Editing The Road to Reality Study Notes (section)

===3.1 A Pythagorean catastrophe?===
We now switch over to the idea of ‘number’, and the layers of generality that lie beneath integers.  The Pythagoreans solved the question, by using proof by contradiction, of attempting to find a rational number (fraction) whose square is precisely 2.  There does not exist such a number within the confines are integers and rationals, which was troubling for them at the time since it was desired to have all of geometry be described by these types of physical numbers. 
 
Penrose gives the proof by contradiction for the Pythagorean question above and explains the necessity of identifying the precise assumptions that go into a proof.  He explains that there exist other generalities than that was originally used in the proof, whereby these precise assumptions must be used to judge the logic.  The Pythagoreans used integers and rationals to explain the existence of the real numbers.