Introduction: irrational numbers and the infinite

The discovery of irrational numbers in ancient Greece is attributed to the School of Pythagoras in 500–400 BCE. The discovery seems to have caused a type of crisis in Greek science and mathematics, which had until then considered that measurement must consist in counting natural units and considering their ratios—that is, it was considered that measurement is a discrete process, or a process of counting. This was the view of Pythagoras, who regarded nature as an expression of whole numbers [5, pp. 154–155]. Sir Karl Popper regarded the discovery of irrational numbers as leading to major philosophical adjustments in Greek thought. He writes [12, p. 75]:

I think it is fair to say that the problems raised by the discovery of irrational numbers are essentially unresolved to this day. For, once we admit the notion of irrational numbers, we admit entities whose description essentially requires an infinite sequence of discrete and independent pieces of information.