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Total Functional Programming
 Journal of Universal Computer Science
, 2004
"... We now define the notion, already discussed, of an effectively calculable function of positive integers by identifying it with the notion of a recursive function of positive integers (or of a lambdadefinable function of positive integers). The phrase in parentheses refers to the apparatus which Chur ..."
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Cited by 29 (1 self)
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We now define the notion, already discussed, of an effectively calculable function of positive integers by identifying it with the notion of a recursive function of positive integers (or of a lambdadefinable function of positive integers). The phrase in parentheses refers to the apparatus which Church had developed to investigate this and other problems in the foundations of mathematics: the calculus of lambda conversion. Both the Thesis and the lambda calculus have been of seminal influence on the development of Computing Science. The main subject of this article is the lambda calculus but I will begin with a brief sketch of the emergence of the Thesis. The epistemological status of Church’s Thesis is not immediately clear from the above quotation and remains a matter of debate, as is explored in other papers of this volume. My own view, which I will state but not elaborate here, is that the thesis is empirical because it relies for its significance on a claim about what can be calculated by mechanisms. This becomes clearer in
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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Cited by 6 (2 self)
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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.
Cauchy nets in the constructive theory of apartness spaces
 Scientiae Mathematicae Japonicae
, 2002
"... Abstract. A notion of Cauchy net is introduced into the constructive theory of apartness spaces. It is shown that for a sequence in a metric space this notion is equivalent to the standard metric notion of Cauchy sequence. Applications of this notion are then given, culminating in a generalisation o ..."
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Cited by 2 (0 self)
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Abstract. A notion of Cauchy net is introduced into the constructive theory of apartness spaces. It is shown that for a sequence in a metric space this notion is equivalent to the standard metric notion of Cauchy sequence. Applications of this notion are then given, culminating in a generalisation of Bishop’s Lemma on locatedness. 1 Introduction Axioms for a constructive theory of apartness between sets were introduced in [12], where the particular example of a uniform space was discussed in detail. In the present paper we discuss Cauchy and convergent sequences in the framework of that theory. By constructive mathematics we mean mathematics developed with intuitionistic logic
Did Brouwer Really Believe That?
, 2007
"... This article is a commentary on remarks made in a recent book [12] that perpetuate several myths about Brouwer and intuitionism. The footnote on page 279 of [12] is an unfortunate, historically and factually inaccurate, blemish on an otherwise remarkable book. In that footnote, in which Ok discusses ..."
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This article is a commentary on remarks made in a recent book [12] that perpetuate several myths about Brouwer and intuitionism. The footnote on page 279 of [12] is an unfortunate, historically and factually inaccurate, blemish on an otherwise remarkable book. In that footnote, in which Ok discusses Brouwer (who, incidentally, was normally known not as “Jan ” but as “Bertus”, a shortening of his second name, Egbertus), 1 he says:...later in his career, he [Brouwer] became the most forceful proponent of the socalled intuitionist philosophy of mathematics, which not only forbids the use of the Axiom of Choice but also rejects the axiom that a proposition is either true or false (thereby disallowing the method of proof by contradiction). The consequences of taking this position are dire. For instance, an intuitionist would not accept the existence of an irrational number! In fact, in his later years, Brouwer did not view the Brouwer Fixed Point Theorem as a theorem. These sentences contain a number of outdated but still common misconceptions
Separatedness in Constructive Topology
 DOCUMENTA MATH.
, 2003
"... We discuss three natural, classically equivalent, Hausdorff separation properties for topological spaces in constructive mathematics. Using Brouwerian examples, we show that our results are the best possible in our constructive framework. ..."
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We discuss three natural, classically equivalent, Hausdorff separation properties for topological spaces in constructive mathematics. Using Brouwerian examples, we show that our results are the best possible in our constructive framework.