Search results for 'Mathematical notation' (try it on Scholar)

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  1. William James Meyers (1975). A Mathematical Theory of Parenthesis, Free Notations. Państwowe Wydawn. Naukowe.score: 48.0
     
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  2. W. Douglas Maurer (1999). The Influence of the Computer Upon Mathematical Notation. Semiotica 125 (1-3):165-168.score: 45.0
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  3. W. E. Underwood (forthcoming). Symbolic Configurations and Two-Dimensional Mathematical Notation. Semiotics:523-532.score: 45.0
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  4. Yanjie Zhao (forthcoming). What is Mathematical Notation. Semiotics:257-273.score: 45.0
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  5. Yanjie Zhao (forthcoming). What Is Mathematical Notation: An Interdisciplinary Approach1. Semiotics:257-273.score: 45.0
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  6. Robert Feys (1969). Dictionary of Symbols of Mathematical Logic. Amsterdam, North-Holland Pub. Co..score: 39.0
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  7. Ivor Bulmer-Thomas (1985). Boethian Number Theory Michael Masi: Boethian Number Theory: A Translation of the De Institutione Arithmetica (with Introduction and Notes). (Studies in Classical Antiquity, 6.) Pp. 198; 8 Figures with Mathematical Diagrams and Musical Notation in Text. Amsterdam: Editions Rodopi, 1983. Paper, Fl. 60. [REVIEW] The Classical Review 35 (01):86-87.score: 36.0
  8. Toshiyasu Arai (2002). Buchholz Wilfried. Notation Systems for Infinitary Derivations. Archive for Mathematical Logic, Vol. 30 No. 5–6 (1991), Pp. 277–296. Buchholz Wilfried. Explaining Gentzen's Consistency Proof Within Infinitary Proof Theory. Computational Logic and Proof Theory, 5th Kurt Gödel Colloquium, KGC'97, Vienna, Austria, August 25–29, 1997, Proceedings, Edited by Gottlob Georg, Leitsch Alexander, and Mundici Daniele, Lecture Notes in Computer Science, Vol. 1289, Springer, Berlin, Heidelberg, New York, Etc., 1997 ... [REVIEW] Bulletin of Symbolic Logic 8 (3):437-439.score: 36.0
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  9. Gregory Landini (2012). Frege's Notations: What They Are and How They Mean. Palgrave Macmillan.score: 33.0
     
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  10. Michael Kohlhase, Adaptation of Notations in Living Mathematical Documents.score: 31.0
    Notations are central for understanding mathematical discourse. Readers would like to read notations that transport the meaning well and prefer notations that are familiar to them. Therefore, authors optimize the choice of notations with respect to these two criteria, while at the same time trying to remain consistent over the document and their own prior publications. In print media where notations are fixed at publication time, this is an over-constrained problem. In living documents notations can be adapted at reading (...)
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  11. Michael Kohlhase, Notations for Living Mathematical Documents.score: 31.0
    Notations are central for understanding mathematical discourse. Readers would like to read notations that transport the meaning well and prefer notations that are familiar to them. Therefore, authors optimize the choice of notations with respect to these two criteria, while at the same time trying to remain consistent over the document and their own prior publications. In print media where notations are fixed at publication time, this is an over-constrained problem. In living documents notations can be adapted at reading (...)
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  12. Madeline M. Muntersbjorn (1999). Naturalism, Notation, and the Metaphysics of Mathematics. Philosophia Mathematica 7 (2):178-199.score: 30.0
    The instability inherent in the historical inventory of mathematical objects challenges philosophers. Naturalism suggests we can construct enduring answers to ontological questions through an investigation of the processes whereby mathematical objects come into existence. Patterns of historical development suggest that mathematical objects undergo an intelligible process of reification in tandem with notational innovation. Investigating changes in mathematical languages is a necessary first step towards a viable ontology. For this reason, scholars should not modernize historical texts without (...)
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  13. P. B. Andrews (2002). An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof. Kluwer Academic Publishers.score: 27.0
    This introduction to mathematical logic starts with propositional calculus and first-order logic. Topics covered include syntax, semantics, soundness, completeness, independence, normal forms, vertical paths through negation normal formulas, compactness, Smullyan's Unifying Principle, natural deduction, cut-elimination, semantic tableaux, Skolemization, Herbrand's Theorem, unification, duality, interpolation, and definability. The last three chapters of the book provide an introduction to type theory (higher-order logic). It is shown how various mathematical concepts can be formalized in this very expressive formal language. This expressive (...) facilitates proofs of the classical incompleteness and undecidability theorems which are very elegant and easy to understand. The discussion of semantics makes clear the important distinction between standard and nonstandard models which is so important in understanding puzzling phenomena such as the incompleteness theorems and Skolem's Paradox about countable models of set theory. Some of the numerous exercises require giving formal proofs. A computer program called ETPS which is available from the web facilitates doing and checking such exercises. Audience: This volume will be of interest to mathematicians, computer scientists, and philosophers in universities, as well as to computer scientists in industry who wish to use higher-order logic for hardware and software specification and verification. (shrink)
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  14. Eszter Szabó Attila Krajcsi (2012). The Role of Number Notation: Sign-Value Notation Number Processing is Easier Than Place-Value. Frontiers in Psychology 3.score: 27.0
    Number notations can influence the way numbers are handled in computations; however, the role of notation itself in mental processing has not been examined directly. From a mathematical point of view, it is believed that place-value number notation systems, such as the Indo-Arabic numbers, are superior to sign-value systems, such as the Roman numbers. However, sign-value notation might have sufficient efficiency; for example, sign-value notations were common in flourishing cultures, such as in ancient Egypt. Herein we (...)
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  15. Attila Krajcsi & Eszter Szabó (2012). The Role of Number Notation: Sign-Value Notation Number Processing is Easier Than Place-Value. Frontiers in Psychology 3.score: 27.0
    Number notations can influence the way numbers are handled in computations; however, the role of notation itself in mental processing has not been examined directly. From a mathematical point of view, it is believed that place-value number notation systems, such as the Indo-Arabic numbers, are superior to sign-value systems, such as the Roman numbers. However, sign-value notation might have sufficient efficiency; for example, sign-value notations were common in flourishing cultures, such as in ancient Egypt. Herein we (...)
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  16. Thomas Mormann (2005). Mathematical Metaphors in Natorp’s Neo-Kantian Epistemology and Philosophy of Science. In Falk Seeger, Johannes Lenard & Michael H. G. Hoffmann (eds.), Activity and Sign. Grounding Mathematical Education. Springer.score: 21.0
    A basic thesis of Neokantian epistemology and philosophy of science contends that the knowing subject and the object to be known are only abstractions. What really exists, is the relation between both. For the elucidation of this “knowledge relation ("Erkenntnisrelation") the Neokantians of the Marburg school used a variety of mathematical metaphors. In this con-tribution I reconsider some of these metaphors proposed by Paul Natorp, who was one of the leading members of the Marburg school. It is shown that (...)
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  17. 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.score: 21.0
    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 (...)
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  18. Stephen Cole Kleene (1967/2002). Mathematical Logic. Dover Publications.score: 21.0
    Undergraduate students with no prior classroom instruction in mathematical logic will benefit from this evenhanded multipart text by one of the centuries greatest authorities on the subject. Part I offers an elementary but thorough overview of mathematical logic of first order. The treatment does not stop with a single method of formulating logic; students receive instruction in a variety of techniques, first learning model theory (truth tables), then Hilbert-type proof theory, and proof theory handled through derived rules. Part (...)
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  19. Jan Von Plato (2007). In the Shadows of the Löwenheim-Skolem Theorem: Early Combinatorial Analyses of Mathematical Proofs. Bulletin of Symbolic Logic 13 (2):189-225.score: 21.0
    The Löwenheim-Skolem theorem was published in Skolem's long paper of 1920, with the first section dedicated to the theorem. The second section of the paper contains a proof-theoretical analysis of derivations in lattice theory. The main result, otherwise believed to have been established in the late 1980s, was a polynomial-time decision algorithm for these derivations. Skolem did not develop any notation for the representation of derivations, which makes the proofs of his results hard to follow. Such a formal (...) is given here by which these proofs become transparent. A third section of Skolem's paper gives an analysis for derivations in plane projective geometry. To clear a gap in Skolem's result, a new conservativity property is shown for projective geometry, to the effect that a proper use of the axiom that gives the uniqueness of connecting lines and intersection points requires a conclusion with proper cases (logically, a disjunction in a positive part) to be proved. The forgotten parts of Skolem's first paper on the Löwenheim-Skolem theorem are the perhaps earliest combinatorial analyses of formal mathematical proofs, and at least the earliest analyses with profound results. (shrink)
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  20. Helen De Cruz & Johan De Smedt (2013). Mathematical Symbols as Epistemic Actions. Synthese 190 (1):3-19.score: 18.0
    Recent experimental evidence from developmental psychology and cognitive neuroscience indicates that humans are equipped with unlearned elementary mathematical skills. However, formal mathematics has properties that cannot be reduced to these elementary cognitive capacities. The question then arises how human beings cognitively deal with more advanced mathematical ideas. This paper draws on the extended mind thesis to suggest that mathematical symbols enable us to delegate some mathematical operations to the external environment. In this view, mathematical symbols (...)
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  21. Valeria Giardino (2010). Intuition and Visualization in Mathematical Problem Solving. Topoi 29 (1):29-39.score: 18.0
    In this article, I will discuss the relationship between mathematical intuition and mathematical visualization. I will argue that in order to investigate this relationship, it is necessary to consider mathematical activity as a complex phenomenon, which involves many different cognitive resources. I will focus on two kinds of danger in recurring to visualization and I will show that they are not a good reason to conclude that visualization is not reliable, if we consider its use in (...) practice. Then, I will give an example of mathematical reasoning with a figure, and show that both visualization and intuition are involved. I claim that mathematical intuition depends on background knowledge and expertise, and that it allows to see the generality of the conclusions obtained by means of visualization. (shrink)
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  22. Helen De Cruz & Johan De Smedt (2010). The Innateness Hypothesis and Mathematical Concepts. Topoi 29 (1):3-13.score: 18.0
    In historical claims for nativism, mathematics is a paradigmatic example of innate knowledge. Claims by contemporary developmental psychologists of elementary mathematical skills in human infants are a legacy of this. However, the connection between these skills and more formal mathematical concepts and methods remains unclear. This paper assesses the current debates surrounding nativism and mathematical knowledge by teasing them apart into two distinct claims. First, in what way does the experimental evidence from infants, nonhuman animals and neuropsychology (...)
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  23. Giuseppe Longo & Arnaud Viarouge (2010). Mathematical Intuition and the Cognitive Roots of Mathematical Concepts. Topoi 29 (1):15-27.score: 18.0
    The foundation of Mathematics is both a logico-formal issue and an epistemological one. By the first, we mean the explicitation and analysis of formal proof principles, which, largely a posteriori, ground proof on general deduction rules and schemata. By the second, we mean the investigation of the constitutive genesis of concepts and structures, the aim of this paper. This “genealogy of concepts”, so dear to Riemann, Poincaré and Enriques among others, is necessary both in order to enrich the foundational analysis (...)
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  24. José Ferreirós (2009). Hilbert, Logicism, and Mathematical Existence. Synthese 170 (1):33 - 70.score: 18.0
    David Hilbert’s early foundational views, especially those corresponding to the 1890s, are analysed here. I consider strong evidence for the fact that Hilbert was a logicist at that time, following upon Dedekind’s footsteps in his understanding of pure mathematics. This insight makes it possible to throw new light on the evolution of Hilbert’s foundational ideas, including his early contributions to the foundations of geometry and the real number system. The context of Dedekind-style logicism makes it possible to offer a new (...)
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  25. Marianna Antonutti Marfori (2010). Informal Proofs and Mathematical Rigour. Studia Logica 96 (2):261-272.score: 18.0
    The aim of this paper is to provide epistemic reasons for investigating the notions of informal rigour and informal provability. I argue that the standard view of mathematical proof and rigour yields an implausible account of mathematical knowledge, and falls short of explaining the success of mathematical practice. I conclude that careful consideration of mathematical practice urges us to pursue a theory of informal provability.
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  26. Michał Walicki (2012). Introduction to Mathematical Logic. World Scientific.score: 18.0
    A history of logic -- Patterns of reasoning -- A language and its meaning -- A symbolic language -- 1850-1950 mathematical logic -- Modern symbolic logic -- Elements of set theory -- Sets, functions, relations -- Induction -- Turning machines -- Computability and decidability -- Propositional logic -- Syntax and proof systems -- Semantics of PL -- Soundness and completeness -- First order logic -- Syntax and proof systems of FOL -- Semantics of FOL -- More semantics -- Soundness (...)
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  27. W. V. Quine (1951). Mathematical Logic. Cambridge, Harvard University Press.score: 18.0
    INTRODUCTION MATHEMATICAL logic differs from the traditional formal logic so markedly in method, and so far surpasses it in power and subtlety, ...
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  28. Margaret Catherine Morrison (2006). Scientific Understanding and Mathematical Abstraction. Philosophia 34 (3):337-353.score: 18.0
    This paper argues for two related theses. The first is that mathematical abstraction can play an important role in shaping the way we think about and hence understand certain phenomena, an enterprise that extends well beyond simply representing those phenomena for the purpose of calculating/predicting their behaviour. The second is that much of our contemporary understanding and interpretation of natural selection has resulted from the way it has been described in the context of statistics and mathematics. I argue for (...)
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  29. Gualtiero Piccinini (2003). Alan Turing and the Mathematical Objection. Minds and Machines 13 (1):23-48.score: 18.0
    This paper concerns Alan Turing’s ideas about machines, mathematical methods of proof, and intelligence. By the late 1930s, Kurt Gödel and other logicians, including Turing himself, had shown that no finite set of rules could be used to generate all true mathematical statements. Yet according to Turing, there was no upper bound to the number of mathematical truths provable by intelligent human beings, for they could invent new rules and methods of proof. So, the output of a (...)
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  30. Hao Wang (1981/1993). Popular Lectures on Mathematical Logic. Dover Publications.score: 18.0
    Noted logician and philosopher addresses various forms of mathematical logic, discussing both theoretical underpinnings and practical applications. After historical survey, lucid treatment of set theory, model theory, recursion theory and constructivism and proof theory. Place of problems in development of theories of logic, logic’s relationship to computer science, more. Suitable for readers at many levels of mathematical sophistication. 3 appendixes. Bibliography. 1981 edition.
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  31. Richard Tieszen (2010). Mathematical Problem-Solving and Ontology: An Exercise. [REVIEW] Axiomathes 20 (2-3):295-312.score: 18.0
    In this paper the reader is asked to engage in some simple problem-solving in classical pure number theory and to then describe, on the basis of a series of questions, what it is like to solve the problems. In the recent philosophy of mind this “what is it like” question is one way of signaling a turn to phenomenological description. The description of what it is like to solve the problems in this paper, it is argued, leads to several morals (...)
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  32. Imre Lakatos (1976). Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge University Press.score: 18.0
    Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or (...)
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  33. A. Prestel (2011). Mathematical Logic and Model Theory: A Brief Introduction. Springer.score: 18.0
    Therefore, the text is divided into three parts: an introduction into mathematical logic (Chapter 1), model theory (Chapters 2 and 3), and the model theoretic ...
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  34. David Sherry (2009). The Role of Diagrams in Mathematical Arguments. Foundations of Science 14 (1-2):59-74.score: 18.0
    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 (...)
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  35. Mauro Dorato (2012). Mathematical Biology and the Existence of Biological Laws. In DieksD (ed.), Probabilities, Laws and Structure. Springer.score: 18.0
    An influential position in the philosophy of biology claims that there are no biological laws, since any apparently biological generalization is either too accidental, fact-like or contingent to be named a law, or is simply reducible to physical laws that regulate electrical and chemical interactions taking place between merely physical systems. In the following I will stress a neglected aspect of the debate that emerges directly from the growing importance of mathematical models of biological phenomena. My main aim is (...)
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  36. Jennifer Wilson Mulnix (2008). Reliabilism, Intuition, and Mathematical Knowledge. Filozofia 62 (8):715-723.score: 18.0
    It is alleged that the causal inertness of abstract objects and the causal conditions of certain naturalized epistemologies precludes the possibility of mathematical know- ledge. This paper rejects this alleged incompatibility, while also maintaining that the objects of mathematical beliefs are abstract objects, by incorporating a naturalistically acceptable account of ‘rational intuition.’ On this view, rational intuition consists in a non-inferential belief-forming process where the entertaining of propositions or certain contemplations results in true beliefs. This view is free (...)
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  37. Christian Hennig (2010). Mathematical Models and Reality: A Constructivist Perspective. [REVIEW] Foundations of Science 15 (1):29-48.score: 18.0
    To explore the relation between mathematical models and reality, four different domains of reality are distinguished: observer-independent reality (to which there is no direct access), personal reality, social reality and mathematical/formal reality. The concepts of personal and social reality are strongly inspired by constructivist ideas. Mathematical reality is social as well, but constructed as an autonomous system in order to make absolute agreement possible. The essential problem of mathematical modelling is that within mathematics there is agreement (...)
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  38. John N. Crossley (ed.) (1972/1990). What is Mathematical Logic? Dover Publications.score: 18.0
    This lively introduction to mathematical logic, easily accessible to non-mathematicians, offers an historical survey, coverage of predicate calculus, model theory, Godel’s theorems, computability and recursivefunctions, consistency and independence in axiomatic set theory, and much more. Suggestions for Further Reading. Diagrams.
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  39. Hannes Leitgeb (2013). Scientific Philosophy, Mathematical Philosophy, and All That. Metaphilosophy 44 (3):267-275.score: 18.0
    This article suggests that scientific philosophy, especially mathematical philosophy, might be one important way of doing philosophy in the future. Along the way, the article distinguishes between different types of scientific philosophy; it mentions some of the scientific methods that can serve philosophers; it aims to undermine some worries about mathematical philosophy; and it tries to make clear why in certain cases the application of mathematical methods is necessary for philosophical progress.
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  40. Gerhard Nuffer (2009). Stalnaker on Mathematical Information. Southern Journal of Philosophy 47 (2):187-204.score: 18.0
    Robert Stalnaker has argued that mathematical information is information about the sentences and expressions of mathematics. I argue that this metalinguistic account is open to a variant of Alonzo Church's translation objection and that Stalnaker's attempt to get around this objection is not successful. If correct, this tells not only against Stalnaker's account of mathematical truths, but against any metalinguistic account of truths that are both necessary and informative.
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  41. Haskell B. Curry (1963/1977). Foundations of Mathematical Logic. Dover Publications.score: 18.0
    Comprehensive account of constructive theory of first-order predicate calculus. Covers formal methods including algorithms and epi-theory, brief treatment of Markov’s approach to algorithms, elementary facts about lattices and similar algebraic systems, more. Philosophical and reflective as well as mathematical. Graduate-level course. 1963 ed. Exercises.
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  42. Jessica Carter (2013). Handling Mathematical Objects: Representations and Context. Synthese 190 (17):3983-3999.score: 18.0
    This article takes as a starting point the current popular anti realist position, Fictionalism, with the intent to compare it with actual mathematical practice. Fictionalism claims that mathematical statements do purport to be about mathematical objects, and that mathematical statements are not true. Considering these claims in the light of mathematical practice leads to questions about how mathematical objects are handled, and how we prove that certain statements hold. Based on a case study on (...)
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  43. Emil Leon Post (1941). The Two-Valued Iterative Systems of Mathematical Logic. London, H. Milford, Oxford University Press.score: 18.0
    INTRODUCTION In ita original form the present paper was presented to the American Mathematical Society, April 2k,, as a companion piece to the writer's ...
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  44. 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.score: 18.0
    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 (...)
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  45. Andrei Rodin (2010). How Mathematical Concepts Get Their Bodies. Topoi 29 (1):53-60.score: 18.0
    When the traditional distinction between a mathematical concept and a mathematical intuition is tested against examples taken from the real history of mathematics one can observe the following interesting phenomena. First, there are multiple examples where concepts and intuitions do not well fit together; some of these examples can be described as “poorly conceptualised intuitions” while some others can be described as “poorly intuited concepts”. Second, the historical development of mathematics involves two kinds of corresponding processes: poorly conceptualised (...)
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  46. Humphrey P. Polanen Van Petel (2006). Universal Grammar as a Theory of Notation. Axiomathes 16 (4):460-485.score: 18.0
    What is common to all languages is notation, so Universal Grammar can be understood as a system of notational types. Given that infants acquire language, it can be assumed to arise from some a priori mental structure. Viewing language as having the two layers of calculus and protocol, we can set aside the communicative habits of speakers. Accordingly, an analysis of notation results in the three types of Identifier, Modifier and Connective. Modifiers are further interpreted as Quantifiers and (...)
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  47. Roman Murawski (2010). Philosophy of Mathematics in the Warsaw Mathematical School. Axiomathes 20 (2-3):279-293.score: 18.0
    The aim of this paper is to present and discuss the philosophical views concerning mathematics of the founders of the so called Warsaw Mathematical School, i.e., Wacław Sierpiński, Zygmunt Janiszewski and Stefan Mazurkiewicz. Their interest in the philosophy of mathematics and their philosophical papers will be considered. We shall try to answer the question whether their philosophical views influenced their proper mathematical investigations. Their views towards set theory and its rôle in mathematics will be emphasized.
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  48. Jean Paul van Bendegem & Bart van Kerkhove (2009). Mathematical Arguments in Context. Foundations of Science 14 (1-2):45-57.score: 18.0
    Except in very poor mathematical contexts, mathematical arguments do not stand in isolation of other mathematical arguments. Rather, they form trains of formal and informal arguments, adding up to interconnected theorems, theories and eventually entire fields. This paper critically comments on some common views on the relation between formal and informal mathematical arguments, most particularly applications of Toulmin’s argumentation model, and launches a number of alternative ideas of presentation inviting the contextualization of pieces of mathematical (...)
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  49. Charles S. Chihara (1990). Constructibility and Mathematical Existence. Oxford University Press.score: 18.0
    Chihara here develops a mathematical system in which there are no existence assertions but only assertions of the constructibility of certain sorts of things. He utilizes this system in the analysis of the nature of mathematics, and discusses many recent works in the philosophy of mathematics from the viewpoint of the constructibility theory developed. This innovative analysis will appeal to mathematicians and philosophers of logic, mathematics, and science.
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