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
    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.  2
    Stathis Livadas (2009). The Leap From the Ego of Temporal Consciousness to the Phenomenology of Mathematical Continuum. Manuscrito 32 (2):321-356.
    This article attempts to link the notion of absolute ego as the ultimate subjectivity of consciousness in continental tradition with a phenomenology of Mathematical Continuum as this term is generally established following Cantor’s pioneering ideas on the properties and cardinalities of sets. My motivation stems mainly from the inherent ambiguities underlying the nature and properties of this fundamental mathematical notion which, in my view, cannot be resolved in principle by the analytical means of any formal language not (...)
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  3. Solomon Feferman, Is the Continuum Hypothesis a Definite Mathematical Problem?
    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
     
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  5.  34
    Eduardo Castro (2013). Defending the Indispensability Argument: Atoms, Infinity and the Continuum. Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 44 (1):41-61.
    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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  6.  2
    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.
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  7.  2
    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.
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  8.  1
    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.
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  9. James Cummings (1995). Gitik Moti and Magidor Menachem. The Singular Cardinal Hypothesis Revisited. 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 Etc. 1992, Pp. 243–279. [REVIEW] Journal of Symbolic Logic 60 (1):339-340.
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  10. Gabriel Debs (2001). Dougherty Randall and Kechris Alexander S.. The Complexity of Antidifferentiation. Advances in Mathematics, Vol. 88 , Pp. 145–169.Beleznay Ferenc and Foreman Matthew. The Collection of Distal Flows is Not Borel. American Journal of Mathematics, Vol. 117 , Pp. 203–239.Beleznay Ferenc and Foreman Matthew. The Complexity of the Collection of Measure-Distal Transformations. Ergodic Theory and Dynamical Systems, Vol. 16 , Pp. 929–962.Becker Howard. Pointwise Limits of Subsequences and Sets. Fundamenta Mathematicae, Vol. 128 , Pp. 159–170.Becker Howard, Kahane Sylvain, and Louveau Alain. Some Complete Sets in Harmonic Analysis. Transactions of the American Mathematical Society, Vol. 339 , Pp. 323–336.Kaufman Robert. PCA Sets and Convexity Fundamenta Mathematicae, Vol. 163 , Pp. 267–275).Becker Howard. Descriptive Set Theoretic Phenomena in Analysis and Topology. Set Theory of the Continuum, Edited by Judah H., Just W., and Woodin H., Mathematical Sciences Research Institute Publications, Vol. [REVIEW] Bulletin of Symbolic Logic 7 (3):385-388.
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  11. Natasha Dobrinen (2014). Cummings James and Schimmerling Ernest, Editors. Lecture Note Series of the London Mathematical Society, Vol. 406. Cambridge University Press, New York, Xi + 419 Pp.Larson Paul B., Lumsdaine Peter, and Yin Yimu. An Introduction to Pmax Forcing. Pp. 5–23.Thomas Simon and Schneider Scott. Countable Borel Equivalence Relations. Pp. 25–62.Farah Ilijas and Wofsey Eric. Set Theory and Operator Algebras. Pp. 63–119.Moore Justin and Milovich David. A Tutorial on Set Mapping Reflection. Pp. 121–144.Pestov Vladimir G. And Kwiatkowska Aleksandra. An Introduction to Hyperlinear and Sofic Groups. Pp. 145–185.Neeman Itay and Unger Spencer. Aronszajn Trees and the SCH. Pp. 187–206.Eisworth Todd, Tatch Moore Justin, and Milovich David. Iterated Forcing and the Continuum Hypothesis. Pp. 207–244.Gitik Moti and Unger Spencer. Short Extender Forcing. Pp. 245–263.Kechris Alexander S. And Tucker-Drob Robin D.. The Complexity of Classification Problems in Ergodic Theory. Pp. 265–299.Magidor Menachem and Lamb. [REVIEW] Bulletin of Symbolic Logic 20 (1):94-97.
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  12. Stephen J. Garland (1975). Kunen Kenneth. Indescribability and the Continuum. Axiomatic Set Theory, Proceedings of Symposia in Pure Mathematics, Vol. 13 Part 1, American Mathematical Society, Providence, Rhode Island, 1971, Pp. 199–203. [REVIEW] Journal of Symbolic Logic 40 (4):632.
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  13. S. C. Kleene (1948). Reviews. Kurt Gödel. What is Cantor's Continuum Problem? The American Mathematical Monthly, Vol. 54 , Pp. 515–525. Journal of Symbolic Logic 13 (2):116-117.
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  14. Claude Laflamme (2001). Bartoszynski Tomek. On the Structure of Measurable Filters on a Countable Set. Real Analysis Exchange, Vol. 17 No. 2 , Pp. 681–701.Bartoszynski Tomek and Shelah Saharon. Intersection of Archive for Mathematical Logic, Vol. 31 , 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, Etc., 1992, Pp. 175–201.Bartoszynski Tomek, Goldstern Martin, Judah Haim, and Shelah Saharon. All Meager Filters May Be Null. Proceedings of the American Mathematical Society, Vol. 117 , Pp. 515–521.Bartoszyński Tomek. Remarks on the Intersection of Filters. Topology and its Applications, Vol. 84 , Pp. 139–143. [REVIEW] Bulletin of Symbolic Logic 7 (3):388-389.
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  15. J. W. Robbin (1968). Scott Dana. A Proof of the Independence of the Continuum Hypothesis. Mathematical Systems Theory, Vol. 1 , Pp. 89–111. Journal of Symbolic Logic 33 (2):293.
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  16. 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.
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  17. C. Smoryński (1979). Hilbert David. Mathematical Problems. Lecture Delivered Before the International Congress of Mathematicians at Paris in 1900. A Reprint of 1084 . Mathematical Developments Arising From Hilbert Problems, Proceedings of the Symposium in Pure Mathematics of the American Mathematical Society, Held at Northern Illinois University, De Kalb, Illinois, May 1974, Edited by Browder Felix E., Proceedings of Symposia in Pure Mathematics, Vol. 28, American Mathematical Society, Providence 1976, Pp. 1–34.Martin Donald A.. Hilbert's First Problem: The Continuum Hypothesis. A Reprint of 1084 . Mathematical Developments Arising From Hilbert Problems, Proceedings of the Symposium in Pure Mathematics of the American Mathematical Society, Held at Northern Illinois University, De Kalb, Illinois, May 1974, Edited by Browder Felix E., Proceedings of Symposia in Pure Mathematics, Vol. 28, American Mathematical Society, Providence 1976, Pp. 81–92.Kreisel G.. What Have We Learnt From Hilbert's Second Problem? A. [REVIEW] Journal of Symbolic Logic 44 (1):116-119.
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  18. Leslie H. Tharp (1970). Karp Carol. A Proof of the Relative Consistency of the Continuum Hypothesis. Sets, Models and Recursion Theory, Proceedings of the Summer School in Mathematical Logic and Tenth Logic Colloquium, Leicester, August-September 1965, Edited by Crossley John N., Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Company, Amsterdam, and Humanities Press, New York, 1967, Pp. 1–32. [REVIEW] Journal of Symbolic Logic 35 (2):344-345.
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  19.  8
    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.
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  20.  44
    Matthew E. Moore (2007). The Genesis of the Peircean Continuum. Transactions of the Charles S. Peirce Society 43 (3):425 - 469.
    : 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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  21.  16
    Richard Jozsa (1986). An Approach to the Modelling of the Physical Continuum. British Journal for the Philosophy of Science 37 (4):395-404.
    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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  22.  21
    Anne Newstead & James Franklin (2008). On the Reality of the Continuum Discussion Note: A Reply to Ormell, 'Russell's Moment of Candour', "Philosophy". Philosophy 83 (323):117-127.
    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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  23.  48
    Anne Newstead & Franklin James (2008). On the Reality of the Continuum. Philosophy 83 (1):117-28.
    In a recent article, 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 of study. In practice, this means excluding (...)
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  24.  6
    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.
    In this paper we study some statements similar to the Partition Principle and the Trichotomy. We prove some relationships between these statements, the Axiom of Choice, and the Generalized Continuum Hypothesis. We also prove some independence results. MSC: 03E25, 03E50, 04A25, 04A50.
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  25.  97
    John L. Bell (2000). Hermann Weyl on Intuition and the Continuum. Philosophia Mathematica 8 (3):259-273.
    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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  26.  43
    Janet Folina (2008). Intuition Between the Analytic-Continental Divide: Hermann Weyl's Philosophy of the Continuum. Philosophia Mathematica 16 (1):25-55.
    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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  27.  10
    Paul J. Cohen (1966). Set Theory and the Continuum Hypothesis. New York, W. A. Benjamin.
  28.  21
    Richard Tieszen (2000). The Philosophical Background of Weyl's Mathematical Constructivism. Philosophia Mathematica 8 (3):274-301.
    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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  29.  15
    Stuart Elden (2001). The Place of Geometry: Heidegger's Mathematical Excursus on Aristotle. Heythrop Journal 42 (3):311–328.
    ‘The Place of Geometry’ discusses the excursus on mathematics from Heidegger's 1924–25 lecture course on Platonic dialogues, which has been published as Volume 19 of the Gesamtausgabe as Plato's Sophist, as a starting point for an examination of geometry in Euclid, Aristotle and Descartes. One of the crucial points Heidegger makes is that in Aristotle there is a fundamental difference between arithmetic and geometry, because the mode of their connection is different. The units of geometry are positioned, the units of (...)
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  30.  35
    Paul Thompson (1998). The Nature and Role of Intuition in Mathematical Epistemology. Philosophia 26 (3-4):279-319.
    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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  31.  3
    Colin Rittberg (2015). How Woodin Changed His Mind: New Thoughts on the Continuum Hypothesis. Archive for History of Exact Sciences 69 (2):125-151.
    The Continuum Problem has inspired set theorists and philosophers since the days of Cantorian set theory. In the last 15 years, W. Hugh Woodin, a leading set theorist, has not only taken it upon himself to engage in this question, he has also changed his mind about the answer. This paper illustrates Woodin’s solutions to the problem, starting in Sect. 3 with his 1999–2004 argument that Cantor’s hypothesis about the continuum was incorrect. From 2010 onwards, Woodin presents a (...)
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  32.  4
    José Enrique García Pascua (2003). Aquiles, la Tortuga y el infinito. Revista de Filosofía (Madrid) 28 (2):215-236.
    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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  33.  3
    José García García (1994). A. Bowie: estética y subjetividad. Logos 28 (2):337-348.
    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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  34.  9
    Jakob Kellner & Saharon Shelah (2012). Creature Forcing and Large Continuum: The Joy of Halving. Archive for Mathematical Logic 51 (1-2):49-70.
    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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  35.  20
    Stephen Pollard (2007). Mathematical Determinacy and the Transferability of Aboutness. Synthese 159 (1):83-98.
    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 (...)
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  36.  6
    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;.
  37.  11
    Sara Negri & Daniele Soravia (1999). The Continuum as a Formal Space. Archive for Mathematical Logic 38 (7):423-447.
    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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  38.  9
    Ramez L. Sami (1989). Turing Determinacy and the Continuum Hypothesis. Archive for Mathematical Logic 28 (3):149-154.
    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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  39.  4
    András Hajnal (1956). On a Consistency Theorem Connected with the Generalized Continuum Problem. Mathematical Logic Quarterly 2 (8‐9):131-136.
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  40.  3
    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.
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  41.  13
    Colin Hamlin (forthcoming). Towards a Theory of Universes: Structure Theory and the Mathematical Universe Hypothesis. Synthese:1-21.
    The maturation of the physical image has made apparent the limits of our scientific understanding of fundamental reality. These limitations serve as motivation for a new form of metaphysical inquiry that restricts itself to broadly scientific methods. Contributing towards this goal we combine the mathematical universe hypothesis as developed by Max Tegmark with the axioms of Stewart Shapiro’s structure theory. The result is a theory we call the Theory of the Structural Multiverse. The focus is on informal theory development (...)
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  42. Solomon Feferman, Conceptual Structuralism and the Continuum.
    • 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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  43.  22
    Peter Roeper (2006). The Aristotelian Continuum. A Formal Characterization. Notre Dame Journal of Formal Logic 47 (2):211-232.
    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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  44.  5
    Carl J. Posy (2000). Epistemology, Ontology and the Continuum. In Emily Grosholz & Herbert Breger (eds.), The Growth of Mathematical Knowledge. Kluwer Academic Publishers 199--219.
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  45.  42
    Rudolf Carnap (1952). The Continuum of Inductive Methods. [Chicago]University of Chicago Press.
  46.  7
    Walter A. Carnielli, Itala M. L. D'ottaviano & Brazilian Conference on Mathematical Logic (1999). Advances in Contemporary Logic and Computer Science Proceedings of the Eleventh Brazilian Conference on Mathematical Logic, May 6-10, 1996, Salvador, Bahia, Brazil. [REVIEW] Monograph Collection (Matt - Pseudo).
    This volume presents the proceedings from the Eleventh Brazilian Logic Conference on Mathematical Logic held by the Brazilian Logic Society (co-sponsored by the Centre for Logic, Epistemology and the History of Science, State University of Campinas, Sao Paulo) in Salvador, Bahia, Brazil. The conference and the volume are dedicated to the memory of professor Mario Tourasse Teixeira, an educator and researcher who contributed to the formation of several generations of Brazilian logicians. Contributions were made from leading Brazilian logicians and (...)
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  47.  11
    Simon D. Smith (2015). Kant’s Mathematical Sublime and the Role of the Infinite: Reply to Crowther. Kantian Review 20 (1):99-120.
    This paper offers an analysis of KantNature is thus sublime in those of its appearances the intuition of which brings with them the idea of its infinitys interpretation of this species of aesthetic experience, and I reject his interpretation as not being reflective of Kant’s actual position. I go on to show that the experience of the mathematical sublime is necessarily connected with the progression of the imagination in its move towards the infinite.
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  48.  9
    Kajsa Bråting (2012). Ambiguities of Fundamental Concepts in Mathematical Analysis During the Mid-Nineteenth Century. Foundations of Science 17 (4):301-320.
    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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  49.  15
    Alexander Abian (1973). The Consistency of the Continuum Hypothesis Via Synergistic Models. Mathematical Logic Quarterly 19 (13):193-198.
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  50.  5
    Roy O. Davies (1962). Equivalence to the Continuum Hypothesis of a Certain Proposition of Elementary Plane Geometry. Mathematical Logic Quarterly 8 (2):109-111.
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