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Peter Schuster [81]Peter M. Schuster [5]Peter-Klaus Schuster [1]Peters Schuster [1]
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Peter Schuster
University of Leeds
  1.  14
    Optimization of Multiple Criteria: Pareto Efficiency and Fast Heuristics Should Be More Popular Than They Are.Peter Schuster - 2013 - Complexity 18 (2):5-7.
  2.  21
    Strong Continuity Implies Uniform Sequential Continuity.Douglas Bridges, Hajime Ishihara, Peter Schuster & Luminiţa Vîţa - 2005 - Archive for Mathematical Logic 44 (7):887-895.
    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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  3. Binary Refinement Implies Discrete Exponentiation.Peter Aczel, Laura Crosilla, Hajime Ishihara, Erik Palmgren & Peter Schuster - 2006 - Studia Logica 84 (3):361-368.
    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 prove that the (...)
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  4.  28
    On Constructing Completions.Laura Crosilla, Hajime Ishihara & Peter Schuster - 2005 - Journal of Symbolic Logic 70 (3):969-978.
    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 and (...)
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  5.  22
    Unique Solutions.Peter Schuster - 2006 - Mathematical Logic Quarterly 52 (6):534-539.
    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. (...)
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  6.  15
    Ebola-Challenge and Revival of Theoretical Epidemiology: Why Extrapolations From Early Phases of Epidemics Are Problematic.Peter Schuster - 2015 - Complexity 20 (5):7-12.
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  7.  7
    Compactness Under Constructive Scrutiny.Hajime Ishihara & Peter Schuster - 2004 - Mathematical Logic Quarterly 50 (6):540-550.
    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 constructive implication that every sequentially compact separable metric space is totally bounded. As a by-product, (...)
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  8.  14
    Quasi-Apartness and Neighbourhood Spaces.Hajime Ishihara, Ray Mines, Peter Schuster & Luminiţa Vîţă - 2006 - Annals of Pure and Applied Logic 141 (1):296-306.
    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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  9.  15
    How Does Complexity Arise in Evolution:Nature's Recipe for Mastering Scarcity, Abundance, and Unpredictability.Peter Schuster - 1996 - Complexity 2 (1):22-30.
  10.  21
    The Fan Theorem and Unique Existence of Maxima.Josef Berger, Douglas Bridges & Peter Schuster - 2006 - Journal of Symbolic Logic 71 (2):713 - 720.
    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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  11.  20
    The Kripke Schema in Metric Topology.Robert Lubarsky, Fred Richman & Peter Schuster - 2012 - Mathematical Logic Quarterly 58 (6):498-501.
    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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  12.  23
    Nonlinear Dynamics From Physics to Biology.Peter Schuster - 2007 - Complexity 12 (4):9-11.
  13.  8
    The End of Moore's Law: Living Without an Exponential Increase in the Efficiency of Computational Facilities.Peter Schuster - 2016 - Complexity 21 (S1):6-9.
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  14.  17
    Some Forms of Excluded Middle for Linear Orders.Peter Schuster & Daniel Wessel - 2019 - Mathematical Logic Quarterly 65 (1):105-107.
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  15.  68
    A Revival of the Landscape Paradigm: Large Scale Data Harvesting Provides Access to Fitness Landscapes.Peter Schuster - 2012 - Complexity 17 (5):6-10.
  16.  30
    Classifying Dini's Theorem.Josef Berger & Peter Schuster - 2006 - Notre Dame Journal of Formal Logic 47 (2):253-262.
    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 analogue of the fan theorem, weak König's lemma, in (...)
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  17.  53
    A Silent Revolution in Mathematics.Peter Schuster - 2013 - Complexity 18 (6):7-10.
  18.  10
    Corrigendum to “Unique Solutions”.Peter Schuster - 2007 - Mathematical Logic Quarterly 53 (2):214-214.
  19.  14
    Is There a Newton of the Blade of Grass?Peter Schuster - 2011 - Complexity 16 (6):5-9.
  20.  19
    “Less is More” and the Art of Modeling Complex Phenomena: Simplification May but Need Not Be the Key to Handle Large Networks.Peter Schuster - 2005 - Complexity 11 (2):11-13.
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  21.  34
    Apartness, Topology, and Uniformity: A Constructive View.Douglas Bridges, Peter Schuster & Luminiţa Vîţă - 2002 - Mathematical Logic Quarterly 48 (4):16-28.
    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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  22.  11
    Formal Zariski Topology: Positivity and Points.Peter Schuster - 2006 - Annals of Pure and Applied Logic 137 (1):317-359.
    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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  23.  16
    A Constructive Look at Generalised Cauchy Reals.Peter M. Schuster - 2000 - Mathematical Logic Quarterly 46 (1):125-134.
    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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  24.  53
    Origins of Life: Concepts, Data, and Debates.Peter Schuster - 2010 - Complexity 15 (3):7-10.
  25.  90
    Recycling and Growth in Early Evolution and Today.Peter Schuster - 2014 - Complexity 19 (2):6-9.
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  26.  65
    Generation of Information and Complexity: Different Forms of Learning and Innovation: A Simple Mechanism of Learning.Peter Schuster - 2005 - Complexity 10 (4):12-14.
  27.  7
    Approximating Beppo Levi's "Principio di Approssimazione".Riccardo Bruni & Peter Schuster - forthcoming - Association for Symbolic Logic: The Bulletin of Symbolic Logic.
    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 original intention. (...)
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  28.  23
    A Predicative Completion of a Uniform Space.Josef Berger, Hajime Ishihara, Erik Palmgren & Peter Schuster - 2012 - Annals of Pure and Applied Logic 163 (8):975-980.
  29.  14
    The Commons' Tragicomedy: Self‐Governance Doesn't Come Easily.Peter Schuster - 2005 - Complexity 10 (6):10-12.
  30.  26
    Countable Choice as a Questionable Uniformity Principle.Peter M. Schuster - 2004 - Philosophia Mathematica 12 (2):106-134.
    The form of nominalism known as 'mathematical fictionalism' is examined and found wanting, mainly on grounds that go back to an early antinominalist work of Rudolf Carnap that has unfortunately not been paid sufficient attention by more recent writers.
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  31.  49
    Linear Independence Without Choice.Douglas Bridges, Fred Richman & Peter Schuster - 1999 - Annals of Pure and Applied Logic 101 (1):95-102.
    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 (...)
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  32.  19
    Lethal Mutagenesis, Error Thresholds, and the Fight Against Viruses: Rigorous Modeling is Facilitated by a Firm Physical Background.Peter Schuster - 2011 - Complexity 17 (2):5-9.
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  33.  34
    A Beginning of the End of the Holism Versus Reductionism Debate?: Molecular Biology Goes Cellular and Organismic.Peter Schuster - 2007 - Complexity 13 (1):10-13.
  34.  39
    Too Simple Solutions of Hard Problems.Peter M. Schuster - 2010 - Nordic Journal of Philosophical Logic 6 (2):138-146.
    Even after yet another grand conjecture has been proved or refuted, any omniscience principle that had trivially settled this question is just as little acceptable as before. The significance of th...
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  35.  22
    Untamable Curiosity, Innovation, Discovery, and Bricolage: Are We Doomed to Progress to Ever Increasing Complexity?Peter Schuster - 2006 - Complexity 11 (5):9-11.
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  36.  19
    From Sets and Types to Topology and Analysis: Towards Practicable Foundations for Constructive Mathematics.Laura Crosilla & Peter Schuster (eds.) - 2005 - Oxford University Press.
    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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  37.  45
    Power Laws in Biology: Between Fundamental Regularities and Useful Interpolation Rules.Peter Schuster - 2011 - Complexity 16 (3):6-9.
  38.  48
    Free Will, Information, Quantum Mechanics, and Biology.Peter Schuster - 2009 - Complexity 15 (1):8-10.
  39.  35
    Are There Recipes for How to Handle Complexity?Peter Schuster - 2008 - Complexity 14 (1):8-12.
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  40.  37
    Designing Living Matter. Can We Do Better Than Evolution?Peter Schuster - 2013 - Complexity 18 (6):21-33.
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  41.  49
    Are There Enough Injective Sets?Peter Aczel, Benno Berg, Johan Granström & Peter Schuster - 2013 - Studia Logica 101 (3):467-482.
    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 (...)
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  42.  9
    Quo Vadit Complexity.Peter Schuster - 2001 - Complexity 7 (1):3-4.
  43.  24
    Evolution and Design: The Darwinian View of Evolution is a Scientific Fact and Not an Ideology.Peter Schuster - 2005 - Complexity 11 (1):12-15.
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  44.  15
    Wolfgang P. Müller, Die Abtreibung: Anfänge der Kriminalisierung, 1140–1650. Cologne, Weimar, and Vienna: Böhlau, 2000. Paper. Pp. viii, 355. €45.Peter Schuster - 2004 - Speculum 79 (2):530-531.
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  45.  21
    Networks in Biology: Handling Biological Complexity Requires Novel Inputs Into Network Theory.Peter Schuster - 2011 - Complexity 16 (4):6-9.
  46.  14
    Models: From Exploration to Prediction: Bad Reputation of Modeling in Some Disciplines Results From Nebulous Goals.Peter Schuster - 2016 - Complexity 21 (1):6-9.
  47.  18
    Editorial Remarks.Peter Schuster - 2008 - Complexity 13 (6):11-11.
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  48.  12
    Complexity has Come of Age.Alfred Hübler & Peter Schuster - 2016 - Complexity 21 (S2):6-6.
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  49.  9
    Finite Methods in Mathematical Practice.Peter Schuster & Laura Crosilla - 2014 - In Godehard Link (ed.), Formalism and Beyond: On the Nature of Mathematical Discourse. De Gruyter. pp. 351-410.
    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 is laid on (...)
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  50.  14
    On the Contrapositive of Countable Choice.Hajime Ishihara & Peter Schuster - 2011 - Archive for Mathematical Logic 50 (1-2):137-143.
    We show that in elementary analysis (EL) the contrapositive of countable choice is equivalent to double negation elimination for ${\Sigma_{2}^{0}}$ -formulas. By also proving a recursive adaptation of this equivalence in Heyting arithmetic (HA), we give an instance of the conservativity of EL over HA with respect to recursive functions and predicates. As a complement, we prove in HA enriched with the (extended) Church thesis that every decidable predicate is recursive.
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