2. Geometric Shapes are Simple and Useful but Unreal

Geometry deals with points that have no size, lines that have no width, surfaces that have no thickness, and bodies that are regular. In the objective world all points have a size, all lines have a width, all planes have a thickness, no bodies are regular. Therefore, the points, lines, planes and bodies studied in geometry do not exist objectively, but are symbols and logical constructs imagined by human being.

Humans create logical constructs for simplicity. All concrete things are infinitely complex, subjective things are much simpler. When creating logical constructs, there is always a process from simple to complex. The logical constructs created at the beginning are all very simple. Geometric forms are simple and useful but unreal logical constructs.

For example, the surface area of a cube is equal to the length of the sides squared times 6, which is very simple and very precise. But if you want to calculate the surface area of an irregular rock, can you figure it out? There is no real cube in the objective world. A regular cube is a logical construct created in human mind and can only exist in the subjective world. The cube in the objective world is not a regular cube, all have various bumps or defects. Humans create regular cubes for the purpose of simplification, an abstraction from the real cube.

Some real things are useful, some real things are not useful. All we want is useful things. For example, a park wants to paint a huge stone, they need to calculate the surface area of the stone, and then decide how many barrels of paint to buy. They don't need a precise number, just a rough estimate. So they can simplify the irregular shape of the stone into regular polyhedron, which make it easier to calculate. The surface area of this regular polyhedron is a simple and useful but unreal number.

3. Maths and Geometry are the Systematization of Logical Constructs

Numbers, dots, lines, planes and bodies are not objects of the objective world, but symbols and logical constructs created by human being. People continued to construct and deduce these logical constructs, making them complicate and orderly in order to form subjects like maths and geometry. The content of these subjects is the systematic content of subjective logical constructs.

When human first created symbols, the purpose was to simulate the real things. Later, the symbols gradually developed, and the symbols themselves constituted a complex and orderly system. The process in which these symbols are processed and transformed by people's thought games often has nothing to do with the external things that were initially imitated. Maths and geometry are such systems of symbols. Points, lines, planes, and bodies can find their external counterparts at the beginning, but many of the geometry propositions they constitute have nothing to do with external things. Natural numbers can be said to be description of the number of external things, but the relationship between external things and decimal, fraction, irrational number, set, calculus and other mathematical content is becoming less and less.