I remember, well and clear, the first sign of it. The first time I feel it. One day, I opened the book and trying to explain hyperbola equations and properties. It was obvious, clear, something you can see in your head, imagine it, and enjoy by reading it. That was my starting point in the world of Mathematics.
At that time, I didn’t know what it is, why I see it, I feel it, but its beauties were enough to make a tough decision: That’s my road. There I belong. Later, I learn it’s called “Abstract Thinking”.
The following years were critical. It started with a feeling but ended by a lot of research papers and books around, different visions, approaches, and aspects, re-and-re analyze almost everything.
The core of Mathematics is Reduction. Which in its heart: Language. A language of lazy humans who likes to simplify complicated things, and accomplish goals with the least possible of work. One of my professors told me a joke:“The mathematicians are smart because they are lazy.” I couldn’t agree more.
The upper element is “Freedom” – Literally, I mean it – you have to be free in interpreting concepts in different terms. When you have that sense, your mathematical maturity will develop over time. By the end, you will see the “empty space” as “Product of Identity”.
Sets, Differential Calculus, Abstract Algebra, Analysis, Geometry, Numbers, Matrices, and lastly but not last Topology. How these things at first thought looks separated, but deeply, is connected with elegance.
To answer a mathematical question, you need two things: knowledge, and derivation. Started as a beginner, you don’t know enough about the subject. You need to study it in-depth, looking for sources of derivation. For example, if you need to show that the summation of two even numbers is even, you cannot show it without knowing the definition of “even number”. Does this enough? Absolutely not! Even you know the definition, you have to know how to derive the conclusion from it. Someone could write:
x+y=2k+2u ; where
k,u are natural numbers.