Plato's Idealism has been heavily criticized. It is now accepted that ideas do not exist in the objective world, it is just like the God doesn't exist in the objective world, but concepts do exist. The so called "do exist" is existence as logical constructs in the subjective world. The fact that concepts and ideas do not exist in the objective world does not mean that concepts are not important. On the contrary, concepts are very important and are the basis of all science and philosophy. The richness and development of concepts often represent the level of development of science and philosophy. Since idea is the same as concept in the subjective world, Plato's Objective Idealism is of great value to the development of philosophy.

Frege's concept is just Plato's idea. “And we are driven to the conclusion that number is neither spatial and physical, like Mill's piles of pebbles and gingersnaps, nor yet subjective like ideas, but non-sensible and objective." Frege said. "Now objectivity cannot, of course, be based on any sense-impression, which as an affection of our mind is entirely subjective, but only, so far as I can see, on the reason. "[6]

Frege argued that concepts had nothing to do with psychology. He said, "We suppose, it would seem, that concepts sprout in the individual mind like leaves on a tree, and we think to discover their nature by studying their birth: we seek to define them psychologically, in terms of the nature of the human mind. But this account makes everything subjective, and if we follow it through to the end, does away with truth."[7] Frege's bias was completely wrong, his error was the same as Plato's essentially.

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, "