A irracionalidade numérica na filosofia da matemática de Wittgenstein

Abstract

The objective of this work is to understand “numerical irrationality” from the perspective of Wittgenstein's philosophy of mathematics. In the Wittgensteinian conception, an irrational number cannot be understood as just one other type of number within the set of Real Numbers. For Wittgenstein, we can even use “numerical irrationality” for calculation, the mistake is in giving the same treatment similar to that of an integer to a “number” which would be “irrational”. To introduce theses questions, it was necessary to understand how Wittgenstein construes mathematical rules, separating them into geometric mathematical rules and arithmetic mathematical rules, in order to point out the main themes involved in this approach. We emphasize the Greek way of understanding the issue, as it is precisely the Greek understanding which comes closest to the way Wittgenstein understands “numerical irrationality”. The example which shows the entire problem of our work is the relationship between “the side of the square” and the “diagonal of the square”, which, after applying the Euclid algorithm, does not yield a common standard, as the algorithm enters into a loop. This looping of Euclid's algorithm is both a demonstration of “numerical irrationality” and provides a method of approximation of , which does not produce an integer in its result but pairs of upper and lower bounds to the geometrical magnitude.