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  1. Dr Afsar Abbas, Mathematics as an Exact and Precise Language of Nature.
    One of the outstanding problems of philosophy of science and mathematics today is whether there is just "one" unique mathematics or the same can be bifurcated into "pure" and "applied" categories. A novel solution for this problem is offered here. This will allow us to appreciate the manner in which mathematics acts as an exact and precise language of nature. This has significant implications for Artificial Intelligence.
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  2. Evandro Agazzi (1997). The Relation of Mathematics to the Other Sciences. In Evandro Agazzi & György Darvas (eds.), Philosophy of Mathematics Today. Kluwer 235--259.
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  3. I. A. Akchurin, M. F. Vedenov & Iu V. Sachkov (1966). Methodological Problems of Mathematical Modeling in Natural Science. Russian Studies in Philosophy 5 (2):23-34.
    The constantly accelerating progress of contemporary natural science is indissolubly associated with the development and use of mathematics and with the processes of mathematical modeling of the phenomena of nature. The essence of this diverse and highly fertile interaction of mathematics and natural science and the dialectics of this interaction can only be disclosed through analysis of the nature of theoretical notions in general. Today, above all in the ranks of materialistically minded researchers, it is generally accepted that theory possesses (...)
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  4. Abraham Akkerman (1994). Sameness of Age Cohorts in the Mathematics of Population Growth. British Journal for the Philosophy of Science 45 (2):679-691.
    The axiom of extensionality of set theory states that any two classes that have identical members are identical. Yet the class of persons age i at time t and the class of persons age i + 1 at t + l, both including same persons, possess different demographic attributes, and thus appear to be two different classes. The contradiction could be resolved by making a clear distinction between age groups and cohorts. Cohort is a multitude of individuals, which is constituted (...)
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  5. Peressini Anthony (1997). Troubles with Indispensability: Applying Pure Mathematics in Physical Theory. Philosophia Mathematica 5 (3).
  6. William Aspray & Philip Kitcher (1988). History and Philosophy of Modern Mathematics. Monograph Collection (Matt - Pseudo).
  7. Jody Azzouni (2000). Applying Mathematics. The Monist 83 (2):209-227.
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  8. Jody Azzouni (1999). Comments on Shapiro. Journal of Philosophy 96 (10):541 - 544.
  9. Kazimierz Badziag (1967). B. Réponses de l'Enquête Sur l'Enseignement de Mathématique Et de Physique B. Replies on the Teaching of Mathematics and Physics Reply to the Questionnaire. Dialectica 21 (1‐4):157-158.
  10. Francis Bailly (2010). Mathematics and the Natural Sciences: The Physical Singularity of Life. Imperial College Press.
    This book identifies the organizing concepts of physical and biological phenomena by an analysis of the foundations of mathematics and physics.
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  11. Alan Baker (2005). Are There Genuine Mathematical Explanations of Physical Phenomena? Mind 114 (454):223-238.
    Many explanations in science make use of mathematics. But are there cases where the mathematical component of a scientific explanation is explanatory in its own right? This issue of mathematical explanations in science has been for the most part neglected. I argue that there are genuine mathematical explanations in science, and present in some detail an example of such an explanation, taken from evolutionary biology, involving periodical cicadas. I also indicate how the answer to my title question impacts on broader (...)
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  12. Alan Baker & Mark Colyvan (2011). Indexing and Mathematical Explanation. Philosophia Mathematica 19 (3):323-334.
    We discuss a recent attempt by Chris Daly and Simon Langford to do away with mathematical explanations of physical phenomena. Daly and Langford suggest that mathematics merely indexes parts of the physical world, and on this understanding of the role of mathematics in science, there is no need to countenance mathematical explanation of physical facts. We argue that their strategy is at best a sketch and only looks plausible in simple cases. We also draw attention to how frequently Daly and (...)
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  13. Mark Balaguer (2002). Review: Stewart Shapiro, Thinking About Mathematics. The Philosophy of Mathematics. [REVIEW] Bulletin of Symbolic Logic 8 (1):89-91.
  14. Aristides Baltas (1995). Do Mathematics Constitute a Scientific Continent? Neusis 3:97-108.
  15. S. Bangu (2006). Steiner on the Applicability of Mathematics and Naturalism. Philosophia Mathematica 14 (1):26-43.
    Steiner defines naturalism in opposition to anthropocentrism, the doctrine that the human mind holds a privileged place in the universe. He assumes the anthropocentric nature of mathematics and argues that physicists' employment of mathematically guided strategies in the discovery of quantum mechanics challenges scientists' naturalism. In this paper I show that Steiner's assumption about the anthropocentric character of mathematics is questionable. I draw attention to mathematicians' rejection of what Maddy calls ‘definabilism’, a methodological maxim governing the development of mathematics. I (...)
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  16. Sorin Bangu (2016). On The Unreasonable Effectiveness of Mathematics in the Natural Sciences. In Ippoliti, Sterpetti & Nickles (eds.), Models and Inferences in Science. Springer 11-29.
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  17. Sorin Bangu (2012). The Applicability of Mathematics in Science: Indispensability and Ontology. Palgrave Macmillan.
  18. Sorin Bangu (2009). Wigner's Puzzle for Mathematical Naturalism. International Studies in the Philosophy of Science 23 (3):245-263.
    I argue that a recent version of the doctrine of mathematical naturalism faces difficulties arising in connection with Wigner's old puzzle about the applicability of mathematics to natural science. I discuss the strategies to solve the puzzle and I show that they may not be available to the naturalist.
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  19. Sorin Bangu (2008). Reifying Mathematics? Prediction and Symmetry Classification. Studies in History and Philosophy of Science Part B 39 (2):239-258.
    In this paper I reconstruct and critically examine the reasoning leading to the famous prediction of the ‘omega minus’ particle by M. Gell-Mann and Y. Ne’eman (in 1962) on the basis of a symmetry classification scheme. While the peculiarity of this prediction has occasionally been noticed in the literature, a detailed treatment of the methodological problems it poses has not been offered yet. By spelling out the characteristics of this type of prediction, I aim to underscore the challenges raised by (...)
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  20. Sorin Ioan Bangu (2008). Inference to the Best Explanation and Mathematical Realism. Synthese 160 (1):13-20.
    Arguing for mathematical realism on the basis of Field’s explanationist version of the Quine–Putnam Indispensability argument, Alan Baker has recently claimed to have found an instance of a genuine mathematical explanation of a physical phenomenon. While I agree that Baker presents a very interesting example in which mathematics plays an essential explanatory role, I show that this example, and the argument built upon it, begs the question against the mathematical nominalist.
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  21. Sam Baron (2016). Mathematical Explanation and Epistemology: Please Mind the Gap. Ratio 29 (2):149-167.
    This paper draws together two strands in the debate over the existence of mathematical objects. The first strand concerns the notion of extra-mathematical explanation: the explanation of physical facts, in part, by facts about mathematical objects. The second strand concerns the access problem for platonism: the problem of how to account for knowledge of mathematical objects. I argue for the following conditional: if there are extra-mathematical explanations, then the core thesis of the access problem is false. This has implications for (...)
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  22. Jeffrey A. Barrett (1995). Review of I. Ekeland, The Broken Dice, and Other Mathematical Tales of Chance. [REVIEW] Philosophia Mathematica 3 (3):310-313.
  23. John D. Barrow (2004). Mathematical Explanation. In John Cornwell (ed.), Explanations: Styles of Explanation in Science. Oxford University Press 81--109.
  24. Robert Batterman (2010). On the Explanatory Role of Mathematics in Empirical Science. British Journal for the Philosophy of Science 61 (1):1-25.
    This paper examines contemporary attempts to explicate the explanatory role of mathematics in the physical sciences. Most such approaches involve developing so-called mapping accounts of the relationships between the physical world and mathematical structures. The paper argues that the use of idealizations in physical theorizing poses serious difficulties for such mapping accounts. A new approach to the applicability of mathematics is proposed.
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  25. Nora Berenstain (2016). The Applicability of Mathematics to Physical Modality. Synthese:1-17.
    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 (...)
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  26. Sandy Berkovski (2002). Surprising User-Friendliness. Logique Et Analyse 45 (179-180):283-297.
    Some theorists are bewildered by the effectiveness of mathematical concepts. For example, Steiner attempts to show that there can be no rational explanation of mathematical applicability in physics. Others (notably Penrose) are concerned primarily with the unexpected effectiveness within mathematics. Both views consist of two parts: a puzzle and a positive solution. I defend their paradoxical parts against the sceptics who do not believe that the very problem of effectiveness is a genuine one. Utilising Horwich’s theory of surprise, I argue (...)
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  27. H. Billinge (2000). Applied Constructive Mathematics: On Hellman's 'Mathematical Constructivism in Spacetime'. British Journal for the Philosophy of Science 51 (2):299-318.
    claims that constructive mathematics is inadequate for spacetime physics and hence that constructive mathematics cannot be considered as an alternative to classical mathematics. He also argues that the contructivist must be guilty of a form of a priorism unless she adopts a strong form of anti-realism for science. Here I want to dispute both claims. First, even if there are non-constructive results in physics this does not show that adequate constructive alternatives could not be formulated. Secondly, the constructivist adopts a (...)
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  28. James Robert Brown (1997). What is Applied Mathematics? Foundations of Science 2 (1):21-37.
    A number of issues connected with the nature of applied mathematics are discussed. Among the claims are these: mathematics "hooks onto" the world by providing models or representations, not by describing the world; classic platonism is to be preferred to structuralism; and several issues in the philosophy of science are intimately connected to the nature of applied mathematics.
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  29. O. Bueno (2012). An Easy Road to Nominalism. Mind 121 (484):967-982.
    In this paper, I provide an easy road to nominalism which does not rely on a Field-type nominalization strategy for mathematics. According to this proposal, applications of mathematics to science, and alleged mathematical explanations of physical phenomena, only emerge when suitable physical interpretations of the mathematical formalism are advanced. And since these interpretations are rarely distinguished from the mathematical formalism, the impression arises that mathematical explanations derive from the mathematical formalism alone. I correct this misimpression by pointing out, in the (...)
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  30. Otávio Bueno (2011). An Inferential Conception of the Application of Mathematics. Noûs 45 (2):345 - 374.
    A number of people have recently argued for a structural approach to accounting for the applications of mathematics. Such an approach has been called "the mapping account". According to this view, the applicability of mathematics is fully accounted for by appreciating the relevant structural similarities between the empirical system under study and the mathematics used in the investigation ofthat system. This account of applications requires the truth of applied mathematical assertions, but it does not require the existence of mathematical objects. (...)
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  31. Otávio Bueno & Steven French (2012). Can Mathematics Explain Physical Phenomena? British Journal for the Philosophy of Science 63 (1):85-113.
    Batterman raises a number of concerns for the inferential conception of the applicability of mathematics advocated by Bueno and Colyvan. Here, we distinguish the various concerns, and indicate how they can be assuaged by paying attention to the nature of the mappings involved and emphasizing the significance of interpretation in this context. We also indicate how this conception can accommodate the examples that Batterman draws upon in his critique. Our conclusion is that ‘asymptotic reasoning’ can be straightforwardly accommodated within the (...)
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  32. John P. Burgess (1992). How Foundational Work in Mathematics Can Be Relevant to Philosophy of Science. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1992:433 - 441.
    Foundational work in mathematics by some of the other participants in the symposium helps towards answering the question whether a heterodox mathematics could in principle be used as successfully as is orthodox mathematics in scientific applications. This question is turn, it will be argued, is relevant to the question how far current science is the way it is because the world is the way it is, and how far because we are the way we are, which is a central question, (...)
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  33. O. Chateaubriand (2008). Mathematics and Logic: Response to Mark Wilson. Manuscrito 31 (1):355-359.
    Mark Wilson argues that in order to make physical first-order properties suitable for inclusion in the bottom levels of a logical hierarchy of properties, their proper treatment must take into account the methods of applied mathematics. I agree that the methods of applied mathematics are essential for studying physical properties, and in my response focus on the nature of the logical hierarchy and on the requirements of classical logic.Mark Wilson argumenta que um tratamento adequado para tornar as propriedades físicas de (...)
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  34. Maria Luisa Dalla Chiara (1985). Some Foundational Problems in Mathematics Suggested by Physics. Synthese 62 (2):303 - 315.
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  35. Mark Colyvan, Applying Inconsistent Mathematics.
    At various times, mathematicians have been forced to work with inconsistent mathematical theories. Sometimes the inconsistency of the theory in question was apparent (e.g. the early calculus), while at other times it was not (e.g. pre-paradox na¨ıve set theory). The way mathematicians confronted such difficulties is the subject of a great deal of interesting work in the history of mathematics but, apart from the crisis in set theory, there has been very little philosophical work on the topic of inconsistent mathematics. (...)
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  36. Mark Colyvan (2001). The Miracle of Applied Mathematics. Synthese 127 (3):265-277.
    Mathematics has a great variety ofapplications in the physical sciences.This simple, undeniable fact, however,gives rise to an interestingphilosophical problem:why should physical scientistsfind that they are unable to evenstate their theories without theresources of abstract mathematicaltheories? Moreover, theformulation of physical theories inthe language of mathematicsoften leads to new physical predictionswhich were quite unexpected onpurely physical grounds. It is thought by somethat the puzzles the applications of mathematicspresent are artefacts of out-dated philosophical theories about thenature of mathematics. In this paper I argue (...)
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  37. David Corfield, Reflections on Michael Friedman's Dynamics of Reason.
    Friedman's rich account of the way the mathematical sciences ideally are transformed affords mathematics a more influential role than is common in the philosophy of science. In this paper I assess Friedman's position and argue that we can improve on it by pursuing further the parallels between mathematics and science. We find a richness to the organisation of mathematics similar to that Friedman finds in physics.
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  38. C. A. Coulson (1953). The Spirit of Applied Mathematics. Oxford, Clarendon Press.
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  39. Jacquette Dale (2006). Applied Mathematics in the Sciences. Croatian Journal of Philosophy 17:237-267.
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  40. 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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  41. Kevin J. Davey (2003). Problems in Applying Mathematics: On the Inferential and Representational Limits of Mathematics in Physics. Dissertation, University of Pittsburgh
    It is often supposed that we can use mathematics to capture the time evolution of any physical system. By this, I mean that we can capture the basic truths about the time evolution of a physical system with a set of mathematical assertions, which can then be used as premises in arbitrary mathematical arguments to deduce more complex properties of the system. ;I would like to argue that this picture of the role of mathematics in physics is incorrect. Specifically, I (...)
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  42. Boudewijn de Bruin (2010). Explaining Games: The Epistemic Programme in Game Theory. Springer.
    Contents. Introduction. 1. Preliminaries. 2. Normal Form Games. 3. Extensive Games. 4. Applications of Game Theory. 5. The Methodology of Game Theory. Conclusion. Appendix. Bibliography. Index. Does game theory—the mathematical theory of strategic interaction—provide genuine explanations of human behaviour? Can game theory be used in economic consultancy or other normative contexts? Explaining Games: The Epistemic Programme in Game Theory—the first monograph on the philosophy of game theory—is an attempt to combine insights from epistemic logic and the philosophy of science to (...)
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  43. Boudewijn de Bruin (2009). Overmathematisation in Game Theory: Pitting the Nash Equilibrium Refinement Programme Against the Epistemic Programme. Studies in History and Philosophy of Science Part A 40 (3):290-300.
    The paper argues that the Nash Equilibrium Refinement Programme in game theory was less successful than its competitor, the Epistemic Programme (Interactive Epistemology). The prime criterion of success is the extent to which the programmes were able to reach the key objective guiding non-cooperative game theory for much of the 20th century, namely, to develop a complete characterisation of the strategic rationality of economic agents in the form of the ultimate game theoretic solution concept for any normal form and extensive (...)
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  44. William Dembski, The Last Magic.
    If mathematics is about finding solutions to well-defined problems, then philosophy is about finding problems in what previously we thought were well-settled solutions. Mark Steiner's The Applicability of Mathematics As a Philosophical Problem mirrors both sides of this statement, admitting that mathematics is the key to solving problems in the physical sciences, but also asserting that this very applicability of mathematics to physics constitutes a problem.
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  45. Ken Dennis (1995). A Logical Critique of Mathematical Formalism in Economics. Journal of Economic Methodology 2 (2):181-200.
    Mathematical economic theory is lacking in logical rigour. Even if the mathematics used in constructing formal economic theory is rigorous as pure mathematics, economic theory possesses both mathematical and non-mathematical components. But mathematical reductionism fails to formalize the non-mathematical components of economic theory, and the method of numerics (outlined in this paper) shows how, in simple cases, the two components of economic theory can be formally identified, distinguished, and integrated. However, the real challenge to formalizing economic theory points not to (...)
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  46. Nicolas Fillion & Sorin Bangu (2015). Numerical Methods, Complexity, and Epistemic Hierarchies. Philosophy of Science 82 (5):941-955.
    Modern mathematical sciences are hard to imagine without appeal to efficient computational algorithms. We address several conceptual problems arising from this interaction by outlining rival but complementary perspectives on mathematical tractability. More specifically, we articulate three alternative characterizations of the complexity hierarchy of mathematical problems that are themselves based on different understandings of computational constraints. These distinctions resolve the tension between epistemic contexts in which exact solutions can be found and the ones in which they cannot; however, contrary to a (...)
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  47. Louk Fleischhacker (1995). Beyond Structure: The Power and Limitations of Mathematical Thought in Common Sense, Science, and Philosophy. Peter Lang.
  48. James Franklin (2014). Global and Local. Mathematical Intelligencer 36 (4).
    The global/local contrast is ubiquitous in mathematics. This paper explains it with straightforward examples. It is possible to build a circular staircase that is rising at any point (locally) but impossible to build one that rises at all points and comes back to where it started (a global restriction). Differential equations describe the local structure of a process; their solution describes the global structure that results. The interplay between global and local structure is one of the great themes of mathematics, (...)
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  49. James Franklin (1988). Mathematics, The Computer Revolution and the Real World. Philosophica 42:79-92.
    The philosophy of mathematics has largely abandoned foundational studies, but is still fixated on theorem proving, logic and number theory, and on whether mathematical knowledge is certain. That is not what mathematics looks like to, say, a knot theorist or an industrial mathematical modeller. The "computer revolution" shows that mathematics is a much more direct study of the world, especially its structural aspects.
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  50. Steven French (2000). The Reasonable Effectiveness of Mathematics: Partial Structures and the Application of Group Theory to Physics. Synthese 125 (1-2):103 - 120.
    Wigner famously referred to the `unreasonable effectiveness' of mathematics in its application to science. Using Wigner's own application of group theory to nuclear physics, I hope to indicate that this effectiveness can be seen to be not so unreasonable if attention is paid to the various idealising moves undertaken. The overall framework for analysing this relationship between mathematics and physics is that of da Costa's partial structures programme.
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