Scanning the issue...
“Scanning the Issue” G. Toussaint, editor of the volulme “Computational Geometry” .
Proceedings of the IEEE, sept. 1992, vol.80, no.9.
• Scientific tools have always influenced mathematics: abacus, ruler and compass (of different kinds), bar and joint mechanisms,...
• Classically, lots of mathematical contents are tool-driven: for instance, elementary euclidian geometry.
• The introduction of new exploration tools qualitatively modifies our conception of the mathematical world: the medium is the message.
•In some sense we can say Euclid himself conceived it (the geometry) as an exploration of the power of some geometric artifacts:
--operational model, equivalent models,
--number of steps,
--limits,
--paradigmatic problems,
--etc.
•It is rather unnatural to explore the geometric world from the point of view of the euclidian tools.
• Computers deeply influence mathematics (and conversely) and math teaching
• For instance as an advantageous simulation for some previous tools (such as paper and pencil, logartihmic tables, rule and compass)
• But the less important consequence is this computational trasposition of some classical tools.
• Much more important are the changing mathematic interests that spring around this new tool.
For instance, regarding the connection to geometry and school teaching:
--ICMI Study: “School Mathematics in the 1990's” (Kuwait, 1986)
“even if the students will not have to deal with computers till they leave school, it will be necessary to rethink the curriculum, because of the changes in interests that computer have brought. Let us mention here just three of them:
a) Algorithms, b) Discrete mathematics, c) Symbol manipulation.”
“it seems appropriate, in the last (secondary school) grades, to bring attention to issues such as algorithmic efficiency, and to distinguish between those, say, with polynomial and with not polynomial complexities"
“the solution obtained by the application of a mechanical procedure could be aesthetically less satisfactory than one of geometric style, but, are there objections other than aesthetical?”
--Following this recommendation we will be, probably, closer to the espirit of Euclides.
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