We all know Pythagorean theorem, $a^2+b^2 = c^2 $
Im reading John Stillwell, Mathematics and its history at the moment, and during the greek antiquity they had some trouble by interpretating $\sqrt{2}$, if $a=b=1$.
One could not accept (?) that a number which is irrational, would be the distance between two points. The author writes about the 'tension' between arithmetic and geometry.
Can anybody tell me what the tension exactly is?
See page 3 :
The Pythagorean theorem was the first hint of a hidden, deeper relationship between arithmetic and geometry, and it has continued to hold a key position between these two realms throughout the history of mathematics. This has sometimes been a position of cooperation and sometimes one of conflict, as followed the discovery that $\sqrt 2$ is irrational.
It is often the case that new ideas emerge from such areas of tension, resolving the conflict and allowing previously irreconcilable ideas to interact fruitfully. The tension between arithmetic and geometry [emphasis added] is, without doubt, the most profound in mathematics, and it has led to the most profound theorems.
In Greek antiquity, their earlier theory about geometry held that any two magnitudes were commensurable.
But if one believes the Pythagorean theorem, they can show that the hypotenuse of a $45-45-90$ triangle with leg magnitude $1$ is not commensurable to the legs.
So the Pythagorean theorem (which could be viewed as an algebraic theorem relating areas or side lengths) did not match up with the geometric intuition of the time. Thus, they were forced to grapple with incommensurable magnitudes.