Why is the axiomatic thinking so important

Why is the axiomatic thinking so important

 I believe that many people do not understand Euclid's greatness, and perhaps some people do not agree with my high praise for him, because in our life, Euclidean geometry is only a simple knowledge used to solve some plane geometry problems. However, from the perspective of philosophy, the geometric system created by Euclid creates the possibility for our thinking to transcend the real world.

It is no exaggeration to say that without the axiomatic thinking and methods put forward by Euclid in geometry, the development of science can only stay at the level of using the known to deduce the known, while Euclid uses the points, lines, surfaces, and their relations that do not exist in the real world to transcend the confinement of our senses and deduce the unknown from the known. We all know that the rapid development of human society today depends on the ability to deduce the unknown from the known.

For example, there is a strange assumption in Einstein's general theory of Relativity: this space is four-dimensional and can be bent. But can this be imagined by the human mind? The answer is No. Humans can easily imagine curved lines existing in two-dimensional space and build curved planes existing in three-dimensional space in the brain, but most people cannot form a cognition of curved space in the brain. Because we live in three-dimensional space, the limit that our eyes can see and the limit that our brain can recognize is the three-dimensional level. Just as two-dimensional worms cannot imagine our three-dimensional world, this is the so-called sensory channel confinement.

So far, although the level of human science and technology has not yet reached the level that can verify whether space can be bent, Einstein's general theory of relativity is still used as an axiom by many theoretical physicists in the scientific community. Because from a logical point of view, the final result must be correct under the premise that the first principle and the derivation process are correct. In other words, human beings can only understand four-dimensional space, but cannot exist in four-dimensional space.

This is the deep reason why mathematics and geometry are called divine knowledge. Axiomatic thinking can transcend the confinement of our senses and deduce a new world by means of logical reasoning. In other words, if you don't understand geometry, have no mathematical thinking, or even lack pure logical thinking, you can only live in the world you can see before your eyes. But the world is too narrow. Whether it is personal development or human progress, we need to constantly break the material constraints and find a way forward in the unknown future. In essence, geometry is a philosophy, but it also contains a certain world outlook.

Once a young man wanted to learn from Euclid. He asked Euclid a question: "what is the use of learning geometry?"

This problem is a typical oriental mode of thinking. It attaches importance to practicality, wants to apply what it has learned, and integrates knowledge with practice. Euclid was furious when he heard this question and said, "it's an insult to me that you want to learn useful things from me. You can learn useful things from craftsmen. How can you learn useful things from me?"