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Implementation of Intuitionistic Type Theory and Realizability Theory
, 1995
"... Writing correct programming code is necessary in computer system development, where complete testing is not possible. Intuitionistic type theory leads to a mechanical generation of correct code by using specifications. The idea is that the specification of a program is its type, and the specificatio ..."
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Writing correct programming code is necessary in computer system development, where complete testing is not possible. Intuitionistic type theory leads to a mechanical generation of correct code by using specifications. The idea is that the specification of a program is its type, and the specification can be expressed by logical statements called wellformed formulas (wffs) and therefore proved by using mathematical axioms and inference rules of logic. Then, using the correspondences propositions are types are specifications and proofs are programs are values [16], a proof can be translated into a correct programming code. The fundamental idea of realizability theory is that a proof can be translated into not only correct, but also minimal programming code, which contains only computational values. Based on these theories, a realizability algorithm developed by John Hatcliff defines how the translation can be done. We analyzed Hatcliff's algorithm and implemented it in a system. System ...
Reality and virtual reality in mathematics
"... This article introduces three of the twentieth century's main philosophies ofmathematics andarguesthat ofthose three, one describesmathematical reality, the \reality " of the other two being merely virtual. What are mathematical objects, really? What, for example, is that thing that we ..."
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This article introduces three of the twentieth century's main philosophies ofmathematics andarguesthat ofthose three, one describesmathematical reality, the \reality &quot; of the other two being merely virtual. What are mathematical objects, really? What, for example, is that thing that we call \the number one&quot;, or \the set of all positive whole numbers&quot;, or \the shortest path between two points on the surface of a sphere&quot;? Most mathematicians (let alone most people) would ¯nd little interest in such questions, since they are totally preoccupied with the practice of their discipline rather than with questions about its meaning. In this essay I shall outline three 1 of the standard philosophical approaches to the meaning of mathematics and present a case that one of those three represents the reality of mathematics, each of the other two amounting to virtual reality. The ¯rst approach that I want to mention is known as platonism. The platonist mathematician believes that mathematical objects do exist, in perfect
Constructive Mathematics and Quantum Physics
, 1999
"... This paper is dedicated to the memory of Prof. Gottfried T. Ru ttimann ..."
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This paper is dedicated to the memory of Prof. Gottfried T. Ru ttimann
Running Head: Students, Functions, and Curriculum
"... recommendations stated here are those of the author and do not necessarily reflect official positions of NSF. Thompson Students, Functions, and Curriculum Someone, I cannot remember who, paraphrased Winston Churchill by saying that mathematics and mathematics education are two disciplines separated ..."
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recommendations stated here are those of the author and do not necessarily reflect official positions of NSF. Thompson Students, Functions, and Curriculum Someone, I cannot remember who, paraphrased Winston Churchill by saying that mathematics and mathematics education are two disciplines separated by a common subject. The mathematician is primarily concerned with doing mathematics at a high level of abstraction. The mathematics educator is primarily concerned with what it is that one does when doing mathematics and what kinds of experiences are propitious for a person’s later successes. Until recently mathematics education research has focused predominantly on the learning and teaching of early mathematics in the school curriculum, so it is natural that practicing mathematicians have found it difficult to relate to mathematics education research. I suspect that the current interest in calculus reform [21, 63] and the broader rethinking of the undergraduate curriculum, together with the advent of the AMS/MAA Joint Committee on Research in Undergraduate Mathematics Education, will lead to a wider recognition that mathematics and mathematics education