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Substitution for FraenkelMostowski foundations
"... A fundamental and unanalysed logical concept is substitution. This seemingly innocuous operation — substituting a variable for a term or valuating a variable to an element of a domain — is hard to characterise other than by concrete constructions. It is widely viewed as a technicality to be dispense ..."
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A fundamental and unanalysed logical concept is substitution. This seemingly innocuous operation — substituting a variable for a term or valuating a variable to an element of a domain — is hard to characterise other than by concrete constructions. It is widely viewed as a technicality to be dispensed with on the way to studying other things. Discussions of computer science foundations, and of the philosophy of logic, have largely ignored it. We show that FraenkelMostowski set theory gives a model of variables and substitution as constructions on sets. Thus models of variables and substitution are exhibited as constructions in a foundational universe, just like models of arithmetic (the ordinals) and other mathematical entities. The door is open for classes of denotations in which variables, substitution, and evaluations are constructed directly in sets and studied independently of syntax, in ways which would previously have not been possible.
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.
The Society for the Study of Artificial Intelligence and
, 2008
"... Proceedings of the ..."
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KRDB RESEARCH CENTRE KNOWLEDGE REPRESENTATION
, 2010
"... An empirical assessment of the use of foundational ontologies in ontology development ..."
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An empirical assessment of the use of foundational ontologies in ontology development
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.