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Douglas S. Bridges - One of the best experts on this subject based on the ideXlab platform.
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the pseudocompactness of 0 1 is equivalent to the uniform continuity theorem
Journal of Symbolic Logic, 2007Co-Authors: Douglas S. Bridges, Hannes DienerAbstract:We prove Constructively that, in order to derive the uniform continuity theorem for pointwise continuous mappings from a compact metric space into a metric space, it is necessary and sufficient to prove any of a number of equivalent conditions, such as that every pointwise continuous mapping of [0, 1] into R is bounded. The proofs are analytic, making no use of, for example, fan-theoretic ideas. In any variety of Constructive Mathematics, the status of the (classically valid) uniform continuity theorem, UCT Every pointwise continuous mapping of a compact (that is, com plete, totally bounded) metric space into a metric space is uniformly continuous, reveals a great deal about the kind of constructivity that characterises the variety. For example, UCT holds in intuitionistic Mathematics as a result of Brouwer's fan theorem, whereas in the recursive Constructive Mathematics of the Markov school there is an example of a bounded, pointwise continuous function from [0,1] to [0,1] that fails to be uniformly continuous [9, Chapters 3 and 5]. It is therefore a problem of some interest in Constructive reverse Mathematics to discover logical or analytic statements that are equivalent, Constructively, to UCT. Berger [4, 3, 5, 6] and Loeb [11] have made important contributions in connection with this problem. The work of the former deals primarily with the connection between UCT and versions of Brouwer's fan theorem. Loeb, on the other hand, works within a strict formal system (whereas we work more informally), and her continuous functions are equipped with stronger information than ours. In this paper we work in Bishop-style Constructive Mathematics (BISH: mathe matics with intuitionistic logic and some appropriate set-theoretic foundation such as Aczel's CZF [1]), in which it is well known that a uniformly continuous function from a compact metric space into R is bounded (and actually has a supremum and an infimum). We shall prove a strong converse: if every pointwise continuous mapping of [0,1] into R is bounded, then UCT holds. We also show that UCT is equivalent to other statements, including 'every pointwise continuous mapping of [0,1] into R is Lebesgue integrable'. We begin with some technical material that will enable us to connect pointwise continuous, real-valued functions on [0,1] with such functions on Cantor space 2N. Received November 16, 2006. ? 2007. Association for Symbolic Logic 0022-4812/07/7204-0020/$1.60
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Constructive Mathematics and quantum physics
International Journal of Theoretical Physics, 2000Co-Authors: Douglas S. Bridges, Karl SvozilAbstract:We discuss some aspects of quantum logic within Bishop's ConstructiveMathematics. In particular, we present a set of axioms that abstracts theConstructive properties of the lattices of subspaces and projections on a Hilbertspace.
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Can Constructive Mathematics be Applied in Physics?
Journal of Philosophical Logic, 1999Co-Authors: Douglas S. BridgesAbstract:The nature of modern Constructive Mathematics, and its applications, actual and potential, to classical and quantum physics, are discussed.
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Constructive Mathematics a foundation for computable analysis
Theoretical Computer Science, 1999Co-Authors: Douglas S. BridgesAbstract:This paper introduces Bishop's Constructive Mathematics, which can be regarded as the Constructive core of Mathematics and whose theorems can be translated into many formal systems of computable analysis. The real numbers are presented using a set of Constructive axioms, from which are derived some elementary properties of the real line R, including its completeness.
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Constructive Mathematics in theory and programming practice
Philosophia Mathematica, 1999Co-Authors: Douglas S. Bridges, Steeve ReevesAbstract:The first part of the paper introduces the varieties of modern Constructive Mathematics, concentrating on Bishop’s Constructive Mathematics (BISH). It gives a sketch of both Myhill’s axiomatic system for BISH and a Constructive axiomatic development of the real line R. The second part of the paper focusses on the relation between Constructive Mathematics and programming, with emphasis on Martin-Lof’s theory of types as a formal system for BISH. 1 What is Constructive Mathematics? The story of modern Constructive Mathematics begins with the publication, in 1907, of L.E.J. Brouwer’s doctoral dissertation Over de Grondslagen der Wiskunde [18], in which he gave the first exposition of his philosophy of intuitionism (a general philosophy, not merely one for Mathematics). According to Brouwer, Mathematics is a creation of the human mind, and precedes logic: the logic we use in Mathematics grows from mathematical practice, and is not some a priori given before mathematical activity can be undertaken. It is not difficult to see how, with this view of Mathematics as a strictly creative activity, Brouwer came to the view that the phrase “there exists” should be interpreted strictly and uniquely as “there can be constructed” or, in more modern parlance, “we can compute”. In turn, this interpretation of existence led Brouwer to reject the unbridled use of the Law of Excluded Middle (LEM), P ∨¬P, in mathematical arguments. For example, consider the following statement, the Limited Principle of Omniscience (LPO): ∀a ∈ {0, 1} (a = 0 ∨ a 6= 0) . (1) Here, N = {0, 1, 2, . . .} is the set of natural numbers, {0, 1} is the set of all
Hajime Ishihara - One of the best experts on this subject based on the ideXlab platform.
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consistency of the intensional level of the minimalist foundation with church s thesis and axiom of choice
Archive for Mathematical Logic, 2018Co-Authors: Hajime Ishihara, Maria Emilia Maietti, Samuele Maschio, Thomas StreicherAbstract:Consistency with the formal Church’s thesis, for short CT, and the axiom of choice, for short AC, was one of the requirements asked to be satisfied by the intensional level of a two-level foundation for Constructive Mathematics as proposed by Maietti and Sambin (in Crosilla, Schuster (eds) From sets and types to topology and analysis: practicable foundations for Constructive Mathematics, Oxford University Press, Oxford, 2005). Here we show that this is the case for the intensional level of the two-level Minimalist Foundation, for short MF, completed in 2009 by the second author. The intensional level of MF consists of an intensional type theory a la Martin-Lof, called mTT. The consistency of mTT with CT and AC is obtained by showing the consistency with the formal Church’s thesis of a fragment of intensional Martin-Lof’s type theory, called $$\mathbf{MLtt}_1$$ , where mTT can be easily interpreted. Then to show the consistency of $$\mathbf{MLtt}_1$$ with CT we interpret it within Feferman’s predicative theory of non-iterative fixpoints $$\widehat{ID_1}$$ by extending the well known Kleene’s realizability semantics of intuitionistic arithmetics so that CT is trivially validated. More in detail the fragment $$\mathbf{MLtt}_1$$ we interpret consists of first order intensional Martin-Lof’s type theory with one universe and with explicit substitution rules in place of usual equality rules preserving type constructors (hence without the so called $$\xi $$ -rule which is not valid in our realizability semantics). A key difficulty encountered in our interpretation was to use the right interpretation of lambda abstraction in the applicative structure of natural numbers in order to model all the equality rules of $$\mathbf{MLtt}_1$$ correctly. In particular the universe of $$\mathbf{MLtt}_1$$ is modelled by means of $$\widehat{ID_1}$$ -fixpoints following a technique due first to Aczel and used by Feferman and Beeson.
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reverse Mathematics in bishop s Constructive Mathematics
Philosophia Scientiæ. Travaux d'histoire et de philosophie des sciences, 2006Co-Authors: Hajime IshiharaAbstract:We will overview the results in an informal approach to Constructive reverse Mathematics, that is reverse Mathematics in Bishop’s Constructive Mathematics, especially focusing on compactness properties and continuous properties.
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sequentially continuity in Constructive Mathematics
Discrete Mathematics & Theoretical Computer Science, 2001Co-Authors: Hajime IshiharaAbstract:The classical validity of many important theorems of functional analysis, such as the Banach-Steinhaus theorem, the open mapping theorem and the closed graph theorem, depends on Baire’s theorem about complete metric spaces, which is an indispensable tool in this area. A form of Baire’s theorem has a Constructive proof [5, Theorem 1.3], but its classical equivalent, if a complete metric space is the union of a sequence of its subsets, then the closure of at least one set in the sequence must have nonempty interior which is used in the standard argument to prove that the above theorems have no known Constructive proof. If we could prove the Baire’s theorem of the above form, we would have the following forms of Constructive versions of Banach’s inverse mapping theorem, the open mapping theorem, the closed graph theorem, the Banach-Steinhaus theorem and the Hellinger-Toeplits theorem: Theorem 1 (Banach’s inverse mapping theorem) LetT be a one-one continuous linear mapping of a separable Banach space E onto a Banach space F. Then T-1 is continuous. Theorem 2 (The open mapping theorem) Let T be a continuous linear mapping of a Banach space E onto a Banach space F such that ker(T) is located1. Then T is open. Theorem 3 (The closed graph theorem) Let T be a linear mapping of a Banach space E into a Banach space F such that graph(T) is closed and separable. Then T is continuous. Theorem 4 (The Banach-Steinhaus theorem) Let {Tm} be a sequence of continuous linear mappings from a separable Banach space E into a normed space F such that $$ Tx: = \mathop {\lim }\limits_{m \to \infty } {T_m}x $$ exists for all x ∈ E. Then T is continuous. Theorem 5 (The Hellinger-Toeplitz theorem) Let T be a linear mapping from a Banach space E into a separable normed space with the following property: if f is a normable2 linear functional f on F, and {xn} converges to 0 in E, then f (Txn) → 0. Then T is continuous.
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continuity properties in Constructive Mathematics
Journal of Symbolic Logic, 1992Co-Authors: Hajime IshiharaAbstract:The purpose of this paper is an axiomatic study of the interrelations between certain continuity properties. We deal with principles which are equivalent to the statements "every mapping is sequentially nondiscontinuous", "every sequentially nondiscontinuous mapping is sequentially continuous", and "every sequentially continuous mapping is continuous". As corollaries, we show that every mapping of a complete separable space is continuous in Constructive recursive Mathematics (the Kreisel-LacombeSchoenfield-Tsejtin theorem) and in intuitionism. As early as 1954 Markov had obtained the first continuity result in Constructive recursive Mathematics that every mapping from R to R is sequentially nondiscontinuous. (We say that a mapping between metric spaces is sequentially nondiscontinuous if xn -+ x as n -+ o and d(f(xn), f(x)) ? 6 for all n imply 6 0 there exists 6 > 0 such that d(x, y) < 6 implies d(f(x), f(y)) < E for all y E X.) For applications of these results to concrete mathematical problems, see [2, Chapter XVI]. Beeson [1] carefully studied the KLST theorem, and showed that Markov's principle MP implies KLST, but the converse is unprovable, in the intuitionistic formal system of Heyting's arithmetic HA with Church's thesis CT. (Constructive recursive Mathematics could be formalized relative to the system HA with CT and MP.) On the other hand, Troelstra [12] proved results corresponding to the KLST theorem and to Orevkov's theorem in intuitionism, which is inconsistent with CT. Received February 5, 1991; revised May 3, 1991. 1991 Mathematics Subject Classification. Primary 03F65; Secondary 46S30.
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continuity and nondiscontinuity in Constructive Mathematics
Journal of Symbolic Logic, 1991Co-Authors: Hajime IshiharaAbstract:The purpose of this paper is an axiomatic study of the interrelations between certain continuity properties. We show that every mapping is sequentially continuous if and only if it is sequentially nondiscontinuous and strongly extensional, and that “every mapping is strongly extensional”, “every sequentially nondiscontinuous mapping is sequentially continuous”, and a weak version of Markov's principle are equivalent. Also, assuming a consequence of Church's thesis, we prove a version of the Kreisel-Lacombe-Shoenfield-Tseĭtin theorem.
Thierry Coquand - One of the best experts on this subject based on the ideXlab platform.
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lorenzen and Constructive Mathematics
2021Co-Authors: Thierry CoquandAbstract:The goal of this paper is to present a short survey of some of Lorenzen’s contributions to Constructive Mathematics, and its influence on recent developments in mathematical logic and Constructive algebra. We also present some work in measure theory which uses these contributions in an essential way.
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revisiting zariski main theorem from a Constructive point of view
Journal of Algebra, 2014Co-Authors: Maria Emilia Alonso, Thierry Coquand, Henri LombardiAbstract:This paper deals with the Peskine version of Zariski Main Theorem published in 1965 and discusses some applications. It is written in the style of Bishop's Constructive Mathematics. Being Constructive, each proof in this paper can be interpreted as an algorithm for constructing explicitly the conclusion from the hypothesis. The main non-Constructive argument in the proof of Peskine is the use of minimal prime ideals. Essentially we substitute this point by two dynamical arguments; one about gcd's, using subresultants, and another using our notion of strong transcendence. In particular we obtain algorithmic versions for the Multivariate Hensel Lemma and the structure theorem of quasi-finite algebras. (C) 2014 Elsevier Inc. All rights reserved.
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recursive functions and Constructive Mathematics
CONSTRUCTIVITY AND COMPUTABILITY IN HISTORICAL AND PHILOSOPHICAL PERSPECTIVE, 2014Co-Authors: Thierry CoquandAbstract:The goal of this paper is to discuss the following question: is the theory of recursive functions needed for a rigorous development of Constructive Mathematics? I will try to present the point of view of Constructive Mathematics on this question. The plan is the following: I first explain the gradual loss of appreciation of constructivity after 1936, clearly observed by Heyting and Skolem, in connection with the development of recursivity. There is an important change in 1967, publication of Bishop’s book, and the (re)discovery that the theory of recursive functions is actually not needed for a rigorous development of Constructive Mathematics. I then end with a presentation of the current view of Constructive Mathematics: Mathematics done using intuitionistic logic, view which, surprisingly, does not rely on any explicit notion of algorithm.
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formal topology and Constructive Mathematics the gelfand and stone yosida representation theorems
arXiv: Functional Analysis, 2008Co-Authors: Thierry Coquand, Bas SpittersAbstract:We present a Constructive proof of the Stone-Yosida representation theorem for Riesz spaces motivated by considerations from formal topology. This theorem is used to derive a representation theorem for f-algebras. In turn, this theorem implies the Gelfand representation theorem for C*-algebras of operators on Hilbert spaces as formulated by Bishop and Bridges. Our proof is shorter, clearer, and we avoid the use of approximate eigenvalues.
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Constructive Mathematics and functional programming abstract
European Symposium on Programming, 2008Co-Authors: Thierry CoquandAbstract:Around thirty years ago, P. Martin-Lof [12] suggested that the intuitionistic theory of types, originally designed as a formal system for Constructive Mathematics, could be viewed as a programming language. The conclusion of this paper stresses the mutual benefit of relating Constructive Mathematics and computer programming. In one direction one gets a precise system of notations for both statements and proofs, and one obtains the computerization of abstract intuitionistic Mathematics that was asked by Bishop [2]. In the other direction, computer programming “gets access to the whole conceptual apparatus of pure Mathematics”.
Hannes Diener - One of the best experts on this subject based on the ideXlab platform.
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seemingly impossible theorems in Constructive Mathematics
arXiv: Logic, 2019Co-Authors: Hannes Diener, Matthew HendtlassAbstract:We prove some Constructive results that on first and maybe even on second glance seem impossible.
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sequences of real functions on 0 1 in Constructive reverse Mathematics
Annals of Pure and Applied Logic, 2009Co-Authors: Hannes Diener, Iris LoebAbstract:Abstract We give an overview of the role of equicontinuity of sequences of real-valued functions on [ 0 , 1 ] and related notions in classical Mathematics, intuitionistic Mathematics, Bishop’s Constructive Mathematics, and Russian recursive Mathematics. We then study the logical strength of theorems concerning these notions within the programme of Constructive Reverse Mathematics. It appears that many of these theorems, like a version of Ascoli’s Lemma, are equivalent to fan-theoretic principles.
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the pseudocompactness of 0 1 is equivalent to the uniform continuity theorem
Journal of Symbolic Logic, 2007Co-Authors: Douglas S. Bridges, Hannes DienerAbstract:We prove Constructively that, in order to derive the uniform continuity theorem for pointwise continuous mappings from a compact metric space into a metric space, it is necessary and sufficient to prove any of a number of equivalent conditions, such as that every pointwise continuous mapping of [0, 1] into R is bounded. The proofs are analytic, making no use of, for example, fan-theoretic ideas. In any variety of Constructive Mathematics, the status of the (classically valid) uniform continuity theorem, UCT Every pointwise continuous mapping of a compact (that is, com plete, totally bounded) metric space into a metric space is uniformly continuous, reveals a great deal about the kind of constructivity that characterises the variety. For example, UCT holds in intuitionistic Mathematics as a result of Brouwer's fan theorem, whereas in the recursive Constructive Mathematics of the Markov school there is an example of a bounded, pointwise continuous function from [0,1] to [0,1] that fails to be uniformly continuous [9, Chapters 3 and 5]. It is therefore a problem of some interest in Constructive reverse Mathematics to discover logical or analytic statements that are equivalent, Constructively, to UCT. Berger [4, 3, 5, 6] and Loeb [11] have made important contributions in connection with this problem. The work of the former deals primarily with the connection between UCT and versions of Brouwer's fan theorem. Loeb, on the other hand, works within a strict formal system (whereas we work more informally), and her continuous functions are equipped with stronger information than ours. In this paper we work in Bishop-style Constructive Mathematics (BISH: mathe matics with intuitionistic logic and some appropriate set-theoretic foundation such as Aczel's CZF [1]), in which it is well known that a uniformly continuous function from a compact metric space into R is bounded (and actually has a supremum and an infimum). We shall prove a strong converse: if every pointwise continuous mapping of [0,1] into R is bounded, then UCT holds. We also show that UCT is equivalent to other statements, including 'every pointwise continuous mapping of [0,1] into R is Lebesgue integrable'. We begin with some technical material that will enable us to connect pointwise continuous, real-valued functions on [0,1] with such functions on Cantor space 2N. Received November 16, 2006. ? 2007. Association for Symbolic Logic 0022-4812/07/7204-0020/$1.60
Iris Loeb - One of the best experts on this subject based on the ideXlab platform.
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sequences of real functions on 0 1 in Constructive reverse Mathematics
Annals of Pure and Applied Logic, 2009Co-Authors: Hannes Diener, Iris LoebAbstract:Abstract We give an overview of the role of equicontinuity of sequences of real-valued functions on [ 0 , 1 ] and related notions in classical Mathematics, intuitionistic Mathematics, Bishop’s Constructive Mathematics, and Russian recursive Mathematics. We then study the logical strength of theorems concerning these notions within the programme of Constructive Reverse Mathematics. It appears that many of these theorems, like a version of Ascoli’s Lemma, are equivalent to fan-theoretic principles.
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factoring out intuitionistic theorems continuity principles and the uniform continuity theorem
Conference on Computability in Europe, 2008Co-Authors: Iris LoebAbstract:We prove the equivalence between some intuitionistic theorems and the conjunction of a continuity principle and a compactness principle over Bishop's Constructive Mathematics within the programme of Constructive Reverse Mathematics. To clarify our line of thought, we first point out the relation between quasi-equicontinuity, quasi-uniform convergence, and the continuity principle saying that the limit of a convergent sequence of continuous functions is again continuous. Finally, as a spin-off, we conclude that we have found a new, more economic proof of the statement that every convergent sequence of functions on a compact metric space converges uniformly.