First, the interpretation of mathematical intuition

Mathematical intuition: a form of thinking that uses relevant knowledge chunks and image intuition to analyze and reason current problems keenly, and can quickly find the direction or way to solve problems.

It is a form of psychological activity that directly reflects the structural relationship of mathematical objects, and it is a direct understanding or insight of mathematical objects by the human brain. Simply put, mathematical intuition is the direct understanding and insight of mathematical objects (structures and their relationships) by the human brain.

Second, the basic form of mathematical intuitive thinking

1. Intuitive concept: an intuitive model or spatial graph, which is represented by graphs, regular formulas expressed by words and symbols, etc. It has a similar function to the concept in logical thinking, so it is called intuitive concept (direct thinking for short).

It's vague. Therefore, it is difficult to communicate with others in mathematical intuitive thinking with direct thinking as the thinking cell.

2. Intuitive reasoning: the transformation from one direct idea to another is driven by imagination. We call the movement process from imagination to direct thinking intuitive reasoning. For example, Newton and Leibniz discovered the process of calculus.

3. Intuitive judgment: the brain's quick and keen recognition of mathematical objects and their structures, direct understanding of the essence and comprehensive overall judgment are a leap-forward way of thinking. For example, Lobachevsky and Riemann created non-Euclidean geometry.

4, intuitive inspiration: that is, inspiration. It is manifested as a sudden understanding of the problems that people have been exploring for a long time but failed to solve, and it is also a kind of "enlightenment" when they don't understand.

Features: Sudden, fuzzy, accidental and illogical. Such as Descartes' discovery, analytic geometry and Poincare's discovery of fuchs function transformation method.

Differences and characteristics of intuitive thinking

I. Differences

Mathematical intuition is different from perceptual intuition;

Physical objects, perception, common sense.

Mathematical object, mathematical intuition, mathematical knowledge.

Second, the characteristics of intuitive thinking

Generally speaking, intuitive thinking refers to a form of thinking that directly understands the essence of things without being bound by some fixed logical rules. There are two forms: intuition and inspiration. It has the following characteristics: the suddenness of cognition, the mutation of cognitive process and the breakthrough of cognitive achievements.