One of the founding principles of mathematics is The ‘Elements’ – a book written by Euclid of Alexandria. It describes basic rules of mathematics – axioms, which are so fundamental that can’t be proven. He uses these axioms to prove other results or theorem.
One of the most striking results is Euclid’s fifth axiom also called the parallel lines postulate. It is stated as – ‘If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.’
These axioms described mathematical reality till the 19th century, till mathematicians such as Gauss challenged the fifth postulate. Bernhard Riemann, a German mathematician came up with a new branch of math – Riemannian geometry which deals with curved space and mathematical quantities called tensors that describe the magnitude and extent of this curvature.
This geometry was vital to the development of the General Theory of Relativity by Albert Einstein which predicted that mass bends light. The shortcoming of the special theory was that it was unable to deal with non-inertial frames of motion (reference points in space that have a net acceleration) and Riemannian geometry was able to circumvent that by allowing the discussion of the curved areas in space-time continuum. Riemannian geometry is still an exciting avenue of research.
In the face of all this one must not forget Euclid’s contribution to math and that all except the fifth axiom remain unchallenged. This goes on to say that math moves from concrete to abstract and trying and failing is better than not trying at all.