36,227 research outputs found
Manufacturing a mathematical group: a study in heuristics
I examine the way a relevant conceptual novelty in mathematics, that is, the notion of group, has been constructed in order to show the kinds of heuristic reasoning that enabled its manufacturing. To this end, I examine salient aspects of the works of Lagrange, Cauchy, Galois and Cayley (Sect. 2). In more detail, I examine the seminal idea resulting from Lagrange’s heuristics and how Cauchy, Galois and Cayley develop it. This analysis shows us how new mathematical entities are generated, and also how what counts as a solution to a problem is shaped and changed. Finally, I argue that this case study shows us that we have to study inferential micro-structures (Sect. 3), that is, the ways similarities and regularities are sought, in order to understand how theoretical novelty is constructed and heuristic reasoning is put forwar
The history of the concept of function and some educational implications
Several fields of mathematics deal directly or indirectly with functions: mathematical
analysis considers functions of one, two, or n variables, studying their properties as well as
those of their derivatives; the theories of differential and integral equations aim at solving
equations in which the unknowns are functions; functional analysis works with spaces made
up of functions; and numerical analysis studies the processes of controlling the errors in the
evaluation of all different kinds of functions. Other fields of mathematics deal with concepts
that constitute generalizations or outgrowths of the notion of function; for example, algebra
considers operations and relations, and mathematical logic studies recursive functions.
It has long been argued that functions should constitute a fundamental concept in secondary
school mathematics (Klein, 1908/1945) and the most recent curriculum orientations clearly
emphasize the importance of functions (National Council of Teachers of Mathematics, 1989).
Depending on the dominant mathematical viewpoint, the notion of function can be regarded
in a number of different ways, each with different educational implications.
This paper reviews some of the more salient aspects of the history of the concept of
function,1 looks at its relationship with other sciences, and discusses its use in the study of
real world situations. Finally, the problem of a didactical approach is considered, giving
special attention to the nature of the working concept underlying the activities of students and
the role of different forms of representation
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An Ontology for Grounding Vague Geographic Terms
Many geographic terms, such as “river” and “lake”, are vague, with no clear boundaries of application. In particular, the spatial extent of such features is often vaguely carved out of a continuously varying observable domain. We present a means of defining vague terms using standpoint semantics, a refinement of the
philosophical idea of supervaluation semantics. Such definitions can be grounded in actual data by geometric analysis and segmentation of the data set. The issues
raised by this process with regard to the nature of boundaries and domains of logical quantification are discussed. We describe a prototype implementation of a system capable of segmenting attributed polygon data into geographically significant regions and evaluating queries involving vague geographic feature terms
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