Mathematician and Politician-an anology

Consider the situation we know that the sum of the angles of a traingle is 180 degree. This theorem is from Eucledean plane geometry in which sides of traingle are straight line segments.

But our earth being spherical we can not draw straight line segment any where on the surface of the earth. So there exist other geometries known as Non Euclidean Geometries to deal with such situations. Important among these goemetries are 1)Riemannian geometry 2)Bolyai-Labachovsky geometry.

In one of thes geometries the sum of the angles of traingle is less than 180 degree and in other it is greater than 180 degree . Wonder is How mathematics being an exact subject can give different results about the same property of traingle. why this happens?

But this will focus that there is nothing like exact or absolute truth in mathematics, but validity of any result depends on the postulate (i.e axioms) chosen while starting the study of branch of Mathematics in which that result lies. In other words, any result or propositoon in mathematics has to be consistent with the axioms on which it is based.

Based on this concept we have following good humourous story

There were two freinds. One was a mathematican and the other a politician. They were fast friends from their childhool and they had maintained their friendship thorughout, even though they belonged to different professions.

Once the politician told his mathematical frined: "we are birds of the same feather. We both talk nonsencse. The day before yesterday I came to meet you in your school. you were teaching in a class. I did not want to disturb you; therefore I did not call you outside. But your voice being loud, I could hear from outside what you were teaching. You were telling your students that the 'sum of the angles of a traingle is 180 degree. We did study some thing like this, but I do not remember exactly what it was. So I took it that what you were teaching in the class was correct. Yesterday also I came to meet you and that time also you were teaching. But his time I heard you telling your students that the ' sum of the angles of a traingle is less than 180 degree, and in the other class you tell that it is less than 180 degree. . Mathematics being an exact subject, only one of these two statements can and must be true. So your case is like only. We politicians also tell one thing one platform and exactly opposite thing on the other platform. So I say that we both are birds of the same feather; we both talk nonsense".

To this mathematician replied: " Yes, my dear friend, we both talk nonsense. But there is a difference. You talk inconsistent nonsense, while we talk consistent nonsense. Our statements, though they may look contradictory, have to consistent with the axioms with which we start our subject.