1) ? Intuitive geometric symmetry (point symmetry, translational symmetry)-all symmetries (groups)-group homomorphism and groups (categories).

2) ? Scalar-directed scalar (vector)-single parameter mapping (tensor)-mapping (functor) between categories.

3) ? Cartesian coordinate system-Cartesian coordinate system-manifold-linear space-set.

4) ? Intuitive invariance-topological invariance.

5) ? Differential-external differential.

6) ? Pythagorean Theorem-Metrics-metric tensor.

7) ? Integer-fraction-negative number-irrational number-complex number-transcendental number.

8) ? Tangent vector-tangent space-tangent follower-fiber bundle.

9) ? Covector-cospace-section.

History of mathematics development:

Mathematics [English: Mathematics, derived from the ancient Greek μ θ η μ α (má th ē ma); Often abbreviated as math or maths], it is a discipline that studies concepts such as quantity, structure, change, space and information.

Mathematics is a universal means for human beings to strictly describe the abstract structure and mode of things, and can be applied to any problem in the real world. All mathematical objects are artificially defined in essence. In this sense, mathematics belongs to formal science, not natural science. Different mathematicians and philosophers have a series of views on the exact scope and definition of mathematics.

Mathematics plays an irreplaceable role in the development of human history and social life, and it is also an indispensable basic tool for studying and studying modern science and technology.