What is the problem with Euclidean geometry?

Question

I read somewhere that Euclid's element is a little outdated (sounds normal since it's super old) then i read on wikipedia:

"..Well-known modern axiomatizations of Euclidean geometry are those of Alfred Tarski, George Birkhoff and David Hilbert".. so since there are 3 (and I guess more) "new geometries" I think they are somehow better then classic euclidean geometry (with this I mean the geometry explained by Euclid) so I was wondering what the problem is with the old geometry. Can you guys explain me in easy words please since I don't know more than high school level geometry?

Thank you!

Answer 1

Hilbert and others showed that Euclid's axioms were incomplete and that further axioms were needed to prove the standard results rigorously.

What I don't know is whether or not the increased rigor has allowed any new results to be deduced.

Answer 2

One of the most important aspects of Euclid's elements is that it is an axiomatic logical system, i.e., built by proofs from the ground up. In fact it was the first one. Nothing is true unless assumed or proven via inference (e.g., by constructing and relating quantities). In fact much of Euclid's Elements (c. 3rd century BCE) was nothing new even to the Greeks alone. The Pythagorean theorem (c. 5th century BCE but discovered earlier elsewhere) and Thales' theorem (c. 6th century BCE) predate the Elements by centuries. I'd even argue that a lot of Euclidean geometry can be intuitively seen to be true without rigorous proofs. However, none of these theorems were proven meticulously from a foundation of axioms until Euclid.

So, it is not to say that we are at a total loss if we don't approach logic from the ground up. I'm sure many professional mathematicians are ignorant of minor foundational details that their work is contingent on. But, the boon of axiomatic approaches is that it can show what faulty assumptions we've made (as intuition can be wrong) and whether we're limiting ourselves by overlooking possibilities that could be valid (e.g., alternative non-Euclidean geometries). This is one reason why it is important for the axioms of Euclidean geometry to be well formulated. Hilbert's, Tarski's, etc. formulations of geometry rectify some of the missing pieces that Euclidean geometry was missing; namely the uncertainty of whether the parallel postulate could be proven from other axioms.