1, Aesthetics: Mathematical objects are symmetrical, harmonious and concise in form, always giving people the impression of beauty and feeling of beauty.
For example, geometry often gives people an intuitive aesthetic image, which is beautiful, symmetrical and beyond reproach;
Arithmetic and algebra also have many subjects:
Such as (a+b) c = a c+b c;
a+b=b+a
These formulas and rules are very symmetrical and harmonious, and also give people a sense of beauty.
But the beauty of appearance is not necessarily true and correct.
For example: sin(A+B)=sinA+sinB Good "symmetry", "harmony" and "beauty"! But it is wrong, just as it is beautiful but poisonous.
2, beauty: many things in mathematics, only by recognizing its correctness, can we feel its "beauty".
There are many ugly examples, such as the root formula of quadratic equation, which is asymmetrical, unharmonious and ugly in any way. However, when we really understand it and use it, we will feel its value and beauty. This formula tells us a lot of information: it means that it has two roots, and a≠0 and △ will show the number of roots and the properties of the equation. ...
3. Wonderful: Wonderful feelings need to be cultivated, and wonderful feelings often come from things that are "unexpected" but "reasonable". This is the case when the height of a triangle intersects at a point; After the two cylinders intersect vertically, the section unfolds, and the curve corresponding to the section line turns out to be a sine curve, which is "unexpected" compared with the original guess of a broken arc. After analysis, it is proved that it is indeed sinusoidal and "reasonable", and a wonderful feeling arises spontaneously.
4. Perfection: Mathematics always tries to be perfect. This is the highest "quality" of mathematics and the highest spiritual "realm". The establishment of Euclid's axiomatic system of geometry and the proof of "1+ 1" are typical examples of pursuing mathematical perfection.