I am reading A Very Short Introduction to Mathematics by Tim Gowers. In there he talks about how is is better to think abstractly about math. Think of math as a chess game. Think of the mathematical objects as game pieces (rooks, knights, queens) - these pieces have certain rules of behavior. The queen can move in any direction and number of squares. The pawn can move one space. Integers must follow certain behavior rules.
So, it is better to follow the rules of the game and treat the game as an abstraction then to think sit down and get caught up in "what is an integer", "what is meant by infinity", "how can something go to 0 an infinite number of times". Instead just treat it as a box. These are the rules. These are the objects. Given this situation, what is true?
This is the best way to think about math, according to Tim Gowers.
I have to say that this idea has helped me a lot. I am definitely one of those people who has never been comfortable with thinking of math abstractly. I have wanted to know how it all fit and how it worked. I wanted to see and completely be able to grab hold of 12 dimensions. But I give up. I do see how this has made math very hard for me.
Cheer Tim Gowers!