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Contributions to a science of contemporary mathematics, preprint; current draft at http:// www.math.vt.edu/people/quinn
"... Abstract. This essay provides a description of modern mathematical practice, with emphasis on differences between this and practices in the nineteenth century, and in other sciences. Roughly, modern practice is well adapted to the structure of the subject and, within this constraint, much better ad ..."
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Abstract. This essay provides a description of modern mathematical practice, with emphasis on differences between this and practices in the nineteenth century, and in other sciences. Roughly, modern practice is well adapted to the structure of the subject and, within this constraint, much better adapted to the strengths and weaknesses of human cognition. These adaptations greatly increased the effectiveness of mathematical methods and enabled sweeping developments in the twentieth century. The subject is approached in a bottomup ‘scientific ’ way, finding patterns in concrete microlevel observations and being eventually lead by these to understanding at macro levels. The complex and intenselydisciplined technical details of modern practice are fully represented. Finding accurate commonalities that transcend technical detail is certainly a challenge, but any account that shies away from this cannot be complete. As in all sciences, the final result is complex, highly nuanced, and has many surprises. A particular objective is to provide a resource for mathematics education. Elementary education remains modeled on the mathematics of the nineteenth century and before, and outcomes have not changed much either. Modern methodologies might lead to educational gains similar to those seen in professional practice. This draft is about 90 % complete, and comments are welcome. 1.
Unified Proof Style for Teaching Mathematics
"... Structured derivations were introduced by Back and von Wright as an extension of the calculational proof style originally proposed by E.W. Dijkstra and his colleagues. Structured derivations added nested subderivations and inherited assumptions to the original calculational style. This paper introdu ..."
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Structured derivations were introduced by Back and von Wright as an extension of the calculational proof style originally proposed by E.W. Dijkstra and his colleagues. Structured derivations added nested subderivations and inherited assumptions to the original calculational style. This paper introduces a further extension of the structured derivation format, and gives a precise syntax and semantics for the extended proof style. The extension provides a unification of the tree main proof styles used in mathematics today: Hilbertstyle forward chaining proofs, Gentzenstyle backward chaining proofs and algebraic derivations and calculations (in particular, Dijkstra’s calculational proof style). Each of these proof styles can be directly modelled as an extended structured derivation. Even more importantly, the three proof styles can be freely intermixed in a single structured derivation, allowing different proof styles to be used in different parts of the derivation, each time choosing the proof style that is most suitable for the (sub)problem at hand. We describe here (extended) structured derivations, feature by feature, and
THE NATURE OF CONTEMPORARY CORE MATHEMATICS
, 2010
"... Abstract. The goal of this essay is a description of modern mathematical practice, with emphasis on differences between this and practices in the nineteenth century. I explain how and why these differences greatly increased the effectiveness of mathematical methods and enabled sweeping developments ..."
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Abstract. The goal of this essay is a description of modern mathematical practice, with emphasis on differences between this and practices in the nineteenth century. I explain how and why these differences greatly increased the effectiveness of mathematical methods and enabled sweeping developments in the twentieth century. A particular concern is the significance for mathematics education: elementary education remains modeled on the mathematics of the nineteenth century and before, and use of modern methodologies might give advantages similar to those seen in mathematics. This draft is about 90 % complete, and comments are welcome. 1.