Axiomatic thinking refers to the idea of establishing mathematical theories on the premise of certain propositions and using them only without using other assumptions. Basic ideas supporting modern mathematics. As early as the 3rd century BC, the Greek mathematician Euclid used the method of logical reasoning to sort out the previous research results in geometry into "Elements of Geometry" on the premise of several propositions that have been proved by repeated practice and that there is no need to prove them. These propositions are called axioms or assumptions.
In the process of studying the axioms in the Elements of Geometry, especially in the process of establishing non-Euclidean geometry, later generations gradually changed from taking axioms as self-evident ideas to taking axioms as the premise of a theory. This change means the formation of axiomatic thought.
At the beginning of the 20th century, German mathematician Hilbert first established a strict axiomatic system of Euclidean geometry with modern axiomatic thought. Since the 1960s, many mathematicians have advocated introducing axiomatic ideas into middle school mathematics, which is reflected in some new textbooks. China also permeated the axiomatic thought in middle school geometry textbooks.
The application of axiomatic thought
1, systematicness and accuracy: Axiomatic thinking can systematically and accurately describe and analyze complex problems and theoretical systems. By defining axioms and deduction rules, a strict logical system can be established, which makes the deduction and proof of the theory more clear and accurate.
2. Theoretical construction and development: Axiomatic thought is the foundation of scientific theory construction and development. Through the axiomatic method, we can deduce and prove the existing theories and discover the internal structure and laws of the theories. At the same time, axiomatic thought can also be used to construct a new theoretical framework, thus promoting the development and progress of the discipline.
3. Logical analysis and reasoning: Axiomatic thinking can help us carry out logical analysis and reasoning. By defining axioms and deducing rules, complex problems can be logically decomposed and reasoned, and accurate conclusions can be drawn. Axiomatic thought can also help us find logical contradictions and loopholes, thus improving and perfecting the theory.