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: 528.0
    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. Solomon Feferman, Is the Continuum Hypothesis a Definite Mathematical Problem?score: 486.0
    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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  3. 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: 450.0
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  4. Gordon Locke (2000). Peirce's Metaphysics: Evolution, Synechism, and the Mathematical Conception of the Continuum. Transactions of the Charles S. Peirce Society 36 (1):133 - 147.score: 444.0
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  5. Stathis Livadas (2009). The Leap From the Ego of Temporal Consciousness to the Phenomenology of Mathematical Continuum. Manuscrito 32 (2).score: 435.0
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  6. Eduardo Castro (2013). Defending the Indispensability Argument: Atoms, Infinity and the Continuum. Journal for General Philosophy of Science 44 (1):41-61.score: 432.0
    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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  7. 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: 405.0
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  8. 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: 405.0
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  9. 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: 405.0
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  10. 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: 405.0
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  11. Matthew E. Moore (2007). The Genesis of the Peircean Continuum. Transactions of the Charles S. Peirce Society 43 (3):425 - 469.score: 306.0
    : 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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  12. Richard Jozsa (1986). An Approach to the Modelling of the Physical Continuum. British Journal for the Philosophy of Science 37 (4):395-404.score: 306.0
    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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  13. Anne Newstead & Franklin James (2008). On the Reality of the Continuum. Philosophy 83 (01):117-28.score: 297.0
    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. Anne Newstead & James Franklin (2008). On the Reality of the Continuum Discussion Note: A Reply to Ormell, 'Russell's Moment of Candour', "Philosophy&Quot;. Philosophy 83 (323):117 - 127.score: 285.0
    This paper discusses an argument for the reality of the classical mathematical continuum. An inference to the best explanation type of argument is used to defend the idea that real numbers exist even when they cannot be constructively specified as with the "indefinable numbers".
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  15. John L. Bell (2000). Hermann Weyl on Intuition and the Continuum. Philosophia Mathematica 8 (3):259-273.score: 270.0
    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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  16. Janet Folina (2008). Intuition Between the Analytic-Continental Divide: Hermann Weyl's Philosophy of the Continuum. Philosophia Mathematica 16 (1):25-55.score: 270.0
    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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  17. Paul J. Cohen (1966). Set Theory and the Continuum Hypothesis. New York, W. A. Benjamin.score: 264.0
  18. Paul Thompson (1998). The Nature and Role of Intuition in Mathematical Epistemology. Philosophia 26 (3-4):279-319.score: 261.0
    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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  19. Richard Tieszen (2000). The Philosophical Background of Weyl's Mathematical Constructivism. Philosophia Mathematica 8 (3):274-301.score: 261.0
    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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  20. José García García (1994). A. Bowie: estética y subjetividad. Logos 28 (2):337-348.score: 261.0
    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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  21. José Enrique García Pascua (2003). Aquiles, la Tortuga y el infinito. Revista de Filosofía (Madrid) 28 (2):215-236.score: 261.0
    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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  22. 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: 250.0
    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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  23. Jakob Kellner & Saharon Shelah (2012). Creature Forcing and Large Continuum: The Joy of Halving. Archive for Mathematical Logic 51 (1-2):49-70.score: 243.0
    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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  24. Arthur L. Rubin & Jean E. Rubin (1993). Weak Forms of the Axiom of Choice and the Generalized Continuum Hypothesis. Mathematical Logic Quarterly 39 (1):7-22.score: 237.0
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  25. Stephen Pollard (2007). Mathematical Determinacy and the Transferability of Aboutness. Synthese 159 (1):83 - 98.score: 234.0
    Competent speakers of natural languages can borrow reference from one another. You can arrange for your utterances of ‘Kirksville’ to refer to the same thing as my utterances of ‘Kirksville’. We can then talk about the same thing when we discuss Kirksville. In cases like this, you borrow “aboutness” from me by borrowing reference. Now suppose I wish to initiate a line of reasoning applicable to any prime number. I might signal my intention by saying, “Let p be any (...)
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  26. Kurt Gödel (1940). The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory. Princeton University Press;.score: 228.0
  27. Sara Negri & Daniele Soravia (1999). The Continuum as a Formal Space. Archive for Mathematical Logic 38 (7):423-447.score: 225.0
    A constructive definition of the continuum based on formal topology is given and its basic properties studied. A natural notion of Cauchy sequence is introduced and Cauchy completeness is proved. Other results include elementary proofs of the Baire and Cantor theorems. From a classical standpoint, formal reals are seen to be equivalent to the usual reals. Lastly, the relation of real numbers as a formal space to other approaches to constructive real numbers is determined.
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  28. Ramez L. Sami (1989). Turing Determinacy and the Continuum Hypothesis. Archive for Mathematical Logic 28 (3):149-154.score: 225.0
    From the hypothesis that all Turing closed games are determined we prove: (1) the Continuum Hypothesis and (2) every subset of ℵ1 is constructible from a real.
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  29. András Hajnal (1956). On a Consistency Theorem Connected with the Generalized Continuum Problem. Mathematical Logic Quarterly 2 (8‐9):131-136.score: 219.0
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  30. Stephen H. Hechler (1973). Powers of Singular Cardinals and a Strong Form of The Negation of The Generalized Continuum Hypothesis. Mathematical Logic Quarterly 19 (3‐6):83-84.score: 219.0
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  31. Solomon Feferman, Conceptual Structuralism and the Continuum.score: 216.0
    • This comes from my general view of the nature of mathematics, that it is humanly based and that it deals with more or less clear conceptions of mathematical structures; for want of a better word, I call that view conceptual structuralism.
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  32. Peter Roeper (2006). The Aristotelian Continuum. A Formal Characterization. Notre Dame Journal of Formal Logic 47 (2):211-232.score: 216.0
    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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  33. Carl J. Posy (2000). Epistemology, Ontology and the Continuum. In. In Emily Grosholz & Herbert Breger (eds.), The Growth of Mathematical Knowledge. Kluwer Academic Publishers. 199--219.score: 216.0
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  34. Paul Feyerabend (1983). Some Observations on Aristotle's Theory of Mathematics and of the Continuum. Midwest Studies in Philosophy 8 (1):67-88.score: 215.0
  35. John L. Bell (2005). Divergent Conceptions of the Continuum in 19th and Early 20th Century Mathematics and Philosophy. Axiomathes 15 (1):63-84.score: 215.0
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  36. P. Cassou-Nogues (2013). Albert Lautman. Mathematics, Ideas and the Physical Real. Simon B. Duffy, Trans. London and New York: Continuum, 2011. 978-1-4411-2344-2 (Pbk); 978-1-44114656-4 (Hbk); 978-1-44114433-1 (Pdf E-Bk); 978-1-44114654-0 (Epub E-Bk). Pp. Xlii + 310. [REVIEW] Philosophia Mathematica 21 (3):411-416.score: 215.0
  37. Rudolf Carnap (1952). The Continuum of Inductive Methods. [Chicago]University of Chicago Press.score: 198.0
  38. Raymond Louis Wilder (1965/2012). Introduction to the Foundations of Mathematics: Second Edition. Dover Publications, Inc..score: 195.0
    This_classic undergraduate text_elegantly acquaints students with the_fundamental concepts and methods of mathematics. In addition to introducing_many noteworthy historical figures_from the 18th through the mid-20th centuries, it examines_the axiomatic method, set theory, infinite sets, the linear continuum and the real number system, groups, intuitionism,_formal systems, mathematical logic, and other topics.
     
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  39. Kajsa Bråting (2012). Ambiguities of Fundamental Concepts in Mathematical Analysis During the Mid-Nineteenth Century. Foundations of Science 17 (4):301-320.score: 192.0
    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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  40. Alexander Abian (1973). The Consistency of the Continuum Hypothesis Via Synergistic Models. Mathematical Logic Quarterly 19 (13):193-198.score: 189.0
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  41. Roy O. Davies (1962). Equivalence to the Continuum Hypothesis of a Certain Proposition of Elementary Plane Geometry. Mathematical Logic Quarterly 8 (2):109-111.score: 189.0
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  42. Keith J. Devlin (1980). Concerning the Consistency of the Souslin Hypothesis with the Continuum Hypothesis. Annals of Mathematical Logic 19 (1-2):115-125.score: 189.0
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  43. Rolf Schock (1971). On the Axiom of Choice and the Continuum Hypothesis. Mathematical Logic Quarterly 17 (1):35-37.score: 189.0
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  44. Frederick Bagemihl (1959). Some Results Connected with the Continuum Hypothesis. Mathematical Logic Quarterly 5 (7‐13):97-116.score: 189.0
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  45. Jacques Grassin (1981). Δ11‐Good Inductive Definitions Over The Continuum. Mathematical Logic Quarterly 27 (1):11-16.score: 189.0
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  46. James Robert Brown (2008). Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures. Routledge.score: 183.0
    1. Introduction : the mathematical image -- 2. Platonism -- 3. Picture-proofs and Platonism -- 4. What is applied mathematics? -- 5. Hilbert and Gödel -- 6. Knots and notation -- 7. What is a definition? -- 8. Constructive approaches -- 9. Proofs, pictures and procedures in Wittgenstein -- 10. Computation, proof and conjecture -- 11. How to refute the continuum hypothesis -- 12. Calling the bluff.
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  47. Stojan Obradović & Slobodan Ninković (2009). The Heuristic Function of Mathematics in Physics and Astronomy. Foundations of Science 14 (4):351-360.score: 183.0
    This paper considers the role of mathematics in the process of acquiring new knowledge in physics and astronomy. The defining of the notions of continuum and discreteness in mathematics and the natural sciences is examined. The basic forms of representing the heuristic function of mathematics at theoretical and empirical levels of knowledge are studied: deducing consequences from the axiomatic system of theory, the method of generating mathematical hypotheses, “pure” proofs for the existence of objects and processes, mathematical (...)
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  48. Gualtiero Piccinini (2003). Alan Turing and the Mathematical Objection. Minds and Machines 13 (1):23-48.score: 180.0
    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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  49. Janet Folina (1994). Poincare on Mathematics, Intuition and the Foundations of Science. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1994:217 - 226.score: 180.0
    In his first philosophy book, Science and Hypothesis, Poincare provides a picture in which the different sciences are arranged in a hierarchy. Arithmetic is the most general of all the sciences because it is presupposed by all the others. Next comes mathematical magnitude, or the analysis of the continuum, which presupposes arithmetic; and so on. Poincare's basic view was that experiment in science depends on fixing other concepts first. More generally, certain concepts must be fixed before others: hence (...)
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  50. Geoffrey K. Pullum (2011). On the Mathematical Foundations of Syntactic Structures. Journal of Logic, Language and Information 20 (3):277-296.score: 180.0
    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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