2 - Irrational Numbers and Dynamical Systems

Summary

All our philosophy is a correction of the common usage of words

Georg Christoph Lichtenberg (1742–1799)

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]:

My thesis here is that Plato's central philosophical doctrine, the so-called Theory of Forms or Ideas, cannot be properly understood except … in the context of the critical problem situation in Greek science (mainly in the theory of matter) which developed as the result of the discovery of the irrationality of the square root of 2.

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.