Results 11  20
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40
Intuitionism As Generalization
 Philosophia Math
, 1990
"... ms that hold only in computational models. For example, Brouwer proved that every totally defined function on the real line is continuous. A theory where this is provable cannot refer to the classical universe, so we have to consider whether we like its models better than the classical model. But an ..."
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ms that hold only in computational models. For example, Brouwer proved that every totally defined function on the real line is continuous. A theory where this is provable cannot refer to the classical universe, so we have to consider whether we like its models better than the classical model. But any theorem in constructive mathematics is a theorem in classical mathematics, so in this case it is not a question of choosing between models, but of deciding whether it is worthwhile to talk about computational models in addition to the classical model. By comparing mathematical realism with intuitionism from an informal axiomatic point of view, we can steer clear of most of the metaphysical problems involved in analyzing these notions from the ground up, and concentrate on what may be termed the purely mathematical aspects. In particular we won't have to consider whether intuitionists hold that a theorem "isn't true until it's known to be true", as Blais 1 <F
Constructive Mathematics, in Theory and Programming Practice
, 1997
"... 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 pap ..."
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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 MartinLof's theory of types as a formal system for BISH.
On weak Markov's principle
 MLQ MATH. LOG. Q
, 2002
"... We show that the socalled weak Markov's principle (WMP) which states that every pseudopositive real number is positive is underivable in T # :=EHA # +AC. Since T # allows to formalize (at least large parts of) Bishop's constructive mathematics this makes it unlikely that WMP can be proved ..."
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We show that the socalled weak Markov's principle (WMP) which states that every pseudopositive real number is positive is underivable in T # :=EHA # +AC. Since T # allows to formalize (at least large parts of) Bishop's constructive mathematics this makes it unlikely that WMP can be proved within the framework of Bishopstyle mathematics (which has been open for about 20 years). The underivability even holds if the ine#ective schema of full comprehension (in all types) for negated formulas (in particular for #free formulas) is added which allows to derive the law of excluded middle for such formulas.
A direct proof of the equivalence between Brouwer’s fan theorem and König’s lemma with a uniqueness hypothesis
 Journal of Universal Computer Science
, 2005
"... Abstract: From results of Ishihara it is known that the weak (that is, binary) form of König’s lemma (WKL) implies Brouwer’s fan theorem (Fan). Moreover, Berger and Ishihara [MLQ 2005] have shown that a weakened form WKL! of WKL, where as an additional hypothesis it is required that in an effective ..."
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Abstract: From results of Ishihara it is known that the weak (that is, binary) form of König’s lemma (WKL) implies Brouwer’s fan theorem (Fan). Moreover, Berger and Ishihara [MLQ 2005] have shown that a weakened form WKL! of WKL, where as an additional hypothesis it is required that in an effective sense infinite paths are unique, is equivalent to Fan. The proof that WKL! implies Fan is done explicitely. The other direction (Fan implies WKL!) is far less directly proved; the emphasis is rather to provide a fair number of equivalents to Fan, and to do the proofs economically by giving a circle of implications. Here we give a direct construction. Moreover, we go one step further and formalize the equivalence proof (in the Minlog proof assistant). Since the statements of both Fan and WKL! have computational content, we can automatically extract terms from the two proofs. It turns out that these terms express in a rather perspicuous way the informal constructions. Key Words: Brouwer’s fan theorem, König’s lemma Category: F.1
Computable versions of Baire’s category theorem
 MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE 2001
, 2001
"... We study different computable versions of Baire’s Category Theorem in computable analysis. Similarly, as in constructive analysis, different logical forms of this theorem lead to different computational interpretations. We demonstrate that, analogously to the classical theorem, one of the computab ..."
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We study different computable versions of Baire’s Category Theorem in computable analysis. Similarly, as in constructive analysis, different logical forms of this theorem lead to different computational interpretations. We demonstrate that, analogously to the classical theorem, one of the computable versions of the theorem can be used to construct interesting counterexamples, such as a computable but nowhere differentiable function.
Operator Methods, Abelian Processes and Dynamic Conditioning
, 2007
"... A mathematical framework for Continuous Time Finance based on operator algebraic methods offers a new direct and entirely constructive perspective on the field. It also leads to new numerical analysis techniques which can take advantage of the emerging massively parallel GPU architectures which are ..."
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A mathematical framework for Continuous Time Finance based on operator algebraic methods offers a new direct and entirely constructive perspective on the field. It also leads to new numerical analysis techniques which can take advantage of the emerging massively parallel GPU architectures which are uniquely suited to execute large matrix manipulations. This is partly a review paper as it covers and expands on the mathematical framework underlying a series of more applied articles. In addition, this article also presents a few key new theorems that make the treatment selfcontained. Stochastic processes with continuous time and continuous space variables are defined constructively by establishing new convergence estimates for Markov chains on simplicial sequences. We emphasize high precision computability by numerical linear algebra methods as opposed to the ability of arriving to analytically closed form expressions in terms of special functions. Path dependent processes adapted to a given Markov filtration are associated to an operator algebra. If this algebra is commutative, the corresponding process is named Abelian, a concept which provides a far reaching extension of the notion of stochastic integral. We recover the classic CameronDysonFeynmanGirsanovItoKacMartin theorem as a particular case of a broadly general blockdiagonalization algorithm.
Locating Subsets Of A Hilbert Space
 Proceedings of the American Mathematical Society, 129(5):1385–1390, 2001. B.: Constructive Results on Operator Algebras
, 1998
"... . This paper deals with locatedness of convex subsets in inner product and Hilbert spaces which plays a crucial role in the constructive validity of many important theorems of analysis. 1. Introduction In Bishop's constructive mathematics, the framework of this paper, locatedness of subsets (especi ..."
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. This paper deals with locatedness of convex subsets in inner product and Hilbert spaces which plays a crucial role in the constructive validity of many important theorems of analysis. 1. Introduction In Bishop's constructive mathematics, the framework of this paper, locatedness of subsets (especially convex subsets) of normed spaces plays a crucial role in the validity of many important theorems of analysis such as the HahnBanach and separation theorems [1, Chapter 7.4], [7], the open and unopen mapping theorems [5], and the existence theorems of Minkowski functionals [9]. (Recall that subset C of a normed space X is located if (x; C) := inffkx yk : y 2 Cg exists for each x 2 X.) Richman [10] extended the denition of weakly totally boundedness, which was rst dened in [8] for separable Hilbert spaces, to inner product spaces which are not necessarily separable as follows: a subset C of an inner product space X is weakly totally bounded if for each x 2 X , fhx; yi : y 2 Cg i...
Quotient topologies in constructive set theory and type theory
, 2005
"... The standard construction of quotient spaces in topology uses full separation and power sets. We show how to make this construction using only the (generalised) predicative methods available in constructive type theory and constructive set theory. MSC: primary 03F65, 54B15, secondary 03B15, 03E70. K ..."
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The standard construction of quotient spaces in topology uses full separation and power sets. We show how to make this construction using only the (generalised) predicative methods available in constructive type theory and constructive set theory. MSC: primary 03F65, 54B15, secondary 03B15, 03E70. Keywords: quotient spaces, constructive topology, MartinLöf type
Locating the range of an operator on a Hilbert space
 Bull. London Math. Soc
, 1992
"... In classical operator theory it is taken for granted that we can project onto the closure of the range of an operator T on a Hilbert space H. In a constructive development of operator theory, to which this note is a contribution, this projection exists if and only if ran(r), the range of T, is locat ..."
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In classical operator theory it is taken for granted that we can project onto the closure of the range of an operator T on a Hilbert space H. In a constructive development of operator theory, to which this note is a contribution, this projection exists if and only if ran(r), the range of T, is located, in the sense that the distance
Resolution of the uniform lower bound problem in constructive analysis
, 2007
"... constructive analysis ..."