Search results for 'Numerical mathematics' (try it on Scholar)

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  1. Hans-Christoph Nuerk Korbinian Moeller, Laura Martignon, Silvia Wessolowski, Joachim Engel (2011). Effects of Finger Counting on Numerical Development – The Opposing Views of Neurocognition and Mathematics Education. Frontiers in Psychology 2.score: 192.0
    Usually children learn the basic principles of number and arithmetic by the help of finger-based representations. However, whether the reliance on finger-based representations is only beneficial or whether it may even become detrimental is the subject of an ongoing debate between neuro-cognitive and mathematics education researchers. From the neuro-cognitive perspective finger counting provides multi-sensory input conveying both cardinal and ordinal aspects of numbers. Recent data indicate that children with good finger-based numerical representations show better arithmetic skills and that (...)
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  2. Laurence Rousselle & Marie-Pascale Noël (2007). Basic Numerical Skills in Children with Mathematics Learning Disabilities: A Comparison of Symbolic Vs Non-Symbolic Number Magnitude Processing. Cognition 102 (3):361-395.score: 120.0
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  3. John Myhill (1972). Review: Errett Bishop, Foundations of Constructive Analysis; Errett Bishop, A. Kino, J. Myhill, R. E. Vesley, Mathematics as a Numerical Language. [REVIEW] Journal of Symbolic Logic 37 (4):744-747.score: 120.0
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  4. Korbinian Moeller, Laura Martignon, Silvia Wessolowski, Joachim Engel & Hans-Christoph Nuerk (2011). Effects of Finger Counting on Numerical Development–the Opposing Views of Neurocognition and Mathematics Education. Frontiers in Psychology 2.score: 120.0
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  5. Tobias U. Hauser, Stephanie Rotzer, Roland H. Grabner, Susan Mérillat & Lutz Jäncke (2013). Enhancing Performance in Numerical Magnitude Processing and Mental Arithmetic Using Transcranial Direct Current Stimulation (tDCS). Frontiers in Human Neuroscience 7.score: 102.0
    The ability to accurately process numerical magnitudes and solve mental arithmetic is of highest importance for schooling and professional career. Although impairments in these domains in disorders such as developmental dyscalculia (DD) are highly detrimental, remediation is still sparse. In recent years, transcranial brain stimulation methods such as transcranial Direct Current Stimulation (tDCS) have been suggested as a treatment for various neurologic and neuropsychiatric disorders. The posterior parietal cortex (PPC) is known to be crucially involved in numerical magnitude (...)
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  6. Han Geurdes, The Construction of Transfinite Equivalence Algorithms.score: 86.0
    Context: Consistency of mathematical constructions in numerical analysis and the application of computerized proofs in the light of the occurrence of numerical chaos in simple systems. Purpose: To show that a computer in general and a numerical analysis in particular can add its own peculiarities to the subject under study. Hence the need of thorough theoretical studies on chaos in numerical simulation. Hence, a questioning of what e.g. a numerical disproof of a theorem in physics (...)
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  7. Fenna van Nes (2011). Mathematics Education and Neurosciences: Towards Interdisciplinary Insights Into the Development of Young Children's Mathematical Abilities. Educational Philosophy and Theory 43 (1):75-80.score: 84.0
    The Mathematics Education and Neurosciences project is an interdisciplinary research program that bridges mathematics education research with neuroscientific research. The bidirectional collaboration will provide greater insight into young children's (aged four to six years) mathematical abilities. Specifically, by combining qualitative ‘design research’ with quantitative ‘experimental research’, we aim to come to a more thorough understanding of prerequisites that are involved in the development of early spatial and number sense. The mathematics education researchers are concerned with kindergartner's spatial (...)
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  8. Vitor G. Haase Annelise Júlio-Costa, Andressa M. Antunes, Júlia B. Lopes-Silva, Bárbara C. Moreira, Gabrielle S. Vianna, Guilherme Wood, Maria R. S. Carvalho (2013). Count on Dopamine: Influences of COMT Polymorphisms on Numerical Cognition. Frontiers in Psychology 4.score: 84.0
    Catechol-O-methyltransferase (COMT) is an enzyme that is particularly important for the metabolism of dopamine. Functional polymorphisms of COMT have been implicated in working memory and numerical cognition. This is an exploratory study that aims at investigating associations between COMT polymorphisms, working memory and numerical cognition. Elementary school children from 2th to 6th grades were divided into two groups according to their COMT val158met polymorphism (homozygous for valine allele [n= 61] versus heterozygous plus methionine homozygous children or met+ group (...)
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  9. Giuseppe Longo & Arnaud Viarouge (2010). Mathematical Intuition and the Cognitive Roots of Mathematical Concepts. Topoi 29 (1):15-27.score: 82.0
    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 (...)
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  10. Mauro Engelmann (2010). As Filosofias da Matemática de Wittgenstein: Intensionalismo Sistêmico e a Aplicação de um Novo Método (Sobre o Desenvolvimento da Filosofia da Matemática de Wittgenstein). Doispontos 6 (2).score: 72.0
    This essay intends to identify intentionalism (infinity given by rules, not by extensions) and the idea of multiple complete mathematical systems (several “mathematics”) as the central characteristics of Wittgenstein’s philosophy of mathematics. We intend to roughly show how these ideas come up, interact to each other, how they develop and, in the end, how they are abandoned in the late period. According to the Tractatus Logico-Philosophicus, infinities can only be given by rules and there is a single (...) system (the number’s essence is the general idea of ordering). Intentionalism is up to at least 1933, but the idea of a single system is abandoned in 1929-30 (already in the Philosophische Bemerkungen). In its place one finds the idea of multiple, independent and complete numerical systems. This idea will engender some key moves in Wittgenstein’s philosophy of Mathematics. The notion of “seeing an aspect” from the Big Typescript, of instance, comes up so as to explain such systems. From 1934 onwards, Wittgenstein gradually abandons intentionalism and the idea of multiple, independent and complete systems. In his late philosophy, both ideas are used only as instruments to dissolve philosophical prose regarding mathematics. (shrink)
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  11. M. Giaquinto (2007). Visual Thinking in Mathematics: An Epistemological Study. Oxford University Press.score: 66.0
    Visual thinking -- visual imagination or perception of diagrams and symbol arrays, and mental operations on them -- is omnipresent in mathematics. Is this visual thinking merely a psychological aid, facilitating grasp of what is gathered by other means? Or does it also have epistemological functions, as a means of discovery, understanding, and even proof? By examining the many kinds of visual representation in mathematics and the diverse ways in which they are used, Marcus Giaquinto argues that visual (...)
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  12. Nathan U. Salmon (2005). Metaphysics, Mathematics, and Meaning. Oxford University Press.score: 66.0
    Metaphysics, Mathematics, and Meaning brings together Nathan Salmon's influential papers on topics in the metaphysics of existence, non-existence, and fiction; modality and its logic; strict identity, including personal identity; numbers and numerical quantifiers; the philosophical significance of Godel's Incompleteness theorems; and semantic content and designation. Including a previously unpublished essay and a helpful new introduction to orient the reader, the volume offers rich and varied sustenance for philosophers and logicians.
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  13. Beckett Sterner (forthcoming). Well-Structured Biology: Numerical Taxonomy's Epistemic Vision for Systematics. In Andrew Hamilton (ed.), Patterns in Nature. University of California Press.score: 66.0
    What does it look like when a group of scientists set out to re-envision an entire field of biology in symbolic and formal terms? I analyze the founding and articulation of Numerical Taxonomy between 1950 and 1970, the period when it set out a radical new approach to classification and founded a tradition of mathematics in systematic biology. I argue that introducing mathematics in a comprehensive way also requires re-organizing the daily work of scientists in the field. (...)
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  14. D. F. M. Strauss (2010). The Significance of a Non-Reductionist Ontology for the Discipline of Mathematics: A Historical and Systematic Analysis. [REVIEW] Axiomathes 20 (1):19-52.score: 66.0
    A Christian approach to scholarship, directed by the central biblical motive of creation, fall and redemption and guided by the theoretical idea that God subjected all of creation to His Law-Word, delimiting and determining the cohering diversity we experience within reality, in principle safe-guards those in the grip of this ultimate commitment and theoretical orientation from absolutizing or deifying anything within creation. In this article my over-all approach is focused on the one-sided legacy of mathematics, starting with Pythagorean arithmeticism (...)
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  15. Felice L. Bedford (2001). Generality, Mathematical Elegance, and Evolution of Numerical/Object Identity. Behavioral and Brain Sciences 24 (4):654-655.score: 66.0
    Object identity, the apprehension that two glimpses refer to the same object, is offered as an example of combining generality, mathematics, and evolution. We argue that it applies to glimpses in time (apparent motion), modality (ventriloquism), and space (Gestalt grouping); that it has a mathematically elegant solution of nested geometries (Euclidean, Similarity, Affine, Projective, Topology); and that it is evolutionarily sound despite our Euclidean world. [Shepard].
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  16. Sarah Wu, Hitha Amin, Maria Barth, Vanessa Malcarne & Vinod Menon (2012). Math Anxiety in Second and Third Graders and Its Relation to Mathematics Achievement. Frontiers in Psychology 3.score: 66.0
    Although the detrimental effects of math anxiety in adults are well understood, few studies have examined how it affects younger children who are beginning to learn math in a formal academic setting. Here, we examine the relationship between math anxiety and math achievement in 2nd and 3rd graders. In response to the need for a grade-appropriate measure of assessing math anxiety in this group we first describe the development of Scale for Early Mathematics Anxiety (SEMA), a new measure for (...)
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  17. Amélie Lubin, Sandrine Rossi, Gregory Simon, Céline Lanoë, Gaëlle Leroux, Nicolas Poirel, Arlette Pineau & Olivier Houdé (2013). Numerical Transcoding Proficiency in 10-Year-Old Schoolchildren is Associated with Gray Matter Inter-Individual Differences: A Voxel-Based Morphometry Study. Frontiers in Psychology 4.score: 66.0
    Are individual differences in numerical performance sustained by variations in grey matter volume in schoolchildren? To our knowledge, this challenging question for neuroeducation has not yet been investigated in typical development. We used the Voxel-Based Morphometry method to search for possible structural brain differences between two groups of 10-year-old schoolchildren (N=22) whose performance differed only in numerical transcoding between analog and symbolic systems. The results indicated that children with low numerical proficiency have less grey matter volume in (...)
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  18. Daniel Ansari Filip Van Opstal, Seppe Santens (2012). The Numerate Brain: Recent Findings and Theoretical Reviews on the Neurocognitive Foundations of Number Processing. Frontiers in Human Neuroscience 6.score: 60.0
    The Numerate Brain: Recent Findings and Theoretical Reviews on the Neurocognitive Foundations of Number Processing.
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  19. Filip Van Opstal, Seppe Santens & Daniel Ansari (2012). The Numerate Brain: Recent Findings and Theoretical Reviews on the Neurocognitive Foundations of Number Processing. Frontiers in Human Neuroscience 6.score: 60.0
    The Numerate Brain: Recent Findings and Theoretical Reviews on the Neurocognitive Foundations of Number Processing.
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  20. Elizabeth S. Spelke, Sex Differences in Intrinsic Aptitude for Mathematics and Science?score: 54.0
    This article considers 3 claims that cognitive sex differ- ences account for the differential representation of men and women in high-level careers in mathematics and sci- ence: (a) males are more focused on objects from the beginning of life and therefore are predisposed to better learning about mechanical systems; (b) males have a pro- file of spatial and numerical abilities producing greater aptitude for mathematics; and (c) males are more variable in their cognitive abilities and therefore predominate (...)
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  21. Marcus Giaquinto (2011). Visual Thinking in Mathematics. OUP Oxford.score: 54.0
    Visual thinking - visual imagination or perception of diagrams and symbol arrays, and mental operations on them - is omnipresent in mathematics. Is this visual thinking merely a psychological aid, facilitating grasp of what is gathered by other means? Or does it also have epistemological functions, as a means of discovery, understanding, and even proof? By examining the many kinds of visual representation in mathematics and the diverse ways in which they are used, Marcus Giaquinto argues that visual (...)
     
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  22. Davide Rizza (2006). Measurement-Theoretic Observations on Field's Instrumentalism and the Applicability of Mathematics. Abstracta 2 (2):148-171.score: 54.0
    In this paper I examine Field’s account of the applicability of mathematics from a measurementtheoretic perspective. Within this context, I object to Field’s instrumentalism, arguing that it depends on an incomplete analysis of applicability. I show in particular that, once the missing piece of analysis is provided, the role played by numerical entities in basic empirical theories must be revised: such revision implies that instrumentalism should be rejected and mathematical entities be regarded not merely as useful tools but (...)
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  23. P. Rambaud (1989). Farmers and Their Self-Defined Numerical Mathematical Languages. Cahiers Internationaux de Sociologie 87:197-221.score: 50.0
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  24. Lance J. Rips, Amber Bloomfield & Jennifer Asmuth (2008). From Numerical Concepts to Concepts of Number. Behavioral and Brain Sciences 31 (6):623-642.score: 48.0
    Many experiments with infants suggest that they possess quantitative abilities, and many experimentalists believe that these abilities set the stage for later mathematics: natural numbers and arithmetic. However, the connection between these early and later skills is far from obvious. We evaluate two possible routes to mathematics and argue that neither is sufficient: (1) We first sketch what we think is the most likely model for infant abilities in this domain, and we examine proposals for extrapolating the natural (...)
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  25. Christian Agrillo (2013). One Vs. Two Non-Symbolic Numerical Systems? Looking to the ATOM Theory for Clues to the Mystery. Frontiers in Human Neuroscience 7.score: 44.0
    One vs. two non-symbolic numerical systems? Looking to the ATOM theory for clues to the mystery.
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  26. Andrea Bender & Sieghard Beller (2013). Of Adding Oranges and Apples: How Non-Abstract Representations May Foster Abstract Numerical Cognition. Frontiers in Human Neuroscience 7:903.score: 44.0
    Of adding oranges and apples: how non-abstract representations may foster abstract numerical cognition.
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  27. Charles Echelbarger (2013). Hume on the Objects of Mathematics. The European Legacy 18 (4):432-443.score: 42.0
    In this essay, I argue that Hume?s theory of Quantitative and Numerical Philosophical Relations can be interpreted in a way which allows mathematical knowledge to be about a body of objective and necessary truths, while preserving Hume?s nominalism and the basic principles of his theory of ideas. Attempts are made to clear up a number of obscure points about Hume?s claims concerning the abstract sciences of Arithmetic and Algebra by means of re-examining what he says and what he could (...)
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  28. Susan C. Hale (1990). Elementarity and Anti-Matter in Contemporary Physics: Comments on Michael D. Resnik's "Between Mathematics and Physics". PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1990:379 - 383.score: 42.0
    I point out that conceptions of particles as mathematical, or quasi mathematical, entities have a longer history than Resnik notices. I argue that Resnik's attack on the distinction between mathematical and physical entities is not deep enough. The crucial problem for this distinction finds its locus in the numerical indeterminancy of elementary particles. This problem, traced by Heisenberg, emerges from the discovery of antimatter.
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  29. S. Dehaene (1992). Varieties of Numerical Abilities. Cognition 44 (1-2):1-42.score: 42.0
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  30. Dfm Strauss (2011). Bernays, Dooyeweerd and Gödel – the Remarkable Convergence in Their Reflections on the Foundations of Mathematics. South African Journal of Philosophy 30 (1).score: 42.0
    In spite of differences the thought of Bernays, Dooyeweerd and Gödel evinces a remarkable convergence. This is particularly the case in respect of the acknowledgement of the difference between the discrete and the continuous, the foundational position of number and the fact that the idea of continuity is derived from space (geometry – Bernays). What is furthermore similar is the recognition of what is primitive (and indefinable) as well as the account of the coherence of what is unique, such as (...)
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  31. Fraser MacBride (2006). What Constitutes the Numerical Diversity of Mathematical Objects? Analysis 66 (289):63–69.score: 40.0
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  32. Stephanie Bugden & Daniel Ansari (2011). Individual Differences in Children's Mathematical Competence Are Related to the Intentional but Not Automatic Processing of Arabic Numerals. Cognition 118 (1):32-44.score: 40.0
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  33. Andrzej Blikle (1964). Review: A. A. Markov, Mathematical Logic and Numerical Analysis. [REVIEW] Journal of Symbolic Logic 29 (4):209-209.score: 40.0
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  34. D. Haskell, G. Hjorth, C. Jockusch, A. Kanamori, H. J. Keisler, V. McGee & T. Pitassi (2000). Structures and the Hyperarithmetical Hierarchy. Knight has Directed or Co-Directed Seven Doctoral Dissertations in Mathematics and One in Electrical Engineering. She Served on Selection Panels for the NSF Postdoctoral Fellowships, on Program Committees of Numerous Meetings, and as an Editor of The Journal of Symbolic Logic (1989-1995). [REVIEW] Bulletin of Symbolic Logic 6 (1).score: 40.0
     
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  35. Franfoise Monnoyeur Broitman (2013). The Indefinite within Descartes' Mathematical Physics. Eidos 19 (19):107-122.score: 38.0
    Descartes' philosophy contains an intriguing notion of the infinite, a concept labeled by the philosopher as indefinite. Even though Descartes clearly defined this term on several occasions in the correspondence with his contemporaries, as well as in his Principles of Philosophy, numerous problems about its meaning have arisen over the years. Most commentators reject the view that the indefinite could mean a real thing and, instead, identify it with an Aristotelian potential infinite. In the first part of this article, I (...)
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  36. Ursula Anderson & Sara Cordes (2013). 1 < 2 and 2 < 3: Non-Linguistic Appreciations of Numerical Order. Frontiers in Psychology 4.score: 38.0
    Ordinal understanding is involved in understanding social hierarchies, series of actions, and everyday events. Moreover, an appreciation of numerical order is critical to understanding to number at a highly abstract, conceptual level. In this paper, we review the findings concerning the development and expression of ordinal numerical knowledge in preverbal human infants in light of literature about the same cognitive abilities in nonhuman animals. We attempt to reconcile seemingly contradictory evidence, provide new directions for prospective research, and evaluate (...)
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  37. Manuela Piazza, Pierre Pica, Véronique Izard, Elizabeth Spelke & Stanislas Dehaene (2013). Education Enhances the Acuity of the Nonverbal Approximate Number System. Psychological Science 24 (4):p.score: 36.0
    All humans share a universal, evolutionarily ancient approximate number system (ANS) that estimates and combines the numbers of objects in sets with ratio-limited precision. Interindividual variability in the acuity of the ANS correlates with mathematical achievement, but the causes of this correlation have never been established. We acquired psychophysical measures of ANS acuity in child and adult members of an indigene group in the Amazon, the Mundurucú, who have a very restricted numerical lexicon and highly variable access to (...) education. By comparing Mundurucú subjects with and without access to schooling, we found that education significantly enhances the acuity with which sets of concrete objects are estimated. These results indicate that culture and education have an important effect on basic number perception. We hypothesize that symbolic and nonsymbolic numerical thinking mutually enhance one another over the course of mathematics instruction. (shrink)
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  38. Pierre Pica, Stanislas Dehaene, Elizabeth Spelke & Véronique Izard (2008). Log or Linear? Distinct Intuitions of the Number Scale in Western and Amazonian Indigene Cultures. Science 320 (5880):1217-1220.score: 36.0
    The mapping of numbers onto space is fundamental to measurement and to mathematics. Is this mapping a cultural invention or a universal intuition shared by all humans regardless of culture and education? We probed number-space mappings in the Mundurucu, an Amazonian indigene group with a reduced numerical lexicon and little or no formal education. At all ages, the Mundurucu mapped symbolic and nonsymbolic numbers onto a logarithmic scale, whereas Western adults used linear mapping with small or symbolic numbers (...)
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  39. Karl Menger (1954). On Variables in Mathematics and in Natural Science. British Journal for the Philosophy of Science 5 (18):134-142.score: 36.0
    Attempting to answer the question "what is a variable?," menger discusses the following topics: (1) numerical variables and variables in the sense of the logicians, (2) variable quantities, (3) scientific variable quantities, (4) functions, And (5) variable quantities and functions in pure and applied analysis. (staff).
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  40. Stephen Wolfram, Mathematics by Computer.score: 36.0
    The most elementary way to think about Mathctrtati ca is as an enhance calculator — a calculator that does not only numerical computation but also algebraic computation and graphics. Matltcmatica can function much like a standard calt".1a- tor. you type in a question, you get back an answer. But Mat/tctttadca ga's turthcr I ue an ordinary calculator. You can type in questions that require answers that arc longer than a calculator can handle. For example, Matltcmatictt can giv; you thc (...)
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  41. Catherine Rowett (2013). Philosophy's Numerical Turn: Why the Pythagoreans' Interest in Numbers is Truly Awesome. In Dirk Obbink & David Sider (eds.), Doctrine and Doxography: Studies on Heraclitus and Pythagoras. De Gruyter. 3-32.score: 36.0
    Philosophers are generally somewhat wary of the hints of number mysticism in the reports about the beliefs and doctrines of the so-called Pythagoreans. It's not clear how much Pythagoras himself (as opposed to his later followers) indulged in speculation about numbers, or in more serious mathematics. But the Pythagoreans whom Aristotle discusses in the Metaphysics had some elaborate stories to tell about how the universe could be explained in terms of numbers—not just its physics but perhaps morality too. Was (...)
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  42. Gerardo Santana Trujillo (2012). Las Matemáticas en el Pensamiento de Vilém Flusser. Flusser Studies 13.score: 36.0
    This paper aims to establish the importance of mathematical thinking in the work of Vilém Flusser. For this purpose highlights the concept of escalation of abstraction with which the Czech German philosopher finishes by reversing the top of the traditional pyramid of knowledge, we know from Plato and Aristotle. It also assumes the implicit cultural revolution in the refinement of the numerical element in a process of gradual abandonment of purely alphabetic code, highlights the new key code, together with (...)
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  43. Marian Mrozek & Jacek Urbaniec (1997). Evolution of Mathematical Proof. Foundations of Science 2 (1):77-85.score: 34.0
    The authors present the main ideas of the computer-assisted proof of Mischaikow and Mrozek that chaos is really present in the Lorenz equations. Methodological consequences of this proof are examined. It is shown that numerical calculations can constitute an essential part of mathematical proof not only in the discrete mathematics but also in the mathematics of continua.
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  44. Sophie Roux (2010). Forms of Mathematization (14th-17th Centuries). Early Science and Medicine 15 (4):319-337.score: 34.0
    According to a grand narrative that long ago ceased to be told, there was a seventeenth century Scientific Revolution, during which a few heroes conquered nature thanks to mathematics. This grand narrative began with the exhibition of quantitative laws that these heroes, Galileo and Newton for example, had disclosed: the law of falling bodies, according to which the speed of a falling body is proportional to the square of the time that has elapsed since the beginning of its fall; (...)
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  45. David Barner (2008). In Defense of Intuitive Mathematical Theories as the Basis for Natural Number. Behavioral and Brain Sciences 31 (6):643-644.score: 34.0
    Though there are holes in the theory of how children move through stages of numerical competence, the current approach offers the most promising avenue for characterizing changes in competence as children confront new mathematical concepts. Like the science of mathematics, children's discovery of number is rooted in intuitions about sets, and not purely in analytic truths.
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  46. Mark Balaguer, Fictionalism in the Philosophy of Mathematics. Stanford Encyclopedia of Philosophy.score: 30.0
    Mathematical fictionalism (or as I'll call it, fictionalism) is best thought of as a reaction to mathematical platonism. Platonism is the view that (a) there exist abstract mathematical objects (i.e., nonspatiotemporal mathematical objects), and (b) our mathematical sentences and theories provide true descriptions of such objects. So, for instance, on the platonist view, the sentence ‘3 is prime’ provides a straightforward description of a certain object—namely, the number 3—in much the same way that the sentence ‘Mars is red’ provides a (...)
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  47. Danielle Macbeth (2007). Striving for Truth in the Practice of Mathematics: Kant and Frege. Grazer Philosophische Studien 75 (1):65-92.score: 30.0
    My aim is to understand the practice of mathematics in a way that sheds light on the fact that it is at once a priori and capable of extending our knowledge. The account that is sketched draws first on the idea, derived from Kant, that a calculation or demonstration can yield new knowledge in virtue of the fact that the system of signs it employs involves primitive parts (e.g., the ten digits of arithmetic or the points, lines, angles, and (...)
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  48. Jonathan Lear (1982). Aristotle's Philosophy of Mathematics. Philosophical Review 91 (2):161-192.score: 30.0
    Whether aristotle wrote a work on mathematics as he did on physics is not known, and sources differ. this book attempts to present the main features of aristotle's philosophy of mathematics. methodologically, the presentation is based on aristotle's "posterior analytics", which discusses the nature of scientific knowledge and procedure. concerning aristotle's views on mathematics in particular, they are presented with the support of numerous references to his extant works. his criticism of his predecessors is added at the (...)
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  49. Lieven Decock (2008). The Conceptual Basis of Numerical Abilities: One-to-One Correspondence Versus the Successor Relation. Philosophical Psychology 21 (4):459 – 473.score: 30.0
    In recent years, neologicists have demonstrated that Hume's principle, based on the one-to-one correspondence relation, suffices to construct the natural numbers. This formal work is shown to be relevant for empirical research on mathematical cognition. I give a hypothetical account of how nonnumerate societies may acquire arithmetical knowledge on the basis of the one-to-one correspondence relation only, whereby the acquisition of number concepts need not rely on enumeration (the stable-order principle). The existing empirical data on the role of the one-to-one (...)
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  50. George R. Exner (1997). An Accompaniment to Higher Mathematics. Springer.score: 30.0
    This text prepares undergraduate mathematics students to meet two challenges in the study of mathematics, namely, to read mathematics independently and to understand and write proofs. The book begins by teaching how to read mathematics actively, constructing examples, extreme cases, and non-examples to aid in understanding an unfamiliar theorem or definition (a technique famililar to any mathematician, but rarely taught); it provides practice by indicating explicitly where work with pencil and paper must interrupt reading. The book (...)
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