The story goes for that when Euclid wrote his book "The Elements" that it seemed to bother him and every mathematician after him that he didn't spend as much time and care proving his fifth postulate:
If the sum of the two interior angles equals 180°, the lines are parallel and will never intersect.So for years mathematicians worked on this postulate, trying to prove it in anyway possible. Then, around the end of the 18th century, a mathematician came up with an idea to disprove one of the other four. Honestly, this is akin to proving stories from any holy text are wrong. It's said that as a good mathematician, Einstein had read The Elements and carried around a copy. That book is what a lot of modern math is based on. But, some of the postulates were not always correct.
In the 1820's Nikolai Lobachevsky devolved an imaginary geometry in which two parallel lines would eventually intersect. This proposal helped support the idea of Hyperbolic geometry. Janos Bolyai developed trigonometry, circles and spheres that could exist outside of 2 dimensional Euclidean geometry. And Georg Riemann proposed the basis for spherical and elliptical geometry. This is important because these ideas show themselves in how light moves across a warped space-time. It's important because spherical geometry helps circumnavigate the globe. It's important because we can describe fractal math as Fractal Geometry.
Later, in the early 1900's, logic became an obsession with the world as a whole. A detective was created that solved mystery with logic, and he became very popular. A mathematician wrote three famous books, depending on what world you live in. In the world of pop culture, Lewis Carrol wrote the Alice books, but as a mathematician he wrote a book on linear logic that a teacher of mine swore was the greatest book on linear logic ever written. The puzzles are what you expect; just completely off the wall strange. Lots of fun though, you should check it out if you like logic puzzles.
During this time, many mathematicians began to experiment with logic and developed their own rules for logic that went against Aristotle's principles. My favorite is Kurt Godel who seems to be intent on finding paradoxes in everything. Logic created the computing languages and strange counting systems (Binary, Hexadecimal) that we know and love today. To go deeper into both non-Greek systems would find us neck deep in mathematical theory. So I'll leave it for another time.