The purpose of abstraction is to simplify. Concrete things are complex and difficult to deal with logically. After simplification they are easy to deal with logically, and easy to be used. Only after they can be dealt logically, can we play complex thought games. Every discipline is based on abstract concepts. But the concepts of most disciplines retain more attributes after abstraction, so they are not easy to analyze and study, while numbers, dot, line, plane and body are the results that retain few attributes after abstraction. They are simpler than concepts of other disciplines and easier to analyze and study. Therefore, rich and colorful thought games were played even in ancient times.

Ⅴ. Maths and Geometry

1. Numbers are Abstract Logical Constructs

What is a chicken plus a dog? Is it equal to two chickens or two dogs? Obviously neither. But we can say that 1 plus 1 is 2. Because we have abstracted them before we do this operation. What do we mean by 1? It is a logical construct created by abstracting things such as a chicken, a dog, a person, and an earth. When I need to add a chicken and a dog, I have to first abstract away their personalities and retain their commonness, such as they are both small animals, so that I can add them.

"Never let us take a description of the origin of an idea for a definition."Frege said, "but psychology should not imagine that it can contribute anything whatever to the foundation of arithmetic."[8] Frege did not know that many of the most basic concepts must be defined by association in mind. For example if a child does not know what is 1, adult will say to him, "Look, child, this number is called 1. What is 1? I have a pen, an eraser, a chair, these are all 1. Got it?" The child says, "I understand." By saying this the child means that he has created an association in his mind. Next time he sees 1, he will think of a pen, an eraser and a chair. That's how the concept of 1 is defined. Frege devoted 24 pages of his Arithmetical Foundation