O conceito de medida, o continuum e o discreto

Abstract

The word “measure” has been used throughout the history of humanity in almost every sector of human activity. It is not surprising that any changes in the way things are measured, also cause impacts in the developent of science and technologies. The refinements in the measurement criteria, in a broad sense, occur thanks to existing technologies, or they emerge from some well-defined rule, written in some language, with some scientific or practical purpose. Whereas, in a remote past, the diameter of the Earth was measured based on similarities of triangles, in modern times technologies made these measures much more precise. Even so a consensus is not yet reached, given that the mathematical continuum imposes restrictions on measurement reality. For example, there is no way to use in its fullness in laboratories, since approximations are necessary, taking into account that it is an irrational number with infinite decimal places. This characterizes a seesaw, in which, on one hand, there are the practical measures in the reality we live in, and, on the other, the theoretical measures. The goal in this article is to expose that, on one hand, from the perspective of mathematics, some examples characterize the relation between continuum and the discrete, in the measured aspect. On the other hand, we show that this relationship can indicate contradictions, in a stage of interactions between the practical and the theoretical world, if no careful reading happens. Apart from a historical digression with examples, it is shown that something similar occurs with the concept of measure when it is seen as an amount of information, called entropy by C. E. Shannon. There is also care to be taken regarding entropy, seen from the point of view of discrete models, and their extension to continuous models, differential entropy. While on the discrete side the amount of information is positive, the differential entropy, on the continuous side, can be negative, positive or arbitrarily large.