Search results for 'the mathematical Continuum' (try it on Scholar)

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  1. Anne Newstead (2001). Aristotle and Modern Mathematical Theories of the Continuum. In Demetra Sfendoni-Mentzou & James Brown (eds.), Aristotle and Contemporary Philosophy of Science. Peter Lang.score: 122.0
    This paper is on Aristotle's conception of the continuum. It is argued that although Aristotle did not have the modern conception of real numbers, his account of the continuum does mirror the topology of the real number continuum in modern mathematics especially as seen in the work of Georg Cantor. Some differences are noted, particularly as regards Aristotle's conception of number and the modern conception of real numbers. The issue of whether Aristotle had the notion of open (...)
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  2. Eduardo Castro (2013). Defending the Indispensability Argument: Atoms, Infinity and the Continuum. Journal for General Philosophy of Science 44 (1):41-61.score: 120.0
    This paper defends the Quine-Putnam mathematical indispensability argument against two objections raised by Penelope Maddy. The objections concern scientific practices regarding the development of the atomic theory and the role of applied mathematics in the continuum and infinity. I present two alternative accounts by Stephen Brush and Alan Chalmers on the atomic theory. I argue that these two theories are consistent with Quine’s theory of scientific confirmation. I advance some novel versions of the indispensability argument. I argue that (...)
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  3. Solomon Feferman, Is the Continuum Hypothesis a Definite Mathematical Problem?score: 108.0
    The purpose of this article is to explain why I believe that the Continuum Hypothesis (CH) is not a definite mathematical problem. My reason for that is that the concept of arbitrary set essential to its formulation is vague or underdetermined and there is no way to sharpen it without violating what it is supposed to be about. In addition, there is considerable circumstantial evidence to support the view that CH is not definite.
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  4. Giuseppe Longo (1999). The Mathematical Continuum, From Intuition to Logic. In Jean Petitot, Franscisco J. Varela, Barnard Pacoud & Jean-Michel Roy (eds.), Naturalizing Phenomenology. Stanford University Press.score: 90.0
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  5. José García García (1994). A. Bowie: estética y subjetividad. Logos: Anales Del Seminario de Metafísica 28 (2):337-348.score: 87.0
    This paper shows an analysis of some found solutions for the famous aporia of the race between Achilles and the Tortoise. As an introduction, we present the mechanical solution, to establish that it is not in the field of matters of fact where you can resolve a purely rational problem like the one raised by Zeno of Elea. And so, the main part of the article is dedicated to the mathematical solutions, which face the problem under the point of (...)
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  6. Stathis Livadas (2009). The Leap From the Ego of Temporal Consciousness to the Phenomenology of Mathematical Continuum. Manuscrito 32 (2).score: 87.0
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  7. José Enrique García Pascua (2003). Aquiles, la Tortuga y el infinito. Revista de Filosofía (Madrid) 28 (2):215-236.score: 87.0
    This paper shows an analysis of some found solutions for the famous aporia of the race between Achilles and the Tortoise. As an introduction, we present the mechanical solution, to establish that it is not in the field of matters of fact where you can resolve a purely rational problem like the one raised by Zeno of Elea. And so, the main part of the article is dedicated to the mathematical solutions, which face the problem under the point of (...)
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  8. Claude Laflamme (2001). Bartoszynski Tomek. On the Structure of Measurable Filters on a Countable Set. Real Analysis Exchange, Vol. 17 No. 2 (1992), Pp. 681–701. Bartoszynski Tomek and Shelah Saharon. Intersection of Archive for Mathematical Logic, Vol. 31 (1992), Pp. 221–226. Bartoszynski Tomek and Judah Haim. Measure and Category—Filters on Ω. Set Theory of the Continuum, Edited by Judah H., Just W., and Woodin H., Mathematical Sciences Research Institute Publications, Vol. 26, Springer-Verlag, New York, Berlin, Heidelberg ... [REVIEW] Bulletin of Symbolic Logic 7 (3):388-389.score: 81.0
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  9. Juris Steprāns (2005). Shelah S.. On Cardinal Invariants of the Continuum. Axiomatic Set Theory, Translated and Edited by Martin DA, Baumgartner J., and Shelah S., Contemporary Mathematics, Vol. 31. American Mathematical Society, Providence, 1984, Pp. 183–207. [REVIEW] Bulletin of Symbolic Logic 11 (3):451-453.score: 81.0
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  10. Heike Mildenberger (2002). Blass Andreas. Simple Cardinal Characteristics of the Continuum. Set Theory of the Reals, Edited by Judah Haim, Israel Mathematical Conference Proceedings, Vol. 6, Gelbart Research Institute for Mathematical Sciences, Bar-Ilan University, Ramat-Gan 1993, Distributed by the American Mathematical Society, Providence, Pp. 63–90. [REVIEW] Bulletin of Symbolic Logic 8 (4):552-553.score: 81.0
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  11. Joel W. Robbin (1997). Sprinkle H D. A Development of Cardinals in “The Consistency of the Continuum Hypothesis.” Proceedings of the American Mathematical Society, Vol. 7 (1956), Pp. 289–291. [REVIEW] Journal of Symbolic Logic 31 (4):663-663.score: 81.0
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  12. Peter Roeper (2006). The Aristotelian Continuum. A Formal Characterization. Notre Dame Journal of Formal Logic 47 (2):211-232.score: 80.0
    While the classical account of the linear continuum takes it to be a totality of points, which are its ultimate parts, Aristotle conceives of it as continuous and infinitely divisible, without ultimate parts. A formal account of this conception can be given employing a theory of quantification for nonatomic domains and a theory of region-based topology.
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  13. Anne Newstead & Franklin James (2008). On the Reality of the Continuum. Philosophy 83 (01):117-28.score: 75.0
    In a recent article (‘The Continuum: Russell’s Moment of Candour’), Christopher Ormell argues against the traditional mathematical view that the real numbers form an uncountably infinite set. He rejects the conclusion of Cantor’s diagonal argument for the higher, non-denumerable infinity of the real numbers. He does so on the basis that the classical conception of a real number is mys- terious, ineffable, and epistemically suspect. Instead, he urges that mathematics should admit only ‘well-defined’ real numbers as proper objects (...)
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  14. Matthew E. Moore (2007). The Genesis of the Peircean Continuum. Transactions of the Charles S. Peirce Society 43 (3):425 - 469.score: 74.0
    : In the Cambridge Conferences Lectures of 1898 Peirce defines a continuum as a "collection of so vast a multitude" that its elements "become welded into one another." He links the transinfinity (the "vast multitude") of a continuum to the confusion of its elements by a line of mathematical reasoning closely related to Cantor's Theorem. I trace the mathematical and philosophical roots of this conception of continuity, and examine its unresolved tensions, which arise mainly from difficulties (...)
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  15. Richard Jozsa (1986). An Approach to the Modelling of the Physical Continuum. British Journal for the Philosophy of Science 37 (4):395-404.score: 74.0
    We describe a way of constructing models for the continuum which does not require an underlying structure of points. With a condition of spatial homogeneity the models have the mathematical structure of a sheaf.
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  16. Kajsa Bråting (2012). Ambiguities of Fundamental Concepts in Mathematical Analysis During the Mid-Nineteenth Century. Foundations of Science 17 (4):301-320.score: 72.0
    In this paper we consider the major development of mathematical analysis during the mid-nineteenth century. On the basis of Jahnke’s (Hist Math 20(3):265–284, 1993 ) distinction between considering mathematics as an empirical science based on time and space and considering mathematics as a purely conceptual science we discuss the Swedish nineteenth century mathematician E.G. Björling’s general view of real- and complexvalued functions. We argue that Björling had a tendency to sometimes consider mathematical objects in a naturalistic way. One (...)
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  17. John L. Bell (2000). Hermann Weyl on Intuition and the Continuum. Philosophia Mathematica 8 (3):259-273.score: 66.0
    Hermann Weyl, one of the twentieth century's greatest mathematicians, was unusual in possessing acute literary and philosophical sensibilities—sensibilities to which he gave full expression in his writings. In this paper I use quotations from these writings to provide a sketch of Weyl's philosophical orientation, following which I attempt to elucidate his views on the mathematical continuum, bringing out the central role he assigned to intuition.
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  18. Sheldon R. Smith (2007). Continuous Bodies, Impenetrability, and Contact Interactions: The View From the Applied Mathematics of Continuum Mechanics. British Journal for the Philosophy of Science 58 (3):503 - 538.score: 66.0
    Many philosophers have claimed that there is a tension between the impenetrability of matter and the possibility of contact between continuous bodies. This tension has led some to claim that impenetrable continuous bodies could not ever be in contact, and it has led others to posit certain structural features to continuous bodies that they believe would resolve the tension. Unfortunately, such philosophical discussions rarely borrow much from the investigation of actual matter. This is probably largely because actual matter is not (...)
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  19. Janet Folina (2008). Intuition Between the Analytic-Continental Divide: Hermann Weyl's Philosophy of the Continuum. Philosophia Mathematica 16 (1):25-55.score: 66.0
    Though logical positivism is part of Kant's complex legacy, positivists rejected both Kant's theory of intuition and his classification of mathematical knowledge as synthetic a priori. This paper considers some lingering defenses of intuition in mathematics during the early part of the twentieth century, as logical positivism was born. In particular, it focuses on the difficult and changing views of Hermann Weyl about the proper role of intuition in mathematics. I argue that it was not intuition in general, but (...)
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  20. Paul J. Cohen (1966). Set Theory and the Continuum Hypothesis. New York, W. A. Benjamin.score: 64.0
  21. Paul Thompson (1998). The Nature and Role of Intuition in Mathematical Epistemology. Philosophia 26 (3-4):279-319.score: 63.0
    Great intuitions are fundamental to conjecture and discovery in mathematics. In this paper, we investigate the role that intuition plays in mathematical thinking. We review key events in the history of mathematics where paradoxes have emerged from mathematicians' most intuitive concepts and convictions, and where the resulting difficulties led to heated controversies and debates. Examples are drawn from Riemannian geometry, set theory and the analytic theory of the continuum, and include the Continuum Hypothesis, the Tarski-Banach Paradox, and (...)
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  22. Richard Tieszen (2000). The Philosophical Background of Weyl's Mathematical Constructivism. Philosophia Mathematica 8 (3):274-301.score: 63.0
    Weyl's inclination toward constructivism in the foundations of mathematics runs through his entire career, starting with Das Kontinuum. Why was Weyl inclined toward constructivism? I argue that Weyl's general views on foundations were shaped by a type of transcendental idealism in which it is held that mathematical knowledge must be founded on intuition. Kant and Fichte had an impact on Weyl but HusserFs transcendental idealism was even more influential. I discuss Weyl's views on vicious circularity, existence claims, meaning, the (...)
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  23. Gualtiero Piccinini (2003). Alan Turing and the Mathematical Objection. Minds and Machines 13 (1):23-48.score: 62.0
    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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  24. Geoffrey K. Pullum (2011). On the Mathematical Foundations of Syntactic Structures. Journal of Logic, Language and Information 20 (3):277-296.score: 62.0
    Chomsky’s highly influential Syntactic Structures ( SS ) has been much praised its originality, explicitness, and relevance for subsequent cognitive science. Such claims are greatly overstated. SS contains no proof that English is beyond the power of finite state description (it is not clear that Chomsky ever gave a sound mathematical argument for that claim). The approach advocated by SS springs directly out of the work of the mathematical logician Emil Post on formalizing proof, but few linguists are (...)
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  25. Carlo Cellucci (2000). The Growth of Mathematical Knowledge: An Open World View. In Emily Grosholz & Herbert Breger (eds.), The Growth of Mathematical Knowledge, pp. 153-176. Kluwer. 153--176.score: 60.0
    In his book The Value of Science Poincaré criticizes a certain view on the growth of mathematical knowledge: “The advance of science is not comparable to the changes of a city, where old edifices are pitilessly torn down to give place to new ones, but to the continuous evolution of zoological types which develop ceaselessly and end by becoming unrecognizable to the common sight, but where an expert eye finds always traces of the prior work of the centuries past” (...)
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  26. Philipp Bagus & David Howden (2012). The Continuing Continuum Problem of Deposits and Loans. Journal of Business Ethics 106 (3):295-300.score: 60.0
    Barnett and Block (J Bus Ethics 18(2):179–194, 2011 ) argue that one cannot distinguish between deposits and loans due to the continuum problem of maturities and because future goods do not exist—both essential characteristics that distinguish deposit from loan contracts. In a similar way but leading to opposite conclusions (Cachanosky, forthcoming) maintains that both maturity mismatching and fractional reserve banking are ethically justified as these contracts are equivalent. We argue herein that the economic and legal differences between genuine deposit (...)
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  27. Andrew Aberdein (2012). The Parallel Structure of Mathematical Reasoning. In Alison Pease & Brendan Larvor (eds.), Proceedings of the Symposium on Mathematical Practice and Cognition Ii: A Symposium at the Aisb/Iacap World Congress 2012. Society for the Study of Artificial Intelligence and the Simulation of Behaviour. 7--14.score: 60.0
    This paper proposes an account of mathematical reasoning as parallel in structure: the arguments which mathematicians use to persuade each other of their results comprise the argumentational structure; the inferential structure is composed of derivations which offer a formal counterpart to these arguments. Some conflicts about the foundations of mathematics correspond to disagreements over which steps should be admissible in the inferential structure. Similarly, disagreements over the admissibility of steps in the argumentational structure correspond to different views about (...) practice. The latter steps may be analysed in terms of argumentation schemes. Three broad types of scheme are distinguished, a distinction which is then used to characterize and evaluate four contrasting approaches to mathematical practice. (shrink)
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  28. Mary Grace Neville (2007). Positive Deviance on the Ethical Continuum. Proceedings of the International Association for Business and Society 18:72-75.score: 59.0
    Increasingly, stories are emerging about businesses that engage in ethical behaviors above and beyond mere compliance with regulations. These positive deviations along the ethical continuum provide an opportunity to explore how some companies’ business philosophy leads them to pursue an array of outcomes beyond the bottom line. This paper presents a case study of Green Mountain Coffee Roasters, the leading ethical company in the U.S. as rated by Forbes magazine, exploring the company culture and operating philosophy from a perspective (...)
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  29. Jacques Hadamard (1945/1996). The Mathematician's Mind: The Psychology of Invention in the Mathematical Field. Princeton University Press.score: 58.0
    Fifty years ago when Jacques Hadamard set out to explore how mathematicians invent new ideas, he considered the creative experiences of some of the greatest thinkers of his generation, such as George Polya, Claude Le;vi-Strauss, and Albert Einstein. It appeared that inspiration could strike anytime, particularly after an individual had worked hard on a problem for days and then turned attention to another activity. In exploring this phenomenon, Hadamard produced one of the most famous and cogent cases for the existence (...)
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  30. Christopher Clarke (forthcoming). Multi-Level Selection and the Explanatory Value of Mathematical Decompositions. British Journal for the Philosophy of Science.score: 57.0
    Do multi-level selection explanations of the evolution of social traits deepen the understanding provided by single-level explanations? Central to the former is a mathematical theorem, the multi-level Price decomposition. I build a framework through which to understand the explanatory role of such non-empirical decompositions in scientific practice. Applying this general framework to the present case places two tasks on the agenda. The first task is to distinguish the various ways of suppressing within-collective variation in fitness, and moreover to evaluate (...)
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  31. Jakob Kellner & Saharon Shelah (2012). Creature Forcing and Large Continuum: The Joy of Halving. Archive for Mathematical Logic 51 (1-2):49-70.score: 57.0
    For ${f,g\in\omega^\omega}$ let ${c^\forall_{f,g}}$ be the minimal number of uniform g-splitting trees needed to cover the uniform f-splitting tree, i.e., for every branch ν of the f-tree, one of the g-trees contains ν. Let ${c^\exists_{f,g}}$ be the dual notion: For every branch ν, one of the g-trees guesses ν(m) infinitely often. We show that it is consistent that ${c^\exists_{f_\epsilon,g_\epsilon}{=}c^\forall_{f_\epsilon,g_\epsilon}{=}\kappa_\epsilon}$ for continuum many pairwise different cardinals ${\kappa_\epsilon}$ and suitable pairs ${(f_\epsilon,g_\epsilon)}$ . For the proof we introduce a new mixed-limit creature (...)
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  32. Joseph F. Mucci (1974). On the Mathematical Form of de Broglie's Cyclical Action Integral. Foundations of Physics 4 (1):91-95.score: 56.0
    Mathematical expressions for the entropyS, the average information gained per trial (Ī) from information theory, and the de Broglie cyclical action integralA from his reinterpretation of wave mechanics are shown to be similar. The importance of this observation in our understanding ofS andĪ is considered. Furthermore, the similarity in the mathematical form of these functions indicates a possible route to further interpretation of de Broglie'sA and the nature of his “hidden thermostat.”.
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  33. Joseph S. Fulda (1992). The Mathematical Pull of Temptation. Mind 101 (402):305-307.score: 56.0
    Argues that the mathematical structure of a tempting or, more generally, risk-taking situation may prove far more dispositive of the choice made than either character or the lure/pull of the subject/object of temptation/risk-taking. -/- Briefly discusses some implications of this.
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  34. H. S. Arsen (2012). A Case For The Utility Of The Mathematical Intermediates. Philosophia Mathematica 20 (2):200-223.score: 56.0
    Many have argued against the claim that Plato posited the mathematical objects that are the subjects of Metaphysics M and N. This paper shifts the burden of proof onto these objectors to show that Plato did not posit these entities. It does so by making two claims: first, that Plato should posit the mathematical Intermediates because Forms and physical objects are ill suited in comparison to Intermediates to serve as the objects of mathematics; second, that their utility, combined (...)
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  35. Gregory H. Moore (2011). Early History of the Generalized Continuum Hypothesis: 1878—1938. Bulletin of Symbolic Logic 17 (4):489-532.score: 56.0
    This paper explores how the Generalized Continuum Hypothesis (GCH) arose from Cantor's Continuum Hypothesis in the work of Peirce, Jourdain, Hausdorff, Tarski, and how GCH was used up to Gödel's relative consistency result.
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  36. Alberto Vanzo (2005). Kant's Treatment of the Mathematical Antinomies in the First Critique and in the Prolegomena: A Comparison. Croatian Journal of Philosophy 5 (3):505-531.score: 56.0
    This paper discusses an apparent contrast between Kant’s accounts of the mathematical antinomies in the first Critique and in the Prolegomena. The Critique claims that the antitheses are infinite judgements. The Prolegomena seem to claim that they are negative judgements. For the Critique, theses and antitheses are false because they presuppose that the world has a determinate magnitude, and this is not the case. For the Prolegomena, theses and antitheses are false because they presuppose an inconsistent notion of world. (...)
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  37. Hubert C. Kennedy (1963). The Mathematical Philosophy of Giuseppe Peano. Philosophy of Science 30 (3):262-266.score: 56.0
    Because Bertrand Russell adopted much of the logical symbolism of Peano, because Russell always had a high regard for the great Italian mathematician, and because Russell held the logicist thesis so strongly, many English-speaking mathematicians have been led to classify Peano as a logicist, or at least as a forerunner of the logicist school. An attempt is made here to deny this by showing that Peano's primary interest was in axiomatics, that he never used the mathematical logic developed by (...)
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  38. Adrian Heathcote (2014). On the Exhaustion of Mathematical Entities by Structures. Axiomathes 24 (2):167-180.score: 56.0
    There has been considerable discussion in the literature of one kind of identity problem that mathematical structuralism faces: the automorphism problem, in which the structure is unable to individuate the mathematical entities in its domain. Shapiro (Philos Math 16(3):285–309, 2008) has partly responded to these concerns. But I argue here that the theory faces an even more serious kind of identity problem, which the theory can’t overcome staying within its remit. I give two examples to make the point.
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  39. J. B. Paris (1994). The Uncertain Reasoner's Companion: A Mathematical Perspective. Cambridge University Press.score: 56.0
    Reasoning under uncertainty, that is, making judgements with only partial knowledge, is a major theme in artificial intelligence. Professor Paris provides here an introduction to the mathematical foundations of the subject. It is suited for readers with some knowledge of undergraduate mathematics but is otherwise self-contained, collecting together the key results on the subject, and formalising within a unified framework the main contemporary approaches and assumptions. The author has concentrated on giving clear mathematical formulations, analyses, justifications and consequences (...)
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  40. Max Tegmark (2008). The Mathematical Universe. Foundations of Physics 38 (2):101-150.score: 56.0
    I explore physics implications of the External Reality Hypothesis (ERH) that there exists an external physical reality completely independent of us humans. I argue that with a sufficiently broad definition of mathematics, it implies the Mathematical Universe Hypothesis (MUH) that our physical world is an abstract mathematical structure. I discuss various implications of the ERH and MUH, ranging from standard physics topics like symmetries, irreducible representations, units, free parameters, randomness and initial conditions to broader issues like consciousness, parallel (...)
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  41. Toke Knudsen (2012). A Survey of the Mathematical Tradition of a Subcontinent. Metascience 21 (2):309-311.score: 56.0
    A survey of the mathematical tradition of a subcontinent Content Type Journal Article Category Book Review Pages 1-3 DOI 10.1007/s11016-011-9608-3 Authors Toke Knudsen, Department of Mathematics, Computer Science, and Statistics, SUNY Oneonta, Fitzelle Hall 234, Oneonta, NY 13820, USA Journal Metascience Online ISSN 1467-9981 Print ISSN 0815-0796.
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  42. Richard G. Lanzara (1994). Weber's Law Modeled by the Mathematical Description of a Beam Balance. .score: 56.0
    A beam balance is analyzed as a model that describes Weber's law. The mathematical derivations of the torques on a beam balance produce a description that is strictly compatible with that law. The natural relationship of the beam balance model to Weber's law provides for an intuitive understanding of the relationship of Weber's law to sensory and receptor systems. Additionally, this model may offer a simple way to compute perturbations that result from unequal effects on coupled steady state systems. (...)
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  43. Elizabeth de Freitas (2013). The Mathematical Event: Mapping the Axiomatic and the Problematic in School Mathematics. Studies in Philosophy and Education 32 (6):581-599.score: 56.0
    Traditional philosophy of mathematics has been concerned with the nature of mathematical objects rather than events. This traditional focus on reified objects is reflected in dominant theories of learning mathematics whereby the learner is meant to acquire familiarity with ideal mathematical objects, such as number, polygon, or tangent. I argue that the concept of event—rather than object—better captures the vitality of mathematics, and offers new ways of thinking about mathematics education. In this paper I draw on two different (...)
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  44. Hidemi Takahashi (2011). The Mathematical Sciences in Syriac: From Sergius of Resh-'Aina and Severus Sebokht to Barhebraeus and Patriarch Ni 'Matallah. Annals of Science 68 (4):477-491.score: 56.0
    Summary Syriac translations and Syriac scholars played an important role in the transmission of the sciences, including the mathematical sciences, from the Greek to the Arabic world. Relatively little, unfortunately, remains of the translations and original mathematical works of earlier Syriac scholars, but some materials have survived, and further glimpses of what once existed may be gained from works of later authors. The paper will provide an overview of the earlier materials that have survived or are known to (...)
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  45. I. Grattan-Guinness (ed.) (1994). Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences. Routledge.score: 56.0
    The Companion Encyclopedia is the first comprehensive work to cover all the principal lines and themes of the history and philosophy of mathematics from ancient times up to the twentieth century. In 176 articles contributed by 160 authors of 18 nationalities, the work describes and analyzes the variety of theories, proofs, techniques, and cultural and practical applications of mathematics. The work's aim is to recover our mathematical heritage and show the importance of mathematics today by treating its interactions with (...)
     
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  46. Giuseppe Longo & Arnaud Viarouge (2010). Mathematical Intuition and the Cognitive Roots of Mathematical Concepts. Topoi 29 (1):15-27.score: 54.0
    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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  47. Helen De Cruz & Johan De Smedt (2010). The Innateness Hypothesis and Mathematical Concepts. Topoi 29 (1):3-13.score: 54.0
    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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  48. Imre Lakatos (1976). Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge University Press.score: 54.0
    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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  49. David Sherry (2009). The Role of Diagrams in Mathematical Arguments. Foundations of Science 14 (1-2):59-74.score: 54.0
    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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  50. Geoffrey Hellman & Stewart Shapiro (2013). The Classical Continuum Without Points. Review of Symbolic Logic 6 (3):488-512.score: 54.0
    We develop a point-free construction of the classical one-dimensional continuum, with an interval structure based on mereology and either a weak set theory or a logic of plural quantification. In some respects, this realizes ideas going back to Aristotle, although, unlike Aristotle, we make free use of contemporary . Also, in contrast to intuitionistic analysis, smooth infinitesimal analysis, and Eret Bishopgunky lineindecomposabilityCantor structure of ℝ as a complete, separable, ordered field. We also present some simple topological models of our (...)
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