Some beautiful theorems in mathematics have such characteristics: they are easy to be summarized from facts, but the proofs are extremely profound. -Gauss

As long as a branch of science can raise a large number of questions, it is full of vitality, and no questions indicate the termination or decline of independent development. Just as every career of human beings is to achieve some ultimate goal, mathematical research needs problems. Solving problems exercises the strength of researchers. By solving problems, he discovered new methods and new ideas, and broadened his horizons.

1900 speech at the international congress of mathematicians in Paris-D. Hilbert

In mathematics, the main tools for us to find truth are induction and simulation. Laplace

Mathematics is a variety of proof skills. -Wittgenstein

Start with the simplest. -Polya

New mathematical methods and concepts are often more important than solving mathematical problems themselves.

It is not intuitive when the number is missing, and it is difficult to be nuanced when the number is missing.

There are two necessary processes to lay a good foundation of mathematics: first, learning and accepting "from thin to thick"; Re-digest and refine "from coarse to fine"-China

I always try my best to get rid of that heavy and monotonous calculation. -Napier Thinking begins with doubt and surprise. -Aristotle

Less is better, but better.

Half the proof is equal to 0. Gauss

The problem is the core of mathematics. -P.R. halmos

Without bold speculation, there will be no great discovery. newton

The creation of mathematics can never be achieved by inference alone. First of all, it is usually some vague guesses, trying to figure out the possible popularization, and then drawing an uncertain conclusion. Then sort out the train of thought until you see the clues of the facts, and it often takes a lot of effort to put everything into logical proof. This process is not achieved overnight. It is common to experience many failures and setbacks, guess and ponder again and again, and waste months in temptation. Halmos

Although we are not allowed to see through the secrets of nature, so as to know the real reason of the phenomenon, some fictitious assumptions may still occur to explain many phenomena.

Because the structure of the universe is God's most perfect and wise creation, nothing will happen if there is no certain maximum or minimum law in the universe. —— Euler (leonhard euler 1707- 1783)

Use a single curve, like the one drawn for cotton price, to describe the effect in the most complicated music performance-in my opinion, this is an excellent proof of mathematical ability. -Kelvin (Lord Kelvin 1824- 1907)

Any promotion is just a hypothesis and plays a necessary role. No one denies it, but it must be proved. -poincare (Henry poincare 1854- 19 12)

Mathematical method is the essence of mathematics. Mathematicians are people who can fully understand mathematical methods. Hardenbergh

The only way to learn mathematics is to do it. Halmos

Don't ignore analogy, it can lead us to discover.

Analogy is another rich source of discovery. -Polya

The method lies entirely in properly sorting out and classifying the things that we must pay attention to, so as to make them orderly. Descartes

Observation is the source of wealth invented by the subjective physical world, just as it is the source of wealth invented by the perceptual phenomenon world.

There is a simple saying about the psychological process of mathematicians, which is generally recognized in the history of science: observation plays an important role in the psychological process of mathematicians and plays a huge role. Emmett

Observation may lead to discovery.

Observation will reveal some laws, patterns or rules. -Polya

Imagination is more important than knowledge. -Einstein

The motive force of mathematical invention is not reasoning, but imagination. -De Morgan

Non-mathematical induction plays an indispensable role in mathematics learning. -I. Shure

What is observed is only the outline and conjecture of experimental nature, not proof. -Polya

Most mathematical creations are the result of intuition, which is a direct perception or quick understanding of facts to some extent, and has nothing to do with any lengthy or formal reasoning process. -William. Lucas