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  1. Viewing-as Explanations and Ontic Dependence.William D’Alessandro - 2020 - Philosophical Studies 177 (3):769-792.
    According to a widespread view in metaphysics and philosophy of science, all explanations involve relations of ontic dependence between the items appearing in the explanandum and the items appearing in the explanans. I argue that a family of mathematical cases, which I call “viewing-as explanations”, are incompatible with the Dependence Thesis. These cases, I claim, feature genuine explanations that aren’t supported by ontic dependence relations. Hence the thesis isn’t true in general. The first part of the paper defends this claim (...)
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  • Functional Explanation in Mathematics.Matthew Inglis & Juan Pablo Mejía Ramos - 2019 - Synthese 198 (26):6369-6392.
    Mathematical explanations are poorly understood. Although mathematicians seem to regularly suggest that some proofs are explanatory whereas others are not, none of the philosophical accounts of what such claims mean has become widely accepted. In this paper we explore Wilkenfeld’s suggestion that explanations are those sorts of things that generate understanding. By considering a basic model of human cognitive architecture, we suggest that existing accounts of mathematical explanation are all derivable consequences of Wilkenfeld’s ‘functional explanation’ proposal. We therefore argue that (...)
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  • Arithmetic, Set Theory, Reduction and Explanation.William D’Alessandro - 2018 - Synthese 195 (11):5059-5089.
    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 (...)
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  • Towards a Knowledge-Based Account of Understanding.Christoph9 Kelp - 2016 - In S. Grimm, C. Baumberger & S. Ammon (eds.), Explaining Understanding. Routledge.
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  • 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 (...)
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  • Mathematical Concepts and Definitions.Jamie Tappenden - 2008 - In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice. Oxford University Press. pp. 256--275.
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  • What Are Mathematical Coincidences ?M. Lange - 2010 - Mind 119 (474):307-340.
    Although all mathematical truths are necessary, mathematicians take certain combinations of mathematical truths to be ‘coincidental’, ‘accidental’, or ‘fortuitous’. The notion of a ‘ mathematical coincidence’ has so far failed to receive sufficient attention from philosophers. I argue that a mathematical coincidence is not merely an unforeseen or surprising mathematical result, and that being a misleading combination of mathematical facts is neither necessary nor sufficient for qualifying as a mathematical coincidence. I argue that although the components of a mathematical coincidence (...)
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  • Depth — A Gaussian Tradition in Mathematics.Jeremy Gray - 2015 - Philosophia Mathematica 23 (2):177-195.
    Mathematicians use the word ‘deep’ to convey a high appreciation of a concept, theorem, or proof. This paper investigates the extent to which the term can be said to have an objective character by examining its first use in mathematics. It was a consequence of Gauss's work on number theory and the agreement among his successors that specific parts of Gauss's work were deep, on grounds that indicate that depth was a structural feature of mathematics for them. In contrast, French (...)
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  • Depth and Explanation in Mathematics.Marc Lange - 2015 - Philosophia Mathematica 23 (2):196-214.
    This paper argues that in at least some cases, one proof of a given theorem is deeper than another by virtue of supplying a deeper explanation of the theorem — that is, a deeper account of why the theorem holds. There are cases of scientific depth that also involve a common abstract structure explaining a similarity between two otherwise unrelated phenomena, making their similarity no coincidence and purchasing depth by answering why questions that separate, dissimilar explanations of the two phenomena (...)
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  • 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 (...)
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  • Explanation, Existence and Natural Properties in Mathematics – A Case Study: Desargues’ Theorem.Marc Lange - 2015 - Dialectica 69 (4):435-472.
  • The Philosophy of Mathematical Practice.Paolo Mancosu (ed.) - 2008 - Oxford, England: Oxford University Press.
    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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  • Because Without Cause: Non-Causal Explanations in Science and Mathematics.Marc Lange - 2016 - Oxford, England: Oxford University Press USA.
    Not all scientific explanations work by describing causal connections between events or the world's overall causal structure. In addition, mathematicians regard some proofs as explaining why the theorems being proved do in fact hold. This book proposes new philosophical accounts of many kinds of non-causal explanations in science and mathematics.
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  • Explanatory Unification and the Causal Structure of the World.Philip Kitcher - 1989 - In Philip Kitcher & Wesley Salmon (eds.), Scientific Explanation. Minneapolis: University of Minnesota Press. pp. 410-505.
  • 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 paper focuses on (...)
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  • Explanatory Anti-Psychologism Overturned by Lay and Scientific Case Classifications.Jonathan Waskan, Ian Harmon, Zachary Horne, Joseph Spino & John Clevenger - 2014 - Synthese 191 (5):1-23.
    Many philosophers of science follow Hempel in embracing both substantive and methodological anti-psychologism regarding the study of explanation. The former thesis denies that explanations are constituted by psychological events, and the latter denies that psychological research can contribute much to the philosophical investigation of the nature of explanation. Substantive anti-psychologism is commonly defended by citing cases, such as hyper-complex descriptions or vast computer simulations, which are reputedly generally agreed to constitute explanations but which defy human comprehension and, as a result, (...)
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  • Functional Explaining: A New Approach to the Philosophy of Explanation.Daniel A. Wilkenfeld - 2014 - Synthese 191 (14):3367-3391.
    In this paper, I argue that explanations just ARE those sorts of things that, under the right circumstances and in the right sort of way, bring about understanding. This raises the question of why such a seemingly simple account of explanation, if correct, would not have been identified and agreed upon decades ago. The answer is that only recently has it been made possible to analyze explanation in terms of understanding without the risk of collapsing both to merely phenomenological states. (...)
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  • A Mathematician's Apology.Godfrey Harold Hardy - 1969 - Cambridge University Press.
    G. H. Hardy was one of this century's finest mathematical thinkers, renowned among his contemporaries as a 'real mathematician ... the purest of the pure'. He was also, as C. P. Snow recounts in his Foreword, 'unorthodox, eccentric, radical, ready to talk about anything'. This 'apology', written in 1940 as his mathematical powers were declining, offers a brilliant and engaging account of mathematics as very much more than a science; when it was first published, Graham Greene hailed it alongside Henry (...)
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  • An Introduction to the Philosophy of Mathematics.Mark Colyvan - 2012 - Cambridge University Press.
    Machine generated contents note: 1. Mathematics and its philosophy; 2. The limits of mathematics; 3. Plato's heaven; 4. Fiction, metaphor, and partial truths; 5. Mathematical explanation; 6. The applicability of mathematics; 7. Who's afraid of inconsistent mathematics?; 8. A rose by any other name; 9. Epilogue: desert island theorems.
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  • Scientific Explanation and the Sense of Understanding.J. D. Trout - 2002 - Philosophy of Science 69 (2):212-233.
    Scientists and laypeople alike use the sense of understanding that an explanation conveys as a cue to good or correct explanation. Although the occurrence of this sense or feeling of understanding is neither necessary nor sufficient for good explanation, it does drive judgments of the plausibility and, ultimately, the acceptability, of an explanation. This paper presents evidence that the sense of understanding is in part the routine consequence of two well-documented biases in cognitive psychology: overconfidence and hindsight. In light of (...)
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  • Mathematical Explanation.Mark Steiner - 1978 - Philosophical Studies 34 (2):135 - 151.
  • The Role of Explanation in Understanding.Kareem Khalifa - 2013 - British Journal for the Philosophy of Science 64 (1):161-187.
    Peter Lipton has argued that understanding can exist in the absence of explanation. We argue that this does not denigrate explanation's importance to understanding. Specifically, we show that all of Lipton's examples are consistent with the idea that explanation is the ideal of understanding, i.e. other modes of understanding ought to be assessed by how well they replicate the understanding provided by a good and correct explanation. We defend this idea by showing that for all of Lipton's examples of non-explanatory (...)
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  • Explanation, Independence and Realism in Mathematics.Michael D. Resnik & David Kushner - 1987 - British Journal for the Philosophy of Science 38 (2):141-158.
  • The Goal of Explanation.Stephen R. Grimm - 2010 - Studies in History and Philosophy of Science Part A 41 (4):337-344.
    I defend the claim that understanding is the goal of explanation against various persistent criticisms, especially the criticism that understanding is not truth-connected in the appropriate way, and hence is a merely psychological state. Part of the reason why understanding has been dismissed as the goal of explanation, I suggest, is because the psychological dimension of the goal of explanation has itself been almost entirely neglected. In turn, the psychological dimension of understanding—the Aha! experience, the sense that a certain explanation (...)
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  • Understanding and the Norm of Explanation.John Turri - 2015 - Philosophia 43 (4):1171-1175.
    I propose and defend the hypothesis that understanding is the norm of explanation. On this proposal, an explanation should express understanding. I call this the understanding account of explanation. The understanding account is supported by social and introspective observations. It is also supported by the relationship between knowledge and understanding, on the one hand, and assertion and explanation, on the other.
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  • 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.
     
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  • Mathematical Explanation: Why It Matters.Paolo Mancosu - 2008 - In The Philosophy of Mathematical Practice. Oxford University Press. pp. 134--149.
  • Mathematical Fit: A Case Study.Manya Raman-Sundström & Lars-Daniel Öhman - forthcoming - Philosophia Mathematica:nkw015.
    Mathematicians routinely pass judgements on mathematical proofs. A proof might be elegant, cumbersome, beautiful, or awkward. Perhaps the highest praise is that a proof is right, that is, that the proof fits the theorem in an optimal way. It is also common to judge that one proof fits better than another, or that a proof does not fit a theorem at all. This paper attempts to clarify the notion of mathematical fit. We suggest six criteria that distinguish proofs as being (...)
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  • Mathematical Depth.Alasdair Urquhart - 2015 - Philosophia Mathematica 23 (2):233-241.
    The first part of the paper is devoted to surveying the remarks that philosophers and mathematicians such as Maddy, Hardy, Gowers, and Zeilberger have made about mathematical depth. The second part is devoted to the question of whether we can make the notion precise by a more formal proof-theoretical approach. The idea of measuring depth by the depth and bushiness of the proof is considered, and compared to the related notion of the depth of a chess combination.
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  • Explanation in Mathematics: Proofs and Practice.William D'Alessandro - 2019 - Philosophy Compass 14 (11).
    Mathematicians distinguish between proofs that explain their results and those that merely prove. This paper explores the nature of explanatory proofs, their role in mathematical practice, and some of the reasons why philosophers should care about them. Among the questions addressed are the following: what kinds of proofs are generally explanatory (or not)? What makes a proof explanatory? Do all mathematical explanations involve proof in an essential way? Are there really such things as explanatory proofs, and if so, how do (...)
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  • 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.
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  • The Unsolvability of The Quintic: A Case Study in Abstract Mathematical Explanation.Christopher Pincock - 2015 - Philosophers' Imprint 15.
    This paper identifies one way that a mathematical proof can be more explanatory than another proof. This is by invoking a more abstract kind of entity than the topic of the theorem. These abstract mathematical explanations are identified via an investigation of a canonical instance of modern mathematics: the Galois theory proof that there is no general solution in radicals for fifth-degree polynomial equations. I claim that abstract explanations are best seen as describing a special sort of dependence relation between (...)
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  • Proof Style and Understanding in Mathematics I: Visualization, Unification and Axiom Choice.Jamie Tappenden - unknown
    Mathematical investigation, when done well, can confer understanding. This bare observation shouldn’t be controversial; where obstacles appear is rather in the effort to engage this observation with epistemology. The complexity of the issue of course precludes addressing it tout court in one paper, and I’ll just be laying some early foundations here. To this end I’ll narrow the field in two ways. First, I’ll address a specific account of explanation and understanding that applies naturally to mathematical reasoning: the view proposed (...)
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  • Explanatory Proofs in Mathematics.Erik Weber & Liza Verhoeven - 2002 - Logique Et Analyse 179:299-307.