Results for 'Mathematical explanation'

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  1.  80
    Mathematical Explanations of Physical Phenomena.Sorin Bangu - 2021 - Australasian Journal of Philosophy 99 (4):669-682.
    Can there be mathematical explanations of physical phenomena? In this paper, I suggest an affirmative answer to this question. I outline a strategy to reconstruct several typical examples of such explanations, and I show that they fit a common model. The model reveals that the role of mathematics is explicatory. Isolating this role may help to re-focus the current debate on the more specific question as to whether this explicatory role is, as proposed here, also an explanatory one.
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  2. Mathematical Explanation by Law.Sam Baron - 2019 - British Journal for the Philosophy of Science 70 (3):683-717.
    Call an explanation in which a non-mathematical fact is explained—in part or in whole—by mathematical facts: an extra-mathematical explanation. Such explanations have attracted a great deal of interest recently in arguments over mathematical realism. In this article, a theory of extra-mathematical explanation is developed. The theory is modelled on a deductive-nomological theory of scientific explanation. A basic DN account of extra-mathematical explanation is proposed and then redeveloped in the light (...)
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  3.  34
    Mathematical Explanations of Physical Phenomena.Sorin Bangu - 2021 - Australasian Journal of Philosophy 99 (4):669-682.
    ABSTRACT Can there be mathematical explanations of physical phenomena? In this paper, I suggest an affirmative answer to this question. I outline a strategy to reconstruct several typical examples of such explanations, and I show that they fit a common model. The model reveals that the role of mathematics is explicatory. Isolating this role may help to re-focus the current debate on the more specific question as to whether this explicatory role is, as proposed here, also an explanatory one.
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  4. 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, (...)
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  5. Against Mathematical Explanation.Mark Zelcer - 2013 - 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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  6. Mathematical Explanation beyond Explanatory Proof.William D’Alessandro - 2017 - British Journal for the Philosophy of Science 71 (2):581-603.
    Much recent work on mathematical explanation has presupposed that the phenomenon involves explanatory proofs in an essential way. I argue that this view, ‘proof chauvinism’, is false. I then look in some detail at the explanation of the solvability of polynomial equations provided by Galois theory, which has often been thought to revolve around an explanatory proof. The article concludes with some general worries about the effects of chauvinism on the theory of mathematical explanation. 1Introduction (...)
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  7. Mathematical Explanation: A Contextual Approach.Sven Delarivière, Joachim Frans & Bart Van Kerkhove - 2017 - Journal of Indian Council of Philosophical Research 34 (2):309-329.
    PurposeIn this article, we aim to present and defend a contextual approach to mathematical explanation.MethodTo do this, we introduce an epistemic reading of mathematical explanation.ResultsThe epistemic reading not only clarifies the link between mathematical explanation and mathematical understanding, but also allows us to explicate some contextual factors governing explanation. We then show how several accounts of mathematical explanation can be read in this approach.ConclusionThe contextual approach defended here clears up the (...)
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  8. 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.
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  9. Explaining Mathematical Explanation.Sam Baron - 2016 - Philosophical Quarterly 66 (264):458-480.
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  10. Are mathematical explanations causal explanations in disguise?A. Jha, Douglas Campbell, Clemency Montelle & Phillip L. Wilson - 2024 - Philosophy of Science (NA):1-19.
    There is a major debate as to whether there are non-causal mathematical explanations of physical facts that show how the facts under question arise from a degree of mathematical necessity considered stronger than that of contingent causal laws. We focus on Marc Lange’s account of distinctively mathematical explanations to argue that purported mathematical explanations are essentially causal explanations in disguise and are no different from ordinary applications of mathematics. This is because these explanations work not by (...)
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  11. Platonism and Intra-mathematical Explanation.Sam Baron - forthcoming - Philosophical Quarterly.
    I introduce an argument for Platonism based on intra-mathematical explanation: the explanation of one mathematical fact by another. The argument is important for two reasons. First, if the argument succeeds then it provides a basis for Platonism that does not proceed via standard indispensability considerations. Second, if the argument fails it can only do so for one of three reasons: either because there are no intra-mathematical explanations, or because not all explanations are backed by dependence (...)
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  12. Magicicada, Mathematical Explanation and Mathematical Realism.Davide Rizza - 2011 - Erkenntnis 74 (1):101-114.
    Baker claims to provide an example of mathematical explanation of an empirical phenomenon which leads to ontological commitment to mathematical objects. This is meant to show that the positing of mathematical entities is necessary for satisfactory scientific explanations and thus that the application of mathematics to science can be used, at least in some cases, to support mathematical realism. In this paper I show that the example of explanation Baker considers can actually be given (...)
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  13.  74
    Do mathematical explanations have instrumental value?Rebecca Lea Morris - 2019 - Synthese (2):1-20.
    Scientific explanations are widely recognized to have instrumental value by helping scientists make predictions and control their environment. In this paper I raise, and provide a first analysis of, the question whether explanatory proofs in mathematics have analogous instrumental value. I first identify an important goal in mathematical practice: reusing resources from existing proofs to solve new problems. I then consider the more specific question: do explanatory proofs have instrumental value by promoting reuse of the resources they contain? In (...)
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  14. Mathematical Explanations Of Empirical Facts, And Mathematical Realism.Aidan Lyon - 2012 - Australasian Journal of Philosophy 90 (3):559-578.
    A main thread of the debate over mathematical realism has come down to whether mathematics does explanatory work of its own in some of our best scientific explanations of empirical facts. Realists argue that it does; anti-realists argue that it doesn't. Part of this debate depends on how mathematics might be able to do explanatory work in an explanation. Everyone agrees that it's not enough that there merely be some mathematics in the explanation. Anti-realists claim there is (...)
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  15. Mathematical Explanations in Evolutionary Biology or Naturalism? A Challenge for the Statisticalist.Fabio Sterpetti - 2021 - Foundations of Science 27 (3):1073-1105.
    This article presents a challenge that those philosophers who deny the causal interpretation of explanations provided by population genetics might have to address. Indeed, some philosophers, known as statisticalists, claim that the concept of natural selection is statistical in character and cannot be construed in causal terms. On the contrary, other philosophers, known as causalists, argue against the statistical view and support the causal interpretation of natural selection. The problem I am concerned with here arises for the statisticalists because the (...)
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  16. Mathematical explanation.Mark Steiner - 1978 - Philosophical Studies 34 (2):135 - 151.
  17. 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 (...)
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  18.  78
    Mathematical Explanation and Epistemology: Please Mind the Gap.Sam Baron - 2015 - 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 (...)
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  19. 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 (...)
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  20. Mathematics, explanation, and scientific knowledge.Mark Steiner - 1978 - Noûs 12 (1):17-28.
  21. 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 (...)
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  22.  71
    Mathematical Explanations that are Not Proofs.Marc Lange - 2018 - Erkenntnis 83 (6):1285-1302.
    Explanation in mathematics has recently attracted increased attention from philosophers. The central issue is taken to be how to distinguish between two types of mathematical proofs: those that explain why what they prove is true and those that merely prove theorems without explaining why they are true. This way of framing the issue neglects the possibility of mathematical explanations that are not proofs at all. This paper addresses what it would take for a non-proof to explain. The (...)
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  23. Modality and constitution in distinctively mathematical explanations.Mark Povich - 2020 - European Journal for Philosophy of Science 10 (3):1-10.
    Lange argues that some natural phenomena can be explained by appeal to mathematical, rather than natural, facts. In these “distinctively mathematical” explanations, the core explanatory facts are either modally stronger than facts about ordinary causal law or understood to be constitutive of the physical task or arrangement at issue. Craver and Povich argue that Lange’s account of DME fails to exclude certain “reversals”. Lange has replied that his account can avoid these directionality charges. Specifically, Lange argues that in (...)
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  24. Mathematical explanation and the theory of why-questions.David Sandborg - 1998 - British Journal for the Philosophy of Science 49 (4):603-624.
    Van Fraassen and others have urged that judgements of explanations are relative to why-questions; explanations should be considered good in so far as they effectively answer why-questions. In this paper, I evaluate van Fraassen's theory with respect to mathematical explanation. I show that his theory cannot recognize any proofs as explanatory. I also present an example that contradicts the main thesis of the why-question approach—an explanation that appears explanatory despite its inability to answer the why-question that motivated (...)
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  25. 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 (...)
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  26.  72
    Mathematical Explanation and the Biological Optimality Fallacy.Samantha Wakil & James Justus - 2017 - Philosophy of Science 84 (5):916-930.
    Pure mathematics can play an indispensable role explaining empirical phenomena if recent accounts of insect evolution are correct. In particular, the prime life cycles of cicadas and the geometric structure of honeycombs are taken to undergird an inference to the best explanation about mathematical entities. Neither example supports this inference or the mathematical realism it is intended to establish. Both incorrectly assume that facts about mathematical optimality drove selection for the respective traits and explain why they (...)
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  27. 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 (...)
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  28. Mathematical Explanation: A Pythagorean Proposal.Samuel Baron - forthcoming - British Journal for the Philosophy of Science.
    Mathematics appears to play an explanatory role in science. This, in turn, is thought to pave a way toward mathematical Platonism. A central challenge for mathematical Platonists, however, is to provide an account of how mathematical explanations work. I propose a property-based account: physical systems possess mathematical properties, which either guarantee the presence of other mathematical properties and, by extension, the physical states that possess them; or rule out other mathematical properties, and their associated (...)
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  29. Explanatory Information in Mathematical Explanations of Physical Phenomena.Manuel Barrantes - 2020 - Australasian Journal of Philosophy 98 (3):590-603.
    In this paper I defend an intermediate position between the ‘bare mathematical results’ view and the ‘transmission’ view of mathematical explanations of physical phenomena (MEPPs). I argue that, in MEPPs, it is not enough to deduce the explanandum from the generalizations cited in the explanans. Rather, we must add information regarding why those generalizations obtain. However, I also argue that it is not necessary to provide explanatory proofs of the mathematical theorems that represent those generalizations. I illustrate (...)
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  30.  97
    Mathematical explanations of the rainbow.Christopher Pincock - 2011 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 42 (1):13-22.
    Explanations of three different aspects of the rainbow are considered. The highly mathematical character of these explanations poses some interpretative questions concerning what the success of these explanations tells us about rainbows. I develop a proposal according to which mathematical explanations can highlight what is relevant about a given phenomenon while also indicating what is irrelevant to that phenomenon. This proposal is related to the extensive work by Batterman on asymptotic explanation with special reference to Batterman’s own (...)
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  31. Comparing Mathematical Explanations.Isaac Wilhelm - 2023 - British Journal for the Philosophy of Science 74 (1):269-290.
    Philosophers have developed several detailed accounts of what makes some mathematical proofs explanatory. Significantly less attention has been paid, however, to what makes some proofs more explanatory than other proofs. That is problematic, since the reasons for thinking that some proofs explain are also reasons for thinking that some proofs are more explanatory than others. So in this paper, I develop an account of comparative explanation in mathematics. I propose a theory of the `at least as explanatory as' (...)
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  32. Optimisation and mathematical explanation: doing the Lévy Walk.Sam Baron - 2014 - Synthese 191 (3).
    The indispensability argument seeks to establish the existence of mathematical objects. The success of the indispensability argument turns on finding cases of genuine extra- mathematical explanation. In this paper, I identify a new case of extra- mathematical explanation, involving the search patterns of fully-aquatic marine predators. I go on to use this case to predict the prevalence of extra- mathematical explanation in science.
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  33. Mathematical explanation: Why it matters.Paolo Mancosu - 2008 - In The Philosophy of Mathematical Practice. Oxford University Press. pp. 134--149.
  34. Mathematical explanation: Problems and prospects.Paolo Mancosu - 2001 - Topoi 20 (1):97-117.
  35. The Narrow Ontic Counterfactual Account of Distinctively Mathematical Explanation.Mark Povich - 2021 - British Journal for the Philosophy of Science 72 (2):511-543.
    An account of distinctively mathematical explanation (DME) should satisfy three desiderata: it should account for the modal import of some DMEs; it should distinguish uses of mathematics in explanation that are distinctively mathematical from those that are not (Baron [2016]); and it should also account for the directionality of DMEs (Craver and Povich [2017]). Baron’s (forthcoming) deductive-mathematical account, because it is modelled on the deductive-nomological account, is unlikely to satisfy these desiderata. I provide a counterfactual (...)
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  36.  19
    Mathematical explanation.John D. Barrow - 2004 - In John Cornwell (ed.), Explanations: Styles of Explanation in Science. Oxford University Press. pp. 81--109.
  37.  33
    Mathematical Explanations: An Analysis Via Formal Proofs and Conceptual Complexity.Francesca Poggiolesi - forthcoming - Philosophia Mathematica:nkad023.
    This paper studies internal (or intra-)mathematical explanations, namely those proofs of mathematical theorems that seem to explain the theorem they prove. The goal of the paper is a rigorous analysis of these explanations. This will be done in two steps. First, we will show how to move from informal proofs of mathematical theorems to a formal presentation that involves proof trees, together with a decomposition of their elements; secondly we will show that those mathematical proofs that (...)
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  38.  47
    Mathematical Explanation in Practice.Ellen Lehet - 2021 - Axiomathes 31 (5):553-574.
    The connection between understanding and explanation has recently been of interest to philosophers. Inglis and Mejía-Ramos (Synthese, 2019) propose that within mathematics, we should accept a functional account of explanation that characterizes explanations as those things that produce understanding. In this paper, I start with the assumption that this view of mathematical explanation is correct and consider what we can consequently learn about mathematical explanation. I argue that this view of explanation suggests that (...)
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  39.  25
    Mathematical Explanation as Part of an (Im) perfect Scientific Explanation: An Analysis of Two Examples.Vladimir Drekalović - 2019 - Filozofia Nauki 28 (4):23-41.
    Alan Baker argues that mathematical objects play an indispensable explanatory role in science. There are several examples cited in the literature as solid candidates for such a role. We discuss two such examples and show that they are very different in their strength and (im)perfection, although both are recognized by the scientific community as examples of the best scientific explanations of particular phenomena. More specifically, it will be shown that the explanation of the cicada case has serious shortcomings (...)
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  40. Platonic Relations and Mathematical Explanations.Robert Knowles - 2021 - Philosophical Quarterly 71 (3):623-644.
    Some scientific explanations appear to turn on pure mathematical claims. The enhanced indispensability argument appeals to these ‘mathematical explanations’ in support of mathematical platonism. I argue that the success of this argument rests on the claim that mathematical explanations locate pure mathematical facts on which their physical explananda depend, and that any account of mathematical explanation that supports this claim fails to provide an adequate understanding of mathematical explanation.
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  41.  25
    Distinctively mathematical explanation and the problem of directionality: A quasi-erotetic solution.Travis L. Holmes - 2021 - Studies in History and Philosophy of Science Part A 87 (C):13-21.
    The increasing preponderance of opinion that some natural phenomena can be explained mathematically has inspired a search for a viable account of distinctively mathematical explanation. Among the desiderata for an adequate account is that it should solve the problem of directionality and the reversals of distinctively mathematical explanations should not count as members among the explanatory fold but any solution must also avoid the exclusion of genuine explanations. In what follows, I introduce and defend what I refer (...)
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  42.  13
    Mathematical explanation and indispensability arguments.Simon Langford Chris Daly - 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.
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  43. 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 (...)
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  44. Mathematics, explanation and reductionism: exposing the roots of the Egyptianism of European civilization.Arran Gare - 2005 - Cosmos and History 1 (1):54-89.
    We have reached the peculiar situation where the advance of mainstream science has required us to dismiss as unreal our own existence as free, creative agents, the very condition of there being science at all. Efforts to free science from this dead-end and to give a place to creative becoming in the world have been hampered by unexamined assumptions about what science should be, assumptions which presuppose that if creative becoming is explained, it will be explained away as an illusion. (...)
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  45.  58
    Mathematical Explanations in Euler’s Königsberg.Tim Räz - unknown
    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 (...)
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  46. Using corpus linguistics to investigate mathematical explanation.Juan Pablo Mejía Ramos, Lara Alcock, Kristen Lew, Paolo Rago, Chris Sangwin & Matthew Inglis - 2019 - In Eugen Fischer & Mark Curtis (eds.), Methodological Advances in Experimental Philosophy. London: Bloomsbury Academic. pp. 239–263.
    In this chapter we use methods of corpus linguistics to investigate the ways in which mathematicians describe their work as explanatory in their research papers. We analyse use of the words explain/explanation (and various related words and expressions) in a large corpus of texts containing research papers in mathematics and in physical sciences, comparing this with their use in corpora of general, day-to-day English. We find that although mathematicians do use this family of words, such use is considerably less (...)
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  47.  23
    Mathematical explanation and indispensability.Susan Vineberg - 2018 - Theoria: Revista de Teoría, Historia y Fundamentos de la Ciencia 33 (2):233-247.
    This paper discusses Baker’s Enhanced Indispensability Argument for mathematical realism on the basis of the indispensable role mathematics plays in scientific explanations of physical facts, along with various responses to it. I argue that there is an analogue of causal explanation for mathematics which, of several basic types of explanation, holds the most promise for use in the EIA. I consider a plausible case where mathematics plays an explanatory role in this sense, but argue that such use (...)
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  48.  24
    Mathematical Explanation: Epistemic Aims and Diverging Assessments.Joachim Frans & Bart Van Kerkhove - 2023 - Global Philosophy 33 (2):1-26.
    Mathematicians suggest that some proofs are valued for their explanatory value. This has led to a philosophical debate about the distinction between explanatory and non-explanatory proofs. In this paper, we explore whether contrasting views about the explanatory value of proof are possible and how to understand these diverging assessments. By considering an epistemic and contextual conception of explanation, we can make sense of disagreements about explanatoriness in mathematics by identifying differences in the background knowledge, skill corpus, or epistemic aims (...)
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  49. Mathematical explanation doesn't require mathematical truth.Mary Leng - 2020 - In Shamik Dasgupta, Brad Weslake & Ravit Dotan (eds.), Current Controversies in Philosophy of Science. London: Routledge.
     
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  50. Not so distinctively mathematical explanations: topology and dynamical systems.Aditya Jha, Douglas Campbell, Clemency Montelle & Phillip L. Wilson - 2022 - Synthese 200 (3):1-40.
    So-called ‘distinctively mathematical explanations’ (DMEs) are said to explain physical phenomena, not in terms of contingent causal laws, but rather in terms of mathematical necessities that constrain the physical system in question. Lange argues that the existence of four or more equilibrium positions of any double pendulum has a DME. Here we refute both Lange’s claim itself and a strengthened and extended version of the claim that would pertain to any n-tuple pendulum system on the ground that such (...)
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