Mathematical abstraction mainly includes: abstracting mathematical concepts and the relationship between concepts from the relationship between quantities and the relationship between graphs. General laws and structures are abstracted from the specific background of things and expressed in mathematical language.
For example, in the teaching of the concept of function monotonicity, combined with examples, experience the abstract process from concrete intuitive description to formal symbol expression, deepen the understanding of the concept of function monotonicity, and realize the necessity of expressing mathematical definition with symbols.
Mathematical abstraction can basically be divided into four types:
1, weak abstraction
That is, a certain feature (profile) is abstracted from the prototype, so that the connotation of the prototype is reduced, the structure is weakened and the extension is expanded, and the original structure becomes a special case of the latter. The key of weak abstraction is to identify essential attributes or characteristics from many attributes or characteristics of mathematical objects, and to find similar things from seemingly different similar mathematical objects. This law of abstract thinking can be called "the law of feature separation and generalization".
2. Strong abstraction
That is, by introducing new features into the prototype, the connotation of the prototype is increased, the structure is strengthened, and the extension is contracted, making the latter a special case of the former. The key to strong abstraction is to connect some seemingly unrelated mathematical concepts, introduce new relational structures, and designate emerging attributes as features. This law of abstract thinking can be called "the law of qualitative relations".
3. Conformational abstraction
That is to say, according to the logical needs of mathematical development, a completely idealized mathematical object that can not be directly extracted by the realistic prototype is conceived and added to a mathematical structure system as a new element to make it complete, that is, it runs unimpeded in this structure system.
For example, in the theory of real variable functions, the introduction of Lebesgue integrable function and square integrable function makes 1. And 1. Z becomes a complete space. This law of abstract thinking can be called "the law of adding new elements to complete".
4. Axiomatic abstraction
That is, according to the needs of mathematical development, a completely idealized new axiom (or basic law) is conceived to eliminate mathematical paradox and restore the harmony and unity of the whole mathematical theoretical system. Non-Euclidean geometry parallel axiom and non-Archimedean axiom are the products of axiom abstraction. The law of this abstract thinking can be called:'. Axiom updates the law of harmony.