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32Evolution and design: The Darwinian view of evolution is a scientific fact and not an ideologyComplexity 11 (1): 12-15. 2005.
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19The end of Moore's law: Living without an exponential increase in the efficiency of computational facilitiesComplexity 21 (S1): 6-9. 2016.
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51Countable choice as a questionable uniformity principlePhilosophia Mathematica 12 (2): 106-134. 2004.Should weak forms of the axiom of choice really be accepted within constructive mathematics? A critical view of the Brouwer-Heyting-Kolmogorov interpretation, accompanied by the intention to include nondeterministic algorithms, leads us to subscribe to Richman's appeal for dropping countable choice. As an alternative interpretation of intuitionistic logic, we propose to renew dialogue semantics.
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15Compactness under constructive scrutinyMathematical Logic Quarterly 50 (6): 540-550. 2004.How are the various classically equivalent definitions of compactness for metric spaces constructively interrelated? This question is addressed with Bishop-style constructive mathematics as the basic system – that is, the underlying logic is the intuitionistic one enriched with the principle of dependent choices. Besides surveying today's knowledge, the consequences and equivalents of several sequential notions of compactness are investigated. For instance, we establish the perhaps unexpected co…Read more
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The Weak Koenig Lemma, Brouwer's Fan Theorem, De Morgan's Law, and Dependent ChoiceReports on Mathematical Logic 63-86. 2012.
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26Formal Zariski topology: positivity and pointsAnnals of Pure and Applied Logic 137 (1-3): 317-359. 2006.The topic of this article is the formal topology abstracted from the Zariski spectrum of a commutative ring. After recollecting the fundamental concepts of a basic open and a covering relation, we study some candidates for positivity. In particular, we present a coinductively generated positivity relation. We further show that, constructively, the formal Zariski topology cannot have enough points
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42The Fan Theorem and Unique Existence of MaximaJournal of Symbolic Logic 71 (2). 2006.The existence and uniqueness of a maximum point for a continuous real—valued function on a metric space are investigated constructively. In particular, it is shown, in the spirit of reverse mathematics, that a natural unique existence theorem is equivalent to the fan theorem
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11Approximating beppo levi’s principio di approssimazioneBulletin of Symbolic Logic 20 (2): 141-169. 2014.We try to recast in modern terms a choice principle conceived by Beppo Levi, who called it the Approximation Principle. Up to now, there was almost no discussion about Levi’s contribution, due to the quite obscure formulation of AP the author has chosen. After briefly reviewing the historical and philosophical surroundings of Levi’s proposal, we undertake our own attempt at interpreting AP. The idea underlying the principle, as well as the supposed faithfulness of our version to Levi’s original …Read more
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51Too simple solutions of hard problemsNordic Journal of Philosophical Logic 6 (2): 138-146. 2010.Volume 6, Issue 2, 2001, Page 138-146.
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17Concepts of Proof in Mathematics, Philosophy, and Computer Science (edited book)De Gruyter. 2016.
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23On the contrapositive of countable choiceArchive for Mathematical Logic 50 (1-2): 137-143. 2011.We show that in elementary analysis (EL) the contrapositive of countable choice is equivalent to double negation elimination for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Sigma_{2}^{0}}$$\end{document}-formulas. By also proving a recursive adaptation of this equivalence in Heyting arithmetic (HA), we give an …Read more
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56Linear independence without choiceAnnals of Pure and Applied Logic 101 (1): 95-102. 1999.The notions of linear and metric independence are investigated in relation to the property: if U is a set of n+1 independent vectors, and X is a set of n independent vectors, then adjoining some vector in U to X results in a set of n+1 independent vectors. It is shown that this property holds in any normed linear space. A related property – that finite-dimensional subspaces are proximinal – is established for strictly convex normed spaces over the real or complex numbers. It follows that metric …Read more
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34The Kripke schema in metric topologyMathematical Logic Quarterly 58 (6): 498-501. 2012.A form of Kripke's schema turns out to be equivalent to each of the following two statements from metric topology: every open subspace of a separable metric space is separable; every open subset of a separable metric space is a countable union of open balls. Thus Kripke's schema serves as a point of reference for classifying theorems of classical mathematics within Bishop-style constructive reverse mathematics
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12A direct proof of Wiener's theoremIn S. Barry Cooper (ed.), How the World Computes, . pp. 293--302. 2012.
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83Are There Enough Injective Sets?Studia Logica 101 (3): 467-482. 2013.The axiom of choice ensures precisely that, in ZFC, every set is projective: that is, a projective object in the category of sets. In constructive ZF (CZF) the existence of enough projective sets has been discussed as an additional axiom taken from the interpretation of CZF in Martin-Löf’s intuitionistic type theory. On the other hand, every non-empty set is injective in classical ZF, which argument fails to work in CZF. The aim of this paper is to shed some light on the problem whether there ar…Read more
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30Untamable curiosity, innovation, discovery, and bricolage: Are we doomed to progress to ever increasing complexity?Complexity 11 (5): 9-11. 2006.
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32Quasi-apartness and neighbourhood spacesAnnals of Pure and Applied Logic 141 (1): 296-306. 2006.We extend the concept of apartness spaces to the concept of quasi-apartness spaces. We show that there is an adjunction between the category of quasi-apartness spaces and the category of neighbourhood spaces, which indicates that quasi-apartness is a more natural concept than apartness. We also show that there is an adjoint equivalence between the category of apartness spaces and the category of Grayson’s separated spaces
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33Strong continuity implies uniform sequential continuityArchive for Mathematical Logic 44 (7): 887-895. 2005.Uniform sequential continuity, a property classically equivalent to sequential continuity on compact sets, is shown, constructively, to be a consequence of strong continuity on a metric space. It is then shown that in the case of a separable metric space, uniform sequential continuity implies strong continuity if and only if one adopts a certain boundedness principle that, although valid in the classical, recursive and intuitionistic setting, is independent of Heyting arithmetic.
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33A continuity principle, a version of Baire's theorem and a boundedness principleJournal of Symbolic Logic 73 (4): 1354-1360. 2008.We deal with a restricted form WC-N' of the weak continuity principle, a version BT' of Baire's theorem, and a boundedness principle BD-N. We show, in the spirit of constructive reverse mathematics, that WC-N'. BT' + ¬LPO and BD-N + ¬LPO are equivalent in a constructive system, where LPO is the limited principle of omniscience
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35A predicative completion of a uniform spaceAnnals of Pure and Applied Logic 163 (8): 975-980. 2012.
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38Unique solutionsMathematical Logic Quarterly 52 (6): 534-539. 2006.It is folklore that if a continuous function on a complete metric space has approximate roots and in a uniform manner at most one root, then it actually has a root, which of course is uniquely determined. Also in Bishop's constructive mathematics with countable choice, the general setting of the present note, there is a simple method to validate this heuristic principle. The unique solution even becomes a continuous function in the parameters by a mild modification of the uniqueness hypothesis. …Read more
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8FrontmatterIn Peter Schuster & Dieter Probst (eds.), Concepts of Proof in Mathematics, Philosophy, and Computer Science, De Gruyter. 2016.
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59Classifying Dini's TheoremNotre Dame Journal of Formal Logic 47 (2): 253-262. 2006.Dini's theorem says that compactness of the domain, a metric space, ensures the uniform convergence of every simply convergent monotone sequence of real-valued continuous functions whose limit is continuous. By showing that Dini's theorem is equivalent to Brouwer's fan theorem for detachable bars, we provide Dini's theorem with a classification in the recently established constructive reverse mathematics propagated by Ishihara. As a complement, Dini's theorem is proved to be equivalent to the an…Read more
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13Approximating beppo Levi's "principio di approssimazione"Association for Symbolic Logic: The Bulletin of Symbolic Logic. forthcoming.We try to recast in modern terms a choice principle conceived by Beppo Levi. who called it the Approximation Principle (AP). Up to now. there was almost no discussion about Levi's contribution. due to the quite obscure formulation of AP the author has chosen. After briefly reviewing the historical and philosophical surroundings of Levi's proposal. we undertake our own attempt at interpreting AP. The idea underlying the principle. as well as the supposed faithfulness of our version to Levi's orig…Read more
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University of LeedsRegular Faculty
Leeds, West Yorkshire, United Kingdom of Great Britain and Northern Ireland
Areas of Interest
Logic and Philosophy of Logic |
Medieval and Renaissance Philosophy |