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Profile: Peter Schuster (University of Leeds)
  1.  48
    Peter Schuster (2005). Generation of Information and Complexity: Different Forms of Learning and Innovation: A Simple Mechanism of Learning. Complexity 10 (4):12-14.
  2. Peter Schuster (2012). Optimization of Multiple Criteria: Pareto Efficiency and Fast Heuristics Should Be More Popular Than They Are. Complexity 18 (2):5-7.
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  3.  1
    Riccardo Bruni & Peter Schuster (forthcoming). Approximating Beppo Levi's "Principio di Approssimazione". 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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  4.  27
    Peter Schuster (2011). Power Laws in Biology: Between Fundamental Regularities and Useful Interpolation Rules. Complexity 16 (3):6-9.
  5.  4
    Laura Crosilla, Hajime Ishihara & Peter Schuster (2005). On Constructing Completions. 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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  6.  30
    Peter Aczel, Laura Crosilla, Hajime Ishihara, Erik Palmgren & Peter Schuster (2006). Binary Refinement Implies Discrete Exponentiation. 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 re.nement principle implies all the instances of the exponentiation axiom in which the basis is a discrete set. In particular binary re.nement implies that the class of detachable subsets of a set form a set. Binary re.nement was originally extracted from the fullness axiom, an equivalent of subset collection, as a principle that was su.cient to prove that the (...)
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  7.  8
    Peter Schuster (2010). Origins of Life: Concepts, Data, and Debates. Complexity 15 (3):7-10.
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  8.  2
    Hajime Ishihara, Ray Mines, Peter Schuster & Luminiţa Vîţă (2006). Quasi-Apartness and Neighbourhood Spaces. 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.  3
    Peter Schuster (1996). How Does Complexity Arise in Evolution:Nature's Recipe for Mastering Scarcity, Abundance, and Unpredictability. Complexity 2 (1):22-30.
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  10.  4
    Josef Berger, Hajime Ishihara, Erik Palmgren & Peter Schuster (2012). A Predicative Completion of a Uniform Space. Annals of Pure and Applied Logic 163 (8):975-980.
  11.  14
    Peter Schuster (2007). A Beginning of the End of the Holism Versus Reductionism Debate?: Molecular Biology Goes Cellular and Organismic. Complexity 13 (1):10-13.
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  12.  3
    Peter Schuster (2006). Unique Solutions. 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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  13.  2
    Peter Schuster (2011). Lethal Mutagenesis, Error Thresholds, and the Fight Against Viruses: Rigorous Modeling is Facilitated by a Firm Physical Background. Complexity 17 (2):5-9.
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  14.  3
    Douglas Bridges, Hajime Ishihara, Peter Schuster & Luminiţa Vîţa (2005). Strong Continuity Implies Uniform Sequential Continuity. Archive for Mathematical Logic 44 (7):887-895.
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  15.  8
    Peter Schuster (2008). Are There Recipes for How to Handle Complexity? Complexity 14 (1):8-12.
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  16.  2
    Hajime Ishihara & Peter Schuster (2004). Compactness Under Constructive Scrutiny. 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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  17.  3
    Peter Schuster (2007). Nonlinear Dynamics From Physics to Biology. Complexity 12 (4):9-11.
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  18.  9
    Peter Schuster (2006). Untamable Curiosity, Innovation, Discovery, and Bricolage: Are We Doomed to Progress to Ever Increasing Complexity? Complexity 11 (5):9-11.
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  19.  12
    Josef Berger & Peter Schuster (2006). Classifying Dini's Theorem. 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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  20.  1
    Peter Schuster (2007). Corrigendum to “Unique Solutions”. Mathematical Logic Quarterly 53 (2):214-214.
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  21.  10
    Josef Berger, Douglas Bridges & Peter Schuster (2006). The Fan Theorem and Unique Existence of Maxima. 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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  22.  7
    Peter Schuster (2008). Editorial Remarks. Complexity 13 (6):11-11.
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  23.  1
    Peter Schuster (2005). The Commons' Tragicomedy: Self‐Governance Doesn't Come Easily. Complexity 10 (6):10-12.
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  24.  10
    Peter Schuster (2005). Evolution and Design: The Darwinian View of Evolution is a Scientific Fact and Not an Ideology. Complexity 11 (1):12-15.
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  25.  8
    Peter Schuster (2011). Networks in Biology: Handling Biological Complexity Requires Novel Inputs Into Network Theory. Complexity 16 (4):6-9.
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  26.  4
    Peter Schuster (2004). The Disaster of Central Control. Complexity 9 (4):13-14.
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  27. Peter Schuster (2006). Formal Zariski Topology: Positivity and Points. 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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  28.  2
    Douglas Bridges, Peter Schuster & Luminiţa Vîţă (2002). Apartness, Topology, and Uniformity: A Constructive View. Mathematical Logic Quarterly 48 (S1):16-28.
    The theory of apartness spaces, and their relation to topological spaces and uniform spaces , 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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  29.  3
    Peter Schuster (2013). Recycling and Growth in Early Evolution and Today. Complexity 19 (2):6-9.
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  30.  5
    Robert S. Lubarsky, Fred Richman & Peter Schuster (2012). The Kripke Schema in Metric Topology. 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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  31.  7
    Peter Schuster (2009). Free Will, Information, Quantum Mechanics, and Biology. Complexity 15 (1):8-10.
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  32.  7
    Peter Schuster (2012). A Revival of the Landscape Paradigm: Large Scale Data Harvesting Provides Access to Fitness Landscapes. Complexity 17 (5):6-10.
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  33.  14
    Peter Aczel, Benno Berg, Johan Granström & Peter Schuster (2013). Are There Enough Injective Sets? 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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  34.  5
    Peter Schuster (2013). A Silent Revolution in Mathematics. Complexity 18 (6):7-10.
  35.  5
    Peter Schuster (2011). Is There a Newton of the Blade of Grass? Complexity 16 (6):5-9.
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  36.  4
    Peter Schuster (2013). Designing Living Matter. Can We Do Better Than Evolution? Complexity 18 (6):21-33.
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  37.  10
    Hajime Ishihara & Peter Schuster (2008). A Continuity Principle, a Version of Baire's Theorem and a Boundedness Principle. Journal of Symbolic Logic 73 (4):1354-1360.
    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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  38.  1
    Peter Schuster (2005). “Less is More” and the Art of Modeling Complex Phenomena: Simplification May but Need Not Be the Key to Handle Large Networks. Complexity 11 (2):11-13.
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  39.  3
    Peter Schuster (1999). Die Modernität der Ehre. Historische Zweifel. Ethik Und Sozialwissenschaften 10.
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  40.  3
    Douglas Bridges, Fred Richman & Peter Schuster (1999). Linear Independence Without Choice. 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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  41.  3
    Hajime Ishihara & Peter Schuster (2011). On the Contrapositive of Countable Choice. 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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  42.  1
    Peter Schuster & Alfred Hübler (2004). Good‐Bye and Thank You to Our Complexity‐at‐Large Editors. Complexity 9 (6):3-3.
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  43.  3
    Peter Aczel, Benno van den Berg, Johan Granström & Peter Schuster (2013). Are There Enough Injective Sets? Studia Logica 101 (3):467-482.
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  44.  9
    Peter M. Schuster (2004). Countable Choice as a Questionable Uniformity Principle. 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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  45.  2
    Peter Schuster (2004). Chemical Reaction Kinetics is Back: Attempts to Deal with Complexity in Biology: Developing a Quantitative Molecular View to Understanding Life. Complexity 10 (1):14-16.
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  46.  2
    Peter Schuster (2010). Contingeny and Memory in Evolution. Complexity 15 (6):7-10.
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  47.  2
    Josef Berger, Dirk Pattinson, Peter Schuster & Júlia Zappe (2008). Editorial: Math. Log. Quart. 1/2008. Mathematical Logic Quarterly 54 (1):4-4.
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  48.  2
    Peter Schuster (2014). Are Computer Scientists the Sutlers of Modern Biology?: Bioinformatics is Indispensible for Progress in Molecular Life Sciences but Does Not Get Credit for its Contributions. Complexity 19 (4):10-14.
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  49.  1
    Peter Schuster (forthcoming). How Do RNA Molecules and Viruses Explore Their Worlds? Complexity.
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  50.  3
    Peter M. Schuster (2001). Too Simple Solutions of Hard Problems. 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 the constructive enterprise is therefore not affected by any gain of knowledge. In particular, there is no need to adapt weak counterexamples to mathematical progress.
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