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  1. 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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  2. Peressini Anthony (1997). Troubles with Indispensability: Applying Pure Mathematics in Physical Theory. Philosophia Mathematica 5 (3).
  3. 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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  4. 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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  5. 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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  6. 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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  7. Sorin Bangu (2012). The Applicability of Mathematics in Science: Indispensability and Ontology. Palgrave Macmillan.
  8. 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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  9. 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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  10. 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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  11. 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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  12. 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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  13. 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 (reality of spacetime, the quantum state) are intimately connected to the nature of applied mathematics.
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  14. 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 (Field 1980). 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, (...)
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  15. 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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  16. 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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  17. 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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  18. O. Chateaubriand (2008). Mathematics and Logic: Response to Mark Wilson. Manuscrito 31 (1).
  19. Maria Luisa Dalla Chiara (1985). Some Foundational Problems in Mathematics Suggested by Physics. Synthese 62 (2):303 - 315.
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  20. 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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  21. 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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  22. 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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  23. C. A. Coulson (1953). The Spirit of Applied Mathematics. Oxford, Clarendon Press.
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  24. 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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  25. 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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  26. 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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  27. 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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  28. 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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  29. Louk Fleischhacker (1995). Beyond Structure: The Power and Limitations of Mathematical Thought in Common Sense, Science, and Philosophy. Peter Lang.
  30. 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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  31. Axel Gelfert (2011). Mathematical Formalisms in Scientific Practice: From Denotation to Model-Based Representation. Studies in History and Philosophy of Science 42 (2):272-286.
    The present paper argues that ‘mature mathematical formalisms’ play a central role in achieving representation via scientific models. A close discussion of two contemporary accounts of how mathematical models apply—the DDI account (according to which representation depends on the successful interplay of denotation, demonstration and interpretation) and the ‘matching model’ account—reveals shortcomings of each, which, it is argued, suggests that scientific representation may be ineliminably heterogeneous in character. In order to achieve a degree of unification that is compatible with successful (...)
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  32. Emily R. Grosholz (1986). A Case Study in the Application of Mathematics to Physics: Descartes' Principles of Philosophy, Part II. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1986:116 - 124.
    The question of how and why mathematics can be applied to physical reality should be approached through the history of science, as a series of case studies which may reveal both generalizable patterns and salient differences in the grounds and nature of that application from era to era. The present examination of Descartes' Principles of Philosophy Part II, reveals a deep ambiguity in the relation of Euclidean geometry to res extensa, and a tension between geometrical form and 'common motion of (...)
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  33. Ian Hacking (2011). Why is There Philosophy of Mathematics AT ALL? South African Journal of Philosophy 30 (1):1-15.
    Mathematics plays an inordinate role in the work of many of famous Western philosophers, from the time of Plato, through Husserl and Wittgenstein, and even to the present. Why? This paper points to the experience of learning or making mathematics, with an emphasis on proof. It distinguishes two sources of the perennial impact of mathematics on philosophy. They are classified as Ancient and Enlightenment. Plato is emblematic of the former, and Kant of the latter. The Ancient fascination arises from the (...)
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  34. Philip Hugly & Charles Sayward (1990). Tractatus 6.2–6.22. Philosophical Investigations 13 (2):126-136.
    It is argued that Wittgenstein’s remarks 6.2-6.22 Tractatus fare well when one focuses on non-quantificational arithmetic, but they are problematic when one moves to quantificational arithmetic.
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  35. Dale Jacquette (2006). Applied Mathematics in the Sciences. Croatian Journal of Philosophy 6 (2):237-267.
    A complete philosophy of mathematics must address Paul Benacerraf’s dilemma. The requirements of a general semantics for the truth of mathematical theorems that coheres also with the meaning and truth conditions for non-mathematical sentences, according to Benacerraf, should ideally be coupled with an adequate epistemology for the discovery of mathematical knowledge. Standard approaches to the philosophy of mathematics are criticized against their own merits and against the background of Benacerraf’s dilemma, particularly with respect to the problem of understanding the distinction (...)
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  36. Doug Jesseph (2008). Review of Emily R. Grosholz, Representation and Productive Ambiguity in Mathematics and the Sciences. [REVIEW] Notre Dame Philosophical Reviews 2008 (5).
  37. Philip E. B. Jourdain (1915). The Purely Ordinal Conceptions of Mathematics and Their Significance for Mathematical Physics. The Monist 25 (1):140-144.
  38. David Liggins (2014). Abstract Expressionism and the Communication Problem. British Journal for the Philosophy of Science 65 (3):599-620.
    Some philosophers have recently suggested that the reason mathematics is useful in science is that it expands our expressive capacities. Of these philosophers, only Stephen Yablo has put forward a detailed account of how mathematics brings this advantage. In this article, I set out Yablo’s view and argue that it is implausible. Then, I introduce a simpler account and show it is a serious rival to Yablo’s. 1 Introduction2 Yablo’s Expressionism3 Psychological Objections to Yablo’s Expressionism4 Introducing Belief Expressionism5 Objections and (...)
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  39. 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.
  40. Penelope J. Maddy (2001). Some Naturalistic Reflections on Set Theoretic Method. Topoi 20 (1):17-27.
    My ultimate goal in this paper is to illuminate, from a naturalistic point of view, the significance of the application of mathematics in the natural sciences for the practice of contemporary set theory.
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  41. Carlos Madrid (2009). Do Mathematical Models Represent the World? : The Case of Quantum Mathematical Models. In González Recio & José Luis (eds.), Philosophical Essays on Physics and Biology. G. Olms.
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  42. Russell Marcus (2007). Numbers Without Science. Dissertation, The Graduate School and University Center of the City University of New York
    Numbers without Science opposes the Quine-Putnam indispensability argument, seeking to undermine the argument and reduce its profound influence. Philosophers rely on indispensability to justify mathematical knowledge using only empiricist epistemology. I argue that we need an independent account of our knowledge of mathematics. The indispensability argument, in broad form, consists of two premises. The major premise alleges that we are committed to mathematical objects if science requires them. The minor premise alleges that science in fact requires mathematical objects. The most (...)
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  43. Nicholas Maxwell (2010). Wisdom Mathematics. Friends of Wisdom Newsletter (6):1-6.
    For over thirty years I have argued that all branches of science and scholarship would have both their intellectual and humanitarian value enhanced if pursued in accordance with the edicts of wisdom-inquiry rather than knowledge-inquiry. I argue that this is true of mathematics. Viewed from the perspective of knowledge-inquiry, mathematics confronts us with two fundamental problems. (1) How can mathematics be held to be a branch of knowledge, in view of the difficulties that view engenders? What could mathematics be knowledge (...)
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  44. Ulrich Meyer (2004). How to Apply Mathematics. Erkenntnis 61 (1):17-28.
    This paper presents a novel account of applied mathematics. It shows how we can distinguish the physical content from the mathematical form of a scientific theory even in cases where the mathematics applied is indispensable and cannot be eliminated by paraphrase.
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  45. V. V. Nalimov (1974). Logical Foundations of Applied Mathematics. Synthese 27 (1-2):211 - 250.
    In applied problems mathematics is used as language or as a metalanguage on which metatheories are built, E.G., Mathematical theory of experiment. The structure of pure mathematics is grammar of the language. As opposed to pure mathematics, In applied problems we must keep in mind what underlies the sign system. Optimality criteria-Axioms of applied mathematics-Prove mutually incompatible, They form a mosaic and not mathematical structures which, According to bourbaki, Make mathematics a unified science. One of the peculiarities of applied mathematical (...)
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  46. Anthony Peressini (1999). Applying Pure Mathematics. Philosophy of Science 66 (3):13.
    Much of the current thought concerning mathematical ontology and epistemology follows Quine and Putnam in looking to the indispensable application of mathematics in science. A standard assumption of the indispensability approach is some version of confirmational holism, i.e., that only "sufficiently large" sets of beliefs "face the tribunal of experience." In this paper I develop and defend a distinction between a pure mathematical theory and a mathematized scientific theory in which it is applied. This distinction allows for the possibility that (...)
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  47. C. Pincock (2005). Torsten Wilholt, Number and Reality: A Philosphical Investigation of the Applicability of Mathenatics. Philosophia Mathematica 13 (3):329-337.
  48. Chistroper Pincock (2004). A New Perspective on the Problem of Applying Mathematics. Philosophia Mathematica 12 (2):135-161.
    The form of nominalism known as 'mathematical fictionalism' is examined and found wanting, mainly on grounds that go back to an early antinominalist work of Rudolf Carnap that has unfortunately not been paid sufficient attention by more recent writers.
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  49. Christopher Pincock, I.
    Most contemporary philosophy of mathematics focuses on a small segment of mathematics, mainly the natural numbers and foundational disciplines like set theory. While there are good reasons for this approach, in this paper I will examine the philosophical problems associated with the area of mathematics known as applied mathematics.
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  50. Christopher Pincock, The Applicability of Mathematics.
    Depending on how it is clarified, the applicability of mathematics can lie anywhere on a spectrum from the completely trivial to the utterly mysterious. At the one extreme, it is obvious that mathematics is used outside of mathematics in cases which range from everyday calculations like the attempt to balance one s checkbook through the most demanding abstract modeling of subatomic particles. The techniques underlying these applications are perfectly clear to those who have mastered them and there seems to be (...)
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