26 found
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  1.  33
    A Language and Axioms for Explicit Mathematics.Solomon Feferman, J. N. Crossley, Maurice Boffa, Dirk van Dalen & Kenneth Mcaloon - 1984 - Journal of Symbolic Logic 49 (1):308-311.
  2.  93
    Brouwer and Weyl: The Phenomenology and Mathematics of the Intuitive Continuum.Mark van Atten, Dirk van Dalen & Richard Tieszen - 2002 - Philosophia Mathematica 10 (2):203-226.
    Brouwer and Weyl recognized that the intuitive continuum requires a mathematical analysis of a kind that set theory is not able to provide. As an alternative, Brouwer introduced choice sequences. We first describe the features of the intuitive continuum that prompted this development, focusing in particular on the flow of internal time as described in Husserl's phenomenology. Then we look at choice sequences and their logic. Finally, we investigate the differences between Brouwer and Weyl, and argue that Weyl's conception of (...)
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  3. Brouwer and Weyl: The phenomenology and mathematics of the intuitive continuumt.Mark van Atten, Dirk van Dalen & Richard Tieszen - 2002 - Philosophia Mathematica 10 (2):203-226.
    Brouwer and Weyl recognized that the intuitive continuum requires a mathematical analysis of a kind that set theory is not able to provide. As an alternative, Brouwer introduced choice sequences. We first describe the features of the intuitive continuum that prompted this development, focusing in particular on the flow of internal time as described in Husserl's phenomenology. Then we look at choice sequences and their logic. Finally, we investigate the differences between Brouwer and Weyl, and argue that Weyl's conception of (...)
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  4. Hermann Weyl's intuitionistic mathematics.Dirk van Dalen - 1995 - Bulletin of Symbolic Logic 1 (2):145-169.
    Dedicated to Dana Scott on his sixtieth birthday.It is common knowledge that for a short while Hermann Weyl joined Brouwer in his pursuit of a revision of mathematics according to intuitionistic principles. There is, however, little in the literature that sheds light on Weyl's role and in particular on Brouwer's reaction to Weyl's allegiance to the cause of intuitionism. This short episode certainly raises a number of questions: what made Weyl give up his own program, spelled out in “Das Kontinuum”, (...)
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  5. Zermelo and the Skolem paradox.Dirk Van Dalen & Heinz-Dieter Ebbinghaus - 2000 - Bulletin of Symbolic Logic 6 (2):145-161.
    On October 4, 1937, Zermelo composed a small note entitled “Der Relativismus in der Mengenlehre und der sogenannte Skolemsche Satz” in which he gives a refutation of “Skolem's paradox”, i.e., the fact that Zermelo-Fraenkel set theory—guaranteeing the existence of uncountably many sets—has a countable model. Compared with what he wished to disprove, the argument fails. However, at a second glance, it strongly documents his view of mathematics as based on a world of objects that could only be grasped adequately by (...)
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  6.  60
    From Brouwerian counter examples to the creating subject.Dirk van Dalen - 1999 - Studia Logica 62 (2):305-314.
    The original Brouwerian counter examples were algorithmic in nature; after the introduction of choice sequences, Brouwer devised a version which did not depend on algorithms. This is the origin of the creating subject technique. The method allowed stronger refutations of classical principles. Here it is used to show that negative dense subsets of the continuum are indecomposable.
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  7.  88
    Arguments for the continuity principle.Mark van Atten & Dirk van Dalen - 2002 - Bulletin of Symbolic Logic 8 (3):329-347.
    There are two principles that lend Brouwer's mathematics the extra power beyond arithmetic. Both are presented in Brouwer's writings with little or no argument. One, the principle of bar induction, will not concern us here. The other, the continuity principle for numbers, occurs for the first time in print in [4]. It is formulated and immediately applied to show that the set of numerical choice sequences is not enumerable. In fact, the idea of the continuity property can be dated fairly (...)
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  8.  17
    Intuitionistic Logic.Dirk van Dalen - 2017 - In Lou Goble (ed.), The Blackwell Guide to Philosophical Logic. Oxford, UK: Blackwell. pp. 224–257.
    There are basically two ways to view intuitionistic logic: as a philosophical‐foundational issue in mathematics; or as a technical discipline within mathematical logic. Considering first the philosophical aspects, for they will provide the motivation for the subject, this chapter follows L. E. J. Brouwer, the founding father of intuitionism. Although Brouwer himself contributed little to intuitionistic logic as seen from textbooks and papers, he did point the way for his successors.
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  9.  70
    How connected is the intuitionistic continuum?Dirk van Dalen - 1997 - Journal of Symbolic Logic 62 (4):1147-1150.
  10.  10
    Brouwer and Fraenkel on Intuitionism.Dirk Van Dalen - 2000 - Bulletin of Symbolic Logic 6 (3):284-310.
    In the present paper the story is told of the brief and far from tranquil encounter of L.E.J. Brouwer and A. Fraenkel. The relationship which started in perfect harmony, ended in irritation and reproaches.The mutual appreciation at the outset is beyond question. All the more deplorable is the sudden outbreak of an emotional disagreement in 1927. Looking at the Brouwer–Fraenkel episode, one should keep in mind that at that time the so-called Grundlagenstreit was in full swing. An emotional man like (...)
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  11.  67
    Brouwer and Fraenkel on intuitionism.Dirk Van Dalen - 2000 - Bulletin of Symbolic Logic 6 (3):284-310.
    In the present paper the story is told of the brief and far from tranquil encounter of L.E.J. Brouwer and A. Fraenkel. The relationship which started in perfect harmony, ended in irritation and reproaches.The mutual appreciation at the outset is beyond question. All the more deplorable is the sudden outbreak of an emotional disagreement in 1927. Looking at the Brouwer–Fraenkel episode, one should keep in mind that at that time the so-called Grundlagenstreit was in full swing. An emotional man like (...)
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  12.  79
    Zermelo and the Skolem Paradox.Dirk Van Dalen & Heinz-Dieter Ebbinghaus - 2000 - Bulletin of Symbolic Logic 6 (2):145-161.
    On October 4, 1937, Zermelo composed a small note entitled “Der Relativismus in der Mengenlehre und der sogenannte Skolemsche Satz”(“Relativism in Set Theory and the So-Called Theorem of Skolem”) in which he gives a refutation of “Skolem's paradox”, i.e., the fact that Zermelo-Fraenkel set theory—guaranteeing the existence of uncountably many sets—has a countable model. Compared with what he wished to disprove, the argument fails. However, at a second glance, it strongly documents his view of mathematics as based on a world (...)
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  13.  36
    Variants of Rescher's semantics for preference logic and some completeness theorems.Dirk van Dalen - 1974 - Studia Logica 33 (2):163-181.
  14.  14
    L.E.J. Brouwer: Topologist, Intuitionist, Philosopher: How Mathematics is Rooted in Life.Dirk van Dalen - 2012 - Springer.
    Dirk van Dalen’s biography studies the fascinating life of the famous Dutch mathematician and philosopher Luitzen Egbertus Jan Brouwer. Brouwer belonged to a special class of genius; complex and often controversial and gifted with a deep intuition, he had an unparalleled access to the secrets and intricacies of mathematics. Most mathematicians remember L.E.J. Brouwer from his scientific breakthroughs in the young subject of topology and for the famous Brouwer fixed point theorem. Brouwer’s main interest, however, was in the foundation of (...)
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  15.  13
    Computer Science Logic.Dirk van Dalen & Marc Bezem (eds.) - 1997 - Springer.
    The related fields of fractal image encoding and fractal image analysis have blossomed in recent years. This book, originating from a NATO Advanced Study Institute held in 1995, presents work by leading researchers. It is developing the subjects at an introductory level, but it also has some recent and exciting results in both fields. The book contains a thorough discussion of fractal image compression and decompression, including both continuous and discrete formulations, vector space and hierarchical methods, and algorithmic optimizations. The (...)
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  16. Dedicated to Dana Scott on his sixtieth birthday.Dirk van Dalen - 1995 - Bulletin of Symbolic Logic 1 (2).
  17. Eine Bemerkung zum Aufsatz „Der Fundamentalsatz der Algebra und der Intuitionismus “von H. Kneser.Dirk van Dalen - 1985 - Archive for Mathematical Logic 25 (1):43-44.
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  18.  17
    Fans Generated by Nondeterministic Automata.Dirk van Dalen - 1968 - Mathematical Logic Quarterly 14 (18):273-278.
  19.  23
    Fans Generated by Nondeterministic Automata.Dirk van Dalen - 1968 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 14 (18):273-278.
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  20. Heinz-Dieter Ebbinghaus. Zermelo and the Skolem Paradox.Dirk Van Dalen - 2000 - Bulletin of Symbolic Logic 1 (2):145-161.
  21.  7
    Intuitionism.Dirk van Dalen & Mark van Atten - 2006 - In Dale Jacquette (ed.), A Companion to Philosophical Logic. Oxford, UK: Blackwell. pp. 511–530.
    This chapter contains sections titled: Logic: The Proof Interpretation Analysis: Choice Sequences Further Semantics.
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  22.  32
    Reducibilities in intuitionistic topology.Dirk Van Dalen - 1968 - Journal of Symbolic Logic 33 (3):412-417.
  23.  36
    Fourman M. P. and Scott D. S.. Sheaves and logic. Applications of sheaves, Proceedings of the Research Symposium on Applications of Sheaf Theory to Logic, Algebra, and Analysis, Durham, July 9–21, 1977, edited by Fourman M. P., Mulvey C. J., and Scott D. S., Lecture notes in mathematics, vol. 753, Springer-Verlag, Berlin, Heidelberg, and New York, 1979, pp. 302–401. [REVIEW]Dirk van Dalen - 1983 - Journal of Symbolic Logic 48 (4):1201-1203.
  24.  26
    Review: Elliott Mendelson, Introduction to Mathematical Logic. [REVIEW]Dirk van Dalen - 1969 - Journal of Symbolic Logic 34 (1):110-111.
  25.  13
    Review: M. P. Fourman, D. S. Scott, C. J. Mulvey, Sheaves and Logic. [REVIEW]Dirk van Dalen - 1983 - Journal of Symbolic Logic 48 (4):1201-1203.
  26.  15
    Arguments for the Continuity Principle. [REVIEW]Mark van Atten & Dirk van Dalen - 2002 - Bulletin of Symbolic Logic 8 (3):329-347.
    There are two principles that lend Brouwer's mathematics the extra power beyond arithmetic. Both are presented in Brouwer's writings with little or no argument. One, the principle of bar induction, will not concern us here. The other, the continuity principle for numbers, occurs for the first time in print in [4]. It is formulated and immediately applied to show that the set of numerical choice sequences is not enumerable. In fact, the idea of the continuity property can be dated fairly (...)
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