引子  封闭运算   对称性  数形结合   拓扑性    运算层次   可测层级  变换过渡   逻辑   表示论

The humanity of the subject of mathematical study is primarily manifested in the following aspects.

Mathematics as a Product of Human Thought: Mathematics is a creation of human thought, not only focusing on concepts such as quantity, structure, change, space, and information, but also representing the pinnacle of human wisdom. From the ancient Greek philosopher Plato viewing mathematics as the highest ideal of culture to the widespread application of modern mathematics in various fields, it all embodies mathematics as a part of human spiritual creation.

The Humanistic Spirit in Mathematics: The humanistic spirit embedded in mathematical research includes self-motivation and self-perfection, a spirit of truth-seeking, exploration, and discovery, a materialistic and dialectical approach, a spirit of innovation and progress, as well as dedication to public welfare and the spirit of unity and cooperation. These spiritual qualities not only drive the development of the field of mathematics, but also influence the culture and values of human society.

Mathematical structures refer to the types of mathematical objects imposed on sets, which make the sets easier to operate on or endow them with special meanings. Common mathematical structures include:

Measure: Used to measure the size or number of elements in a set.Algebraic structures: Including groups, rings, fields, etc., which define the rules of operations on sets.

Topology: The study of continuity and limit properties between sets and their subsets.

Metric structure (geometry): Involves the measurement and properties of geometric spaces.

Order: Defines the arrangement or sequence of elements in a set.

Equivalence relation: Used to partition a set into equivalence classes.

The confrontation between logic and empiricism is primarily evident in the process of theoretical construction and verification. Logic emphasizes the accuracy and rationality of reasoning, while empiricism focuses on validating theories through actual data. In the confrontation between logic and empiricism, operational invariants may serve as an important reference point or standard. Logic may concern itself with how to conduct reasoning without altering the operational invariants, while empiricism may concern itself with verifying whether these operational invariants remain constant in actual situations.

附记：大道至简之把所有学科都读成历史谈学会欣赏说优选聊逻辑与历史顺序的非同一性

引子   映射   矛盾    本质    转换    无穷势    连续统

没有数学,我们无法看透哲学的深度;没有哲学,人们也无法看透数学的深度;而若没有两者,人们就什么也看不透。

按语：就全息的本质而言，在无限论域中，部分可以等于全体，而无限的实现中，全体仅仅属于一部分。

------全息科学之相容性构造范畴悖论最大化