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  1. Michael A. Arbib (1990). A Piagetian Perspective on Mathematical Construction. 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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  2. Conrad Asmus (2009). Jody Azzouni. Tracking Reason: Proof, Consequence and Truth. Philosophia Mathematica 17 (3):369-377.
    (No abstract is available for this citation).
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  3. Steve Awodey & A. W. Carus, The Turning Point and the Revolution: Philosophy of Mathematics in Logical Empiricism From Tractatus on Logical Syllogism.
    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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  4. Carla Bagnoli (2004). Introduction. Croatian Journal of Philosophy 4 (3):311-316.
    This volume collects articles in realism, anti-realism, and constructivism.
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  5. Alan Baker (2007). Is There a Problem of Induction for Mathematics? In M. Potter (ed.), Mathematical Knowledge. Oxford University Press. 57-71.
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  6. Claus Beisbart (2008). Review of M. Wille, Mathematics and the Synthetic A Priori: Epistemological Investigations Into the Status of Mathematical Axioms. [REVIEW] Philosophia Mathematica 16 (1):130-132.
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  7. David Bell & W. D. Hart (1979). The Epistemology of Abstract Objects: Access and Inference. Proceedings of the Aristotelian Society 53:153-165.
  8. Kajsa Bråting & Johanna Pejlare (2008). Visualizations in Mathematics. 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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  9. Otávio Bueno (2008). Truth and Proof. Manuscrito 31 (1).
    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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  10. Paola Cantù & De Zan Mauro (2009). Life and Works of Giovanni Vailati. 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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  11. Carlo Cellucci (2000). The Growth of Mathematical Knowledge: An Open World View. In Emily Grosholz & Herbert Breger (eds.), The Growth of Mathematical Knowledge, pp. 153-176. Kluwer. 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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  12. Julian C. Cole (2013). Towards an Institutional Account of the Objectivity, Necessity, and Atemporality of Mathematics. 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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  13. Chris Daly & David Liggins (2014). Nominalism, Trivialist Platonism and Benacerraf's Dilemma. 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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  14. Helen De Cruz & Johan De Smedt (2013). Mathematical Symbols as Epistemic Actions. 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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  15. Francisco Antonio Doria (2007). Informal Versus Formal Mathematics. 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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  16. Kenny Easwaran (2009). Probabilistic Proofs and Transferability. 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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  17. Philip A. Ebert & Marcus Rossberg (2009). Ed Zalta's Version of Neo-Logicism: A Friendly Letter of Complaint. In Hannes Leitgeb & Alexander Hieke (eds.), Reduction – Abstraction – Analysis. Ontos. 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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  18. Editor (1973). Is Mathematics an “Anomaly” in the Theory of “Scientific Revolutions” ? Philosophia Mathematica (1):92-101.
  19. S. Feferman (2006). Are There Absolutely Unsolvable Problems? Godel's Dichotomy. 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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  20. J. Ferreiros (1999). Matemáticas y Platonismo(S). Gaceta de la Real Sociedad Matemática Española 2 (446):473.
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  21. Jose Ferreiros (2010). Mathematical Knowledge and the Interplay of Practices. In Mauricio Suárez, M. Dorato & M. Rédei (eds.), EPSA Philosophical Issues in the Sciences · Launch of the European Philosophy of Science Association. Springer. 55--64.
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  22. Curtis Franks (2009). The Autonomy of Mathematical Knowledge: Hilbert's Program Revisited. Cambridge University Press.
    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.
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  23. S. Friederich (2011). Motivating Wittgenstein's Perspective on Mathematical Sentences as Norms. 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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  24. Haim Gaifman, Some Thoughts and a Proposal in the Philosophy of Mathematics.
    The paper outlines a project in the philosophy of mathematics based on a proposed view of the nature of mathematical reasoning. It also contains a brief evaluative overview of the discipline and some historical observations; here it points out and illustrates the division between the philosophical dimension, where questions of realism and the status of mathematics are treated, and the more descriptive and looser dimension of epistemic efficiency, which has to do with ways of organizing the mathematical material. The paper’s (...)
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  25. Mihai Ganea (2008). Epistemic Optimism. Philosophia Mathematica 16 (3):333-353.
    Michael Dummett's argument for intuitionism can be criticized for the implicit reliance on the existence of what might be called absolutely undecidable statements. Neil Tennant attacks epistemic optimism, the view that there are no such statements. I expose what seem serious flaws in his attack, and I suggest a way of defending the use of classical logic in arithmetic that circumvents the issue of optimism. I would like to thank an anonymous referee for helpful comments. CiteULike Connotea Del.icio.us What's this?
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  26. M. Giaquinto (2007). Visual Thinking in Mathematics: An Epistemological Study. Oxford University Press.
    Visual thinking -- visual imagination or perception of diagrams and symbol arrays, and mental operations on them -- is omnipresent in mathematics. Is this visual thinking merely a psychological aid, facilitating grasp of what is gathered by other means? Or does it also have epistemological functions, as a means of discovery, understanding, and even proof? By examining the many kinds of visual representation in mathematics and the diverse ways in which they are used, Marcus Giaquinto argues that visual thinking in (...)
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  27. V. Giardino (2007). Gabriele Lolli. Fenomenologia Della Dimostrazione. Turin: Il Mulino, 2005. ISBN 88-339-1588-3. Pp. 182. Philosophia Mathematica 15 (1):132-134.
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  28. Valeria Giardino (2010). Intuition and Visualization in Mathematical Problem Solving. Topoi 29 (1):29-39.
    In this article, I will discuss the relationship between mathematical intuition and mathematical visualization. I will argue that in order to investigate this relationship, it is necessary to consider mathematical activity as a complex phenomenon, which involves many different cognitive resources. I will focus on two kinds of danger in recurring to visualization and I will show that they are not a good reason to conclude that visualization is not reliable, if we consider its use in mathematical practice. Then, I (...)
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  29. Eduard Glas (1989). Testing the Philosophy of Mathematics in the History of Mathematics. Studies in History and Philosophy of Science Part A 20 (2):157-174.
    Recent philosophical accounts of mathematics increasingly focus on the quasi-Empirical rather than the formal aspects of the field, The praxis of how mathematics is done rather than the idealized logical structure and foundations of the theory. The ultimate test of any philosophy of mathematics, However idealized, Is its ability to account adequately for the factual development of the subject in real time. As a text case, The works and views of felix klein (1849-1925) were studied. Major advances in mathematics turn (...)
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  30. Eduard Glas (1989). Testing the Philosophy of Mathematics in the History of Mathematics. Studies in History and Philosophy of Science Part A 20 (1):115-131.
    Recent philosophical accounts of mathematics increasingly focus on the quasi-Empirical rather than the formal aspects of the field, The praxis of how mathematics is done rather than the idealized logical structure and foundations of the theory. The ultimate test of any philosophy of mathematics, However idealized, Is its ability to account adequately for the factual development of the subject in real time. As a text case, The works and views of felix klein (1849-1925) were studied. Major advances in mathematics turn (...)
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  31. Emily Grosholz & Herbert Breger (eds.) (2000). The Growth of Mathematical Knowledge. Kluwer Academic Publishers.
    This book draws its inspiration from Hilbert, Wittgenstein, Cavaillès and Lakatos and is designed to reconfigure contemporary philosophy of mathematics by making the growth of knowledge rather than its foundations central to the study of mathematical rationality, and by analyzing the notion of growth in historical as well as logical terms. Not a mere compendium of opinions, it is organised in dialogical forms, with each philosophical thesis answered by one or more historical case studies designed to support, complicate or question (...)
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  32. Jacques Hadamard (2008/1954). An Essay on the Psychology of Invention in the Mathematical Field. Read Books.
    We are republishing these classic works in affordable, high quality, modern editions, using the original text and artwork.
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  33. Jacques Hadamard (1980). In Particular and in Retrospect: “The Psychology of Invention in the Mathematical Field” (a Sum-Up). Philosophia Mathematica (1):29-38.
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  34. Jacques Hadamard (1945/1996). The Mathematician's Mind: The Psychology of Invention in the Mathematical Field. Princeton University Press.
    Fifty years ago when Jacques Hadamard set out to explore how mathematicians invent new ideas, he considered the creative experiences of some of the greatest thinkers of his generation, such as George Polya, Claude Le;vi-Strauss, and Albert Einstein. It appeared that inspiration could strike anytime, particularly after an individual had worked hard on a problem for days and then turned attention to another activity. In exploring this phenomenon, Hadamard produced one of the most famous and cogent cases for the existence (...)
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  35. David S. Henley (1995). Syntax-Directed Discovery in Mathematics. Erkenntnis 43 (2):241 - 259.
    It is shown how mathematical discoveries such as De Moivre's theorem can result from patterns among the symbols of existing formulae and that significant mathematical analogies are often syntactic rather than semantic, for the good reason that mathematical proofs are always syntactic, in the sense of employing only formal operations on symbols. This radically extends the Lakatos approach to mathematical discovery by allowing proof-directed concepts to generate new theorems from scratch instead of just as evolutionary modifications to some existing theorem. (...)
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  36. Jan Heylen (2013). Modal-Epistemic Arithmetic and the Problem of Quantifying In. Synthese 190 (1):89-111.
    The subject of this article is Modal-Epistemic Arithmetic (MEA), a theory introduced by Horsten to interpret Epistemic Arithmetic (EA), which in turn was introduced by Shapiro to interpret Heyting Arithmetic. I will show how to interpret MEA in EA such that one can prove that the interpretation of EA is MEA is faithful. Moreover, I will show that one can get rid of a particular Platonist assumption. Then I will discuss models for MEA in light of the problems of logical (...)
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  37. Jan Heylen (2010). Descriptions and Unknowability. Analysis 70 (1):50-52.
    (No abstract is available for this citation).
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  38. Thomas Hofweber (2001). Review of "Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures" by James Robert Brown. [REVIEW] British Journal for the Philosophy of Science 52 (2):413-416.
  39. F. Janet (2007). Review of J. Norman, After Euclid: Visual Reasoning and the Epistemology of Diagrams. [REVIEW] Philosophia Mathematica 15 (1):116-121.
  40. C. S. Jenkins (2008). Grounding Concepts: An Empirical Basis for Arithmetical Knowledge. OUP Oxford.
    Grounding Concepts tackles the issue of arithmetical knowledge, developing a new position which respects three intuitions which have appeared impossible to satisfy simultaneously: a priorism, mind-independence realism, and empiricism. -/- Drawing on a wide range of philosophical influences, but avoiding unnecessary technicality, a view is developed whereby arithmetic can be known through the examination of empirically grounded concepts. These are concepts which, owing to their relationship to sensory input, are non-accidentally accurate representations of the mind-independent world. Examination of such concepts (...)
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  41. C. S. Jenkins (2005). Knowledge of Arithmetic. British Journal for the Philosophy of Science 56 (4):727-747.
    The goal of the research programme I describe in this article is a realist epistemology for arithmetic which respects arithmetic's special epistemic status (the status usually described as a prioricity) yet accommodates naturalistic concerns by remaining fundamentally empiricist. I argue that the central claims which would allow us to develop such an epistemology are (i) that arithmetical truths are known through an examination of our arithmetical concepts; (ii) that (at least our basic) arithmetical concepts are accurate mental representations of elements (...)
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  42. Ivan Kasa (2010). On Field's Epistemological Argument Against Platonism. Studia Logica 96 (2):141-147.
    Hartry Field's formulation of an epistemological argument against platonism is only valid if knowledge is constrained by a causal clause. Contrary to recent claims (e.g. in Liggins (2006), Liggins (2010)), Field's argument therefore fails the very same criterion usually taken to discredit Benacerraf's earlier version.
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  43. Mikhail G. Katz & Thomas Mormann, Infinitesimals and Other Idealizing Completions in Neo-Kantian Philosophy of Mathematics.
    We seek to elucidate the philosophical context in which the so-called revolution of rigor in inifinitesimal calculus and mathematical analysis took place. Some of the protagonists of the said revolution were Cauchy, Cantor, Dedekind, and Weierstrass. The dominant current of philosophy in Germany at that time was neo-Kantianism. Among its various currents, the Marburg school (Cohen, Natorp, Cassirer, and others) was the one most interested in matters scientific and mathematical. Our main thesis is that Marburg Neo-Kantian philosophy formulated a sophisticated (...)
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  44. Wojciech Krysztofiak (2012). Indexed Natural Numbers in Mind: A Formal Model of the Basic Mature Number Competence. [REVIEW] Axiomathes 22 (4):433-456.
    The paper undertakes three interdisciplinary tasks. The first one consists in constructing a formal model of the basic arithmetic competence, that is, the competence sufficient for solving simple arithmetic story-tasks which do not require any mathematical mastery knowledge about laws, definitions and theorems. The second task is to present a generalized arithmetic theory, called the arithmetic of indexed numbers (INA). All models of the development of counting abilities presuppose the common assumption that our simple, folk arithmetic encoded linguistically in the (...)
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  45. Imre Lakatos (1976). A Renaissance of Empiricism in the Recent Philosophy of Mathematics. British Journal for the Philosophy of Science 27 (3):201-223.
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  46. Mary Leng, Alexander Paseau & Michael D. Potter (eds.) (2007). Mathematical Knowledge. Oxford University Press.
    What is the nature of mathematical knowledge? Is it anything like scientific knowledge or is it sui generis? How do we acquire it? Should we believe what mathematicians themselves tell us about it? Are mathematical concepts innate or acquired? Eight new essays offer answers to these and many other questions. Written by some of the world's leading philosophers of mathematics, psychologists, and mathematicians, Mathematical Knowledge gives a lively sense of the current state of debate in this fascinating field. Contents 1. (...)
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  47. Øystein Linnebo (2006). Epistemological Challenges to Mathematical Platonism. Philosophical Studies 129 (3):545-574.
    Since Benacerraf’s “Mathematical Truth” a number of epistemological challenges have been launched against mathematical platonism. I first argue that these challenges fail because they unduely assimilate mathematics to empirical science. Then I develop an improved challenge which is immune to this criticism. Very roughly, what I demand is an account of how people’s mathematical beliefs are responsive to the truth of these beliefs. Finally I argue that if we employ a semantic truth-predicate rather than just a deflationary one, there surprisingly (...)
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  48. Benedikt Löwe & Thomas Müller (2008). Mathematical Knowledge is Context Dependent. Grazer Philosophische Studien 76 (1):91-107.
    We argue that mathematical knowledge is context dependent. Our main argument is that on pain of distorting mathematical practice, one must analyse the notion of having available a proof, which supplies justification in mathematics, in a context dependent way.
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  49. Russell Marcus (2013). Intrinsic Explanation and Field's Dispensabilist Strategy. International Journal of Philosophical Studies 21 (2):163 - 183.
    Philosophy of mathematics for the last half-century has been dominated in one way or another by Quine’s indispensability argument. The argument alleges that our best scientific theory quantifies over, and thus commits us to, mathematical objects. In this paper, I present new considerations which undermine the most serious challenge to Quine’s argument, Hartry Field’s reformulation of Newtonian Gravitational Theory.
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  50. Boaz Miller (2009). What Does It Mean That PRIMES is in P: Popularization and Distortion Revisited. Social Studies of Science 39 (2):257-288.
    In August 2002, three Indian computer scientists published a paper, ‘PRIMES is in P’, online. It presents a ‘deterministic algorithm’ which determines in ‘polynomial time’ if a given number is a prime number. The story was quickly picked up by the general press, and by this means spread through the scientific community of complexity theorists, where it was hailed as a major theoretical breakthrough. This is although scientists regarded the media reports as vulgar popularizations. When the paper was published in (...)
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