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  1. Andrew Aberdein (2014). Commentary On: Michel Dufour's "Argument and Explanation in Mathematics". 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
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  2. M. S. Akperov (1992). Filosofskie Problemy Matematiki.
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  3. William Aspray & Philip Kitcher (1988). History and Philosophy of Modern Mathematics. Monograph Collection (Matt - Pseudo).
  4. Jeremy Avigad (2010). Understanding, Formal Verification, and the Philosophy of Mathematics. Journal of the Indian Council of Philosophical Research 27:161-197.
    The philosophy of mathematics has long been concerned with deter- mining the means that are appropriate for justifying claims of mathemat- ical knowledge, and the metaphysical considerations that render them so. But, as of late, many philosophers have called attention to the fact that a much broader range of normative judgments arise in ordinary math- ematical practice; for example, questions can be interesting, theorems important, proofs explanatory, concepts powerful, and so on. The as- sociated values are often loosely classied as (...)
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  5. Alan Baker (2010). Mathematical Induction and Explanation. Analysis 70 (4):681-689.
  6. Alan Baker (2005). Are There Genuine Mathematical Explanations of Physical Phenomena? Mind 114 (454):223-238.
    Many explanations in science make use of mathematics. But are there cases where the mathematical component of a scientific explanation is explanatory in its own right? This issue of mathematical explanations in science has been for the most part neglected. I argue that there are genuine mathematical explanations in science, and present in some detail an example of such an explanation, taken from evolutionary biology, involving periodical cicadas. I also indicate how the answer to my title question impacts on broader (...)
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  7. Alan Baker & Mark Colyvan (2011). Indexing and Mathematical Explanation. Philosophia Mathematica 19 (3):323-334.
    We discuss a recent attempt by Chris Daly and Simon Langford to do away with mathematical explanations of physical phenomena. Daly and Langford suggest that mathematics merely indexes parts of the physical world, and on this understanding of the role of mathematics in science, there is no need to countenance mathematical explanation of physical facts. We argue that their strategy is at best a sketch and only looks plausible in simple cases. We also draw attention to how frequently Daly and (...)
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  8. Sorin Bangu (2013). Indispensability and Explanation. British Journal for the Philosophy of Science 64 (2):255-277.
    The question as to whether there are mathematical explanations of physical phenomena has recently received a great deal of attention in the literature. The answer is potentially relevant for the ontology of mathematics; if affirmative, it would support a new version of the indispensability argument for mathematical realism. In this article, I first review critically a few examples of such explanations and advance a general analysis of the desiderata to be satisfied by them. Second, in an attempt to strengthen the (...)
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  9. Sorin Ioan Bangu (2008). Inference to the Best Explanation and Mathematical Realism. Synthese 160 (1):13-20.
    Arguing for mathematical realism on the basis of Field’s explanationist version of the Quine–Putnam Indispensability argument, Alan Baker has recently claimed to have found an instance of a genuine mathematical explanation of a physical phenomenon. While I agree that Baker presents a very interesting example in which mathematics plays an essential explanatory role, I show that this example, and the argument built upon it, begs the question against the mathematical nominalist.
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  10. Sam Baron (2016). Mathematical Explanation and Epistemology: Please Mind the Gap. Ratio 29 (2):149-167.
    This paper draws together two strands in the debate over the existence of mathematical objects. The first strand concerns the notion of extra-mathematical explanation: the explanation of physical facts, in part, by facts about mathematical objects. The second strand concerns the access problem for platonism: the problem of how to account for knowledge of mathematical objects. I argue for the following conditional: if there are extra-mathematical explanations, then the core thesis of the access problem is false. This has implications for (...)
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  11. John D. Barrow (2004). Mathematical Explanation. In John Cornwell (ed.), Explanations: Styles of Explanation in Science. Oxford University Press 81--109.
  12. Jacob Busch (2011). Scientific Realism and the Indispensability Argument for Mathematical Realism: A Marriage Made in Hell. International Studies in the Philosophy of Science 25 (4):307-325.
    An emphasis on explanatory contribution is central to a recent formulation of the indispensability argument for mathematical realism. Because scientific realism is argued for by means of inference to the best explanation, it has been further argued that being a scientific realist entails a commitment to IA and thus to mathematical realism. It has, however, gone largely unnoticed that the way that IBE is argued to be truth conducive involves citing successful applications of IBE and tracing this success over time. (...)
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  13. Jacob Busch & Joe Morrison (2016). Should Scientific Realists Be Platonists? Synthese 193 (2):435-449.
    Enhanced indispensability arguments claim that Scientific Realists are committed to the existence of mathematical entities due to their reliance on Inference to the best explanation. Our central question concerns this purported parity of reasoning: do people who defend the EIA make an appropriate use of the resources of Scientific Realism to achieve platonism? We argue that just because a variety of different inferential strategies can be employed by Scientific Realists does not mean that ontological conclusions concerning which things we should (...)
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  14. Justin Clarke-Doane (2008). Multiple Reductions Revisited. Philosophia Mathematica 16 (2):244-255.
    Paul Benacerraf's argument from multiple reductions consists of a general argument against realism about the natural numbers (the view that numbers are objects), and a limited argument against reductionism about them (the view that numbers are identical with prima facie distinct entities). There is a widely recognized and severe difficulty with the former argument, but no comparably recognized such difficulty with the latter. Even so, reductionism in mathematics continues to thrive. In this paper I develop a difficulty for Benacerraf's argument (...)
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  15. Michael Detlefsen (1988). Fregean Hierarchies and Mathematical Explanation. International Studies in the Philosophy of Science 3 (1):97 – 116.
    There is a long line of thinkers in the philosophy of mathematics who have sought to base an account of proof on what might be called a 'metaphysical ordering' of the truths of mathematics. Use the term 'metaphysical' to describe these orderings is intended to call attention to the fact that they are regarded as objective and not subjective and that they are conceived primarily as orderings of truths and only secondarily as orderings of beliefs. -/- I describe and consider (...)
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  16. Steven Ericsson-Zenith (forthcoming). Explaining Experience In Nature: The Foundations Of Logic And Apprehension. Institute for Advanced Science & Engineering.
    At its core this book is concerned with logic and computation with respect to the mathematical characterization of sentient biophysical structure and its behavior. -/- Three related theories are presented: The first of these provides an explanation of how sentient individuals come to be in the world. The second describes how these individuals operate. And the third proposes a method for reasoning about the behavior of individuals in groups. -/- These theories are based upon a new explanation of experience in (...)
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  17. Don Fallis (2002). What Do Mathematicians Want? Probabilistic Proofs and the Epistemic Goals of Mathematicians. Logique Et Analyse 45.
    Several philosophers have used the framework of means/ends reasoning to explain the methodological choices made by scientists and mathematicians (see, e.g., Goldman 1999, Levi 1962, Maddy 1997). In particular, they have tried to identify the epistemic objectives of scientists and mathematicians that will explain these choices. In this paper, the framework of means/ends reasoning is used to study an important methodological choice made by mathematicians. Namely, mathematicians will only use deductive proofs to establish the truth of mathematical claims. In this (...)
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  18. James Franklin (2014). Aristotelian Realist Philosophy of Mathematics. Palgrave MacMillan.
    An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, are parts (...)
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  19. 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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  20. 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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  21. Jan Heylen (forthcoming). The Epistemic Significance of Numerals. Synthese:1-27.
    The central topic of this article is de re knowledge about natural numbers and its relation with names for numbers. It is held by several prominent philosophers that numerals are eligible for existential quantification in epistemic contexts, whereas other names for natural numbers are not. In other words, numerals are intimately linked with de re knowledge about natural numbers, whereas the other names for natural numbers are not. In this article I am looking for an explanation of this phenomenon. It (...)
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  22. Kate Hodesdon (2014). Mathematical Representation: Playing a Role. Philosophical Studies 168 (3):769-782.
    The primary justification for mathematical structuralism is its capacity to explain two observations about mathematical objects, typically natural numbers. Non-eliminative structuralism attributes these features to the particular ontology of mathematics. I argue that attributing the features to an ontology of structural objects conflicts with claims often made by structuralists to the effect that their structuralist theses are versions of Quine’s ontological relativity or Putnam’s internal realism. I describe and argue for an alternative explanation for these features which instead explains the (...)
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  23. Miguel Hoeltje, Benjamin Schnieder & Alex Steinberg (2013). Explanation by Induction? Synthese 190 (3):509-524.
    Philosophers of mathematics commonly distinguish between explanatory and non-explanatory proofs. An important subclass of mathematical proofs are proofs by induction. Are they explanatory? This paper addresses the question, based on general principles about explanation. First, a recent argument for a negative answer is discussed and rebutted. Second, a case is made for a qualified positive take on the issue.
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  24. John Kadvany (2003). Letters. Philosophia Mathematica 11 (3):364-364.
    A brief correction to a review of my book Imre Lakatos and the Guises of Reason published in Philosophia Mathematica, regarding the role of George Polya's notion of heuristic in Lakatos' Proofs and Refutations.
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  25. Robert Kraut (2001). Metaphysical Explanation and the Philosophy of Mathematics: Reflections on Jerrold Katz's Realistic Rationalism. Philosophia Mathematica 9 (2):154-183.
    Mathematical practice prompts theories about aprioricity, necessity, abstracta, and non-causal epistemic connections. But it is not clear what to count as the data: mathematical necessity or the appearance of mathematical necessity, abstractness or apparent abstractness, a prioricity or apparent aprioricity. Nor is it clear whether traditional metaphysical theories provide explanation or idle redescription. This paper suggests that abstract objects, rather than doing explanatory work, provide codifications of the data to be explained. It also suggests that traditional rivals—conceptualism, nominalism, realism—engage different (...)
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  26. Imre Lakatos (ed.) (1976). Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge University Press.
    Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre (...)
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  27. M. Lange (2010). What Are Mathematical Coincidences ? Mind 119 (474):307-340.
    Although all mathematical truths are necessary, mathematicians take certain combinations of mathematical truths to be ‘coincidental’, ‘accidental’, or ‘fortuitous’. The notion of a ‘ mathematical coincidence’ has so far failed to receive sufficient attention from philosophers. I argue that a mathematical coincidence is not merely an unforeseen or surprising mathematical result, and that being a misleading combination of mathematical facts is neither necessary nor sufficient for qualifying as a mathematical coincidence. I argue that although the components of a mathematical coincidence (...)
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  28. Marc Lange (2013). What Makes a Scientific Explanation Distinctively Mathematical? British Journal for the Philosophy of Science 64 (3):485-511.
    Certain scientific explanations of physical facts have recently been characterized as distinctively mathematical –that is, as mathematical in a different way from ordinary explanations that employ mathematics. This article identifies what it is that makes some scientific explanations distinctively mathematical and how such explanations work. These explanations are non-causal, but this does not mean that they fail to cite the explanandum’s causes, that they abstract away from detailed causal histories, or that they cite no natural laws. Rather, in these explanations, (...)
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  29. Marc Lange (2009). Dimensional Explanations. Noûs 43 (4):742-775.
  30. Marc Lange (2009). Why Proofs by Mathematical Induction Are Generally Not Explanatory. Analysis 69 (2):203-211.
    Philosophers who regard some mathematical proofs as explaining why theorems hold, and others as merely proving that they do hold, disagree sharply about the explanatory value of proofs by mathematical induction. I offer an argument that aims to resolve this conflict of intuitions without making any controversial presuppositions about what mathematical explanations would be.
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  31. Uri D. Leibowitz & Neil Sinclair (eds.) (2016). Explanation in Ethics and Mathematics: Debunking and Dispensability. Oxford University Press Uk.
    How far should our realism extend? For many years philosophers of mathematics and philosophers of ethics have worked independently to address the question of how best to understand the entities apparently referred to by mathematical and ethical talk. But the similarities between their endeavours are not often emphasised. This book provides that emphasis. In particular, it focuses on two types of argumentative strategies deployed in both areas. The first aims to put pressure on realism by emphasising the redundancy of mathematical (...)
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  32. Gianluca Longa (2016). Review of Daniele Molinini, Che cos'è una spiegazione matematica. [REVIEW] Lo Sguardo. Rivista di Filosofia 20:325-327.
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  33. Paolo Mancosu (2011). Explanation in Mathematics. 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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  34. Paolo Mancosu (2008). Mathematical Explanation: Why It Matters. In The Philosophy of Mathematical Practice. OUP Oxford 134--149.
  35. Paolo Mancosu (2001). Mathematical Explanation: Problems and Prospects. Topoi 20 (1):97-117.
  36. Paolo Mancosu & Johannes Hafner (2008). Unification and Explanation: A Case Study From Real Algebraic Geometry. In The Philosophy of Mathematical Practice. OUP Oxford 151--178.
  37. 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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  38. Daniele Molinini (2014). Che Cos'è una Spiegazione Matematica. Carocci.
    Può la matematica spiegare il mondo che ci circonda, o addirittura sé stessa? Possono i numeri, e più in generale le teorie matematiche, dirci perché alcuni fenomeni naturali e sociali avvengono o perché alcuni risultati matematici siano da considerarsi veri? Che cosa si intende esattamente per spiegazione matematica? Attraverso numerosi esempi, l’autore offre una risposta a queste domande e illustra le principali posizioni filosofiche elaborate per la nozione di spiegazione matematica, nozione che è alla base di dibattiti riguardanti aree diverse (...)
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  39. Joe Morrison (2012). Evidential Holism and Indispensability Arguments. Erkenntnis 76 (2):263-278.
    The indispensability argument is a method for showing that abstract mathematical objects exist. Various versions of this argument have been proposed. Lately, commentators seem to have agreed that a holistic indispensability argument will not work, and that an explanatory indispensability argument is the best candidate. In this paper I argue that the dominant reasons for rejecting the holistic indispensability argument are mistaken. This is largely due to an overestimation of the consequences that follow from evidential holism. Nevertheless, the holistic indispensability (...)
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  40. Fabrice Pataut (2016). Comments on “Parsimony and Inference to the Best Mathematical Explanation”. Synthese 193 (2):351-363.
    The author of “Parsimony and inference to the best mathematical explanation” argues for platonism by way of an enhanced indispensability argument based on an inference to yet better mathematical optimization explanations in the natural sciences. Since such explanations yield beneficial trade-offs between stronger mathematical existential claims and fewer concrete ontological commitments than those involved in merely good mathematical explanations, one must countenance the mathematical objects that play a theoretical role in them via an application of the relevant mathematical results. The (...)
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  41. Christopher Pincock (2015). The Unsolvability of The Quintic: A Case Study in Abstract Mathematical Explanation. Philosophers' Imprint 15 (3).
    This paper identifies one way that a mathematical proof can be more explanatory than another proof. This is by invoking a more abstract kind of entity than the topic of the theorem. These abstract mathematical explanations are identified via an investigation of a canonical instance of modern mathematics: the Galois theory proof that there is no general solution in radicals for fifth-degree polynomial equations. I claim that abstract explanations are best seen as describing a special sort of dependence relation between (...)
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  42. Christopher Pincock & Paolo Mancosu, Mathemetical Explanation. Oxford Bibliographies in Philosophy.
  43. Christopher Pincock & Paolo Mancosu, Mathemetical Explanation. Oxford Bibliographies in Philosophy.
  44. Tim Räz, Mathematical Explanations in Euler’s Königsberg.
    I examine Leonhard Euler’s original solution to the Königsberg bridges problem. Euler’s solution can be interpreted as both an explanation within mathematics and a scientific explanation using mathematics. At the level of pure mathematics, Euler proposes three different solutions to the Königsberg problem. The differences between these solutions can be fruitfully explicated in terms of explanatory power. In the scientific version of the explanation, mathematics aids by representing the explanatorily salient causal structure of Königsberg. Based on this analysis, I defend (...)
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  45. Michael D. Resnik & David Kushner (1987). Explanation, Independence and Realism in Mathematics. British Journal for the Philosophy of Science 38 (2):141-158.
  46. Juha Saatsi (2012). Mathematics and Program Explanations. Australasian Journal of Philosophy 90 (3):579-584.
    Aidan Lyon has recently argued that some mathematical explanations of empirical facts can be understood as program explanations. I present three objections to his argument.
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  47. Mark Steiner (1978). Mathematical Explanation. Philosophical Studies 34 (2):135 - 151.
  48. Mark Steiner (1978). Mathematics, Explanation, and Scientific Knowledge. Noûs 12 (1):17-28.
  49. Jean Paul Van Bendegem (2006). Review of P. Mancosu, K. F. Jørgensen, and S. A. Pedersen (Eds.), Visualization, Explanation and Reasoning Styles in Mathematics. [REVIEW] Philosophia Mathematica 14 (3):378-391.
  50. Mark Zelcer (2013). Against Mathematical Explanation. Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 44 (1):173-192.
    Lately, philosophers of mathematics have been exploring the notion of mathematical explanation within mathematics. This project is supposed to be analogous to the search for the correct analysis of scientific explanation. I argue here that given the way philosophers have been using “ explanation,” the term is not applicable to mathematics as it is in science.
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