False paradoxes: the first faces of the infinity concept in the context of mathematical science

Abstract

The paper presents the results of a theoretical research that studied the infinity and the relation of this mathematical concept with the false paradoxes given by Zeno, contrary to atomistic conception of time and space. More specifically, we studied the paradoxes of Achilles Dichotomy, who argue against the hypothesis that space is infinitely divided, and the Stadium and Arrow paradoxes, which question the possibility of a segment being formed by an infinite of divisions. Although nowadays we are used to deal daily, even intuitively, with the idea of speed and movement, these are undoubtedly abstract concepts. This is due to the Zeno’s Paradoxes importance: by exposing a first systematic thinking about the assumption. The Arrow and Stadium Paradoxes are, in fact, real, if time is composed of indivisible minimum units and space by discrete points. In contrast, if time and space are considered continuous, the Achilles Dichotomy arises. Thus, Zeno’s thoughts surround on all sides the idea of movement and speed, coming up controversies that sometimes go unnoticed by the eyes already used to observe the movement. Through dialectics, starting from the apparently consistent premises and arriving at absurd conclusions, Zeno presented arguments to prove the fragility of the multiplicity and divisibility concepts, adopted by the Pythagorean School. These paradoxes, based on Parmenides philosophy, present situations to support the movement impossibility, considering it an illusion of the perception of the sensitive world and not the truth of the intelligible world, which characterizes the being as unique, immutable, infinite and immovable.

Keywords

• educação matemática
• história da matemática
• infinito
• paradoxo.