I believe math is one of the hardest things to define. It involves so many different areas that each type of problem is unique. The best way for me to generalize what math is to say that math is a system to simplify problems into something more understandable. This can be taken in many different ways; however, what I get out of this definition is that we have a more complex form of some problem. For example, one of the most basic forms of mathematics is addition. If we are given 25+10, we can therefore simplify to 35 so that we better understand what 25+10 actually means. I don't believe there is one proper term to define mathematics but there are many ways to describe math. The simplest form is to use numbers, letters, and symbols to help understand the world around us.
As far as mathematical discoveries I cannot think of many. However, I tend to think about laws and other rules that we have came across our classes throughout our schooling.
I think one of the biggest discoveries in mathematics was the discovery of Euclidean Geometry. This was the turning point that made sense of every day life and objects that existed in the universe. It helped humans, and still helps humans understand the world. It is the most commonly used geometry by almost everyone and intrigues me how these rules and laws still are proven to be true.
In Discrete Mathematics, I think of Euler and the Eulerian Circuit. I like puzzles and other elements that involve problem solving and figuring things out, such as riddles. Therefore I think of the Eulerian Circuit as a fascinating puzzle that requires deep thinking and interests me.
The Fibonacci Sequence and variations of it are also interesting to me because I find it fun to discover the pattern and how to get to the next term. When first learning about the Fibonacci numbers I was well engaged and later after learning more complex sequences really enjoyed solving the patterns and finding the equations at which they changed by.
Lastly, I know most of these are kind of far fetched due to little knowledge I do have about famous discoveries, I really find the biggest discoveries are related to finding best ways to teach students. There are so many variations on how and why we teach the subjects the way we do and I think, with more studies and research, students should be encouraged to think differently. I think allowing students to find new patterns and ways to solve problems is the coolest thing about math. There isn't always one way to do something.