After the 19th century, a lot of Mathematicians came to the conclusion that Euclidean Geometry is “incomplete”. The reason was Euclid’s 5th axiom or as also called the parallel postulate, was not that obvious. After doubting this axiom, different geometries, non-Euclidean geometries, were born.
However, my point for this article is that I don’t quite understand why mathematicians called it “incomplete”. There’s no mathematics without postulates at all.
I can list some postulates that are true for me and I make my assumptions (not necessarily the rest of the world agrees), and based on my truth I can develop my theorems. There’s no problem in that, at all.
Euclid developed a geometry based on his axioms. Based on his truth, based on how he saw the plane or space. Mathematics might be sometimes nothing more than an agreement. We create some tools (axioms), the way we want, those tools can be anything, sometimes we might even need a dictionary to define the symbols we might use. That’s alright.
However, after we are done with that, the truth of mathematics will appear. The power of proofs will shine. One cannot call something incomplete because of the definitions or axioms. What should be questioned is actually the theorems and their proofs, if something there is incomplete.
If we doubt axioms, we start a new path of creating something new, not destroying an already developed work.
For me, Euclidean Geometry contributed a lot to my mathematical and creative thinking. At the beginning, it was easy to understand since the concepts made sense. In my High School years, I did Advanced Euclidean Geometry as well and it was one of the best decisions on where to invest my time.
I can certainly say that without doing Geometry at such a high level, I wouldn’t have been able to think the same way I do today. I wouldn’t have practiced the ability to move objects in my head or solving complicated problems using only basic theorems and definitions which usually took me 3 to 4 pages for the proof.