - 2000 - In Emily Grosholz & Herbert Breger (eds.), The Growth of Mathematical Knowledge, pp. 153-176. Kluwer Academic Publishers. pp. 153--176.
    In his book The Value of Science Poincaré criticizes a certain view on the growth of mathematical knowledge: “The advance of science is not comparable to the changes of a city, where old edifices are pitilessly torn down to give place to new ones, but to the continuous evolution of zoological types which develop ceaselessly and end by becoming unrecognizable to the common sight, but where an expert eye finds always traces of the prior work of the centuries past” (Poincaré (...)
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  22. Ethics and Mathematics: The Reliability Challenge. Clarke-Doane - manuscript
    It is sometimes alleged that “the reliability challenge” to moral realism is equally compelling against mathematical realism. This allegation is of interest. The reliability challenge to moral realism is increasingly taken to be the most serious challenge to moral realism. However, the specific considerations that are said to motivate it – such as considerations of rational dubitability and evolutionary influence – are widely held not to motivate an analogous challenge to mathematical realism. If it turned out that, in fact, they (...)
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  23. Debunking Arguments: Mathematics, Logic, and Modal Security.Justin Clarke-Doane - forthcoming - In Robert Richards and Michael Ruse (ed.), Cambridge Handbook of Evolutionary Ethics. Cambridge University Press.
    I discuss the structure of genealogical debunking arguments. I argue that they undermine our mathematical beliefs if they undermine our moral beliefs. The contrary appearance stems from a confusion of arithmetic truths with (first-order) logical truths, or from a confusion of reliability with justification. I conclude with a discussion of the cogency of debunking arguments, in light of the above. Their cogency depends on whether information can undermine all of our beliefs of a kind, F, without giving us direct reason (...)
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  24. Objectivity and Reliability.Justin Clarke-Doane - 2017 - Canadian Journal of Philosophy 47 (6):841-855.
    Scanlon’s Being Realistic about Reasons (BRR) is a beautiful book – sleek, sophisticated, and programmatic. One of its key aims is to demystify knowledge of normative and mathematical truths. In this article, I develop an epistemological problem that Scanlon fails to explicitly address. I argue that his “metaphysical pluralism” can be understood as a response to that problem. However, it resolves the problem only if it undercuts the objectivity of normative and mathematical inquiry.
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  25. Towards an Institutional Account of the Objectivity, Necessity, and Atemporality of Mathematics.Julian C. Cole - 2013 - Philosophia Mathematica 21 (1):9-36.
    I contend that mathematical domains are freestanding institutional entities that, at least typically, are introduced to serve representational functions. In this paper, I outline an account of institutional reality and a supporting metaontological perspective that clarify the content of this thesis. I also argue that a philosophy of mathematics that has this thesis as its central tenet can account for the objectivity, necessity, and atemporality of mathematics.
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  26. The Transformational Epistemology of Radical Constructivism: A Tribute to Ernst von Glasersfeld.J. Confrey - 2011 - Constructivist Foundations 6 (2):177-182.
    Problem: What is it that Ernst von Glasersfeld brought to mathematics education with radical constructivism? Method: Key ideas in the author’s early thinking are related to ideas that are central in constructivism, with the aim of showing their importance in math education. Results: The author’s initial thinking about constructivism began with Toulmin’s view of thinking as evolving. Ernst showed how Piaget’s genetic epistemology implied an epistemology that was not about ontology. Continuing with an analysis of the way radical and trivial (...)
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  27. Nominalism, Trivialist Platonism and Benacerraf's Dilemma.Chris Daly & David Liggins - 2014 - Analysis 74 (2):224-231.
    In his stimulating new book The Construction of Logical Space , Agustín Rayo offers a new account of mathematics, which he calls ‘Trivialist Platonism’. In this article, we take issue with Rayo’s case for Trivialist Platonism and his claim that the view overcomes Benacerraf’s dilemma. Our conclusion is that Rayo has not shown that Trivialist Platonism has any advantage over nominalism.
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  28. Autonomy Platonism and the Indispensability Argument. By Russell Marcus. Lanham, Md.: Lexington Books, 2015. Pp. Xii + 247. [REVIEW]Nicholas Danne - 2017 - Metaphilosophy 48 (4):591-594.
  29. Empiricism in Arithmetic and Analysis.E. B. Davies - 2003 - Philosophia Mathematica 11 (1):53-66.
    We discuss the philosophical status of the statement that (9n – 1) is divisible by 8 for various sizes of the number n. We argue that even this simple problem reveals deep tensions between truth and verification. Using Gillies's empiricist classification of theories into levels, we propose that statements in arithmetic should be classified into three different levels depending on the sizes of the numbers involved. We conclude by discussing the relationship between the real number system and the physical continuum.
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  30. Mathematical Symbols as Epistemic Actions.De Cruz Helen & De Smedt Johan - 2013 - Synthese 190 (1):3-19.
    Recent experimental evidence from developmental psychology and cognitive neuroscience indicates that humans are equipped with unlearned elementary mathematical skills. However, formal mathematics has properties that cannot be reduced to these elementary cognitive capacities. The question then arises how human beings cognitively deal with more advanced mathematical ideas. This paper draws on the extended mind thesis to suggest that mathematical symbols enable us to delegate some mathematical operations to the external environment. In this view, mathematical symbols are not only used to (...)
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  31. ‘Chasing’ The Diagram - The Use of Visualizations in Algebraic Reasoning.Silvia De Toffoli - 2017 - Review of Symbolic Logic 10 (1):158-186.
    The aim of this article is to investigate the roles of commutative diagrams (CDs) in a specific mathematical domain, and to unveil the reasons underlying their effectiveness as a mathematical notation; this will be done through a case study. It will be shown that CDs do not depict spatial relations, but represent mathematical structures. CDs will be interpreted as a hybrid notation that goes beyond the traditional bipartition of mathematical representations into diagrammatic and linguistic. It will be argued that one (...)
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  32. Forms and Roles of Diagrams in Knot Theory.Silvia De Toffoli & Valeria Giardino - 2014 - Erkenntnis 79 (4):829-842.
    The aim of this article is to explain why knot diagrams are an effective notation in topology. Their cognitive features and epistemic roles will be assessed. First, it will be argued that different interpretations of a figure give rise to different diagrams and as a consequence various levels of representation for knots will be identified. Second, it will be shown that knot diagrams are dynamic by pointing at the moves which are commonly applied to them. For this reason, experts must (...)
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  33. Proof: Its Nature and Significance.Michael Detlefsen - 2008 - In Bonnie Gold & Roger Simons (eds.), Proof and Other Dilemmas: Mathematics and Philosophy. Mathematical Association of America. pp. 1.
    I focus on three preoccupations of recent writings on proof. -/- I. The role and possible effects of empirical reasoning in mathematics. Do recent developments (specifically, the computer-assisted proof of the 4CT) point to something essentially new as regards the need for and/or effects of using broadly empirical and inductive reasoning in mathematics? In particular, should we see such things as the computer-assisted proof of the 4CT as pointing to the existence of mathematical truths of which we cannot have a (...)
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  34. Purity as an Ideal of Proof.Michael Detlefsen - 2008 - In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice. Oxford University Press. pp. 179-197.
    Various ideals of purity are surveyed and discussed. These include the classical Aristotelian ideal, as well as certain neo-classical and contemporary ideals. The focus is on a type of purity ideal I call topical purity. This is purity which emphasizes a certain symmetry between the conceptual resources used to prove a theorem and those needed for the clarification of its content. The basic idea is that the resources of proof ought ideally to be restricted to those which determine its content.
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  35. What Does Gödel's Second Theorem Say.Michael Detlefsen - 2001 - Philosophia Mathematica 9 (1):37-71.
    We consider a seemingly popular justification (we call it the Re-flexivity Defense) for the third derivability condition of the Hilbert-Bernays-Löb generalization of Godel's Second Incompleteness Theorem (G2). We argue that (i) in certain settings (rouglily, those where the representing theory of an arithmetization is allowed to be a proper subtheory of the represented theory), use of the Reflexivity Defense to justify the tliird condition induces a fourth condition, and that (ii) the justification of this fourth condition faces serious obstacles. We (...)
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  36. Brouwerian Intuitionism.Michael Detlefsen - 1990 - Mind 99 (396):501-534.
    The aims of this paper are twofold: firstly, to say something about that philosophy of mathematics known as 'intuitionism' and, secondly, to fit these remarks into a more general message for the philosophy of mathematics as a whole. What I have to say on the first score can, without too much inaccuracy, be compressed into two theses. The first is that the intuitionistic critique of classical mathematics can be seen as based primarily on epistemological rather than on meaning-theoretic considerations. The (...)
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  37. 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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  38. Informal Versus Formal Mathematics.Francisco Doria - 2007 - Synthese 154 (3):401-415.
    We discuss Kunen’s algorithmic implementation of a proof for the Paris–Harrington theorem, and the author’s and da Costa’s proposed “exotic” formulation for the P = NP hypothesis. Out of those two examples we ponder the relation between mathematics within an axiomatic framework, and intuitive or informal mathematics.
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  39. Réalisme et antimécanisme chez K. Gödel.Jacques Dubucs - 1986 - Dialectica 40 (4):297-308.
    RésuméSelon K. Gödel, les restrictions méthodologiques des constructivistes sont aberrantes, et le point de vue réaliste est beaucoup plus fécond: il est hi‐même parvenu à ses résultats logiques fondarnentaux en donnant une place à la notion de vérite, qui n'est pas de type combinatoire. Le premier théorème d'incomplétude est hi‐même un argument décisif en faveur du réalisme, si l'on accepte le principe d'«accessibilité» en vertu duquel nous pouvons décider de toutes les propriétés de nos constructions intellectuelles. Par ailleurs, il existe (...)
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  40. Probabilistic Proofs and Transferability.Kenny Easwaran - 2009 - Philosophia Mathematica 17 (3):341-362.
    In a series of papers, Don Fallis points out that although mathematicians are generally unwilling to accept merely probabilistic proofs, they do accept proofs that are incomplete, long and complicated, or partly carried out by computers. He argues that there are no epistemic grounds on which probabilistic proofs can be rejected while these other proofs are accepted. I defend the practice by presenting a property I call ‘transferability’, which probabilistic proofs lack and acceptable proofs have. I also consider what this (...)
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  41. A Framework for Implicit Definitions and the A Priori.Philip A. Ebert - 2016 - In Philip A. Ebert & Marcus Rossberg (eds.), Abstractionism. Oxford: Oxford University Press. pp. 133--160.
  42. Introduction to Abstractionism.Philip A. Ebert & Marcus Rossberg - 2016 - In Philip A. Ebert & Marcus Rossberg (eds.), Abstractionism. Oxford: Oxford University Press. pp. 3-33.
  43. Ed Zalta's Version of Neo-Logicism: A Friendly Letter of Complaint.Philip A. Ebert & Marcus Rossberg - 2009 - In Hannes Leitgeb & Alexander Hieke (eds.), Reduction – Abstraction – Analysis. Ontos. pp. 11--305.
    In this short letter to Ed Zalta we raise a number of issues with regards to his version of Neo-Logicism. The letter is, in parts, based on a longer manuscript entitled “What Neo-Logicism could not be” which is in preparation. A response by Ed Zalta to our letter can be found on his website: http://mally.stanford.edu/publications.html (entry C3).
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  44. Is Mathematics an “Anomaly” in the Theory of “Scientific Revolutions” ? Editor - 1973 - Philosophia Mathematica (1):92-101.
  45. From Abacus to Algorism: Theory and Practice in Medieval Arithmetic.Gillian Evans - 1977 - British Journal for the History of Science 10 (2):114-131.
    Even at the level of the most elementary arithmetical operations, procedures and practices change. During the twelfth and thirteenth centuries an unusually well documented development took place: at the beginning of the period the authors of elementary manuals of computation taught the use of the abacus, whereas at the end they described the method of calculation which came to be known as the algorism. Their ideas about number, however, were still largely drawn from Boethius's rendering of Nicomachus of Gerasa's Introduction (...)
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  46. Are There Absolutely Unsolvable Problems? Godel's Dichotomy.S. Feferman - 2006 - Philosophia Mathematica 14 (2):134-152.
    This is a critical analysis of the first part of Go¨del’s 1951 Gibbs lecture on certain philosophical consequences of the incompleteness theorems. Go¨del’s discussion is framed in terms of a distinction between objective mathematics and subjective mathematics, according to which the former consists of the truths of mathematics in an absolute sense, and the latter consists of all humanly demonstrable truths. The question is whether these coincide; if they do, no formal axiomatic system (or Turing machine) can comprehend the mathematizing (...)
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  47. Matemáticas y Platonismo(S).J. Ferreiros - 1999 - Gaceta de la Real Sociedad Matemática Española 2 (446):473.
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  48. Mathematical Knowledge and the Interplay of Practices.Jose Ferreiros - 2010 - In Mauricio Suárez, M. Dorato & M. Rédei (eds.), EPSA Philosophical Issues in the Sciences · Launch of the European Philosophy of Science Association. Springer. pp. 55--64.
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  49. The Autonomy of Mathematical Knowledge: Hilbert's Program Revisited.Curtis Franks - 2010 - Cambridge University Press.
    Most scholars think of David Hilbert's program as the most demanding and ideologically motivated attempt to provide a foundation for mathematics, and because they see technical obstacles in the way of realizing the program's goals, they regard it as a failure. Against this view, Curtis Franks argues that Hilbert's deepest and most central insight was that mathematical techniques and practices do not need grounding in any philosophical principles. He weaves together an original historical account, philosophical analysis, and his own development (...)
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  50. Motivating Wittgenstein's Perspective on Mathematical Sentences as Norms.S. Friederich - 2011 - Philosophia Mathematica 19 (1):1-19.
    The later Wittgenstein’s perspective on mathematical sentences as norms is motivated for sentences belonging to Hilbertian axiomatic systems where the axioms are treated as implicit definitions. It is shown that in this approach the axioms are employed as norms in that they function as standards of what counts as using the concepts involved. This normative dimension of their mode of use, it is argued, is inherited by the theorems derived from them. Having been motivated along these lines, Wittgenstein’s perspective on (...)
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