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  1. 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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  2. 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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  3. 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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  4. 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 interactions) have (...)
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  5. O. Bueno & S. French (2012). Can Mathematics Explain Physical Phenomena? British Journal for the Philosophy of Science 63 (1):85-113.
    Batterman ([2010]) raises a number of concerns for the inferential conception of the applicability of mathematics advocated by Bueno and Colyvan ([2011]). 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 (...)
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  6. Christopher Clarke (forthcoming). Multi-Level Selection and the Explanatory Value of Mathematical Decompositions. British Journal for the Philosophy of Science.
    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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  7. 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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  8. 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.
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  9. 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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  10. Marc Lange (2012). Abstraction and Depth in Scientific Explanation. [REVIEW] Philosophy and Phenomenological Research 84 (2):483-491.
  11. M. 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.
  12. J. B. Paris (1994). The Uncertain Reasoner's Companion: A Mathematical Perspective. Cambridge University Press.
    Reasoning under uncertainty, that is, making judgements with only partial knowledge, is a major theme in artificial intelligence. Professor Paris provides here an introduction to the mathematical foundations of the subject. It is suited for readers with some knowledge of undergraduate mathematics but is otherwise self-contained, collecting together the key results on the subject, and formalising within a unified framework the main contemporary approaches and assumptions. The author has concentrated on giving clear mathematical formulations, analyses, justifications and consequences of the (...)
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  13. Christopher Pincock, Mathematical Contributions to Scientific Explanation.
    After reviewing some different indispensability arguments, I distinguish several different ways in which mathematics can make an important contribution to a scientific explanation. Once these contributions are highlighted it will be possible to see that indispensability arguments have little chance of convincing us of the existence of abstract objects, even though they may give us good reason to accept the truth of some mathematical claims. However, in the concluding part of this paper, I argue that even though there is a (...)
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  14. T. Raz (2013). On the Application of the Honeycomb Conjecture to the Bee's Honeycomb. Philosophia Mathematica 21 (3):351-360.
    In a recent paper, Aidan Lyon and Mark Colyvan have proposed an explanation of the structure of the bee's honeycomb based on the mathematical Honeycomb Conjecture. This explanation has instantly become one of the standard examples in the philosophical debate on mathematical explanations of physical phenomena. In this critical note, I argue that the explanation is not scientifically adequate. The reason for this is that the explanation fails to do justice to the essentially three-dimensional structure of the bee's honeycomb.
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  15. Alexander Reutlinger (forthcoming). Are Causal Facts Really Explanatorily Emergent? Ladyman and Ross on Higher-Level Causal Facts and Renormalization Group Explanation. Synthese.
    In their Every Thing Must Go, Ladyman and Ross defend a novel version of Neo- Russellian metaphysics of causation, which falls into three claims: (1) there are no fundamental physical causal facts (orthodox Russellian claim), (2) there are higher-level causal facts of the special sciences, and (3) higher-level causal facts are explanatorily emergent. While accepting claims (1) and (2), I attack claim (3). Ladyman and Ross argue that higher-level causal facts are explanatorily emergent, because (a) certain aspects of these higher-level (...)
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  16. Alexander Reutlinger (forthcoming). Why Is There Universal Macro-Behavior? Renormalization Group Explanation As Non-Causal Explanation. Philosophy of Science.
    Renormalization group (RG) methods are an established strategy to explain how it is possible that microscopically different systems exhibit virtually the same macro behavior when undergoing phase-transitions. I argue – in agreement with Robert Batterman – that RG explanations are non-causal explanations. However, Batterman misidentifies the reason why RG explanations are non-causal: it is not the case that an explanation is non- causal if it ignores causal details. I propose an alternative argument, according to which RG explanations are non-causal explanations (...)
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  17. Juha Saatsi (2011). The Enhanced Indispensability Argument: Representational Versus Explanatory Role of Mathematics in Science. British Journal for the Philosophy of Science 62 (1):143-154.
    The Enhanced Indispensability Argument (Baker [ 2009 ]) exemplifies the new wave of the indispensability argument for mathematical Platonism. The new wave capitalizes on mathematics' role in scientific explanations. I will criticize some analyses of mathematics' explanatory function. In turn, I will emphasize the representational role of mathematics, and argue that the debate would significantly benefit from acknowledging this alternative viewpoint to mathematics' contribution to scientific explanations and knowledge.
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  18. Amy M. Schmitter (2000). Mind and Sign: Method and the Interpretation of Mathematics in Descartes's Early Work. Canadian Journal of Philosophy 30 (3):371-411.
  19. David Sherry (2009). The Role of Diagrams in Mathematical Arguments. Foundations of Science 14 (1-2):59-74.
    Recent accounts of the role of diagrams in mathematical reasoning take a Platonic line, according to which the proof depends on the similarity between the perceived shape of the diagram and the shape of the abstract object. This approach is unable to explain proofs which share the same diagram in spite of drawing conclusions about different figures. Saccheri’s use of the bi-rectangular isosceles quadrilateral in Euclides Vindicatus provides three such proofs. By forsaking abstract objects it is possible to give a (...)
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  20. Mark Zelcer (2013). Against Mathematical Explanation. Journal for General Philosophy of Science 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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