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Summary One of the lines of reasoning in support of mathematical platonism employs the fact that mathematical theories find applications in sciences which, at least prima facie, concern themselves with the physical world. From the indispensability of mathematics in science the argument moves to the indispensability of reference to mathematical objects in science. Further on, since we, supposedly, have good reasons to accept the existence of objects our best scientific theories have to refer to, we should accept the existence of such mathematical objects, on a par with the existence of electrons and other invisible entities postulated by such scientific theories. Accordingly, the argument has been attacked on different grounds. Some deny the indispensability of mathematics in science, some claim that indispensability of mathematical theories is not the same as the indispensability of reference to mathematical objects, some insist that this approach doesn't make justice to the difference between a priori mathematical knowledge and a posteriori scientific knowledge, some worry that applied mathematics is only a part of theoretical mathematics and some suggest that best scientific theories don't have to be our guide to metaphysics.
Key works Loci classici are Quine 1953QUINE 1981Putnam 1975 and Putnam 1971. Further considerations can be found for instance in Parsons 1980Chihara 1973 and   Maddy 1992. Field 1980 is directed at showing the dispensability of mathematics in science. An extensive defence of the indispensability argument have been mounted by Colyvan 2001.
Introductions Start with Colyvan 2008 (and references therein).
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  1. added 2020-06-16
    Explanation in Mathematics.Paolo Mancosu - 2011 - In Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy.
    The philosophical analysis of mathematical explanations concerns itself with two different, although connected, areas of investigation. The first area addresses the problem of whether mathematics can play an explanatory role in the natural and social sciences. The second deals with the problem of whether mathematical explanations occur within mathematics itself. Accordingly, this entry surveys the contributions to both areas, it shows their relevance to the history of philosophy and science, it articulates their connection, and points to the philosophical pay-offs to (...)
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  2. added 2020-04-17
    Justification and Explanation in Mathematics and Morality.Justin Clarke-Doane - 2015 - Oxford Studies in Metaethics 10.
    In his influential book, The Nature of Morality, Gilbert Harman writes: “In explaining the observations that support a physical theory, scientists typically appeal to mathematical principles. On the other hand, one never seems to need to appeal in this way to moral principles.” What is the epistemological relevance of this contrast, if genuine? This chapter argues that ethicists and philosophers of mathematics have misunderstood it. They have confused what the chapter calls the justificatory challenge for realism about an area, D—the (...)
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  3. added 2020-03-30
    How Can Mathematical Objects Be Real but Mind-Dependent?Hazhir Roshangar - manuscript
    Taking mathematics as a language based on empirical experience, I argue for an account of mathematics in which its objects are abstracta that describe and communicate the structure of reality based on some of our ancestral interactions with their environment. I argue that mathematics as a language is mostly invented, and it is mind-dependent in a specific sense. However, the bases of mathematics will characterize it as a real, non-fictional science of structures.
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  4. added 2020-02-06
    How to Make Reflectance a Surface Property.Nicholas Danne - 2020 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 70:19-27.
    Reflectance physicalists define reflectance as the intrinsic disposition of a surface to reflect light at a given efficiency per wavelength. I criticize a leading account of dispositional reflectance for failing to account for what I call 'harmonic dispersion', the inverse relationship of a light pulse's duration to its bandwidth. I argue that harmonic dispersion renders reflectance defined in terms of light pulses an extrinsic disposition. Reflectance defined as the per-wavelength efficiency to reflect the superimposed, infinite-duration, Fourier harmonics of pulses can (...)
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  5. added 2019-11-14
    Infinite Lies and Explanatory Ties: Idealization in Phase Transitions.Sam Baron - 2019 - Synthese 196 (5):1939-1961.
    Infinite idealizations appear in our best scientific explanations of phase transitions. This is thought by some to be paradoxical. In this paper I connect the existing literature on the phase transition paradox to work on the concept of indispensability, which arises in discussions of realism and anti-realism within the philosophy of science and the philosophy of mathematics. I formulate a version of the phase transition paradox based on the idea that infinite idealizations are explanatorily indispensable to our best science, and (...)
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  6. added 2019-11-14
    How Mathematics Can Make a Difference.Sam Baron, Mark Colyvan & David Ripley - 2017 - Philosophers' Imprint 17.
    Standard approaches to counterfactuals in the philosophy of explanation are geared toward causal explanation. We show how to extend the counterfactual theory of explanation to non-causal cases, involving extra-mathematical explanation: the explanation of physical facts by mathematical facts. Using a structural equation framework, we model impossible perturbations to mathematics and the resulting differences made to physical explananda in two important cases of extra-mathematical explanation. We address some objections to our approach.
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  7. added 2019-07-24
    Mathematics and Explanatory Generality: Nothing but Cognitive Salience.Juha Saatsi & Robert Knowles - forthcoming - Erkenntnis:1-19.
    We demonstrate how real progress can be made in the debate surrounding the enhanced indispensability argument. Drawing on a counterfactual theory of explanation, well-motivated independently of the debate, we provide a novel analysis of ‘explanatory generality’ and how mathematics is involved in its procurement. On our analysis, mathematics’ sole explanatory contribution to the procurement of explanatory generality is to make counterfactual information about physical dependencies easier to grasp and reason with for creatures like us. This gives precise content to key (...)
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  8. added 2019-07-22
    The Applicability of Mathematics and the Indispensability Arguments.Michele Ginammi - 2016 - Lato Sensu, Revue de la Société de Philosophie des Sciences 3 (1).
    In this paper I will take into examination the relevance of the main indispensability arguments for the comprehension of the applicability of mathematics. I will conclude not only that none of these indispensability arguments are of any help for understanding mathematical applicability, but also that these arguments rather require a preliminary analysis of the problems raised by the applicability of mathematics in order to avoid some tricky difficulties in their formulations. As a consequence, we cannot any longer consider the applicability (...)
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  9. added 2019-07-05
    Optimal Representations and the Enhanced Indispensability Argument.Manuel Barrantes - 2019 - Synthese 196 (1):247-263.
    The Enhanced Indispensability Argument appeals to the existence of Mathematical Explanations of Physical Phenomena to justify mathematical Platonism, following the principle of Inference to the Best Explanation. In this paper, I examine one example of a MEPP—the explanation of the 13-year and 17-year life cycle of magicicadas—and argue that this case cannot be used defend the EIA. I then generalize my analysis of the cicada case to other MEPPs, and show that these explanations rely on what I will call ‘optimal (...)
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  10. added 2019-06-11
    Importance and Explanatory Relevance: The Case of Mathematical Explanations.Gabriel Târziu - 2018 - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 49 (3):393-412.
    A way to argue that something plays an explanatory role in science is by linking explanatory relevance with importance in the context of an explanation. The idea is deceptively simple: a part of an explanation is an explanatorily relevant part of that explanation if removing it affects the explanation either by destroying it or by diminishing its explanatory power, i.e. an important part is an explanatorily relevant part. This can be very useful in many ontological debates. My aim in this (...)
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  11. added 2019-06-11
    Mathematical Explanations and the Piecemeal Approach to Thinking About Explanation.Gabriel Târziu - 2018 - Logique Et Analyse 61 (244):457-487.
    A new trend in the philosophical literature on scientific explanation is that of starting from a case that has been somehow identified as an explanation and then proceed to bringing to light its characteristic features and to constructing an account for the type of explanation it exemplifies. A type of this approach to thinking about explanation – the piecemeal approach, as I will call it – is used, among others, by Lange (2013) and Pincock (2015) in the context of their (...)
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  12. added 2019-06-11
    Can We Have Mathematical Understanding of Physical Phenomena?Gabriel Târziu - 2018 - Theoria : An International Journal for Theory, History and Fundations of Science 33 (1):91-109.
    Can mathematics contribute to our understanding of physical phenomena? One way to try to answer this question is by getting involved in the recent philosophical dispute about the existence of mathematical explanations of physical phenomena. If there is such a thing, given the relation between explanation and understanding, we can say that there is an affirmative answer to our question. But what if we do not agree that mathematics can play an explanatory role in science? Can we still consider that (...)
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  13. added 2019-06-06
    Unifying Scientific Theories: Physical Concepts and Mathematical Structures.Andrew Wayne - 2002 - Canadian Journal of Philosophy 32 (1):117-138.
    Philosophers of science have long been concerned with these questions. In the 1980s, influential work by Clark Glymour, Michael Friedman, John Watkins, and Philip Kitcher articulated general accounts of theory unification that attempted to underwrite a connection between unification, truth, and understanding. According to the ‘unifiers,’ as we may call them, a theory is unified to the extent that it has a small theoretical structure relative to the domain of phenomena it covers, and there are general syntactic criteria that allow (...)
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  14. added 2019-06-05
    The Eleatic and the Indispensabilist.Russell Marcus - 2015 - Theoria : An International Journal for Theory, History and Fundations of Science 30 (3):415-429.
    The debate over whether we should believe that mathematical objects exist quickly leads to the question of how to determine what we should believe. Indispensabilists claim that we should believe in the existence of mathematical objects because of their ineliminable roles in scientific theory. Eleatics argue that only objects with causal properties exist. Mark Colyvan’s recent defenses of Quine’s indispensability argument against some contemporary eleatics attempt to provide reasons to favor the indispensabilist’s criterion. I show that Colyvan’s argument is not (...)
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  15. added 2019-06-05
    On Frege's Alleged Indispensability Argument.Pieranna Garavaso - 2005 - Philosophia Mathematica 13 (2):160-173.
    The expression ‘indispensability argument’ denotes a family of arguments for mathematical realism supported among others by Quine and Putnam. More and more often, Gottlob Frege is credited with being the first to state this argument in section 91 of the _Grundgesetze der Arithmetik_. Frege's alleged indispensability argument is the subject of this essay. On the basis of three significant differences between Mark Colyvan's indispensability arguments and Frege's applicability argument, I deny that Frege presents an indispensability argument in that very often (...)
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  16. added 2019-06-05
    Practical Reason and Mathematical Argument.John O'Neill - 1998 - Studies in History and Philosophy of Science Part A 29 (2):195-205.
  17. added 2019-06-05
    Realism in Mathematics. Penelope Maddy.Jill Dieterle & Stewart Shapiro - 1993 - Philosophy of Science 60 (4):659-660.
  18. added 2019-05-28
    The Enhanced Indispensability Argument, the Circularity Problem, and the Interpretability Strategy.Jan Heylen & Lars Arthur Tump - forthcoming - Synthese:1-13.
    Within the context of the Quine–Putnam indispensability argument, one discussion about the status of mathematics is concerned with the ‘Enhanced Indispensability Argument’, which makes explicit in what way mathematics is supposed to be indispensable in science, namely explanatory. If there are genuine mathematical explanations of empirical phenomena, an argument for mathematical platonism could be extracted by using inference to the best explanation. The best explanation of the primeness of the life cycles of Periodical Cicadas is genuinely mathematical, according to Baker (...)
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  19. added 2019-03-01
    An Introduction to the Philosophy of Mathematics, by Colyvan Mark: Cambridge: Cambridge University Press, 2012, Pp. X + 188, AU$46.95. [REVIEW]Zach Weber - 2013 - Australasian Journal of Philosophy 91 (4):828-828.
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  20. added 2019-02-01
    Quine's Weak and Strong Indispensability Argument.Lieven Decock - 2002 - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 33 (2):231-250.
    Quine's views on indispensability arguments in mathematics are scrutinised. A weak indispensability argument is distinguished from a strong indispensability thesis. The weak argument is the combination of the criterion of ontological commitment, holism and a mild naturalism. It is used to refute nominalism. Quine's strong indispensability thesis claims that one should consider all and only the mathematical entities that are really indispensable. Quine has little support for this thesis. This is even clearer if one takes into account Maddy's critique of (...)
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  21. added 2019-01-10
    Quine and the Incoherence of the Indispensability Argument.Michael J. Shaffer - 2019 - Logos and Episteme 10 (2):207-213.
    It is an under-appreciated fact that Quine's rejection of the analytic/synthetic distinction, when coupled with some other plausible and related views, implies that there are serious difficulties in demarcating empirical theories from pure mathematical theories within the Quinean framework. This is a serious problem because there seems to be a principled difference between the two disciplines that cannot apparently be captured in the orthodox Quienan framework. For the purpose of simplicity let us call this Quine's problem of demarcation. In this (...)
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  22. added 2018-10-25
    Infinitesimal Idealization, Easy Road Nominalism, and Fractional Quantum Statistics.Elay Shech - 2019 - Synthese 196 (5):1963-1990.
    It has been recently debated whether there exists a so-called “easy road” to nominalism. In this essay, I attempt to fill a lacuna in the debate by making a connection with the literature on infinite and infinitesimal idealization in science through an example from mathematical physics that has been largely ignored by philosophers. Specifically, by appealing to John Norton’s distinction between idealization and approximation, I argue that the phenomena of fractional quantum statistics bears negatively on Mary Leng’s proposed path to (...)
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  23. added 2018-10-09
    Response to Colyvan.Joseph Melia - 2002 - Mind 111 (441):75-80.
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  24. added 2018-09-06
    Confirmational Holism and its Mathematical (W)Holes.Anthony Peressini - 2008 - Studies in History and Philosophy of Science Part A 39 (1):102-111.
    I critically examine confirmational holism as it pertains to the indispensability arguments for mathematical Platonism. I employ a distinction between pure and applied mathematics that grows out of the often overlooked symbiotic relationship between mathematics and science. I argue that this distinction undercuts the notion that mathematical theories fall under the holistic scope of the confirmation of our scientific theories.Keywords: Confirmational holism; Indispensability argument; Mathematics; Application; Science.
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  25. added 2018-08-06
    Clarificando o Suporte do Argumento Melhorado da Indispensabilidade Matemática.Eduardo Castro - 2017 - Argumentos 17 (9):57-71.
    The enhanced mathematical indispensability argument, proposed by Alan Baker (2005), argues that we must commit to mathematical entities, because mathematical entities play an indispensable explanatory role in our best scientific theories. This article clarifies the doctrines that support this argument, namely, the doctrines of naturalism and confirmational holism.
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  26. added 2018-02-17
    Platonism, Naturalism, and Mathematical Knowledge.James Robert Brown - 2011 - Routledge.
    This study addresses a central theme in current philosophy: Platonism vs Naturalism and provides accounts of both approaches to mathematics, crucially discussing Quine, Maddy, Kitcher, Lakoff, Colyvan, and many others. Beginning with accounts of both approaches, Brown defends Platonism by arguing that only a Platonistic approach can account for concept acquisition in a number of special cases in the sciences. He also argues for a particular view of applied mathematics, a view that supports Platonism against Naturalist alternatives. Not only does (...)
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  27. added 2018-02-16
    Are There Genuine Physical Explanations of Mathematical Phenomena?Brdford Skow - 2015 - British Journal for the Philosophy of Science 66 (1):69-93.
    There are lots of arguments for, or justifications of, mathematical theorems that make use of principles from physics. Do any of these constitute explanations? On the one hand, physical principles do not seem like they should be explanatorily relevant; on the other, some particular examples of physical justifications do look explanatory. In this article, I defend the idea that physical justifications can and do explain mathematical facts. 1 Physical Arguments for Mathematical Truths2 Preview3 Mathematical Facts4 Purity5 Doubts about Purity: I6 (...)
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  28. added 2017-11-28
    What We Talk About When We Talk About Numbers.Richard Pettigrew - 2018 - Annals of Pure and Applied Logic 169 (12):1437-1456.
    In this paper, I describe and motivate a new species of mathematical structuralism, which I call Instrumental Nominalism about Set-Theoretic Structuralism. As the name suggests, this approach takes standard Set-Theoretic Structuralism of the sort championed by Bourbaki and removes its ontological commitments by taking an instrumental nominalist approach to that ontology of the sort described by Joseph Melia and Gideon Rosen. I argue that this avoids all of the problems that plague other versions of structuralism.
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  29. added 2017-11-20
    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.
  30. added 2017-10-04
    Rejecting Mathematical Realism While Accepting Interactive Realism.Seungbae Park - 2018 - Analysis and Metaphysics 17:7-21.
    Indispensablists contend that accepting scientific realism while rejecting mathematical realism involves a double standard. I refute this contention by developing an enhanced version of scientific realism, which I call interactive realism. It holds that interactively successful theories are typically approximately true, and that the interactive unobservable entities posited by them are likely to exist. It is immune to the pessimistic induction while mathematical realism is susceptible to it.
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  31. added 2017-03-11
    In Defense of Mathematical Inferentialism.Seungbae Park - 2017 - Analysis and Metaphysics 16:70-83.
    I defend a new position in philosophy of mathematics that I call mathematical inferentialism. It holds that a mathematical sentence can perform the function of facilitating deductive inferences from some concrete sentences to other concrete sentences, that a mathematical sentence is true if and only if all of its concrete consequences are true, that the abstract world does not exist, and that we acquire mathematical knowledge by confirming concrete sentences. Mathematical inferentialism has several advantages over mathematical realism and fictionalism.
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  32. added 2017-02-14
    Liu Hui's Theories of Mathematics.R. Mei - 1996 - Boston Studies in the Philosophy of Science 179:243-254.
  33. added 2017-02-13
    The Pure and the Applied: Bourbakism Comes to Mathematical Economics.E. Roy Weintraub & Philip Mirowski - 1994 - Science in Context 7 (2):245-272.
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  34. added 2017-02-13
    Perennial Philosophy: Evidence From the Mathematical and Physical Sciences.Alan M. Laibelman - 1992 - Ultimate Reality and Meaning 15:216.
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  35. added 2017-02-12
    Practice, Constraint, and Mathematical Concepts.Mark C. R. Smith - 2012 - Philosophia Scientiae 16 (1):15-28.
    Dans cet article je propose d'exprimer et de défendre une conception des pratiques et du domaine de discours mathématiques qui soit sensible, d'une part, au pluralisme des relations entre pratiques inférentielles et intérêts, et d'autre part, à la structure objective et déterminante des concepts mathématiques. J'ébauche tout d'abord une caractérisation générale des pratiques, pour ensuite préciser certains phénomènes propres aux pratiques mathématiques. Suit un recensement des idées qui se dégagent des arguments pluralistes, et de celles qui sont à retenir. Mais (...)
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  36. added 2017-02-11
    Mathematics Without Truth (a Reply to Maddy).H. Field - 1990 - Pacific Philosophical Quarterly 71 (3):206-222.
    This paper elaborates on the fictionalist conception of mathematics, and on how it accommodates the obvious fact that mathematical claims are important in application to the physical world. It also replies to Maddy's argument that fictionalism does not have the epistemological advantage over Platonism that it appears to have; the reply involves a discussion of whether mathematics should be regarded as conservative over second order physical theories as well as first order ones.
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  37. added 2017-02-10
    Naturalising Mathematics: A Critical Look at the Quine-Maddy Debate.Marianna Antonutti Marfori - 2012 - Disputatio 4 (32):323-342.
    This paper considers Maddy’s strategy for naturalising mathematics in the context of Quine’s scientific naturalism. The aim of this proposal is to account for the acceptability of mathematics on scientific grounds without committing to revisionism about mathematical practice entailed by the Quine-Putnam indispensability argument. It has been argued that Maddy’s mathematical naturalism makes inconsistent assumptions on the role of mathematics in scientific explanations to the effect that it cannot distinguish mathematics from pseudo-science. I shall clarify Maddy’s arguments and show that (...)
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  38. added 2017-02-01
    Margaret Morrison, Critical Discussion of Unifying Scientific Theories. Physical Concepts and Mathematical Structures.F. A. Muller - 2001 - Erkenntnis 55 (1):132-143.
  39. added 2017-01-29
    Parsimony and Inference to the Best Mathematical Explanation.Alan Baker - 2016 - Synthese 193 (2).
    Indispensability-based arguments for mathematical platonism are typically motivated by drawing an analogy between abstract mathematical objects and concrete scientific posits. In this paper, I argue that mathematics can sometimes help to reduce our concrete ontological, ideological, and structural commitments. My focus is on optimization explanations, and in particular the case study involving periodical cicadas. I argue that in this case, stronger mathematical apparatus yields explanations that have fewer concrete commitments. The nominalist cannot accept these more parsimonious explanations without embracing the (...)
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  40. added 2017-01-27
    Is Indispensability Still a Problem for Fictionalism?Susan Vineberg - 2008 - ProtoSociology 25:128-142.
    For quite some time the indispensability arguments of Quine and Putnam were considered a formidable obstacle to anyone who would reject the existence of mathematical objects.1 Various attempts to respond to the indispensability arguments were developed, most notably by Chihara and Field.2 Field tried to defend mathematical fictionalism, according to which the existential assertions of mathematics are false, by showing that the mathematics used in applications is in fact dispensable. Chihara suggested, on the other hand, that mathematics makes true existential (...)
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  41. added 2017-01-27
    The Mathematical Science of Christopher Wren. [REVIEW]John Hendry - 1983 - British Journal for the History of Science 16 (3):291-292.
  42. added 2017-01-26
    Revealing the Face of Isis.J. L. Usó-Doménech & J. Nescolarde-Selva - 2014 - Foundations of Science 19 (3):311-318.
    This reply to Gash’s (Found Sci 2014) commentary on Nescolarde-Selva and Usó-Doménech (Found Sci 2014b) answers the questions raised and at the same time opens up new questions.
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  43. added 2017-01-26
    Reviewed Work(S): An Introduction to the Philosophy of Mathematics by Mark Colyvan.Review by: Richard Pettigrew - 2013 - Bulletin of Symbolic Logic 19 (3):396-397,.
  44. added 2017-01-26
    Reply to Colyvan.Joseph Melia - 2002 - Mind 111:75-9.
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  45. added 2017-01-26
    Mathematics in Science: The Role of the History of Science in Communicating the Significance of Mathematical Formalism in Science.Kevin C. de Berg - 1992 - Science & Education 1 (1):77-87.
  46. added 2017-01-26
    The Mystery of the Commons: On the Indispensability of Civic Rhetoric.Manfred Stanley - 1983 - Social Research 50.
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  47. added 2017-01-25
    On Tins and Tin-Openers.Michael Liston - 2009 - In Henk W. de Regt (ed.), Epsa Philosophy of Science: Amsterdam 2009. Springer. pp. 151--160.
    Most science requires applied mathematics. This truism underlies the Quine-Putnam indispensability argument: we cannot be mathematical nominalists without rejecting whole swaths of good science that are seamlessly linked with mathematics. One style of response accepts the challenge head-on and attempts to show how to do science without mathematics. There is some consensus that the response fails because the nominalistic apparatus deployed either is not extendible to all of mathematical physics or is merely a deft reconstrual equivalent to standard mathematics. A (...)
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  48. added 2017-01-24
    The Applicability of Mathematics in Science: Indispensability and Ontology.Penelope Rush - 2013 - International Studies in the Philosophy of Science 27 (2):219-222.
  49. added 2017-01-24
    Quine's Master Argument.John F. Fox - 2010 - Logique Et Analyse 53 (212):429-447.
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  50. added 2017-01-23
    Remarks on Mark Colyvan on Mathematical Explanation.Fabrice Pataut - unknown
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