La cronotopía, antes y después de la geometría

Abstract

The paper develops a historical account of a few famous geometry programs, starting with Pythagoras’ Program or the Crotona Program. Other
programs will be identified with the name of a person or of a city, depending on the mathematician who proposed it or the city where he lived
or worked when the given program was proposed. The most famous is the Erlangen Program. After the theory of relativity, no geometry program should separate spatial from temporal aspects. Thus, instead of space regions or intervals and time lapses or intervals, we should rather think and speak of spatio-temporal intervals. For this reason, the central object of study for future geometry programs should be called “The Chronotope”, a word that highlights the fusion of time and space in a new 4-dimensional construct. “Chronotopy” should be the corresponding name for the theoretical study of the logical and metric aspects of the Chronotope and its extensions to higher dimensions. Chronotopy spreads out into four intertwined disciplines: Chronology, Chronometry, Topology, and Topometry. Chronotopy should be included in primary, secondary, and tertiary Mathematics Education, and also as a central line of mathematical research, following the process of exploration, conceptualization, formulation, and testing of conjectures, reasoning and proving,
then returning to visualization and embodiment of some aspects of the solutions. This new vision of what was called “Geometry” is called “the Bogota Program” and proposes to strengthen spatio-temporal thinking and reasoning from early education to advanced research endeavors.