Matemática vs. Filosofia: Por Que Platão é Mané e Archimedes Gênio

Excelente artigo de Mike Alder mostrando que a má compreensão da geometria euclidiana está na base do erro filosófico básico do platonismo: This pretty much does for Platonism as far as mathematicians are concerned. Axioms stopped being self-evident truths as soon as the work was read and understood. Instead they were simply postulates, and they might be interpreted as true statements about the world, perhaps in several different ways. Or they might not be interpreted at all. Platonism died for mathematicians some centuries ago, and simply looks silly. Mathematics doesn’t give truths, it gives consequences. The axiom of parallels is merely the postulate that the space in which we are working is flat. This tells us nothing about whether the space we live in really is flat – maybe it is and maybe it isn’t. We would need to find out by observation, and Gauss, who grasped the point immediately, suggested putting three telescopes on different mountain peaks and measuring the sum of the angles of the triangle so formed. If it came to 180 degrees, space was flat, at least up to the limits of accuracy of the measurements. If more, we lived in Riemannian space, if less then in a Lobachevskian space. Reason alone couldn’t possibly tell us which.