About this topic

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 2001, Baker 2005, Pincock 2007, and Baker 2009.

Introductions For  general overviews on the subject, see Mancosu 2011Pincock & Mancosu 2012, and Molinini 2014.
Related categories

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  1. Abstract Versus Causal Explanations?Reutlinger Alexander & Andersen Holly - 2016 - International Studies in the Philosophy of Science 30 (2):129-146.
    In the recent literature on causal and non-causal scientific explanations, there is an intuitive assumption according to which an explanation is non-causal by virtue of being abstract. In this context, to be ‘abstract’ means that the explanans in question leaves out many or almost all causal microphysical details of the target system. After motivating this assumption, we argue that the abstractness assumption, in placing the abstract and the causal character of an explanation in tension, is misguided in ways that are (...)
  2. Explanation Beyond Causation: Philosophical Perspectives on Non-Causal Explanations.Reutlinger Alexander & Juha Saatsi (eds.) - forthcoming - Oxford University Press.
    Explanations are important to us in many contexts: in science, mathematics, philosophy, and also in everyday and juridical contexts. But what is an explanation? In the philosophical study of explanation, there is a long-standing, in uential tradition that links explanation intimately to causation: we often explain by providing accurate information about the causes of the phenomenon to be explained. Such causal accounts have been the received view of the nature of explanation, particularly in philosophy of science, since the 1980s. However, (...)
  3. Complements, Not Competitors: Causal and Mathematical Explanations.Holly Andersen - 2017 - British Journal for the Philosophy of Science:axw023.
    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 (...)
  4. Science-Driven Mathematical Explanation.Alan Baker - 2012 - 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 (...)
  5. Mathematical Induction and Explanation.Alan Baker - 2010 - Analysis 70 (4):681-689.
  6. Mathematical Explanation in Science.Alan Baker - 2009 - 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 (...)
  7. Are There Genuine Mathematical Explanations of Physical Phenomena?Alan Baker - 2005 - 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 (...)
  8. Indexing and Mathematical Explanation.Alan Baker & Mark Colyvan - 2011 - 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 (...)
  9. Indispensability and Explanation.Sorin Bangu - 2013 - 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 (...)
  10. Inference to the Best Explanation and Mathematical Realism.Sorin Bangu - 2008 - 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.
  11. Mathematical Explanation and Epistemology: Please Mind the Gap.Sam Baron - 2016 - 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 (...)
  12. Time Enough for Explanation.Sam Baron & Mark Colyvan - 2016 - Journal of Philosophy 113 (2):61-88.
    The present paper advances an analogy between cases of extra-mathematical explanation and cases of what might be termed ‘extra-logical explanation’: the explanation of a physical fact by a logical fact. A particular case of extra-logical explanation is identified that arises in the philosophical literature on time travel. This instance of extra-logical explanation is subsequently shown to be of a piece with cases of extra-mathematical explanation. Using this analogy, we argue extra-mathematical explanation is part of a broader class of non-causal explanation. (...)
  13. Mathematical Explanation.John D. Barrow - 2004 - In John Cornwell (ed.), Explanations: Styles of Explanation in Science. Oxford University Press. pp. 81--109.
  14. On the Explanatory Role of Mathematics in Empirical Science.Robert Batterman - 2010 - 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.
  15. The Devil in the Details: Asymptotic Reasoning in Explanation, Reduction, and Emergence.Robert W. Batterman - 2001 - 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.
  16. The Applicability of Mathematics to Physical Modality.Nora Berenstain - 2017 - Synthese 194 (9):3361-3377.
    This paper argues that scientific realism commits us to a metaphysical determination relation between the mathematical entities that are indispensible to scientific explanation and the modal structure of the empirical phenomena those entities explain. The argument presupposes that scientific realism commits us to the indispensability argument. The viewpresented here is that the indispensability of mathematics commits us not only to the existence of mathematical structures and entities but to a metaphysical determination relation between those entities and the modal structure of (...)
  17. Explanation of Molecular Processes Without Tracking Mechanism Operation.Ingo Brigandt - 2018 - Philosophy of Science 85.
    Philosophical discussions of systems biology have enriched the notion of mechanistic explanation by pointing to the role of mathematical modeling. However, such accounts still focus on explanation in terms of tracking a mechanism's operation across time (by means of mental or computational simulation). My contention is that there are explanations of molecular systems where the explanatory understanding does not consist in tracking a mechanism's operation and productive continuity. I make this case by a discussion of bifurcation analysis in dynamical systems, (...)
  18. Systems Biology and the Integration of Mechanistic Explanation and Mathematical Explanation.Ingo Brigandt - 2013 - 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 have failed to address how a mathematical model could contribute to such explanations. I discuss how mathematical (...)
  19. Explanation in Biology: Reduction, Pluralism, and Explanatory Aims.Ingo Brigandt - 2011 - Science & 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 (...)
  20. Can Mathematics Explain Physical Phenomena?Otávio Bueno & Steven French - 2012 - 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 (...)
  21. Scientific Realism and the Indispensability Argument for Mathematical Realism: A Marriage Made in Hell.Jacob Busch - 2011 - 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. (...)
  22. Should Scientific Realists Be Platonists?Jacob Busch & Joe Morrison - 2016 - 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 (...)
  23. Problemas para a Explicação Matemática.Eduardo Castro - 2017 - Revista Portuguesa de Filosofia 73 (3-4):1437-1462.
    Mathematical proofs aim to establish the truth of mathematical propositions by means of logical rules. Some recent literature in philosophy of mathematics alleges that some mathematical proofs also reveal why the proved mathematical propositions are true. These mathematical proofs are called explanatory mathematical proofs. In this paper, I present and discuss some salient problems around mathematical explanation: the existence problem, the normative problem, the explanandum problems of truth value and psychological value, the logical structure problem, the regress problem and the (...)
  24. The Nature of Mathematical Explanation.Carlo Cellucci - 2008 - Studies in History and Philosophy of Science Part A 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 (...)
  25. Multi-Level Selection and the Explanatory Value of Mathematical Decompositions.Christopher Clarke - 2016 - British Journal for the Philosophy of Science 67 (4):1025-1055.
    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 (...)
  26. The Directionality of Distinctively Mathematical Explanations.Carl F. Craver & Mark Povich - 2017 - Studies in History and Philosophy of Science Part A 63:31-38.
    In “What Makes a Scientific Explanation Distinctively Mathematical?” (2013b), Lange uses several compelling examples to argue that certain explanations for natural phenomena appeal primarily to mathematical, rather than natural, facts. In such explanations, the core explanatory facts are modally stronger than facts about causation, regularity, and other natural relations. We show that Lange's account of distinctively mathematical explanation is flawed in that it fails to account for the implicit directionality in each of his examples. This inadequacy is remediable in each (...)
  27. Arithmetic, Set Theory, Reduction and Explanation.William D'Alessandro - forthcoming - Synthese:1-31.
    Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important particular case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In defense (...)
  28. Mathematical Explanation and Indispensability Arguments.Chris Daly & Simon Langford - 2009 - 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.
  29. Fregean Hierarchies and Mathematical Explanation.Michael Detlefsen - 1988 - 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 (...)
  30. Argument and Explanation in Mathematics.Michel Dufour - 2013 - In Dima Mohammed and Marcin Lewiński (ed.), Virtues of Argumentation. Proceedings of the 10th International Conference of the Ontario Society for the Study of Argumentation (OSSA), 22-26 May 2013. pp. pp. 1-14..
    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.
  31. Inertia, the Communication of Motion, and Kant's Third Law of Mechanics.Howard Duncan - 1984 - Philosophy of Science 51 (1):93-119.
    In Kant's Metaphysical Foundations of Natural Science are found a dynamist reduction of matter and an account of the communication of motion by impact. One would expect to find an analysis of the causal mechanism involved in the communication of motion between bodies given in terms of the fundamental dynamical nature of bodies. However, Kant's analysis, as given in the discussion of his third law of mechanics (an action-reaction law) is purely kinematical, invoking no causal mechanisms at all, let alone (...)
  32. Mechanisms Meet Structural Explanation.Laura Felline - 2018 - Synthese 195 (1):99-114.
    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 mechanistic explanation 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 (...)
  33. Review of R. Batterman: The Devil in the Details: Asymptotic Reasoning in Explanation, Reduction and Emergence. [REVIEW]Laura Felline - 2010 - APhEx – Portale Italiano di Filosofia Analitica 2:99-109.
  34. Mechanistic Explanation and Explanatory Proofs in Mathematics.Joachim Frans & Erik Weber - 2014 - 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 (...)
  35. Symmetries and Explanatory Dependencies in Physics.Steven French & Juha Saatsi - 2018 - In Alexander Reutlinger & Juha Saatsi (eds.), Explanation Beyond Causation: Philosophical Perspectives on Non-Causal Explanations. Oxford: Oxford University Press. pp. 185-205.
    Many important explanations in physics are based on ideas and assumptions about symmetries, but little has been said about the nature of such explanations. This chapter aims to fill this lacuna, arguing that various symmetry explanations can be naturally captured in the spirit of the counterfactual-dependence account of Woodward, liberalized from its causal trappings. From the perspective of this account symmetries explain by providing modal information about an explanatory dependence, by showing how the explanandum would have been different, had the (...)
  36. On the Epistemological Significance of the Hungarian Project.Michèle Friend - 2015 - 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 (...)
  37. Using Mathematics to Explain a Scientific Theory.Michèle Friend & Daniele Molinini - 2016 - Philosophia Mathematica 24 (2):185-213.
    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 (...)
  38. Proclus' Account of Explanatory Demonstrations in Mathematics and its Context.Orna Harari - 2008 - 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 (...)
  39. Tuples All the Way Down?Simon Thomas Hewitt - 2018 - Thought: A Journal of Philosophy 7 (3):161-169.
  40. Mathematical Representation: Playing a Role.Kate Hodesdon - 2014 - 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 (...)
  41. Explanation by Induction?Miguel Hoeltje, Benjamin Schnieder & Alex Steinberg - 2013 - 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.
  42. Explanatory Abstractions.Lina Jansson & Juha Saatsi - 2016 - British Journal for the Philosophy of Science:axx016.
    A number of philosophers have recently suggested that some abstract, plausibly non-causal and/or mathematical, explanations explain in a way that is radically dif- ferent from the way causal explanation explain. Namely, while causal explanations explain by providing information about causal dependence, allegedly some abstract explanations explain in a way tied to the independence of the explanandum from the microdetails, or causal laws, for example. We oppose this recent trend to regard abstractions as explanatory in some sui generis way, and argue (...)
  43. The Varieties of Mathematical Explanation.Hafner Johannes & Paolo Mancosu - 2005 - In Paolo Mancosu (ed.), Visualization, Explanation and Reasoning Styles in Mathematics. Dordrecht: Springer. pp. 215-250.
  44. Was Regression to the Mean Really the Solution to Darwin’s Problem with Heredity? [REVIEW]Adam Krashniak & Ehud Lamm - 2017 - Biology and Philosophy (5):1-10.
    Statistical reasoning is an integral part of modern scientific practice. In The Seven Pillars of Statistical Wisdom Stephen Stigler presents seven core ideas, or pillars, of statistical thinking and the historical developments of each of these pillars, many of which were concurrent with developments in biology. Here we focus on Stigler’s fifth pillar, regression, and his discussion of how regression to the mean came to be thought of as a solution to a challenge for the theory of natural selection. Stigler (...)
  45. Aspects of Mathematical Explanation: Symmetry, Unity, and Salience.Marc Lange - 2014 - 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 (...)
  46. What Makes a Scientific Explanation Distinctively Mathematical?Marc Lange - 2013 - 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, (...)
  47. Abstraction and Depth in Scientific Explanation. [REVIEW]Marc Lange - 2012 - Philosophy and Phenomenological Research 84 (2):483-491.
  48. Dimensional Explanations.Marc Lange - 2009 - Noûs 43 (4):742-775.
  49. Why Proofs by Mathematical Induction Are Generally Not Explanatory.Marc Lange - 2009 - 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.
  50. What Good Is an Explanation?Peter Lipton - 2001 - In G. Hon & S. Rakover (eds.), Explanation. Springer Verlag. pp. 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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