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Summary

Mathematical explanations are explanations in which mathematics plays a fundamental role. The expression ‘mathematical explanation’ (ME) has two distinct, although connected, meanings: in relation to pure mathematics ME denotes proofs that are able not only to demonstrate the truth of a given mathematical statement, but also to explain why the statement is true, whereas in connection with empirical sciences ME refers to explanations of non-mathematical facts (physical, biological, social, psychological) justified by recourse to mathematics. 

Although the concept of ME has been the subject of analysis at least since Aristotle’s distinction between apodeixis tou oti and apodeixis tou dioti (Post. An. I.13), and has been dealt with a few times over the course of the development of Western thought (e.g. Descartes, Newton, and Bolzano), it is only since the 1970s that an intense philosophical debate has sprung up regarding the nature of ME. This debate, linked to the gradual diffusion of Quinean epistemology (Steiner 1978) and the development of the anti-foundationalist philosophy of mathematics (the so-called ‘maverick’ tradition, Cellucci 2008), centers on the following questions: Do mathematical explanations exist? If mathematical explanations exist, can they be reduced to a single model or are they heterogeneous among themselves? What implications does the comprehension of the concept of mathematical explanation have for some of the most important problems of the contemporary philosophy of science (e.g. indispensability arguments, inference to the best explanation, and the theory of scientific explanation)? 

Key works

The key works about mathematical explanation within mathematics are Steiner 1978 (for criticisms of the model proposed by Steiner see Resnik & Kushner 1987, Weber & Verhoeven 2002, and Mancosu & JØrgensen 2006), Kitcher 1983, and Kitcher 1989 (a careful analysis of the limitations of the model proposed by Kitcher can be found in Mancosu & Hafner 2008). Regarding the notion of mathematical explanation in natural sciences, see Batterman 2002, Baker 2005, Pincock 2007, and Baker 2009.

Introductions For  general overviews on the subject, see Mancosu 2011Pincock & Mancosu 2012, and Molinini 2014.
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  1. Holly Andersen (forthcoming). Complements, Not Competitors: Causal and Mathematical Explanations. British Journal for the Philosophy of Science.
    A finer-grained delineation of a given explanandum reveals a nexus of closely related causal and non- causal explanations, complementing one another in ways that yield further explanatory traction on the phenomenon in question. By taking a narrower construal of what counts as a causal explanation, a new class of distinctively mathematical explanations pops into focus; Lange’s characterization of distinctively mathematical explanations can be extended to cover these. This new class of distinctively mathematical explanations is illustrated with the Lotka-Volterra equations. There (...)
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  2. Alan Baker (2012). Science-Driven Mathematical Explanation. Mind 121 (482):243-267.
    Philosophers of mathematics have become increasingly interested in the explanatory role of mathematics in empirical science, in the context of new versions of the Quinean ‘Indispensability Argument’ which employ inference to the best explanation for the existence of abstract mathematical objects. However, little attention has been paid to analysing the nature of the explanatory relation involved in these mathematical explanations in science (MES). In this paper, I attack the only articulated account of MES in the literature (an account sketched by (...)
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  3. Alan Baker (2010). Mathematical Induction and Explanation. Analysis 70 (4):681-689.
  4. Alan Baker (2009). Mathematical Explanation in Science. British Journal for the Philosophy of Science 60 (3):611-633.
    Does mathematics ever play an explanatory role in science? If so then this opens the way for scientific realists to argue for the existence of mathematical entities using inference to the best explanation. Elsewhere I have argued, using a case study involving the prime-numbered life cycles of periodical cicadas, that there are examples of indispensable mathematical explanations of purely physical phenomena. In this paper I respond to objections to this claim that have been made by various philosophers, and I discuss (...)
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  5. 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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  6. 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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  7. 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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  8. Sorin 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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  9. 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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  10. Sam Baron & Mark Colyvan (forthcoming). Time Enough for Explanation. Journal of Philosophy.
  11. John D. Barrow (2004). Mathematical Explanation. In John Cornwell (ed.), Explanations: Styles of Explanation in Science. Oxford University Press 81--109.
  12. Robert Batterman (2010). On the Explanatory Role of Mathematics in Empirical Science. British Journal for the Philosophy of Science 61 (1):1-25.
    This paper examines contemporary attempts to explicate the explanatory role of mathematics in the physical sciences. Most such approaches involve developing so-called mapping accounts of the relationships between the physical world and mathematical structures. The paper argues that the use of idealizations in physical theorizing poses serious difficulties for such mapping accounts. A new approach to the applicability of mathematics is proposed.
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  13. Robert W. Batterman (2002). The Devil in the Details: Asymptotic Reasoning in Explanation, Reduction, and Emergence. Oxford University Press.
    Robert Batterman examines a form of scientific reasoning called asymptotic reasoning, arguing that it has important consequences for our understanding of the scientific process as a whole. He maintains that asymptotic reasoning is essential for explaining what physicists call universal behavior. With clarity and rigor, he simplifies complex questions about universal behavior, demonstrating a profound understanding of the underlying structures that ground them. This book introduces a valuable new method that is certain to fill explanatory gaps across disciplines.
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  14. Ingo Brigandt (2013). Explanation in Biology: Reduction, Pluralism, and Explanatory Aims. Science and Education 22 (1):69-91.
    This essay analyzes and develops recent views about explanation in biology. Philosophers of biology have parted with the received deductive-nomological model of scientific explanation primarily by attempting to capture actual biological theorizing and practice. This includes an endorsement of different kinds of explanation (e.g., mathematical and causal-mechanistic), a joint study of discovery and explanation, and an abandonment of models of theory reduction in favor of accounts of explanatory reduction. Of particular current interest are philosophical accounts of complex explanations that appeal (...)
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  15. Ingo Brigandt (2013). Systems Biology and the Integration of Mechanistic Explanation and Mathematical Explanation. Studies in History and Philosophy of Biological and Biomedical Sciences 44 (4):477-492.
    The paper discusses how systems biology is working toward complex accounts that integrate explanation in terms of mechanisms and explanation by mathematical models—which some philosophers have viewed as rival models of explanation. Systems biology is an integrative approach, and it strongly relies on mathematical modeling. Philosophical accounts of mechanisms capture integrative in the sense of multilevel and multifield explanations, yet accounts of mechanistic explanation (as the analysis of a whole in terms of its structural parts and their qualitative (...)
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  16. Otávio Bueno & Steven French (2012). Can Mathematics Explain Physical Phenomena? British Journal for the Philosophy of Science 63 (1):85-113.
    Batterman raises a number of concerns for the inferential conception of the applicability of mathematics advocated by Bueno and Colyvan. Here, we distinguish the various concerns, and indicate how they can be assuaged by paying attention to the nature of the mappings involved and emphasizing the significance of interpretation in this context. We also indicate how this conception can accommodate the examples that Batterman draws upon in his critique. Our conclusion is that ‘asymptotic reasoning’ can be straightforwardly accommodated within the (...)
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  17. 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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  18. 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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  19. Carlo Cellucci (2008). The Nature of Mathematical Explanation. Studies in History and Philosophy of Science 39 (2):202-210.
    Although in the past three decades interest in mathematical explanation revived, recent literature on the subject seems to neglect the strict connection between explanation and discovery. In this paper I sketch an alternative approach that takes such connection into account. My approach is a revised version of one originally considered by Descartes. The main difference is that my approach is in terms of the analytic method, which is a method of discovery prior to axiomatized mathematics, whereas Descartes’s approach is in (...)
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  20. Christopher Clarke (forthcoming). Multi-Level Selection and the Explanatory Value of Mathematical Decompositions. British Journal for the Philosophy of Science:axv008.
    Do multi-level selection explanations of the evolution of social traits deepen the understanding provided by single-level explanations? Central to the former is a mathematical theorem, the multi-level Price decomposition. I build a framework through which to understand the explanatory role of such non-empirical decompositions in scientific practice. Applying this general framework to the present case places two tasks on the agenda. The first task is to distinguish the various ways of suppressing within-collective variation in fitness, and moreover to evaluate their (...)
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  21. Chris Daly & Simon Langford (2009). Mathematical Explanation and Indispensability Arguments. Philosophical Quarterly 59 (237):641-658.
    We defend Joseph Melia's thesis that the role of mathematics in scientific theory is to 'index' quantities, and that even if mathematics is indispensable to scientific explanations of concrete phenomena, it does not explain any of those phenomena. This thesis is defended against objections by Mark Colyvan and Alan Baker.
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  22. 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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  23. Michel Dufour, Argument and Explanation in Mathematics.
    Are there arguments in mathematics? Are there explanations in mathematics? Are there any connections between argument, proof and explanation? Highly controversial answers and arguments are reviewed. The main point is that in the case of a mathematical proof, the pragmatic criterion used to make a distinction between argument and explanation is likely to be insufficient for you may grant the conclusion of a proof but keep on thinking that the proof is not explanatory.
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  24. Laura Felline (forthcoming). Mechanisms Meet Structural Explanation. Synthese:1-16.
    This paper investigates the relationship between Structural Explanation and the New Mechanistic account of explanation. The aim of this paper is twofold: firstly, to argue that some phenomena in the domain of fundamental physics, although mechanically brute, are structurally explained; and secondly, by elaborating on the contrast between SE and ME, to better clarify some features of SE. Finally, this paper will argue that, notwithstanding their apparently antithetical character, SE and ME can be reconciled within a unified account of general (...)
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  25. Laura Felline (2010). Review of R. Batterman: The Devil in the Details: Asymptotic Reasoning in Explanation, Reduction and Emergence. [REVIEW] APhEx – Portale Italiano di Filosofia Analitica 2:99-109.
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  26. Joachim Frans & Erik Weber (2014). Mechanistic Explanation and Explanatory Proofs in Mathematics. Philosophia Mathematica 22 (2):231-248.
    Although there is a consensus among philosophers of mathematics and mathematicians that mathematical explanations exist, only a few authors have proposed accounts of explanation in mathematics. These accounts fit into the unificationist or top-down approach to explanation. We argue that these models can be complemented by a bottom-up approach to explanation in mathematics. We introduce the mechanistic model of explanation in science and discuss the possibility of using this model in mathematics, arguing that using it does not presuppose a Platonist (...)
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  27. Michèle Friend (2015). On the Epistemological Significance of the Hungarian Project. Synthese 192 (7):2035-2051.
    There are three elements in this paper. One is what we shall call ‘the Hungarian project’. This is the collected work of Andréka, Madarász, Németi, Székely and others. The second is Molinini’s philosophical work on the nature of mathematical explanations in science. The third is my pluralist approach to mathematics. The theses of this paper are that the Hungarian project gives genuine mathematical explanations for physical phenomena. A pluralist account of mathematical explanation can help us with appreciating the significance of (...)
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  28. Michèle Friend & Daniele Molinini (forthcoming). Using Mathematics to Explain a Scientific Theory. Philosophia Mathematica:nkv022.
    We answer three questions: 1. Can we give a wholly mathematical explanation of a physical phenomenon? 2. Can we give a wholly mathematical explanation for a whole physical theory? 3. What is gained or lost in giving a wholly, or partially, mathematical explanation of a phenomenon or a scientific theory? To answer these questions we look at a project developed by Hajnal Andréka, Judit Madarász, István Németi and Gergely Székely. They, together with collaborators, present special relativity theory in a three-sorted (...)
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  29. Orna Harari (2008). Proclus' Account of Explanatory Demonstrations in Mathematics and its Context. Archiv für Geschichte der Philosophie 90 (2):137-164.
    I examine the question why in Proclus' view genetic processes provide demonstrative explanations, in light of the interpretation of Aristotle's theory of demonstration in late antiquity. I show that in this interpretation mathematics is not an explanatory science in the strict sense because its objects, being immaterial, do not admit causal explanation. Placing Proclus' account of demonstrative explanation in this context, I argue that this account is aimed at answering the question whether mathematical proofs provide causal explanation as opposed to (...)
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  30. 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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  31. 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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  32. 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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  33. Marc Lange (2014). Aspects of Mathematical Explanation: Symmetry, Unity, and Salience. Philosophical Review 123 (4):485-531.
    Unlike explanation in science, explanation in mathematics has received relatively scant attention from philosophers. Whereas there are canonical examples of scientific explanations, there are few examples that have become widely accepted as exhibiting the distinction between mathematical proofs that explain why some mathematical theorem holds and proofs that merely prove that the theorem holds without revealing the reason why it holds. This essay offers some examples of proofs that mathematicians have considered explanatory, and it argues that these examples suggest a (...)
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  34. 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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  35. Marc Lange (2012). Abstraction and Depth in Scientific Explanation. [REVIEW] Philosophy and Phenomenological Research 84 (2):483-491.
  36. Marc Lange (2009). Dimensional Explanations. Noûs 43 (4):742-775.
  37. 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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  38. Peter Lipton (2001). What Good Is an Explanation? In G. Hon & S. Rakover (eds.), Explanation. Springer Netherlands 43-59.
    We are addicted to explanation, constantly asking and answering why-questions. But what does an explanation give us? I will consider some of the possible goods, intrinsic and instrumental, that explanations provide. The name for the intrinsic good of explanation is `understanding', but what is this? In the first part of this paper I will canvass various conceptions of understanding, according to which explanations provide reasons for belief, make familiar, unify, show to be necessary, or give causes. Three general features of (...)
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  39. Michael Liston (2013). Christopher Pincock. Mathematics and Scientific Representation. Oxford University Press, 2012. ISBN 978-0-19-975710-7. Pp. Xv + 330. [REVIEW] Philosophia Mathematica 21 (3):371-385.
  40. 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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  41. Aidan Lyon & Mark Colyvan (2008). The Explanatory Power of Phase Spaces. Philosophia Mathematica 16 (2):227-243.
    David Malament argued that Hartry Field's nominalisation program is unlikely to be able to deal with non-space-time theories such as phase-space theories. We give a specific example of such a phase-space theory and argue that this presentation of the theory delivers explanations that are not available in the classical presentation of the theory. This suggests that even if phase-space theories can be nominalised, the resulting theory will not have the explanatory power of the original. Phase-space theories thus raise problems for (...)
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  42. 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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  43. Paolo Mancosu (2008). Mathematical Explanation: Why It Matters. In The Philosophy of Mathematical Practice. OUP Oxford 134--149.
  44. Paolo Mancosu (ed.) (2008/2011). The Philosophy of Mathematical Practice. OUP Oxford.
    There is an urgent need in philosophy of mathematics for new approaches which pay closer attention to mathematical practice. This book will blaze the trail: it offers philosophical analyses of important characteristics of contemporary mathematics and of many aspects of mathematical activity which escape purely formal logical treatment.
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  45. Paolo Mancosu (2001). Mathematical Explanation: Problems and Prospects. Topoi 20 (1):97-117.
  46. Paolo Mancosu (1999). Bolzano and Cournot on Mathematical Explanation/Bolzano Et Cournot À Propos de l'Explication Mathématique. Revue d'Histoire des Sciences 52 (3):429-456.
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  47. 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.
  48. Paolo Mancosu & Klaus JØrgensen (2006). Paolo Mancosu, Klaus Frovin JØrgensen, and Stig Andur Pedersen, Eds. Visualization, Explanation and Reasoning Stryles in Mathematics. Synthese Library, Vol. 327. Dordrecht: Springer, 2005. ISBN 1-4020-3334-6 ; 1-4020-3335-4 . Pp. X + 300. [REVIEW] Philosophia Mathematica 14 (2):265.
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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. Ernan McMullin (2011). Kepler: Moving the Earth. Hopos: The Journal of the International Society for the History of Philosophy of Science 1 (1):3-22.
    The discrepancy between the Aristotelian and the Ptolemaic astronomies led many medievals to regard the latter (and mathematical astronomy generally) as no more than a calculational device. This was the challenge that Copernicus and Kepler had to meet: How was one to show that a mathematically expressed astronomy could indicate that the earth really moves? Copernicus pointed to features of the planetary motions that he could explain but that Ptolemy could not. Kepler went much further. His account of the planetary (...)
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