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36From sets and types to topology and analysis: towards practicable foundations for constructive mathematics (edited book)Oxford University Press. 2005.This edited collection bridges the foundations and practice of constructive mathematics and focuses on the contrast between the theoretical developments, which have been most useful for computer science (ie: constructive set and type theories), and more specific efforts on constructive analysis, algebra and topology. Aimed at academic logician, mathematicians, philosophers and computer scientists with contributions from leading researchers, it is up to date, highly topical and broad in scope
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3PrefaceIn Ulrich Berger, Hannes Diener, Peter Schuster & Monika Seisenberger (eds.), Logic, Construction, Computation, De Gruyter. pp. 2-4. 2012.
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5Logic, Construction, Computation (edited book)De Gruyter. 2012.Over the last few decades the interest of logicians and mathematicians in constructive and computational aspects of their subjects has been steadily growing, and researchers from disparate areas realized that they can benefit enormously from the mutual exchange of techniques concerned with those aspects. A key figure in this exciting development is the logician and mathematician Helmut Schwichtenberg to whom this volume is dedicated on the occasion of his 70th birthday and his turning emeritus. …Read more
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24How does complexity arise in evolution:Nature's recipe for mastering scarcity, abundance, and unpredictabilityComplexity 2 (1): 22-30. 1996.
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60On constructing completionsJournal of Symbolic Logic 70 (3): 969-978. 2005.The Dedekind cuts in an ordered set form a set in the sense of constructive Zermelo—Fraenkel set theory. We deduce this statement from the principle of refinement, which we distill before from the axiom of fullness. Together with exponentiation, refinement is equivalent to fullness. None of the defining properties of an ordering is needed, and only refinement for two—element coverings is used. In particular, the Dedekind reals form a set; whence we have also refined an earlier result by Aczel an…Read more
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23A generalized cut characterization of the fullness axiom in CZFLogic Journal of the IGPL 21 (1): 63-76. 2013.In the present note, we study a generalization of Dedekind cuts in the context of constructive Zermelo–Fraenkel set theory CZF. For this purpose, we single out an equivalent of CZF's axiom of fullness and show that it is sufficient to derive that the Dedekind cuts in this generalized sense form a set. We also discuss the instance of this equivalent of fullness that is tantamount to the assertion that the class of Dedekind cuts in the rational numbers, in the customary constructive sense includin…Read more
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144Binary Refinement Implies Discrete ExponentiationStudia Logica 84 (3): 361-368. 2006.Working in the weakening of constructive Zermelo-Fraenkel set theory in which the subset collection scheme is omitted, we show that the binary refinement principle implies all the instances of the exponentiation axiom in which the basis is a discrete set. In particular binary refinement implies that the class of detachable subsets of a set form a set. Binary refinement was originally extracted from the fullness axiom, an equivalent of subset collection, as a principle that was sufficient to prov…Read more
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43Apartness, Topology, and Uniformity: a Constructive ViewMathematical Logic Quarterly 48 (4): 16-28. 2002.The theory of apartness spaces, and their relation to topological spaces (in the point–set case) and uniform spaces (in the set–set case), is sketched. New notions of local decomposability and regularity are investigated, and the latter is used to produce an example of a classically metrisable apartness on R that cannot be induced constructively by a uniform structure.
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39Mathesis Universalis, Computability and Proof (edited book)Springer Verlag. 2019.In a fragment entitled Elementa Nova Matheseos Universalis Leibniz writes “the mathesis [...] shall deliver the method through which things that are conceivable can be exactly determined”; in another fragment he takes the mathesis to be “the science of all things that are conceivable.” Leibniz considers all mathematical disciplines as branches of the mathesis and conceives the mathesis as a general science of forms applicable not only to magnitudes but to every object that exists in our imaginat…Read more
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61st EHoP Conference, Graz, Austria, September 18-21, 2006: proceedings of the First European History of Physics (EHoP) Conference of the History of Physics Section of the Austrian Physical Society (OEPG) in conjunction with the History of Physics Group of the European Physical Society (EPS) and the History of Physics Group of the Institute of Physics (IOP) (edited book, review)Living Edition. 2008.
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4The roots of physics in Europe: Echophysics, Pöllau/Austria, 2010: proceedings of the first joint European Symposium on the History of Physics, held under the auspices of the first European Centre for the History of Physics: Echophysics, Poellau Castle, Styria/Austria, May 28-29, 2010 (edited book, review)Living Edition. 2013.
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2Syntax for Semantics: Krull’s Maximal Ideal TheoremIn Gerhard Heinzmann & Gereon Wolters (eds.), Paul Lorenzen -- Mathematician and Logician, Springer Verlag. pp. 77-102. 2021.Krull’s Maximal Ideal Theorem is one of the most prominent incarnations of the Axiom of Choice in ring theory. For many a consequence of AC, constructive counterparts are well within reach, provided attention is turned to the syntactical underpinning of the problem at hand. This is one of the viewpoints of the revised Hilbert Programme in commutative algebra, which will here be carried out for MIT and several related classical principles.
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18The Jacobson Radical of a Propositional TheoryBulletin of Symbolic Logic 28 (2): 163-181. 2022.Alongside the analogy between maximal ideals and complete theories, the Jacobson radical carries over from ideals of commutative rings to theories of propositional calculi. This prompts a variant of Lindenbaum’s Lemma that relates classical validity and intuitionistic provability, and the syntactical counterpart of which is Glivenko’s Theorem. The Jacobson radical in fact turns out to coincide with the classical deductive closure. As a by-product we obtain a possible interpretation in logic of t…Read more
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11This book bridges the gaps between logic, mathematics and computer science by delving into the theory of well-quasi orders, also known as wqos. This highly active branch of combinatorics is deeply rooted in and between many fields of mathematics and logic, including proof theory, commutative algebra, braid groups, graph theory, analytic combinatorics, theory of relations, reverse mathematics and subrecursive hierarchies. As a unifying concept for slick finiteness or termination proofs, wqos have…Read more
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22Some forms of excluded middle for linear ordersMathematical Logic Quarterly 65 (1): 105-107. 2019.The intersection of a linearly ordered set of total subrelations of a total relation with range 2 need not be total, constructively.
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5Proof and Computation (edited book)World Scientific. 1995.Proceedings of the NATO Advanced Study Institute on Proof and Computation, held in Marktoberdorf, Germany, July 20 - August 1, 1993.
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32Finite Methods in Mathematical PracticeIn Godehard Link (ed.), Formalism and Beyond: On the Nature of Mathematical Discourse, De Gruyter. pp. 351-410. 2014.In the present contribution we look at the legacy of Hilbert's programme in some recent developments in mathematics. Hilbert's ideas have seen new life in generalised and relativised forms by the hands of proof theorists and have been a source of motivation for the so--called reverse mathematics programme initiated by H. Friedman and S. Simpson. More recently Hilbert's programme has inspired T. Coquand and H. Lombardi to undertake a new approach to constructive algebra in which strong emphasis i…Read more
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23A Constructive Look at Generalised Cauchy RealsMathematical Logic Quarterly 46 (1): 125-134. 2000.We investigate how nonstandard reals can be established constructively as arbitrary infinite sequences of rationals, following the classical approach due to Schmieden and Laugwitz. In particular, a total standard part map into Richman's generalised Dedekind reals is constructed without countable choice
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16Eliminating disjunctions by disjunction eliminationBulletin of Symbolic Logic 23 (2): 181-200. 2017.Completeness and other forms of Zorn’s Lemma are sometimes invoked for semantic proofs of conservation in relatively elementary mathematical contexts in which the corresponding syntactical conservation would suffice. We now show how a fairly general syntactical conservation theorem that covers plenty of the semantic approaches follows from an utmost versatile criterion for conservation given by Scott in 1974.To this end we work with multi-conclusion entailment relations as extending single-concl…Read more
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From Sets and Types to Topology and Analysis: Towards Practicable Foundations for Constructive MathematicsBulletin of Symbolic Logic 12 (4): 611-612. 2006.
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7How complexity originates: Examples from history reveal additional roots to complexityComplexity 21 (S2): 7-12. 2016.
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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 |