to trying to define 1[9], which he left unclear.
Frege said, "No, sensations are absolutely no concern of arithmetic. No more are mental pictures, formed from the amalgamated traces of earlier sense-impressions. All these phases of consciousness are characteristically fluctuating and indefinite, in strong contrast to the definiteness and fixity of the concepts and objects of mathematics."[10] Frege did not understand that the subjective products of abstraction can be certain and stable. Frege kept to fundamental principles: always to separate sharply the psychological from the logical, the subjective from the objective.[11] He regarded this as an important principle of logic. Later western language philosophers regarded Frege's principle as the standard.
Abstract processes can only be carried out in the subjective world. In the objective world, there is no abstract number 1, only a specific person, a book, a table, etc. No two leaves are the same in objective world, and one is never equal to one. In the abstract subjective world, 1 can be equal to 1. Because every time we talk about 1 in maths, we're talking about a logical construct that doesn't have specific properties.
The natural numbers studied in maths do not exist objectively, but are logical constructs of human imagination. Other numbers are also logical constructs, integers, fractions, decimals, real numbers and imaginary numbers are all logical constructs created by human being.
Chess is a game with a board and pieces. Among the pieces there are rook, horse, soldier, king and queen, etc. These pieces are fake. They are simulations of real things like chariots and horses. Numbers, points, lines, planes and bodies are highly abstract concepts, just like the chess pieces.
The rules of the game of chess are simulation of the objective rules, but these simulations is far from the real rules, can be said completely innovation. Under these innovative rules, a variety of games can be played. Maths and geometry are like chess, playing thought games using innovative rules rather than objective rules. Philosophy, like maths and geometry, is a game of thought.